From ca2b527863109d51aa7142e2358a22042b4ef141 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Tue, 26 Apr 2022 13:41:57 +0200 Subject: [PATCH 1/8] adaptions to reflect the adapted nuisance estimation for the PLR IV-type score --- doc/examples/double_ml_bonus_data.ipynb | 8 ++++---- doc/guide/basics.rst | 4 ++-- doc/guide/learners.rst | 4 ++-- doc/guide/scores.rst | 4 ++-- 4 files changed, 10 insertions(+), 10 deletions(-) diff --git a/doc/examples/double_ml_bonus_data.ipynb b/doc/examples/double_ml_bonus_data.ipynb index f440d85a..69c35ab0 100644 --- a/doc/examples/double_ml_bonus_data.ipynb +++ b/doc/examples/double_ml_bonus_data.ipynb @@ -98,7 +98,7 @@ " 'max_features': 'sqrt',\n", " 'max_depth': 5}\n", "\n", - "dml_plr_rf.set_ml_nuisance_params('ml_g', 'tg', pars)\n", + "dml_plr_rf.set_ml_nuisance_params('ml_l', 'tg', pars)\n", "dml_plr_rf.set_ml_nuisance_params('ml_m', 'tg', pars)" ] }, @@ -153,7 +153,7 @@ " 'dml2')\n", "\n", "# set some hyperparameters for the learners\n", - "dml_plr_lasso.set_ml_nuisance_params('ml_g', 'tg', {'alpha': 0.0005})\n", + "dml_plr_lasso.set_ml_nuisance_params('ml_l', 'tg', {'alpha': 0.0005})\n", "dml_plr_lasso.set_ml_nuisance_params('ml_m', 'tg', {'alpha': 0.0026})" ] }, @@ -265,7 +265,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, @@ -279,7 +279,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.8.6" + "version": "3.9.7" } }, "nbformat": 4, diff --git a/doc/guide/basics.rst b/doc/guide/basics.rst index 85ad4de9..e7476a63 100644 --- a/doc/guide/basics.rst +++ b/doc/guide/basics.rst @@ -176,7 +176,7 @@ other half of observations indexed with :math:`i \in I` .. ipython:: python - def non_orth_score(y, d, g_hat, m_hat, smpls): + def non_orth_score(y, d, l_hat, g_hat, m_hat, smpls): u_hat = y - g_hat psi_a = -np.multiply(d, d) psi_b = np.multiply(d, u_hat) @@ -222,7 +222,7 @@ other half of observations indexed with :math:`i \in I` .. jupyter-execute:: - non_orth_score = function(y, d, g_hat, m_hat, smpls) { + non_orth_score = function(y, d, l_hat, g_hat, m_hat, smpls) { u_hat = y - g_hat psi_a = -1*d*d psi_b = d*u_hat diff --git a/doc/guide/learners.rst b/doc/guide/learners.rst index e3ca2d25..cf51aa0f 100644 --- a/doc/guide/learners.rst +++ b/doc/guide/learners.rst @@ -81,7 +81,7 @@ Without further specification of the hyperparameters, default values are used. T dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, RandomForestRegressor(), RandomForestRegressor()) - dml_plr_obj.set_ml_nuisance_params('ml_g', 'd', {'n_estimators': 10}); + dml_plr_obj.set_ml_nuisance_params('ml_l', 'd', {'n_estimators': 10}); print(dml_plr_obj.fit().summary) Setting treatment-variable-specific or fold-specific hyperparameters: @@ -163,7 +163,7 @@ In this case the tuning should be done externally and the parameters can then be ml_g = Lasso() ml_m = Lasso() dml_plr_obj = dml.DoubleMLPLR(dml_data, ml_g, ml_m) - dml_plr_obj.set_ml_nuisance_params('ml_g', 'd', {'alpha': ml_g_tune.alpha_}); + dml_plr_obj.set_ml_nuisance_params('ml_l', 'd', {'alpha': ml_g_tune.alpha_}); dml_plr_obj.set_ml_nuisance_params('ml_m', 'd', {'alpha': ml_m_tune.alpha_}); print(dml_plr_obj.params) print(dml_plr_obj.fit().summary) diff --git a/doc/guide/scores.rst b/doc/guide/scores.rst index c74e2634..f5849849 100644 --- a/doc/guide/scores.rst +++ b/doc/guide/scores.rst @@ -280,7 +280,7 @@ can be obtained with import numpy as np - def non_orth_score(y, d, g_hat, m_hat, smpls): + def non_orth_score(y, d, l_hat, g_hat, m_hat, smpls): u_hat = y - g_hat psi_a = -np.multiply(d, d) psi_b = np.multiply(d, u_hat) @@ -290,7 +290,7 @@ can be obtained with .. jupyter-execute:: - non_orth_score = function(y, d, g_hat, m_hat, smpls) { + non_orth_score = function(y, d, l_hat, g_hat, m_hat, smpls) { u_hat = y - g_hat psi_a = -1*d*d psi_b = d*u_hat From 0c5731dc975bdad5fab7fa25943b31e7ed49fe7d Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Tue, 26 Apr 2022 14:09:40 +0200 Subject: [PATCH 2/8] another adaption due to the IV-type score adaption --- doc/guide/learners.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/guide/learners.rst b/doc/guide/learners.rst index cf51aa0f..2609c87a 100644 --- a/doc/guide/learners.rst +++ b/doc/guide/learners.rst @@ -257,7 +257,7 @@ Without further specification of the hyperparameters, default values are used. T set.seed(3141) ml_g = lrn("regr.ranger") dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g , ml_m) - dml_plr_obj$set_ml_nuisance_params("ml_g", "d", list("num.trees"=10)) + dml_plr_obj$set_ml_nuisance_params("ml_l", "d", list("num.trees"=10)) dml_plr_obj$fit() dml_plr_obj$summary() @@ -293,7 +293,7 @@ Setting treatment-variable-specific or fold-specific hyperparameters: # Set globally params = list("num.trees"=10) - dml_plr_obj$set_ml_nuisance_params("ml_g", "d", params=params) + dml_plr_obj$set_ml_nuisance_params("ml_l", "d", params=params) dml_plr_obj$set_ml_nuisance_params("ml_m", "d", params=params) dml_plr_obj$learner dml_plr_obj$params @@ -314,7 +314,7 @@ The following example illustrates how to set parameters for each fold. # Set values for each fold params_exact = rep(list(rep(list(params), n_folds)), n_rep) - dml_plr_obj$set_ml_nuisance_params("ml_g", "d", params=params_exact, + dml_plr_obj$set_ml_nuisance_params("ml_l", "d", params=params_exact, set_fold_specific=TRUE) dml_plr_obj$set_ml_nuisance_params("ml_m", "d", params=params_exact, set_fold_specific=TRUE) From 118faaafeefaa0e19a68b937b034e33bbb306527 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Thu, 28 Apr 2022 14:05:36 +0200 Subject: [PATCH 3/8] major update of the basics of DML user guide --- doc/guide/basics.rst | 614 ++++++++++++++++++++++++++++++++----------- 1 file changed, 461 insertions(+), 153 deletions(-) diff --git a/doc/guide/basics.rst b/doc/guide/basics.rst index e7476a63..86ead55e 100644 --- a/doc/guide/basics.rst +++ b/doc/guide/basics.rst @@ -17,14 +17,14 @@ We consider the following partially linear model .. math:: - y_i = \theta d_i + g_0(x_i) + \zeta_i, & &\zeta_i \sim \mathcal{N}(0,1), + y_i = \theta_0 d_i + g_0(x_i) + \zeta_i, & &\zeta_i \sim \mathcal{N}(0,1), d_i = m_0(x_i) + v_i, & &v_i \sim \mathcal{N}(0,1), with covariates :math:`x_i \sim \mathcal{N}(0, \Sigma)`, where :math:`\Sigma` is a matrix with entries -:math:`\Sigma_{kj} = 0.7^{|j-k|}`. We are interested in performing valid inference on the causal parameter :math:`\theta`. -The true parameter :math:`\theta` is set to :math:`0.5` in our simulation experiment. +:math:`\Sigma_{kj} = 0.7^{|j-k|}`. We are interested in performing valid inference on the causal parameter +:math:`\theta_0`. The true parameter :math:`\theta_0` is set to :math:`0.5` in our simulation experiment. The nuisance functions are given by @@ -48,8 +48,8 @@ The nuisance functions are given by np.random.seed(1234) n_rep = 1000 - n_obs = 200 - n_vars = 150 + n_obs = 500 + n_vars = 20 alpha = 0.5 data = list() @@ -65,8 +65,8 @@ The nuisance functions are given by library(DoubleML) set.seed(1234) n_rep = 1000 - n_obs = 200 - n_vars = 150 + n_obs = 500 + n_vars = 20 alpha = 0.5 data = list() @@ -76,104 +76,32 @@ The nuisance functions are given by } -OLS estimation -++++++++++++++ - -For comparison we run a simple OLS regression of :math:`Y` on :math:`D` and :math:`X`. -As we will see in the following, due to the considered high-dimensional setting the variance of the unregularized OLS -estimates is higher in comparison to the double machine learning estimates and therefore the estimates are also less -efficient. - -.. tabbed:: Python - - .. The following block makes sure that the seaborn graphics are rendered appropriately but does not need to be shown - - .. ipython:: python - :suppress: - - import seaborn as sns - sns.set() - - .. ipython:: python - - from sklearn.linear_model import LinearRegression - import matplotlib.pyplot as plt - import seaborn as sns - colors = sns.color_palette() - - def est_ols(y, X): - ols = LinearRegression(fit_intercept=True) - results = ols.fit(X, y) - theta = results.coef_[0] - return theta - - # to speed up the illustration we hard-code the simulation results - theta_ols = np.array([0.59318776, 0.45971453, 0.46241367, 0.48078614, 0.47250707, 0.48543412, 0.70513768, 0.68020653, 0.51909541, 0.47951409, 0.37821017, 0.38242628, 0.55801763, 0.49086914, 0.84131015, 0.7710256 , 0.56847739, 0.45701804, 0.52761673, 0.6346921 , 0.54605122, 0.5723868 , 0.54334723, 0.65323525, 0.61125249, 0.43241426, 0.43104578, 0.45377385, 0.50867609, 0.14461668, 0.40294401, 0.43876645, 0.71483579, 0.4399299 , 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this_theta)) - theta_ols[i_rep] = this_theta - - ax = sns.kdeplot(theta_ols, shade=True, color=colors[0]) - @savefig ols.png width=5in - ax.axvline(0.5, color='k', label='True theta'); - -.. tabbed:: R - - .. jupyter-execute:: - - library(ggplot2) - - est_ols = function(df) { - ols = stats::lm(y ~ 1 +., df) - theta = coef(ols)["d"] - return(theta) - } - - # to speed up the illustration we hard-code the simulation results - theta_ols = c(0.607697047, 0.517030100, 0.589652643, 0.472635435, 0.664240859, 0.493210806, 0.539281370, 0.314045648, 0.610190305, 0.381273649, 0.449613657, 0.591410296, 0.475418067, 0.453225281, 0.626975360, 0.133881932, 0.688402884, 0.584272744, 0.400995390, 0.508011546, 0.652163232, 0.450246441, 0.481371297, 0.387925003, 0.339319744, 0.455211505, 0.393108971, 0.508044365, 0.570043632, 0.504138457, 0.649995372, 0.573181791, 0.450204168, 0.580155929, 0.559124755, 0.319166483, 0.619605243, 0.559989398, 0.497594152, 0.516462187, 0.261960663, 0.204280903, 0.669016644, 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0.7044627148, 0.5141194878, 0.4610210390, 0.4283283243, 0.2124464880, 0.3241674624, 0.8986303116, 0.6650904154, 0.4358467530, 0.3623206466, 0.3086154653, 0.4073734322, 0.3954349869, 0.7140152024, 0.5475629076, 0.3772126235, 0.6549230244, 0.7222120900, 0.6293562288, 0.7074974891, 0.3862122773, 0.5190522868, 0.5086677172, 0.4566653024, 0.6198970108, 0.5534357746, 0.6403650132, 0.4776773409, 0.7993807655, 0.3316596983, 0.5347903151, -0.0073145945, 0.7505758880, 0.7721696602, 0.4786098286, 0.7733355763, 0.4292071198, 0.5524794380, 0.4738381907, 0.4532365578, 0.3864576996, 0.3154620027, 0.4125880055, 0.7336049906, 0.7191266612, 0.4174040603, 0.3966765475, 0.4200732061, 0.5158521385, 0.5179938216, 0.6002159767, 0.6655745967, 0.3359239334, 0.2815784591, 0.3850891300, 0.7365577578, 0.3621555499, 0.6444221888, 0.5666878213, 0.5194972086, 0.7835955228, 0.8471366245, 0.6936365628, 0.2793957788, 0.5781979405, 0.3851868684, 0.5080473573, 0.5638570327, 0.2292095223, 0.5795689538, 0.4331280258, 0.4737607122, 0.5558744047, 0.6336470554, 0.5559408316, 0.4499369967, 0.6496618608, 0.6065969525, 0.3587967152, 0.6153089390, 0.5826552712, 0.6817230843, 0.4902681535, 0.4304772497, 0.3568752494, 0.4414055507, 0.5421596269, 0.6616140050, 0.5645489367, 0.4600724670, 0.5220926673, 0.2790394247, 0.7390774505, 0.3820212110, 0.5281986424, 0.6567182815, 0.4522734618, 0.3908002136, 0.7182619596, 0.6261977128, 0.2436503360, 0.3906712092, 0.4287648988, 0.5611551354, 0.7583334768, 0.3711041209) - # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset - #for (i_rep in seq_len(n_rep)) { - for (i_rep in seq_len(1)) { - df = data[[i_rep]] - this_theta = est_ols(df) - print(abs(theta_ols[i_rep] - this_theta)) - theta_ols[i_rep] = this_theta - } - - g_ols = ggplot(data.frame(theta_ols), aes(x = theta_ols)) + - geom_density(fill = "dark blue", alpha = 0.3, color = "dark blue") + - geom_vline(aes(xintercept = alpha), col = "black") + - xlim(c(0.08, 0.75)) + xlab("") + ylab("") + theme_minimal() - g_ols - - Regularization bias in simple ML-approaches +++++++++++++++++++++++++++++++++++++++++++ Naive inference that is based on a direct application of machine learning methods to estimate the causal -parameter, :math:`\theta`, is generally invalid. The use of machine learning methods introduces a bias that arises due to +parameter, :math:`\theta_0`, is generally invalid. The use of machine learning methods introduces a bias that arises due to regularization. A simple ML approach is given by randomly splitting the sample into two parts. On the auxiliary sample indexed by :math:`i \in I^C` the nuisance function :math:`g_0(X)` is estimated with an ML method, for example a random forest learner. -Given the estimate :math:`\hat{g}_0(X)`, the final estimate of :math:`\theta` is obtained as (:math:`n=N/2`) using the +Given the estimate :math:`\hat{g}_0(X)`, the final estimate of :math:`\theta_0` is obtained as (:math:`n=N/2`) using the other half of observations indexed with :math:`i \in I` .. math:: - \hat{\theta} = \left(\frac{1}{n} \sum_{i\in I} D_i^2\right)^{-1} \frac{1}{n} \sum_{i\in I} D_i (Y_i - \hat{g}_0(X_i)). + \hat{\theta}_0 = \left(\frac{1}{n} \sum_{i\in I} D_i^2\right)^{-1} \frac{1}{n} \sum_{i\in I} D_i (Y_i - \hat{g}_0(X_i)). .. tabbed:: Python + .. The following block makes sure that the seaborn graphics are rendered appropriately but does not need to be shown + + .. ipython:: python + :suppress: + + import seaborn as sns + sns.set() + .. ipython:: python def non_orth_score(y, d, l_hat, g_hat, m_hat, smpls): @@ -189,14 +117,21 @@ other half of observations indexed with :math:`i \in I` from sklearn.ensemble import RandomForestRegressor from sklearn.base import clone import numpy as np + + from scipy import stats + import matplotlib.pyplot as plt + import seaborn as sns + face_colors = sns.color_palette('pastel') + edge_colors = sns.color_palette('dark') + np.random.seed(1111) - learner = RandomForestRegressor(n_estimators=100, max_features=n_vars, max_depth=5, min_samples_leaf=2) - ml_m = clone(learner) - ml_g = clone(learner) + ml_m = RandomForestRegressor(n_estimators=378, max_features=20, max_depth=3, min_samples_leaf=6) + ml_g = RandomForestRegressor(n_estimators=132, max_features=12, max_depth=5, min_samples_leaf=1) # to speed up the illustration we hard-code the simulation results - theta_nonorth = np.array([0.46220293, 0.36751016, 0.33633589, 0.40576358, 0.41204023, 0.44099791, 0.29077441, 0.51049579, 0.40872897, 0.34517717, 0.56135355, 0.20250422, 0.30014152, 0.3121669 , 0.48135903, 0.4250507 , 0.37715598, 0.37129675, 0.48191319, 0.45303646, 0.23189062, 0.47421963, 0.46100559, 0.51690426, 0.37912305, 0.3164967 , 0.5018225 , 0.34575523, 0.40272864, 0.31201177, 0.2868938 , 0.43818704, 0.5791864 , 0.37342275, 0.61239568, 0.35811336, 0.29545835, 0.49732933, 0.40667421, 0.32322828, 0.51692541, 0.2692527 , 0.14121754, 0.33655586, 0.33198124, 0.25389419, 0.38976939, 0.32516571, 0.24778959, 0.45216823, 0.62536581, 0.33626136, 0.48605982, 0.43367237, 0.22609483, 0.45695586, 0.51988 , 0.36181456, 0.3017518 , 0.35912742, 0.33766141, 0.28139109, 0.33423182, 0.28849494, 0.35598731, 0.30291489, 0.26696673, 0.3965472 , 0.30285673, 0.38740067, 0.38927095, 0.37276621, 0.24221399, 0.2478911 , 0.19369063, 0.49629465, 0.24420607, 0.46295666, 0.38999345, 0.28754147, 0.40750691, 0.3474621 , 0.2714744 , 0.40731088, 0.40115083, 0.3567057 , 0.39989027, 0.49022217, 0.25400176, 0.53195003, 0.37218689, 0.47224868, 0.29372151, 0.29748552, 0.15382988, 0.31676935, 0.4371993 , 0.58477685, 0.47081611, 0.24718853, 0.36163077, 0.24769679, 0.35941911, 0.3296207 , 0.30717881, 0.26779407, 0.3010835 , 0.31544093, 0.53523355, 0.32454461, 0.3090798 , 0.59057928, 0.40864869, 0.30559666, 0.51896589, 0.2602647 , 0.44938393, 0.40882983, 0.39964357, 0.18228496, 0.26144559, 0.40876431, 0.27486033, 0.41148144, 0.40479421, 0.37690703, 0.43214482, 0.41183592, 0.30884357, 0.4395378 , 0.30733564, 0.28518014, 0.51267318, 0.418407 , 0.21566437, 0.38225971, 0.44524558, 0.44356552, 0.30699452, 0.28843287, 0.29729574, 0.40998509, 0.3152678 , 0.37034559, 0.32783926, 0.21489544, 0.30194505, 0.33168137, 0.57711369, 0.42932541, 0.30693366, 0.47730183, 0.31486424, 0.49108048, 0.37679733, 0.13837656, 0.38903388, 0.27232008, 0.25654145, 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0.55846382, 0.60565034, 0.5072831 , 0.54395348, 0.40379622, 0.46622254, 0.6354662 , 0.42312813, 0.4616702 , 0.53952288, 0.4644455 , 0.53320828, 0.47985077, 0.67884889, 0.68665247, 0.55675144, 0.49726657, 0.51153323, 0.35450556, 0.38406104, 0.54364234, 0.54481311, 0.50429127, 0.51665527, 0.50536631, 0.62471313, 0.54995188, 0.51069711, 0.59585771, 0.49361382, 0.56525314, 0.53472606, 0.52108574, 0.59277627, 0.52030413, 0.53303672, 0.506324, 0.58464649, 0.51682546, 0.53908917, 0.48472873, 0.59931961, 0.52987924, 0.51830952, 0.5981733 , 0.52570183, 0.46157526, 0.33681896, 0.52118572, 0.55196885, 0.436666, 0.60214705, 0.47801228, 0.63529308, 0.66420412, 0.47089482, 0.50299033, 0.54153378, 0.50012678, 0.55512458, 0.61867588, 0.61297676, 0.51619218, 0.7108226 , 0.52405154, 0.50545777, 0.58881996, 0.6855725 , 0.44158941, 0.56506293, 0.57571301, 0.46155939, 0.55610752, 0.63709365, 0.53899487, 0.48867157, 0.57918563, 0.45980471, 0.56143977, 0.37378031, 0.63367835, 0.38939997, 0.55374046, 0.55710308, 0.33720147, 0.49855745, 0.54127316, 0.55849223, 0.42419311, 0.54393195, 0.41161067, 0.67767974, 0.41019723, 0.58881843, 0.61116325, 0.50547189, 0.54250286, 0.61225748, 0.44507568, 0.58904454, 0.53349416, 0.55814679, 0.58270325, 0.52712488, 0.56809314, 0.42769283, 0.58359583, 0.46196793, 0.48111024, 0.45197845, 0.43420935, 0.56175895, 0.53589292, 0.52315139, 0.50566554, 0.62762057, 0.56084874, 0.63904796, 0.5943957 , 0.53875507, 0.46986491, 0.6426977 , 0.49275138, 0.57931751, 0.70266163, 0.47655526, 0.62173036, 0.48717578, 0.59845574, 0.60794702, 0.42043813, 0.51655123, 0.4353051 , 0.47904624, 0.57368244, 0.5999047 , 0.52617388, 0.46936766, 0.45231895, 0.52393147, 0.43778395, 0.47623361, 0.47047228, 0.54637139, 0.53675411, 0.58225605, 0.44181779, 0.5102228 , 0.53705222, 0.65732853, 0.62801308, 0.38570668, 0.44181883, 0.65367556, 0.66954032, 0.53737963, 0.51781873, 0.55671749, 0.55356872, 0.49045887, 0.50322435, 0.58473624, 0.64025277, 0.50574165, 0.61981544, 0.61961095, 0.54121279, 0.47724547, 0.44325164, 0.53215914, 0.53575759, 0.50360814, 0.52205319, 0.60251245, 0.45673497, 0.54048138, 0.533345, 0.45471002, 0.54831181, 0.51394181, 0.46073889, 0.48973606, 0.51447332, 0.58756885, 0.45003506, 0.52646263, 0.58968246, 0.57353131, 0.55404332, 0.67903712, 0.43332497, 0.46782929, 0.55690244, 0.54962445, 0.37052726, 0.53694322, 0.57987237, 0.55581497, 0.58648345, 0.57742465, 0.61295249, 0.47082473, 0.49805351, 0.62943103, 0.4802072 , 0.61564795, 0.48670492, 0.55891522, 0.51289051, 0.46746993, 0.46179283, 0.52611937, 0.47259637, 0.46293598, 0.51716995, 0.43636162, 0.62709106, 0.62078752, 0.57865712, 0.57993246, 0.49595949, 0.5951574 , 0.60691815, 0.51491792, 0.5696144 , 0.4335185 , 0.61645309, 0.66783008, 0.41726308, 0.52357785, 0.57782111, 0.38525498, 0.40030394, 0.44383597, 0.48282351, 0.55315702, 0.52740055, 0.54191889, 0.58805125, 0.50005222, 0.62269219, 0.56192058, 0.69720681, 0.6420251 , 0.57585553, 0.65266588, 0.56394678, 0.55881803, 0.58853661, 0.62358828, 0.58900213, 0.55180601, 0.61334419, 0.47432699, 0.69636555, 0.47580728, 0.55062803, 0.5751468 , 0.40882965, 0.54733972, 0.53255764, 0.49895855, 0.5789291 , 0.47475823, 0.51362334, 0.50007997, 0.62200651, 0.65925014, 0.51606822, 0.49451512, 0.52548882, 0.41514725, 0.48751079, 0.61703613, 0.65643943, 0.62289755, 0.53872061, 0.54484886, 0.57697036, 0.59413314, 0.56346526, 0.53203294, 0.40909514, 0.50064258, 0.52035007, 0.5012625 , 0.48578658, 0.55824125, 0.36505325, 0.55176115, 0.41277624, 0.54335046, 0.58947315, 0.48733078, 0.63269652, 0.45560179, 0.67663234, 0.62937034, 0.53462866, 0.59090083, 0.41680354, 0.49839325, 0.47130792, 0.64484127, 0.67686348, 0.50108123, 0.48438192, 0.38428532, 0.43414261, 0.65013365, 0.5098531 , 0.60004169, 0.5697646 , 0.63651472, 0.50115709, 0.50382518, 0.53465802, 0.54041583, 0.59443663, 0.68394535, 0.43136825, 0.52093863, 0.492323, 0.60204015, 0.58571954, 0.48222725, 0.39901315, 0.50874368, 0.54478095, 0.41794544, 0.61308008, 0.56279119, 0.44958539, 0.53686155, 0.51535318, 0.4157253 , 0.52779584, 0.4433108 , 0.64116852, 0.41727246, 0.52716229, 0.66511726, 0.51072705, 0.4799285 , 0.4626921 , 0.43598565, 0.6236199 , 0.58418263, 0.54486924, 0.50675202, 0.4970176 , 0.61610926, 0.63488907, 0.63780451, 0.43754633, 0.44503914, 0.57447591, 0.56007183, 0.44319286, 0.51329395, 0.49871485, 0.47726431, 0.50767185, 0.49482428, 0.60439789, 0.4615146 , 0.53859255, 0.62038444, 0.64913913, 0.49418167, 0.60124017, 0.47822072, 0.49744256, 0.58586246, 0.50871457, 0.46203378, 0.56520742, 0.53317718, 0.55889281, 0.38172112, 0.45594551, 0.60580421, 0.52110075, 0.51444763, 0.52045491, 0.47283489, 0.34755781, 0.49803659, 0.48277783, 0.56328869, 0.45957583, 0.51915878, 0.60958301, 0.53878142, 0.50927979, 0.48057417, 0.44326046, 0.63453822, 0.62161229, 0.54641439, 0.54541847, 0.47026904, 0.60681138, 0.4997663 , 0.35607407, 0.55529754, 0.46402998, 0.3376591 , 0.51921144, 0.41938588, 0.548278, 0.51927345, 0.51137058, 0.66853675, 0.42879294, 0.52635556, 0.57270967, 0.41102776, 0.63381592, 0.47499351, 0.62242358, 0.54527768, 0.54904066, 0.52769862, 0.50747416, 0.62760339, 0.52289013, 0.37655823, 0.60151613, 0.63345199, 0.533801, 0.58577084]) + se_nonorth = np.array([0.04752632, 0.05252437, 0.04081372, 0.04851484, 0.04749561, 0.03818379, 0.04720593, 0.04623519, 0.04438108, 0.04704747, 0.04582967, 0.04103284, 0.04095206, 0.04473644, 0.0474447 , 0.04428884, 0.05291583, 0.04548479, 0.04127787, 0.04264639, 0.04136002, 0.04768211, 0.05077625, 0.0432164 , 0.04778945, 0.03571416, 0.0475896 , 0.04516676, 0.04991276, 0.05196731, 0.04048258, 0.03857071, 0.03671693, 0.03770704, 0.05089652, 0.0461722 , 0.03836501, 0.04226271, 0.05511125, 0.04450304, 0.0418925 , 0.03910182, 0.0449247 , 0.04629896, 0.04367623, 0.04376256, 0.04025187, 0.04698785, 0.0388972 , 0.05079093, 0.04002389, 0.04597267, 0.04720004, 0.0450238 , 0.0398305 , 0.04065346, 0.04406542, 0.04115013, 0.04865718, 0.04056625, 0.03925187, 0.0421328 , 0.04681513, 0.0439763 , 0.05035432, 0.04201041, 0.04668544, 0.04394981, 0.04567135, 0.04550151, 0.03709427, 0.04533645, 0.04038179, 0.04737091, 0.04547688, 0.04493072, 0.04029656, 0.0453583 , 0.04700267, 0.04726174, 0.04940269, 0.04655482, 0.04319775, 0.04573236, 0.04239855, 0.04643579, 0.0377732 , 0.04382505, 0.05463616, 0.03968628, 0.04575842, 0.04779906, 0.04300628, 0.05252161, 0.04393881, 0.04544886, 0.04236938, 0.03688514, 0.04746812, 0.0378483 , 0.05071514, 0.04644373, 0.05289415, 0.04719282, 0.04062532, 0.03272569, 0.04314814, 0.04073433, 0.04090295, 0.04302725, 0.05074448, 0.03953333, 0.04706719, 0.04636343, 0.04401789, 0.05329037, 0.04354488, 0.04171928, 0.04274639, 0.04327196, 0.05557301, 0.04892522, 0.04641841, 0.04540333, 0.03970085, 0.04825305, 0.04346595, 0.04280488, 0.04588877, 0.05058765, 0.04157508, 0.04732415, 0.04712535, 0.03734686, 0.04341058, 0.04598636, 0.05144396, 0.04306507, 0.04984114, 0.04101317, 0.04112485, 0.04503048, 0.03760539, 0.04674369, 0.04421478, 0.04468265, 0.03989578, 0.04154939, 0.04059497, 0.04524515, 0.04567546, 0.04919117, 0.04402552, 0.04256382, 0.04039857, 0.04127004, 0.04250842, 0.04436529, 0.04374075, 0.04671761, 0.04513487, 0.04592405, 0.04222837, 0.04096571, 0.04748011, 0.04810947, 0.04291903, 0.04158746, 0.05036586, 0.04856613, 0.04024124, 0.04255147, 0.05180867, 0.04174598, 0.03891627, 0.04144685, 0.04489533, 0.044931 , 0.04161421, 0.04375287, 0.03767374, 0.04130609, 0.04236881, 0.04844647, 0.05030521, 0.03866574, 0.03888635, 0.04925452, 0.04878495, 0.04701012, 0.06132132, 0.05171138, 0.05270967, 0.05174661, 0.04853573, 0.04839576, 0.04838823, 0.04515296, 0.04383871, 0.04472405, 0.04274227, 0.04431837, 0.05238373, 0.03970582, 0.05159934, 0.04642126, 0.04256806, 0.05103923, 0.04887934, 0.04601176, 0.0462473 , 0.04795098, 0.04403834, 0.0425341 , 0.04602572, 0.03943239, 0.04502544, 0.04592906, 0.04898315, 0.04152593, 0.04151051, 0.04045435, 0.04717776, 0.03928857, 0.04215977, 0.04481115, 0.04533248, 0.04506638, 0.03780037, 0.04497642, 0.03946211, 0.03870278, 0.04645436, 0.04390321, 0.0429915 , 0.03877647, 0.04317605, 0.04447082, 0.04380389, 0.0429053 , 0.04753429, 0.04872603, 0.04279851, 0.04675329, 0.04925358, 0.04557681, 0.04613846, 0.04374635, 0.04513399, 0.044832 , 0.04005348, 0.04870915, 0.04604291, 0.04508179, 0.04285389, 0.04576191, 0.04165158, 0.04003255, 0.04034477, 0.04507987, 0.04382144, 0.03962126, 0.04643146, 0.04497413, 0.03977343, 0.04236554, 0.0476213 , 0.04659588, 0.04008921, 0.04770813, 0.04437344, 0.04840868, 0.04700976, 0.04226155, 0.03719109, 0.0449964 , 0.04613542, 0.04166904, 0.04180454, 0.03868801, 0.04794871, 0.050236 , 0.038375 , 0.04443406, 0.04488851, 0.0480011 , 0.04926898, 0.04058049, 0.04281985, 0.05338039, 0.04353161, 0.03515195, 0.04341719, 0.0480715 , 0.04142801, 0.04610912, 0.04653238, 0.04254146, 0.04605789, 0.041817 , 0.04159624, 0.04436698, 0.04346619, 0.04216123, 0.04861165, 0.03987033, 0.03852878, 0.05134763, 0.04642145, 0.05162566, 0.05261917, 0.04179967, 0.04888241, 0.05090512, 0.04474789, 0.04407596, 0.05172393, 0.04751141, 0.03935585, 0.04963206, 0.03907181, 0.04180968, 0.0409687 , 0.04472983, 0.04861775, 0.0436817 , 0.04685306, 0.04050731, 0.04378468, 0.04514822, 0.04459432, 0.04066988, 0.05043818, 0.0472485 , 0.0411004 , 0.05049794, 0.04683268, 0.0513245 , 0.04593956, 0.03999829, 0.04140311, 0.0450592 , 0.04604977, 0.04726002, 0.04946856, 0.04769101, 0.04216129, 0.04838215, 0.04368365, 0.04347249, 0.0438762 , 0.0407478 , 0.05192566, 0.04615664, 0.04734136, 0.04603765, 0.04313318, 0.04724211, 0.04168363, 0.04094983, 0.0372976 , 0.04829181, 0.0446341 , 0.04769589, 0.04983229, 0.04015445, 0.04259574, 0.03966965, 0.04393717, 0.03807938, 0.04714134, 0.04191703, 0.03853675, 0.04181588, 0.04050847, 0.05163033, 0.04446326, 0.04501493, 0.04730852, 0.04157382, 0.03881549, 0.04233757, 0.04605436, 0.04617815, 0.05141122, 0.04477161, 0.05493223, 0.03734768, 0.04321097, 0.04090805, 0.05053544, 0.04355217, 0.04057866, 0.0399262 , 0.04216453, 0.05066079, 0.04253812, 0.05315858, 0.04169033, 0.04204156, 0.04193905, 0.04167116, 0.05251147, 0.05098957, 0.04557835, 0.0469838 , 0.04635001, 0.05180438, 0.06008725, 0.05141524, 0.04317844, 0.04091926, 0.04801724, 0.04130072, 0.04178246, 0.045185 , 0.04053058, 0.04737165, 0.0458431 , 0.0424302 , 0.04922552, 0.03756568, 0.04577134, 0.04048251, 0.044854 , 0.04179815, 0.04468901, 0.0570919 , 0.04727883, 0.04391833, 0.04408565, 0.03926577, 0.04921733, 0.0518598 , 0.04868963, 0.04498284, 0.04055751, 0.04909337, 0.04020206, 0.04224491, 0.04949091, 0.04230575, 0.04603067, 0.04389281, 0.04015306, 0.04804292, 0.04533735, 0.04336376, 0.04668769, 0.04377271, 0.04340193, 0.04902237, 0.05078105, 0.04378715, 0.04524027, 0.05113331, 0.0434951 , 0.04721075, 0.04122844, 0.03932327, 0.04374526, 0.04669035, 0.04187597, 0.04693448, 0.0426543 , 0.04738004, 0.04837465, 0.03902091, 0.04042924, 0.04449191, 0.04539387, 0.0526423 , 0.04508715, 0.0448976 , 0.04888229, 0.04107251, 0.0466375 , 0.04372284, 0.04710068, 0.05160466, 0.04580307, 0.04740671, 0.04718352, 0.05165969, 0.05177518, 0.04273175, 0.04386661, 0.04451535, 0.05204564, 0.04576443, 0.04588544, 0.04735161, 0.04342469, 0.04442042, 0.04698244, 0.04611934, 0.04691028, 0.0409842 , 0.03810167, 0.04575679, 0.04909961, 0.0454315 , 0.0427315 , 0.04633341, 0.04573215, 0.03881894, 0.0470943 , 0.04260926, 0.0464628 , 0.05029253, 0.04798311, 0.04606002, 0.04309083, 0.04775266, 0.05008584, 0.04025573, 0.0430455 , 0.04014136, 0.04486536, 0.04627781, 0.04798719, 0.03881474, 0.05517719, 0.04318348, 0.04514346, 0.04135471, 0.04287774, 0.04317367, 0.04828672, 0.03960944, 0.05080501, 0.04688066, 0.03913251, 0.05097918, 0.04619538, 0.04290995, 0.04692129, 0.04394257, 0.04717779, 0.04703665, 0.04442 , 0.04803709, 0.05008089, 0.0492336 , 0.04947701, 0.04524332, 0.04498547, 0.0481187 , 0.05048569, 0.04471597, 0.03972998, 0.04268966, 0.05178417, 0.04283741, 0.03857392, 0.04135066, 0.0456479 , 0.04507646, 0.04263503, 0.05813869, 0.04155548, 0.04791942, 0.04709474, 0.0460457 , 0.04312264, 0.05437283, 0.04088724, 0.05058245, 0.04867745, 0.04425616, 0.04100133, 0.03590292, 0.0444132 , 0.03899607, 0.04688414, 0.03770601, 0.04247705, 0.05147905, 0.04803248, 0.04426342, 0.05230619, 0.04657694, 0.04252559, 0.04361401, 0.05345193, 0.04602656, 0.04810443, 0.04183419, 0.04012391, 0.0407237 , 0.05076034, 0.04640053, 0.0449217 , 0.04254334, 0.0488389 , 0.04312078, 0.04395541, 0.04736764, 0.05048539, 0.05125508, 0.04774661, 0.04646159, 0.04833585, 0.04269022, 0.04084439, 0.03721398, 0.03979135, 0.03962285, 0.04042207, 0.03933019, 0.04165832, 0.04007042, 0.03939831, 0.04617874, 0.04997446, 0.0485718 , 0.04318544, 0.0456973 , 0.04705564, 0.04719067, 0.04101112, 0.04300064, 0.05181371, 0.04190036, 0.04324893, 0.04671167, 0.04282735, 0.04375806, 0.04936542, 0.04855079, 0.04841259, 0.04907001, 0.04262872, 0.04786209, 0.04330581, 0.04838113, 0.04345308, 0.04631977, 0.04332576, 0.04825251, 0.04271991, 0.04771019, 0.04091959, 0.04464475, 0.04717248, 0.04601467, 0.05137321, 0.04633675, 0.04536597, 0.04559622, 0.0439799 , 0.04061692, 0.05049085, 0.04156028, 0.03761666, 0.04067874, 0.05075575, 0.04185722, 0.03855531, 0.05061284, 0.03804943, 0.0445161 , 0.04970717, 0.0492808 , 0.04092466, 0.04726328, 0.04916354, 0.04789055, 0.04637153, 0.04378265, 0.04045451, 0.04161615, 0.04225726, 0.03974276, 0.04122544, 0.04129785, 0.04783874, 0.04517845, 0.0474952 , 0.04806748, 0.04253693, 0.04535949, 0.0469902 , 0.05479359, 0.04487014, 0.05138728, 0.04738945, 0.04542307, 0.04370665, 0.03788868, 0.04488763, 0.04978662, 0.04543908, 0.04260334, 0.04680035, 0.04977666, 0.04518287, 0.05047852, 0.04451354, 0.04672927, 0.0461525 , 0.04004093, 0.05181208, 0.03881654, 0.04644726, 0.04902551, 0.0388593 , 0.04177445, 0.03758699, 0.04566246, 0.0505544 , 0.0468971 , 0.03927459, 0.0488062 , 0.042157 , 0.05119987, 0.05186735, 0.04075175, 0.04237306, 0.04548427, 0.04691215, 0.04066061, 0.04071047, 0.0410111 , 0.04652544, 0.03910531, 0.04045403, 0.04553288, 0.04408173, 0.04414605, 0.04027273, 0.04410796, 0.04777986, 0.04729922, 0.04583422, 0.04112711, 0.04720161, 0.04526709, 0.05064341, 0.04711904, 0.04717546, 0.04783787, 0.04000833, 0.04300354, 0.04214608, 0.04428618, 0.04996382, 0.04440921, 0.04582499, 0.04314297, 0.04184367, 0.0431282 , 0.04207993, 0.04819509, 0.04354023, 0.04200337, 0.04735732, 0.04437863, 0.04245207, 0.03971191, 0.035791 , 0.0504074 , 0.04056927, 0.05130085, 0.04418714, 0.04128746, 0.04495476, 0.04674706, 0.04491591, 0.04365693, 0.04704554, 0.04046825, 0.04227887, 0.04637332, 0.04543442, 0.04557147, 0.04245467, 0.04896935, 0.05097535, 0.05108846, 0.04589934, 0.04796408, 0.04678668, 0.04427926, 0.03908375, 0.04111836, 0.04338717, 0.04695997, 0.04225651, 0.03826845, 0.04402799, 0.04661007, 0.04384848, 0.04797444, 0.04768971, 0.048136 , 0.04301756, 0.04471567, 0.04199196, 0.04553344, 0.04615266, 0.05142863, 0.04415546, 0.04175551, 0.04490888, 0.0455252 , 0.04554681, 0.04598016, 0.04836324, 0.04980829, 0.04725647, 0.04770914, 0.03977447, 0.04205265, 0.05306058, 0.04422718, 0.04203523, 0.04359023, 0.04355804, 0.04075618, 0.04375993, 0.04617599, 0.04208219, 0.04199941, 0.0409864 , 0.04707944, 0.04216365, 0.05933166, 0.04383176, 0.04528336, 0.04310086, 0.04630154, 0.04377311, 0.04063786, 0.0354718 , 0.05156779, 0.03882174, 0.0435703 , 0.04106655, 0.03698484, 0.04070984, 0.05121285, 0.05164795, 0.04328943, 0.04556888, 0.0468324 , 0.05077517, 0.04482054, 0.04279588, 0.04729371, 0.05248564, 0.04496889, 0.04524623, 0.0391199 , 0.04279905, 0.03831627, 0.04998856, 0.04125977, 0.04932733, 0.0410571 , 0.04878879, 0.04462988, 0.04413883, 0.05073432, 0.04503141, 0.04450833, 0.0489624 , 0.04622872, 0.04575345, 0.04430143, 0.05610003, 0.04686195, 0.04166272, 0.03959071, 0.03956318, 0.04595394, 0.04153185, 0.04176013, 0.04323898, 0.04279855, 0.04505489, 0.04003259, 0.04134781, 0.04285591, 0.04886803, 0.0543866 , 0.04069868, 0.04278918, 0.04668385, 0.04295076, 0.04597368, 0.04101972, 0.04297835, 0.03651599, 0.0462089 , 0.04459918, 0.04383747, 0.05220179, 0.05259663, 0.04050992, 0.04628942, 0.05149538, 0.04670411, 0.04981779, 0.04639563, 0.049215 , 0.04375691, 0.03838808, 0.04651726, 0.04386707, 0.04959572, 0.04719161, 0.04231231, 0.04785895, 0.03947752, 0.04925271, 0.04616122, 0.03791607, 0.04701615, 0.04272888, 0.05732519, 0.04058686, 0.04179869, 0.05142427, 0.0407993 , 0.04770618, 0.04845589, 0.04515179, 0.0399421 , 0.0493371 , 0.04494348, 0.04021067, 0.04180386, 0.0474344 , 0.04993469, 0.04580776, 0.04513645, 0.04563901, 0.04458455, 0.04670768, 0.0463566 , 0.05037703, 0.04200332, 0.04363388, 0.04013193, 0.03818682, 0.04708778, 0.05157659, 0.0438879 , 0.05193443, 0.04417957, 0.0443017 , 0.04444654, 0.04645802, 0.04208909, 0.04790682, 0.04689531, 0.04812119, 0.04436477, 0.04683151, 0.04755189, 0.0445168 , 0.04429936, 0.04130354, 0.04965516, 0.04633012, 0.04019696, 0.04630375, 0.0372558 , 0.04502071, 0.04541922, 0.04117667, 0.04044187, 0.0503596 , 0.04525318, 0.04296958, 0.04744989, 0.04244503, 0.04556984, 0.04867774, 0.03669044, 0.04222442, 0.04536255, 0.0436591 , 0.04574726, 0.05396777, 0.04394745, 0.0479081 , 0.04133669, 0.04493979, 0.04831306, 0.04299433, 0.04304778, 0.04357632, 0.05194496, 0.04887002, 0.04466858, 0.0503679 , 0.05755543, 0.04690671]) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for i_rep in range(n_rep): @@ -210,13 +145,25 @@ other half of observations indexed with :math:`i \in I` score=non_orth_score) obj_dml_plr_nonorth.fit() this_theta = obj_dml_plr_nonorth.coef[0] + this_se = obj_dml_plr_nonorth.se[0] # we show that the loaded result matches the just computed print(np.abs(theta_nonorth[i_rep] - this_theta)) + print(np.abs(se_nonorth[i_rep] - this_se)) theta_nonorth[i_rep] = this_theta - - ax = sns.kdeplot(theta_nonorth, shade=True, color=colors[1]) + se_nonorth[i_rep] = this_se + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_nonorth - alpha)/se_nonorth, + color=face_colors[0], edgecolor = edge_colors[0], + stat='density', bins=30, label='Non-orthogonal ML'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); @savefig nonorth.png width=5in - ax.axvline(0.5, color='k', label='True theta'); + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); .. tabbed:: R @@ -233,21 +180,28 @@ other half of observations indexed with :math:`i \in I` .. jupyter-execute:: + library(ggplot2) library(mlr3) library(mlr3learners) library(data.table) + lgr::get_logger("mlr3")$set_threshold("warn") + options(repr.plot.width=5, repr.plot.height=4) + set.seed(1111) - learner = lrn("regr.ranger", num.trees=100, mtry=n_vars, min.node.size=2, max.depth=5) - ml_m = learner$clone() - ml_g = learner$clone() + ml_m = lrn("regr.ranger", num.trees = 378, max.depth = 3, mtry = 20, min.node.size = 6) + ml_g = lrn("regr.ranger", num.trees = 132, max.depth = 5, mtry = 12, min.node.size = 1) # to speed up the illustration we hard-code the simulation results - theta_nonorth = theta_nonorth = c(0.335140973, 0.419119658, 0.543364882, 0.465531595, 0.489423796, 0.580690927, 0.446899757, 0.272292296, 0.240826014, 0.365608710, 0.268312379, 0.354473099, 0.385506555, 0.615247222, 0.463371670, 0.410046844, 0.544959995, 0.489229912, 0.457527626, 0.614892818, 0.338072758, 0.394127768, 0.276357084, 0.433449188, 0.249084049, 0.528947106, 0.451120107, 0.327303152, 0.546307385, 0.395934562, 0.301388718, 0.239982004, 0.530202292, 0.169856178, 0.322835980, 0.065608845, 0.154554350, 0.430897491, 0.424487779, 0.324143676, 0.389261597, 0.272135966, 0.351868551, 0.256792473, 0.504886282, 0.474738538, 0.384538245, 0.483251852, 0.363105174, 0.216916587, 0.294053708, 0.479088598, 0.269570359, 0.377226775, 0.437593329, 0.225371600, 0.470220118, 0.580390685, 0.304737611, 0.305733674, 0.354465325, 0.436580536, 0.445360886, 0.368723539, 0.157181386, 0.155664995, 0.464140152, 0.480734551, 0.362887864, 0.337695182, 0.642324202, 0.310090948, 0.349702691, 0.305153800, 0.316219058, 0.381051853, 0.320910647, 0.469491087, 0.424923136, 0.429822780, 0.275509449, 0.323183059, 0.555469839, 0.191757655, 0.236563643, 0.431688361, 0.428128667, 0.352089478, 0.099148330, 0.444152504, 0.394307625, 0.255398535, 0.438946752, 0.253313216, 0.112699448, 0.477861515, 0.458580858, 0.583866474, 0.340893368, 0.591232974, 0.346205252, 0.543514451, 0.460809329, 0.231994475, 0.284534860, 0.328665723, 0.316603063, 0.364736497, 0.326819423, 0.360088787, 0.437105307, 0.340068129, 0.437790456, 0.293644690, 0.256570227, 0.391332290, 0.379705449, 0.345781442, 0.192708760, 0.495761272, 0.357445752, 0.289328987, 0.271279462, 0.246514418, 0.321481385, 0.547511147, 0.323289625, 0.421790101, 0.361704227, 0.183561036, 0.240601590, 0.373376260, 0.418223589, 0.234931680, 0.298725451, 0.334964767, 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0.258199280, 0.430870900, 0.325387453, 0.366884342, 0.388394987, 0.332531847, 0.456149765, 0.480522342, 0.198383637, 0.200533915, 0.269881924, 0.339918216, 0.453854936, 0.473168756, 0.460025119, 0.284768638, 0.360963578, 0.210442939, 0.299954368, 0.389141673, 0.361433498, 0.443818155, 0.380874367, 0.315574954, 0.514698977, 0.369786833, 0.342736404, 0.496349510, 0.539101645, 0.301455561, 0.273359613, 0.470802282, 0.435284567, 0.515617885, 0.284359396, 0.311403925, 0.324005567, - 0.42139206, 0.25604031, 0.36201789, 0.10846016, 0.41784835, 0.34505290, 0.35388876, 0.74150699, 0.16133690, 0.29388264, 0.47603740, 0.26673708, 0.36747984, 0.32719126, 0.42509806, 0.35977158, 0.33368013, 0.50825836, 0.43838805, 0.44730481, 0.34552361, 0.35458993, 0.23337476, 0.26513718, 0.47182588, 0.44153584, 0.47829997, 0.26140131, 0.25369408, 0.38922793, 0.15108407, 0.21647681, 0.43478364, 0.38368258, 0.36937969, 0.39530397, 0.47783846, 0.47734570, 0.23662392, 0.44670463, 0.48737923, 0.48664646, 0.37433932, 0.40340501, 0.42819694, 0.50312924, 0.26157946, 0.31198968, 0.39955912, 0.33634231, 0.33144015, 0.31181757, 0.33995580, 0.43817849, 0.30326375, 0.31102104, 0.32270600, 0.35016055, 0.40434170, 0.46299218, 0.48458540, 0.32291477, 0.38090308, 0.37289522, 0.49450332, 0.37622246, 0.26345370, 0.21200119, 0.20610146, 0.40286913, 0.30702282, 0.24507251, 0.34157918, 0.34007658, 0.32387305, 0.35049316, 0.37436835, 0.34540160, 0.48407376, 0.46762908, 0.37074398, 0.30789111, 0.35586455, 0.34952500, 0.50174516, 0.37325996, 0.27937008, 0.40428039, 0.33630248, 0.29045981, 0.36996854, 0.38606302, 0.29149853, 0.47210709, 0.51180091, 0.29678938, 0.35412552, 0.39716491, 0.28452926, 0.26660583, 0.46892162, 0.30673355, 0.45047690, 0.42939064, 0.45773370, 0.32853933, 0.46402431, 0.31983450, 0.29105138, 0.18750465, 0.32012184, 0.37348488, 0.37709558, 0.30038336, 0.35647914, 0.51807477, 0.20440505, 0.25672821, 0.27799653, 0.35506732, 0.34781610, 0.50966358, 0.35900531, 0.51515573, 0.41425917, 0.32255047, 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0.0471457232, 0.0432730238, 0.0420443922, 0.0425626359, 0.0400757175, 0.0428709057, 0.0523370930, 0.0457095456, 0.0387461239, 0.0454164368, 0.0474067583, 0.0422056453, 0.0423306811, 0.0458742009, 0.0418851150, 0.0375060607, 0.0476444692, 0.0473158918, 0.0437627534, 0.0489777285, 0.0459813410, 0.0485886882, 0.0473717508, 0.0406818520, 0.0462999317, 0.0363856680, 0.0450027066, 0.0492219529, 0.0390316111, 0.0461459314, 0.0509053393, 0.0460205187, 0.0428744223, 0.0421943189, 0.0403611049, 0.0441886496, 0.0496008183, 0.0422805921, 0.0532239884, 0.0416174164, 0.0459029114, 0.0398305637, 0.0467203692, 0.0451460843, 0.0544602119, 0.0420129382, 0.0461996776, 0.0444963448, 0.0491157891, 0.0470482264, 0.0367606777, 0.0444011869, 0.0476917130, 0.0387451532, 0.0484612156, 0.0520540431, 0.0426581207, 0.0530953686, 0.0488003880, 0.0452046344, 0.0456663570, 0.0504463665, 0.0395335748, 0.0418899004, 0.0404035187, 0.0448713245, 0.0476916901, 0.0461400539, 0.0403482903, 0.0420793960, 0.0469157779, 0.0426540095, 0.0411240546, 0.0475006544, 0.0442164510, 0.0427828038, 0.0493846273, 0.0438576577, 0.0406313613, 0.0465298653, 0.0445994443, 0.0463059937, 0.0438992617, 0.0468660278, 0.0411646643, 0.0516189392, 0.0417182243, 0.0447124494, 0.0460886082, 0.0503461423, 0.0495413210, 0.0418845428, 0.0470049760, 0.0398121373, 0.0485289895, 0.0503722920, 0.0435886254, 0.0489487204, 0.0494519494, 0.0438545065, 0.0397148739, 0.0520638856, 0.0406954281, 0.0474046990, 0.0460098363, 0.0407100521, 0.0434347753, 0.0377814424, 0.0450191260, 0.0532696689, 0.0428635977, 0.0462511433, 0.0451252219, 0.0595109881, 0.0415188135, 0.0462933305, 0.0399142707, 0.0457308366, 0.0458449235, 0.0466075330, 0.0459267034, 0.0452804927, 0.0450031724, 0.0405805835, 0.0481327093, 0.0446886594, 0.0407773776, 0.0428498566, 0.0418913126, 0.0416114384, 0.0482691372, 0.0468687012, 0.0475622762, 0.0451409043, 0.0440544641, 0.0466109319, 0.0444994155, 0.0395735845, 0.0489770988, 0.0513474350, 0.0476471824, 0.0474166294, 0.0422779747, 0.0483946630, 0.0460189236, 0.0433891859, 0.0480850303, 0.0505456668, 0.0461110607, 0.0392525139, 0.0440482730, 0.0510736230, 0.0430907766, 0.0513039372, 0.0437664660, 0.0478224785, 0.0387364981, 0.0391934132, 0.0443478705, 0.0468773935, 0.0450421815, 0.0494911337, 0.0413486799, 0.0422647277, 0.0385349835, 0.0561999522, 0.0380104667, 0.0524001385, 0.0497509208, 0.0494165088, 0.0506288203, 0.0499250336, 0.0483102620, 0.0444882253, 0.0432652527, 0.0489046969, 0.0570060453, 0.0359903313, 0.0503936218, 0.0444652977, 0.0449598097, 0.0525180496, 0.0383749194, 0.0522110450, 0.0430210793, 0.0405819589, 0.0393943644, 0.0488834320, 0.0437871919, 0.0419881386, 0.0502366835, 0.0483239506, 0.0454876973, 0.0393104070, 0.0387805425, 0.0507050119, 0.0417782241, 0.0455323571, 0.0439461195, 0.0398420253, 0.0418836739, 0.0478967241, 0.0437425739, 0.0448702453, 0.0396011810, 0.0437846518, 0.0474342375) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for (i_rep in seq_len(n_rep)) { for (i_rep in seq_len(1)) { @@ -260,22 +214,33 @@ other half of observations indexed with :math:`i \in I` apply_cross_fitting=FALSE) obj_dml_plr_nonorth$fit() this_theta = obj_dml_plr_nonorth$coef + this_se = obj_dml_plr_nonorth$se print(abs(theta_nonorth[i_rep] - this_theta)) + print(abs(se_nonorth[i_rep] - this_se)) theta_nonorth[i_rep] = this_theta + se_nonorth[i_rep] = this_se } - g_nonorth = ggplot(data.frame(theta_nonorth), aes(x = theta_nonorth)) + - geom_density(fill = "dark orange", alpha = 0.3, color = "dark orange") + - geom_vline(aes(xintercept = alpha), col = "black") + - xlim(c(0.08, 0.75)) + xlab("") + ylab("") + theme_minimal() + g_nonorth = ggplot(data.frame(theta_rescaled=(theta_nonorth - alpha)/se_nonorth)) + + geom_histogram(aes(y=..density.., x=theta_rescaled, colour = "Non-orthogonal ML", fill="Non-orthogonal ML"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Non-orthogonal ML", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", "N(0, 1)"='black')) + + scale_fill_manual(name='',, + breaks=c("Non-orthogonal ML", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() g_nonorth -The regularization bias in the simple ML-approach is caused by the slow convergence of :math:`\hat{\theta}` +The regularization bias in the simple ML-approach is caused by the slow convergence of :math:`\hat{\theta}_0` .. math:: - |\sqrt{n} (\hat{\theta} - \theta) | \rightarrow_{P} \infty + |\sqrt{n} (\hat{\theta}_0 - \theta_0) | \rightarrow_{P} \infty i.e., slower than :math:`1/\sqrt{n}`. The driving factor is the bias that arises by learning :math:`g` with a random forest or any other ML technique. @@ -283,7 +248,7 @@ A heuristic illustration is given by .. math:: - \sqrt{n}(\hat{\theta} - \theta) = \underbrace{\left(\frac{1}{n} \sum_{i\in I} D_i^2\right)^{-1} \frac{1}{n} \sum_{i\in I} D_i \zeta_i}_{=:a} + \sqrt{n}(\hat{\theta}_0 - \theta_0) = \underbrace{\left(\frac{1}{n} \sum_{i\in I} D_i^2\right)^{-1} \frac{1}{n} \sum_{i\in I} D_i \zeta_i}_{=:a} + \underbrace{\left(\frac{1}{n} \sum_{i\in I} D_i^2\right)^{-1} \frac{1}{n} \sum_{i\in I} D_i (g_0(X_i) - \hat{g}_0(X_i))}_{=:b}. :math:`a` is approximately Gaussian under mild conditions. @@ -299,7 +264,7 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate .. math:: - \check{\theta} = \left(\frac{1}{n} \sum_{i\in I} \hat{V}_i D_i\right)^{-1} \frac{1}{n} \sum_{i\in I} \hat{V}_i (Y_i - \hat{g}_0(X_i)). + \check{\theta}_0 = \left(\frac{1}{n} \sum_{i\in I} \hat{V}_i D_i\right)^{-1} \frac{1}{n} \sum_{i\in I} \hat{V}_i (Y_i - \hat{g}_0(X_i)). .. tabbed:: Python @@ -309,7 +274,8 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate np.random.seed(2222) # to speed up the illustration we hard-code the simulation results - theta_orth_nosplit = np.array([0.28310852, 0.2242779 , 0.21826119, 0.27480264, 0.2632724 , 0.26145622, 0.35155547, 0.29961815, 0.26635995, 0.2916099 , 0.26905356, 0.20612577, 0.26398208, 0.24738297, 0.31720634, 0.33410743, 0.22858844, 0.27328181, 0.2883969 , 0.32534549, 0.23480727, 0.27738886, 0.24176323, 0.30223882, 0.25703061, 0.24784644, 0.31566354, 0.27268702, 0.26854096, 0.17321738, 0.25972241, 0.22442174, 0.29994908, 0.32036162, 0.32746914, 0.24166733, 0.19887599, 0.30592848, 0.22804781, 0.23233486, 0.30921309, 0.23463971, 0.29075226, 0.1809537 , 0.27576646, 0.27086624, 0.24678385, 0.20240443, 0.25876583, 0.25828577, 0.34559146, 0.26086781, 0.33256474, 0.29259962, 0.25623473, 0.26968172, 0.26378077, 0.19208867, 0.24181579, 0.29534961, 0.2733621 , 0.22095648, 0.28348618, 0.30043176, 0.25261295, 0.26165325, 0.28380731, 0.23151629, 0.25394419, 0.23464047, 0.24771 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0.2239705 , 0.20986456, 0.2981559 , 0.22735091, 0.2385243 , 0.28349625, 0.26244455, 0.22222827, 0.25961584, 0.23382021, 0.28271075, 0.27364524, 0.22208577, 0.22338154, 0.25133704, 0.2927259 , 0.27249243, 0.19810369, 0.29739084, 0.23977764, 0.33163001, 0.20838183, 0.32524989, 0.22693308, 0.24781074, 0.26384576, 0.24114416, 0.22417866, 0.26879515, 0.2671183 , 0.27647383, 0.21137266, 0.25714512, 0.31201653, 0.28097195, 0.23299879, 0.2663719 , 0.28909349, 0.2835265 , 0.25468493, 0.24928326, 0.26264776, 0.28256157, 0.19424368, 0.2629451 , 0.29564766, 0.23307118, 0.21502833, 0.28635937, 0.28488341, 0.2700886 , 0.28704458, 0.27922478, 0.17263791, 0.28635712, 0.23075424, 0.17741433, 0.23449618, 0.18159064, 0.25603823, 0.24429592, 0.252812 , 0.27307543, 0.34339119, 0.252644 , 0.29226866, 0.19520718, 0.27469428, 0.28431326, 0.2082317 , 0.28421696, 0.3070267 , 0.30473623, 0.27110717, 0.28557713, 0.25306611, 0.20115915, 0.28109259, 0.23066984, 0.29306542, 0.23246166, 0.26619353, 0.26780462, 0.21972068, 0.21848152, 0.33710686, 0.2530146 , 0.25794811, 0.26789696, 0.20139948, 0.31620582, 0.18905824, 0.30757573, 0.25823602, 0.32225168, 0.29416294, 0.33819316, 0.33667058, 0.24208325, 0.24359405, 0.26374286, 0.28239228, 0.23335644, 0.27874058, 0.30653043, 0.26781161, 0.2588007 , 0.26839054, 0.28338128, 0.28230784, 0.28294972, 0.26593038, 0.33051548, 0.31535588, 0.22534891, 0.29542099, 0.25698647, 0.30864245, 0.21753042, 0.25400367, 0.22775759, 0.19603169, 0.23960378, 0.31154354, 0.27754087, 0.20881702, 0.28891996, 0.28937791, 0.29420089, 0.26498318, 0.30885406, 0.26559376, 0.27107953, 0.31337713, 0.21710285, 0.24758666, 0.24602542, 0.2719711 , 0.29087773, 0.22520773, 0.22748796, 0.20393595, 0.28756757, 0.14159469, 0.30246539, 0.20772598, 0.2444757 , 0.25797414, 0.27913936, 0.13832664, 0.15671679, 0.3150215 , 0.21286612, 0.30024509, 0.25108497, 0.33896346, 0.23340797, 0.32430489, 0.27986208, 0.24811693, 0.18176737, 0.23715368, 0.19188467, 0.28837014, 0.28031912, 0.26374323, 0.2776042 , 0.19989007, 0.27678714, 0.29959389, 0.19771432, 0.2368231 , 0.36130171, 0.23823292, 0.22617482, 0.24363743, 0.28811019, 0.31579602, 0.25930383, 0.22536299, 0.29135006, 0.28875076]) + theta_orth_nosplit = np.array([0.5269861 , 0.48682698, 0.45808479, 0.46591054, 0.47453753, 0.474066 , 0.50998524, 0.45472461, 0.51296185, 0.57402367, 0.49932393, 0.51328342, 0.53728227, 0.38413749, 0.46069765, 0.47095928, 0.49359267, 0.44035397, 0.50447968, 0.47268968, 0.46074218, 0.41151753, 0.434847 , 0.4873961 , 0.46688001, 0.43024809, 0.49051175, 0.45356252, 0.45427558, 0.50022074, 0.48977533, 0.56098269, 0.38961805, 0.41125901, 0.4748894 , 0.42412004, 0.44705749, 0.5231581 , 0.49033896, 0.48503933, 0.43015338, 0.44154682, 0.4902393 , 0.38245209, 0.51450476, 0.47647825, 0.50007773, 0.47903055, 0.49418268, 0.56080375, 0.45953682, 0.46652166, 0.51456846, 0.50829186, 0.54538745, 0.41911111, 0.50936297, 0.53669094, 0.42254389, 0.4544877 , 0.50019279, 0.52443831, 0.52700521, 0.48787306, 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0.47329737, 0.48512522, 0.45297114, 0.56179915, 0.47435093, 0.4592582 , 0.39308681, 0.51746997, 0.55920064, 0.53658168, 0.49889319, 0.46111205, 0.44999989, 0.40907121, 0.46986319, 0.43990175, 0.55234788, 0.53661098, 0.49330377, 0.4040682 , 0.48534139, 0.42742481, 0.51077351, 0.49533248, 0.42045776, 0.5344603 , 0.40972033, 0.39091224, 0.5215851 , 0.53294953, 0.48338073, 0.47730069, 0.44219701, 0.47943696, 0.56002162, 0.54204664, 0.47793652, 0.44526255, 0.5103111 , 0.46524098, 0.42589576, 0.46350273, 0.43878478, 0.50774426, 0.45034939, 0.47611014, 0.53052248, 0.49346943, 0.51199455, 0.48761231, 0.43072246, 0.55977859, 0.46252589, 0.51027744, 0.49178512, 0.43658063, 0.44140418, 0.54593446, 0.5099492 , 0.48637227, 0.44615891, 0.52222773, 0.49006639, 0.44954301, 0.42483585, 0.45181009, 0.42576419, 0.50461041, 0.51571916, 0.46911646, 0.49450105, 0.49478691, 0.50042925, 0.52024835, 0.55977333, 0.46990362, 0.52257744, 0.46704933, 0.46301227, 0.46681985, 0.40915423, 0.39569977, 0.52492256, 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0.41400792, 0.52320126, 0.45159391, 0.51923563, 0.42497262, 0.52047988, 0.53274828, 0.40228563, 0.45588327, 0.48982012, 0.47352543, 0.47223994, 0.53485516, 0.4118618 , 0.51781447, 0.46538265, 0.52966071, 0.52424753, 0.47217289, 0.46273616, 0.51869698, 0.46317502, 0.48169949, 0.41855489, 0.44349482, 0.54201979, 0.47137472, 0.52945612, 0.44409319, 0.50464629, 0.43547169, 0.47253531, 0.43410852, 0.39608272, 0.51737693, 0.54404677, 0.44838106, 0.47019746, 0.51934514, 0.48196157, 0.5892373 , 0.46130456, 0.48244219, 0.50055673, 0.46595818, 0.49014524, 0.50559026, 0.55906078, 0.43110781, 0.49156007, 0.46765774, 0.52517817, 0.51003126, 0.42910519, 0.4654511 , 0.44930796, 0.52347481, 0.4193886 , 0.48611425, 0.45503422, 0.44758526, 0.51913489, 0.48615427, 0.45915381, 0.48966807, 0.41589988, 0.52822353, 0.49645715, 0.48918748, 0.47859385, 0.48490774, 0.39952061, 0.53175053, 0.50365121, 0.42264625, 0.45147569, 0.50496877, 0.48535063, 0.50744424, 0.51465332, 0.42414122, 0.45978169, 0.46411468, 0.49467868, 0.54409124, 0.47484253, 0.5068238 , 0.49396259, 0.53964916, 0.51172738, 0.47625784, 0.44923915, 0.48536552, 0.53069993, 0.49840198, 0.51187068, 0.46036831, 0.59581793, 0.48621146, 0.51410491, 0.4376168 , 0.52375321, 0.50083837, 0.46521782, 0.43536398, 0.47399854, 0.57235617, 0.48849083, 0.49901722, 0.50293966, 0.57851074, 0.39839549, 0.53103419, 0.54299908, 0.43289476, 0.46317743, 0.50137498, 0.51942289, 0.48557863, 0.4876492 , 0.48626463, 0.43484103, 0.4987074 , 0.56855659, 0.47981524, 0.41360043, 0.48092136, 0.44830716, 0.43753123, 0.45849459, 0.38110897, 0.47937367, 0.49034043, 0.46447731, 0.46973607, 0.49351532, 0.41452509, 0.42686193, 0.42527102, 0.53671829, 0.45457996, 0.46858151, 0.46619171, 0.47110253, 0.52061093, 0.57677477, 0.4699085 , 0.4496484 , 0.51996 , 0.46174003, 0.56715301, 0.42983828, 0.47521886, 0.52808329, 0.44851982, 0.4396226 , 0.43644067, 0.46111939, 0.46671984, 0.47274524, 0.45005976, 0.47461079, 0.44815431, 0.50703117, 0.45689512, 0.55516936, 0.55672735, 0.45293766, 0.48078288, 0.53844884, 0.43451484, 0.4775106 , 0.47762549, 0.50073684, 0.45568321, 0.50863967, 0.46257647, 0.55713696, 0.42439858, 0.53324603, 0.50664103, 0.42667372, 0.48713298, 0.43443033, 0.41249548, 0.43178557, 0.49643472, 0.5357908 , 0.49244275, 0.48560644, 0.57357695, 0.47801147, 0.4656959 , 0.45555179, 0.38060909, 0.44908041, 0.51529283, 0.51116703, 0.51160535, 0.56733604, 0.4637596 , 0.5161148 , 0.49774216, 0.46608273, 0.45297828, 0.35560759, 0.54295982, 0.45633101, 0.47249958, 0.43947559, 0.54984426, 0.43906653, 0.53537798, 0.45456857, 0.48796028, 0.52143596, 0.49293529, 0.53605175, 0.40139111, 0.50600271, 0.51908615, 0.47864096, 0.48689579, 0.50902385, 0.50369694, 0.43217983, 0.50677829, 0.49677785, 0.45289065, 0.41667242, 0.40712091, 0.45834197, 0.5057925 , 0.47651157, 0.54407932, 0.50013731, 0.48122896, 0.51997758, 0.47265008, 0.52992725, 0.40527474, 0.52039882, 0.48131845, 0.4510595 , 0.40413769, 0.52876278, 0.47489569, 0.4596825 , 0.5314986 , 0.42287487, 0.52030867, 0.49836773, 0.42014062, 0.49254797, 0.538642 , 0.42061216, 0.46500315, 0.43892295, 0.43702694, 0.49656952, 0.42301431, 0.51570935, 0.37316705, 0.5071268 , 0.52374101, 0.46331057, 0.48839092, 0.45366578, 0.42403456, 0.55836914, 0.48513398, 0.50529783, 0.45320212, 0.4491557 , 0.53586077, 0.57710138, 0.49492567, 0.47280252, 0.43089228, 0.53389067, 0.48039643, 0.46776854, 0.51406094, 0.42801787, 0.52341148, 0.46519127, 0.43181267, 0.56571248, 0.52404914, 0.49590624, 0.52889378, 0.52266558, 0.49889508, 0.4612402 , 0.47452792, 0.46355367, 0.54064643, 0.5086885 , 0.43543993, 0.47346238, 0.42796893, 0.47182714, 0.4295712 , 0.4643917 , 0.49052014, 0.57363413, 0.48587792, 0.45667329, 0.52436139, 0.39282397, 0.49423503, 0.48071968, 0.44336415, 0.47717448, 0.46242517, 0.51544736, 0.4538936 , 0.48770825, 0.39656014, 0.53140325, 0.52410917, 0.47079746, 0.46092059, 0.5299388 , 0.41879821, 0.49509749, 0.45823221, 0.41639722, 0.48395687, 0.4033108 , 0.47893983, 0.48929843, 0.46444785, 0.43743348, 0.47776494, 0.4557678 , 0.53800864, 0.46523766, 0.48527377, 0.45602462, 0.46100225, 0.57398326, 0.53971796, 0.55100335, 0.49459865, 0.47626581, 0.46692972, 0.36834415, 0.54171753, 0.51124387, 0.46717469, 0.51236106, 0.48676235, 0.45806207, 0.5146505]) + se_orth_nosplit = np.array([0.03300109, 0.03481441, 0.03023695, 0.03309708, 0.03608485, 0.03213616, 0.03648971, 0.03603703, 0.03801085, 0.03446007, 0.03402841, 0.03384802, 0.02981774, 0.03169046, 0.03356073, 0.03099595, 0.03922716, 0.03448312, 0.03044878, 0.03433093, 0.03455709, 0.04124444, 0.03382217, 0.03354967, 0.03725346, 0.02786431, 0.03397265, 0.03386515, 0.03639662, 0.03304494, 0.03120514, 0.03220385, 0.03133162, 0.03302797, 0.03650748, 0.03161229, 0.03712612, 0.0344661 , 0.03387415, 0.03423439, 0.03472111, 0.03380087, 0.0349508 , 0.03290457, 0.03236628, 0.03372614, 0.03360537, 0.03333013, 0.03117967, 0.03528887, 0.03265259, 0.03079767, 0.03340276, 0.03616061, 0.03364504, 0.0375724 , 0.03255368, 0.03167439, 0.03308361, 0.03185402, 0.03473072, 0.03695189, 0.03641993, 0.0330973 , 0.03363454, 0.0345028 , 0.03639678, 0.03246405, 0.03399911, 0.03587028, 0.03256571, 0.0306337 , 0.031742 , 0.03721978, 0.03176932, 0.03141078, 0.03409519, 0.03513022, 0.03262367, 0.03497397, 0.0360815 , 0.03600124, 0.03243677, 0.0331164 , 0.03149059, 0.03691718, 0.03196211, 0.03608905, 0.03755199, 0.03560633, 0.03318104, 0.03333913, 0.03130253, 0.03838744, 0.03212319, 0.03383321, 0.03205457, 0.03356666, 0.03544808, 0.03182409, 0.03454725, 0.03517473, 0.03614165, 0.03405882, 0.03397554, 0.0320861 , 0.0347675 , 0.03705034, 0.03187931, 0.0335782 , 0.0360765 , 0.03155507, 0.03304993, 0.03317409, 0.03580993, 0.03796723, 0.03467349, 0.03234889, 0.03437548, 0.0342068 , 0.03577822, 0.03535941, 0.03214434, 0.03227508, 0.03595557, 0.0353463 , 0.03429855, 0.03149308, 0.03357296, 0.03478293, 0.03668304, 0.03814905, 0.03689617, 0.03031397, 0.0356352 , 0.03752262, 0.03552895, 0.03582576, 0.03441394, 0.03186472, 0.03521422, 0.03531235, 0.03257761, 0.03639319, 0.03429881, 0.03462154, 0.03072465, 0.03311946, 0.0308567 , 0.03208536, 0.0337658 , 0.03710826, 0.03553074, 0.03607779, 0.03132417, 0.03221319, 0.03311266, 0.03093447, 0.03409707, 0.03604383, 0.0341283 , 0.03440112, 0.03587337, 0.03311389, 0.03730926, 0.03545398, 0.03334088, 0.03262428, 0.03348249, 0.03625748, 0.033844 , 0.03630546, 0.0344714 , 0.03217738, 0.03130945, 0.03101241, 0.03133432, 0.03678077, 0.03668157, 0.03587959, 0.02945016, 0.0339122 , 0.0336982 , 0.03594457, 0.03729026, 0.0333612 , 0.02820126, 0.04043288, 0.03398932, 0.03383912, 0.03688906, 0.03529162, 0.03308525, 0.03710377, 0.03697566, 0.03545765, 0.0393287 , 0.03511189, 0.03734153, 0.0309365 , 0.03599573, 0.03340091, 0.03631366, 0.03222798, 0.03391268, 0.03210976, 0.03517214, 0.03514128, 0.0362618 , 0.03526742, 0.03506047, 0.03744133, 0.03687915, 0.0312038 , 0.03342389, 0.0322474 , 0.03504425, 0.03462889, 0.03503286, 0.03175144, 0.03223341, 0.03574766, 0.03786785, 0.03079034, 0.0384242 , 0.03443336, 0.03509659, 0.0331189 , 0.03253633, 0.03212039, 0.03106236, 0.03329483, 0.03644602, 0.02921328, 0.03308668, 0.03062067, 0.03640539, 0.03360491, 0.03389683, 0.03850131, 0.03817785, 0.03238483, 0.03239908, 0.03544456, 0.03526306, 0.03552583, 0.03512413, 0.03581594, 0.03142119, 0.03688495, 0.03302685, 0.03583831, 0.03554739, 0.03517519, 0.03342819, 0.03240985, 0.0306706 , 0.03516093, 0.03546941, 0.03305869, 0.03444686, 0.03196067, 0.03258359, 0.03225225, 0.03405359, 0.0314135 , 0.03757875, 0.03694918, 0.03637657, 0.03603543, 0.03546733, 0.03470401, 0.03565601, 0.03408186, 0.03189548, 0.03375535, 0.03530298, 0.03328103, 0.03603692, 0.03215662, 0.03455473, 0.03628099, 0.03542867, 0.03459071, 0.03740486, 0.03241454, 0.03699095, 0.03362525, 0.03453697, 0.03560096, 0.03032834, 0.03261881, 0.03519571, 0.03175207, 0.03087034, 0.03661209, 0.0384522 , 0.03349561, 0.03298505, 0.03541992, 0.03483835, 0.0326382 , 0.0331954 , 0.03040188, 0.03141423, 0.03178553, 0.03399286, 0.03670659, 0.03529897, 0.03445289, 0.03910183, 0.03693579, 0.03745123, 0.03544073, 0.03308966, 0.03547556, 0.03707074, 0.03525615, 0.03456536, 0.03722953, 0.03312706, 0.03419826, 0.03419267, 0.03646339, 0.03257063, 0.03177225, 0.03429715, 0.03310622, 0.0358156 , 0.03475818, 0.03237383, 0.03740262, 0.03248381, 0.03041655, 0.03560246, 0.035361 , 0.03596175, 0.03756068, 0.03343584, 0.02914506, 0.03257972, 0.03186928, 0.03444595, 0.03448274, 0.03487147, 0.03307192, 0.03687837, 0.03554501, 0.02969871, 0.02882754, 0.03339066, 0.03207911, 0.03308664, 0.03293925, 0.03573058, 0.02838616, 0.03435363, 0.0370067 , 0.03200265, 0.03037309, 0.03357999, 0.03656119, 0.03843627, 0.03333474, 0.03880846, 0.03059737, 0.03275688, 0.03285129, 0.03220145, 0.03253442, 0.03618327, 0.03462011, 0.03432366, 0.03459524, 0.03157359, 0.03489503, 0.03187288, 0.03052811, 0.03633338, 0.03112468, 0.0328625 , 0.03208614, 0.03383455, 0.03118336, 0.03521309, 0.03375108, 0.03258716, 0.03324911, 0.03317217, 0.03288663, 0.03633928, 0.03324796, 0.02999422, 0.03228744, 0.03726359, 0.03703165, 0.03518771, 0.0349349 , 0.03320673, 0.03414009, 0.03372712, 0.031291 , 0.03871908, 0.0389572 , 0.03480945, 0.03369572, 0.03286408, 0.03432919, 0.03617588, 0.03514486, 0.03164417, 0.03464502, 0.03206587, 0.03142204, 0.03520723, 0.03901041, 0.03472864, 0.03446797, 0.03601697, 0.03129906, 0.03326078, 0.03360178, 0.03040458, 0.03195887, 0.03420034, 0.0362044 , 0.03443136, 0.03663701, 0.0338398 , 0.03190395, 0.03857493, 0.03287289, 0.03439351, 0.03784665, 0.03712634, 0.0350762 , 0.03203719, 0.03249487, 0.03143156, 0.03528639, 0.0366816 , 0.03070296, 0.03364303, 0.03233427, 0.03063859, 0.03310549, 0.03643439, 0.0330483 , 0.03227323, 0.0332104 , 0.03490121, 0.03534769, 0.03526284, 0.0350183 , 0.03454025, 0.03610774, 0.03448665, 0.0351776 , 0.03492046, 0.03587773, 0.03598657, 0.0330358 , 0.03617942, 0.03409603, 0.03405978, 0.03243784, 0.0357111 , 0.03422289, 0.03395089, 0.03419511, 0.03751485, 0.03752663, 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0.03403197, 0.03686333, 0.03104238, 0.03510361, 0.03461223, 0.03273386, 0.03506437, 0.03576537, 0.03456747, 0.03157396, 0.0356792 , 0.03312858, 0.03533342, 0.03395183, 0.02952398, 0.03133257, 0.0290001 , 0.03676953, 0.03198111, 0.03659722, 0.03297687, 0.03075361, 0.03692363, 0.03547212, 0.03423718, 0.03911962, 0.0337903 , 0.03549648, 0.03114438, 0.03639941, 0.03452909, 0.03333842, 0.03325164, 0.0318147 , 0.03348661, 0.03410641, 0.03684512, 0.03423216, 0.03254072, 0.03454452, 0.0342963 , 0.03554913, 0.03560372, 0.03737662, 0.03501553, 0.03416058, 0.03649484, 0.0353373 , 0.03306659, 0.03383092, 0.03258973, 0.03128851, 0.03535036, 0.0326183 , 0.03313489, 0.03298149, 0.03001876, 0.03292446, 0.0336928 , 0.0353293 , 0.03914796, 0.03681156, 0.03548131, 0.03490516, 0.03623682, 0.03389732, 0.03374424, 0.03564595, 0.0317539 , 0.03420668, 0.03417477, 0.03212741, 0.03260072, 0.03469001, 0.03336305, 0.03773065, 0.03327173, 0.03285666, 0.03506289, 0.03237553, 0.03862934, 0.03316271, 0.03686143, 0.03178891, 0.03681042, 0.03392235, 0.03511499, 0.03274074, 0.03035431, 0.03218739, 0.02936936, 0.03737124, 0.03797012, 0.03170762, 0.03751291, 0.03129583, 0.03430933, 0.03584066, 0.03263332, 0.03347521, 0.0301352 , 0.03149596, 0.03259497, 0.02944742, 0.03743508, 0.02899995, 0.03271523, 0.03931528, 0.03301583, 0.03106761, 0.03268265, 0.0339521 , 0.03190526, 0.03722284, 0.03506228, 0.03461957, 0.03004975, 0.03152583, 0.03188864, 0.03137891, 0.03125911, 0.03443212, 0.03086839, 0.03683474, 0.0362767 , 0.03311872, 0.03572921, 0.03316816, 0.0323083 , 0.03521522, 0.03557253, 0.03365509, 0.03403009, 0.0326287 , 0.03244894, 0.03690506, 0.03537427, 0.03649063, 0.03128687, 0.03721945, 0.03366335, 0.03434353, 0.03885856, 0.03119781, 0.03581426, 0.03252629, 0.03185808, 0.03493191, 0.03395895, 0.03227437, 0.03889401, 0.03115375, 0.03044907, 0.03352168, 0.03725254, 0.04023851, 0.03216466, 0.03268951, 0.03385814, 0.03274238, 0.03196035, 0.0368327 , 0.03088809, 0.03402669, 0.03434839, 0.03245973, 0.03077444, 0.03989451, 0.03465213, 0.03461663, 0.03282729, 0.03283131, 0.02965922, 0.03205531, 0.0327355 , 0.03385212, 0.03641985, 0.03535839, 0.03367922, 0.03383552, 0.03400515, 0.03173495, 0.03555596, 0.03687678, 0.03612996, 0.0335418 , 0.03272114, 0.03485328, 0.03215544, 0.03069584, 0.03557681, 0.03713931, 0.03404416, 0.03707127, 0.03633673, 0.03394292, 0.03453973, 0.03580137, 0.03147122, 0.03635144, 0.03505335, 0.03346194, 0.0352425 , 0.03309578, 0.03361246, 0.03162954, 0.03824726, 0.03128456, 0.03308664, 0.03249983, 0.0362276 , 0.03332539, 0.03361665, 0.03060977, 0.03693433, 0.0366655 , 0.03647699, 0.03065926, 0.03489498, 0.03484303, 0.03671298, 0.03384923, 0.0325788 , 0.04077159, 0.03860608, 0.03624336, 0.03847722, 0.03230998, 0.0378641 , 0.0311665 , 0.03302566, 0.03502139, 0.03389091, 0.03040683, 0.0327089 , 0.02989121, 0.03733698, 0.03252497, 0.0368373 , 0.03775841, 0.03408025, 0.03216691, 0.03484384, 0.03315263, 0.03539599, 0.03259381, 0.03414304, 0.03295092, 0.03846051, 0.03206971, 0.03048838, 0.03200569, 0.03212656, 0.03333666, 0.0348813 , 0.03645685, 0.03917926, 0.03368409, 0.03389568, 0.03244002, 0.03411523, 0.03402238, 0.03634798, 0.03779468, 0.03357757, 0.03237467, 0.03858004, 0.03477383, 0.03394397, 0.03230739, 0.03672567, 0.03409324, 0.03521135, 0.03204122, 0.03561428, 0.03498935, 0.03859978, 0.03507295, 0.03702111, 0.03092078, 0.03511057, 0.02930761, 0.03544484, 0.03255327, 0.03181242, 0.02968715, 0.04256435, 0.03930061, 0.03385781, 0.03310617, 0.03527292, 0.03542688, 0.03528373, 0.03279445, 0.03181911, 0.03577111, 0.03326912, 0.03262289, 0.03629811, 0.0378563 , 0.0312053 , 0.03712152, 0.03346831, 0.03766047, 0.03423268, 0.03298717, 0.03066189, 0.03542913, 0.03547632, 0.03402202, 0.03417251, 0.03692622, 0.036344 , 0.03403163, 0.03608709, 0.03406768, 0.03759235, 0.03547541, 0.03092644, 0.03172698, 0.03815645, 0.0333857 , 0.03187081, 0.03629718, 0.03323217, 0.03266878, 0.03316033, 0.03248826, 0.03524635, 0.03389906, 0.03489943, 0.02924824, 0.03481939, 0.03539659, 0.03357566, 0.03399703, 0.03150613, 0.0364799 , 0.03248547, 0.03047435, 0.03421486, 0.03462341, 0.03470768, 0.03761934, 0.03315931, 0.03509053, 0.03477462, 0.0304219 , 0.03633647, 0.03796296, 0.03736833, 0.03386945, 0.03103569, 0.03055042, 0.03651902, 0.03561151, 0.03592187, 0.03560313, 0.03095704, 0.03515765, 0.03537821, 0.03516444, 0.03394301, 0.03365487, 0.03376338, 0.03625588, 0.03294306, 0.03143146, 0.03241328, 0.03014768, 0.03647779, 0.03977323, 0.03352695, 0.03299449, 0.03759643, 0.03312409, 0.031813 , 0.03387483, 0.03599901, 0.03765133, 0.03177601, 0.03316265, 0.03215408, 0.03199014, 0.03368082, 0.03376967, 0.0316642 , 0.03014013, 0.03350298, 0.03401597, 0.03083467, 0.03712924, 0.03491432, 0.0317063 , 0.03464181, 0.03251904, 0.03534762, 0.03439422, 0.03835411, 0.03623541, 0.0340211 , 0.03545453, 0.03167305, 0.03432224, 0.03212673, 0.03596564, 0.03338769, 0.03552314, 0.03368746, 0.03383131, 0.03705007, 0.03340853, 0.03626988, 0.03358266, 0.03560516, 0.03447281, 0.03138491, 0.03209008, 0.0345985 , 0.03430723, 0.0344173 , 0.03037132, 0.03406071, 0.03662991, 0.03487076, 0.03266249, 0.03427314, 0.03168054, 0.03632044, 0.03633239, 0.03903545, 0.03464841, 0.03560016, 0.03340148, 0.03447032, 0.03320291, 0.02977214, 0.03428828, 0.03499615, 0.03676726, 0.03597865, 0.0333828 , 0.03563908, 0.03401826, 0.03394263]) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for i_rep in range(n_rep): @@ -323,13 +289,25 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate apply_cross_fitting=False) obj_dml_plr_orth_nosplit.fit() this_theta = obj_dml_plr_orth_nosplit.coef[0] + this_se = obj_dml_plr_orth_nosplit.se[0] # we show that the loaded result matches the just computed print(np.abs(theta_orth_nosplit[i_rep] - this_theta)) + print(np.abs(se_orth_nosplit[i_rep] - this_se)) theta_orth_nosplit[i_rep] = this_theta - - ax = sns.kdeplot(theta_orth_nosplit, shade=True, color=colors[2]) + se_orth_nosplit[i_rep] = this_se + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_orth_nosplit - alpha)/se_orth_nosplit, + color=face_colors[1], edgecolor = edge_colors[1], + stat='density', bins=30, label='Double ML (no sample splitting)'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); @savefig orth_nosplit.png width=5in - ax.axvline(0.5, color='k', label='True theta'); + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); .. tabbed:: R @@ -340,10 +318,14 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate set.seed(2222) # to speed up the illustration we hard-code the simulation results - theta_orth_nosplit = c(0.24404981, 0.24200765, 0.27908831, 0.25502043, 0.32824182, 0.26923132, 0.24104406, 0.20969115, 0.32043631, 0.19615339, 0.26752732, 0.28054535, 0.20565819, 0.23094579, 0.30052261, 0.22506210, 0.30899243, 0.33166896, 0.29901923, 0.21645924, 0.19363222, 0.25194347, 0.20642525, 0.24555231, 0.22453258, 0.30131307, 0.23932457, 0.24134974, 0.32292519, 0.24820202, 0.29051209, 0.19341549, 0.25560635, 0.24635088, 0.25058896, 0.19513198, 0.23325175, 0.25732967, 0.25630722, 0.23618513, 0.17672957, 0.21211532, 0.26970909, 0.23979100, 0.24278216, 0.31524081, 0.21496712, 0.23975950, 0.22430487, 0.25032633, 0.31171158, 0.28673753, 0.23617691, 0.30279132, 0.25171843, 0.21937215, 0.24240521, 0.23905356, 0.18858651, 0.20627836, 0.18322578, 0.25066665, 0.27527609, 0.27380098, 0.16270080, 0.22131889, 0.22989677, 0.25631445, 0.30572004, 0.28186739, 0.28453591, 0.31102750, 0.28063015, 0.25876131, 0.29582343, 0.24814810, 0.35109199, 0.25108377, 0.22487320, 0.30794975, 0.23894645, 0.26638033, 0.23674057, 0.20538273, 0.19608607, 0.33079935, 0.19395816, 0.21966814, 0.18736528, 0.21029872, 0.29902640, 0.26053716, 0.25603558, 0.21075900, 0.20782755, 0.28236968, 0.25193825, 0.28224143, 0.23540541, 0.26424208, 0.29290447, 0.24258066, 0.27822969, 0.20114274, 0.23819599, 0.25825846, 0.30886038, 0.21687679, 0.26809187, 0.29897851, 0.28606557, 0.25426056, 0.17297763, 0.22302356, 0.24706098, 0.24361315, 0.20613373, 0.20395747, 0.26581544, 0.27803372, 0.28981682, 0.28375231, 0.26902671, 0.22007559, 0.21972414, 0.22691012, 0.23230200, 0.19660759, 0.22971657, 0.24840646, 0.27392351, 0.23431585, 0.23109943, 0.22423472, 0.17548714, 0.27187182, 0.23154822, 0.23570134, 0.20276194, 0.20357300, 0.29668416, 0.23894281, 0.29103412, 0.26916173, 0.20825452, 0.27174241, 0.28702397, 0.26323370, 0.20993333, 0.22861364, 0.28225493, 0.25282767, 0.25466747, 0.20584595, 0.25209117, 0.26527392, 0.24857230, 0.30070528, 0.29237504, 0.28291041, 0.26424070, 0.20749123, 0.33763761, 0.17805976, 0.23412056, 0.31256005, 0.24081702, 0.24795085, 0.29925883, 0.25657520, 0.17849887, 0.27372626, 0.23904392, 0.22488373, 0.24942245, 0.29803576, 0.23674057, 0.27779304, 0.25638963, 0.22914358, 0.22338097, 0.23024613, 0.22301133, 0.29410433, 0.30106813, 0.27284832, 0.24042849, 0.24791942, 0.25071209, 0.25156142, 0.23516980, 0.24695011, 0.25794186, 0.27033305, 0.27898959, 0.22825721, 0.26305627, 0.28653743, 0.24861610, 0.29348304, 0.22902869, 0.30333429, 0.26935249, 0.19811759, 0.27767260, 0.19347321, 0.33130437, 0.25953422, 0.26075776, 0.17054191, 0.22197221, 0.26321300, 0.27792267, 0.20024470, 0.31713220, 0.23528659, 0.28108999, 0.21161821, 0.18928942, 0.25071159, 0.32293010, 0.18651541, 0.20749911, 0.22686192, 0.30217232, 0.27595429, 0.26376699, 0.28513188, 0.29005017, 0.22022413, 0.18666795, 0.18394129, 0.22996187, 0.29361925, 0.24450357, 0.29357722, 0.22286200, 0.28265768, 0.22675825, 0.27649261, 0.28951131, 0.28271866, 0.17806041, 0.21163621, 0.30509409, 0.26342174, 0.32677576, 0.24797358, 0.23200651, 0.27904545, - 0.26504825, 0.19402431, 0.26306555, 0.31078689, 0.24485790, 0.26520010, 0.21393591, 0.25235737, 0.23623305, 0.21269470, 0.36499369, 0.21848834, 0.28736613, 0.25397938, 0.25896726, 0.14757031, 0.15458744, 0.25966165, 0.29306232, 0.29959702, 0.34040452, 0.24615155, 0.16853516, 0.28525494, 0.23482806, 0.23951146, 0.26795442, 0.26138905, 0.28521849, 0.30644821, 0.22308750, 0.20662353, 0.26295613, 0.23490771, 0.20767176, 0.25672486, 0.23762899, 0.24988185, 0.21882009, 0.32730028, 0.31310884, 0.25821877, 0.23069627, 0.22345636, 0.19806570, 0.29756479, 0.21450103, 0.27358563, 0.24101731, 0.23376150, 0.22875492, 0.30380284, 0.27886358, 0.25644140, 0.27983788, 0.29336975, 0.20603135, 0.23124943, 0.25156872, 0.30053623, 0.27771452, 0.19847193, 0.25991230, 0.25088226, 0.25607595, 0.29055568, 0.20725805, 0.20844012, 0.26323932, 0.26251734, 0.29311885, 0.29691861, 0.28589456, 0.28678294, 0.23915972, 0.26246128, 0.24366650, 0.25425399, 0.31767331, 0.25261640, 0.21726079, 0.30237963, 0.22177598, 0.19588260, 0.23795208, 0.26311371, 0.28360425, 0.21153313, 0.17872878, 0.23587613, 0.25150900, 0.25867927, 0.26743636, 0.19428422, 0.24070189, 0.26749686, 0.21719999, 0.26282296, 0.23008293, 0.28084342, 0.22708225, 0.26011481, 0.25512443, 0.24929613, 0.26494690, 0.32442653, 0.24635223, 0.23136118, 0.29793829, 0.22573153, 0.20318348, 0.26693828, 0.24952603, 0.27215040, 0.26322105, 0.22329061, 0.27891631, 0.27127217, 0.27125980, 0.27248351, 0.22775018, 0.25404143, 0.25163026, 0.30982078, 0.20748986, 0.27073473, 0.23114844, 0.20224835, 0.30671944, 0.19242940, 0.24885178, 0.22356225, 0.29483575, 0.25740248, 0.18617692, 0.24523393, 0.25576838, 0.35176784, 0.25000509, 0.29496086, 0.18229425, 0.31624955, 0.21398406, 0.31201873, 0.27873516, 0.29288879, 0.21909350, 0.33838391, 0.28401419, 0.28814919, 0.25356693, 0.28313276, 0.29096662, 0.24443356, 0.27196073, 0.22707857, 0.23169359, 0.23623725, 0.29841002, 0.19643420, 0.22633211, 0.28084501, 0.26918329, 0.36923637, 0.21106338, 0.27083820, 0.23144420, 0.31389920, 0.22205859, 0.22125093, 0.26983642, 0.26984812, 0.27143384, 0.22440664, 0.24382201, 0.29489124, 0.27572998, 0.23011901, 0.21742103, 0.24521077, 0.24721123, 0.29630124, 0.29493903, 0.28724526, 0.27999346, 0.24368907, 0.22984274, 0.17392723, 0.27441598, 0.25581560, 0.22590773, 0.28188015, 0.22709537, 0.31067150, 0.32512125, 0.23057776, 0.30177850, 0.23242622, 0.28326166, 0.29190056, 0.25107901, 0.23130449, 0.24929647, 0.19796943, 0.26118980, 0.28361317, 0.28984450, 0.24239718, 0.23415392, 0.27891457, 0.31708859, 0.25101173, 0.24828103, 0.26219254, 0.26629355, 0.30090049, 0.21231956, 0.28541164, 0.31735485, 0.23788811, 0.21526438, 0.22865108, 0.26906438, 0.21838840, 0.26907421, 0.26570784, 0.20624892, 0.30029962, 0.34755127, 0.31459386, 0.28203083, 0.26903597, 0.29458419, 0.23009954, 0.24293185, 0.29547741, 0.22776509, 0.25928743, 0.27324063, 0.19938081, 0.34164972, 0.27462158, 0.22935702, 0.32378804, 0.25564062, 0.23829308, 0.17257210, 0.27254683, 0.20348844, 0.31883511, - 0.30223972, 0.28047451, 0.25603642, 0.27022816, 0.28062654, 0.25154276, 0.25218666, 0.26535623, 0.21176103, 0.21301494, 0.26220526, 0.27431630, 0.27729789, 0.24779401, 0.24868582, 0.18360005, 0.26786428, 0.24958852, 0.22578138, 0.27438830, 0.32905933, 0.31888969, 0.24873045, 0.28213371, 0.20563072, 0.24977431, 0.30339369, 0.24085435, 0.17003776, 0.25333376, 0.28797501, 0.30977813, 0.25259406, 0.30296617, 0.26677033, 0.22128846, 0.30456640, 0.24690890, 0.30914977, 0.33190582, 0.26943920, 0.32514576, 0.25079613, 0.28332654, 0.22076585, 0.27260810, 0.21275210, 0.32582630, 0.28814724, 0.22820021, 0.23679701, 0.26929291, 0.20674902, 0.21578408, 0.23102393, 0.19422745, 0.26693484, 0.24910152, 0.25459030, 0.28373457, 0.23307690, 0.22377395, 0.22574906, 0.29702891, 0.25427968, 0.21206913, 0.24253494, 0.23148299, 0.28956044, 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0.0351824195, 0.0367480774, 0.0377885216, 0.0347934045, 0.0343594004, 0.0360172437, 0.0332004741, 0.0328187484, 0.0319095283, 0.0341301662, 0.0368152226, 0.0325636887, 0.0278098444, 0.0338527584, 0.0357619561, 0.0344976289, 0.0308734936, 0.0335201837, 0.0345951360, 0.0338154139, 0.0384176165, 0.0339703122, 0.0321008854, 0.0304343495, 0.0383744231, 0.0326511530, 0.0357007692, 0.0364832862, 0.0336478745, 0.0354983104, 0.0349903528, 0.0323750578, 0.0362311983, 0.0317867627, 0.0348228931, 0.0347552913, 0.0354843736, 0.0366361339, 0.0326680627, 0.0325121978, 0.0373359480, 0.0332577694, 0.0359011959, 0.0326710008, 0.0374441123, 0.0294243636, 0.0337145872, 0.0380173731, 0.0366694190, 0.0328193935, 0.0379898684, 0.0322195023, 0.0337454946, 0.0345276309, 0.0364909653, 0.0334296165, 0.0358988742, 0.0331649512, 0.0331349341, 0.0355790826, 0.0355390276, 0.0340805740, 0.0325635992, 0.0345979215, 0.0348568556, 0.0353007498, 0.0358776752, 0.0365003484, 0.0361202812, 0.0321241206, 0.0391502926, 0.0309165817, 0.0379365085, 0.0340129424, 0.0325177785, 0.0332456390, 0.0333622897, 0.0346441767, 0.0351135120, 0.0344107385, 0.0363620369, 0.0319351166, 0.0312223492, 0.0360603563, 0.0381125251, 0.0332026160, 0.0352838661, 0.0346451541, 0.0303747736, 0.0335054672, 0.0314421632, 0.0329926348, 0.0368155443, 0.0340350988, 0.0346156710, 0.0359462684) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for (i_rep in seq_len(n_rep)){ for (i_rep in seq_len(1)) { @@ -356,19 +338,30 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate apply_cross_fitting=FALSE) obj_dml_plr_orth_nosplit$fit() this_theta = obj_dml_plr_orth_nosplit$coef + this_se = obj_dml_plr_orth_nosplit$se print(abs(theta_orth_nosplit[i_rep] - this_theta)) + print(abs(se_orth_nosplit[i_rep] - this_se)) theta_orth_nosplit[i_rep] = this_theta + se_orth_nosplit[i_rep] = this_se } - g_nosplit = ggplot(data.frame(theta_orth_nosplit), aes(x = theta_orth_nosplit)) + - geom_density(fill = "dark green", alpha = 0.3, color = "dark green") + - geom_vline(aes(xintercept = alpha), col = "black") + - xlim(c(0.08, 0.75)) + xlab("") + ylab("") + theme_minimal() + g_nosplit = ggplot(data.frame(theta_rescaled=(theta_orth_nosplit - alpha)/se_orth_nosplit), aes(x = theta_rescaled)) + + geom_histogram(aes(y=..density.., x=theta_rescaled, colour = "Double ML (no sample splitting)", fill="Double ML (no sample splitting)"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Double ML (no sample splitting)", "N(0, 1)"), + values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"='black')) + + scale_fill_manual(name='',, + breaks=c("Double ML (no sample splitting)", "N(0, 1)"), + values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() g_nosplit If the nuisance models :math:`\hat{g}_0()` and :math:`\hat{m}()` are estimated on the whole dataset, which is also used for obtaining -the final estimate :math:`\check{\theta}`, another bias is observed. +the final estimate :math:`\check{\theta}_0`, another bias is observed. .. _bias_overfitting: @@ -376,7 +369,7 @@ Sample splitting to remove bias induced by overfitting ++++++++++++++++++++++++++++++++++++++++++++++++++++++ Using sample splitting, i.e., estimate the nuisance models :math:`\hat{g}_0()` and :math:`\hat{m}()` on one part of the -data (training data) and estimate :math:`\check{\theta}` on the other part of the data (test data), overcomes the bias +data (training data) and estimate :math:`\check{\theta}_0` on the other part of the data (test data), overcomes the bias induced by overfitting. We can exploit the benefits of cross-fitting by switching the role of the training and test sample. Cross-fitting performs well empirically because the entire sample can be used for estimation. @@ -388,7 +381,8 @@ Cross-fitting performs well empirically because the entire sample can be used fo np.random.seed(3333) # to speed up the illustration we hard-code the simulation results - theta_dml = np.array([0.52231688, 0.43821994, 0.36741092, 0.52884091, 0.5015594 , 0.58615797, 0.50844654, 0.54049318, 0.54273396, 0.53888897, 0.507104 , 0.40035828, 0.48716359, 0.46810072, 0.64261576, 0.71117208, 0.42699728, 0.4835898 , 0.49635877, 0.50369895, 0.5347252 , 0.49725455, 0.53288677, 0.58781331, 0.58458898, 0.577733 , 0.63314775, 0.56648841, 0.51656325, 0.5853161 , 0.55397083, 0.49646073, 0.70452214, 0.57921975, 0.66033219, 0.4141339 , 0.39301364, 0.59563687, 0.48364131, 0.48707462, 0.59045756, 0.50181798, 0.55173923, 0.37780184, 0.49094078, 0.56072755, 0.51914519, 0.47253627, 0.56482564, 0.45755482, 0.58579367, 0.45175687, 0.55837968, 0.59198114, 0.54112865, 0.51840052, 0.45087786, 0.43342863, 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0.48614475, 0.49863181, 0.61503884, 0.44949508, 0.55972135, 0.49525966, 0.49261208, 0.44111368, 0.63441355, 0.48637754, 0.55398729, 0.56778945, 0.45902377, 0.57463649, 0.46121876, 0.54155362, 0.44478171, 0.52435051, 0.57087088, 0.39245217, 0.46396707, 0.50510056, 0.5069028 , 0.48665451, 0.57847942, 0.42323615, 0.54759841, 0.51696255, 0.55060358, 0.54442527, 0.47507197, 0.48206163, 0.56839492, 0.52930626, 0.48166739, 0.43443332, 0.47844387, 0.54589331, 0.48463549, 0.55026587, 0.46140555, 0.53735227, 0.45285953, 0.47713501, 0.45401454, 0.35975943, 0.51008005, 0.55417445, 0.49993366, 0.48587355, 0.55409922, 0.51759863, 0.6103252 , 0.47978687, 0.54027196, 0.53353782, 0.48636421, 0.49506053, 0.52526545, 0.58294162, 0.45855799, 0.52657642, 0.51304469, 0.52025121, 0.53871685, 0.45504542, 0.46904091, 0.49470649, 0.55199598, 0.46424461, 0.50388651, 0.46500381, 0.48168773, 0.52653742, 0.48911706, 0.48392117, 0.49352537, 0.45431985, 0.56327712, 0.52790318, 0.48400641, 0.49237972, 0.53373982, 0.39831133, 0.5649761 , 0.54902235, 0.44074566, 0.42049476, 0.51954156, 0.52258857, 0.54230792, 0.50280809, 0.4339837 , 0.4644633 , 0.50787032, 0.52874887, 0.59047196, 0.51215975, 0.52840121, 0.51081136, 0.56514492, 0.54241922, 0.53519724, 0.52820358, 0.55715823, 0.58547163, 0.49777178, 0.52623281, 0.5017119 , 0.64372458, 0.52223454, 0.51593815, 0.44428961, 0.56491759, 0.50354986, 0.46730887, 0.47892029, 0.47908061, 0.57468579, 0.50815634, 0.50228529, 0.52348798, 0.61573092, 0.39695719, 0.55025551, 0.54751588, 0.46280727, 0.47667546, 0.53997524, 0.52529117, 0.53178652, 0.51157481, 0.51605013, 0.47286783, 0.52629508, 0.5846704 , 0.49872661, 0.45332425, 0.52898154, 0.4699794 , 0.46390095, 0.47759878, 0.37733012, 0.5115532 , 0.50444157, 0.51921322, 0.53001455, 0.50901359, 0.48552478, 0.43383384, 0.44819809, 0.55747646, 0.47719987, 0.52906354, 0.53946325, 0.49667595, 0.53863836, 0.60556717, 0.50571002, 0.4651874 , 0.56184654, 0.48148176, 0.60553531, 0.44449289, 0.50566925, 0.55150338, 0.5001125 , 0.44491697, 0.45301713, 0.49467688, 0.48813413, 0.49126065, 0.49358158, 0.51385847, 0.47953825, 0.5431249 , 0.45075482, 0.59074403, 0.56949778, 0.49203902, 0.52315143, 0.57593538, 0.48795262, 0.48162209, 0.52078148, 0.4909698 , 0.45673381, 0.53827103, 0.48172874, 0.58061229, 0.47593299, 0.54073734, 0.54522708, 0.46184543, 0.4959692 , 0.4294827 , 0.44999313, 0.44973668, 0.51974583, 0.51672712, 0.53010395, 0.5228019 , 0.59312345, 0.48991575, 0.50370248, 0.48362954, 0.40305167, 0.46827572, 0.53782401, 0.54927567, 0.55919428, 0.58823779, 0.49755018, 0.52863269, 0.5288442 , 0.48776318, 0.46883583, 0.36395434, 0.59900297, 0.47637107, 0.51653852, 0.460052 , 0.59300112, 0.43400939, 0.56324665, 0.4900747 , 0.5231264 , 0.55965705, 0.53133956, 0.54956469, 0.41375333, 0.48080361, 0.60415333, 0.48909132, 0.50683815, 0.52202015, 0.53511728, 0.48063877, 0.53188573, 0.51275626, 0.49709038, 0.45734633, 0.48069221, 0.45463481, 0.55533079, 0.50718593, 0.57828931, 0.53999175, 0.54793958, 0.52846782, 0.49284515, 0.57232448, 0.42503111, 0.57343691, 0.51694705, 0.46771321, 0.43041661, 0.59017731, 0.49683349, 0.51239223, 0.50168615, 0.4435032 , 0.56883681, 0.50566309, 0.4768778 , 0.52026538, 0.56956521, 0.41806343, 0.49501286, 0.47926238, 0.46166467, 0.49539291, 0.43903866, 0.54666996, 0.36408008, 0.51138189, 0.56013817, 0.47949159, 0.51620916, 0.46406434, 0.4321351 , 0.59831181, 0.50361498, 0.53678082, 0.47096002, 0.43574422, 0.56583098, 0.59492964, 0.50920948, 0.51323477, 0.44414516, 0.57889363, 0.49414783, 0.48518923, 0.54112235, 0.44818199, 0.55585544, 0.50517962, 0.42871447, 0.52354361, 0.55310075, 0.5062023 , 0.54722905, 0.52960227, 0.50620311, 0.47954884, 0.49630517, 0.47788121, 0.58900933, 0.53264215, 0.47771333, 0.51253034, 0.45793071, 0.49573853, 0.44799847, 0.53553779, 0.51327999, 0.61805459, 0.51404724, 0.48044965, 0.5548255 , 0.42701429, 0.46496555, 0.52774306, 0.48027187, 0.49042958, 0.47982127, 0.55689975, 0.46786333, 0.49741252, 0.4305642 , 0.53657994, 0.55687243, 0.50280181, 0.50661455, 0.54592754, 0.46576354, 0.54996808, 0.48235281, 0.43544111, 0.4979282 , 0.40185842, 0.47833626, 0.52129907, 0.46030088, 0.44938231, 0.50064939, 0.46586232, 0.55550797, 0.49609405, 0.49965942, 0.47790909, 0.45629575, 0.60067607, 0.55270055, 0.52972619, 0.52151004, 0.47740249, 0.51620499, 0.40906359, 0.58986775, 0.55217603, 0.49252861, 0.53500691, 0.52813659, 0.48438568, 0.51182359]) + se_dml = np.array([0.04451577, 0.04490176, 0.04013621, 0.04605524, 0.04977187, 0.04212784, 0.05068112, 0.05198796, 0.05196444, 0.04506378, 0.04616982, 0.04480436, 0.03900256, 0.0408141 , 0.04463835, 0.04262014, 0.05274251, 0.04588425, 0.0406925 , 0.04619202, 0.0443178 , 0.05331612, 0.04373689, 0.04470977, 0.04717082, 0.03746608, 0.04401159, 0.04484962, 0.04559701, 0.04350091, 0.03976468, 0.0404016 , 0.04144786, 0.04561865, 0.0477659 , 0.04166419, 0.04830947, 0.04266617, 0.04119545, 0.04471898, 0.04575399, 0.04373017, 0.04500419, 0.0438294 , 0.04052023, 0.04356239, 0.04462905, 0.04127356, 0.04148683, 0.04766906, 0.04087846, 0.03775995, 0.04123315, 0.04630391, 0.04460172, 0.05037454, 0.04571886, 0.04152775, 0.0427847 , 0.04150442, 0.04431639, 0.04820055, 0.04518448, 0.04374162, 0.0444376 , 0.04319458, 0.04870047, 0.03979945, 0.04596467, 0.04931064, 0.03986882, 0.03962937, 0.04125952, 0.04870504, 0.04270478, 0.04403063, 0.04529282, 0.04620437, 0.04520664, 0.04757334, 0.04328208, 0.04642612, 0.04225026, 0.04083541, 0.0420464 , 0.04743644, 0.04094565, 0.04811671, 0.04929223, 0.04734472, 0.04486223, 0.04522005, 0.03971825, 0.0494492 , 0.04246032, 0.04445648, 0.04564281, 0.04151786, 0.04590433, 0.0429455 , 0.0458209 , 0.04451129, 0.04592348, 0.04650847, 0.04314546, 0.04175605, 0.04911152, 0.0449944 , 0.04460767, 0.04476069, 0.04956918, 0.04341115, 0.04423823, 0.0454818 , 0.05354475, 0.04697678, 0.04636224, 0.04538301, 0.04269111, 0.04881574, 0.04682834, 0.05010542, 0.04094056, 0.04192807, 0.04817982, 0.04648762, 0.04513767, 0.04538384, 0.0431342 , 0.04366012, 0.04717716, 0.05104802, 0.05037837, 0.03937307, 0.04616315, 0.04925793, 0.04352705, 0.04513465, 0.04400273, 0.04393063, 0.04720379, 0.04673644, 0.04176819, 0.04758137, 0.04702437, 0.04555107, 0.04090498, 0.04103245, 0.04237081, 0.04192252, 0.04409652, 0.04748513, 0.04606071, 0.0466778 , 0.04083337, 0.04075193, 0.04389173, 0.04114668, 0.04487498, 0.04585767, 0.04381332, 0.04707023, 0.04608198, 0.04299302, 0.04885294, 0.04625258, 0.04341669, 0.04364555, 0.04496738, 0.04771595, 0.04221626, 0.04669934, 0.04655068, 0.04293044, 0.04142471, 0.0430474 , 0.04142686, 0.04910276, 0.05069312, 0.04412301, 0.03741907, 0.04366074, 0.04271793, 0.04822319, 0.04955733, 0.04480288, 0.03854231, 0.05258446, 0.04632841, 0.04218598, 0.04873542, 0.04684362, 0.04173435, 0.04979273, 0.04954611, 0.04809578, 0.04894047, 0.04714015, 0.04727152, 0.0405341 , 0.04869596, 0.04211921, 0.0472672 , 0.03973715, 0.04285718, 0.0441104 , 0.04695732, 0.04738585, 0.04666111, 0.05082641, 0.04619593, 0.04727027, 0.05069427, 0.0456792 , 0.04395541, 0.04032792, 0.04956266, 0.04489709, 0.04551175, 0.04170445, 0.039224 , 0.04439168, 0.05087589, 0.04068655, 0.04844672, 0.04605304, 0.04629987, 0.04351723, 0.04140273, 0.04298938, 0.04106323, 0.04317099, 0.04732946, 0.03829337, 0.04569024, 0.04079556, 0.04742151, 0.04236698, 0.04340556, 0.04842246, 0.04903173, 0.04463866, 0.04369826, 0.04606098, 0.04681235, 0.04796005, 0.04444611, 0.05009542, 0.04329401, 0.0476929 , 0.04254179, 0.04640316, 0.04698205, 0.04594029, 0.04440063, 0.04371973, 0.04043747, 0.04419555, 0.04642175, 0.04144921, 0.04500867, 0.04081082, 0.04333657, 0.04190279, 0.04592907, 0.04117577, 0.04978934, 0.04488768, 0.04813121, 0.04600955, 0.04739863, 0.0458207 , 0.04700503, 0.04251042, 0.04407827, 0.04381095, 0.04798204, 0.04381264, 0.0468246 , 0.04432791, 0.04262607, 0.04803382, 0.04367965, 0.04473193, 0.05151782, 0.0431806 , 0.04946927, 0.04764608, 0.04325926, 0.0458524 , 0.03946785, 0.04038225, 0.04750486, 0.03900081, 0.04092891, 0.0480587 , 0.04586711, 0.04201325, 0.04718183, 0.04768575, 0.04396952, 0.04126091, 0.04288731, 0.04131418, 0.04292477, 0.04131691, 0.04881642, 0.04769117, 0.04634296, 0.04856059, 0.05022532, 0.04751964, 0.04985441, 0.04781258, 0.04623542, 0.04786554, 0.04871963, 0.04515905, 0.0444008 , 0.05050105, 0.04165176, 0.04565505, 0.04683316, 0.04405835, 0.04556649, 0.04277391, 0.04445551, 0.04347322, 0.04471067, 0.04621331, 0.04364481, 0.04606537, 0.04067207, 0.04011425, 0.04845894, 0.04581275, 0.04729834, 0.04866388, 0.0434395 , 0.03920529, 0.041447 , 0.0425844 , 0.0455108 , 0.04510086, 0.04611818, 0.04471245, 0.04859071, 0.04859655, 0.03961201, 0.04108481, 0.04350163, 0.04066479, 0.04299716, 0.0409928 , 0.04570243, 0.04068177, 0.0424811 , 0.04828524, 0.04412345, 0.03944542, 0.04338805, 0.04922771, 0.04946275, 0.0447216 , 0.04872858, 0.04079523, 0.04283997, 0.04124148, 0.04565558, 0.04281041, 0.04773227, 0.04550097, 0.04508202, 0.04749466, 0.03991025, 0.04691873, 0.04008758, 0.03827076, 0.04817591, 0.04113973, 0.04271469, 0.04142675, 0.04383673, 0.04241571, 0.0444138 , 0.0448699 , 0.0445637 , 0.04551745, 0.04266308, 0.04128645, 0.05029066, 0.04258456, 0.04092529, 0.04228388, 0.04809675, 0.04779026, 0.04547699, 0.04681882, 0.04515109, 0.04455849, 0.04062707, 0.04394847, 0.0540176 , 0.04906232, 0.04280255, 0.04570146, 0.0404382 , 0.04646957, 0.0494561 , 0.04502105, 0.04290089, 0.04731206, 0.04166547, 0.04091599, 0.04691376, 0.05123364, 0.04381634, 0.0444253 , 0.04774744, 0.04107327, 0.04970158, 0.04363839, 0.04146108, 0.0400315 , 0.04564745, 0.04812489, 0.04502683, 0.04618436, 0.04718696, 0.04238817, 0.04991695, 0.04687108, 0.04568382, 0.05013433, 0.05055857, 0.04523953, 0.04260869, 0.04038252, 0.03702686, 0.04697013, 0.04766749, 0.04014567, 0.04355803, 0.04262662, 0.04035628, 0.04161503, 0.04735591, 0.0419495 , 0.04501374, 0.04442854, 0.04369903, 0.04695287, 0.04540954, 0.04597675, 0.04189571, 0.04684482, 0.04469593, 0.04726894, 0.04466175, 0.04596659, 0.04904686, 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0.05189341, 0.04170926, 0.04438121, 0.04457078, 0.04517643, 0.04370614, 0.04958972, 0.03840254, 0.04512002, 0.04392756, 0.04209291, 0.0404631 , 0.04973605, 0.04544987, 0.04430782, 0.04150844, 0.04293343, 0.03735777, 0.04090707, 0.04377457, 0.04167597, 0.04646467, 0.04530597, 0.04724711, 0.04549568, 0.04503617, 0.04175847, 0.04705591, 0.04651767, 0.04638402, 0.04362851, 0.04381222, 0.04696647, 0.04563065, 0.04006881, 0.04642041, 0.04673795, 0.04505947, 0.04894623, 0.04850348, 0.04274634, 0.04661226, 0.04584635, 0.04418679, 0.0488062 , 0.04684457, 0.04166846, 0.04580169, 0.04385596, 0.04443812, 0.042286 , 0.04960309, 0.04097018, 0.04379288, 0.04165084, 0.04515601, 0.04370915, 0.04254403, 0.03945375, 0.04632602, 0.04639047, 0.04663681, 0.04179178, 0.04725791, 0.04757171, 0.04843659, 0.04273204, 0.04256972, 0.05294427, 0.05185989, 0.04727349, 0.05511793, 0.04127009, 0.05043235, 0.04191125, 0.04420794, 0.04700199, 0.04715707, 0.03940633, 0.04295491, 0.04048252, 0.04686483, 0.04432732, 0.05075038, 0.04767598, 0.04395612, 0.04282439, 0.04487319, 0.04284997, 0.04689691, 0.04344958, 0.04622157, 0.04383957, 0.04808749, 0.04143248, 0.04020974, 0.041259 , 0.04111953, 0.04542849, 0.04617783, 0.04547408, 0.05126034, 0.0449876 , 0.04431241, 0.04331047, 0.04359685, 0.04466332, 0.04352393, 0.05053184, 0.0420588 , 0.04557359, 0.04991296, 0.04589503, 0.04366415, 0.04336119, 0.05020386, 0.04424407, 0.04738682, 0.0428142 , 0.04654414, 0.04670968, 0.04839783, 0.04567707, 0.04619739, 0.04063572, 0.04939731, 0.03943886, 0.04608255, 0.04359433, 0.04050661, 0.04111041, 0.05775102, 0.04982569, 0.04418107, 0.04213239, 0.04606388, 0.04622949, 0.04454934, 0.04437447, 0.04153468, 0.04723632, 0.04337555, 0.04531656, 0.04531844, 0.04965025, 0.03908481, 0.04886395, 0.04144512, 0.05106899, 0.0435134 , 0.04010173, 0.04324871, 0.04741957, 0.04429757, 0.04478984, 0.04471098, 0.05048376, 0.0455334 , 0.04490975, 0.0461487 , 0.04498832, 0.05002286, 0.04224944, 0.03989775, 0.04229135, 0.05001072, 0.0444921 , 0.04369312, 0.049683 , 0.04144563, 0.04192496, 0.04427042, 0.04268237, 0.04406869, 0.04276288, 0.04455397, 0.0400342 , 0.04535496, 0.04483139, 0.04496589, 0.04388483, 0.04325281, 0.04742232, 0.04147194, 0.03935482, 0.0449111 , 0.04847137, 0.04637344, 0.0476185 , 0.04605648, 0.04594097, 0.04594877, 0.04271411, 0.04803218, 0.04732811, 0.04761602, 0.0447552 , 0.04163056, 0.04239556, 0.04513909, 0.04752108, 0.04820901, 0.04538986, 0.04130778, 0.04395333, 0.04485472, 0.04424779, 0.04422037, 0.04472496, 0.04255319, 0.04775233, 0.04389739, 0.03939557, 0.04467024, 0.03869845, 0.04545562, 0.04902888, 0.04442053, 0.04398383, 0.04517772, 0.0440139 , 0.04175955, 0.04460751, 0.04662963, 0.05069564, 0.04281444, 0.04561765, 0.0429581 , 0.04310135, 0.03930565, 0.04523182, 0.0416166 , 0.03961216, 0.04025454, 0.04639793, 0.04287705, 0.04865216, 0.04496899, 0.04240827, 0.04512669, 0.04363657, 0.04442822, 0.04522617, 0.04841711, 0.049549 , 0.04556039, 0.04758399, 0.04074275, 0.04459929, 0.04159275, 0.04585022, 0.04388692, 0.04480129, 0.04471656, 0.04502996, 0.04695812, 0.04245843, 0.04893408, 0.04471161, 0.04508629, 0.04459937, 0.04035969, 0.04275175, 0.04880425, 0.04554817, 0.04345368, 0.04076855, 0.04290657, 0.04557181, 0.04811059, 0.04359669, 0.04545599, 0.04268864, 0.04507617, 0.04556212, 0.04969935, 0.0472998 , 0.04697147, 0.04470329, 0.04410011, 0.0448672 , 0.04116184, 0.0454671 , 0.04731715, 0.04707245, 0.04644871, 0.04360283, 0.04479812, 0.04705519, 0.0437019]) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for i_rep in range(n_rep): @@ -401,13 +395,25 @@ Cross-fitting performs well empirically because the entire sample can be used fo score='IV-type') obj_dml_plr.fit() this_theta = obj_dml_plr.coef[0] + this_se = obj_dml_plr.se[0] # we show that the loaded result matches the just computed print(np.abs(theta_dml[i_rep] - this_theta)) + print(np.abs(se_dml[i_rep] - this_se)) theta_dml[i_rep] = this_theta - - ax = sns.kdeplot(theta_dml, shade=True, color=colors[3]) + se_dml[i_rep] = this_se + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_dml - alpha)/se_dml, + color=face_colors[2], edgecolor = edge_colors[2], + stat='density', bins=30, label='Double ML with cross-fitting'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); @savefig orth.png width=5in - ax.axvline(0.5, color='k', label='True theta'); + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); .. tabbed:: R @@ -416,10 +422,14 @@ Cross-fitting performs well empirically because the entire sample can be used fo set.seed(3333) # to speed up the illustration we hard-code the simulation results - theta_dml = c(0.42913806, 0.43316824, 0.61369310, 0.47420288, 0.56612731, 0.48624066, 0.48266824, 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0.0403181647, 0.0434331567, 0.0420806305, 0.0423684465, 0.0456064497, 0.0410769126, 0.0434588066, 0.0489372916) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for (i_rep in seq_len(n_rep)) { @@ -432,14 +442,25 @@ Cross-fitting performs well empirically because the entire sample can be used fo score='IV-type') obj_dml_plr$fit() this_theta = obj_dml_plr$coef + this_se = obj_dml_plr$se print(abs(theta_dml[i_rep] - this_theta)) + print(abs(se_dml[i_rep] - this_se)) theta_dml[i_rep] = this_theta + se_dml[i_rep] = this_se } - g_dml = ggplot(data.frame(theta_dml), aes(x = theta_dml)) + - geom_density(fill = "dark red", alpha = 0.3, color = "dark red") + - geom_vline(aes(xintercept = alpha), col = "black") + - xlim(c(0.08, 0.75)) + xlab("") + ylab("") + theme_minimal() + g_dml = ggplot(data.frame(theta_rescaled=(theta_dml - alpha)/se_dml), aes(x = theta_rescaled)) + + geom_histogram(aes(y=..density.., x=theta_rescaled, colour = "Double ML with cross-fitting", fill="Double ML with cross-fitting"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Double ML with cross-fitting", "N(0, 1)"), + values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"='black')) + + scale_fill_manual(name='',, + breaks=c("Double ML with cross-fitting", "N(0, 1)"), + values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() g_dml @@ -450,7 +471,7 @@ To illustrate the benefits of the auxiliary prediction step in the DML framework .. math:: - \sqrt{n}(\check{\theta} - \theta) = a^* + b^* + c^* + \sqrt{n}(\check{\theta}_0 - \theta_0) = a^* + b^* + c^* Chernozhukov et al. (2018) argues that: @@ -476,29 +497,316 @@ The third term :math:`c^*` vanishes in probability if sample splitting is applie .. ipython:: python - ax = sns.kdeplot(theta_ols, shade=True, label='OLS') - sns.kdeplot(theta_nonorth, shade=True, ax=ax, label='Non-orthogonal ML'); - sns.kdeplot(theta_orth_nosplit, shade=True, label='Double ML (no sample splitting)'); - sns.kdeplot(theta_dml, shade=True, label='Double ML with cross-fitting'); - ax.axvline(0.5, color='k', label='True $\\theta$'); + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_nonorth - alpha)/se_nonorth, + color=face_colors[0], edgecolor = edge_colors[0], + stat='density', bins=30, label='Non-orthogonal ML'); + sns.histplot((theta_orth_nosplit - alpha)/se_orth_nosplit, + color=face_colors[1], edgecolor = edge_colors[1], + stat='density', bins=30, label='Double ML (no sample splitting)'); + sns.histplot((theta_dml - alpha)/se_dml, + color=face_colors[2], edgecolor = edge_colors[2], + stat='density', bins=30, label='Double ML with cross-fitting'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); @savefig comparison.png width=5in - ax.legend(); + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); .. tabbed:: R .. jupyter-execute:: - g_all = ggplot(data.frame(theta_ols, theta_nonorth, theta_orth_nosplit, theta_dml)) + - geom_density(aes(x = theta_ols), fill = "dark blue", alpha = 0.3, color = "dark blue") + - geom_density(aes(x = theta_nonorth), fill = "dark orange", alpha = 0.3, color = "dark orange") + - geom_density(aes(x = theta_orth_nosplit), fill = "dark green", alpha = 0.3, color = "dark green") + - geom_density(aes(x = theta_dml), fill = "dark red", alpha = 0.3, color = "dark red") + - geom_vline(aes(xintercept = alpha), col = "black") + - xlim(c(0.08, 0.75)) + xlab("") + ylab("") + theme_minimal() + g_all = ggplot(data.frame(t_nonorth=(theta_nonorth - alpha)/se_nonorth, + t_orth_nosplit=(theta_orth_nosplit - alpha)/se_orth_nosplit, + t_dml=(theta_dml - alpha)/se_dml)) + + geom_histogram(aes(x = t_nonorth, y=..density.., colour = "Non-orthogonal ML", fill="Non-orthogonal ML"), + bins = 30, alpha = 0.3) + + geom_histogram(aes(x = t_orth_nosplit, y=..density.., colour = "Double ML (no sample splitting)", fill="Double ML (no sample splitting)"), + bins = 30, alpha = 0.3) + + geom_histogram(aes(x = t_dml, y=..density.., colour = "Double ML with cross-fitting", fill="Double ML with cross-fitting"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Non-orthogonal ML", "Double ML (no sample splitting)", "Double ML with cross-fitting", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", + "Double ML (no sample splitting)"="dark orange", + "Double ML with cross-fitting"="dark green", + "N(0, 1)"='black')) + + scale_fill_manual(name='', + breaks=c("Non-orthogonal ML", "Double ML (no sample splitting)", "Double ML with cross-fitting", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", + "Double ML (no sample splitting)"="dark orange", + "Double ML with cross-fitting"="dark green", + "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() g_all +Partialling out score ++++++++++++++++++++++ +Another debiased estimator, based on the partialling-out approach of Robinson(1988), is + +.. math:: + + \check{\theta}_0 = \left(\frac{1}{n} \sum_{i\in I} \hat{V}_i \hat{V}_i \right)^{-1} \frac{1}{n} \sum_{i\in I} \hat{V}_i (Y_i - \hat{\ell}_0(X_i)), + +with :math:`\ell_0(X_i) = E(Y|X)`. +All nuisance parameters for the estimator with ``score='partialling out'`` are conditional mean functions, which can be +directly estimated using ML methods. This is a minor advantage over the estimator with ``score='IV-type'``. +In the following, we repeat the above analysis with ``score='partialling out'``. In a first part of the analysis, we +estimate :math:`\theta_0` without sample splitting. Again we observe a bias from overfitting. + +.. tabbed:: Python + + .. ipython:: python + + import numpy as np + np.random.seed(4444) + + # to speed up the illustration we hard-code the simulation results + theta_orth_po_nosplit = np.array([0.44689745, 0.41560548, 0.39147078, 0.40061244, 0.41388613, 0.40812013, 0.44497593, 0.38585885, 0.44117726, 0.49159354, 0.42731988, 0.43751013, 0.45444596, 0.33483986, 0.39266598, 0.38949641, 0.41844929, 0.37425161, 0.44119905, 0.40405533, 0.38900643, 0.36211315, 0.37546086, 0.41105622, 0.39496921, 0.37376606, 0.42015518, 0.39926076, 0.38283516, 0.42610636, 0.41467175, 0.47398591, 0.33381323, 0.35597262, 0.40394593, 0.37240773, 0.38373405, 0.45020658, 0.43253392, 0.42079326, 0.37306779, 0.37176962, 0.41359098, 0.32905667, 0.43924895, 0.40621988, 0.4248174 , 0.4188524 , 0.42545551, 0.48109574, 0.39236691, 0.41109807, 0.43830774, 0.43131476, 0.46297181, 0.36525 , 0.42785268, 0.45948104, 0.37387788, 0.39193983, 0.41964355, 0.45105777, 0.45068412, 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0.34107974, 0.4503945 , 0.37347764, 0.43770291, 0.45765222, 0.41591249, 0.37818029, 0.40079684, 0.40742744, 0.51493047, 0.38143551, 0.38922804, 0.36482121, 0.48946817, 0.40386025, 0.3789189 , 0.43472849, 0.4482401 , 0.41958351, 0.3874319 , 0.46551354, 0.40688943, 0.42076722, 0.42261826, 0.42613403, 0.3982167 , 0.40560043, 0.38510455, 0.41363841, 0.4153734 , 0.42633414, 0.48195624, 0.4064136 , 0.39629634, 0.37622216, 0.44147214, 0.42028914, 0.35511551, 0.32997176, 0.37204305, 0.34571186, 0.38392038, 0.454553 , 0.42857666, 0.37738172, 0.38752155, 0.39888209, 0.33003166, 0.39749669, 0.40623056, 0.38995359, 0.43574707, 0.39400018, 0.38395351, 0.42308851, 0.45226606, 0.39925256, 0.41512287, 0.38956918, 0.36903949, 0.39506798, 0.38304656, 0.45834436, 0.44365015, 0.3559809 , 0.49703991, 0.36755582, 0.32405813, 0.35703684, 0.45291725, 0.3862536 , 0.45430432, 0.43151156, 0.42199184, 0.45506641, 0.41894509, 0.40927734, 0.44103929, 0.41244317, 0.45798097, 0.41423993, 0.41234892, 0.49257489, 0.39316852, 0.45133683, 0.48647005, 0.52262288, 0.39411164, 0.4361213 , 0.47251784, 0.41942088, 0.38252744, 0.48580277, 0.41970908, 0.41279836, 0.4496232 , 0.39192501, 0.44793215, 0.39777389, 0.36491141, 0.42293433, 0.38311846, 0.34729731, 0.3467867 , 0.38890129, 0.41369155, 0.43718281, 0.42255496, 0.42606418, 0.34113915, 0.40785283, 0.4266964 , 0.48600678, 0.38993538, 0.37674472, 0.36469012, 0.41382762, 0.3910651 , 0.36058839, 0.4385373 , 0.3853683 , 0.45285794, 0.43859701, 0.35940378, 0.35532133, 0.36747323, 0.31895662, 0.4765736 , 0.44085975, 0.43925062, 0.43305578, 0.38225082, 0.40558291, 0.42308383, 0.44251497, 0.38819576, 0.48809178, 0.39663077, 0.43415774, 0.44212896, 0.46864931, 0.41112727, 0.46702856, 0.35242806, 0.43235724, 0.43946028, 0.41192496, 0.3837241 , 0.38252215, 0.40869806, 0.51874593, 0.3617031 , 0.45676473, 0.34502171, 0.42263263, 0.46617327, 0.44260223, 0.38736024, 0.38962609, 0.41169203, 0.42714373, 0.35953918, 0.44465238, 0.41616539, 0.40898319, 0.35428957, 0.39896774, 0.46760866, 0.35534866, 0.39458018, 0.3010404 , 0.4275348 , 0.4130533 , 0.42218207, 0.50383664, 0.35943386, 0.46271252, 0.43461421, 0.44607091, 0.38827735, 0.4151206 , 0.39796933, 0.38299143, 0.37773728, 0.39394795, 0.46817041, 0.37619518, 0.37979198, 0.39579081, 0.38947461, 0.49225938, 0.32224241, 0.45937019, 0.40716829, 0.41395486, 0.35955957, 0.41686161, 0.37045689, 0.35787435, 0.46819524, 0.42714861, 0.4317218 , 0.41438206, 0.37601954, 0.4035717 , 0.38687642, 0.46099666, 0.39478873, 0.47577063, 0.46146113, 0.43162587, 0.37506128, 0.367823 , 0.3814589 , 0.43190693, 0.37833206, 0.44924794, 0.43123303, 0.43457722, 0.45822097, 0.42546258, 0.44599857, 0.43560714, 0.4018572 , 0.35588031, 0.39884516, 0.38828861, 0.3938863 , 0.36650029, 0.38365921, 0.45357156, 0.40473875, 0.38083154, 0.34806574, 0.45759315, 0.392613 , 0.44443932, 0.3859657 , 0.42588092, 0.37958914, 0.43936553, 0.40565913, 0.37050438, 0.38075554, 0.41480746, 0.41046031, 0.44809874, 0.40125569, 0.37153855, 0.34580786, 0.39599036, 0.40912106, 0.40286777, 0.4204081 , 0.40193548, 0.38921222, 0.36955462, 0.43778725, 0.38803946, 0.40488867, 0.39969558, 0.41182898, 0.41815582, 0.39436057, 0.40924522, 0.34340729, 0.36349859, 0.32419746, 0.38380303, 0.40018357, 0.45952972, 0.38707899, 0.35145758, 0.38427105, 0.39348254, 0.41127523, 0.4128339 , 0.40017793, 0.44410231, 0.40519121, 0.43557319, 0.39673401, 0.42723438, 0.46481697, 0.46732216, 0.35065925, 0.44945291, 0.38482921, 0.38124517, 0.40981892, 0.43832261, 0.43373603, 0.41552012, 0.38114908, 0.46441949, 0.43537922, 0.44091101, 0.41058189, 0.38814734, 0.37535326, 0.43135988, 0.4162225 , 0.41918978, 0.42145809, 0.39896498, 0.37387455, 0.43116882, 0.3833774 , 0.38335557, 0.40718161, 0.45818585, 0.39677352, 0.35869148, 0.43742879, 0.4388087 , 0.39325689, 0.40411463, 0.34076527, 0.4002913 , 0.38073684, 0.42822759, 0.41065555, 0.36612017, 0.43793757, 0.38045494, 0.40073521, 0.43634931, 0.37228829, 0.38076058, 0.44028763, 0.40618622, 0.39886155, 0.39606878, 0.40121651, 0.42329785, 0.38391397, 0.48701045, 0.40852587, 0.38811837, 0.33700027, 0.43337934, 0.47638898, 0.45862387, 0.43101757, 0.39399701, 0.37816887, 0.34243431, 0.40166362, 0.3677345 , 0.48000176, 0.46025401, 0.42700476, 0.34592185, 0.41396936, 0.3651648 , 0.44434418, 0.42944586, 0.37357513, 0.45325424, 0.35135126, 0.34060331, 0.44891731, 0.45647405, 0.40584386, 0.40792945, 0.37417043, 0.40859987, 0.48744066, 0.45568601, 0.40720023, 0.38534174, 0.44418675, 0.40091154, 0.36556 , 0.40113015, 0.37258915, 0.44318192, 0.38091808, 0.40725597, 0.44666564, 0.42190333, 0.44481681, 0.41919932, 0.37032402, 0.48364551, 0.38748608, 0.44018272, 0.42203535, 0.37242747, 0.38219231, 0.46776887, 0.43337044, 0.42124322, 0.37882104, 0.44778111, 0.41846436, 0.37811701, 0.37155185, 0.39007756, 0.36378231, 0.42661165, 0.43822261, 0.40546708, 0.41911667, 0.4260626 , 0.42256451, 0.44192597, 0.47842638, 0.39611512, 0.44491116, 0.40365296, 0.39359214, 0.40003074, 0.34958918, 0.34758949, 0.45646971, 0.38078347, 0.42067093, 0.39756817, 0.41426262, 0.45689868, 0.42263988, 0.48026677, 0.44894483, 0.35304089, 0.40763657, 0.35102165, 0.40311662, 0.44168856, 0.42674332, 0.40034222, 0.37597472, 0.37988866, 0.49597058, 0.36669922, 0.50231546, 0.47759904, 0.41977028, 0.3902988 , 0.35337019, 0.29725319, 0.37085959, 0.40683181, 0.33962811, 0.3775262 , 0.40378711, 0.37522767, 0.43806618, 0.37608875, 0.38363954, 0.4213465 , 0.37685846, 0.43864085, 0.42072523, 0.41303825, 0.44085664, 0.44935243, 0.3955478 , 0.41700251, 0.42745432, 0.40076353, 0.44415625, 0.40592704, 0.42403648, 0.43164961, 0.43741185, 0.4556666 , 0.35447542, 0.36179497, 0.37609757, 0.41936549, 0.45255993, 0.39765959, 0.41473308, 0.45325442, 0.43660725, 0.41725395, 0.42584886, 0.35245624, 0.40770218, 0.34849235, 0.41924673, 0.42967222, 0.4688549 , 0.41986642, 0.5394729 , 0.41234063, 0.38889672, 0.4133863 , 0.48377739, 0.3985232 , 0.44467283, 0.40187751, 0.41207328, 0.38259148, 0.49482289, 0.40994346, 0.43842771, 0.46299092, 0.35792853, 0.43628973, 0.38300048, 0.4402988 , 0.37719632, 0.447099 , 0.46458036, 0.34070484, 0.38668328, 0.41877418, 0.39920467, 0.41402592, 0.46197827, 0.34998859, 0.44292424, 0.39554279, 0.45806025, 0.44549445, 0.41144303, 0.39757833, 0.44460934, 0.39282869, 0.40982084, 0.3594684 , 0.38035559, 0.46361533, 0.39866406, 0.45390917, 0.37363494, 0.42751835, 0.36809349, 0.3981875 , 0.38283569, 0.33765472, 0.445369 , 0.46432175, 0.3828271 , 0.40032819, 0.43784922, 0.40907506, 0.50620526, 0.39825537, 0.40494115, 0.42097051, 0.39813231, 0.42767821, 0.43632554, 0.48342796, 0.37171266, 0.4231365 , 0.39791793, 0.4475876 , 0.43187838, 0.36713241, 0.39348117, 0.37435029, 0.44677536, 0.3560875 , 0.41850455, 0.38217649, 0.37817251, 0.44770328, 0.43056134, 0.38866372, 0.42418107, 0.35271324, 0.44943607, 0.41989644, 0.4117774 , 0.40604735, 0.40890907, 0.33590615, 0.46407173, 0.43129958, 0.36415985, 0.38422561, 0.43792973, 0.41323115, 0.4359505 , 0.43611463, 0.36418954, 0.39182405, 0.39411455, 0.41871595, 0.46333644, 0.402323 , 0.44001463, 0.40980919, 0.46406623, 0.43420349, 0.40457103, 0.38281922, 0.4200635 , 0.45682402, 0.42409818, 0.43678774, 0.39269752, 0.51152078, 0.41304056, 0.44903095, 0.3799896 , 0.44601481, 0.42465758, 0.3995831 , 0.38351877, 0.39567814, 0.49077009, 0.4166894 , 0.42924459, 0.43517778, 0.49714309, 0.34822792, 0.45457007, 0.45281455, 0.37328413, 0.39353403, 0.43454553, 0.44074761, 0.4115068 , 0.41406096, 0.42452023, 0.3787139 , 0.43965875, 0.49540995, 0.41198105, 0.35261123, 0.41107871, 0.38119491, 0.37193428, 0.38633685, 0.32842662, 0.40204225, 0.42665654, 0.40166691, 0.40939109, 0.41806659, 0.3609013 , 0.37036398, 0.36511387, 0.45810674, 0.38218874, 0.40843999, 0.39675316, 0.41089759, 0.45017994, 0.48048135, 0.40035762, 0.38592168, 0.43293865, 0.38960311, 0.49343706, 0.36434481, 0.4042647 , 0.4543836 , 0.37766804, 0.37252239, 0.38376966, 0.4007445 , 0.40013226, 0.40464522, 0.39352434, 0.40585953, 0.38774478, 0.43955992, 0.38933552, 0.47584388, 0.4760254 , 0.38309879, 0.4010732 , 0.46586454, 0.37250797, 0.41343097, 0.41691298, 0.42565652, 0.39245694, 0.43781405, 0.39532442, 0.46479819, 0.37892093, 0.4500263 , 0.42672327, 0.36790712, 0.41575054, 0.36628736, 0.35657307, 0.37455605, 0.42584967, 0.44626351, 0.42206467, 0.40746804, 0.49480752, 0.40585813, 0.39371666, 0.38483937, 0.32532322, 0.37839034, 0.44029757, 0.43886105, 0.44089243, 0.48483716, 0.38290252, 0.44274085, 0.4272902 , 0.40004465, 0.37967661, 0.29926117, 0.46973071, 0.39067255, 0.40449049, 0.36781328, 0.46677963, 0.37546023, 0.455158 , 0.3808844 , 0.41382398, 0.45051137, 0.4279048 , 0.47032358, 0.34036995, 0.43181476, 0.44090817, 0.41262752, 0.42174053, 0.43467423, 0.43022294, 0.37072532, 0.43068552, 0.42300783, 0.38696066, 0.36009265, 0.34119649, 0.39879691, 0.4365699 , 0.4120221 , 0.47382627, 0.42169809, 0.40599731, 0.44671891, 0.40248377, 0.44969546, 0.35117912, 0.44837504, 0.4161964 , 0.37582456, 0.35299654, 0.447547 , 0.39787134, 0.3803233 , 0.45412813, 0.36251435, 0.45113205, 0.42711722, 0.35638102, 0.42258747, 0.45780058, 0.35630224, 0.39713166, 0.37472678, 0.37831073, 0.43158708, 0.36181206, 0.45090422, 0.32068871, 0.43781083, 0.44439732, 0.4035753 , 0.40448936, 0.37632135, 0.35958719, 0.46351404, 0.40932618, 0.4341114 , 0.38818929, 0.38353028, 0.45386913, 0.50000539, 0.42122936, 0.40654102, 0.36816651, 0.44763332, 0.41004633, 0.39898167, 0.44444691, 0.36822278, 0.4553816 , 0.3905132 , 0.36842322, 0.48547593, 0.44972635, 0.41782122, 0.45257978, 0.45659295, 0.42285171, 0.39763381, 0.41796586, 0.39381587, 0.46934179, 0.42922762, 0.38053444, 0.40544273, 0.36006186, 0.39863556, 0.3719792 , 0.39821203, 0.41790859, 0.49867499, 0.42365679, 0.395948 , 0.43571833, 0.34534889, 0.41997742, 0.41201298, 0.37578671, 0.41356327, 0.39392439, 0.44224059, 0.38535319, 0.41808058, 0.33271179, 0.46387916, 0.44893328, 0.40814264, 0.39180342, 0.43969064, 0.35560315, 0.4266996 , 0.38941229, 0.35687593, 0.40373073, 0.34140252, 0.40307342, 0.42094028, 0.39607329, 0.37907175, 0.41389525, 0.39529958, 0.45090839, 0.40201934, 0.42203752, 0.39398395, 0.39915548, 0.49363392, 0.4614152 , 0.47190431, 0.42186278, 0.40138577, 0.40398946, 0.31354679, 0.4615471 , 0.44052724, 0.40421449, 0.43339062, 0.42131888, 0.39299912, 0.43951129]) + se_orth_po_nosplit = np.array([0.03474502, 0.0373666 , 0.03254624, 0.035711 , 0.03857083, 0.03393274, 0.0382176 , 0.0390209 , 0.03947079, 0.03632686, 0.0365749 , 0.03596462, 0.03283355, 0.03305384, 0.03462809, 0.03429812, 0.04047603, 0.03751935, 0.03350184, 0.03692805, 0.03675086, 0.04425244, 0.03688826, 0.03667372, 0.04111085, 0.03012514, 0.03567342, 0.03630253, 0.03869279, 0.03591116, 0.03291957, 0.03330094, 0.03466816, 0.03615811, 0.03925414, 0.03463743, 0.0403616 , 0.03672733, 0.03673892, 0.03729507, 0.03656263, 0.0361764 , 0.03740284, 0.03674824, 0.03385807, 0.03696247, 0.03624528, 0.03540566, 0.03432279, 0.03781447, 0.03544402, 0.03242579, 0.03509169, 0.03933641, 0.03515456, 0.04092438, 0.03431629, 0.03421128, 0.03484018, 0.03407852, 0.03809609, 0.03848484, 0.0382654 , 0.03495158, 0.03619387, 0.03773587, 0.03797528, 0.03389219, 0.03663497, 0.03785326, 0.03551873, 0.03299111, 0.03513855, 0.03886106, 0.03445817, 0.03464812, 0.0355295 , 0.03802371, 0.0352706 , 0.03641776, 0.03786271, 0.03845134, 0.03586474, 0.03619399, 0.03436212, 0.03985137, 0.03525877, 0.0385634 , 0.04007947, 0.03799021, 0.03552911, 0.03579554, 0.03336541, 0.04141928, 0.03515444, 0.03668795, 0.03444747, 0.03644692, 0.03806484, 0.03519977, 0.03743296, 0.03828006, 0.03827395, 0.03657322, 0.03633241, 0.03422569, 0.03600341, 0.03973577, 0.03454769, 0.03568129, 0.0396222 , 0.03521892, 0.03596001, 0.03624428, 0.03815015, 0.03995477, 0.03826805, 0.03525698, 0.03742405, 0.03782724, 0.03844241, 0.03742421, 0.03438375, 0.03364354, 0.03845537, 0.03885686, 0.03643648, 0.03452413, 0.03600843, 0.0369478 , 0.03951252, 0.04147925, 0.03912529, 0.03211512, 0.03828445, 0.04122697, 0.03767405, 0.03814904, 0.03677246, 0.0348576 , 0.03664641, 0.03738636, 0.03694522, 0.03921987, 0.03652723, 0.03714045, 0.03334889, 0.03591158, 0.03467354, 0.03291863, 0.03650219, 0.03963865, 0.03836493, 0.03835041, 0.03321675, 0.03321789, 0.03548968, 0.03402733, 0.03633222, 0.0378797 , 0.03689956, 0.03718884, 0.03861813, 0.03437033, 0.0396973 , 0.03825877, 0.03611252, 0.03611327, 0.03678618, 0.03914326, 0.03612262, 0.03907715, 0.03660782, 0.03543502, 0.034366 , 0.03436277, 0.03323214, 0.03867857, 0.03855516, 0.03736977, 0.03163678, 0.0357661 , 0.03607132, 0.03876504, 0.04041912, 0.03634519, 0.03088368, 0.04378558, 0.03802604, 0.03509891, 0.03971062, 0.03689964, 0.03480082, 0.03983675, 0.03928926, 0.03992703, 0.04197775, 0.03720813, 0.04007649, 0.03307689, 0.03943535, 0.03518234, 0.03879254, 0.03356806, 0.03555913, 0.03467602, 0.03840389, 0.03858417, 0.03910251, 0.03937124, 0.03695466, 0.03822676, 0.04013834, 0.03398154, 0.03628905, 0.03605391, 0.03737559, 0.03754431, 0.03731786, 0.03372294, 0.03382883, 0.03872675, 0.0408826 , 0.03332793, 0.04086265, 0.03681539, 0.03775453, 0.03515851, 0.03418413, 0.03516572, 0.03302308, 0.03497077, 0.0376036 , 0.03127416, 0.03481038, 0.032526 , 0.03874042, 0.03619953, 0.03620319, 0.04098181, 0.04187841, 0.03502884, 0.03437674, 0.03790772, 0.03717506, 0.03745607, 0.03696607, 0.03843035, 0.03415598, 0.03953016, 0.03580827, 0.03824459, 0.03737082, 0.03784694, 0.03611907, 0.03467281, 0.03198953, 0.03810093, 0.03684028, 0.03511122, 0.03606661, 0.03401353, 0.03538895, 0.0342976 , 0.03729535, 0.03238319, 0.03912026, 0.03972276, 0.03902827, 0.03950146, 0.03682319, 0.03569845, 0.039429 , 0.03579186, 0.03471518, 0.03556643, 0.03674656, 0.03592277, 0.0395152 , 0.03486726, 0.03607796, 0.03755308, 0.03898473, 0.03776401, 0.04209196, 0.03472871, 0.03780762, 0.03604197, 0.03741701, 0.03777826, 0.03220387, 0.03437665, 0.03660562, 0.03364242, 0.03439096, 0.03905695, 0.0408217 , 0.03625206, 0.03527866, 0.03767158, 0.03844056, 0.03399458, 0.03530452, 0.03274726, 0.03356248, 0.03429015, 0.03689297, 0.03870866, 0.0376923 , 0.03621653, 0.04218239, 0.04052051, 0.03984298, 0.03756918, 0.03609664, 0.03872729, 0.04016341, 0.03726113, 0.03766057, 0.03861554, 0.03569534, 0.03717902, 0.03770473, 0.04025156, 0.03445772, 0.03484697, 0.0370166 , 0.0351166 , 0.0365101 , 0.0374368 , 0.0346713 , 0.03894735, 0.03490064, 0.03174345, 0.0385112 , 0.03815683, 0.03793529, 0.04031503, 0.0362836 , 0.03127785, 0.03453122, 0.03460265, 0.03765777, 0.03644693, 0.03743668, 0.03402657, 0.04032537, 0.03817767, 0.03306188, 0.031091 , 0.03503291, 0.03468745, 0.03430702, 0.03550759, 0.03727183, 0.02990325, 0.03680624, 0.03968029, 0.03608235, 0.03198201, 0.0366191 , 0.03998343, 0.04061129, 0.03567003, 0.04076703, 0.03249735, 0.03515599, 0.03485176, 0.035669 , 0.03549834, 0.03879247, 0.03661692, 0.03747362, 0.03750218, 0.03330484, 0.03813348, 0.033797 , 0.03391037, 0.04017033, 0.03433249, 0.03476942, 0.03520731, 0.03512518, 0.03375186, 0.03708254, 0.03703487, 0.03503332, 0.03616182, 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0.03911459, 0.03689581, 0.03241027, 0.03787878, 0.03603995, 0.04165727, 0.04056771, 0.03741212, 0.04157788, 0.0383887 , 0.03631563, 0.03781401, 0.03979479, 0.03416006, 0.0341921 , 0.03618848, 0.03964278, 0.03551569, 0.03615303, 0.03525878, 0.03574053, 0.03594098, 0.04178191, 0.03929774, 0.03581196, 0.04015341, 0.03319067, 0.03153384, 0.03607218, 0.03691251, 0.03548286, 0.0346934 , 0.03886053, 0.03477746, 0.03550772, 0.03587398, 0.03568007, 0.03744178, 0.0349228 , 0.03687551, 0.03503053, 0.03286544, 0.03669241, 0.03896562, 0.0343098 , 0.03777013, 0.03480939, 0.0387271 , 0.0351315 , 0.03754386, 0.03496421, 0.03995719, 0.03707774, 0.0382146 , 0.03683755, 0.03623896, 0.03273244, 0.03571262, 0.03445738, 0.03952536, 0.03657445, 0.03454587, 0.03726281, 0.0382534 , 0.03129206, 0.03768886, 0.03430767, 0.04070874, 0.03799011, 0.03384805, 0.0375251 , 0.03590798, 0.03801495, 0.03763615, 0.03625218, 0.04115623, 0.03998008, 0.03596863, 0.03880255, 0.03579289, 0.03743173, 0.03760444, 0.03482858, 0.03092043, 0.03764129, 0.03918522, 0.03405991, 0.03793179, 0.03773335, 0.03510268, 0.0355164 , 0.03812738, 0.03702729, 0.03428584, 0.0385688 , 0.0361883 , 0.0374408 , 0.03668985, 0.03242703, 0.03392123, 0.0317929 , 0.04031644, 0.03506659, 0.04009986, 0.03490808, 0.03239399, 0.03968875, 0.03832643, 0.03697817, 0.04172446, 0.03641767, 0.03802434, 0.03432554, 0.0393749 , 0.03819441, 0.03672693, 0.03547376, 0.03470635, 0.03660539, 0.03643943, 0.03919675, 0.0361405 , 0.0357092 , 0.0378333 , 0.03754306, 0.0394014 , 0.03973461, 0.04056114, 0.03716176, 0.03680801, 0.03956678, 0.03753543, 0.03598327, 0.03675005, 0.03547101, 0.03492406, 0.03793042, 0.03602637, 0.03615463, 0.03547103, 0.03236276, 0.03696476, 0.03716882, 0.03786123, 0.04152208, 0.0399251 , 0.03797253, 0.03609592, 0.03853076, 0.03632484, 0.03520405, 0.0367671 , 0.03447667, 0.03734966, 0.03661531, 0.03458445, 0.03463842, 0.0371716 , 0.03547479, 0.04045816, 0.03632047, 0.03529348, 0.03648633, 0.03464473, 0.04066408, 0.03632672, 0.03852298, 0.03489493, 0.03893466, 0.03510144, 0.03716704, 0.03528462, 0.03274645, 0.0337673 , 0.03198426, 0.0395838 , 0.04101707, 0.03277293, 0.04074065, 0.03251893, 0.03694768, 0.03948034, 0.03528972, 0.03545367, 0.03282461, 0.03354317, 0.0345222 , 0.03136711, 0.04038137, 0.03186262, 0.03507791, 0.0414906 , 0.03570211, 0.03408069, 0.03395696, 0.03572162, 0.03459173, 0.04038807, 0.03742919, 0.03650051, 0.03371639, 0.03380714, 0.03407761, 0.0336904 , 0.03439459, 0.03640594, 0.03345293, 0.04019585, 0.04134589, 0.03516821, 0.0378304 , 0.03598071, 0.03453574, 0.03846343, 0.03718089, 0.03722151, 0.0362938 , 0.03440824, 0.0354824 , 0.03956783, 0.03707031, 0.03814283, 0.03349067, 0.04074838, 0.03481607, 0.03672595, 0.03990626, 0.03476584, 0.03966015, 0.03477156, 0.034339 , 0.03762172, 0.0370995 , 0.03470849, 0.04224772, 0.03291124, 0.03226651, 0.03696301, 0.03999309, 0.04208041, 0.03438727, 0.03539256, 0.03644022, 0.03480234, 0.03583202, 0.03857345, 0.03350579, 0.03714886, 0.03717429, 0.03530833, 0.03353564, 0.04307491, 0.03645676, 0.03686309, 0.0367092 , 0.03505246, 0.03179146, 0.03502657, 0.03505816, 0.03535941, 0.03823574, 0.03782026, 0.03577473, 0.03670732, 0.03679464, 0.03476963, 0.03889374, 0.03949057, 0.03865546, 0.03631931, 0.03409329, 0.03739669, 0.03325993, 0.03221564, 0.03833878, 0.03957746, 0.03584435, 0.03997407, 0.03960826, 0.03756598, 0.03733641, 0.0386708 , 0.03431129, 0.03770288, 0.03696621, 0.03523463, 0.0371088 , 0.03490614, 0.035653 , 0.03504013, 0.04009002, 0.03475992, 0.03571412, 0.03399289, 0.03837119, 0.03626078, 0.03563635, 0.03259052, 0.03989942, 0.03941204, 0.03900715, 0.03197333, 0.0384862 , 0.03788241, 0.04003726, 0.03603506, 0.03401377, 0.04334113, 0.04094369, 0.04066732, 0.04200563, 0.03444455, 0.04031428, 0.03129243, 0.03565401, 0.03737025, 0.03639366, 0.03277422, 0.03440789, 0.03223552, 0.03970053, 0.0346974 , 0.04086497, 0.03948381, 0.0368716 , 0.03481614, 0.03764524, 0.03437654, 0.03768604, 0.03475278, 0.0377687 , 0.0356684 , 0.04110741, 0.03516462, 0.03384485, 0.03392398, 0.03614618, 0.03474271, 0.03807704, 0.03889803, 0.04126355, 0.03592061, 0.03792865, 0.03459002, 0.03689696, 0.03525553, 0.03818139, 0.03975973, 0.03604507, 0.03658735, 0.04199084, 0.0359869 , 0.03620933, 0.03487707, 0.04071032, 0.03694701, 0.03910624, 0.03560849, 0.03885618, 0.03690596, 0.04243034, 0.03618007, 0.0397088 , 0.03370807, 0.03754302, 0.03168435, 0.03774717, 0.03630268, 0.03329076, 0.03146299, 0.04447259, 0.0410724 , 0.03736169, 0.0336196 , 0.03825899, 0.03744144, 0.04050642, 0.03529018, 0.03470333, 0.03860355, 0.03561639, 0.03627923, 0.03813162, 0.04057664, 0.03332849, 0.03969077, 0.03634978, 0.03917337, 0.03650846, 0.03467217, 0.03198052, 0.0386248 , 0.03941164, 0.0363901 , 0.03570452, 0.03906684, 0.03918833, 0.03749422, 0.03694331, 0.03682521, 0.04132099, 0.03731989, 0.03337376, 0.03387718, 0.04181991, 0.03646947, 0.03333412, 0.03860092, 0.03489202, 0.03532625, 0.03593162, 0.03604625, 0.03726524, 0.03723409, 0.03618784, 0.03247795, 0.03773535, 0.0381503 , 0.035616 , 0.03683644, 0.03373601, 0.04030972, 0.0342309 , 0.03353867, 0.03739194, 0.03584757, 0.03754637, 0.03965457, 0.03587568, 0.03843132, 0.03723711, 0.03364555, 0.03801765, 0.03954501, 0.03919557, 0.03598286, 0.0331916 , 0.03302435, 0.03921672, 0.03782883, 0.03787895, 0.03720195, 0.03426919, 0.03843508, 0.03765756, 0.03658322, 0.03653467, 0.03603962, 0.03587782, 0.03827373, 0.03529358, 0.03287978, 0.03425231, 0.0324074 , 0.03792556, 0.04154412, 0.03515388, 0.03634466, 0.0400113 , 0.034952 , 0.03427303, 0.03561047, 0.03870101, 0.04051633, 0.03391229, 0.03566566, 0.03549207, 0.03462774, 0.0346016 , 0.03661904, 0.03340004, 0.03263238, 0.03592759, 0.03652783, 0.03363493, 0.04014916, 0.0373243 , 0.03569296, 0.03765387, 0.03373767, 0.03786649, 0.03665141, 0.040085 , 0.03916567, 0.03762358, 0.03805482, 0.03526314, 0.03691169, 0.03471357, 0.03976527, 0.03574635, 0.03773984, 0.03642111, 0.03648961, 0.03905214, 0.03505632, 0.03830454, 0.03571615, 0.0380485 , 0.03712205, 0.03357085, 0.03427145, 0.03841388, 0.03599436, 0.03768774, 0.03368026, 0.03557021, 0.04037147, 0.03826648, 0.03545627, 0.0361974 , 0.03457355, 0.03951696, 0.03935793, 0.04005394, 0.03675155, 0.03694277, 0.03466973, 0.03672045, 0.03379401, 0.03248116, 0.03836208, 0.03720803, 0.03818912, 0.03828179, 0.03432679, 0.03796187, 0.03594795, 0.03584864]) + + # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset + #for i_rep in range(n_rep): + for i_rep in range(1): + (x, y, d) = data[i_rep] + obj_dml_data = DoubleMLData.from_arrays(x, y, d) + obj_dml_plr_orth_nosplit = DoubleMLPLR(obj_dml_data, + ml_g, ml_m, + n_folds=1, + score='partialling out', + apply_cross_fitting=False) + obj_dml_plr_orth_nosplit.fit() + this_theta = obj_dml_plr_orth_nosplit.coef[0] + this_se = obj_dml_plr_orth_nosplit.se[0] + # we show that the loaded result matches the just computed + print(np.abs(theta_orth_po_nosplit[i_rep] - this_theta)) + print(np.abs(se_orth_po_nosplit[i_rep] - this_se)) + theta_orth_po_nosplit[i_rep] = this_theta + se_orth_po_nosplit[i_rep] = this_se + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_orth_po_nosplit - alpha)/se_orth_po_nosplit, + color=face_colors[1], edgecolor = edge_colors[1], + stat='density', bins=30, label='Double ML (no sample splitting)'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); + @savefig orth_po_nosplit.png width=5in + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); + +.. tabbed:: R + + .. jupyter-execute:: + + library(data.table) + lgr::get_logger("mlr3")$set_threshold("warn") + set.seed(4444) + + # to speed up the illustration we hard-code the simulation results + theta_orth_po_nosplit = c(0.424894688, 0.415979258, 0.358214049, 0.401787264, 0.352802605, 0.374900867, 0.485234930, 0.401148984, 0.369991876, 0.475517512, 0.445076802, 0.373960370, 0.387824181, 0.342148215, 0.402114616, 0.351833721, 0.381125616, 0.461414960, 0.427616126, 0.418759210, 0.358268211, 0.401264980, 0.407889106, 0.393857741, 0.426785803, 0.368804415, 0.416975212, 0.484760806, 0.425441264, 0.390564373, 0.429884259, 0.403958684, 0.507137102, 0.439465474, 0.388448386, 0.383863300, 0.430680807, 0.442740900, 0.395929332, 0.438879887, 0.416105186, 0.463420783, 0.304417989, 0.467550408, 0.484586052, 0.470996524, 0.460477341, 0.401223146, 0.411119154, 0.383637282, 0.425053123, 0.466271156, 0.469006334, 0.363616413, 0.417576737, 0.513733653, 0.418133692, 0.348599183, 0.394726181, 0.384581212, 0.392746793, 0.365197270, 0.428143024, 0.404918669, 0.440169667, 0.394103063, 0.378160715, 0.395855561, 0.480792301, 0.382743709, 0.451031703, 0.455251318, 0.381408527, 0.381300737, 0.410594718, 0.417900354, 0.431650650, 0.427987923, 0.388522594, 0.404514383, 0.383377328, 0.407637457, 0.422540318, 0.396505143, 0.417604299, 0.433979261, 0.398505303, 0.438577405, 0.386354014, 0.496864697, 0.470265900, 0.398362215, 0.378489890, 0.397782706, 0.352943887, 0.458299569, 0.405409413, 0.411557225, 0.452976954, 0.399677608, 0.383131470, 0.359409202, 0.386673278, 0.403451635, 0.402101397, 0.345926992, 0.425853359, 0.414767703, 0.413546600, 0.425961817, 0.393060787, 0.423209103, 0.382419733, 0.439528814, 0.346969958, 0.406054155, 0.434916933, 0.413842955, 0.420869241, 0.396018540, 0.440530908, 0.463801605, 0.441943536, 0.392670005, 0.412549373, 0.418128097, 0.370135392, 0.433372541, 0.413511986, 0.417984837, 0.433422575, 0.395192998, 0.355008154, 0.481853182, 0.395635319, 0.415828051, 0.465411478, 0.377717947, 0.393839458, 0.416500090, 0.397860190, 0.365627420, 0.454536636, 0.375311328, 0.430895740, 0.381281709, 0.383331630, 0.459377553, 0.452957283, 0.435474077, 0.475775653, 0.418168929, 0.491069389, 0.367996790, 0.512591597, 0.413127082, 0.347189024, 0.371585245, 0.452311983, 0.387491017, 0.502240559, 0.420601769, 0.427401406, 0.392831214, 0.468965141, 0.415459641, 0.470943789, 0.433053303, 0.434626699, 0.492090441, 0.419246226, 0.418893912, 0.393065697, 0.388770405, 0.441625509, 0.412995748, 0.371077872, 0.429367874, 0.377048620, 0.372883541, 0.373757424, 0.370734880, 0.440019202, 0.446060065, 0.368311775, 0.358840911, 0.410658460, 0.434416070, 0.357136936, 0.405721983, 0.426689088, 0.461247188, 0.404427560, 0.486848662, 0.421560795, 0.448725346, 0.395054774, 0.416209154, 0.348607332, 0.366021039, 0.394709281, 0.430889800, 0.344462363, 0.477618944, 0.459546440, 0.339655440, 0.379558250, 0.474168015, 0.442126248, 0.434445686, 0.384672615, 0.434607996, 0.428411546, 0.379106440, 0.429089990, 0.441497373, 0.472405799, 0.420758530, 0.343717385, 0.366743818, 0.428770078, 0.455448048, 0.393978437, 0.346644741, 0.413489758, 0.465933477, 0.410372147, 0.380400118, 0.389768437, 0.433616164, 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0.434156073, 0.336363317, 0.462242061) + se_orth_po_nosplit = c(0.0369249937, 0.0366232166, 0.0352132615, 0.0357962562, 0.0345049556, 0.0361262917, 0.0385463535, 0.0368034228, 0.0354351281, 0.0333831231, 0.0402495762, 0.0335314336, 0.0343515619, 0.0365670189, 0.0337228740, 0.0333227216, 0.0426300264, 0.0440211182, 0.0382558092, 0.0336980296, 0.0382198398, 0.0392737882, 0.0358866245, 0.0371949430, 0.0334161164, 0.0386112083, 0.0367611265, 0.0367412759, 0.0379716336, 0.0411895083, 0.0381076337, 0.0386184276, 0.0398358082, 0.0387196574, 0.0388327060, 0.0364272833, 0.0339169542, 0.0380975564, 0.0362605086, 0.0404485292, 0.0339254053, 0.0358481896, 0.0398973515, 0.0418559718, 0.0322929223, 0.0391445722, 0.0352535863, 0.0407457193, 0.0381563624, 0.0324255617, 0.0405592833, 0.0369996847, 0.0345523241, 0.0387905589, 0.0386349948, 0.0365253978, 0.0433434719, 0.0398188984, 0.0362117055, 0.0342701997, 0.0400886743, 0.0379609911, 0.0333977515, 0.0393487648, 0.0369779531, 0.0375621374, 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0.0372525184, 0.0372967391, 0.0339701181, 0.0366998335, 0.0384988327, 0.0378874575, 0.0420594119, 0.0363607648, 0.0360699822, 0.0317958512, 0.0404294042, 0.0352864907, 0.0405165125, 0.0399594364, 0.0361487076, 0.0374133456, 0.0374805363, 0.0355464495, 0.0392695227, 0.0337534926, 0.0384322112, 0.0391902532, 0.0375980277, 0.0375924745, 0.0352334383, 0.0370078136, 0.0399579600, 0.0360427313, 0.0382787029, 0.0350579378, 0.0401883559, 0.0319766776, 0.0370449943, 0.0404588657, 0.0407541195, 0.0344746390, 0.0412191332, 0.0358989477, 0.0366126875, 0.0370809996, 0.0399497678, 0.0355534343, 0.0393131128, 0.0370407940, 0.0363187432, 0.0389475031, 0.0378005386, 0.0377157383, 0.0353959562, 0.0385682187, 0.0377720111, 0.0388690038, 0.0391966137, 0.0405436846, 0.0384275227, 0.0342578823, 0.0422762326, 0.0332816830, 0.0386548470, 0.0369299494, 0.0351850600, 0.0363734045, 0.0362109816, 0.0364759838, 0.0368868785, 0.0367005653, 0.0383874488, 0.0342395609, 0.0347803973, 0.0392387853, 0.0402901017, 0.0349486732, 0.0388709549, 0.0385755033, 0.0325223829, 0.0366812686, 0.0342228295, 0.0356556147, 0.0380793542, 0.0375113139, 0.0381656074, 0.0383364452) + + # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset + #for (i_rep in seq_len(n_rep)){ + for (i_rep in seq_len(1)) { + df = data[[i_rep]] + obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") + obj_dml_plr_orth_nosplit = DoubleMLPLR$new(obj_dml_data, + ml_g, ml_m, + n_folds=1, + score='partialling out', + apply_cross_fitting=FALSE) + obj_dml_plr_orth_nosplit$fit() + this_theta = obj_dml_plr_orth_nosplit$coef + this_se = obj_dml_plr_orth_nosplit$se + print(abs(theta_orth_po_nosplit[i_rep] - this_theta)) + print(abs(se_orth_po_nosplit[i_rep] - this_se)) + theta_orth_po_nosplit[i_rep] = this_theta + se_orth_po_nosplit[i_rep] = this_se + } + + g_nosplit = ggplot(data.frame(theta_rescaled=(theta_orth_po_nosplit - alpha)/se_orth_po_nosplit), aes(x = theta_rescaled)) + + geom_histogram(aes(y=..density.., x=theta_rescaled, colour = "Double ML (no sample splitting)", fill="Double ML (no sample splitting)"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Double ML (no sample splitting)", "N(0, 1)"), + values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"='black')) + + scale_fill_manual(name='',, + breaks=c("Double ML (no sample splitting)", "N(0, 1)"), + values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() + g_nosplit + +Using sample splitting, overcomes the bias induced by overfitting. + +.. tabbed:: Python + + .. ipython:: python + + import numpy as np + np.random.seed(5555) + + # to speed up the illustration we hard-code the simulation results + theta_dml_po = np.array([0.52876386, 0.49642496, 0.43558359, 0.48385395, 0.48263744, 0.52065436, 0.51196604, 0.45457013, 0.50236773, 0.60226776, 0.49525345, 0.54929426, 0.55862886, 0.38668434, 0.46651516, 0.48596014, 0.51073451, 0.47555414, 0.56154942, 0.47712008, 0.45908535, 0.47905771, 0.48119272, 0.5039857 , 0.48549178, 0.44828651, 0.51054201, 0.46965257, 0.44771795, 0.54639828, 0.47136481, 0.58571177, 0.40240481, 0.43868009, 0.46543858, 0.44238912, 0.45729493, 0.55316233, 0.49995909, 0.53027911, 0.48020647, 0.5191674 , 0.49229801, 0.37730406, 0.57612697, 0.50399902, 0.49049858, 0.51522608, 0.51054361, 0.4939221 , 0.48388515, 0.49238434, 0.47703209, 0.51727205, 0.61579399, 0.38693497, 0.55208572, 0.48290605, 0.49366392, 0.47067728, 0.48912104, 0.54112017, 0.54915469, 0.48089163, 0.56427844, 0.46009275, 0.4533473 , 0.49321491, 0.56400308, 0.56071557, 0.4453699 , 0.48957027, 0.50948593, 0.47617631, 0.49869524, 0.52732989, 0.49713173, 0.51638675, 0.5039623 , 0.53514343, 0.52809171, 0.48055678, 0.47932269, 0.44400388, 0.43742165, 0.5035799 , 0.55940179, 0.52720895, 0.48898253, 0.55113636, 0.38530608, 0.48212808, 0.56255715, 0.4681873 , 0.50922832, 0.49665169, 0.49973596, 0.50728249, 0.61919775, 0.52980464, 0.4470384 , 0.54970346, 0.4603203 , 0.50395574, 0.44948521, 0.59271845, 0.54610525, 0.48435625, 0.40402406, 0.49332716, 0.42959719, 0.48235224, 0.44457234, 0.43675964, 0.48845666, 0.450786 , 0.47855978, 0.4693617 , 0.48993912, 0.41517384, 0.45892182, 0.60196253, 0.50991046, 0.47869586, 0.39594481, 0.58051677, 0.45685749, 0.45722061, 0.48340673, 0.46928463, 0.50811054, 0.48351326, 0.49455325, 0.47532855, 0.45697168, 0.52864274, 0.51573529, 0.55979803, 0.46307792, 0.49551285, 0.4476106 , 0.47584211, 0.57505639, 0.47849081, 0.5754398 , 0.42882078, 0.43890495, 0.5712939 , 0.45392337, 0.54160894, 0.56094447, 0.47404527, 0.47344072, 0.46475148, 0.5162425 , 0.61577599, 0.48598403, 0.4706845 , 0.43193503, 0.60009588, 0.49000676, 0.46722224, 0.53955457, 0.53058301, 0.49350086, 0.46416138, 0.56295573, 0.50641166, 0.50500217, 0.53639407, 0.53181473, 0.49211802, 0.5137631 , 0.47101913, 0.50834946, 0.49639409, 0.49041039, 0.58718694, 0.5053225 , 0.47442319, 0.45029834, 0.53214004, 0.48029363, 0.43771314, 0.41368455, 0.43934673, 0.42540836, 0.45567131, 0.56527456, 0.50845985, 0.44297089, 0.48081024, 0.49390183, 0.41014255, 0.50802301, 0.4715952 , 0.44784824, 0.52827121, 0.461319 , 0.4661665 , 0.52817668, 0.5774521 , 0.4795185 , 0.51946092, 0.45543458, 0.45574257, 0.48488458, 0.48652943, 0.54044881, 0.55479025, 0.44064243, 0.62755167, 0.42012331, 0.40775333, 0.43366019, 0.52687122, 0.4517471 , 0.53678077, 0.50584558, 0.53164932, 0.53117984, 0.50457742, 0.46806336, 0.52904057, 0.5307677 , 0.54662925, 0.45878668, 0.49324139, 0.574838 , 0.49137141, 0.58268286, 0.5924264 , 0.62310897, 0.47510252, 0.53649311, 0.57229299, 0.51940046, 0.45731134, 0.54884152, 0.48680994, 0.51510473, 0.50922301, 0.51677578, 0.57780639, 0.46816876, 0.44897551, 0.51321758, 0.49450203, 0.42872782, 0.4365137 , 0.4628704 , 0.51399804, 0.55653439, 0.52488332, 0.50600489, 0.48140922, 0.50350718, 0.52654721, 0.59815318, 0.42071962, 0.47969415, 0.4456638 , 0.50211309, 0.49697633, 0.44140602, 0.53930818, 0.47884922, 0.56818849, 0.51630494, 0.48671292, 0.46039979, 0.41114395, 0.39818181, 0.56145449, 0.542998 , 0.50010098, 0.49820963, 0.49003785, 0.50305572, 0.51292791, 0.48933725, 0.45660895, 0.49518821, 0.4849069 , 0.52267743, 0.50392296, 0.58937665, 0.47274635, 0.56755771, 0.38741005, 0.52876889, 0.49566081, 0.50850134, 0.44876604, 0.43954376, 0.49462121, 0.59135693, 0.46764294, 0.58774516, 0.42427519, 0.55204104, 0.59373285, 0.5284568 , 0.47892363, 0.46903582, 0.473618 , 0.53177486, 0.42273053, 0.52900964, 0.48550055, 0.50106813, 0.45814701, 0.4722137 , 0.57443057, 0.39776497, 0.49355381, 0.36617604, 0.50976518, 0.52859814, 0.52722868, 0.61453654, 0.45841707, 0.55796556, 0.55644629, 0.5506848 , 0.4702159 , 0.50083763, 0.53856514, 0.46602807, 0.45204245, 0.48080515, 0.56483207, 0.45621144, 0.42794771, 0.4860271 , 0.46479965, 0.56875931, 0.41161689, 0.55302315, 0.47271993, 0.50344196, 0.42053369, 0.52565845, 0.43104463, 0.43922226, 0.57412536, 0.49435242, 0.5598449 , 0.50651293, 0.4243393 , 0.47901695, 0.42058707, 0.5476903 , 0.50184981, 0.57823467, 0.55067893, 0.51966014, 0.41881399, 0.44953744, 0.46319358, 0.53574066, 0.45995912, 0.47871539, 0.56629616, 0.57595297, 0.53876127, 0.53609786, 0.52978709, 0.53875115, 0.50847709, 0.4149362 , 0.45260669, 0.4536379 , 0.50394067, 0.42305305, 0.41424964, 0.52197237, 0.48594172, 0.45009749, 0.4307894 , 0.56654082, 0.52400503, 0.50464779, 0.4883888 , 0.50770741, 0.44965224, 0.48960481, 0.55406137, 0.47093463, 0.46452095, 0.47775312, 0.49929834, 0.54082214, 0.48656641, 0.44160879, 0.40050997, 0.43967865, 0.52972772, 0.48894344, 0.48959524, 0.45225519, 0.46477098, 0.45606621, 0.5488213 , 0.42547011, 0.49625743, 0.4865749 , 0.49467224, 0.55015222, 0.50699681, 0.47427151, 0.43403248, 0.42448919, 0.39432154, 0.48474832, 0.47421844, 0.53084624, 0.46260523, 0.45064904, 0.48230808, 0.43730637, 0.51041301, 0.47497435, 0.49359209, 0.5595982 , 0.48490794, 0.51250825, 0.46914558, 0.53257313, 0.54640514, 0.55242549, 0.43611181, 0.52036922, 0.47315045, 0.46437181, 0.47094681, 0.52144713, 0.52288962, 0.52173716, 0.48752518, 0.52550315, 0.50604638, 0.52228596, 0.49880518, 0.50851985, 0.4465949 , 0.52202277, 0.50249377, 0.46322091, 0.54615601, 0.50894802, 0.43345177, 0.50954733, 0.46339897, 0.46015515, 0.50448629, 0.51675392, 0.50012197, 0.44156462, 0.53740657, 0.53755287, 0.51308269, 0.49421434, 0.39274913, 0.47712458, 0.47812675, 0.49915688, 0.53254843, 0.42084555, 0.46325418, 0.49086648, 0.45019164, 0.5420365 , 0.47279955, 0.48982796, 0.54589376, 0.51462906, 0.43819174, 0.50983744, 0.50781399, 0.49550464, 0.46471352, 0.58631511, 0.47949176, 0.51639418, 0.39523312, 0.51664646, 0.5448198 , 0.56068757, 0.50757457, 0.47905316, 0.45066173, 0.41872143, 0.50006031, 0.42814963, 0.57007205, 0.58021248, 0.53159123, 0.42537663, 0.51625064, 0.47240764, 0.55303755, 0.48823736, 0.42029816, 0.53373549, 0.44653573, 0.43038701, 0.57681681, 0.54862281, 0.5205924 , 0.50207264, 0.46024779, 0.49115148, 0.58285578, 0.56729901, 0.50065596, 0.43313588, 0.53193915, 0.46971056, 0.46810937, 0.54975298, 0.44592224, 0.5273588 , 0.45066263, 0.49022521, 0.54036456, 0.52373846, 0.51341698, 0.49523554, 0.46354297, 0.59599975, 0.46715844, 0.51362765, 0.49410913, 0.44571835, 0.41595069, 0.57077163, 0.52015364, 0.48560591, 0.48571281, 0.51829923, 0.47543306, 0.46038 , 0.44356321, 0.46757523, 0.46887054, 0.55768647, 0.54349256, 0.47197434, 0.51751084, 0.51076787, 0.49690516, 0.60331217, 0.57465717, 0.51494865, 0.56925775, 0.47763364, 0.46602821, 0.5187162 , 0.45728743, 0.37034602, 0.54977269, 0.4680515 , 0.5262913 , 0.48117704, 0.49573812, 0.54406615, 0.51055961, 0.56104265, 0.5592123 , 0.41827194, 0.527187 , 0.46604307, 0.49439706, 0.53662182, 0.51926518, 0.47345959, 0.47946106, 0.47932428, 0.59299744, 0.42411538, 0.60750248, 0.56642017, 0.51119554, 0.49077337, 0.40567808, 0.37298201, 0.48068202, 0.4868408 , 0.41970932, 0.45143647, 0.48364514, 0.4945257 , 0.49703403, 0.49551628, 0.44295951, 0.53390061, 0.43523352, 0.49790843, 0.53749617, 0.46802951, 0.55940488, 0.52359195, 0.50736714, 0.52502905, 0.48463152, 0.45303868, 0.57144172, 0.48372973, 0.53818662, 0.50312434, 0.50637077, 0.54574298, 0.41583979, 0.48042361, 0.50222613, 0.47359926, 0.56900104, 0.46190257, 0.55831883, 0.56685946, 0.52938241, 0.49552427, 0.50929147, 0.47357133, 0.49662637, 0.4343192 , 0.53597924, 0.53930657, 0.53026288, 0.4861995 , 0.61926515, 0.52733928, 0.45598938, 0.53494507, 0.52942118, 0.47078208, 0.53457652, 0.48507914, 0.50306637, 0.4412238 , 0.59952472, 0.52392443, 0.54847361, 0.56994672, 0.41829809, 0.5814434 , 0.45197069, 0.55710022, 0.45528081, 0.5401222 , 0.54592327, 0.44220664, 0.47673524, 0.52873844, 0.4829449 , 0.48515887, 0.54134607, 0.450466 , 0.54837702, 0.47590747, 0.57199309, 0.58851981, 0.47132444, 0.44442932, 0.56183549, 0.48979552, 0.51374854, 0.41011541, 0.46538776, 0.58606274, 0.51545543, 0.54614756, 0.4506355 , 0.49779654, 0.443503 , 0.47268849, 0.4504294 , 0.39291152, 0.54669478, 0.54779101, 0.47582864, 0.50894455, 0.51927094, 0.49861091, 0.59310058, 0.45429373, 0.48678437, 0.50288586, 0.48069622, 0.52760912, 0.5118683 , 0.60196906, 0.4282119 , 0.51393524, 0.5106446 , 0.53389979, 0.53176284, 0.47265592, 0.45720696, 0.42943445, 0.54018803, 0.43633861, 0.51280376, 0.42495233, 0.48423667, 0.53328973, 0.51377185, 0.45810269, 0.49723702, 0.403323 , 0.56444141, 0.51351277, 0.47767768, 0.51641341, 0.4966969 , 0.42828877, 0.55426548, 0.53245294, 0.48623397, 0.42692114, 0.50321609, 0.53440262, 0.54244648, 0.51299351, 0.43263567, 0.47331176, 0.48165027, 0.51232191, 0.55445258, 0.54323231, 0.50930401, 0.50153814, 0.54471426, 0.53923209, 0.47315923, 0.43551507, 0.55701856, 0.55112787, 0.50850312, 0.49295049, 0.47247679, 0.58057907, 0.52169959, 0.51034787, 0.42896117, 0.55294684, 0.52292619, 0.42976747, 0.4706387 , 0.46060276, 0.60014024, 0.51666398, 0.51434187, 0.49136887, 0.61065831, 0.38513663, 0.56736394, 0.54790665, 0.44434231, 0.48441654, 0.57945827, 0.53637528, 0.47735573, 0.52615155, 0.51504026, 0.4641185 , 0.50871316, 0.56002337, 0.4707652 , 0.42641551, 0.48532868, 0.48351299, 0.47063727, 0.46747184, 0.42279864, 0.46458271, 0.51786366, 0.51228997, 0.5033719 , 0.48058348, 0.40904451, 0.43758533, 0.44100579, 0.57999117, 0.4925471 , 0.47551867, 0.5286477 , 0.46000318, 0.5637859 , 0.59928435, 0.48360782, 0.4771193 , 0.53749157, 0.45664404, 0.58401153, 0.46197167, 0.50154002, 0.54169984, 0.47225318, 0.50535922, 0.46600936, 0.48476778, 0.45769416, 0.49335976, 0.47891549, 0.47997919, 0.45763229, 0.53670601, 0.45686323, 0.56551689, 0.59629776, 0.47251306, 0.48778836, 0.56468832, 0.48890077, 0.52307882, 0.53440244, 0.50813894, 0.4428279 , 0.52356287, 0.47703679, 0.60258583, 0.4108778 , 0.54088935, 0.53795036, 0.44628152, 0.48215627, 0.44107261, 0.45252552, 0.46111263, 0.50135487, 0.55187685, 0.51325235, 0.51514277, 0.5680781 , 0.48667023, 0.48555444, 0.45070021, 0.39405688, 0.48145638, 0.54777308, 0.57570516, 0.55441856, 0.58127967, 0.47059829, 0.55769376, 0.52569789, 0.47112601, 0.5010574 , 0.35419379, 0.57555654, 0.47210866, 0.50287819, 0.4258165 , 0.56729575, 0.45470128, 0.55061083, 0.46312785, 0.51492596, 0.51134206, 0.48960366, 0.57966146, 0.37426146, 0.51913096, 0.51783745, 0.51242465, 0.45738121, 0.51659558, 0.48073267, 0.45818103, 0.52604388, 0.53847543, 0.46773059, 0.44172535, 0.43433353, 0.43758119, 0.52672775, 0.50251892, 0.57854698, 0.5175629 , 0.4611426 , 0.52172734, 0.45272314, 0.51589939, 0.43278818, 0.53944443, 0.53054068, 0.49185357, 0.42120363, 0.577586 , 0.52179469, 0.51180071, 0.55495018, 0.42470079, 0.54284353, 0.50912782, 0.43475174, 0.52285215, 0.53892128, 0.46787949, 0.53336718, 0.46930669, 0.45339336, 0.49623817, 0.4311819 , 0.55982476, 0.38516919, 0.49657255, 0.55963216, 0.47214567, 0.5240232 , 0.43767568, 0.46480375, 0.57115072, 0.49141717, 0.49789593, 0.46457368, 0.46656324, 0.55545586, 0.63923336, 0.46879923, 0.45622937, 0.42976341, 0.51129408, 0.49839078, 0.5164966 , 0.57959171, 0.45868351, 0.54262484, 0.49203866, 0.42105985, 0.60347606, 0.55442278, 0.51974864, 0.55806284, 0.56181645, 0.54627102, 0.50413101, 0.48039105, 0.46379635, 0.58390471, 0.5377935 , 0.41095458, 0.42925395, 0.42520725, 0.4893355 , 0.48419668, 0.51066485, 0.46864138, 0.59308306, 0.54484266, 0.4790423 , 0.48557658, 0.43889655, 0.50258992, 0.50661835, 0.45562128, 0.47451614, 0.50141995, 0.52912567, 0.48534654, 0.50892843, 0.43341213, 0.54916986, 0.52695849, 0.47247929, 0.46517683, 0.55822228, 0.43617081, 0.54370429, 0.45345871, 0.4418124 , 0.45103698, 0.3915015 , 0.51655919, 0.51625817, 0.4902453 , 0.47138125, 0.50868646, 0.53437357, 0.5563727 , 0.4870521 , 0.47828182, 0.44788723, 0.48242661, 0.58646635, 0.52495117, 0.53766765, 0.51010016, 0.47590069, 0.49285749, 0.33303451, 0.58635962, 0.47230682, 0.46693197, 0.53479744, 0.49730085, 0.4686777 , 0.52578667]) + se_dml_po = np.array([0.04366143, 0.04632413, 0.03934783, 0.04376407, 0.04776953, 0.04294894, 0.0493605 , 0.05057076, 0.04993363, 0.04528385, 0.04716893, 0.04316392, 0.03844692, 0.04002265, 0.04364729, 0.04180496, 0.04915626, 0.04444379, 0.04258075, 0.04642707, 0.0453024 , 0.05041915, 0.04495728, 0.04627783, 0.04774967, 0.03613853, 0.04456387, 0.0428944 , 0.04773842, 0.04278792, 0.03998002, 0.04199909, 0.04050232, 0.04279711, 0.04826966, 0.04232603, 0.04629067, 0.04520752, 0.04292975, 0.04775048, 0.04387756, 0.04303141, 0.04699689, 0.04426411, 0.04174037, 0.0450418 , 0.04357423, 0.04296971, 0.04115976, 0.04803088, 0.0440271 , 0.03910127, 0.04188743, 0.0461947 , 0.04445704, 0.04758127, 0.04105246, 0.04229529, 0.04348393, 0.03940532, 0.04691201, 0.04712129, 0.04532333, 0.04198115, 0.04467275, 0.0456634 , 0.04796472, 0.04080751, 0.04541844, 0.04754002, 0.04303026, 0.03978123, 0.04099255, 0.04866286, 0.04067465, 0.04196374, 0.04548078, 0.04779556, 0.04321585, 0.04544103, 0.04639701, 0.04444337, 0.04158797, 0.04212771, 0.04294366, 0.04862461, 0.03713917, 0.04505446, 0.04856092, 0.04640594, 0.04342124, 0.04298389, 0.04025997, 0.04828067, 0.04270125, 0.04275123, 0.04282572, 0.04353543, 0.04703092, 0.04467834, 0.04520262, 0.04597939, 0.04693617, 0.04535161, 0.04437884, 0.04265513, 0.04768799, 0.04478721, 0.04296025, 0.04637604, 0.0487638 , 0.04499217, 0.04168946, 0.0439831 , 0.04748039, 0.05013583, 0.05015454, 0.04192372, 0.04506462, 0.04632873, 0.04833074, 0.04748367, 0.04152701, 0.04165487, 0.04654159, 0.04765364, 0.04452404, 0.04139589, 0.0442406 , 0.0457696 , 0.04844991, 0.04943656, 0.04777142, 0.03825993, 0.04597716, 0.04874836, 0.04776631, 0.04595611, 0.0458343 , 0.0428256 , 0.04534578, 0.0448361 , 0.042015 , 0.04811913, 0.0448321 , 0.04292421, 0.03894937, 0.04272008, 0.04455543, 0.03876354, 0.04314453, 0.04628697, 0.04603369, 0.04497386, 0.04267185, 0.04019239, 0.0422623 , 0.04053257, 0.04286978, 0.04745838, 0.04552007, 0.04932073, 0.04500954, 0.04000683, 0.0472705 , 0.04828826, 0.04341315, 0.0464942 , 0.04558246, 0.04395118, 0.04105528, 0.04638659, 0.04586976, 0.04029563, 0.04217629, 0.04204087, 0.04093607, 0.04969833, 0.04664773, 0.04504618, 0.03758201, 0.0452988 , 0.04503191, 0.04843537, 0.04914253, 0.04435506, 0.0368011 , 0.05420537, 0.04624779, 0.04433298, 0.05062355, 0.0447908 , 0.04379796, 0.04982575, 0.04793812, 0.0467464 , 0.0506507 , 0.04515699, 0.04629882, 0.04103533, 0.04999819, 0.04301357, 0.04669165, 0.04087331, 0.04158373, 0.0439232 , 0.04715809, 0.04692125, 0.04897129, 0.04748409, 0.04625044, 0.04551256, 0.04820112, 0.04693338, 0.04282482, 0.04321722, 0.04492408, 0.04680481, 0.04651601, 0.04297949, 0.04045679, 0.04618816, 0.0493394 , 0.03985102, 0.05072911, 0.04725812, 0.0481278 , 0.04495275, 0.04530795, 0.04074308, 0.04334787, 0.04071895, 0.04912917, 0.03774198, 0.04314333, 0.04086531, 0.04824108, 0.04406156, 0.04409377, 0.04891328, 0.0500406 , 0.04516683, 0.0414013 , 0.04620527, 0.04709315, 0.04822646, 0.04682268, 0.05036133, 0.04125324, 0.0461525 , 0.04235533, 0.04557972, 0.04499424, 0.04816397, 0.042596 , 0.04261622, 0.04148597, 0.046121 , 0.04583109, 0.04317212, 0.04462103, 0.04028714, 0.04365713, 0.04248785, 0.0467333 , 0.04079037, 0.04934807, 0.04507899, 0.04644339, 0.04587939, 0.04737139, 0.04552572, 0.04947069, 0.04414528, 0.04099972, 0.04527949, 0.04627014, 0.04386254, 0.04570415, 0.04411444, 0.04289963, 0.04525811, 0.0474116 , 0.04547358, 0.05100277, 0.04175249, 0.0485154 , 0.04771437, 0.04406015, 0.04622295, 0.039592 , 0.04335127, 0.04427896, 0.0393652 , 0.04025816, 0.04790163, 0.04682166, 0.04510704, 0.04299321, 0.0476545 , 0.0473511 , 0.04064169, 0.04439331, 0.04040281, 0.04181763, 0.04290026, 0.04493665, 0.04679808, 0.04864147, 0.04560976, 0.05244334, 0.05009814, 0.04916519, 0.04749958, 0.04397354, 0.04772573, 0.04904464, 0.04965606, 0.04217941, 0.04897094, 0.04123459, 0.0432386 , 0.04709203, 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0.04015547, 0.0433569 , 0.04528929, 0.03860953, 0.04976271, 0.04108088, 0.04336586, 0.05090655, 0.04489043, 0.03832453, 0.0438332 , 0.04216618, 0.04244166, 0.04889756, 0.04376899, 0.04459088, 0.04060368, 0.04171002, 0.04291755, 0.04042948, 0.04301972, 0.04504935, 0.04075044, 0.04658386, 0.04667455, 0.04239227, 0.0454948 , 0.04395952, 0.04566263, 0.04644776, 0.0472576 , 0.04320539, 0.04367673, 0.04372062, 0.04109036, 0.04963561, 0.04485157, 0.04830976, 0.03934099, 0.04930203, 0.04521219, 0.04431363, 0.04630182, 0.04414541, 0.04679306, 0.04193693, 0.04414566, 0.04577935, 0.04780609, 0.04347577, 0.05090181, 0.04015752, 0.03947669, 0.04295765, 0.04977118, 0.05137322, 0.04235594, 0.04365023, 0.04671102, 0.04115597, 0.04226988, 0.04707771, 0.03925437, 0.04513961, 0.0465118 , 0.04131011, 0.04014652, 0.0512401 , 0.04836521, 0.0448319 , 0.04389894, 0.04354019, 0.04021803, 0.04309532, 0.04263222, 0.04396544, 0.04923313, 0.0477914 , 0.04328708, 0.04245669, 0.04302829, 0.04192883, 0.04812229, 0.04801755, 0.04761739, 0.04598512, 0.04248567, 0.04684313, 0.04273867, 0.03934617, 0.04554571, 0.04926563, 0.04315612, 0.05041068, 0.04821278, 0.04494212, 0.04427627, 0.04560213, 0.03987408, 0.04400685, 0.04709236, 0.04158083, 0.0447051 , 0.04390364, 0.04233113, 0.04103328, 0.0488311 , 0.04185222, 0.04171723, 0.04182328, 0.0444572 , 0.04489263, 0.04402228, 0.03943944, 0.04881562, 0.04686757, 0.04589471, 0.03900169, 0.04893455, 0.04732217, 0.04823214, 0.04237029, 0.04492963, 0.04876891, 0.04788519, 0.04728083, 0.05084502, 0.04555633, 0.05153353, 0.04245686, 0.0445393 , 0.04529056, 0.04805109, 0.03909855, 0.0441579 , 0.04009273, 0.04833389, 0.04335762, 0.0487235 , 0.04793003, 0.04744083, 0.04497436, 0.04439482, 0.04417169, 0.04487584, 0.04149336, 0.04581779, 0.04302379, 0.04730023, 0.04074397, 0.0426177 , 0.04144998, 0.04554913, 0.04398114, 0.04368888, 0.04684536, 0.04893886, 0.04441862, 0.04646384, 0.04373368, 0.04583414, 0.04424393, 0.04435307, 0.05151157, 0.04523948, 0.04624434, 0.04763616, 0.04429373, 0.04435428, 0.04066626, 0.04850437, 0.0440769 , 0.04807946, 0.04242926, 0.04600121, 0.04535562, 0.05036463, 0.04495395, 0.04966027, 0.04020707, 0.04710001, 0.03835692, 0.04706426, 0.04517028, 0.04060998, 0.04117483, 0.05759099, 0.05168384, 0.04488953, 0.04298455, 0.05004272, 0.0454869 , 0.0449041 , 0.04490075, 0.04452586, 0.04675966, 0.04110656, 0.04448455, 0.04590486, 0.04961628, 0.04045468, 0.05065138, 0.04290253, 0.04904711, 0.04510714, 0.04169845, 0.04017691, 0.04575501, 0.04605821, 0.04405548, 0.04664584, 0.0478982 , 0.04628937, 0.04424418, 0.04613777, 0.04578426, 0.04725448, 0.04645413, 0.04135783, 0.04062497, 0.04972194, 0.04393031, 0.04443315, 0.04687277, 0.04239731, 0.0421795 , 0.04737418, 0.04182724, 0.04215289, 0.04455361, 0.04328229, 0.03965512, 0.04787494, 0.04640227, 0.04050011, 0.04418661, 0.04282853, 0.05295413, 0.04282882, 0.04227624, 0.04621299, 0.04420618, 0.0464249 , 0.04677917, 0.0447752 , 0.04577487, 0.04751948, 0.04174396, 0.04717273, 0.04712549, 0.04967362, 0.04212043, 0.04126582, 0.0400929 , 0.04729312, 0.04916058, 0.04743923, 0.04674456, 0.0426739 , 0.04807577, 0.0471212 , 0.04660028, 0.04404454, 0.04386943, 0.04671481, 0.04770813, 0.04276558, 0.04236685, 0.04024044, 0.04031839, 0.04509778, 0.05081884, 0.0452145 , 0.04434399, 0.04777426, 0.04267024, 0.03933003, 0.04722241, 0.04739409, 0.05255036, 0.04003435, 0.04533648, 0.04412494, 0.04411879, 0.04261987, 0.04469745, 0.04263345, 0.04239181, 0.04201696, 0.04423246, 0.04341077, 0.04819775, 0.04691498, 0.04237232, 0.04410359, 0.04044761, 0.044337 , 0.04929771, 0.04925044, 0.04694016, 0.0438847 , 0.04710392, 0.0415846 , 0.0441434 , 0.04246665, 0.04861628, 0.04398318, 0.04258132, 0.04593046, 0.04446947, 0.04809013, 0.04357285, 0.04728308, 0.04466791, 0.04391723, 0.04725787, 0.04057481, 0.04136977, 0.05221815, 0.04425619, 0.04415679, 0.0404598 , 0.04464952, 0.04641681, 0.04791448, 0.04399596, 0.04606058, 0.04165912, 0.04747206, 0.04528619, 0.05164746, 0.04341658, 0.04389238, 0.04312489, 0.04597068, 0.04712426, 0.04022928, 0.04500399, 0.04459979, 0.04872266, 0.04756165, 0.04166714, 0.04580468, 0.04350212, 0.0425324]) + + # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset + #for i_rep in range(n_rep): + for i_rep in range(1): + (x, y, d) = data[i_rep] + obj_dml_data = DoubleMLData.from_arrays(x, y, d) + obj_dml_plr = DoubleMLPLR(obj_dml_data, + ml_g, ml_m, + n_folds=2, + score='partialling out') + obj_dml_plr.fit() + this_theta = obj_dml_plr.coef[0] + this_se = obj_dml_plr.se[0] + # we show that the loaded result matches the just computed + print(np.abs(theta_dml_po[i_rep] - this_theta)) + print(np.abs(se_dml_po[i_rep] - this_se)) + theta_dml_po[i_rep] = this_theta + se_dml_po[i_rep] = this_se + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_dml_po - alpha)/se_dml_po, + color=face_colors[2], edgecolor = edge_colors[2], + stat='density', bins=30, label='Double ML with cross-fitting'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); + @savefig orth_po.png width=5in + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); + +.. tabbed:: R + + .. jupyter-execute:: + + set.seed(5555) + + # to speed up the illustration we hard-code the simulation results + theta_dml_po = c(0.528929305, 0.524166106, 0.451625400, 0.481825448, 0.437547797, 0.478983941, 0.595735437, 0.504146789, 0.477701836, 0.555757361, 0.509946294, 0.482621495, 0.484000068, 0.435829918, 0.472128822, 0.437704381, 0.492793231, 0.549994506, 0.501524331, 0.500380684, 0.404144531, 0.464623937, 0.507083497, 0.452636998, 0.511753393, 0.454828747, 0.470530138, 0.573594266, 0.500503122, 0.486072954, 0.490245595, 0.435874202, 0.655562845, 0.564458817, 0.456068556, 0.481776445, 0.521628576, 0.528145239, 0.435524326, 0.541558276, 0.520581239, 0.548347798, 0.410600001, 0.548111626, 0.561417232, 0.565057590, 0.569559383, 0.534560415, 0.489788915, 0.462890711, 0.487090830, 0.530282207, 0.603032550, 0.470690285, 0.497606122, 0.622735251, 0.501934927, 0.448152643, 0.476624907, 0.445380734, 0.481506092, 0.448385435, 0.545291003, 0.502574048, 0.504246410, 0.464174968, 0.468343480, 0.461442733, 0.545334139, 0.474707557, 0.542118443, 0.524144042, 0.423730692, 0.436341238, 0.513615913, 0.500882990, 0.521748352, 0.538908355, 0.447217526, 0.493472118, 0.424747680, 0.497873343, 0.508668557, 0.508476976, 0.508078831, 0.489466840, 0.485841133, 0.467494506, 0.492297379, 0.591710996, 0.589443587, 0.474464990, 0.456152648, 0.530132841, 0.402934007, 0.559060278, 0.452289153, 0.531651817, 0.557275666, 0.493257933, 0.450157556, 0.418314553, 0.494901444, 0.510357009, 0.483434818, 0.421694596, 0.516056398, 0.506513906, 0.494035742, 0.502110446, 0.486391123, 0.502495550, 0.467286445, 0.488338097, 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0.0496876593, 0.0439748868, 0.0427690644, 0.0468111097, 0.0494035213, 0.0425496477, 0.0465137448, 0.0447275277, 0.0484827693, 0.0458997490, 0.0464914386, 0.0433278287, 0.0445023944, 0.0466340547, 0.0441450072, 0.0474577431, 0.0480442403, 0.0471825641, 0.0468976085, 0.0421529837, 0.0496266909, 0.0392661148, 0.0482181540, 0.0436511829, 0.0413371341, 0.0429495849, 0.0420580993, 0.0449661668, 0.0472251303, 0.0461371903, 0.0474402788, 0.0438124402, 0.0414037459, 0.0491058290, 0.0500372334, 0.0450386547, 0.0463601996, 0.0461323055, 0.0416831353, 0.0445322971, 0.0426195463, 0.0432235284, 0.0460557804, 0.0431648518, 0.0468505390, 0.0467809093) + + # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset + #for (i_rep in seq_len(n_rep)) { + for (i_rep in seq_len(1)) { + df = data[[i_rep]] + obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") + obj_dml_plr = DoubleMLPLR$new(obj_dml_data, + ml_g, ml_m, + n_folds=2, + score='partialling out') + obj_dml_plr$fit() + this_theta = obj_dml_plr$coef + this_se = obj_dml_plr$se + print(abs(theta_dml_po[i_rep] - this_theta)) + print(abs(se_dml_po[i_rep] - this_se)) + theta_dml_po[i_rep] = this_theta + se_dml_po[i_rep] = this_se + } + + g_dml = ggplot(data.frame(theta_rescaled=(theta_dml_po - alpha)/se_dml_po), aes(x = theta_rescaled)) + + geom_histogram(aes(y=..density.., x=theta_rescaled, colour = "Double ML with cross-fitting", fill="Double ML with cross-fitting"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Double ML with cross-fitting", "N(0, 1)"), + values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"='black')) + + scale_fill_manual(name='',, + breaks=c("Double ML with cross-fitting", "N(0, 1)"), + values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() + g_dml + +.. Comment to get a new tab + +.. tabbed:: Python + + .. ipython:: python + + plt.figure(constrained_layout=True); + ax = sns.histplot((theta_nonorth - alpha)/se_nonorth, + color=face_colors[0], edgecolor = edge_colors[0], + stat='density', bins=30, label='Non-orthogonal ML'); + sns.histplot((theta_orth_po_nosplit - alpha)/se_orth_po_nosplit, + color=face_colors[1], edgecolor = edge_colors[1], + stat='density', bins=30, label='Double ML (no sample splitting)'); + sns.histplot((theta_dml_po - alpha)/se_dml_po, + color=face_colors[2], edgecolor = edge_colors[2], + stat='density', bins=30, label='Double ML with cross-fitting'); + ax.axvline(0., color='k'); + xx = np.arange(-5, +5, 0.001) + yy = stats.norm.pdf(xx) + ax.plot(xx, yy, color='k', label='$\\mathcal{N}(0, 1)$'); + ax.legend(loc='upper right', bbox_to_anchor=(1.2, 1.0)); + ax.set_xlim([-6., 6.]); + @savefig comparison_po.png width=5in + ax.set_xlabel('$(\hat{\\theta}_0 - \\theta_0)/\hat{\sigma}$'); + +.. tabbed:: R + + .. jupyter-execute:: + + g_all = ggplot(data.frame(t_nonorth=(theta_nonorth - alpha)/se_nonorth, + t_orth_nosplit=(theta_orth_po_nosplit - alpha)/se_orth_po_nosplit, + t_dml=(theta_dml_po - alpha)/se_dml_po)) + + geom_histogram(aes(x = t_nonorth, y=..density.., colour = "Non-orthogonal ML", fill="Non-orthogonal ML"), + bins = 30, alpha = 0.3) + + geom_histogram(aes(x = t_orth_nosplit, y=..density.., colour = "Double ML (no sample splitting)", fill="Double ML (no sample splitting)"), + bins = 30, alpha = 0.3) + + geom_histogram(aes(x = t_dml, y=..density.., colour = "Double ML with cross-fitting", fill="Double ML with cross-fitting"), + bins = 30, alpha = 0.3) + + geom_vline(aes(xintercept = 0), col = "black") + + suppressWarnings(geom_function(fun = dnorm, aes(colour = "N(0, 1)", fill="N(0, 1)"))) + + scale_color_manual(name='', + breaks=c("Non-orthogonal ML", "Double ML (no sample splitting)", "Double ML with cross-fitting", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", + "Double ML (no sample splitting)"="dark orange", + "Double ML with cross-fitting"="dark green", + "N(0, 1)"='black')) + + scale_fill_manual(name='', + breaks=c("Non-orthogonal ML", "Double ML (no sample splitting)", "Double ML with cross-fitting", "N(0, 1)"), + values=c("Non-orthogonal ML"="dark blue", + "Double ML (no sample splitting)"="dark orange", + "Double ML with cross-fitting"="dark green", + "N(0, 1)"=NA)) + + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() + g_all References ++++++++++ -Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2018), Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21: C1-C68. doi:`10.1111/ectj.12097 `_. \ No newline at end of file +Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2018), Double/debiased +machine learning for treatment and structural parameters. The Econometrics Journal, 21: C1-C68. +doi:`10.1111/ectj.12097 `_. + +Robinson, P. M. (1988). Root-N-consistent semi-parametric regression. Econometrica 56, 931-54. +doi:`10.2307/1912705 `_. \ No newline at end of file From d340469c05b3a8bd40bbf011563cc551e1f77880 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Thu, 28 Apr 2022 14:39:31 +0200 Subject: [PATCH 4/8] ignore another doi link in the linkcheck --- doc/conf.py | 1 + 1 file changed, 1 insertion(+) diff --git a/doc/conf.py b/doc/conf.py index da421a86..3b1a2213 100644 --- a/doc/conf.py +++ b/doc/conf.py @@ -129,6 +129,7 @@ 'https://doi.org/10.1093/ectj/utaa001', # Valid DOI; Causes 403 Client Error: Forbidden for url:... 'https://doi.org/10.1111/ectj.12097', # Valid DOI; Causes 403 Client Error: Forbidden for url:... 'https://doi.org/10.2307/2171802', # Valid DOI; Causes 403 Client Error: Forbidden for url:... + 'https://doi.org/10.2307/1912705', # Valid DOI; Causes 403 Client Error: Forbidden for url:... ] # To execute R code via jupyter-execute one needs to install the R kernel for jupyter From 6ad95f9814e8e540d04200aa977261895dd2ca84 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Thu, 5 May 2022 16:07:41 +0200 Subject: [PATCH 5/8] update user guide and example gallery: renaming of learner ml_g to ml_l and additional IV-type score for PLIV --- doc/examples/R_double_ml_pension.ipynb | 78 ++++++++++++++-- doc/examples/R_double_ml_pipeline.ipynb | 46 +++++----- doc/examples/double_ml_bonus_data.ipynb | 44 ++++----- .../py_double_ml_multiway_cluster.ipynb | 12 +-- doc/examples/py_double_ml_pension.ipynb | 89 +++++++++++++++++-- .../r_double_ml_multiway_cluster.ipynb | 12 +-- doc/guide/algorithms.rst | 8 +- doc/guide/basics.rst | 48 +++++----- doc/guide/learners.rst | 60 ++++++------- doc/guide/models.rst | 16 ++-- doc/guide/resampling.rst | 32 +++---- doc/guide/scores.rst | 64 ++++++++----- doc/guide/se_confint.rst | 8 +- doc/guide/sim_inf.rst | 8 +- doc/intro/intro.rst | 16 ++-- doc/workflow/workflow.rst | 16 ++-- 16 files changed, 361 insertions(+), 196 deletions(-) diff --git a/doc/examples/R_double_ml_pension.ipynb b/doc/examples/R_double_ml_pension.ipynb index 199e066a..18be502c 100644 --- a/doc/examples/R_double_ml_pension.ipynb +++ b/doc/examples/R_double_ml_pension.ipynb @@ -2,6 +2,7 @@ "cells": [ { "cell_type": "markdown", + "id": "b736f911", "metadata": { "papermill": { "duration": 0.040312, @@ -18,6 +19,7 @@ }, { "cell_type": "markdown", + "id": "56bfb9d6", "metadata": { "papermill": { "duration": 0.036965, @@ -38,6 +40,7 @@ }, { "cell_type": "markdown", + "id": "f5020e5a", "metadata": { "papermill": { "duration": 0.035502, @@ -57,6 +60,7 @@ { "cell_type": "code", "execution_count": null, + "id": "b5167d86", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:44:30.715038Z", @@ -93,6 +97,7 @@ }, { "cell_type": "markdown", + "id": "38231f8a", "metadata": { "papermill": { "duration": 0.036035, @@ -109,6 +114,7 @@ }, { "cell_type": "markdown", + "id": "9406ba1e", "metadata": { "papermill": { "duration": 0.036944, @@ -125,6 +131,7 @@ }, { "cell_type": "markdown", + "id": "f34e7973", "metadata": { "papermill": { "duration": 0.035954, @@ -142,6 +149,7 @@ { "cell_type": "code", "execution_count": null, + "id": "6182ff56", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:44:31.820240Z", @@ -171,6 +179,7 @@ { "cell_type": "code", "execution_count": null, + "id": "fd3f9739", "metadata": {}, "outputs": [], "source": [ @@ -184,6 +193,7 @@ }, { "cell_type": "markdown", + "id": "24774374", "metadata": { "papermill": { "duration": 0.039347, @@ -201,6 +211,7 @@ { "cell_type": "code", "execution_count": null, + "id": "512a2b1c", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:44:33.016258Z", @@ -229,6 +240,7 @@ }, { "cell_type": "markdown", + "id": "1633f2bc", "metadata": { "papermill": { "duration": 0.038905, @@ -246,6 +258,7 @@ { "cell_type": "code", "execution_count": null, + "id": "b45661c7", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:44:33.843625Z", @@ -270,6 +283,7 @@ }, { "cell_type": "markdown", + "id": "2457ef59", "metadata": { "papermill": { "duration": 0.040553, @@ -287,6 +301,7 @@ { "cell_type": "code", "execution_count": null, + "id": "57d965fb", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:44:34.036419Z", @@ -311,6 +326,7 @@ }, { "cell_type": "markdown", + "id": "bd7bffa3", "metadata": { "papermill": { "duration": 0.040806, @@ -327,6 +343,7 @@ }, { "cell_type": "markdown", + "id": "4d5c996f", "metadata": { "papermill": { "duration": 0.040491, @@ -343,6 +360,7 @@ }, { "cell_type": "markdown", + "id": "89867310", "metadata": { "papermill": { "duration": 0.041623, @@ -359,6 +377,7 @@ }, { "cell_type": "markdown", + "id": "6f5da000", "metadata": { "papermill": { "duration": 0.042605, @@ -375,6 +394,7 @@ }, { "cell_type": "markdown", + "id": "626c4986", "metadata": { "papermill": { "duration": 0.043776, @@ -406,6 +426,7 @@ }, { "cell_type": "markdown", + "id": "3261c60c", "metadata": {}, "source": [ "### The Data Backend: `DoubleMLData`" @@ -413,6 +434,7 @@ }, { "cell_type": "markdown", + "id": "b195be65", "metadata": {}, "source": [ "To start our analysis, we initialize the data backend, i.e., a new instance of a [DoubleMLData](https://docs.doubleml.org/r/stable/reference/DoubleMLData.html) object. Here, we manually implement the regression model by using R's formula interface. A shortcut would be to directly specify the options `polynomial_features` and `instrument` when calling [fetch_401k()](https://docs.doubleml.org/r/stable/reference/fetch_401k.html).$^{**}$\n", @@ -425,6 +447,7 @@ { "cell_type": "code", "execution_count": null, + "id": "e028de75", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:47:34.664674Z", @@ -458,6 +481,7 @@ { "cell_type": "code", "execution_count": null, + "id": "a8726d22", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:47:34.664674Z", @@ -497,6 +521,7 @@ }, { "cell_type": "markdown", + "id": "f7bf775d", "metadata": { "papermill": { "duration": 0.044143, @@ -513,6 +538,7 @@ }, { "cell_type": "markdown", + "id": "e88a8607", "metadata": { "papermill": { "duration": 0.043027, @@ -529,6 +555,7 @@ }, { "cell_type": "markdown", + "id": "3e99a6d2", "metadata": { "papermill": { "duration": 0.044546, @@ -548,6 +575,7 @@ }, { "cell_type": "markdown", + "id": "667de94c", "metadata": {}, "source": [ "To estimate the causal parameter $\\theta_0$ here, we use double machine learning with 3-fold cross-fitting. \n", @@ -562,6 +590,7 @@ { "cell_type": "code", "execution_count": null, + "id": "c3f6a786", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:47:35.302091Z", @@ -587,7 +616,7 @@ "\n", "# Initialize DoubleMLPLR model\n", "dml_plr_lasso = DoubleMLPLR$new(data_dml_base, \n", - " ml_g = lasso,\n", + " ml_l = lasso,\n", " ml_m = lasso_class,\n", " n_folds = 3)\n", "dml_plr_lasso$fit()\n", @@ -597,6 +626,7 @@ { "cell_type": "code", "execution_count": null, + "id": "fde8ed43", "metadata": {}, "outputs": [], "source": [ @@ -607,7 +637,7 @@ "\n", "# Initialize DoubleMLPLR model\n", "dml_plr_lasso = DoubleMLPLR$new(data_dml_flex, \n", - " ml_g = lasso,\n", + " ml_l = lasso,\n", " ml_m = lasso_class,\n", " n_folds = 3)\n", "dml_plr_lasso$fit()\n", @@ -616,6 +646,7 @@ }, { "cell_type": "markdown", + "id": "d922db8d", "metadata": { "papermill": { "duration": 0.046627, @@ -633,6 +664,7 @@ { "cell_type": "code", "execution_count": null, + "id": "9670966e", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:47:59.087352Z", @@ -659,7 +691,7 @@ "\n", "set.seed(123)\n", "dml_plr_forest = DoubleMLPLR$new(data_dml_base,\n", - " ml_g = randomForest,\n", + " ml_l = randomForest,\n", " ml_m = randomForest_class,\n", " n_folds = 3)\n", "dml_plr_forest$fit() \n", @@ -668,6 +700,7 @@ }, { "cell_type": "markdown", + "id": "f2cbd323", "metadata": {}, "source": [ "Now, let's use a regression tree as provided by the R package [rpart](https://github.com/bethatkinson/rpart)." @@ -676,6 +709,7 @@ { "cell_type": "code", "execution_count": null, + "id": "76a4b258", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:49:02.526135Z", @@ -700,7 +734,7 @@ "\n", "set.seed(123)\n", "dml_plr_tree = DoubleMLPLR$new(data_dml_base,\n", - " ml_g = trees,\n", + " ml_l = trees,\n", " ml_m = trees_class,\n", " n_folds = 3)\n", "dml_plr_tree$fit()\n", @@ -709,6 +743,7 @@ }, { "cell_type": "markdown", + "id": "dd6800f9", "metadata": {}, "source": [ "We can also experiment with extreme gradient boosting as provided by [xgboost](https://xgboost.readthedocs.io/en/latest/)." @@ -717,6 +752,7 @@ { "cell_type": "code", "execution_count": null, + "id": "a2a0c914", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:49:04.206928Z", @@ -745,7 +781,7 @@ "\n", "set.seed(123)\n", "dml_plr_boost = DoubleMLPLR$new(data_dml_base,\n", - " ml_g = boost,\n", + " ml_l = boost,\n", " ml_m = boost_class,\n", " n_folds = 3)\n", "dml_plr_boost$fit()\n", @@ -754,6 +790,7 @@ }, { "cell_type": "markdown", + "id": "cce3aca6", "metadata": { "papermill": { "duration": 0.055456, @@ -771,6 +808,7 @@ { "cell_type": "code", "execution_count": null, + "id": "20e3a391", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:49:05.421244Z", @@ -804,6 +842,7 @@ { "cell_type": "code", "execution_count": null, + "id": "50e2a7c9", "metadata": {}, "outputs": [], "source": [ @@ -821,6 +860,7 @@ }, { "cell_type": "markdown", + "id": "6405d1e9", "metadata": { "papermill": { "duration": 0.056069, @@ -837,6 +877,7 @@ }, { "cell_type": "markdown", + "id": "89212f23", "metadata": { "papermill": { "duration": 0.055247, @@ -853,6 +894,7 @@ }, { "cell_type": "markdown", + "id": "7cc1040d", "metadata": { "papermill": { "duration": 0.057453, @@ -872,6 +914,7 @@ }, { "cell_type": "markdown", + "id": "dc13bc3d", "metadata": { "papermill": { "duration": 0.055141, @@ -890,6 +933,7 @@ { "cell_type": "code", "execution_count": null, + "id": "7a2a1a88", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:49:06.055682Z", @@ -922,6 +966,7 @@ { "cell_type": "code", "execution_count": null, + "id": "d9d66ac9", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:49:25.191466Z", @@ -967,6 +1012,7 @@ { "cell_type": "code", "execution_count": null, + "id": "d93208c7", "metadata": {}, "outputs": [], "source": [ @@ -997,6 +1043,7 @@ { "cell_type": "code", "execution_count": null, + "id": "a409c687", "metadata": {}, "outputs": [], "source": [ @@ -1037,6 +1084,7 @@ { "cell_type": "code", "execution_count": null, + "id": "c388da15", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:50:25.095222Z", @@ -1080,6 +1128,7 @@ }, { "cell_type": "markdown", + "id": "16a2088c", "metadata": { "papermill": { "duration": 0.067511, @@ -1097,6 +1146,7 @@ }, { "cell_type": "markdown", + "id": "286f8dfa", "metadata": { "papermill": { "duration": 0.06959, @@ -1113,6 +1163,7 @@ }, { "cell_type": "markdown", + "id": "33169af9", "metadata": { "papermill": { "duration": 0.068243, @@ -1129,6 +1180,7 @@ }, { "cell_type": "markdown", + "id": "197b1a47", "metadata": { "papermill": { "duration": 0.070268, @@ -1154,6 +1206,7 @@ { "cell_type": "code", "execution_count": null, + "id": "803215b0", "metadata": {}, "outputs": [], "source": [ @@ -1171,6 +1224,7 @@ { "cell_type": "code", "execution_count": null, + "id": "fe2617d3", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:50:25.988471Z", @@ -1203,6 +1257,7 @@ { "cell_type": "code", "execution_count": null, + "id": "1beb1a78", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:50:26.199177Z", @@ -1236,6 +1291,7 @@ }, { "cell_type": "markdown", + "id": "a80e742e", "metadata": { "papermill": { "duration": 0.071304, @@ -1253,6 +1309,7 @@ { "cell_type": "code", "execution_count": null, + "id": "e9cee42d", "metadata": { "execution": { "iopub.execute_input": "2021-03-23T13:50:48.876339Z", @@ -1307,6 +1364,7 @@ { "cell_type": "code", "execution_count": null, + "id": "e2595fb1", "metadata": {}, "outputs": [], "source": [ @@ -1346,6 +1404,7 @@ { "cell_type": "code", "execution_count": null, + "id": "19d1ef93", "metadata": {}, "outputs": [], "source": [ @@ -1401,6 +1460,7 @@ { "cell_type": "code", "execution_count": null, + "id": "08267d88", "metadata": {}, "outputs": [], "source": [ @@ -1429,6 +1489,7 @@ }, { "cell_type": "markdown", + "id": "907c3978", "metadata": {}, "source": [ "## Summary of Results" @@ -1436,6 +1497,7 @@ }, { "cell_type": "markdown", + "id": "bdeed622", "metadata": {}, "source": [ "To sum up, let's merge all our results so far and illustrate them in a plot. " @@ -1444,6 +1506,7 @@ { "cell_type": "code", "execution_count": null, + "id": "f7c5acef", "metadata": {}, "outputs": [], "source": [ @@ -1454,6 +1517,7 @@ { "cell_type": "code", "execution_count": null, + "id": "d0658530", "metadata": { "tags": [ "nbsphinx-thumbnail" @@ -1476,6 +1540,7 @@ }, { "cell_type": "markdown", + "id": "62b2d3d5", "metadata": { "papermill": { "duration": 0.079866, @@ -1494,6 +1559,7 @@ }, { "cell_type": "markdown", + "id": "c29ea3cb", "metadata": {}, "source": [ "______\n", @@ -1517,7 +1583,7 @@ "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", - "version": "3.6.1" + "version": "4.0.4" }, "papermill": { "default_parameters": {}, diff --git a/doc/examples/R_double_ml_pipeline.ipynb b/doc/examples/R_double_ml_pipeline.ipynb index b79b08ba..60fac16b 100644 --- a/doc/examples/R_double_ml_pipeline.ipynb +++ b/doc/examples/R_double_ml_pipeline.ipynb @@ -113,9 +113,9 @@ "lgr::get_logger(\"mlr3\")$set_threshold(\"warn\")\n", "\n", "learner_lasso = lrn(\"regr.cv_glmnet\", s=\"lambda.min\")\n", - "ml_g_lasso = learner_lasso$clone()\n", + "ml_l_lasso = learner_lasso$clone()\n", "ml_m_lasso = learner_lasso$clone()\n", - "class(ml_g_lasso)" + "class(ml_l_lasso)" ] }, { @@ -125,7 +125,7 @@ "metadata": {}, "outputs": [], "source": [ - "# Random forest learner for nuisance part ml_g\n", + "# Random forest learner for nuisance part ml_l\n", "learner_forest_regr = lrn(\"regr.ranger\",\n", " num.trees=500, mtry=floor(sqrt(dim_x)),\n", " max.depth=5, min.node.size=2)\n", @@ -136,9 +136,9 @@ " mtry=floor(sqrt(dim_x)),\n", " max.depth=5, min.node.size=2)\n", "\n", - "ml_g_forest = learner_forest_regr$clone()\n", + "ml_l_forest = learner_forest_regr$clone()\n", "ml_m_forest = learner_forest_classif$clone()\n", - "class(ml_g_forest)" + "class(ml_l_forest)" ] }, { @@ -158,7 +158,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_sim = DoubleMLPLR$new(dml_data_sim,\n", - " ml_g=ml_g_lasso,\n", + " ml_l=ml_l_lasso,\n", " ml_m=ml_m_lasso)\n", "obj_dml_plr_sim$fit()\n", "print(obj_dml_plr_sim)" @@ -173,7 +173,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_bonus = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g=ml_g_forest,\n", + " ml_l=ml_l_forest,\n", " ml_m=ml_m_forest)\n", "obj_dml_plr_bonus$fit()\n", "print(obj_dml_plr_bonus)" @@ -205,11 +205,11 @@ "# Lasso learner\n", "library(mlr3pipelines)\n", "pipe_lasso = po(lrn(\"regr.cv_glmnet\"), s = \"lambda.min\")\n", - "ml_g_lasso_pipe = as_learner(pipe_lasso)\n", + "ml_l_lasso_pipe = as_learner(pipe_lasso)\n", "ml_m_lasso_pipe = as_learner(pipe_lasso)\n", "\n", "# Class of the lasso learner\n", - "class(ml_g_lasso_pipe)" + "class(ml_l_lasso_pipe)" ] }, { @@ -219,7 +219,7 @@ "metadata": {}, "outputs": [], "source": [ - "# Random forest learner for nuisance part ml_g\n", + "# Random forest learner for nuisance part ml_l\n", "pipe_forest_regr = po(lrn(\"regr.ranger\"),\n", " num.trees=500, mtry=floor(sqrt(dim_x)),\n", " max.depth=5, min.node.size=2)\n", @@ -230,11 +230,11 @@ " mtry=floor(sqrt(dim_x)),\n", " max.depth=5, min.node.size=2)\n", "\n", - "ml_g_forest_pipe = as_learner(pipe_forest_regr)\n", + "ml_l_forest_pipe = as_learner(pipe_forest_regr)\n", "ml_m_forest_pipe = as_learner(pipe_forest_classif)\n", "\n", "# Class of the random forest learners\n", - "class(ml_g_forest_pipe)\n", + "class(ml_l_forest_pipe)\n", "class(ml_m_forest_pipe)" ] }, @@ -255,7 +255,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_sim_pipe = DoubleMLPLR$new(dml_data_sim,\n", - " ml_g=ml_g_lasso_pipe,\n", + " ml_l=ml_l_lasso_pipe,\n", " ml_m=ml_m_lasso_pipe)\n", "obj_dml_plr_sim_pipe$fit()\n", "print(obj_dml_plr_sim_pipe)" @@ -270,7 +270,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_bonus_pipe = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g=ml_g_forest_pipe,\n", + " ml_l=ml_l_forest_pipe,\n", " ml_m=ml_m_forest_pipe)\n", "obj_dml_plr_bonus_pipe$fit()\n", "print(obj_dml_plr_bonus_pipe)" @@ -410,7 +410,7 @@ "# Initiate new DoubleML object and estimate with graph learner\n", "set.seed(123)\n", "obj_dml_plr_sim_pipe_ensemble = DoubleMLPLR$new(dml_data_sim,\n", - " ml_g = ensemble_pipe_regr,\n", + " ml_l = ensemble_pipe_regr,\n", " ml_m = ensemble_pipe_regr)\n", "obj_dml_plr_sim_pipe_ensemble$fit()\n", "print(obj_dml_plr_sim_pipe_ensemble)" @@ -425,7 +425,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_bonus_pipe_ensemble = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g = ensemble_pipe_regr,\n", + " ml_l = ensemble_pipe_regr,\n", " ml_m = ensemble_pipe_classif)\n", "obj_dml_plr_bonus_pipe_ensemble$fit()\n", "print(obj_dml_plr_bonus_pipe_ensemble)" @@ -495,7 +495,7 @@ "\n", "set.seed(123)\n", "obj_dml_plr_bonus_pipe = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g=ml_g_forest,\n", + " ml_l=ml_l_forest,\n", " ml_m=stacklrn)\n", "obj_dml_plr_bonus_pipe$fit()\n", "print(obj_dml_plr_bonus_pipe)" @@ -607,7 +607,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_bonus_pipe2 = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g=ml_g_lasso,\n", + " ml_l=ml_l_lasso,\n", " ml_m=glrn)\n", "obj_dml_plr_bonus_pipe2$fit()\n", "print(obj_dml_plr_bonus_pipe2)" @@ -640,7 +640,7 @@ "source": [ "set.seed(123)\n", "obj_dml_plr_bonus_pipe3 = DoubleMLPLR$new(dml_data_bonus,\n", - " ml_g=ml_g_lasso,\n", + " ml_l=ml_l_lasso,\n", " ml_m=glrn)\n", "obj_dml_plr_bonus_pipe3$fit()\n", "print(obj_dml_plr_bonus_pipe3)" @@ -711,7 +711,7 @@ "tune_settings = list(terminator = trm(\"evals\", n_evals = 10),\n", " algorithm = tnr(\"grid_search\", resolution = 10),\n", " rsmp_tune = rsmp(\"cv\", folds = 5),\n", - " measure = list(\"ml_g\" = msr(\"regr.mse\"),\n", + " measure = list(\"ml_l\" = msr(\"regr.mse\"),\n", " \"ml_m\" = msr(\"regr.mse\")))" ] }, @@ -725,9 +725,9 @@ "# Initiate new DoubleML object and execute tuning with graph learner\n", "set.seed(123)\n", "obj_dml_plr_sim_pipe_tune = DoubleMLPLR$new(dml_data_sim,\n", - " ml_g=glrn_lasso,\n", + " ml_l=glrn_lasso,\n", " ml_m=glrn_lasso)\n", - "obj_dml_plr_sim_pipe_tune$tune(param_set = list(\"ml_g\" = par_grids,\n", + "obj_dml_plr_sim_pipe_tune$tune(param_set = list(\"ml_l\" = par_grids,\n", " \"ml_m\" = par_grids),\n", " tune_settings=tune_settings)" ] @@ -786,7 +786,7 @@ "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", - "version": "3.6.1" + "version": "4.0.4" } }, "nbformat": 4, diff --git a/doc/examples/double_ml_bonus_data.ipynb b/doc/examples/double_ml_bonus_data.ipynb index 69c35ab0..b32f023f 100644 --- a/doc/examples/double_ml_bonus_data.ipynb +++ b/doc/examples/double_ml_bonus_data.ipynb @@ -78,20 +78,20 @@ "metadata": {}, "outputs": [], "source": [ - "# Set machine learning methods for m & g\n", - "ml_g = RandomForestRegressor()\n", + "# Set machine learning methods for m & l\n", + "ml_l = RandomForestRegressor()\n", "ml_m = RandomForestRegressor()\n", "n_folds = 2\n", "n_rep = 10\n", "\n", "np.random.seed(3141)\n", "dml_plr_rf = dml.DoubleMLPLR(dml_data,\n", - " ml_g,\n", + " ml_l,\n", " ml_m,\n", - " n_folds,\n", - " n_rep,\n", - " 'partialling out',\n", - " 'dml2')\n", + " n_folds=n_folds,\n", + " n_rep=n_rep,\n", + " score='partialling out',\n", + " dml_procedure='dml2')\n", "\n", "# set some hyperparameters for the learners\n", "pars = {'n_estimators': 500,\n", @@ -137,20 +137,20 @@ "metadata": {}, "outputs": [], "source": [ - "# Set machine learning methods for m & g\n", - "ml_g = Lasso()\n", + "# Set machine learning methods for m & l\n", + "ml_l = Lasso()\n", "ml_m = Lasso()\n", "n_folds = 2\n", "n_rep = 10\n", "\n", "np.random.seed(3141)\n", "dml_plr_lasso = dml.DoubleMLPLR(dml_data_lasso,\n", - " ml_g,\n", + " ml_l,\n", " ml_m,\n", - " n_folds,\n", - " n_rep,\n", - " 'partialling out',\n", - " 'dml2')\n", + " n_folds=n_folds,\n", + " n_rep=n_rep,\n", + " score='partialling out',\n", + " dml_procedure='dml2')\n", "\n", "# set some hyperparameters for the learners\n", "dml_plr_lasso.set_ml_nuisance_params('ml_l', 'tg', {'alpha': 0.0005})\n", @@ -191,10 +191,10 @@ "dml_irm_rf = dml.DoubleMLIRM(dml_data,\n", " ml_g,\n", " ml_m,\n", - " n_folds,\n", - " n_rep,\n", - " 'ATE',\n", - " 'dml2')\n", + " n_folds=n_folds,\n", + " n_rep=n_rep,\n", + " score='ATE',\n", + " dml_procedure='dml2')\n", "\n", "# set some hyperparameters for the learners\n", "pars = {'n_estimators': 500,\n", @@ -241,10 +241,10 @@ "dml_irm_lasso = dml.DoubleMLIRM(dml_data_lasso,\n", " ml_g,\n", " ml_m,\n", - " n_folds,\n", - " n_rep,\n", - " 'ATE',\n", - " 'dml2')\n", + " n_folds=n_folds,\n", + " n_rep=n_rep,\n", + " score='ATE',\n", + " dml_procedure='dml2')\n", "\n", "# set some hyperparameters for the learners\n", "dml_irm_lasso.set_ml_nuisance_params('ml_g0', 'tg', {'alpha': 0.0019})\n", diff --git a/doc/examples/py_double_ml_multiway_cluster.ipynb b/doc/examples/py_double_ml_multiway_cluster.ipynb index 73691b5e..f00105bf 100644 --- a/doc/examples/py_double_ml_multiway_cluster.ipynb +++ b/doc/examples/py_double_ml_multiway_cluster.ipynb @@ -190,15 +190,15 @@ }, "outputs": [], "source": [ - "# Set machine learning methods for m, g & r\n", + "# Set machine learning methods for l, m & r\n", "learner = LassoCV()\n", - "ml_g = clone(learner)\n", + "ml_l = clone(learner)\n", "ml_m = clone(learner)\n", "ml_r = clone(learner)\n", "\n", "# initialize the DoubleMLPLIV object\n", "dml_pliv_obj = DoubleMLPLIV(obj_dml_data,\n", - " ml_g, ml_m, ml_r,\n", + " ml_l, ml_m, ml_r,\n", " n_folds=3)" ] }, @@ -472,15 +472,15 @@ }, "outputs": [], "source": [ - "# Set machine learning methods for m & g\n", + "# Set machine learning methods for l, m & r\n", "learner = LassoCV()\n", - "ml_g = clone(learner)\n", + "ml_l = clone(learner)\n", "ml_m = clone(learner)\n", "ml_r = clone(learner)\n", "\n", "# initialize the DoubleMLPLIV object\n", "dml_pliv_obj = DoubleMLPLIV(obj_dml_data,\n", - " ml_g, ml_m, ml_r,\n", + " ml_l, ml_m, ml_r,\n", " n_folds=3)" ] }, diff --git a/doc/examples/py_double_ml_pension.ipynb b/doc/examples/py_double_ml_pension.ipynb index 9ca8c560..d9813d16 100644 --- a/doc/examples/py_double_ml_pension.ipynb +++ b/doc/examples/py_double_ml_pension.ipynb @@ -2,6 +2,7 @@ "cells": [ { "cell_type": "markdown", + "id": "2b2a0fd8", "metadata": {}, "source": [ "# Python: Impact of 401(k) on Financial Wealth" @@ -9,6 +10,7 @@ }, { "cell_type": "markdown", + "id": "a967ed5f", "metadata": {}, "source": [ "In this real-data example, we illustrate how the [DoubleML](https://docs.doubleml.org/stable/index.html) package can be used to estimate the effect of 401(k) eligibility and participation on accumulated assets. The 401(k) data set has been analyzed in several studies, among others [Chernozhukov et al. (2018)](https://arxiv.org/abs/1608.00060).\n", @@ -20,6 +22,7 @@ }, { "cell_type": "markdown", + "id": "40b41785", "metadata": {}, "source": [ "## Data\n", @@ -30,6 +33,7 @@ { "cell_type": "code", "execution_count": null, + "id": "c06e87b2", "metadata": {}, "outputs": [], "source": [ @@ -54,6 +58,7 @@ { "cell_type": "code", "execution_count": null, + "id": "28347df3", "metadata": {}, "outputs": [], "source": [ @@ -64,6 +69,7 @@ { "cell_type": "code", "execution_count": null, + "id": "0ad5caf0", "metadata": {}, "outputs": [], "source": [ @@ -78,6 +84,7 @@ { "cell_type": "code", "execution_count": null, + "id": "11763be2", "metadata": {}, "outputs": [], "source": [ @@ -87,6 +94,7 @@ { "cell_type": "code", "execution_count": null, + "id": "cf8659c7", "metadata": {}, "outputs": [], "source": [ @@ -96,6 +104,7 @@ { "cell_type": "code", "execution_count": null, + "id": "c1d9ce03", "metadata": {}, "outputs": [], "source": [ @@ -104,6 +113,7 @@ }, { "cell_type": "markdown", + "id": "3ea8c00f", "metadata": {}, "source": [ "The data consist of 9,915 observations at the household level drawn from the 1991 Survey of Income and Program Participation (SIPP). All the variables are referred to 1990. We use net financial assets (*net\\_tfa*) as the outcome variable, $Y$, in our analysis. The net financial assets are computed as the sum of IRA balances, 401(k) balances, checking accounts, saving bonds, other interest-earning accounts, other interest-earning assets, stocks, and mutual funds less non mortgage debts. " @@ -111,6 +121,7 @@ }, { "cell_type": "markdown", + "id": "1a4f9dc4", "metadata": {}, "source": [ "Among the $9915$ individuals, $3682$ are eligible to participate in the program. The variable *e401* indicates eligibility and *p401* indicates participation, respectively." @@ -119,6 +130,7 @@ { "cell_type": "code", "execution_count": null, + "id": "1fb9537b", "metadata": {}, "outputs": [], "source": [ @@ -131,6 +143,7 @@ { "cell_type": "code", "execution_count": null, + "id": "3d464563", "metadata": {}, "outputs": [], "source": [ @@ -142,6 +155,7 @@ }, { "cell_type": "markdown", + "id": "b8461a88", "metadata": {}, "source": [ "Eligibility is highly associated with financial wealth:" @@ -150,6 +164,7 @@ { "cell_type": "code", "execution_count": null, + "id": "9ab147b8", "metadata": {}, "outputs": [], "source": [ @@ -159,6 +174,7 @@ }, { "cell_type": "markdown", + "id": "aae566a4", "metadata": {}, "source": [ "As a first estimate, we calculate the unconditional average predictive effect (APE) of 401(k) eligibility on accumulated assets. This effect corresponds to the average treatment effect if 401(k) eligibility would be assigned to individuals in an entirely randomized way. The unconditional APE of e401 is about $19559$:" @@ -167,6 +183,7 @@ { "cell_type": "code", "execution_count": null, + "id": "97d9b759", "metadata": {}, "outputs": [], "source": [ @@ -175,6 +192,7 @@ }, { "cell_type": "markdown", + "id": "491fc066", "metadata": {}, "source": [ "Among the $3682$ individuals that are eligible, $2594$ decided to participate in the program. The unconditional APE of p401 is about $27372$:" @@ -183,6 +201,7 @@ { "cell_type": "code", "execution_count": null, + "id": "7f1660be", "metadata": {}, "outputs": [], "source": [ @@ -191,6 +210,7 @@ }, { "cell_type": "markdown", + "id": "58862aff", "metadata": {}, "source": [ "As discussed, these estimates are biased since they do not account for saver heterogeneity and endogeneity of participation." @@ -198,6 +218,7 @@ }, { "cell_type": "markdown", + "id": "bdb138b7", "metadata": {}, "source": [ "## The `DoubleML` package" @@ -205,6 +226,7 @@ }, { "cell_type": "markdown", + "id": "d871d878", "metadata": {}, "source": [ "Let's use the package [DoubleML](https://docs.doubleml.org/stable/index.html) to estimate the average treatment effect of 401(k) eligibility, i.e. `e401`, and participation, i.e. `p401`, on net financial assets `net_tfa`." @@ -212,6 +234,7 @@ }, { "cell_type": "markdown", + "id": "7c626f03", "metadata": {}, "source": [ "## Estimating the Average Treatment Effect of 401(k) Eligibility on Net Financial Assets" @@ -219,6 +242,7 @@ }, { "cell_type": "markdown", + "id": "8f1f0cbb", "metadata": {}, "source": [ "We first look at the treatment effect of `e401` on net total financial assets. We give estimates of the ATE in the linear model\n", @@ -241,6 +265,7 @@ }, { "cell_type": "markdown", + "id": "59d03336", "metadata": {}, "source": [ "### The Data Backend: `DoubleMLData`" @@ -248,6 +273,7 @@ }, { "cell_type": "markdown", + "id": "2e1fe478", "metadata": {}, "source": [ "To start our analysis, we initialize the data backend, i.e., a new instance of a [DoubleMLData](https://docs.doubleml.org/dev/api/generated/doubleml.DoubleMLData.html#doubleml.DoubleMLData) object. We implement the regression model by using scikit-learn's `PolynomialFeatures` class.\n", @@ -258,6 +284,7 @@ { "cell_type": "code", "execution_count": null, + "id": "a69acf79", "metadata": {}, "outputs": [], "source": [ @@ -275,6 +302,7 @@ { "cell_type": "code", "execution_count": null, + "id": "283cb128", "metadata": {}, "outputs": [], "source": [ @@ -284,6 +312,7 @@ { "cell_type": "code", "execution_count": null, + "id": "4f8f4463", "metadata": {}, "outputs": [], "source": [ @@ -313,6 +342,7 @@ { "cell_type": "code", "execution_count": null, + "id": "4d5885ed", "metadata": {}, "outputs": [], "source": [ @@ -321,6 +351,7 @@ }, { "cell_type": "markdown", + "id": "a85b7f11", "metadata": {}, "source": [ "### Partially Linear Regression Model (PLR)" @@ -328,6 +359,7 @@ }, { "cell_type": "markdown", + "id": "f4794f98", "metadata": {}, "source": [ "We start using lasso to estimate the function $g_0$ and $m_0$ in the following PLR model:" @@ -335,6 +367,7 @@ }, { "cell_type": "markdown", + "id": "77aac812", "metadata": {}, "source": [ "\\begin{eqnarray}\n", @@ -345,6 +378,7 @@ }, { "cell_type": "markdown", + "id": "8aaaa9ff", "metadata": {}, "source": [ "To estimate the causal parameter $\\theta_0$ here, we use double machine learning with 3-fold cross-fitting. \n", @@ -357,6 +391,7 @@ { "cell_type": "code", "execution_count": null, + "id": "6481a766", "metadata": { "scrolled": true }, @@ -372,7 +407,7 @@ "np.random.seed(123)\n", "# Initialize DoubleMLPLR model\n", "dml_plr_lasso = dml.DoubleMLPLR(data_dml_base,\n", - " ml_g = lasso,\n", + " ml_l = lasso,\n", " ml_m = lasso_class,\n", " n_folds = 3)\n", "\n", @@ -383,6 +418,7 @@ { "cell_type": "code", "execution_count": null, + "id": "355e78ca", "metadata": {}, "outputs": [], "source": [ @@ -390,7 +426,7 @@ "\n", "np.random.seed(123)\n", "dml_plr_lasso = dml.DoubleMLPLR(data_dml_flex,\n", - " ml_g = lasso,\n", + " ml_l = lasso,\n", " ml_m = lasso_class,\n", " n_folds = 3)\n", "\n", @@ -402,6 +438,7 @@ }, { "cell_type": "markdown", + "id": "92407d1e", "metadata": {}, "source": [ "Alternatively, we can repeat this procedure with other machine learning methods, for example a random forest learner as provided by the [RandomForestRegressor](https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestRegressor.html) and [RandomForestClassifier](https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html) class in [scikit-learn](https://scikit-learn.org). " @@ -410,6 +447,7 @@ { "cell_type": "code", "execution_count": null, + "id": "8462dd26", "metadata": {}, "outputs": [], "source": [ @@ -421,7 +459,7 @@ "\n", "np.random.seed(123)\n", "dml_plr_forest = dml.DoubleMLPLR(data_dml_base,\n", - " ml_g = randomForest,\n", + " ml_l = randomForest,\n", " ml_m = randomForest_class,\n", " n_folds = 3)\n", "dml_plr_forest.fit(store_predictions=True)\n", @@ -432,6 +470,7 @@ }, { "cell_type": "markdown", + "id": "c2e8de45", "metadata": {}, "source": [ "Now, let's use a regression tree as provided in [scikit-learn](https://scikit-learn.org)'s [DecisionTreeRegressor](https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeRegressor.html) and [DecisionTreeClassifier](https://scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.html)." @@ -440,6 +479,7 @@ { "cell_type": "code", "execution_count": null, + "id": "20957649", "metadata": {}, "outputs": [], "source": [ @@ -451,7 +491,7 @@ "\n", "np.random.seed(123)\n", "dml_plr_tree = dml.DoubleMLPLR(data_dml_base,\n", - " ml_g = trees,\n", + " ml_l = trees,\n", " ml_m = trees_class,\n", " n_folds = 3)\n", "dml_plr_tree.fit(store_predictions=True)\n", @@ -462,6 +502,7 @@ }, { "cell_type": "markdown", + "id": "a187d938", "metadata": {}, "source": [ "We can also experiment with extreme gradient boosting as provided by [xgboost](https://xgboost.readthedocs.io/en/latest/)." @@ -470,6 +511,7 @@ { "cell_type": "code", "execution_count": null, + "id": "988af9d6", "metadata": {}, "outputs": [], "source": [ @@ -482,7 +524,7 @@ "\n", "np.random.seed(123)\n", "dml_plr_boost = dml.DoubleMLPLR(data_dml_base,\n", - " ml_g = boost,\n", + " ml_l = boost,\n", " ml_m = boost_class,\n", " n_folds = 3)\n", "dml_plr_boost.fit(store_predictions=True)\n", @@ -493,6 +535,7 @@ }, { "cell_type": "markdown", + "id": "b57d231e", "metadata": {}, "source": [ "Let's sum up the results:" @@ -501,6 +544,7 @@ { "cell_type": "code", "execution_count": null, + "id": "0cc90651", "metadata": {}, "outputs": [], "source": [ @@ -512,6 +556,7 @@ { "cell_type": "code", "execution_count": null, + "id": "20528edc", "metadata": {}, "outputs": [], "source": [ @@ -528,6 +573,7 @@ }, { "cell_type": "markdown", + "id": "446bdae9", "metadata": {}, "source": [ "### Interactive Regression Model (IRM)" @@ -535,6 +581,7 @@ }, { "cell_type": "markdown", + "id": "aa53e2c6", "metadata": {}, "source": [ "Next, we consider estimation of average treatment effects when treatment effects are fully heterogeneous:" @@ -542,6 +589,7 @@ }, { "cell_type": "markdown", + "id": "4ca39012", "metadata": {}, "source": [ "\\begin{eqnarray}\n", @@ -552,6 +600,7 @@ }, { "cell_type": "markdown", + "id": "e61aece3", "metadata": {}, "source": [ "To reduce the disproportionate impact of extreme propensity score weights in the interactive model\n", @@ -561,6 +610,7 @@ { "cell_type": "code", "execution_count": null, + "id": "14b79c21", "metadata": {}, "outputs": [], "source": [ @@ -583,6 +633,7 @@ { "cell_type": "code", "execution_count": null, + "id": "6762686f", "metadata": {}, "outputs": [], "source": [ @@ -614,6 +665,7 @@ { "cell_type": "code", "execution_count": null, + "id": "8589ee4d", "metadata": {}, "outputs": [], "source": [ @@ -645,6 +697,7 @@ { "cell_type": "code", "execution_count": null, + "id": "8ab9a005", "metadata": {}, "outputs": [], "source": [ @@ -677,6 +730,7 @@ { "cell_type": "code", "execution_count": null, + "id": "83a3293b", "metadata": {}, "outputs": [], "source": [ @@ -688,6 +742,7 @@ { "cell_type": "code", "execution_count": null, + "id": "ddf460d6", "metadata": {}, "outputs": [], "source": [ @@ -704,6 +759,7 @@ }, { "cell_type": "markdown", + "id": "541d51d3", "metadata": {}, "source": [ "These estimates that flexibly account for confounding are\n", @@ -712,6 +768,7 @@ }, { "cell_type": "markdown", + "id": "a70f729b", "metadata": {}, "source": [ "## Local Average Treatment Effects of 401(k) Participation on Net Financial Assets" @@ -719,6 +776,7 @@ }, { "cell_type": "markdown", + "id": "f5c32411", "metadata": {}, "source": [ "### Interactive IV Model (IIVM)" @@ -726,6 +784,7 @@ }, { "cell_type": "markdown", + "id": "9ad4bf98", "metadata": {}, "source": [ "In the examples above, we estimated the average treatment effect of *eligibility* on financial asset holdings. Now, we consider estimation of local average treatment effects (LATE) of *participation* using eligibility as an instrument for the participation decision. Under appropriate assumptions, the LATE identifies the treatment effect for so-called compliers, i.e., individuals who would only participate if eligible and otherwise not participate in the program. \n", @@ -742,6 +801,7 @@ { "cell_type": "code", "execution_count": null, + "id": "85c3d0fe", "metadata": {}, "outputs": [], "source": [ @@ -760,6 +820,7 @@ { "cell_type": "code", "execution_count": null, + "id": "9c6f8fc8", "metadata": {}, "outputs": [], "source": [ @@ -778,6 +839,7 @@ { "cell_type": "code", "execution_count": null, + "id": "c2e0942f", "metadata": {}, "outputs": [], "source": [ @@ -802,6 +864,7 @@ }, { "cell_type": "markdown", + "id": "19ddd6ad", "metadata": {}, "source": [ "Again, we repeat the procedure for the other machine learning methods:" @@ -810,6 +873,7 @@ { "cell_type": "code", "execution_count": null, + "id": "438232b9", "metadata": {}, "outputs": [], "source": [ @@ -846,6 +910,7 @@ { "cell_type": "code", "execution_count": null, + "id": "69630a44", "metadata": {}, "outputs": [], "source": [ @@ -882,6 +947,7 @@ { "cell_type": "code", "execution_count": null, + "id": "239364cb", "metadata": {}, "outputs": [], "source": [ @@ -919,6 +985,7 @@ { "cell_type": "code", "execution_count": null, + "id": "9e20803b", "metadata": {}, "outputs": [], "source": [ @@ -930,6 +997,7 @@ { "cell_type": "code", "execution_count": null, + "id": "259eb6eb", "metadata": {}, "outputs": [], "source": [ @@ -939,6 +1007,7 @@ { "cell_type": "code", "execution_count": null, + "id": "87a4af31", "metadata": {}, "outputs": [], "source": [ @@ -955,6 +1024,7 @@ }, { "cell_type": "markdown", + "id": "f80f1c8c", "metadata": {}, "source": [ "## Summary of Results" @@ -962,6 +1032,7 @@ }, { "cell_type": "markdown", + "id": "1e427295", "metadata": {}, "source": [ "To sum up, let's merge all our results so far and illustrate them in a plot. " @@ -970,6 +1041,7 @@ { "cell_type": "code", "execution_count": null, + "id": "75c1862b", "metadata": {}, "outputs": [], "source": [ @@ -981,6 +1053,7 @@ { "cell_type": "code", "execution_count": null, + "id": "dfb07885", "metadata": { "tags": [ "nbsphinx-thumbnail" @@ -1007,6 +1080,7 @@ }, { "cell_type": "markdown", + "id": "4974df69", "metadata": {}, "source": [ "We report results based on four ML methods for estimating the nuisance functions used in\n", @@ -1016,6 +1090,7 @@ }, { "cell_type": "markdown", + "id": "72e515ad", "metadata": {}, "source": [ "______\n", @@ -1029,7 +1104,7 @@ ], "metadata": { "kernelspec": { - "display_name": "Python 3", + "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, @@ -1043,7 +1118,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.7.4" + "version": "3.9.7" } }, "nbformat": 4, diff --git a/doc/examples/r_double_ml_multiway_cluster.ipynb b/doc/examples/r_double_ml_multiway_cluster.ipynb index 0e6cd33b..efd1fdff 100644 --- a/doc/examples/r_double_ml_multiway_cluster.ipynb +++ b/doc/examples/r_double_ml_multiway_cluster.ipynb @@ -175,14 +175,14 @@ "metadata": {}, "outputs": [], "source": [ - "# Set machine learning methods for m, g & r\n", - "ml_g = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", + "# Set machine learning methods for l, m & r\n", + "ml_l = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "ml_m = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "ml_r = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "\n", "# initialize the DoubleMLPLIV object\n", "dml_pliv_obj = DoubleMLPLIV$new(obj_dml_data,\n", - " ml_g, ml_m, ml_r,\n", + " ml_l, ml_m, ml_r,\n", " n_folds=3)" ] }, @@ -451,14 +451,14 @@ "metadata": {}, "outputs": [], "source": [ - "# Set machine learning methods for m, g & r\n", - "ml_g = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", + "# Set machine learning methods for l, m & r\n", + "ml_l = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "ml_m = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "ml_r = lrn(\"regr.cv_glmnet\", nfolds = 10, s = \"lambda.min\")\n", "\n", "# initialize the DoubleMLPLIV object\n", "dml_pliv_obj = DoubleMLPLIV$new(obj_dml_data,\n", - " ml_g, ml_m, ml_r,\n", + " ml_l, ml_m, ml_r,\n", " n_folds=3)" ] }, diff --git a/doc/guide/algorithms.rst b/doc/guide/algorithms.rst index 66c31343..9aec6944 100644 --- a/doc/guide/algorithms.rst +++ b/doc/guide/algorithms.rst @@ -77,11 +77,11 @@ The DML algorithm can be selected via parameter ``dml_procedure='dml1'`` vs. ``d np.random.seed(3141) learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd') - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, dml_procedure='dml1') + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, dml_procedure='dml1') dml_plr_obj.fit(); .. tabbed:: R @@ -95,12 +95,12 @@ The DML algorithm can be selected via parameter ``dml_procedure='dml1'`` vs. ``d lgr::get_logger("mlr3")$set_threshold("warn") learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(3141) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='data.table') obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, dml_procedure="dml1") + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, dml_procedure="dml1") dml_plr_obj$fit() diff --git a/doc/guide/basics.rst b/doc/guide/basics.rst index 86ead55e..c3428e2e 100644 --- a/doc/guide/basics.rst +++ b/doc/guide/basics.rst @@ -104,7 +104,7 @@ other half of observations indexed with :math:`i \in I` .. ipython:: python - def non_orth_score(y, d, l_hat, g_hat, m_hat, smpls): + def non_orth_score(y, d, l_hat, m_hat, g_hat, smpls): u_hat = y - g_hat psi_a = -np.multiply(d, d) psi_b = np.multiply(d, u_hat) @@ -126,12 +126,13 @@ other half of observations indexed with :math:`i \in I` np.random.seed(1111) + ml_l = RandomForestRegressor(n_estimators=132, max_features=12, max_depth=5, min_samples_leaf=1) ml_m = RandomForestRegressor(n_estimators=378, max_features=20, max_depth=3, min_samples_leaf=6) - ml_g = RandomForestRegressor(n_estimators=132, max_features=12, max_depth=5, min_samples_leaf=1) + ml_g = clone(ml_l) # to speed up the illustration we hard-code the simulation results - theta_nonorth = np.array([0.52632753, 0.54450091, 0.66923726, 0.51260459, 0.58655021, 0.42238338, 0.58444181, 0.56730461, 0.618949, 0.63081597, 0.65708607, 0.59712711, 0.53621345, 0.48472368, 0.51958944, 0.53357288, 0.47367291, 0.55942659, 0.47858609, 0.53337773, 0.49257504, 0.46272514, 0.45475363, 0.43459792, 0.69738992, 0.52716308, 0.56226751, 0.44792 , 0.59667564, 0.50763664, 0.4211271 , 0.5733609 , 0.57058201, 0.47007786, 0.55206143, 0.4855682 , 0.47470065, 0.64921184, 0.63133495, 0.58573622, 0.50000705, 0.5727289 , 0.54354782, 0.52841084, 0.52100635, 0.4416036 , 0.48276559, 0.43098207, 0.55875633, 0.68805323, 0.44679974, 0.56261894, 0.57045581, 0.4629181 , 0.6598061 , 0.49830001, 0.52387458, 0.66880133, 0.49474535, 0.57703589, 0.57076417, 0.58603864, 0.55300801, 0.70480402, 0.62472621, 0.49046298, 0.43311798, 0.52923917, 0.6278388 , 0.54277942, 0.45499903, 0.52505168, 0.53237215, 0.55585595, 0.51979766, 0.67231976, 0.55109149, 0.54185552, 0.60075895, 0.54760708, 0.61331265, 0.55037039, 0.5492163 , 0.53902678, 0.41222205, 0.56753453, 0.62997758, 0.57875585, 0.43691693, 0.52066573, 0.47341129, 0.42808255, 0.62071727, 0.50573262, 0.50491432, 0.5580426 , 0.48632197, 0.67055925, 0.53517875, 0.55764907, 0.43080273, 0.58900493, 0.5739211 , 0.63829177, 0.49466558, 0.58512621, 0.55540686, 0.57947496, 0.55240912, 0.45890565, 0.46503817, 0.56075362, 0.43276416, 0.44882857, 0.58238181, 0.5422191 , 0.59003501, 0.496084, 0.56555097, 0.45410121, 0.53870251, 0.64414345, 0.48515644, 0.57521233, 0.46612704, 0.59867503, 0.34863075, 0.43467394, 0.47755725, 0.51982982, 0.49371777, 0.42465747, 0.51187812, 0.50060694, 0.57818785, 0.60258567, 0.65690397, 0.57236611, 0.41869572, 0.47328346, 0.47396537, 0.61894583, 0.50720406, 0.50554704, 0.59806778, 0.52087832, 0.50811361, 0.60012032, 0.53580639, 0.57698226, 0.57092276, 0.49381812, 0.4817072 , 0.49575484, 0.6381777 , 0.61924486, 0.51051105, 0.51368307, 0.44533041, 0.65449358, 0.61141686, 0.45878561, 0.57386337, 0.54739748, 0.50191105, 0.47272935, 0.60296688, 0.56446354, 0.56992489, 0.58921128, 0.52656325, 0.57475984, 0.54327087, 0.52806825, 0.50209864, 0.59104694, 0.46534661, 0.60049126, 0.47334002, 0.52107576, 0.49650274, 0.60740393, 0.59561065, 0.6259308 , 0.34885504, 0.54012387, 0.52770318, 0.55656017, 0.56409892, 0.59827133, 0.49318565, 0.43152159, 0.5156532 , 0.44843868, 0.34185747, 0.52355933, 0.53720761, 0.58116624, 0.57298891, 0.53827502, 0.49953615, 0.48923344, 0.45780343, 0.50543162, 0.54253571, 0.49705922, 0.53491146, 0.52942571, 0.60448499, 0.56464437, 0.44029078, 0.62986432, 0.33214992, 0.4920111 , 0.50365887, 0.56994537, 0.43284289, 0.53052547, 0.45267096, 0.63543567, 0.61143029, 0.58819308, 0.57060566, 0.47072283, 0.57371634, 0.5192611 , 0.54130173, 0.55287277, 0.56783623, 0.57663406, 0.6423188 , 0.60714003, 0.70612328, 0.48594343, 0.56766244, 0.55881674, 0.57292063, 0.40972752, 0.58449475, 0.5660997 , 0.50548568, 0.54001286, 0.52848374, 0.708655, 0.44006166, 0.43420695, 0.5402238 , 0.53604674, 0.56137211, 0.37496873, 0.44549783, 0.4772921 , 0.70858849, 0.6307189 , 0.44788457, 0.44997127, 0.39395763, 0.48406162, 0.51426745, 0.47288861, 0.58690519, 0.50754748, 0.45544924, 0.48199324, 0.37156346, 0.56674501, 0.64549723, 0.69830128, 0.43781527, 0.48179827, 0.49895762, 0.38793322, 0.47032088, 0.59761263, 0.64386114, 0.52746165, 0.60044875, 0.53392011, 0.58111242, 0.52622991, 0.54814684, 0.5521204 , 0.58713028, 0.56390526, 0.61274894, 0.55138326, 0.61017419, 0.48809668, 0.66523225, 0.50945431, 0.49389696, 0.55049743, 0.49029285, 0.50489009, 0.40630758, 0.56436192, 0.52434554, 0.51204277, 0.60070544, 0.4551411 , 0.47651324, 0.53960113, 0.56274511, 0.41342879, 0.3943613 , 0.46394834, 0.53024398, 0.51708571, 0.46525865, 0.53194977, 0.53849591, 0.49994117, 0.47581841, 0.61572919, 0.59611205, 0.52682696, 0.35538809, 0.5381261 , 0.49326596, 0.57011271, 0.58507884, 0.46211342, 0.59720598, 0.59279047, 0.58597674, 0.43145786, 0.51956071, 0.52837217, 0.55362367, 0.53400974, 0.47433081, 0.59293559, 0.46431113, 0.52380309, 0.57597298, 0.57541909, 0.67103911, 0.34278285, 0.56058506, 0.55053381, 0.40513557, 0.42699476, 0.52433699, 0.42203565, 0.43130786, 0.66683369, 0.5788322 , 0.4963422 , 0.49074006, 0.50838229, 0.45458122, 0.40702704, 0.54415849, 0.53812718, 0.61826589, 0.53250998, 0.49475644, 0.51880996, 0.46462693, 0.53718112, 0.55789896, 0.4572382 , 0.61511475, 0.52884488, 0.52028976, 0.56701838, 0.55976025, 0.52125038, 0.61376745, 0.50960714, 0.37932304, 0.63119581, 0.47848638, 0.48856083, 0.59276446, 0.52629694, 0.59530849, 0.4589804 , 0.67763076, 0.31163418, 0.52663268, 0.48316642, 0.50314571, 0.47467997, 0.63180293, 0.42088802, 0.60379662, 0.60904164, 0.51916585, 0.52979153, 0.51979617, 0.59203143, 0.53060133, 0.56879838, 0.49777707, 0.4012822 , 0.58738654, 0.68655488, 0.56704527, 0.5249324 , 0.44152022, 0.57273226, 0.4484671 , 0.53928393, 0.58938908, 0.60041486, 0.55523917, 0.56018467, 0.50105403, 0.51292508, 0.61286158, 0.47776243, 0.52579671, 0.47774149, 0.49451569, 0.46779312, 0.66910488, 0.54441758, 0.4703113 , 0.47178858, 0.65920731, 0.53029139, 0.50764744, 0.49018353, 0.6244943 , 0.53135466, 0.60813009, 0.5388036 , 0.65571997, 0.54466252, 0.64967203, 0.39415302, 0.62414825, 0.60109365, 0.35144364, 0.52220789, 0.5430445 , 0.65287338, 0.53432191, 0.53849406, 0.47408176, 0.57869455, 0.46803112, 0.50697263, 0.55646388, 0.3395878 , 0.48827513, 0.45764797, 0.3905749 , 0.44793202, 0.51525046, 0.53616866, 0.71459544, 0.4754264 , 0.52867616, 0.53437556, 0.59562702, 0.47433736, 0.48383682, 0.68653325, 0.59220446, 0.61687595, 0.46722598, 0.40938717, 0.53999185, 0.41240449, 0.54669903, 0.51821685, 0.39399627, 0.58578038, 0.37025986, 0.48597623, 0.46245557, 0.51143643, 0.47482417, 0.55021875, 0.5706816 , 0.44952284, 0.48579549, 0.55855102, 0.61685545, 0.4316845 , 0.64470136, 0.49930095, 0.46086649, 0.3945979 , 0.54367473, 0.62793152, 0.66439537, 0.60495807, 0.53939032, 0.50123699, 0.48331607, 0.46684808, 0.52089604, 0.51762505, 0.68117756, 0.53410128, 0.4922944 , 0.57508586, 0.47760706, 0.48714391, 0.27885854, 0.48316078, 0.53550691, 0.42021222, 0.42621436, 0.58216478, 0.6434657 , 0.54903173, 0.50882582, 0.53581116, 0.60967509, 0.72811734, 0.53300244, 0.46429436, 0.4682603 , 0.56725131, 0.55938566, 0.47634217, 0.51180058, 0.54440822, 0.49723253, 0.5575793 , 0.49678717, 0.61634438, 0.67425007, 0.54475491, 0.45082149, 0.41768286, 0.58304489, 0.38103786, 0.5086868 , 0.56691891, 0.48175355, 0.60321764, 0.5309256 , 0.60410943, 0.52265228, 0.54921681, 0.57107727, 0.56013254, 0.59456898, 0.47966653, 0.46689563, 0.4431922 , 0.61271605, 0.48510123, 0.56416119, 0.61532589, 0.48616878, 0.55060424, 0.68486338, 0.57923604, 0.51695259, 0.42204912, 0.56573861, 0.49153633, 0.48935159, 0.47706048, 0.43899806, 0.54185621, 0.52202746, 0.56394984, 0.59830286, 0.40655588, 0.6540647 , 0.4870229 , 0.55846382, 0.60565034, 0.5072831 , 0.54395348, 0.40379622, 0.46622254, 0.6354662 , 0.42312813, 0.4616702 , 0.53952288, 0.4644455 , 0.53320828, 0.47985077, 0.67884889, 0.68665247, 0.55675144, 0.49726657, 0.51153323, 0.35450556, 0.38406104, 0.54364234, 0.54481311, 0.50429127, 0.51665527, 0.50536631, 0.62471313, 0.54995188, 0.51069711, 0.59585771, 0.49361382, 0.56525314, 0.53472606, 0.52108574, 0.59277627, 0.52030413, 0.53303672, 0.506324, 0.58464649, 0.51682546, 0.53908917, 0.48472873, 0.59931961, 0.52987924, 0.51830952, 0.5981733 , 0.52570183, 0.46157526, 0.33681896, 0.52118572, 0.55196885, 0.436666, 0.60214705, 0.47801228, 0.63529308, 0.66420412, 0.47089482, 0.50299033, 0.54153378, 0.50012678, 0.55512458, 0.61867588, 0.61297676, 0.51619218, 0.7108226 , 0.52405154, 0.50545777, 0.58881996, 0.6855725 , 0.44158941, 0.56506293, 0.57571301, 0.46155939, 0.55610752, 0.63709365, 0.53899487, 0.48867157, 0.57918563, 0.45980471, 0.56143977, 0.37378031, 0.63367835, 0.38939997, 0.55374046, 0.55710308, 0.33720147, 0.49855745, 0.54127316, 0.55849223, 0.42419311, 0.54393195, 0.41161067, 0.67767974, 0.41019723, 0.58881843, 0.61116325, 0.50547189, 0.54250286, 0.61225748, 0.44507568, 0.58904454, 0.53349416, 0.55814679, 0.58270325, 0.52712488, 0.56809314, 0.42769283, 0.58359583, 0.46196793, 0.48111024, 0.45197845, 0.43420935, 0.56175895, 0.53589292, 0.52315139, 0.50566554, 0.62762057, 0.56084874, 0.63904796, 0.5943957 , 0.53875507, 0.46986491, 0.6426977 , 0.49275138, 0.57931751, 0.70266163, 0.47655526, 0.62173036, 0.48717578, 0.59845574, 0.60794702, 0.42043813, 0.51655123, 0.4353051 , 0.47904624, 0.57368244, 0.5999047 , 0.52617388, 0.46936766, 0.45231895, 0.52393147, 0.43778395, 0.47623361, 0.47047228, 0.54637139, 0.53675411, 0.58225605, 0.44181779, 0.5102228 , 0.53705222, 0.65732853, 0.62801308, 0.38570668, 0.44181883, 0.65367556, 0.66954032, 0.53737963, 0.51781873, 0.55671749, 0.55356872, 0.49045887, 0.50322435, 0.58473624, 0.64025277, 0.50574165, 0.61981544, 0.61961095, 0.54121279, 0.47724547, 0.44325164, 0.53215914, 0.53575759, 0.50360814, 0.52205319, 0.60251245, 0.45673497, 0.54048138, 0.533345, 0.45471002, 0.54831181, 0.51394181, 0.46073889, 0.48973606, 0.51447332, 0.58756885, 0.45003506, 0.52646263, 0.58968246, 0.57353131, 0.55404332, 0.67903712, 0.43332497, 0.46782929, 0.55690244, 0.54962445, 0.37052726, 0.53694322, 0.57987237, 0.55581497, 0.58648345, 0.57742465, 0.61295249, 0.47082473, 0.49805351, 0.62943103, 0.4802072 , 0.61564795, 0.48670492, 0.55891522, 0.51289051, 0.46746993, 0.46179283, 0.52611937, 0.47259637, 0.46293598, 0.51716995, 0.43636162, 0.62709106, 0.62078752, 0.57865712, 0.57993246, 0.49595949, 0.5951574 , 0.60691815, 0.51491792, 0.5696144 , 0.4335185 , 0.61645309, 0.66783008, 0.41726308, 0.52357785, 0.57782111, 0.38525498, 0.40030394, 0.44383597, 0.48282351, 0.55315702, 0.52740055, 0.54191889, 0.58805125, 0.50005222, 0.62269219, 0.56192058, 0.69720681, 0.6420251 , 0.57585553, 0.65266588, 0.56394678, 0.55881803, 0.58853661, 0.62358828, 0.58900213, 0.55180601, 0.61334419, 0.47432699, 0.69636555, 0.47580728, 0.55062803, 0.5751468 , 0.40882965, 0.54733972, 0.53255764, 0.49895855, 0.5789291 , 0.47475823, 0.51362334, 0.50007997, 0.62200651, 0.65925014, 0.51606822, 0.49451512, 0.52548882, 0.41514725, 0.48751079, 0.61703613, 0.65643943, 0.62289755, 0.53872061, 0.54484886, 0.57697036, 0.59413314, 0.56346526, 0.53203294, 0.40909514, 0.50064258, 0.52035007, 0.5012625 , 0.48578658, 0.55824125, 0.36505325, 0.55176115, 0.41277624, 0.54335046, 0.58947315, 0.48733078, 0.63269652, 0.45560179, 0.67663234, 0.62937034, 0.53462866, 0.59090083, 0.41680354, 0.49839325, 0.47130792, 0.64484127, 0.67686348, 0.50108123, 0.48438192, 0.38428532, 0.43414261, 0.65013365, 0.5098531 , 0.60004169, 0.5697646 , 0.63651472, 0.50115709, 0.50382518, 0.53465802, 0.54041583, 0.59443663, 0.68394535, 0.43136825, 0.52093863, 0.492323, 0.60204015, 0.58571954, 0.48222725, 0.39901315, 0.50874368, 0.54478095, 0.41794544, 0.61308008, 0.56279119, 0.44958539, 0.53686155, 0.51535318, 0.4157253 , 0.52779584, 0.4433108 , 0.64116852, 0.41727246, 0.52716229, 0.66511726, 0.51072705, 0.4799285 , 0.4626921 , 0.43598565, 0.6236199 , 0.58418263, 0.54486924, 0.50675202, 0.4970176 , 0.61610926, 0.63488907, 0.63780451, 0.43754633, 0.44503914, 0.57447591, 0.56007183, 0.44319286, 0.51329395, 0.49871485, 0.47726431, 0.50767185, 0.49482428, 0.60439789, 0.4615146 , 0.53859255, 0.62038444, 0.64913913, 0.49418167, 0.60124017, 0.47822072, 0.49744256, 0.58586246, 0.50871457, 0.46203378, 0.56520742, 0.53317718, 0.55889281, 0.38172112, 0.45594551, 0.60580421, 0.52110075, 0.51444763, 0.52045491, 0.47283489, 0.34755781, 0.49803659, 0.48277783, 0.56328869, 0.45957583, 0.51915878, 0.60958301, 0.53878142, 0.50927979, 0.48057417, 0.44326046, 0.63453822, 0.62161229, 0.54641439, 0.54541847, 0.47026904, 0.60681138, 0.4997663 , 0.35607407, 0.55529754, 0.46402998, 0.3376591 , 0.51921144, 0.41938588, 0.548278, 0.51927345, 0.51137058, 0.66853675, 0.42879294, 0.52635556, 0.57270967, 0.41102776, 0.63381592, 0.47499351, 0.62242358, 0.54527768, 0.54904066, 0.52769862, 0.50747416, 0.62760339, 0.52289013, 0.37655823, 0.60151613, 0.63345199, 0.533801, 0.58577084]) - se_nonorth = np.array([0.04752632, 0.05252437, 0.04081372, 0.04851484, 0.04749561, 0.03818379, 0.04720593, 0.04623519, 0.04438108, 0.04704747, 0.04582967, 0.04103284, 0.04095206, 0.04473644, 0.0474447 , 0.04428884, 0.05291583, 0.04548479, 0.04127787, 0.04264639, 0.04136002, 0.04768211, 0.05077625, 0.0432164 , 0.04778945, 0.03571416, 0.0475896 , 0.04516676, 0.04991276, 0.05196731, 0.04048258, 0.03857071, 0.03671693, 0.03770704, 0.05089652, 0.0461722 , 0.03836501, 0.04226271, 0.05511125, 0.04450304, 0.0418925 , 0.03910182, 0.0449247 , 0.04629896, 0.04367623, 0.04376256, 0.04025187, 0.04698785, 0.0388972 , 0.05079093, 0.04002389, 0.04597267, 0.04720004, 0.0450238 , 0.0398305 , 0.04065346, 0.04406542, 0.04115013, 0.04865718, 0.04056625, 0.03925187, 0.0421328 , 0.04681513, 0.0439763 , 0.05035432, 0.04201041, 0.04668544, 0.04394981, 0.04567135, 0.04550151, 0.03709427, 0.04533645, 0.04038179, 0.04737091, 0.04547688, 0.04493072, 0.04029656, 0.0453583 , 0.04700267, 0.04726174, 0.04940269, 0.04655482, 0.04319775, 0.04573236, 0.04239855, 0.04643579, 0.0377732 , 0.04382505, 0.05463616, 0.03968628, 0.04575842, 0.04779906, 0.04300628, 0.05252161, 0.04393881, 0.04544886, 0.04236938, 0.03688514, 0.04746812, 0.0378483 , 0.05071514, 0.04644373, 0.05289415, 0.04719282, 0.04062532, 0.03272569, 0.04314814, 0.04073433, 0.04090295, 0.04302725, 0.05074448, 0.03953333, 0.04706719, 0.04636343, 0.04401789, 0.05329037, 0.04354488, 0.04171928, 0.04274639, 0.04327196, 0.05557301, 0.04892522, 0.04641841, 0.04540333, 0.03970085, 0.04825305, 0.04346595, 0.04280488, 0.04588877, 0.05058765, 0.04157508, 0.04732415, 0.04712535, 0.03734686, 0.04341058, 0.04598636, 0.05144396, 0.04306507, 0.04984114, 0.04101317, 0.04112485, 0.04503048, 0.03760539, 0.04674369, 0.04421478, 0.04468265, 0.03989578, 0.04154939, 0.04059497, 0.04524515, 0.04567546, 0.04919117, 0.04402552, 0.04256382, 0.04039857, 0.04127004, 0.04250842, 0.04436529, 0.04374075, 0.04671761, 0.04513487, 0.04592405, 0.04222837, 0.04096571, 0.04748011, 0.04810947, 0.04291903, 0.04158746, 0.05036586, 0.04856613, 0.04024124, 0.04255147, 0.05180867, 0.04174598, 0.03891627, 0.04144685, 0.04489533, 0.044931 , 0.04161421, 0.04375287, 0.03767374, 0.04130609, 0.04236881, 0.04844647, 0.05030521, 0.03866574, 0.03888635, 0.04925452, 0.04878495, 0.04701012, 0.06132132, 0.05171138, 0.05270967, 0.05174661, 0.04853573, 0.04839576, 0.04838823, 0.04515296, 0.04383871, 0.04472405, 0.04274227, 0.04431837, 0.05238373, 0.03970582, 0.05159934, 0.04642126, 0.04256806, 0.05103923, 0.04887934, 0.04601176, 0.0462473 , 0.04795098, 0.04403834, 0.0425341 , 0.04602572, 0.03943239, 0.04502544, 0.04592906, 0.04898315, 0.04152593, 0.04151051, 0.04045435, 0.04717776, 0.03928857, 0.04215977, 0.04481115, 0.04533248, 0.04506638, 0.03780037, 0.04497642, 0.03946211, 0.03870278, 0.04645436, 0.04390321, 0.0429915 , 0.03877647, 0.04317605, 0.04447082, 0.04380389, 0.0429053 , 0.04753429, 0.04872603, 0.04279851, 0.04675329, 0.04925358, 0.04557681, 0.04613846, 0.04374635, 0.04513399, 0.044832 , 0.04005348, 0.04870915, 0.04604291, 0.04508179, 0.04285389, 0.04576191, 0.04165158, 0.04003255, 0.04034477, 0.04507987, 0.04382144, 0.03962126, 0.04643146, 0.04497413, 0.03977343, 0.04236554, 0.0476213 , 0.04659588, 0.04008921, 0.04770813, 0.04437344, 0.04840868, 0.04700976, 0.04226155, 0.03719109, 0.0449964 , 0.04613542, 0.04166904, 0.04180454, 0.03868801, 0.04794871, 0.050236 , 0.038375 , 0.04443406, 0.04488851, 0.0480011 , 0.04926898, 0.04058049, 0.04281985, 0.05338039, 0.04353161, 0.03515195, 0.04341719, 0.0480715 , 0.04142801, 0.04610912, 0.04653238, 0.04254146, 0.04605789, 0.041817 , 0.04159624, 0.04436698, 0.04346619, 0.04216123, 0.04861165, 0.03987033, 0.03852878, 0.05134763, 0.04642145, 0.05162566, 0.05261917, 0.04179967, 0.04888241, 0.05090512, 0.04474789, 0.04407596, 0.05172393, 0.04751141, 0.03935585, 0.04963206, 0.03907181, 0.04180968, 0.0409687 , 0.04472983, 0.04861775, 0.0436817 , 0.04685306, 0.04050731, 0.04378468, 0.04514822, 0.04459432, 0.04066988, 0.05043818, 0.0472485 , 0.0411004 , 0.05049794, 0.04683268, 0.0513245 , 0.04593956, 0.03999829, 0.04140311, 0.0450592 , 0.04604977, 0.04726002, 0.04946856, 0.04769101, 0.04216129, 0.04838215, 0.04368365, 0.04347249, 0.0438762 , 0.0407478 , 0.05192566, 0.04615664, 0.04734136, 0.04603765, 0.04313318, 0.04724211, 0.04168363, 0.04094983, 0.0372976 , 0.04829181, 0.0446341 , 0.04769589, 0.04983229, 0.04015445, 0.04259574, 0.03966965, 0.04393717, 0.03807938, 0.04714134, 0.04191703, 0.03853675, 0.04181588, 0.04050847, 0.05163033, 0.04446326, 0.04501493, 0.04730852, 0.04157382, 0.03881549, 0.04233757, 0.04605436, 0.04617815, 0.05141122, 0.04477161, 0.05493223, 0.03734768, 0.04321097, 0.04090805, 0.05053544, 0.04355217, 0.04057866, 0.0399262 , 0.04216453, 0.05066079, 0.04253812, 0.05315858, 0.04169033, 0.04204156, 0.04193905, 0.04167116, 0.05251147, 0.05098957, 0.04557835, 0.0469838 , 0.04635001, 0.05180438, 0.06008725, 0.05141524, 0.04317844, 0.04091926, 0.04801724, 0.04130072, 0.04178246, 0.045185 , 0.04053058, 0.04737165, 0.0458431 , 0.0424302 , 0.04922552, 0.03756568, 0.04577134, 0.04048251, 0.044854 , 0.04179815, 0.04468901, 0.0570919 , 0.04727883, 0.04391833, 0.04408565, 0.03926577, 0.04921733, 0.0518598 , 0.04868963, 0.04498284, 0.04055751, 0.04909337, 0.04020206, 0.04224491, 0.04949091, 0.04230575, 0.04603067, 0.04389281, 0.04015306, 0.04804292, 0.04533735, 0.04336376, 0.04668769, 0.04377271, 0.04340193, 0.04902237, 0.05078105, 0.04378715, 0.04524027, 0.05113331, 0.0434951 , 0.04721075, 0.04122844, 0.03932327, 0.04374526, 0.04669035, 0.04187597, 0.04693448, 0.0426543 , 0.04738004, 0.04837465, 0.03902091, 0.04042924, 0.04449191, 0.04539387, 0.0526423 , 0.04508715, 0.0448976 , 0.04888229, 0.04107251, 0.0466375 , 0.04372284, 0.04710068, 0.05160466, 0.04580307, 0.04740671, 0.04718352, 0.05165969, 0.05177518, 0.04273175, 0.04386661, 0.04451535, 0.05204564, 0.04576443, 0.04588544, 0.04735161, 0.04342469, 0.04442042, 0.04698244, 0.04611934, 0.04691028, 0.0409842 , 0.03810167, 0.04575679, 0.04909961, 0.0454315 , 0.0427315 , 0.04633341, 0.04573215, 0.03881894, 0.0470943 , 0.04260926, 0.0464628 , 0.05029253, 0.04798311, 0.04606002, 0.04309083, 0.04775266, 0.05008584, 0.04025573, 0.0430455 , 0.04014136, 0.04486536, 0.04627781, 0.04798719, 0.03881474, 0.05517719, 0.04318348, 0.04514346, 0.04135471, 0.04287774, 0.04317367, 0.04828672, 0.03960944, 0.05080501, 0.04688066, 0.03913251, 0.05097918, 0.04619538, 0.04290995, 0.04692129, 0.04394257, 0.04717779, 0.04703665, 0.04442 , 0.04803709, 0.05008089, 0.0492336 , 0.04947701, 0.04524332, 0.04498547, 0.0481187 , 0.05048569, 0.04471597, 0.03972998, 0.04268966, 0.05178417, 0.04283741, 0.03857392, 0.04135066, 0.0456479 , 0.04507646, 0.04263503, 0.05813869, 0.04155548, 0.04791942, 0.04709474, 0.0460457 , 0.04312264, 0.05437283, 0.04088724, 0.05058245, 0.04867745, 0.04425616, 0.04100133, 0.03590292, 0.0444132 , 0.03899607, 0.04688414, 0.03770601, 0.04247705, 0.05147905, 0.04803248, 0.04426342, 0.05230619, 0.04657694, 0.04252559, 0.04361401, 0.05345193, 0.04602656, 0.04810443, 0.04183419, 0.04012391, 0.0407237 , 0.05076034, 0.04640053, 0.0449217 , 0.04254334, 0.0488389 , 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0.04556888, 0.0468324 , 0.05077517, 0.04482054, 0.04279588, 0.04729371, 0.05248564, 0.04496889, 0.04524623, 0.0391199 , 0.04279905, 0.03831627, 0.04998856, 0.04125977, 0.04932733, 0.0410571 , 0.04878879, 0.04462988, 0.04413883, 0.05073432, 0.04503141, 0.04450833, 0.0489624 , 0.04622872, 0.04575345, 0.04430143, 0.05610003, 0.04686195, 0.04166272, 0.03959071, 0.03956318, 0.04595394, 0.04153185, 0.04176013, 0.04323898, 0.04279855, 0.04505489, 0.04003259, 0.04134781, 0.04285591, 0.04886803, 0.0543866 , 0.04069868, 0.04278918, 0.04668385, 0.04295076, 0.04597368, 0.04101972, 0.04297835, 0.03651599, 0.0462089 , 0.04459918, 0.04383747, 0.05220179, 0.05259663, 0.04050992, 0.04628942, 0.05149538, 0.04670411, 0.04981779, 0.04639563, 0.049215 , 0.04375691, 0.03838808, 0.04651726, 0.04386707, 0.04959572, 0.04719161, 0.04231231, 0.04785895, 0.03947752, 0.04925271, 0.04616122, 0.03791607, 0.04701615, 0.04272888, 0.05732519, 0.04058686, 0.04179869, 0.05142427, 0.0407993 , 0.04770618, 0.04845589, 0.04515179, 0.0399421 , 0.0493371 , 0.04494348, 0.04021067, 0.04180386, 0.0474344 , 0.04993469, 0.04580776, 0.04513645, 0.04563901, 0.04458455, 0.04670768, 0.0463566 , 0.05037703, 0.04200332, 0.04363388, 0.04013193, 0.03818682, 0.04708778, 0.05157659, 0.0438879 , 0.05193443, 0.04417957, 0.0443017 , 0.04444654, 0.04645802, 0.04208909, 0.04790682, 0.04689531, 0.04812119, 0.04436477, 0.04683151, 0.04755189, 0.0445168 , 0.04429936, 0.04130354, 0.04965516, 0.04633012, 0.04019696, 0.04630375, 0.0372558 , 0.04502071, 0.04541922, 0.04117667, 0.04044187, 0.0503596 , 0.04525318, 0.04296958, 0.04744989, 0.04244503, 0.04556984, 0.04867774, 0.03669044, 0.04222442, 0.04536255, 0.0436591 , 0.04574726, 0.05396777, 0.04394745, 0.0479081 , 0.04133669, 0.04493979, 0.04831306, 0.04299433, 0.04304778, 0.04357632, 0.05194496, 0.04887002, 0.04466858, 0.0503679 , 0.05755543, 0.04690671]) + theta_nonorth = np.array([0.52328641, 0.57356639, 0.55810707, 0.5977535 , 0.4813668 , 0.5327912 , 0.61704155, 0.54681571, 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0.41438599, 0.6162567 , 0.59125858, 0.53468529, 0.62485751, 0.5643462 , 0.43432088, 0.55723546, 0.55677339, 0.51964121, 0.59753128, 0.44999261, 0.54992742, 0.51795846, 0.47951257, 0.57153128, 0.57908496, 0.4626688 , 0.45531525, 0.44313973, 0.54603293, 0.57576854, 0.46361253, 0.527298 , 0.5620756 , 0.46704174, 0.44964006, 0.44489873, 0.59324251, 0.38614873, 0.7052925 , 0.54426546, 0.57456808, 0.44467559, 0.53451817, 0.46977995, 0.62371312, 0.44698664, 0.60374147, 0.47923972, 0.59966241, 0.45159563, 0.59206654, 0.54386941, 0.42658765, 0.53008371, 0.55862966, 0.4920328 , 0.56761111, 0.52064894, 0.39681885, 0.44574245, 0.65423631, 0.49700968, 0.52949852, 0.69074303, 0.47083257, 0.56112144, 0.61171242, 0.57201577, 0.55023754, 0.50197115, 0.54440172, 0.41029551, 0.55578038, 0.59709186, 0.50556125, 0.60242079, 0.54916367, 0.51934554, 0.65992837, 0.37152973, 0.61838874, 0.55014278, 0.5717791 , 0.48307382, 0.53725693, 0.56510068, 0.56899223, 0.47274061, 0.38206639, 0.57567921, 0.58460404, 0.69554675, 0.44126231, 0.5453808 , 0.55469192, 0.51558473, 0.45490972, 0.32829398, 0.57063905, 0.61816149, 0.4851176 , 0.62280177, 0.55124556, 0.54876465, 0.62502545, 0.53016613, 0.55607373, 0.42591779, 0.47576693, 0.54436353, 0.61658262, 0.72634035, 0.41672421, 0.52000937, 0.55465022, 0.52901531, 0.55834794, 0.58534343, 0.51694802, 0.43374908, 0.66757089, 0.47552899, 0.50901742, 0.61552002, 0.58569846, 0.49247126, 0.62729266, 0.45454611, 0.57492484, 0.36653027, 0.57255697, 0.57824612, 0.53146156, 0.53596233, 0.53638362, 0.46116763, 0.62344651, 0.52368548, 0.43118748, 0.39730289, 0.58060133, 0.51892256, 0.50985834, 0.52621487, 0.45318262, 0.42636468, 0.55077814, 0.51755157, 0.50558189, 0.48513592, 0.60846418, 0.56548573, 0.63888894, 0.64851154, 0.54867037, 0.46621769, 0.51043987, 0.72938861, 0.56584276, 0.56401497, 0.45178682, 0.73033294, 0.46287789, 0.45124177, 0.50136769, 0.3687888 , 0.55620501, 0.56414095, 0.50050497, 0.51400812, 0.51195696, 0.49870403, 0.54124953, 0.51817863, 0.58274345, 0.36739039, 0.6223105 , 0.49839431, 0.48262867, 0.60799173, 0.48497077, 0.50533284, 0.52068533, 0.51819816, 0.53736352, 0.43353299, 0.49234493, 0.6137744 , 0.50568935, 0.40132373, 0.5764364 , 0.48163201, 0.41512117, 0.47245353, 0.49395644, 0.58230514, 0.58230577, 0.50858737, 0.54478675, 0.49901037, 0.55797797, 0.51010655, 0.53948516, 0.55935642, 0.39606487, 0.59820544, 0.47212327, 0.62012946, 0.66593186, 0.58616071, 0.54776001, 0.4611425 , 0.62522599, 0.3766581 , 0.51283615, 0.44134182, 0.5182332 , 0.60188039, 0.4490443 , 0.46919351, 0.47447443, 0.49600183, 0.57066395, 0.52439289, 0.56334611, 0.56060475, 0.39914239, 0.3675798 , 0.51564721, 0.49865563, 0.50048977, 0.51907243, 0.48807419, 0.4837173 , 0.54264044, 0.56838568, 0.54979317, 0.51717151, 0.47623781, 0.45639949, 0.49258588, 0.59268146, 0.58045385, 0.573669 , 0.63253715, 0.42660048, 0.51028837, 0.52477234, 0.60977474, 0.48255809, 0.50034024, 0.58582749, 0.56229419, 0.55128857, 0.67506051, 0.50764811, 0.59438154, 0.42687047, 0.45088934, 0.56769317, 0.69933473, 0.64133968, 0.58965277, 0.52414937, 0.47795837, 0.51327079, 0.47880271, 0.51766668, 0.56479121, 0.42850267, 0.70736401, 0.36848061, 0.56501167, 0.4173219 , 0.59616084, 0.63402178, 0.59863361, 0.40275257, 0.61610026, 0.55072114, 0.54363882, 0.55507125, 0.40734622, 0.54603062, 0.48360112, 0.37964225, 0.49283026, 0.59018895, 0.59009867, 0.4508418 , 0.56922331, 0.61952373, 0.48797082, 0.53357994, 0.463265 , 0.51038265, 0.59259957, 0.40175384, 0.54355008, 0.48951247, 0.51671369, 0.48761615, 0.45277739, 0.60093516, 0.42003811, 0.51235 , 0.59074681, 0.50383085, 0.30816788, 0.58677263, 0.52795737, 0.55613922, 0.53170224, 0.30069634, 0.57035186, 0.57304229, 0.50054292, 0.47112149, 0.5553012 , 0.45457155, 0.59640452, 0.50753294, 0.60960469, 0.54302708, 0.45963418, 0.64439113, 0.40809175, 0.50408166, 0.57626158, 0.54048658, 0.5797436 , 0.58846209, 0.48535104, 0.47475958, 0.62056536, 0.55518476, 0.5876704 , 0.60219568, 0.61776265, 0.65848052, 0.5299857 , 0.51399049, 0.56802533, 0.509565 , 0.48821165, 0.4238927 , 0.57421096, 0.51616527, 0.65465229, 0.46116782, 0.33355847, 0.58515526, 0.55622069, 0.63010995, 0.49775216, 0.54856983, 0.57677015, 0.52546157, 0.53642151, 0.57683734, 0.51020746, 0.45510774, 0.42469847, 0.42367087, 0.38633576, 0.56000824, 0.51460849, 0.57506362, 0.49294988, 0.64173013, 0.45438165, 0.40507967, 0.58082999, 0.4938575 , 0.53151652, 0.61474763, 0.35312902, 0.53849575, 0.48628075, 0.59266678, 0.52805816, 0.52097269, 0.53184868, 0.63211204, 0.66178091, 0.58451712, 0.49163581, 0.61183509, 0.57385018, 0.56755006, 0.58717042, 0.47939109, 0.46031912, 0.41769586, 0.5317388 , 0.57788374, 0.57080485, 0.49382667, 0.63281403, 0.50398231, 0.60290742, 0.43699714, 0.49636521, 0.5053502 , 0.37261477, 0.48081627, 0.50270047, 0.66441706, 0.42627842, 0.53421282, 0.57808942, 0.44486317, 0.58220927, 0.57082553, 0.52144876, 0.5852739 , 0.60581384, 0.47197106, 0.58540418]) + se_nonorth = np.array([0.04831872, 0.04832813, 0.0440032 , 0.04114123, 0.04031513, 0.04924838, 0.04834419, 0.04662809, 0.04292674, 0.04538775, 0.04924342, 0.04409527, 0.0417032 , 0.05194798, 0.05030047, 0.0367301 , 0.05075158, 0.05003675, 0.04019231, 0.0408118 , 0.03840654, 0.05299583, 0.04520576, 0.04644108, 0.04959359, 0.03872031, 0.04183655, 0.05109119, 0.05050073, 0.04835535, 0.04488573, 0.0381961 , 0.04422992, 0.04346715, 0.05144883, 0.04486199, 0.04513496, 0.04031142, 0.04932358, 0.04871112, 0.0440838 , 0.03665062, 0.04599354, 0.04412655, 0.04248635, 0.04130969, 0.04234649, 0.05137093, 0.04609058, 0.05068716, 0.0416494 , 0.03860875, 0.04452859, 0.04575008, 0.03959878, 0.04713601, 0.04416249, 0.04282636, 0.04742508, 0.04341061, 0.04468596, 0.04133775, 0.05035144, 0.0429771 , 0.04869872, 0.04490033, 0.04602814, 0.04482479, 0.04350321, 0.0507471 , 0.04018135, 0.04230126, 0.03655288, 0.0468958 , 0.04503343, 0.04182587, 0.04241048, 0.04675773, 0.04176107, 0.04655241, 0.049157 , 0.04956828, 0.04646219, 0.04328973, 0.04411666, 0.04837818, 0.04206927, 0.04889302, 0.05010237, 0.04739132, 0.04636621, 0.04739799, 0.04309182, 0.04921978, 0.04310753, 0.04744372, 0.04151646, 0.04221348, 0.04493263, 0.03779441, 0.04688649, 0.03919651, 0.04590451, 0.05016068, 0.03819017, 0.04131799, 0.04337898, 0.0392395 , 0.04348177, 0.04323946, 0.05393595, 0.04038024, 0.04830559, 0.04124088, 0.05180402, 0.05567998, 0.04244997, 0.03911898, 0.04337131, 0.04624824, 0.04949055, 0.04985597, 0.04326196, 0.04212764, 0.04842716, 0.04561389, 0.04082098, 0.04448624, 0.0448946 , 0.05293593, 0.04388315, 0.04321804, 0.04614838, 0.04602541, 0.04429507, 0.04405809, 0.05071437, 0.05146057, 0.04623581, 0.048504 , 0.04321356, 0.04723778, 0.03439781, 0.04850456, 0.04551087, 0.04798238, 0.03911667, 0.03747808, 0.04200668, 0.04340506, 0.04620754, 0.05201033, 0.05035776, 0.0456115 , 0.04366328, 0.04285165, 0.04355332, 0.04149404, 0.04189363, 0.04271938, 0.043017 , 0.04975933, 0.04724617, 0.0478123 , 0.04583418, 0.05493124, 0.04423306, 0.04228908, 0.04539415, 0.04335721, 0.03785391, 0.04805826, 0.04607728, 0.04239644, 0.04415323, 0.04616612, 0.04793052, 0.05040471, 0.04690316, 0.04168859, 0.04188572, 0.04151215, 0.04243537, 0.04459925, 0.05220799, 0.04072488, 0.03941964, 0.05134895, 0.0484997 , 0.04823143, 0.05272294, 0.0518343 , 0.04815165, 0.04984051, 0.0464933 , 0.03990746, 0.05754139, 0.04443048, 0.03948512, 0.04694665, 0.04627509, 0.04513908, 0.05028515, 0.03946741, 0.04535004, 0.04954032, 0.04290176, 0.04387563, 0.05135577, 0.0504392 , 0.04766801, 0.05580917, 0.04468912, 0.04505744, 0.04120763, 0.04365832, 0.05030036, 0.04299801, 0.05215436, 0.04222343, 0.04385739, 0.04116473, 0.04945468, 0.04284718, 0.04603566, 0.0459052 , 0.04613431, 0.03829818, 0.04267634, 0.04598015, 0.04513768, 0.04448734, 0.05130906, 0.04129426, 0.04086385, 0.04564178, 0.04209469, 0.04482055, 0.04366908, 0.04894291, 0.05089028, 0.0456697 , 0.04560245, 0.04475189, 0.04815378, 0.04657151, 0.04587175, 0.03997987, 0.04141135, 0.04238376, 0.04364734, 0.04414603, 0.05422814, 0.05288109, 0.04474988, 0.04839057, 0.03951434, 0.04106973, 0.04353691, 0.04514385, 0.04276905, 0.04152265, 0.04575897, 0.05536061, 0.04523277, 0.04283036, 0.04593579, 0.04794558, 0.04065449, 0.04222009, 0.04894703, 0.04746394, 0.04702815, 0.04692336, 0.03422089, 0.04594671, 0.05291489, 0.04307596, 0.04955153, 0.04553489, 0.04901159, 0.04473089, 0.04589466, 0.0441002 , 0.046049 , 0.04348205, 0.05187483, 0.04383983, 0.04346503, 0.04739405, 0.04447819, 0.04102896, 0.0431398 , 0.04656729, 0.03896816, 0.0542891 , 0.04270792, 0.04289141, 0.05348902, 0.0495009 , 0.04643517, 0.04410614, 0.04200152, 0.04493118, 0.04806057, 0.04190507, 0.04127871, 0.04847289, 0.04842111, 0.04833609, 0.04889407, 0.05076527, 0.04744793, 0.05150318, 0.04888322, 0.04992893, 0.04023113, 0.03881747, 0.0386972 , 0.04694449, 0.03855655, 0.0457303 , 0.04371627, 0.04143969, 0.03938361, 0.04876415, 0.04742728, 0.04700492, 0.04762283, 0.0488795 , 0.04395567, 0.04441107, 0.05068248, 0.04547187, 0.04361415, 0.05412799, 0.04036971, 0.04535149, 0.04536353, 0.03678323, 0.04403829, 0.04812895, 0.0457191 , 0.04258916, 0.04280755, 0.05153953, 0.04413582, 0.04770393, 0.05038132, 0.03757377, 0.04973624, 0.04261597, 0.0456351 , 0.04472504, 0.04771097, 0.04373888, 0.04370413, 0.04552507, 0.04293194, 0.04761842, 0.04287779, 0.04733812, 0.04423761, 0.0448224 , 0.04636959, 0.04296324, 0.0477725 , 0.04653299, 0.04201292, 0.05075975, 0.05017969, 0.04138296, 0.04060744, 0.04932747, 0.0400757 , 0.05220853, 0.03915695, 0.04293621, 0.06503597, 0.03595801, 0.03622248, 0.04329143, 0.0445115 , 0.03907058, 0.04102573, 0.04438729, 0.04948462, 0.04521646, 0.04211678, 0.04829002, 0.05505099, 0.03515793, 0.04173115, 0.04445393, 0.05032834, 0.05006664, 0.04629111, 0.04772484, 0.04092481, 0.04156632, 0.04229721, 0.04327583, 0.04579568, 0.04716264, 0.04178555, 0.04428462, 0.04725546, 0.04493506, 0.05485039, 0.0442419 , 0.03824128, 0.048013 , 0.04400902, 0.03878737, 0.04157463, 0.03909924, 0.04262371, 0.04611657, 0.03955994, 0.04055998, 0.04706018, 0.04562143, 0.04247859, 0.03936683, 0.04726424, 0.04247808, 0.04722227, 0.05177304, 0.05076287, 0.04027302, 0.038523 , 0.04700016, 0.04626175, 0.04963526, 0.04924924, 0.04556549, 0.03992306, 0.04264629, 0.04201707, 0.0421803 , 0.04335506, 0.04803246, 0.04784608, 0.04146418, 0.03996568, 0.05063166, 0.04452316, 0.04521715, 0.04285809, 0.04856424, 0.04510337, 0.04772573, 0.04363936, 0.04424811, 0.04980449, 0.04387433, 0.04367579, 0.04510261, 0.043214 , 0.0401914 , 0.05182074, 0.03951072, 0.0477948 , 0.04673033, 0.04689404, 0.04998034, 0.04552733, 0.04831458, 0.04267216, 0.04826833, 0.04793905, 0.05178392, 0.04410214, 0.03943804, 0.04291292, 0.0406942 , 0.04885506, 0.04308423, 0.04549851, 0.04976944, 0.05076628, 0.04412881, 0.04532336, 0.04897802, 0.0469969 , 0.04439076, 0.04942879, 0.04328135, 0.04629906, 0.0459189 , 0.03818804, 0.04568807, 0.04147959, 0.05170615, 0.05463537, 0.04573667, 0.04900106, 0.04359653, 0.03808469, 0.04496064, 0.03948091, 0.04690949, 0.04154617, 0.0474799 , 0.04454476, 0.04736569, 0.05066139, 0.04207634, 0.0479254 , 0.04132016, 0.04558345, 0.044382 , 0.04258322, 0.04245783, 0.04792109, 0.03685759, 0.04711378, 0.04406799, 0.04270924, 0.05258154, 0.04412806, 0.04270278, 0.04686045, 0.04326495, 0.04031116, 0.04062513, 0.04254267, 0.0404767 , 0.04825754, 0.04323409, 0.05539793, 0.04773133, 0.04162422, 0.04661492, 0.04853648, 0.03979454, 0.04288096, 0.04452497, 0.04612258, 0.04615601, 0.04294166, 0.05242546, 0.04947169, 0.05055515, 0.04491648, 0.04057551, 0.04777092, 0.04561228, 0.04684564, 0.04517448, 0.04342776, 0.04343005, 0.04704644, 0.04112438, 0.03617482, 0.04135897, 0.05115549, 0.04477676, 0.04699584, 0.05039405, 0.04516071, 0.04827367, 0.04621755, 0.04747895, 0.0479186 , 0.0460015 , 0.04309225, 0.04366407, 0.0392787 , 0.03586747, 0.04604356, 0.03884698, 0.05004952, 0.04139646, 0.04689492, 0.04140696, 0.04730347, 0.04878515, 0.04813615, 0.04342726, 0.05762307, 0.04489961, 0.04359375, 0.04612299, 0.04854138, 0.04527062, 0.04472304, 0.0395711 , 0.0405797 , 0.04505481, 0.04728124, 0.04570094, 0.04527361, 0.03862806, 0.04442718, 0.04443652, 0.04895133, 0.04588046, 0.04839031, 0.04469987, 0.05000096, 0.03906171, 0.051967 , 0.03978374, 0.04194069, 0.04661182, 0.03479072, 0.04340331, 0.04006326, 0.05954248, 0.04160275, 0.04321778, 0.04882777, 0.04787445, 0.04925952, 0.05323071, 0.04701163, 0.05337292, 0.04709804, 0.04334403, 0.04061232, 0.04478596, 0.04398654, 0.04274207, 0.05090697, 0.04455623, 0.04872436, 0.0421355 , 0.04898239, 0.04857274, 0.05415483, 0.04700655, 0.04014555, 0.0480144 , 0.03973658, 0.05139387, 0.04163909, 0.04686196, 0.03972785, 0.04952517, 0.04355752, 0.04586151, 0.04503243, 0.04574733, 0.04706773, 0.05015332, 0.04530172, 0.04176983, 0.04060019, 0.04704383, 0.04182002, 0.03868631, 0.04489151, 0.04019074, 0.04062664, 0.04333441, 0.04373353, 0.0441841 , 0.04185141, 0.04855307, 0.03711605, 0.04173668, 0.05377995, 0.04700167, 0.04350333, 0.04455046, 0.04259438, 0.04334646, 0.04146749, 0.04634779, 0.04688937, 0.04008222, 0.0391518 , 0.04666003, 0.04264922, 0.04064489, 0.04484932, 0.04550754, 0.05596455, 0.04654242, 0.04698534, 0.05086802, 0.04602974, 0.04597451, 0.04194012, 0.04508152, 0.0533805 , 0.04400257, 0.04563091, 0.04757315, 0.04725896, 0.04570736, 0.0478959 , 0.04467469, 0.04802436, 0.04367885, 0.04876105, 0.04211308, 0.04510159, 0.04371484, 0.04133507, 0.04158446, 0.04393359, 0.04355243, 0.04995303, 0.04931508, 0.0381846 , 0.04688839, 0.03784551, 0.04850098, 0.05309496, 0.04718066, 0.04085384, 0.04463436, 0.04367085, 0.04746603, 0.047315 , 0.03405395, 0.04403485, 0.0502298 , 0.04417355, 0.03596471, 0.04452519, 0.04578993, 0.04632251, 0.03958637, 0.04449887, 0.03931985, 0.04310929, 0.03709216, 0.04151165, 0.04229465, 0.04899224, 0.0489364 , 0.04226488, 0.03964909, 0.04748648, 0.04178246, 0.05107194, 0.04452271, 0.05032763, 0.04413235, 0.04676899, 0.04810656, 0.04573807, 0.04076975, 0.05570845, 0.0443732 , 0.04552938, 0.0496156 , 0.04152792, 0.04781586, 0.04225201, 0.0495326 , 0.0443289 , 0.05493486, 0.04521976, 0.0468836 , 0.04401311, 0.04912803, 0.04230038, 0.04714092, 0.03973136, 0.04395779, 0.04519541, 0.04936377, 0.04713223, 0.04564903, 0.04156458, 0.04433356, 0.0505021 , 0.04753149, 0.04174502, 0.056807 , 0.0496208 , 0.04767351, 0.04489522, 0.04642526, 0.05266117, 0.04438333, 0.04227131, 0.05097436, 0.04070389, 0.04836683, 0.04708687, 0.05225942, 0.03992956, 0.04902233, 0.04267317, 0.04477181, 0.03611383, 0.05027674, 0.03796292, 0.05075041, 0.04374405, 0.04413964, 0.0406099 , 0.04255497, 0.04242483, 0.04303759, 0.04170372, 0.04663662, 0.04296794, 0.05192534, 0.0449956 , 0.04730325, 0.0450723 , 0.04445592, 0.04766065, 0.0490991 , 0.04479711, 0.05232366, 0.04572643, 0.04847861, 0.04174633, 0.04394398, 0.0432659 , 0.04822833, 0.04773257, 0.04472821, 0.05111633, 0.04862911, 0.04401816, 0.04236195, 0.04526448, 0.05078418, 0.03735922, 0.04876512, 0.04758175, 0.04562471, 0.04335687, 0.04624443, 0.04618243, 0.04639856, 0.03822104, 0.0572194 , 0.04308923, 0.0419536 , 0.03811534, 0.03959254, 0.04130681, 0.05258445, 0.04986443, 0.04372495, 0.05031003, 0.05073988, 0.04547629, 0.050313 , 0.05054227, 0.04252951, 0.05065258, 0.04641125, 0.04230864, 0.04063602, 0.04855955, 0.04236878, 0.04650167, 0.03732677, 0.05326182, 0.04636859, 0.04556033, 0.04708447, 0.04356933, 0.04375771, 0.05121428, 0.04205749, 0.0518704 , 0.04518894, 0.04231454, 0.04398824, 0.05386235, 0.04887486, 0.04226957, 0.03547307, 0.04334838, 0.05197605, 0.04584531, 0.04269476, 0.04454851, 0.04730934, 0.05087434, 0.04269729, 0.04609613, 0.04301385, 0.04510682, 0.04449971, 0.04376506, 0.04954276, 0.04427552, 0.03896203, 0.04919564, 0.04771758, 0.04393024, 0.03870266, 0.0406708 , 0.0469429 , 0.04648953, 0.04738561, 0.04893255, 0.04408171, 0.04506601, 0.05193102, 0.04054061, 0.05030557, 0.04590494, 0.04596106, 0.04486667, 0.04068828, 0.0439199 , 0.0445398 , 0.05113532, 0.04578913, 0.0427432 , 0.04100801, 0.04285793, 0.04261937, 0.04108686, 0.04369689, 0.04362722, 0.03982833, 0.04838574, 0.05134018, 0.04221049, 0.04859673, 0.04196777, 0.05194669, 0.05153446, 0.04606078, 0.04232377, 0.0469268 , 0.04235938, 0.04001685, 0.04145275, 0.05063048, 0.04944555, 0.0413224 , 0.04250659, 0.04647227, 0.04430456, 0.04549796, 0.04603226, 0.04760732, 0.03799237, 0.0473626 , 0.04558186, 0.0486614 , 0.04415898, 0.04871813, 0.04439888, 0.044249 , 0.04096636, 0.04800334, 0.04382408, 0.04466881, 0.04820415, 0.04541524, 0.04297626, 0.04546258, 0.04837869, 0.04856685, 0.04739173, 0.04542921, 0.04363037, 0.04663879, 0.03848235, 0.04428369, 0.04391203, 0.04583622, 0.04338766, 0.04707669, 0.04744157, 0.03843356, 0.04494826, 0.05118331, 0.04920233, 0.04636591, 0.04606445, 0.04217975, 0.04145861, 0.04620102, 0.04249437, 0.03885329, 0.04489372, 0.04626471, 0.04990879, 0.04971651, 0.04811536, 0.0440326 , 0.04365409, 0.04810089, 0.04607049, 0.0445267 , 0.04082594, 0.04242013, 0.04945364, 0.04470348, 0.04007413, 0.04396535, 0.05119831, 0.04607697]) # to run the full simulation uncomment the following line to fit the model for every dataset and not just for the first dataset #for i_rep in range(n_rep): @@ -139,7 +140,7 @@ other half of observations indexed with :math:`i \in I` (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr_nonorth = DoubleMLPLR(obj_dml_data, - ml_m, ml_g, + ml_m, ml_l, ml_g, n_folds=2, apply_cross_fitting=False, score=non_orth_score) @@ -169,7 +170,7 @@ other half of observations indexed with :math:`i \in I` .. jupyter-execute:: - non_orth_score = function(y, d, l_hat, g_hat, m_hat, smpls) { + non_orth_score = function(y, d, l_hat, m_hat, g_hat, smpls) { u_hat = y - g_hat psi_a = -1*d*d psi_b = d*u_hat @@ -190,8 +191,9 @@ other half of observations indexed with :math:`i \in I` set.seed(1111) + ml_l = lrn("regr.ranger", num.trees = 132, max.depth = 5, mtry = 12, min.node.size = 1) ml_m = lrn("regr.ranger", num.trees = 378, max.depth = 3, mtry = 20, min.node.size = 6) - ml_g = lrn("regr.ranger", num.trees = 132, max.depth = 5, mtry = 12, min.node.size = 1) + ml_g = ml_l$clone() # to speed up the illustration we hard-code the simulation results theta_nonorth = c(0.555490805, 0.626761547, 0.527232714, 0.590215088, 0.379737837, 0.398791295, 0.471529799, 0.447306859, 0.398580212, 0.544042875, 0.580504564, 0.474363062, 0.544946070, 0.492569324, 0.476388613, 0.432095547, 0.463801034, 0.538834518, 0.498974469, 0.541736784, 0.556051749, 0.414388248, 0.465484169, 0.523789310, 0.450045410, 0.462671128, 0.483402627, 0.665449409, 0.444034892, 0.571431921, 0.514748934, 0.504854642, 0.543430321, 0.607804819, 0.425343408, 0.461166556, 0.473782998, 0.626301926, 0.554365067, 0.498898748, 0.539603020, 0.588260718, 0.392087266, 0.563189621, 0.622539334, 0.603381790, 0.537399218, 0.450688767, 0.506590096, 0.375848237, 0.518585287, 0.549772758, 0.621927569, 0.488298153, 0.590428253, 0.696371638, 0.627708633, 0.433970065, 0.425033876, 0.537542890, 0.480869086, 0.458919543, 0.653851249, 0.584819357, 0.587246769, 0.459961246, 0.518142266, 0.511701654, 0.620336868, 0.476887219, 0.481538784, 0.616313409, 0.529185572, 0.618412368, 0.522084966, 0.580750958, 0.477682206, 0.574414735, 0.508342507, 0.371303328, 0.527577838, 0.398635240, 0.494730015, 0.508128512, 0.603642166, 0.655167716, 0.601126516, 0.440808052, 0.544282497, 0.708230201, 0.541813291, 0.378489246, 0.437844992, 0.580074282, 0.426458304, 0.644148033, 0.571626341, 0.600921811, 0.710259470, 0.450440901, 0.540628619, 0.431058282, 0.442400358, 0.561393132, 0.523929900, 0.483505805, 0.519636552, 0.607386052, 0.535609002, 0.510519531, 0.490837474, 0.607186868, 0.475418778, 0.614141891, 0.398599358, 0.474117625, 0.532997363, 0.352785069, 0.599932209, 0.506270174, 0.582559300, 0.573352058, 0.688411509, 0.459070744, 0.633930860, 0.531340117, 0.554952484, 0.476316494, 0.586937919, 0.531143264, 0.531667832, 0.604187114, 0.474454410, 0.634159600, 0.515934881, 0.528853550, 0.561666177, 0.632040507, 0.573736861, 0.634677664, 0.545228666, 0.452539377, 0.602514389, 0.406644548, 0.526514048, 0.484831291, 0.515626324, 0.694950769, 0.558903366, 0.543989664, 0.595270602, 0.544302473, 0.701224740, 0.462491671, 0.721976217, 0.509779134, 0.608551613, 0.511985210, 0.610036501, 0.523291208, 0.571238422, 0.486944282, 0.457603820, 0.605811112, 0.602538755, 0.437573404, 0.572777807, 0.540359885, 0.503639372, 0.643482254, 0.542442515, 0.557072215, 0.548365345, 0.520257386, 0.559477646, 0.589800976, 0.620407048, 0.519814608, 0.515998436, 0.361384876, 0.548933980, 0.501600804, 0.627079396, 0.645484925, 0.495473132, 0.442635400, 0.584397481, 0.585125950, 0.352607330, 0.563273699, 0.503440598, 0.488867579, 0.557098343, 0.612293879, 0.463620892, 0.592306697, 0.546759827, 0.643941943, 0.354424800, 0.598255795, 0.484472850, 0.612693579, 0.410380974, 0.598567337, 0.512124207, 0.466188290, 0.416940820, 0.541944143, 0.586775221, 0.546650192, 0.536563665, 0.576996348, 0.503872606, 0.505680665, 0.555680519, 0.518350769, 0.652987312, 0.627428041, 0.552042874, 0.504864880, 0.611606827, 0.517154197, 0.544605688, 0.406896793, 0.597414935, 0.582586483, 0.590689714, 0.495941043, 0.498246261, 0.543623886, 0.555031880, 0.551186152, 0.478571512, 0.472812207, 0.518706192, 0.563640332, 0.554419638, 0.515959815, 0.458494588, 0.549400619, 0.490260796, 0.597798784, 0.510691719, 0.684161072, 0.506404568, 0.397770673, 0.514802604, 0.655332528, 0.472896035, 0.546293061, @@ -208,7 +210,7 @@ other half of observations indexed with :math:`i \in I` df = data[[i_rep]] obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") obj_dml_plr_nonorth = DoubleMLPLR$new(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, ml_g, n_folds=2, score=non_orth_score, apply_cross_fitting=FALSE) @@ -283,7 +285,7 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr_orth_nosplit = DoubleMLPLR(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, ml_g, n_folds=1, score='IV-type', apply_cross_fitting=False) @@ -332,7 +334,7 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate df = data[[i_rep]] obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") obj_dml_plr_orth_nosplit = DoubleMLPLR$new(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, ml_g, n_folds=1, score='IV-type', apply_cross_fitting=FALSE) @@ -390,7 +392,7 @@ Cross-fitting performs well empirically because the entire sample can be used fo (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr = DoubleMLPLR(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, ml_g, n_folds=2, score='IV-type') obj_dml_plr.fit() @@ -437,9 +439,9 @@ Cross-fitting performs well empirically because the entire sample can be used fo df = data[[i_rep]] obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") obj_dml_plr = DoubleMLPLR$new(obj_dml_data, - ml_g, ml_m, - n_folds=2, - score='IV-type') + ml_l, ml_m, ml_g, + n_folds=2, + score='IV-type') obj_dml_plr$fit() this_theta = obj_dml_plr$coef this_se = obj_dml_plr$se @@ -577,7 +579,7 @@ estimate :math:`\theta_0` without sample splitting. Again we observe a bias from (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr_orth_nosplit = DoubleMLPLR(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, n_folds=1, score='partialling out', apply_cross_fitting=False) @@ -627,10 +629,10 @@ estimate :math:`\theta_0` without sample splitting. Again we observe a bias from df = data[[i_rep]] obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") obj_dml_plr_orth_nosplit = DoubleMLPLR$new(obj_dml_data, - ml_g, ml_m, - n_folds=1, - score='partialling out', - apply_cross_fitting=FALSE) + ml_l, ml_m, + n_folds=1, + score='partialling out', + apply_cross_fitting=FALSE) obj_dml_plr_orth_nosplit$fit() this_theta = obj_dml_plr_orth_nosplit$coef this_se = obj_dml_plr_orth_nosplit$se @@ -673,7 +675,7 @@ Using sample splitting, overcomes the bias induced by overfitting. (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr = DoubleMLPLR(obj_dml_data, - ml_g, ml_m, + ml_l, ml_m, n_folds=2, score='partialling out') obj_dml_plr.fit() @@ -720,9 +722,9 @@ Using sample splitting, overcomes the bias induced by overfitting. df = data[[i_rep]] obj_dml_data = double_ml_data_from_data_frame(df, y_col = "y", d_cols = "d") obj_dml_plr = DoubleMLPLR$new(obj_dml_data, - ml_g, ml_m, - n_folds=2, - score='partialling out') + ml_l, ml_m, + n_folds=2, + score='partialling out') obj_dml_plr$fit() this_theta = obj_dml_plr$coef this_se = obj_dml_plr$se diff --git a/doc/guide/learners.rst b/doc/guide/learners.rst index 2609c87a..e4ea0eba 100644 --- a/doc/guide/learners.rst +++ b/doc/guide/learners.rst @@ -53,11 +53,11 @@ for example :py:class:`sklearn.ensemble.RandomForestRegressor`. from sklearn.ensemble import RandomForestRegressor np.random.seed(1234) - ml_g = RandomForestRegressor() + ml_l = RandomForestRegressor() ml_m = RandomForestRegressor() data = make_plr_CCDDHNR2018(alpha=0.5, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd') - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m) dml_plr_obj.fit().summary Without further specification of the hyperparameters, default values are used. To set hyperparameters: @@ -128,17 +128,17 @@ implemented in :class:`sklearn.model_selection.GridSearchCV` or via a randomized import doubleml as dml from sklearn.linear_model import Lasso - ml_g = Lasso() + ml_l = Lasso() ml_m = Lasso() - dml_plr_obj = dml.DoubleMLPLR(dml_data, ml_g, ml_m) - par_grids = {'ml_g': {'alpha': np.arange(0.05, 1., 0.1)}, + dml_plr_obj = dml.DoubleMLPLR(dml_data, ml_l, ml_m) + par_grids = {'ml_l': {'alpha': np.arange(0.05, 1., 0.1)}, 'ml_m': {'alpha': np.arange(0.05, 1., 0.1)}} dml_plr_obj.tune(par_grids, search_mode='grid_search'); print(dml_plr_obj.params) print(dml_plr_obj.fit().summary) np.random.seed(1234) - par_grids = {'ml_g': {'alpha': np.arange(0.05, 1., 0.01)}, + par_grids = {'ml_l': {'alpha': np.arange(0.05, 1., 0.01)}, 'ml_m': {'alpha': np.arange(0.05, 1., 0.01)}} dml_plr_obj.tune(par_grids, search_mode='randomized_search', n_iter_randomized_search=20); print(dml_plr_obj.params) @@ -157,13 +157,13 @@ In this case the tuning should be done externally and the parameters can then be from sklearn.linear_model import LassoCV np.random.seed(1234) - ml_g_tune = LassoCV().fit(dml_data.x, dml_data.y) + ml_l_tune = LassoCV().fit(dml_data.x, dml_data.y) ml_m_tune = LassoCV().fit(dml_data.x, dml_data.d) - ml_g = Lasso() + ml_l = Lasso() ml_m = Lasso() - dml_plr_obj = dml.DoubleMLPLR(dml_data, ml_g, ml_m) - dml_plr_obj.set_ml_nuisance_params('ml_l', 'd', {'alpha': ml_g_tune.alpha_}); + dml_plr_obj = dml.DoubleMLPLR(dml_data, ml_l, ml_m) + dml_plr_obj.set_ml_nuisance_params('ml_l', 'd', {'alpha': ml_l_tune.alpha_}); dml_plr_obj.set_ml_nuisance_params('ml_m', 'd', {'alpha': ml_m_tune.alpha_}); print(dml_plr_obj.params) print(dml_plr_obj.fit().summary) @@ -227,12 +227,12 @@ package for R. # set up a mlr3 learner learner = lrn("regr.ranger") - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(3141) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='data.table') obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) dml_plr_obj$fit() dml_plr_obj$summary() @@ -247,16 +247,16 @@ Without further specification of the hyperparameters, default values are used. T .. jupyter-execute:: set.seed(3141) - ml_g = lrn("regr.ranger", num.trees=10) + ml_l = lrn("regr.ranger", num.trees=10) ml_m = lrn("regr.ranger") obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) dml_plr_obj$fit() dml_plr_obj$summary() set.seed(3141) - ml_g = lrn("regr.ranger") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g , ml_m) + ml_l = lrn("regr.ranger") + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l , ml_m) dml_plr_obj$set_ml_nuisance_params("ml_l", "d", list("num.trees"=10)) dml_plr_obj$fit() dml_plr_obj$summary() @@ -283,13 +283,13 @@ Setting treatment-variable-specific or fold-specific hyperparameters: .. jupyter-execute:: set.seed(3141) - ml_g = lrn("regr.ranger") + ml_l = lrn("regr.ranger") ml_m = lrn("regr.ranger") obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") n_rep = 2 n_folds = 3 - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, n_rep=n_rep, n_folds=n_folds) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_rep=n_rep, n_folds=n_folds) # Set globally params = list("num.trees"=10) @@ -308,9 +308,9 @@ The following example illustrates how to set parameters for each fold. .. jupyter-execute:: learner = lrn("regr.ranger") - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, n_rep=n_rep, n_folds=n_folds) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_rep=n_rep, n_folds=n_folds) # Set values for each fold params_exact = rep(list(rep(list(params), n_folds)), n_rep) @@ -403,11 +403,11 @@ for tuning, each of the two folds would be split up into 5 subfolds and the erro lgr::get_logger("bbotk")$set_threshold("warn") set.seed(1234) - ml_g = lrn("regr.glmnet") + ml_l = lrn("regr.glmnet") ml_m = lrn("regr.glmnet") - dml_plr_obj = DoubleMLPLR$new(dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(dml_data, ml_l, ml_m) - par_grids = list("ml_g" = ParamSet$new(list( + par_grids = list("ml_l" = ParamSet$new(list( ParamDbl$new("lambda", lower = 0.05, upper = 0.1))), "ml_m" = ParamSet$new(list( ParamDbl$new("lambda", lower = 0.05, upper = 0.1)))) @@ -415,7 +415,7 @@ for tuning, each of the two folds would be split up into 5 subfolds and the erro tune_settings = list(terminator = trm("evals", n_evals = 100), algorithm = tnr("grid_search", resolution = 10), rsmp_tune = rsmp("cv", folds = 5), - measure = list("ml_g" = msr("regr.mse"), + measure = list("ml_l" = msr("regr.mse"), "ml_m" = msr("regr.mse"))) dml_plr_obj$tune(param_set=par_grids, tune_settings=tune_settings, tune_on_fold=TRUE) dml_plr_obj$params @@ -444,9 +444,9 @@ external parameter tuning of the nuisance parts. The optimally chosen parameters lgr::get_logger("bbotk")$set_threshold("warn") set.seed(1234) - ml_g = lrn("regr.cv_glmnet", s="lambda.min") + ml_l = lrn("regr.cv_glmnet", s="lambda.min") ml_m = lrn("regr.cv_glmnet", s="lambda.min") - dml_plr_obj = DoubleMLPLR$new(dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(dml_data, ml_l, ml_m) dml_plr_obj$fit() dml_plr_obj$summary() @@ -471,15 +471,15 @@ parameters ``mtry`` and ``max.depth`` of a random forest. Evaluation is based on # set up a mlr3 learner learner = lrn("regr.ranger") - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(3141) obj_dml_data = make_plr_CCDDHNR2018(alpha=0.5) - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) # set up a list of parameter grids - param_grid = list("ml_g" = ParamSet$new(list( + param_grid = list("ml_l" = ParamSet$new(list( ParamInt$new("mtry", lower = 2 , upper = 20), ParamInt$new("max.depth", lower = 2, upper = 5))), "ml_m" = ParamSet$new(list( @@ -488,7 +488,7 @@ parameters ``mtry`` and ``max.depth`` of a random forest. Evaluation is based on tune_settings = list(terminator = mlr3tuning::trm("evals", n_evals = 20), algorithm = tnr("random_search"), rsmp_tune = rsmp("cv", folds = 3), - measure = list("ml_g" = msr("regr.mse"), + measure = list("ml_l" = msr("regr.mse"), "ml_m" = msr("regr.mse"))) dml_plr_obj$tune(param_set=param_grid, tune_settings=tune_settings, tune_on_folds=FALSE) dml_plr_obj$params diff --git a/doc/guide/models.rst b/doc/guide/models.rst index 39f88ac9..3d6521c6 100644 --- a/doc/guide/models.rst +++ b/doc/guide/models.rst @@ -26,12 +26,12 @@ Estimation is conducted via its ``fit()`` method: from sklearn.base import clone learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) np.random.seed(1111) data = make_plr_CCDDHNR2018(alpha=0.5, n_obs=500, dim_x=20, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd') - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m) print(dml_plr_obj.fit()) .. tabbed:: R @@ -45,12 +45,12 @@ Estimation is conducted via its ``fit()`` method: lgr::get_logger("mlr3")$set_threshold("warn") learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(1111) data = make_plr_CCDDHNR2018(alpha=0.5, n_obs=500, dim_x=20, return_type='data.table') obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) dml_plr_obj$fit() print(dml_plr_obj) @@ -77,13 +77,13 @@ Estimation is conducted via its ``fit()`` method: from sklearn.base import clone learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) ml_r = clone(learner) np.random.seed(2222) data = make_pliv_CHS2015(alpha=0.5, n_obs=500, dim_x=20, dim_z=1, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd', z_cols='Z1') - dml_pliv_obj = dml.DoubleMLPLIV(obj_dml_data, ml_g, ml_m, ml_r) + dml_pliv_obj = dml.DoubleMLPLIV(obj_dml_data, ml_l, ml_m, ml_r) print(dml_pliv_obj.fit()) .. tabbed:: R @@ -96,13 +96,13 @@ Estimation is conducted via its ``fit()`` method: library(data.table) learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() ml_r = learner$clone() set.seed(2222) data = make_pliv_CHS2015(alpha=0.5, n_obs=500, dim_x=20, dim_z=1, return_type="data.table") obj_dml_data = DoubleMLData$new(data, y_col="y", d_col = "d", z_cols= "Z1") - dml_pliv_obj = DoubleMLPLIV$new(obj_dml_data, ml_g, ml_m, ml_r) + dml_pliv_obj = DoubleMLPLIV$new(obj_dml_data, ml_l, ml_m, ml_r) dml_pliv_obj$fit() print(dml_pliv_obj) diff --git a/doc/guide/resampling.rst b/doc/guide/resampling.rst index b3181536..455adae8 100644 --- a/doc/guide/resampling.rst +++ b/doc/guide/resampling.rst @@ -28,7 +28,7 @@ implemented in ``DoubleMLPLR``. from sklearn.base import clone learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) np.random.seed(1234) obj_dml_data = make_plr_CCDDHNR2018(alpha=0.5, n_obs=100) @@ -44,7 +44,7 @@ implemented in ``DoubleMLPLR``. library(data.table) learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner + ml_l = learner ml_m = learner data = make_plr_CCDDHNR2018(alpha=0.5, n_obs=100, return_type = "data.table") obj_dml_data = DoubleMLData$new(data, @@ -63,7 +63,7 @@ The default setting is ``n_folds = 5`` and ``n_rep = 1``, i.e., .. ipython:: python - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, n_folds = 5, n_rep = 1) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 5, n_rep = 1) print(dml_plr_obj.n_folds) print(dml_plr_obj.n_rep) @@ -71,7 +71,7 @@ The default setting is ``n_folds = 5`` and ``n_rep = 1``, i.e., .. jupyter-execute:: - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, n_folds = 5, n_rep = 1) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 5, n_rep = 1) print(dml_plr_obj$n_folds) print(dml_plr_obj$n_rep) @@ -133,7 +133,7 @@ It results in :math:`M` random :math:`K`-fold partitions being drawn. .. ipython:: python - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, n_folds = 5, n_rep = 10) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 5, n_rep = 10) print(dml_plr_obj.n_folds) print(dml_plr_obj.n_rep) @@ -141,7 +141,7 @@ It results in :math:`M` random :math:`K`-fold partitions being drawn. .. jupyter-execute:: - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, n_folds = 5, n_rep = 10) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 5, n_rep = 10) print(dml_plr_obj$n_folds) print(dml_plr_obj$n_rep) @@ -234,7 +234,7 @@ initialization of the ``DoubleMLPLR`` object. .. ipython:: python np.random.seed(314) - dml_plr_obj_internal = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, n_folds = 4) + dml_plr_obj_internal = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 4) print(dml_plr_obj_internal.fit().summary) .. tabbed:: R @@ -242,7 +242,7 @@ initialization of the ``DoubleMLPLR`` object. .. jupyter-execute:: set.seed(314) - dml_plr_obj_internal = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, n_folds = 4) + dml_plr_obj_internal = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 4) dml_plr_obj_internal$fit() dml_plr_obj_internal$summary() @@ -253,7 +253,7 @@ and set the partition via the ``set_sample_splitting()`` method. .. ipython:: python - dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, draw_sample_splitting = False) + dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, draw_sample_splitting = False) from sklearn.model_selection import KFold np.random.seed(314) @@ -267,7 +267,7 @@ and set the partition via the ``set_sample_splitting()`` method. .. jupyter-execute:: - dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, draw_sample_splitting = FALSE) + dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, draw_sample_splitting = FALSE) set.seed(314) # set up a task and cross-validation resampling scheme in mlr3 @@ -297,7 +297,7 @@ Note that cross-fitting performs well empirically and is recommended to remove b .. ipython:: python np.random.seed(314) - dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, + dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 2, apply_cross_fitting = False) print(dml_plr_obj_external.fit().summary) @@ -305,7 +305,7 @@ Note that cross-fitting performs well empirically and is recommended to remove b .. jupyter-execute:: - dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, + dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 2, apply_cross_fitting = FALSE) dml_plr_obj_external$fit() dml_plr_obj_external$summary() @@ -318,7 +318,7 @@ via ``set_sample_splitting()`` needs to be applied, like for example: .. ipython:: python np.random.seed(314) - dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, + dml_plr_obj_external = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 2, apply_cross_fitting = False, draw_sample_splitting = False) from sklearn.model_selection import train_test_split @@ -331,7 +331,7 @@ via ``set_sample_splitting()`` needs to be applied, like for example: .. jupyter-execute:: - dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, + dml_plr_obj_external = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 2, apply_cross_fitting = FALSE, draw_sample_splitting = FALSE) @@ -363,7 +363,7 @@ justification, see also :ref:`bias_overfitting`. .. ipython:: python np.random.seed(314) - dml_plr_no_split = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m, + dml_plr_no_split = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m, n_folds = 1, apply_cross_fitting = False) print(dml_plr_obj_external.fit().summary) @@ -372,7 +372,7 @@ justification, see also :ref:`bias_overfitting`. .. jupyter-execute:: - dml_plr_no_split = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m, + dml_plr_no_split = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m, n_folds = 1, apply_cross_fitting = FALSE) set.seed(314) diff --git a/doc/guide/scores.rst b/doc/guide/scores.rst index f5849849..a20c2441 100644 --- a/doc/guide/scores.rst +++ b/doc/guide/scores.rst @@ -62,11 +62,11 @@ implemented in ``DoubleMLPLR``. np.random.seed(3141) learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd') - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m) dml_plr_obj.fit(); print(dml_plr_obj) @@ -81,12 +81,12 @@ implemented in ``DoubleMLPLR``. lgr::get_logger("mlr3")$set_threshold("warn") learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(3141) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='data.table') obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d") - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) dml_plr_obj$fit() print(dml_plr_obj) @@ -131,50 +131,53 @@ Partially linear regression model (PLR) *************************************** For the PLR model implemented in ``DoubleMLPLR`` one can choose between -``score='IV-type'`` and ``score='partialling out'``. +``score='partialling out'`` and ``score='IV-type'``. -``score='IV-type'`` implements the score function: +``score='partialling out'`` implements the score function: .. math:: - \psi(W; \theta, \eta) &:= [Y - D \theta - g(X)] [D - m(X)] + \psi(W; \theta, \eta) &:= [Y - \ell(X) - \theta (D - m(X))] [D - m(X)] - &= - D (D - m(X)) \theta + (Y - g(X)) (D - m(X)) + &= - (D - m(X)) (D - m(X)) \theta + (Y - \ell(X)) (D - m(X)) &= \psi_a(W; \eta) \theta + \psi_b(W; \eta) -with :math:`\eta=(g,m)` and where the components of the linear score are +with :math:`\eta=(\ell,m)` and where the components of the linear score are .. math:: - \psi_a(W; \eta) &= - D (D - m(X)), + \psi_a(W; \eta) &= - (D - m(X)) (D - m(X)), - \psi_b(W; \eta) &= (Y - g(X)) (D - m(X)). + \psi_b(W; \eta) &= (Y - \ell(X)) (D - m(X)). -``score='partialling out'`` implements the score function: +``score='IV-type'`` implements the score function: .. math:: - \psi(W; \theta, \eta) &:= [Y - \ell(X) - \theta (D - m(X))] [D - m(X)] + \psi(W; \theta, \eta) &:= [Y - D \theta - g(X)] [D - m(X)] - &= - (D - m(X)) (D - m(X)) \theta + (Y - \ell(X)) (D - m(X)) + &= - D (D - m(X)) \theta + (Y - g(X)) (D - m(X)) &= \psi_a(W; \eta) \theta + \psi_b(W; \eta) -with :math:`\eta=(\ell,m)` and where the components of the linear score are +with :math:`\eta=(g,m)` and where the components of the linear score are .. math:: - \psi_a(W; \eta) &= - (D - m(X)) (D - m(X)), + \psi_a(W; \eta) &= - D (D - m(X)), - \psi_b(W; \eta) &= (Y - \ell(X)) (D - m(X)). + \psi_b(W; \eta) &= (Y - g(X)) (D - m(X)). Partially linear IV regression model (PLIV) ******************************************* -For the PLIV model implemented in ``DoubleMLPLIV`` -we employ for ``score='partialling out'`` the score function: + +For the PLIV model implemented in ``DoubleMLPLIV`` one can choose between +``score='IV-type'`` and ``score='partialling out'``. + +``score='partialling out'`` implements the score function: .. math:: @@ -192,6 +195,25 @@ with :math:`\eta=(\ell, m, r)` and where the components of the linear score are \psi_b(W; \eta) &= (Y - \ell(X)) (Z - m(X)). +``score='IV-type'`` implements the score function: + +.. math:: + + \psi(W; \theta, \eta) &:= [Y - D \theta - g(X)] [Z - m(X)] + + &= - D (Z - m(X)) \theta + (Y - g(X)) (Z - m(X)) + + &= \psi_a(W; \eta) \theta + \psi_b(W; \eta) + +with :math:`\eta=(g,m)` and where the components of the linear score are + +.. math:: + + \psi_a(W; \eta) &= - D (Z - m(X)), + + \psi_b(W; \eta) &= (Y - g(X)) (Z - m(X)). + + Interactive regression model (IRM) ********************************** @@ -280,7 +302,7 @@ can be obtained with import numpy as np - def non_orth_score(y, d, l_hat, g_hat, m_hat, smpls): + def non_orth_score(y, d, l_hat, m_hat, g_hat, smpls): u_hat = y - g_hat psi_a = -np.multiply(d, d) psi_b = np.multiply(d, u_hat) @@ -290,7 +312,7 @@ can be obtained with .. jupyter-execute:: - non_orth_score = function(y, d, l_hat, g_hat, m_hat, smpls) { + non_orth_score = function(y, d, l_hat, m_hat, g_hat, smpls) { u_hat = y - g_hat psi_a = -1*d*d psi_b = d*u_hat diff --git a/doc/guide/se_confint.rst b/doc/guide/se_confint.rst index 942f2bbc..594e78b7 100644 --- a/doc/guide/se_confint.rst +++ b/doc/guide/se_confint.rst @@ -49,11 +49,11 @@ implemented in ``DoubleMLPLR``. np.random.seed(3141) learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2) - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) data = make_plr_CCDDHNR2018(alpha=0.5, return_type='DataFrame') obj_dml_data = dml.DoubleMLData(data, 'y', 'd') - dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_g, ml_m) + dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m) dml_plr_obj.fit(); .. tabbed:: R @@ -67,12 +67,12 @@ implemented in ``DoubleMLPLR``. lgr::get_logger("mlr3")$set_threshold("warn") learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5) - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() set.seed(3141) obj_dml_data = make_plr_CCDDHNR2018(alpha=0.5) - dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m) + dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m) dml_plr_obj$fit() diff --git a/doc/guide/sim_inf.rst b/doc/guide/sim_inf.rst index 83e1760f..08f1ef91 100644 --- a/doc/guide/sim_inf.rst +++ b/doc/guide/sim_inf.rst @@ -87,9 +87,9 @@ via the option ``method``. dml_data = dml.DoubleMLData.from_arrays(X[:, 10:], y, X[:, :10]) learner = LassoCV() - ml_g = clone(learner) + ml_l = clone(learner) ml_m = clone(learner) - dml_plr = dml.DoubleMLPLR(dml_data, ml_g, ml_m) + dml_plr = dml.DoubleMLPLR(dml_data, ml_l, ml_m) print(dml_plr.fit().bootstrap().confint(joint=True)) print(dml_plr.p_adjust()) @@ -114,9 +114,9 @@ via the option ``method``. dml_data = double_ml_data_from_matrix(X = X[, 11:n_vars], y = y, d = X[,1:10]) learner = lrn("regr.cv_glmnet", s="lambda.min") - ml_g = learner$clone() + ml_l = learner$clone() ml_m = learner$clone() - dml_plr = DoubleMLPLR$new(dml_data, ml_g, ml_m) + dml_plr = DoubleMLPLR$new(dml_data, ml_l, ml_m) dml_plr$fit() dml_plr$bootstrap() diff --git a/doc/intro/intro.rst b/doc/intro/intro.rst index 4672c5a5..c7060152 100644 --- a/doc/intro/intro.rst +++ b/doc/intro/intro.rst @@ -138,11 +138,11 @@ For details on the specification of learners and their hyperparameters we refer from sklearn.linear_model import LassoCV learner = RandomForestRegressor(n_estimators = 500, max_features = 'sqrt', max_depth= 5) - ml_g_bonus = clone(learner) + ml_l_bonus = clone(learner) ml_m_bonus = clone(learner) learner = LassoCV() - ml_g_sim = clone(learner) + ml_l_sim = clone(learner) ml_m_sim = clone(learner) .. tabbed:: R @@ -155,11 +155,11 @@ For details on the specification of learners and their hyperparameters we refer lgr::get_logger("mlr3")$set_threshold("warn") learner = lrn("regr.ranger", num.trees=500, mtry=floor(sqrt(n_vars)), max.depth=5, min.node.size=2) - ml_g_bonus = learner$clone() + ml_l_bonus = learner$clone() ml_m_bonus = learner$clone() learner = lrn("regr.cv_glmnet", s="lambda.min") - ml_g_sim = learner$clone() + ml_l_sim = learner$clone() ml_m_sim = learner$clone() @@ -194,11 +194,11 @@ statistical inference like ``bootstrap()``, ``confint()`` and ``p_adjust()``, fo from doubleml import DoubleMLPLR np.random.seed(3141) - obj_dml_plr_bonus = DoubleMLPLR(dml_data_bonus, ml_g_bonus, ml_m_bonus) + obj_dml_plr_bonus = DoubleMLPLR(dml_data_bonus, ml_l_bonus, ml_m_bonus) obj_dml_plr_bonus.fit(); print(obj_dml_plr_bonus) - obj_dml_plr_sim = DoubleMLPLR(dml_data_sim, ml_g_sim, ml_m_sim) + obj_dml_plr_sim = DoubleMLPLR(dml_data_sim, ml_l_sim, ml_m_sim) obj_dml_plr_sim.fit(); print(obj_dml_plr_sim) @@ -207,11 +207,11 @@ statistical inference like ``bootstrap()``, ``confint()`` and ``p_adjust()``, fo .. jupyter-execute:: set.seed(3141) - obj_dml_plr_bonus = DoubleMLPLR$new(dml_data_bonus, ml_g=ml_g_bonus, ml_m=ml_m_bonus) + obj_dml_plr_bonus = DoubleMLPLR$new(dml_data_bonus, ml_l=ml_l_bonus, ml_m=ml_m_bonus) obj_dml_plr_bonus$fit() print(obj_dml_plr_bonus) - obj_dml_plr_sim = DoubleMLPLR$new(dml_data_sim, ml_g=ml_g_sim, ml_m=ml_m_sim) + obj_dml_plr_sim = DoubleMLPLR$new(dml_data_sim, ml_l=ml_l_sim, ml_m=ml_m_sim) obj_dml_plr_sim$fit() print(obj_dml_plr_sim) diff --git a/doc/workflow/workflow.rst b/doc/workflow/workflow.rst index be7bcad8..7fb46b28 100644 --- a/doc/workflow/workflow.rst +++ b/doc/workflow/workflow.rst @@ -151,14 +151,14 @@ We use a regression learner for the continuous outcome variable net financial as # Random forest learners from sklearn.ensemble import RandomForestClassifier, RandomForestRegressor - ml_g_rf = RandomForestRegressor(n_estimators = 500, max_depth = 7, + ml_l_rf = RandomForestRegressor(n_estimators = 500, max_depth = 7, max_features = 3, min_samples_leaf = 3) ml_m_rf = RandomForestClassifier(n_estimators = 500, max_depth = 5, max_features = 4, min_samples_leaf = 7) # Xgboost learners from xgboost import XGBClassifier, XGBRegressor - ml_g_xgb = XGBRegressor(objective = "reg:squarederror", eta = 0.1, + ml_l_xgb = XGBRegressor(objective = "reg:squarederror", eta = 0.1, n_estimators =35) ml_m_xgb = XGBClassifier(use_label_encoder = False , objective = "binary:logistic", @@ -172,13 +172,13 @@ We use a regression learner for the continuous outcome variable net financial as library(mlr3) library(mlr3learners) # Random forest learners - ml_g_rf = lrn("regr.ranger", max.depth = 7, + ml_l_rf = lrn("regr.ranger", max.depth = 7, mtry = 3, min.node.size =3) ml_m_rf = lrn("classif.ranger", max.depth = 5, mtry = 4, min.node.size = 7) # Xgboost learners - ml_g_xgb = lrn("regr.xgboost", objective = "reg:squarederror", + ml_l_xgb = lrn("regr.xgboost", objective = "reg:squarederror", eta = 0.1, nrounds = 35) ml_m_xgb = lrn("classif.xgboost", objective = "binary:logistic", eval_metric = "logloss", @@ -205,13 +205,13 @@ the dml algorithm (:ref:`DML1 vs. DML2 `) and the score function (:r np.random.seed(123) # Default values dml_plr_tree = DoubleMLPLR(dml_data, - ml_g = ml_g_rf, + ml_l = ml_l_rf, ml_m = ml_m_rf) np.random.seed(123) # Parametrized by user dml_plr_tree = DoubleMLPLR(dml_data, - ml_g = ml_g_rf, + ml_l = ml_l_rf, ml_m = ml_m_rf, n_folds = 3, n_rep = 1, @@ -225,13 +225,13 @@ the dml algorithm (:ref:`DML1 vs. DML2 `) and the score function (:r set.seed(123) # Default values dml_plr_forest = DoubleMLPLR$new(dml_data, - ml_g = ml_g_rf, + ml_l = ml_l_rf, ml_m = ml_m_rf) set.seed(123) # Parametrized by user dml_plr_forest = DoubleMLPLR$new(dml_data, - ml_g = ml_g_rf, + ml_l = ml_l_rf, ml_m = ml_m_rf, n_folds = 3, score = 'partialling out', From 422bdc3ea1e3ae745b55bdc9ddb40df6abb04ed3 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Fri, 20 May 2022 15:29:46 +0200 Subject: [PATCH 6/8] fix order of learners --- doc/guide/basics.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/guide/basics.rst b/doc/guide/basics.rst index c3428e2e..8d934039 100644 --- a/doc/guide/basics.rst +++ b/doc/guide/basics.rst @@ -140,7 +140,7 @@ other half of observations indexed with :math:`i \in I` (x, y, d) = data[i_rep] obj_dml_data = DoubleMLData.from_arrays(x, y, d) obj_dml_plr_nonorth = DoubleMLPLR(obj_dml_data, - ml_m, ml_l, ml_g, + ml_l, ml_m, ml_g, n_folds=2, apply_cross_fitting=False, score=non_orth_score) From c9f2d1814d9db55b3c32a118f7602bf33b9f8129 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Fri, 10 Jun 2022 07:41:29 +0200 Subject: [PATCH 7/8] Apply suggestions from code review Co-authored-by: PhilippBach --- doc/guide/basics.rst | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/doc/guide/basics.rst b/doc/guide/basics.rst index 8d934039..fbce1983 100644 --- a/doc/guide/basics.rst +++ b/doc/guide/basics.rst @@ -231,7 +231,7 @@ other half of observations indexed with :math:`i \in I` scale_color_manual(name='', breaks=c("Non-orthogonal ML", "N(0, 1)"), values=c("Non-orthogonal ML"="dark blue", "N(0, 1)"='black')) + - scale_fill_manual(name='',, + scale_fill_manual(name='', breaks=c("Non-orthogonal ML", "N(0, 1)"), values=c("Non-orthogonal ML"="dark blue", "N(0, 1)"=NA)) + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() @@ -355,7 +355,7 @@ orthogonalized regressor :math:`V = D - m(X)`. We then use the final estimate scale_color_manual(name='', breaks=c("Double ML (no sample splitting)", "N(0, 1)"), values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"='black')) + - scale_fill_manual(name='',, + scale_fill_manual(name='', breaks=c("Double ML (no sample splitting)", "N(0, 1)"), values=c("Double ML (no sample splitting)"="dark orange", "N(0, 1)"=NA)) + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() @@ -459,7 +459,7 @@ Cross-fitting performs well empirically because the entire sample can be used fo scale_color_manual(name='', breaks=c("Double ML with cross-fitting", "N(0, 1)"), values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"='black')) + - scale_fill_manual(name='',, + scale_fill_manual(name='', breaks=c("Double ML with cross-fitting", "N(0, 1)"), values=c("Double ML with cross-fitting"="dark green", "N(0, 1)"=NA)) + xlim(c(-6.0, 6.0)) + xlab("") + ylab("") + theme_minimal() From 87664281bb4b2d009e3af075f7b9255e4a428d66 Mon Sep 17 00:00:00 2001 From: "Malte S. Kurz" Date: Tue, 14 Jun 2022 09:49:04 +0200 Subject: [PATCH 8/8] switch to a different temporary fix for https://github.com/DoubleML/doubleml-docs/issues/45 / https://github.com/scikit-learn/scikit-learn/issues/21997 --- doc/examples/py_double_ml_pension.ipynb | 17 +++++++++++++++++ requirements.txt | 1 - 2 files changed, 17 insertions(+), 1 deletion(-) diff --git a/doc/examples/py_double_ml_pension.ipynb b/doc/examples/py_double_ml_pension.ipynb index d9813d16..deead652 100644 --- a/doc/examples/py_double_ml_pension.ipynb +++ b/doc/examples/py_double_ml_pension.ipynb @@ -91,6 +91,23 @@ "data = fetch_401K(return_type='DataFrame')" ] }, + { + "cell_type": "code", + "execution_count": null, + "id": "853ba4be", + "metadata": {}, + "outputs": [], + "source": [ + "# Temporary fix for https://github.com/DoubleML/doubleml-docs/issues/45 / https://github.com/scikit-learn/scikit-learn/issues/21997\n", + "# Can be removed when scikit-learn version 1.2.0 is released\n", + "dtypes = data.dtypes\n", + "dtypes['nifa'] = 'float64'\n", + "dtypes['net_tfa'] = 'float64'\n", + "dtypes['tw'] = 'float64'\n", + "dtypes['inc'] = 'float64'\n", + "data = data.astype(dtypes)" + ] + }, { "cell_type": "code", "execution_count": null, diff --git a/requirements.txt b/requirements.txt index 5935138e..8a75f569 100644 --- a/requirements.txt +++ b/requirements.txt @@ -1,4 +1,3 @@ -scikit-learn<1.0.0 DoubleML # test