.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/inspection/plot_linear_model_coefficient_interpretation.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py: ================================================================== Common pitfalls in interpretation of coefficients of linear models ================================================================== In linear models, the target value is modeled as a linear combination of the features (see the :ref:linear_model User Guide section for a description of a set of linear models available in scikit-learn). Coefficients in multiple linear models represent the relationship between the given feature, :math:X_i and the target, :math:y, assuming that all the other features remain constant (conditional dependence _). This is different from plotting :math:X_i versus :math:y and fitting a linear relationship: in that case all possible values of the other features are taken into account in the estimation (marginal dependence). This example will provide some hints in interpreting coefficient in linear models, pointing at problems that arise when either the linear model is not appropriate to describe the dataset, or when features are correlated. We will use data from the "Current Population Survey" _ from 1985 to predict wage as a function of various features such as experience, age, or education. .. contents:: :local: :depth: 1 .. GENERATED FROM PYTHON SOURCE LINES 30-39 .. code-block:: default print(__doc__) import numpy as np import scipy as sp import pandas as pd import matplotlib.pyplot as plt import seaborn as sns .. GENERATED FROM PYTHON SOURCE LINES 40-46 The dataset: wages ------------------ We fetch the data from OpenML _. Note that setting the parameter as_frame to True will retrieve the data as a pandas dataframe. .. GENERATED FROM PYTHON SOURCE LINES 46-51 .. code-block:: default from sklearn.datasets import fetch_openml survey = fetch_openml(data_id=534, as_frame=True) .. GENERATED FROM PYTHON SOURCE LINES 52-55 Then, we identify features X and targets y: the column WAGE is our target variable (i.e., the variable which we want to predict). .. GENERATED FROM PYTHON SOURCE LINES 55-58 .. code-block:: default X = survey.data[survey.feature_names] X.describe(include="all") .. raw:: html
EDUCATION SOUTH SEX EXPERIENCE UNION AGE RACE OCCUPATION SECTOR MARR
count 534.000000 534 534 534.000000 534 534.000000 534 534 534 534
unique NaN 2 2 NaN 2 NaN 3 6 3 2
top NaN no male NaN not_member NaN White Other Other Married
freq NaN 378 289 NaN 438 NaN 440 156 411 350
mean 13.018727 NaN NaN 17.822097 NaN 36.833333 NaN NaN NaN NaN
std 2.615373 NaN NaN 12.379710 NaN 11.726573 NaN NaN NaN NaN
min 2.000000 NaN NaN 0.000000 NaN 18.000000 NaN NaN NaN NaN
25% 12.000000 NaN NaN 8.000000 NaN 28.000000 NaN NaN NaN NaN
50% 12.000000 NaN NaN 15.000000 NaN 35.000000 NaN NaN NaN NaN
75% 15.000000 NaN NaN 26.000000 NaN 44.000000 NaN NaN NaN NaN
max 18.000000 NaN NaN 55.000000 NaN 64.000000 NaN NaN NaN NaN

.. GENERATED FROM PYTHON SOURCE LINES 59-62 Note that the dataset contains categorical and numerical variables. We will need to take this into account when preprocessing the dataset thereafter. .. GENERATED FROM PYTHON SOURCE LINES 62-65 .. code-block:: default X.head() .. raw:: html
EDUCATION SOUTH SEX EXPERIENCE UNION AGE RACE OCCUPATION SECTOR MARR
0 8.0 no female 21.0 not_member 35.0 Hispanic Other Manufacturing Married
1 9.0 no female 42.0 not_member 57.0 White Other Manufacturing Married
2 12.0 no male 1.0 not_member 19.0 White Other Manufacturing Unmarried
3 12.0 no male 4.0 not_member 22.0 White Other Other Unmarried
4 12.0 no male 17.0 not_member 35.0 White Other Other Married

.. GENERATED FROM PYTHON SOURCE LINES 66-68 Our target for prediction: the wage. Wages are described as floating-point number in dollars per hour. .. GENERATED FROM PYTHON SOURCE LINES 68-71 .. code-block:: default y = survey.target.values.ravel() survey.target.head() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0 5.10 1 4.95 2 6.67 3 4.00 4 7.50 Name: WAGE, dtype: float64 .. GENERATED FROM PYTHON SOURCE LINES 72-77 We split the sample into a train and a test dataset. Only the train dataset will be used in the following exploratory analysis. This is a way to emulate a real situation where predictions are performed on an unknown target, and we don't want our analysis and decisions to be biased by our knowledge of the test data. .. GENERATED FROM PYTHON SOURCE LINES 77-84 .. code-block:: default from sklearn.model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split( X, y, random_state=42 ) .. GENERATED FROM PYTHON SOURCE LINES 85-90 First, let's get some insights by looking at the variable distributions and at the pairwise relationships between them. Only numerical variables will be used. In the following plot, each dot represents a sample. .. _marginal_dependencies: .. GENERATED FROM PYTHON SOURCE LINES 90-95 .. code-block:: default train_dataset = X_train.copy() train_dataset.insert(0, "WAGE", y_train) _ = sns.pairplot(train_dataset, kind='reg', diag_kind='kde') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_001.png :alt: plot linear model coefficient interpretation :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 96-115 Looking closely at the WAGE distribution reveals that it has a long tail. For this reason, we should take its logarithm to turn it approximately into a normal distribution (linear models such as ridge or lasso work best for a normal distribution of error). The WAGE is increasing when EDUCATION is increasing. Note that the dependence between WAGE and EDUCATION represented here is a marginal dependence, i.e., it describes the behavior of a specific variable without keeping the others fixed. Also, the EXPERIENCE and AGE are strongly linearly correlated. .. _the-pipeline: The machine-learning pipeline ----------------------------- To design our machine-learning pipeline, we first manually check the type of data that we are dealing with: .. GENERATED FROM PYTHON SOURCE LINES 115-118 .. code-block:: default survey.data.info() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none RangeIndex: 534 entries, 0 to 533 Data columns (total 10 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 EDUCATION 534 non-null float64 1 SOUTH 534 non-null category 2 SEX 534 non-null category 3 EXPERIENCE 534 non-null float64 4 UNION 534 non-null category 5 AGE 534 non-null float64 6 RACE 534 non-null category 7 OCCUPATION 534 non-null category 8 SECTOR 534 non-null category 9 MARR 534 non-null category dtypes: category(7), float64(3) memory usage: 17.2 KB .. GENERATED FROM PYTHON SOURCE LINES 119-130 As seen previously, the dataset contains columns with different data types and we need to apply a specific preprocessing for each data types. In particular categorical variables cannot be included in linear model if not coded as integers first. In addition, to avoid categorical features to be treated as ordered values, we need to one-hot-encode them. Our pre-processor will - one-hot encode (i.e., generate a column by category) the categorical columns; - as a first approach (we will see after how the normalisation of numerical values will affect our discussion), keep numerical values as they are. .. GENERATED FROM PYTHON SOURCE LINES 130-143 .. code-block:: default from sklearn.compose import make_column_transformer from sklearn.preprocessing import OneHotEncoder categorical_columns = ['RACE', 'OCCUPATION', 'SECTOR', 'MARR', 'UNION', 'SEX', 'SOUTH'] numerical_columns = ['EDUCATION', 'EXPERIENCE', 'AGE'] preprocessor = make_column_transformer( (OneHotEncoder(drop='if_binary'), categorical_columns), remainder='passthrough' ) .. GENERATED FROM PYTHON SOURCE LINES 144-146 To describe the dataset as a linear model we use a ridge regressor with a very small regularization and to model the logarithm of the WAGE. .. GENERATED FROM PYTHON SOURCE LINES 146-161 .. code-block:: default from sklearn.pipeline import make_pipeline from sklearn.linear_model import Ridge from sklearn.compose import TransformedTargetRegressor model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10 ) ) .. GENERATED FROM PYTHON SOURCE LINES 162-166 Processing the dataset ---------------------- First, we fit the model. .. GENERATED FROM PYTHON SOURCE LINES 166-169 .. code-block:: default _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 170-173 Then we check the performance of the computed model plotting its predictions on the test set and computing, for example, the median absolute error of the model. .. GENERATED FROM PYTHON SOURCE LINES 173-193 .. code-block:: default from sklearn.metrics import median_absolute_error y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(5, 5)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, small regularization') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_002.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 194-208 The model learnt is far from being a good model making accurate predictions: this is obvious when looking at the plot above, where good predictions should lie on the red line. In the following section, we will interpret the coefficients of the model. While we do so, we should keep in mind that any conclusion we draw is about the model that we build, rather than about the true (real-world) generative process of the data. Interpreting coefficients: scale matters --------------------------------------------- First of all, we can take a look to the values of the coefficients of the regressor we have fitted. .. GENERATED FROM PYTHON SOURCE LINES 208-222 .. code-block:: default feature_names = (model.named_steps['columntransformer'] .named_transformers_['onehotencoder'] .get_feature_names(input_features=categorical_columns)) feature_names = np.concatenate( [feature_names, numerical_columns]) coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs .. raw:: html
Coefficients
RACE_Hispanic -0.013564
RACE_Other -0.009120
RACE_White 0.022549
OCCUPATION_Clerical 0.000048
OCCUPATION_Management 0.090531
OCCUPATION_Other -0.025098
OCCUPATION_Professional 0.071967
OCCUPATION_Sales -0.046633
OCCUPATION_Service -0.091050
SECTOR_Construction -0.000180
SECTOR_Manufacturing 0.031273
SECTOR_Other -0.031008
MARR_Unmarried -0.032405
UNION_not_member -0.117154
SEX_male 0.090808
SOUTH_yes -0.033823
EDUCATION 0.054699
EXPERIENCE 0.035005
AGE -0.030867

.. GENERATED FROM PYTHON SOURCE LINES 223-235 The AGE coefficient is expressed in "dollars/hour per living years" while the EDUCATION one is expressed in "dollars/hour per years of education". This representation of the coefficients has the benefit of making clear the practical predictions of the model: an increase of :math:1 year in AGE means a decrease of :math:0.030867 dollars/hour, while an increase of :math:1 year in EDUCATION means an increase of :math:0.054699 dollars/hour. On the other hand, categorical variables (as UNION or SEX) are adimensional numbers taking either the value 0 or 1. Their coefficients are expressed in dollars/hour. Then, we cannot compare the magnitude of different coefficients since the features have different natural scales, and hence value ranges, because of their different unit of measure. This is more visible if we plot the coefficients. .. GENERATED FROM PYTHON SOURCE LINES 235-241 .. code-block:: default coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_003.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 242-253 Indeed, from the plot above the most important factor in determining WAGE appears to be the variable UNION, even if our intuition might tell us that variables like EXPERIENCE should have more impact. Looking at the coefficient plot to gauge feature importance can be misleading as some of them vary on a small scale, while others, like AGE, varies a lot more, several decades. This is visible if we compare the standard deviations of different features. .. GENERATED FROM PYTHON SOURCE LINES 253-263 .. code-block:: default X_train_preprocessed = pd.DataFrame( model.named_steps['columntransformer'].transform(X_train), columns=feature_names ) X_train_preprocessed.std(axis=0).plot(kind='barh', figsize=(9, 7)) plt.title('Features std. dev.') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_004.png :alt: Features std. dev. :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 264-274 Multiplying the coefficients by the standard deviation of the related feature would reduce all the coefficients to the same unit of measure. As we will see :ref:after this is equivalent to normalize numerical variables to their standard deviation, as :math:y = \sum{coef_i \times X_i} = \sum{(coef_i \times std_i) \times (X_i / std_i)}. In that way, we emphasize that the greater the variance of a feature, the larger the weight of the corresponding coefficient on the output, all else being equal. .. GENERATED FROM PYTHON SOURCE LINES 274-285 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0), columns=['Coefficient importance'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_005.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 286-313 Now that the coefficients have been scaled, we can safely compare them. .. warning:: Why does the plot above suggest that an increase in age leads to a decrease in wage? Why the :ref:initial pairplot  is telling the opposite? The plot above tells us about dependencies between a specific feature and the target when all other features remain constant, i.e., **conditional dependencies**. An increase of the AGE will induce a decrease of the WAGE when all other features remain constant. On the contrary, an increase of the EXPERIENCE will induce an increase of the WAGE when all other features remain constant. Also, AGE, EXPERIENCE and EDUCATION are the three variables that most influence the model. Checking the variability of the coefficients -------------------------------------------- We can check the coefficient variability through cross-validation: it is a form of data perturbation (related to resampling _). If coefficients vary significantly when changing the input dataset their robustness is not guaranteed, and they should probably be interpreted with caution. .. GENERATED FROM PYTHON SOURCE LINES 313-335 .. code-block:: default from sklearn.model_selection import cross_validate from sklearn.model_selection import RepeatedKFold cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0) for est in cv_model['estimator']], columns=feature_names ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.xlabel('Coefficient importance') plt.title('Coefficient importance and its variability') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_006.png :alt: Coefficient importance and its variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 336-348 The problem of correlated variables ----------------------------------- The AGE and EXPERIENCE coefficients are affected by strong variability which might be due to the collinearity between the 2 features: as AGE and EXPERIENCE vary together in the data, their effect is difficult to tease apart. To verify this interpretation we plot the variability of the AGE and EXPERIENCE coefficient. .. _covariation: .. GENERATED FROM PYTHON SOURCE LINES 348-358 .. code-block:: default plt.ylabel('Age coefficient') plt.xlabel('Experience coefficient') plt.grid(True) plt.xlim(-0.4, 0.5) plt.ylim(-0.4, 0.5) plt.scatter(coefs["AGE"], coefs["EXPERIENCE"]) _ = plt.title('Co-variations of coefficients for AGE and EXPERIENCE ' 'across folds') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_007.png :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 359-364 Two regions are populated: when the EXPERIENCE coefficient is positive the AGE one is negative and viceversa. To go further we remove one of the 2 features and check what is the impact on the model stability. .. GENERATED FROM PYTHON SOURCE LINES 364-386 .. code-block:: default column_to_drop = ['AGE'] cv_model = cross_validate( model, X.drop(columns=column_to_drop), y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.drop(columns=column_to_drop).std(axis=0) for est in cv_model['estimator']], columns=feature_names[:-1] ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.title('Coefficient importance and its variability') plt.xlabel('Coefficient importance') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_008.png :alt: Coefficient importance and its variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 387-401 The estimation of the EXPERIENCE coefficient is now less variable and remain important for all models trained during cross-validation. .. _scaling_num: Preprocessing numerical variables --------------------------------- As said above (see ":ref:the-pipeline"), we could also choose to scale numerical values before training the model. This can be useful to apply a similar amount regularization to all of them in the Ridge. The preprocessor is redefined in order to subtract the mean and scale variables to unit variance. .. GENERATED FROM PYTHON SOURCE LINES 401-410 .. code-block:: default from sklearn.preprocessing import StandardScaler preprocessor = make_column_transformer( (OneHotEncoder(drop='if_binary'), categorical_columns), (StandardScaler(), numerical_columns), remainder='passthrough' ) .. GENERATED FROM PYTHON SOURCE LINES 411-412 The model will stay unchanged. .. GENERATED FROM PYTHON SOURCE LINES 412-424 .. code-block:: default model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 425-428 Again, we check the performance of the computed model using, for example, the median absolute error of the model and the R squared coefficient. .. GENERATED FROM PYTHON SOURCE LINES 428-447 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, small regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_009.png :alt: Ridge model, small regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 448-449 For the coefficient analysis, scaling is not needed this time. .. GENERATED FROM PYTHON SOURCE LINES 449-459 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_010.png :alt: Ridge model, small regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 460-461 We now inspect the coefficients across several cross-validation folds. .. GENERATED FROM PYTHON SOURCE LINES 461-478 .. code-block:: default cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ for est in cv_model['estimator']], columns=feature_names ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.title('Coefficient variability') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_011.png :alt: Coefficient variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 479-492 The result is quite similar to the non-normalized case. Linear models with regularization --------------------------------- In machine-learning practice, Ridge Regression is more often used with non-negligible regularization. Above, we limited this regularization to a very little amount. Regularization improves the conditioning of the problem and reduces the variance of the estimates. RidgeCV applies cross validation in order to determine which value of the regularization parameter (alpha) is best suited for prediction. .. GENERATED FROM PYTHON SOURCE LINES 492-506 .. code-block:: default from sklearn.linear_model import RidgeCV model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=RidgeCV(alphas=np.logspace(-10, 10, 21)), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 507-508 First we check which value of :math:\alpha has been selected. .. GENERATED FROM PYTHON SOURCE LINES 508-511 .. code-block:: default model[-1].regressor_.alpha_ .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 10.0 .. GENERATED FROM PYTHON SOURCE LINES 512-513 Then we check the quality of the predictions. .. GENERATED FROM PYTHON SOURCE LINES 513-533 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_012.png :alt: Ridge model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 534-536 The ability to reproduce the data of the regularized model is similar to the one of the non-regularized model. .. GENERATED FROM PYTHON SOURCE LINES 536-546 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_013.png :alt: Ridge model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 547-560 The coefficients are significantly different. AGE and EXPERIENCE coefficients are both positive but they now have less influence on the prediction. The regularization reduces the influence of correlated variables on the model because the weight is shared between the two predictive variables, so neither alone would have strong weights. On the other hand, the weights obtained with regularization are more stable (see the :ref:ridge_regression User Guide section). This increased stability is visible from the plot, obtained from data perturbations, in a cross validation. This plot can be compared with the :ref:previous one. .. GENERATED FROM PYTHON SOURCE LINES 560-581 .. code-block:: default cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0) for est in cv_model['estimator']], columns=feature_names ) plt.ylabel('Age coefficient') plt.xlabel('Experience coefficient') plt.grid(True) plt.xlim(-0.4, 0.5) plt.ylim(-0.4, 0.5) plt.scatter(coefs["AGE"], coefs["EXPERIENCE"]) _ = plt.title('Co-variations of coefficients for AGE and EXPERIENCE ' 'across folds') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_014.png :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 582-593 Linear models with sparse coefficients -------------------------------------- Another possibility to take into account correlated variables in the dataset, is to estimate sparse coefficients. In some way we already did it manually when we dropped the AGE column in a previous Ridge estimation. Lasso models (see the :ref:lasso User Guide section) estimates sparse coefficients. LassoCV applies cross validation in order to determine which value of the regularization parameter (alpha) is best suited for the model estimation. .. GENERATED FROM PYTHON SOURCE LINES 593-607 .. code-block:: default from sklearn.linear_model import LassoCV model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=LassoCV(alphas=np.logspace(-10, 10, 21), max_iter=100000), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 608-609 First we verify which value of :math:\alpha has been selected. .. GENERATED FROM PYTHON SOURCE LINES 609-612 .. code-block:: default model[-1].regressor_.alpha_ .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.001 .. GENERATED FROM PYTHON SOURCE LINES 613-614 Then we check the quality of the predictions. .. GENERATED FROM PYTHON SOURCE LINES 614-634 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Lasso model, regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_015.png :alt: Lasso model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 635-636 For our dataset, again the model is not very predictive. .. GENERATED FROM PYTHON SOURCE LINES 636-646 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Lasso model, regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_016.png :alt: Lasso model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 647-672 A Lasso model identifies the correlation between AGE and EXPERIENCE and suppresses one of them for the sake of the prediction. It is important to keep in mind that the coefficients that have been dropped may still be related to the outcome by themselves: the model chose to suppress them because they bring little or no additional information on top of the other features. Additionnaly, this selection is unstable for correlated features, and should be interpreted with caution. Lessons learned --------------- * Coefficients must be scaled to the same unit of measure to retrieve feature importance. Scaling them with the standard-deviation of the feature is a useful proxy. * Coefficients in multivariate linear models represent the dependency between a given feature and the target, **conditional** on the other features. * Correlated features induce instabilities in the coefficients of linear models and their effects cannot be well teased apart. * Different linear models respond differently to feature correlation and coefficients could significantly vary from one another. * Inspecting coefficients across the folds of a cross-validation loop gives an idea of their stability. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 11.356 seconds) .. _sphx_glr_download_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/inspection/plot_linear_model_coefficient_interpretation.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_linear_model_coefficient_interpretation.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_linear_model_coefficient_interpretation.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _