Note
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In this example, we give an overview of the sklearn.compose.TransformedTargetRegressor
. Two examples illustrate the benefit of transforming the targets before learning a linear regression model. The first example uses synthetic data while the second example is based on the Boston housing data set.
# Author: Guillaume Lemaitre <[email protected]> # License: BSD 3 clause from __future__ import print_function, division import numpy as np import matplotlib.pyplot as plt print(__doc__)
from sklearn.datasets import make_regression from sklearn.model_selection import train_test_split from sklearn.linear_model import RidgeCV from sklearn.compose import TransformedTargetRegressor from sklearn.metrics import median_absolute_error, r2_score
A synthetic random regression problem is generated. The targets y
are modified by: (i) translating all targets such that all entries are non-negative and (ii) applying an exponential function to obtain non-linear targets which cannot be fitted using a simple linear model.
Therefore, a logarithmic (np.log1p
) and an exponential function (np.expm1
) will be used to transform the targets before training a linear regression model and using it for prediction.
X, y = make_regression(n_samples=10000, noise=100, random_state=0) y = np.exp((y + abs(y.min())) / 200) y_trans = np.log1p(y)
The following illustrate the probability density functions of the target before and after applying the logarithmic functions.
f, (ax0, ax1) = plt.subplots(1, 2) ax0.hist(y, bins=100, normed=True) ax0.set_xlim([0, 2000]) ax0.set_ylabel('Probability') ax0.set_xlabel('Target') ax0.set_title('Target distribution') ax1.hist(y_trans, bins=100, normed=True) ax1.set_ylabel('Probability') ax1.set_xlabel('Target') ax1.set_title('Transformed target distribution') f.suptitle("Synthetic data", y=0.035) f.tight_layout(rect=[0.05, 0.05, 0.95, 0.95]) X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
At first, a linear model will be applied on the original targets. Due to the non-linearity, the model trained will not be precise during the prediction. Subsequently, a logarithmic function is used to linearize the targets, allowing better prediction even with a similar linear model as reported by the median absolute error (MAE).
f, (ax0, ax1) = plt.subplots(1, 2, sharey=True) regr = RidgeCV() regr.fit(X_train, y_train) y_pred = regr.predict(X_test) ax0.scatter(y_test, y_pred) ax0.plot([0, 2000], [0, 2000], '--k') ax0.set_ylabel('Target predicted') ax0.set_xlabel('True Target') ax0.set_title('Ridge regression \n without target transformation') ax0.text(100, 1750, r'$R^2$=%.2f, MAE=%.2f' % ( r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred))) ax0.set_xlim([0, 2000]) ax0.set_ylim([0, 2000]) regr_trans = TransformedTargetRegressor(regressor=RidgeCV(), func=np.log1p, inverse_func=np.expm1) regr_trans.fit(X_train, y_train) y_pred = regr_trans.predict(X_test) ax1.scatter(y_test, y_pred) ax1.plot([0, 2000], [0, 2000], '--k') ax1.set_ylabel('Target predicted') ax1.set_xlabel('True Target') ax1.set_title('Ridge regression \n with target transformation') ax1.text(100, 1750, r'$R^2$=%.2f, MAE=%.2f' % ( r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred))) ax1.set_xlim([0, 2000]) ax1.set_ylim([0, 2000]) f.suptitle("Synthetic data", y=0.035) f.tight_layout(rect=[0.05, 0.05, 0.95, 0.95])
In a similar manner, the boston housing data set is used to show the impact of transforming the targets before learning a model. In this example, the targets to be predicted corresponds to the weighted distances to the five Boston employment centers.
from sklearn.datasets import load_boston from sklearn.preprocessing import QuantileTransformer, quantile_transform dataset = load_boston() target = np.array(dataset.feature_names) == "DIS" X = dataset.data[:, np.logical_not(target)] y = dataset.data[:, target].squeeze() y_trans = quantile_transform(dataset.data[:, target], output_distribution='normal').squeeze()
A sklearn.preprocessing.QuantileTransformer
is used such that the targets follows a normal distribution before applying a sklearn.linear_model.RidgeCV
model.
f, (ax0, ax1) = plt.subplots(1, 2) ax0.hist(y, bins=100, normed=True) ax0.set_ylabel('Probability') ax0.set_xlabel('Target') ax0.set_title('Target distribution') ax1.hist(y_trans, bins=100, normed=True) ax1.set_ylabel('Probability') ax1.set_xlabel('Target') ax1.set_title('Transformed target distribution') f.suptitle("Boston housing data: distance to employment centers", y=0.035) f.tight_layout(rect=[0.05, 0.05, 0.95, 0.95]) X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
The effect of the transformer is weaker than on the synthetic data. However, the transform induces a decrease of the MAE.
f, (ax0, ax1) = plt.subplots(1, 2, sharey=True) regr = RidgeCV() regr.fit(X_train, y_train) y_pred = regr.predict(X_test) ax0.scatter(y_test, y_pred) ax0.plot([0, 10], [0, 10], '--k') ax0.set_ylabel('Target predicted') ax0.set_xlabel('True Target') ax0.set_title('Ridge regression \n without target transformation') ax0.text(1, 9, r'$R^2$=%.2f, MAE=%.2f' % ( r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred))) ax0.set_xlim([0, 10]) ax0.set_ylim([0, 10]) regr_trans = TransformedTargetRegressor( regressor=RidgeCV(), transformer=QuantileTransformer(output_distribution='normal')) regr_trans.fit(X_train, y_train) y_pred = regr_trans.predict(X_test) ax1.scatter(y_test, y_pred) ax1.plot([0, 10], [0, 10], '--k') ax1.set_ylabel('Target predicted') ax1.set_xlabel('True Target') ax1.set_title('Ridge regression \n with target transformation') ax1.text(1, 9, r'$R^2$=%.2f, MAE=%.2f' % ( r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred))) ax1.set_xlim([0, 10]) ax1.set_ylim([0, 10]) f.suptitle("Boston housing data: distance to employment centers", y=0.035) f.tight_layout(rect=[0.05, 0.05, 0.95, 0.95]) plt.show()
Total running time of the script: ( 0 minutes 1.163 seconds)
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http://scikit-learn.org/stable/auto_examples/compose/plot_transformed_target.html