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Effect of transforming the targets in regression model¶
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 <guillaume.lemaitre@inria.fr>
# License: BSD 3 clause
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
print(__doc__)
Synthetic example¶
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
from sklearn.utils.fixes import parse_version
# `normed` is being deprecated in favor of `density` in histograms
if parse_version(matplotlib.__version__) >= parse_version('2.1'):
density_param = {'density': True}
else:
density_param = {'normed': True}
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, **density_param)
ax0.set_xlim([0, 2000])
ax0.set_ylabel('Probability')
ax0.set_xlabel('Target')
ax0.set_title('Target distribution')
ax1.hist(y_trans, bins=100, **density_param)
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])
Real-world data set¶
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],
n_quantiles=300,
output_distribution='normal',
copy=True).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, **density_param)
ax0.set_ylabel('Probability')
ax0.set_xlabel('Target')
ax0.set_title('Target distribution')
ax1.hist(y_trans, bins=100, **density_param)
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(n_quantiles=300,
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 1.161 seconds)
