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%matplotlib inline

Effect of transforming the targets in regression model¶

In this example, we give an overview of :class:~sklearn.compose.TransformedTargetRegressor. We use two examples to 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 Ames housing data set.

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# Author: Guillaume Lemaitre <[email protected]>
# License: BSD 3 clause

import numpy as np
import matplotlib
import matplotlib.pyplot as plt

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

Synthetic example¶

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# `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 dataset is generated. The targets y are modified by:

translating all targets such that all entries are non-negative (by adding the absolute value of the lowest y) and
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.

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X, y = make_regression(n_samples=10000, noise=100, random_state=0)
y = np.expm1((y + abs(y.min())) / 200)
y_trans = np.log1p(y)

Below we plot the probability density functions of the target before and after applying the logarithmic functions.

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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=0.06, x=0.53)
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 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).

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f, (ax0, ax1) = plt.subplots(1, 2, sharey=True)
# Use linear model
regr = RidgeCV()
regr.fit(X_train, y_train)
y_pred = regr.predict(X_test)
# Plot results
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])
# Transform targets and use same linear model
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 Ames housing data set is used to show the impact of transforming the targets before learning a model. In this example, the target to be predicted is the selling price of each house.

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from sklearn.datasets import fetch_openml
from sklearn.preprocessing import QuantileTransformer, quantile_transform

ames = fetch_openml(name="house_prices", as_frame=True)
# Keep only numeric columns
X = ames.data.select_dtypes(np.number)
# Remove columns with NaN or Inf values
X = X.drop(columns=['LotFrontage', 'GarageYrBlt', 'MasVnrArea'])
y = ames.target
y_trans = quantile_transform(y.to_frame(),
                             n_quantiles=900,
                             output_distribution='normal',
                             copy=True).squeeze()

A :class:~sklearn.preprocessing.QuantileTransformer is used to normalize the target distribution before applying a :class:~sklearn.linear_model.RidgeCV model.

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f, (ax0, ax1) = plt.subplots(1, 2)

ax0.hist(y, bins=100, **density_param)
ax0.set_ylabel('Probability')
ax0.set_xlabel('Target')
ax0.text(s='Target distribution', x=1.2e5, y=9.8e-6, fontsize=12)
ax0.ticklabel_format(axis="both", style="sci", scilimits=(0, 0))

ax1.hist(y_trans, bins=100, **density_param)
ax1.set_ylabel('Probability')
ax1.set_xlabel('Target')
ax1.text(s='Transformed target distribution', x=-6.8, y=0.479, fontsize=12)

f.suptitle("Ames housing data: selling price", y=0.04)
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 transformation results in an increase in $R^2$ and large decrease of the MAE. The residual plot (predicted target - true target vs predicted target) without target transformation takes on a curved, 'reverse smile' shape due to residual values that vary depending on the value of predicted target. With target transformation, the shape is more linear indicating better model fit.

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f, (ax0, ax1) = plt.subplots(2, 2, sharey='row', figsize=(6.5, 8))

regr = RidgeCV()
regr.fit(X_train, y_train)
y_pred = regr.predict(X_test)

ax0[0].scatter(y_pred, y_test, s=8)
ax0[0].plot([0, 7e5], [0, 7e5], '--k')
ax0[0].set_ylabel('True target')
ax0[0].set_xlabel('Predicted target')
ax0[0].text(s='Ridge regression \n without target transformation', x=-5e4,
            y=8e5, fontsize=12, multialignment='center')
ax0[0].text(3e4, 64e4, r'$R^2$=%.2f, MAE=%.2f' % (
    r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred)))
ax0[0].set_xlim([0, 7e5])
ax0[0].set_ylim([0, 7e5])
ax0[0].ticklabel_format(axis="both", style="sci", scilimits=(0, 0))

ax1[0].scatter(y_pred, (y_pred - y_test), s=8)
ax1[0].set_ylabel('Residual')
ax1[0].set_xlabel('Predicted target')
ax1[0].ticklabel_format(axis="both", style="sci", scilimits=(0, 0))

regr_trans = TransformedTargetRegressor(
    regressor=RidgeCV(),
    transformer=QuantileTransformer(n_quantiles=900,
                                    output_distribution='normal'))
regr_trans.fit(X_train, y_train)
y_pred = regr_trans.predict(X_test)

ax0[1].scatter(y_pred, y_test, s=8)
ax0[1].plot([0, 7e5], [0, 7e5], '--k')
ax0[1].set_ylabel('True target')
ax0[1].set_xlabel('Predicted target')
ax0[1].text(s='Ridge regression \n with target transformation', x=-5e4,
            y=8e5, fontsize=12, multialignment='center')
ax0[1].text(3e4, 64e4, r'$R^2$=%.2f, MAE=%.2f' % (
    r2_score(y_test, y_pred), median_absolute_error(y_test, y_pred)))
ax0[1].set_xlim([0, 7e5])
ax0[1].set_ylim([0, 7e5])
ax0[1].ticklabel_format(axis="both", style="sci", scilimits=(0, 0))

ax1[1].scatter(y_pred, (y_pred - y_test), s=8)
ax1[1].set_ylabel('Residual')
ax1[1].set_xlabel('Predicted target')
ax1[1].ticklabel_format(axis="both", style="sci", scilimits=(0, 0))

f.suptitle("Ames housing data: selling price", y=0.035)

plt.show()