Effect of transforming the targets in regression model

In this example, we give an overview of 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.

# Author: Guillaume Lemaitre <guillaume.lemaitre@inria.fr>
# License: BSD 3 clause

import numpy as np
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

Synthetic example

A synthetic random regression dataset is generated. The targets y are modified by:

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

2. 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.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.

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

ax0.hist(y, bins=100, density=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, density=True)
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).

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.

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 QuantileTransformer is used to normalize the target distribution before applying a RidgeCV model.

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

ax0.hist(y, bins=100, density=True)
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=True)
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.

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()