Plot Quantile Regression¶
=================== Quantile regression¶
This example illustrates how quantile regression can predict non-trivial conditional quantiles.
The left figure shows the case when the error distribution is normal, but has non-constant variance, i.e. with heteroscedasticity.
The right figure shows an example of an asymmetric error distribution, namely the Pareto distribution.
Imports for Quantile Regression¶
Standard linear regression (LinearRegression) estimates the conditional mean of the target variable, but the mean can be misleading when the noise distribution is heteroscedastic (variance changes with x) or asymmetric (e.g., heavy-tailed). QuantileRegressor estimates conditional quantiles instead – for instance, the median (50th percentile), or any other quantile like the 5th or 95th percentile.
Why quantiles matter: Predicting the median is more robust to outliers than predicting the mean (MAE vs. MSE). Predicting extreme quantiles (5% and 95%) gives prediction intervals that capture the spread of the data, which is essential for risk assessment in finance, weather forecasting, and supply chain planning. Mathematically, the quantile regression loss is the asymmetric “pinball” loss: for quantile q, overestimates are penalized by (1-q) and underestimates by q, so the optimal prediction is exactly the q-th quantile of the conditional distribution.
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
# %%
# Dataset generation
# ------------------
#
# To illustrate the behaviour of quantile regression, we will generate two
# synthetic datasets. The true generative random processes for both datasets
# will be composed by the same expected value with a linear relationship with a
# single feature `x`.
import numpy as np
rng = np.random.RandomState(42)
x = np.linspace(start=0, stop=10, num=100)
X = x[:, np.newaxis]
y_true_mean = 10 + 0.5 * x
# %%
# We will create two subsequent problems by changing the distribution of the
# target `y` while keeping the same expected value:
#
# - in the first case, a heteroscedastic Normal noise is added;
# - in the second case, an asymmetric Pareto noise is added.
y_normal = y_true_mean + rng.normal(loc=0, scale=0.5 + 0.5 * x, size=x.shape[0])
a = 5
y_pareto = y_true_mean + 10 * (rng.pareto(a, size=x.shape[0]) - 1 / (a - 1))
# %%
# Let's first visualize the datasets as well as the distribution of the
# residuals `y - mean(y)`.
import matplotlib.pyplot as plt
_, axs = plt.subplots(nrows=2, ncols=2, figsize=(15, 11), sharex="row", sharey="row")
axs[0, 0].plot(x, y_true_mean, label="True mean")
axs[0, 0].scatter(x, y_normal, color="black", alpha=0.5, label="Observations")
axs[1, 0].hist(y_true_mean - y_normal, edgecolor="black")
axs[0, 1].plot(x, y_true_mean, label="True mean")
axs[0, 1].scatter(x, y_pareto, color="black", alpha=0.5, label="Observations")
axs[1, 1].hist(y_true_mean - y_pareto, edgecolor="black")
axs[0, 0].set_title("Dataset with heteroscedastic Normal distributed targets")
axs[0, 1].set_title("Dataset with asymmetric Pareto distributed target")
axs[1, 0].set_title(
"Residuals distribution for heteroscedastic Normal distributed targets"
)
axs[1, 1].set_title("Residuals distribution for asymmetric Pareto distributed target")
axs[0, 0].legend()
axs[0, 1].legend()
axs[0, 0].set_ylabel("y")
axs[1, 0].set_ylabel("Counts")
axs[0, 1].set_xlabel("x")
axs[0, 0].set_xlabel("x")
axs[1, 0].set_xlabel("Residuals")
_ = axs[1, 1].set_xlabel("Residuals")
# %%
# With the heteroscedastic Normal distributed target, we observe that the
# variance of the noise is increasing when the value of the feature `x` is
# increasing.
#
# With the asymmetric Pareto distributed target, we observe that the positive
# residuals are bounded.
#
# These types of noisy targets make the estimation via
# :class:`~sklearn.linear_model.LinearRegression` less efficient, i.e. we need
# more data to get stable results and, in addition, large outliers can have a
# huge impact on the fitted coefficients. (Stated otherwise: in a setting with
# constant variance, ordinary least squares estimators converge much faster to
# the *true* coefficients with increasing sample size.)
#
# In this asymmetric setting, the median or different quantiles give additional
# insights. On top of that, median estimation is much more robust to outliers
# and heavy tailed distributions. But note that extreme quantiles are estimated
# by very few data points. 95% quantile are more or less estimated by the 5%
# largest values and thus also a bit sensitive outliers.
#
# In the remainder of this tutorial, we will show how
# :class:`~sklearn.linear_model.QuantileRegressor` can be used in practice and
# give the intuition into the properties of the fitted models. Finally,
# we will compare the both :class:`~sklearn.linear_model.QuantileRegressor`
# and :class:`~sklearn.linear_model.LinearRegression`.
#
# Fitting a `QuantileRegressor`
# -----------------------------
#
# In this section, we want to estimate the conditional median as well as
# a low and high quantile fixed at 5% and 95%, respectively. Thus, we will get
# three linear models, one for each quantile.
#
# We will use the quantiles at 5% and 95% to find the outliers in the training
# sample beyond the central 90% interval.
# %%
from sklearn.linear_model import QuantileRegressor
quantiles = [0.05, 0.5, 0.95]
predictions = {}
out_bounds_predictions = np.zeros_like(y_true_mean, dtype=np.bool_)
for quantile in quantiles:
qr = QuantileRegressor(quantile=quantile, alpha=0)
y_pred = qr.fit(X, y_normal).predict(X)
predictions[quantile] = y_pred
if quantile == min(quantiles):
out_bounds_predictions = np.logical_or(
out_bounds_predictions, y_pred >= y_normal
)
elif quantile == max(quantiles):
out_bounds_predictions = np.logical_or(
out_bounds_predictions, y_pred <= y_normal
)
# %%
# Now, we can plot the three linear models and the distinguished samples that
# are within the central 90% interval from samples that are outside this
# interval.
plt.plot(X, y_true_mean, color="black", linestyle="dashed", label="True mean")
for quantile, y_pred in predictions.items():
plt.plot(X, y_pred, label=f"Quantile: {quantile}")
plt.scatter(
x[out_bounds_predictions],
y_normal[out_bounds_predictions],
color="black",
marker="+",
alpha=0.5,
label="Outside interval",
)
plt.scatter(
x[~out_bounds_predictions],
y_normal[~out_bounds_predictions],
color="black",
alpha=0.5,
label="Inside interval",
)
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
_ = plt.title("Quantiles of heteroscedastic Normal distributed target")
# %%
# Since the noise is still Normally distributed, in particular is symmetric,
# the true conditional mean and the true conditional median coincide. Indeed,
# we see that the estimated median almost hits the true mean. We observe the
# effect of having an increasing noise variance on the 5% and 95% quantiles:
# the slopes of those quantiles are very different and the interval between
# them becomes wider with increasing `x`.
#
# To get an additional intuition regarding the meaning of the 5% and 95%
# quantiles estimators, one can count the number of samples above and below the
# predicted quantiles (represented by a cross on the above plot), considering
# that we have a total of 100 samples.
#
# We can repeat the same experiment using the asymmetric Pareto distributed
# target.
quantiles = [0.05, 0.5, 0.95]
predictions = {}
out_bounds_predictions = np.zeros_like(y_true_mean, dtype=np.bool_)
for quantile in quantiles:
qr = QuantileRegressor(quantile=quantile, alpha=0)
y_pred = qr.fit(X, y_pareto).predict(X)
predictions[quantile] = y_pred
if quantile == min(quantiles):
out_bounds_predictions = np.logical_or(
out_bounds_predictions, y_pred >= y_pareto
)
elif quantile == max(quantiles):
out_bounds_predictions = np.logical_or(
out_bounds_predictions, y_pred <= y_pareto
)
# %%
plt.plot(X, y_true_mean, color="black", linestyle="dashed", label="True mean")
for quantile, y_pred in predictions.items():
plt.plot(X, y_pred, label=f"Quantile: {quantile}")
plt.scatter(
x[out_bounds_predictions],
y_pareto[out_bounds_predictions],
color="black",
marker="+",
alpha=0.5,
label="Outside interval",
)
plt.scatter(
x[~out_bounds_predictions],
y_pareto[~out_bounds_predictions],
color="black",
alpha=0.5,
label="Inside interval",
)
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
_ = plt.title("Quantiles of asymmetric Pareto distributed target")
# %%
# Due to the asymmetry of the distribution of the noise, we observe that the
# true mean and estimated conditional median are different. We also observe
# that each quantile model has different parameters to better fit the desired
# quantile. Note that ideally, all quantiles would be parallel in this case,
# which would become more visible with more data points or less extreme
# quantiles, e.g. 10% and 90%.
#
# Comparing `QuantileRegressor` and `LinearRegression`
# ----------------------------------------------------
#
# In this section, we will linger on the difference regarding the loss functions that
# :class:`~sklearn.linear_model.QuantileRegressor` and
# :class:`~sklearn.linear_model.LinearRegression` are minimizing.
#
# Indeed, :class:`~sklearn.linear_model.LinearRegression` is a least squares
# approach minimizing the mean squared error (MSE) between the training and
# predicted targets. In contrast,
# :class:`~sklearn.linear_model.QuantileRegressor` with `quantile=0.5`
# minimizes the mean absolute error (MAE) instead.
#
# Why does it matter? The loss functions specify what exactly the model is aiming
# to predict, see
# :ref:`user guide on the choice of scoring function<which_scoring_function>`.
# In short, a model minimizing MSE predicts the mean (expectation) and a model
# minimizing MAE predicts the median.
#
# Let's compute the training errors of such models in terms of mean
# squared error and mean absolute error. We will use the asymmetric Pareto
# distributed target to make it more interesting as mean and median are not
# equal.
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_absolute_error, mean_squared_error
linear_regression = LinearRegression()
quantile_regression = QuantileRegressor(quantile=0.5, alpha=0)
y_pred_lr = linear_regression.fit(X, y_pareto).predict(X)
y_pred_qr = quantile_regression.fit(X, y_pareto).predict(X)
print(
"Training error (in-sample performance)\n"
f"{'model':<20} MAE MSE\n"
f"{linear_regression.__class__.__name__:<20} "
f"{mean_absolute_error(y_pareto, y_pred_lr):5.3f} "
f"{mean_squared_error(y_pareto, y_pred_lr):5.3f}\n"
f"{quantile_regression.__class__.__name__:<20} "
f"{mean_absolute_error(y_pareto, y_pred_qr):5.3f} "
f"{mean_squared_error(y_pareto, y_pred_qr):5.3f}"
)
# %%
# On the training set, we see that MAE is lower for
# :class:`~sklearn.linear_model.QuantileRegressor` than
# :class:`~sklearn.linear_model.LinearRegression`. In contrast to that, MSE is
# lower for :class:`~sklearn.linear_model.LinearRegression` than
# :class:`~sklearn.linear_model.QuantileRegressor`. These results confirms that
# MAE is the loss minimized by :class:`~sklearn.linear_model.QuantileRegressor`
# while MSE is the loss minimized
# :class:`~sklearn.linear_model.LinearRegression`.
#
# We can make a similar evaluation by looking at the test error obtained by
# cross-validation.
from sklearn.model_selection import cross_validate
cv_results_lr = cross_validate(
linear_regression,
X,
y_pareto,
cv=3,
scoring=["neg_mean_absolute_error", "neg_mean_squared_error"],
)
cv_results_qr = cross_validate(
quantile_regression,
X,
y_pareto,
cv=3,
scoring=["neg_mean_absolute_error", "neg_mean_squared_error"],
)
print(
"Test error (cross-validated performance)\n"
f"{'model':<20} MAE MSE\n"
f"{linear_regression.__class__.__name__:<20} "
f"{-cv_results_lr['test_neg_mean_absolute_error'].mean():5.3f} "
f"{-cv_results_lr['test_neg_mean_squared_error'].mean():5.3f}\n"
f"{quantile_regression.__class__.__name__:<20} "
f"{-cv_results_qr['test_neg_mean_absolute_error'].mean():5.3f} "
f"{-cv_results_qr['test_neg_mean_squared_error'].mean():5.3f}"
)
# %%
# We reach similar conclusions on the out-of-sample evaluation.