.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/linear_model/plot_quantile_regression.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_linear_model_plot_quantile_regression.py: =================== 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. .. GENERATED FROM PYTHON SOURCE LINES 16-20 .. code-block:: Python # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause .. GENERATED FROM PYTHON SOURCE LINES 21-28 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`. .. GENERATED FROM PYTHON SOURCE LINES 28-35 .. code-block:: Python 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 .. GENERATED FROM PYTHON SOURCE LINES 36-41 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. .. GENERATED FROM PYTHON SOURCE LINES 41-45 .. code-block:: Python 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)) .. GENERATED FROM PYTHON SOURCE LINES 46-48 Let's first visualize the datasets as well as the distribution of the residuals `y - mean(y)`. .. GENERATED FROM PYTHON SOURCE LINES 48-76 .. code-block:: Python 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") .. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_001.png :alt: Dataset with heteroscedastic Normal distributed targets, Dataset with asymmetric Pareto distributed target, Residuals distribution for heteroscedastic Normal distributed targets, Residuals distribution for asymmetric Pareto distributed target :srcset: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 77-112 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. .. GENERATED FROM PYTHON SOURCE LINES 114-133 .. code-block:: Python 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 ) .. GENERATED FROM PYTHON SOURCE LINES 134-137 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. .. GENERATED FROM PYTHON SOURCE LINES 137-163 .. code-block:: Python 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") .. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_002.png :alt: Quantiles of heteroscedastic Normal distributed target :srcset: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 164-178 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. .. GENERATED FROM PYTHON SOURCE LINES 178-195 .. code-block:: Python 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 ) .. GENERATED FROM PYTHON SOURCE LINES 196-223 .. code-block:: Python 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") .. image-sg:: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_003.png :alt: Quantiles of asymmetric Pareto distributed target :srcset: /auto_examples/linear_model/images/sphx_glr_plot_quantile_regression_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 224-248 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 error 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. Let's first 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. .. GENERATED FROM PYTHON SOURCE LINES 248-268 .. code-block:: Python 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( f"""Training error (in-sample performance) {linear_regression.__class__.__name__}: MAE = {mean_absolute_error(y_pareto, y_pred_lr):.3f} MSE = {mean_squared_error(y_pareto, y_pred_lr):.3f} {quantile_regression.__class__.__name__}: MAE = {mean_absolute_error(y_pareto, y_pred_qr):.3f} MSE = {mean_squared_error(y_pareto, y_pred_qr):.3f} """ ) .. rst-class:: sphx-glr-script-out .. code-block:: none Training error (in-sample performance) LinearRegression: MAE = 1.805 MSE = 6.486 QuantileRegressor: MAE = 1.670 MSE = 7.025 .. GENERATED FROM PYTHON SOURCE LINES 269-280 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. .. GENERATED FROM PYTHON SOURCE LINES 280-307 .. code-block:: Python 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( f"""Test error (cross-validated performance) {linear_regression.__class__.__name__}: MAE = {-cv_results_lr["test_neg_mean_absolute_error"].mean():.3f} MSE = {-cv_results_lr["test_neg_mean_squared_error"].mean():.3f} {quantile_regression.__class__.__name__}: MAE = {-cv_results_qr["test_neg_mean_absolute_error"].mean():.3f} MSE = {-cv_results_qr["test_neg_mean_squared_error"].mean():.3f} """ ) .. rst-class:: sphx-glr-script-out .. code-block:: none Test error (cross-validated performance) LinearRegression: MAE = 1.732 MSE = 6.690 QuantileRegressor: MAE = 1.679 MSE = 7.129 .. GENERATED FROM PYTHON SOURCE LINES 308-309 We reach similar conclusions on the out-of-sample evaluation. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.471 seconds) .. _sphx_glr_download_auto_examples_linear_model_plot_quantile_regression.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/linear_model/plot_quantile_regression.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_quantile_regression.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_quantile_regression.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_quantile_regression.zip ` .. include:: plot_quantile_regression.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_