Quantile Regression

New in version 2.0.0.

The script is inspired by this awesome example in sklearn: https://scikit-learn.org/stable/auto_examples/ensemble/plot_gradient_boosting_quantile.html


The feature is only supported using the Python package. In addition, quantile crossing can happen due to limitation in the algorithm.

import argparse
from typing import Dict

import numpy as np
from sklearn.model_selection import train_test_split

import xgboost as xgb

def f(x: np.ndarray) -> np.ndarray:
    """The function to predict."""
    return x * np.sin(x)

def quantile_loss(args: argparse.Namespace) -> None:
    """Train a quantile regression model."""
    rng = np.random.RandomState(1994)
    # Generate a synthetic dataset for demo, the generate process is from the sklearn
    # example.
    X = np.atleast_2d(rng.uniform(0, 10.0, size=1000)).T
    expected_y = f(X).ravel()

    sigma = 0.5 + X.ravel() / 10.0
    noise = rng.lognormal(sigma=sigma) - np.exp(sigma**2.0 / 2.0)
    y = expected_y + noise

    # Train on 0.05 and 0.95 quantiles. The model is similar to multi-class and
    # multi-target models.
    alpha = np.array([0.05, 0.5, 0.95])
    evals_result: Dict[str, Dict] = {}

    X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=rng)
    # We will be using the `hist` tree method, quantile DMatrix can be used to preserve
    # memory.
    # Do not use the `exact` tree method for quantile regression, otherwise the
    # performance might drop.
    Xy = xgb.QuantileDMatrix(X, y)
    # use Xy as a reference
    Xy_test = xgb.QuantileDMatrix(X_test, y_test, ref=Xy)

    booster = xgb.train(
            # Use the quantile objective function.
            "objective": "reg:quantileerror",
            "tree_method": "hist",
            "quantile_alpha": alpha,
            # Let's try not to overfit.
            "learning_rate": 0.04,
            "max_depth": 5,
        # The evaluation result is a weighted average across multiple quantiles.
        evals=[(Xy, "Train"), (Xy_test, "Test")],
    xx = np.atleast_2d(np.linspace(0, 10, 1000)).T
    scores = booster.inplace_predict(xx)
    # dim 1 is the quantiles
    assert scores.shape[0] == xx.shape[0]
    assert scores.shape[1] == alpha.shape[0]

    y_lower = scores[:, 0]  # alpha=0.05
    y_med = scores[:, 1]  # alpha=0.5, median
    y_upper = scores[:, 2]  # alpha=0.95

    # Train a mse model for comparison
    booster = xgb.train(
            "objective": "reg:squarederror",
            "tree_method": "hist",
            # Let's try not to overfit.
            "learning_rate": 0.04,
            "max_depth": 5,
        evals=[(Xy, "Train"), (Xy_test, "Test")],
    xx = np.atleast_2d(np.linspace(0, 10, 1000)).T
    y_pred = booster.inplace_predict(xx)

    if args.plot:
        from matplotlib import pyplot as plt

        fig = plt.figure(figsize=(10, 10))
        plt.plot(xx, f(xx), "g:", linewidth=3, label=r"$f(x) = x\,\sin(x)$")
        plt.plot(X_test, y_test, "b.", markersize=10, label="Test observations")
        plt.plot(xx, y_med, "r-", label="Predicted median")
        plt.plot(xx, y_pred, "m-", label="Predicted mean")
        plt.plot(xx, y_upper, "k-")
        plt.plot(xx, y_lower, "k-")
            xx.ravel(), y_lower, y_upper, alpha=0.4, label="Predicted 90% interval"
        plt.ylim(-10, 25)
        plt.legend(loc="upper left")

if __name__ == "__main__":
    parser = argparse.ArgumentParser()
        help="Specify it to enable plotting the outputs.",
    args = parser.parse_args()

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