本文采用python sklearn库中，作为quantile regression的示例代码。以下为详细解析：
import numpy as np import matplotlib.pyplot as plt from sklearn.ensemble import GradientBoostingRegressor %matplotlib inline np.random.seed(1) #设置随机数生成的种子 def f(x): """The function to predict.""" return x * np.sin(x) #对x取正弦 #---------------------------------------------------------------------- # First the noiseless case X = np.atleast_2d(np.random.uniform(0, 10.0, size=100)).T # 随机采样并转换成数组 # numpy.atleast_2d()其他格式转换成数组 # numpy.random.uniform(low,high,size) 从一个均匀分布[low,high)中随机采样，注意定义域是左闭右开，即包含low，不包含high. # .T 数组转置. X = X.astype(np.float32) #转换数据类型 float32 减少精度 # Observations y = f(X).ravel() #多维数组转1维向量 dy = 1.5 + 1.0 * np.random.random(y.shape) #生成 np.random.random(y.shape) y.shape大小矩阵的随机数 noise = np.random.normal(0, dy) #生成一个正态分布，正态分布标准差（宽度）为dy y += noise y = y.astype(np.float32) # Mesh the input space for evaluations of the real function, the prediction and # its MSE xx = np.atleast_2d(np.linspace(0, 10, 1000)).T #np.linespace 产生从0到10，1000个等差数列中的数字 xx = xx.astype(np.float32) alpha = 0.95 clf = GradientBoostingRegressor(loss='quantile', alpha=alpha, n_estimators=250, max_depth=3, learning_rate=.1, min_samples_leaf=9, min_samples_split=9) #loss: 选择损失函数，默认值为ls(least squres) # learning_rate: 学习率， # alpha ，quantile regression的置信度 # n_estimators: 弱学习器的数目，250 # max_depth: 每一个学习器的最大深度，限制回归树的节点数目，默认为3 # min_samples_split: 可以划分为内部节点的最小样本数，9 # min_samples_leaf: 叶节点所需的最小样本数，9 clf.fit(X, y) # Make the prediction on the meshed x-axis y_upper = clf.predict(xx) #用训练好的分类器去预测xx clf.set_params(alpha=1.0 - alpha) #1-0.95=0.05 再训练一条 clf.fit(X, y) # Make the prediction on the meshed x-axis y_lower = clf.predict(xx) clf.set_params(loss='ls') #改成 ls ：least squares regression 最小二乘回归 clf.fit(X, y) # Make the prediction on the meshed x-axis y_pred = clf.predict(xx) # Plot the function, the prediction and the 95% confidence interval based on # the MSE #matplot画图 y_upper(alpha=0.95)画一条 y_lower(alpha=0.05)画一条 普通的线性回归（prediction）画一条 fig = plt.figure() plt.plot(xx, f(xx), 'g:', label=r'$f(x) = x\,\sin(x)$') plt.plot(X, y, 'b.', markersize=10, label=u'Observations') plt.plot(xx, y_pred, 'r-', label=u'Prediction') plt.plot(xx, y_upper, 'k-') plt.plot(xx, y_lower, 'k-') plt.fill(np.concatenate([xx, xx[::-1]]), np.concatenate([y_upper, y_lower[::-1]]), alpha=.5, fc='b', ec='None', label='95% prediction interval') plt.xlabel('$x$') plt.ylabel('$f(x)$') plt.ylim(-10, 20) plt.legend(loc='upper left') plt.show()
https://scikit-learn.org/stable/auto_examples/ensemble/plot_gradient_boosting_quantile.html#sphx-glr-auto-examples-ensemble-plot-gradient-boosting-quantile-py https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.GradientBoostingRegressor.html
