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保序回归#

在生成的数据上展示保序回归（具有同方差均匀噪声的非线性单调趋势）。

保序回归算法在训练数据上找到函数的非递减近似，同时最小化均方误差。这种非参数模型的优点在于，它除了单调性之外，不对目标函数的形状做任何假设。为了比较，还展示了线性回归。

右侧的图显示了通过阈值点的线性插值得到的模型预测函数。阈值点是训练输入观测值的一个子集，其匹配的目标值由保序非参数拟合计算得出。

# 作者：scikit-learn 开发者
# SPDX-License-Identifier: BSD-3-Clause

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.collections import LineCollection

from sklearn.isotonic import IsotonicRegression
from sklearn.linear_model import LinearRegression
from sklearn.utils import check_random_state

n = 100
x = np.arange(n)
rs = check_random_state(0)
y = rs.randint(-50, 50, size=(n,)) + 50.0 * np.log1p(np.arange(n))

拟合等渗回归和线性回归模型：

ir = IsotonicRegression(out_of_bounds="clip")
y_ = ir.fit_transform(x, y)

lr = LinearRegression()
lr.fit(x[:, np.newaxis], y)  # x needs to be 2d for LinearRegression

LinearRegression()

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绘制结果：

segments = [[[i, y[i]], [i, y_[i]]] for i in range(n)]
lc = LineCollection(segments, zorder=0)
lc.set_array(np.ones(len(y)))
lc.set_linewidths(np.full(n, 0.5))

fig, (ax0, ax1) = plt.subplots(ncols=2, figsize=(12, 6))

ax0.plot(x, y, "C0.", markersize=12)
ax0.plot(x, y_, "C1.-", markersize=12)
ax0.plot(x, lr.predict(x[:, np.newaxis]), "C2-")
ax0.add_collection(lc)
ax0.legend(("Training data", "Isotonic fit", "Linear fit"), loc="lower right")
ax0.set_title("Isotonic regression fit on noisy data (n=%d)" % n)

x_test = np.linspace(-10, 110, 1000)
ax1.plot(x_test, ir.predict(x_test), "C1-")
ax1.plot(ir.X_thresholds_, ir.y_thresholds_, "C1.", markersize=12)
ax1.set_title("Prediction function (%d thresholds)" % len(ir.X_thresholds_))

plt.show()