最小二乘法-非线性拟合-正则化-python实现

需要一些数学推导

Lambda的确定不太清楚，0.001是试出来的

当函数发生变化时，或者拟合函数最高系数变化时，Lambda的取值都应当相应变化

import numpy as np
import matplotlib.pyplot as plt

plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
# 处理中文乱码


def init_fx_data():
    X = np.arange(0, 1, 0.1)
    Y = [np.sin(2.0*np.pi*x) + (np.random.randint(low=-30, high=30) / 100) for x in X]
    return X, Y


def filling_X_Y(xs, ys, m):
    X, Y = [], []
    for x in xs:
        row = [x ** (i + 1) for i in range(m)]
        X.append(row)
    for y in ys:
        Y.append(y)
    return np.array(X), np.array(Y)


def draw_figure(xs, ys, A, n):
    p = plt.figure().add_subplot(111)
    X, Y = np.arange(min(xs), max(xs), 0.01), []
    for i in range(0, len(X)):
        y = 0.0
        for k in range(0, n):
            y += A[k] * X[i] ** (k + 1)
        Y.append(y)
    p.plot(X, Y, color='r', linestyle='-', marker='', label='多项式拟合曲线')
    p.plot(xs, ys, color='b', linestyle='', marker='.', label='曲线真实数据')
    plt.title(s='最小二乘法拟合多项式N={}的函数曲线f(x)'.format(n))
    plt.legend(loc="best")  # 添加默认图例到合适位置
    plt.show()


if __name__ == '__main__':
    x, y = init_fx_data()
    order = 10
    Lambda = 0.001
    X, Y = filling_X_Y(x, y, order)

    W1 = np.linalg.inv(X.T @ X) @ X.T @ Y
    P1 = draw_figure(x, y, W1, order)

    W2 = np.linalg.inv(X.T @ X + Lambda * np.eye(order)) @ X.T @ Y
    P2 = draw_figure(x, y, W2, order)