实现线性不可分logistic逻辑回归

我们目前所学的都是线性回归，例如$$y=w_1{x}_1+w_2{x}_2+b$$
用肉眼来看数据集的话不难发现，线性回归没有用了，那么根据课程所学，我们是不是可以增加 $x_3=x_1{x}_x,x_4=x_1^2,x_5=x_2^2$呢？那么逻辑回归就可以变成
$$y=w_1{x}_1+w_2{x}_2+w_3{x}_3+w_4{x}_4+w_5{x}_5+b$$

import numpy as np
import pandas as pd
from matplotlib import pyplot as pltdef sigmoid(x):return 1/(1+np.exp(-x))def compute_loss(X, y, w, b, lambada):m = X.shape[0]cost = 0.cost_gradient = 0.for i in range(m):z_i = sigmoid((np.dot(X[i], w) + b))cost += -y[i] * np.log(z_i) - (1 - y[i]) * np.log(1 - z_i)cost_gradient += w[i] ** 2return cost / m + lambada * cost_gradient / (2 * m)def compute_gradient_logistic(X, y, w, b, eta, lambada):m, n = X.shapedb_w = np.zeros(n)db_b = 0for i in range(m):z_i = sigmoid((np.dot(X[i], w) + b))err_i = z_i - y[i]for j in range(n):db_w[j] += err_i * X[i][j]db_b += err_ireturn db_w / m, db_b / mdef gradient_descent(X, y, w, b, eta, lambada, iterator):m, n = X.shapefor i in range(iterator):w_tmp = np.copy(w)b_tmp = bdb_w, db_b = compute_gradient_logistic(X, y, w_tmp, b, eta, lambada)db_w += lambada * w / mw = w - eta * db_wb = b - eta * db_breturn w, bif __name__ == '__main__':data = pd.read_csv(r'D:\BaiduNetdiskDownload\data_sets\ex2data2.txt')X_train = data.iloc[:, 0:-1].to_numpy()y_train = data.iloc[:, -1].to_numpy()x1 = (X_train[:, 0] * X_train[:, 1]).reshape(-1, 1)x2 = (X_train[:, 0] ** 2).reshape(-1, 1)x3 = (X_train[:, 1] ** 2).reshape(-1, 1)X_train = np.hstack((X_train, x1, x2, x3))w_tmp = np.zeros_like(X_train[0])b_tmp = 0.alph = 0.1lambada = 0.01iters = 10000w_out, b_out = gradient_descent(X_train, y_train, w_tmp, b_tmp, alph, lambada, iters)count = 0for i in range(X_train.shape[0]):ans = sigmoid(np.dot(X_train[i], w_out) + b_out)prediction = 1 if ans > 0.5 else 0if y_train[i] == prediction:count += 1print('Accuracy = {}'.format(count/X_train.shape[0]))print(w_out, b_out)plt.scatter(X_train[:, 0], X_train[:, 1], c=y_train)# 绘制决策边界x_min, x_max = X_train[:, 0].min() - 0.1, X_train[:, 0].max() + 0.1y_min, y_max = X_train[:, 1].min() - 0.1, X_train[:, 1].max() + 0.1xx, yy = np.meshgrid(np.arange(x_min, x_max, 0.01),np.arange(y_min, y_max, 0.01))# 创建与网格形状匹配的特征grid = np.c_[xx.ravel(), yy.ravel()]print('grid_shape : {}'.format(grid.shape))grid_x1 = (grid[:, 0] * grid[:, 1]).reshape(-1, 1)grid_x2 = (grid[:, 0] ** 2).reshape(-1, 1)grid_x3 = (grid[:, 1] ** 2).reshape(-1, 1)grid_features = np.hstack((grid, grid_x1, grid_x2, grid_x3))# 计算网格点的预测值Z = sigmoid(np.dot(grid_features, w_out) + b_out)Z = Z.reshape(xx.shape)# 绘制决策边界plt.contour(xx, yy, Z, levels=[0.5], colors='g')# 显示图形plt.xlabel('x1')plt.ylabel('x2')plt.title('Decision Boundary')plt.show()

一些图

Accuracy = 0.8376068376068376
然后就是各个参数w1,w2,w3,w4,b
[ 2.12915132 2.82388529 -4.83135528 -8.64819153 -8.31828602] 3.7305124000753627