机器学习笔记 六：逻辑回归中的多分类问题之数字识别

使用逻辑回归来识别手写数字（0到9）

1. 需要用到的函数:

1. sigmoid函数：

2. 逻辑回归的代价函数：

3. 梯度下降算法：

4. 正则化的逻辑回归模型的代价函数：

5. 数字识别原理（向量化标签，1表示真，0表示假）：

2.代码实现

import numpy as np
from scipy.io import loadmat
from scipy.optimize import minimize

""" date: 2022/6/1 author: amyniez name: machine learning one vs all """


# sigmoid函数
def sigmoid(z):
    return 1 / (1 + np.exp(-z))


# 正则化的逻辑回归模型的代价函数J（theta）
def cost(cost_theta, X, y, learningRate):
    cost_theta = np.matrix(cost_theta)
    X = np.matrix(X)
    y = np.matrix(y)
    first = np.multiply(-y, np.log(sigmoid(X * cost_theta.T)))
    second = np.multiply(1 - y, np.log(1 - sigmoid(X * cost_theta.T)))
    reg = (learningRate / (2 * len(X))) * np.sum(np.power(cost_theta[:, 1:cost_theta.shape[1]], 2))
    return np.sum(first - second) / len(X) + reg


# gradient descent function
# 方法一：for循环的梯度函数
def gradient_with_loop(theta, X, y, learningRate):
    theta = np.matrix(theta)
    X = np.matrix(X)
    y = np.matrix(y)

    parameters = int(theta.reval().shape[1])
    gradients = np.zero(parameters)

    error = sigmoid(1 + np.exp(X * theta.T)) - y

    for i in range(parameters):
        term = np.multiply(error, X[:, i])

        if (i == 0):
            gradients[i] = np.sum(term) / len(X)
        else:
            gradients[i] = (np.sum(term) / len(X)) + ((learningRate / len(X)) * theta[:, i])

    return gradients


# 方法二：向量化的梯度函数
def gradient(theta, X, y, learningRate):
    theta = np.matrix(theta)
    X = np.matrix(X)
    y = np.matrix(y)

    parameters = int(theta.ravel().shape[1])
    error = sigmoid(X * theta.T) - y

    grad = ((X.T * error) / len(X)).T + ((learningRate / len(X)) * theta)

    # intercept gradient is not regularized
    grad[0, 0] = np.sum(np.multiply(error, X[:, 0])) / len(X)

    return np.array(grad).ravel()


# 多分类器函数
def one_vs_all(X, y, num_labels, learning_rate):
    rows = X.shape[0]
    params = X.shape[1]

    # k个分类器，n为参数
    # k X (n + 1) array for the parameters of each of the k classifiers
    all_theta = np.zeros((num_labels, params + 1))

    # insert a column of ones at the beginning for the intercept term
    # arr原始数组，可一可多，obj插入元素位置，values是插入内容，axis是按行按列插入（1为列）
    X = np.insert(X, 0, values=np.ones(rows), axis=1)

    # labels are 1-indexed instead of 0-indexed
    for i in range(1, num_labels + 1):
        theta = np.zeros(params + 1)
        y_i = np.array([1 if label == i else 0 for label in y])
        y_i = np.reshape(y_i, (rows, 1))

        # minimize the objective function
        # 高级优化算法，直接计算得到theta参数值
        # fun：传入cost函数，自己写的代价函数（def cost）
        # x0：传入theta的初始值，初始化后的theta
        # args：梯度下降函数的参数（除了theta之外的按顺序写）（def gradient）
        # method：所要使用的优化方法，如TNC、BFGS等
        # jac：传入gradient方法，即计算梯度的方法
        # options：有两个选项，{‘maxiter’:100}可以控制迭代次数；{‘disp’:True}可以打印一些运行细节
        fmin = minimize(fun=cost, x0=theta, args=(X, y_i, learning_rate), method='TNC', jac=gradient)
        # 将优化程序找到的参数分配给参数数组
        all_theta[i - 1, :] = fmin.x

    return all_theta


# 预测函数
def predict_all(X, all_theta):
    rows = X.shape[0]
    params = X.shape[1]
    num_labels = all_theta.shape[0]

    # same as before, insert ones to match the shape
    X = np.insert(X, 0, values=np.ones(rows), axis=1)

    # convert to matrices
    X = np.matrix(X)
    all_theta = np.matrix(all_theta)

    # compute the class probability for each class on each training instance
    # sigmoid函数计算预测概率
    h = sigmoid(X * all_theta.T)

    # create array of the index with the maximum probability
    # 找出最大数的索引，即找出1的索引
    # axis[0]:列方向上搜索，[1]：行方向上搜索
    h_argmax = np.argmax(h, axis=1)

    # because our array was zero-indexed we need to add one for the true label prediction
    # 将0索引转换为1索引
    h_argmax = h_argmax + 1

    return h_argmax


data = loadmat("E:\\Download\\ex3data1.mat")
print(data)
print(data['X'].shape, data['y'].shape)

rows = data['X'].shape[0]
params = data['X'].shape[1]
all_theta = np.zeros((10, params + 1))

X = np.insert(data['X'], 0, values=np.ones(rows), axis=1)
theta = np.zeros(params + 1)
y_0 = np.array([1 if label == 0 else 0 for label in data['y']])
y_0 = np.reshape(y_0, (rows, 1))

print(X.shape, y_0.shape, theta.shape, all_theta.shape)
# 查看有几类标签
print(np.unique(data['y']))

all_theta = one_vs_all(data['X'], data['y'], 10, 1)
print(all_theta)

y_pred = predict_all(data['X'], all_theta)
correct = [1 if a == b else 0 for (a, b) in zip(y_pred, data['y'])]
accuracy = (sum(map(int, correct)) / float(len(correct)))
print ('accuracy = {0}%'.format(accuracy * 100))