1.应用场景、二分类问题

1.预测一个用户是否点击特定的商品
2.判断用户的性别
3.预测用户是否会购买给定的品类
4.判断一条评论是正面的还是负面的

2.sigmoid 函数

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Logistic回归实质上还是线性回归模型，只是在回归的连续值结果上加上了一层函数映射，将特征线性求和，然后使用g(z)作映射，将结果映射到离散值0/1上
（1）首先是映射，将线性模型经过g(z)进行映射

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（2）构造损失函数

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最大似然估计推导出单个样本正确预测的概率

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取似然函数
对于训练数据集，数据特征X={x1,x2,…,xm}，对应分段数据Y={y1,y2,…,ym}；极大似然函数：整个样本的概率分布：

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对数似然函数，将连乘转换为连加求和

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最大似然估计就是寻找使得对数似然函数取最大值的参数，这里可以使用梯度法求解，求得的 就是要求的最佳参数。
（3）使用梯度下降法求解：

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然后是向量化，向量化是使用矩阵计算来代替for循环，以简化计算过程，提高效率。

# -*- coding: utf-8 -*-
from numpy import *
from matplotlib import pyplot as plt


def plot_best_fit(wei, data_set, label):
    weights = wei
    data_set = array(data_set)
    n = shape(data_set)[0]
    xcourd1 = []; ycourd1 = []
    xcourd2 = []; ycourd2 = []
    for i in range(n):
        if int(label[i]) == 1:
            xcourd1.append(data_set[i, 1]); ycourd1.append(data_set[i, 2])
        else:
            xcourd2.append(data_set[i, 1]); ycourd2.append(data_set[i, 2])
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xcourd1, ycourd1, s=30, c='red', marker='s')
    ax.scatter(xcourd2, ycourd2, s=30, c='green')
    x = arange(-3.0, 3.0, 0.1)
    y = (-weights[0] - weights[1]*x)/weights[2]
    ax.plot(x, y)
    plt.xlabel('X1'); plt.ylabel('X2')
    plt.show()

def load_data():
    data_set = []
    label = []
    fr = open('./text.txt')
    for line in fr.readlines():
        line = line.strip().split()
        data_set.append([1.0, float(line[0]), float(line[1])])
        label.append(int(line[2]))
    return data_set, label


def sigmoid(x):
    return 1.0 / (1 + exp(-x))


def train(data_set, label):
    data_matrix = mat(data_set)
    label = mat(label).transpose()
    m, n = shape(data_matrix)
    alpha = 0.001
    max_cycles = 500
    weights = ones((n, 1))
    for k in range(max_cycles):
        h = sigmoid(data_matrix*weights)
        error = label - h
        weights = weights + alpha * data_matrix.transpose() * error
    return weights

# on line to study
def stoc_grad_ascent(data_set, label):
    m, n = shape(data_set)
    alpha = 0.01
    weights = ones(n)
    for i in range(m):
        h = sigmoid(sum(data_set[i]*weights))
        error = label[i] - h
        weights = weights + alpha * error * data_set[i]
    return weights

# on line to study prove
def prove_grad_ascent(data_set, label, num_iter=450):
    m, n = shape(data_set)
    weights = ones(n)
    for j in range(num_iter):
        data_index = range(m)
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.01    # prevent swings
            # choose a random value to prevent periodic swings
            rand_index = int(random.uniform(0, len(data_index)))
            h = sigmoid(sum(data_set[rand_index]*weights))
            error = label[rand_index] -h
            weights = weights + alpha * error * data_set[rand_index]
            del data_index[rand_index]
    return weights

if __name__ == "__main__":
    data_set, label = load_data()
    #print label
    #weights = train(array(data_set), label)
    #weights = stoc_grad_ascent(array(data_set), label)
    weights = prove_grad_ascent(array(data_set), label)
    plot_best_fit(weights, data_set, label)