机器学习实战——Logistic回归

Logistic回归

$S$函数

$$\sigma(z) = \frac{1}{1 + e^{-z}}$$
其中 $$z = w_0x_0 + w_1x_1 + w_2x_2 + \dots + w_nx_n$$
$w$为需求解的回归系数， $x$为特征。
写成向量形式：
$$z = [w_0\ w_1\ w_2\cdots w_n]\left[\begin{matrix}x_0\\ x_1\\ x_2 \\ \vdots \\ x_n\end{matrix}\right] = W\times X^T$$
带入 $S$函数，即变为：
$$\sigma(z) = \frac{1}{1 + e^{-w^Tx}}$$

则在二分类的问题中就有如下的假设：

$$P(y = 1 |x, w) = \sigma_w(x)$$
$$P(y = 0 |x, w) = 1 - \sigma_w(x)$$
将两式和在一起：
$$Cost(\sigma_w(x), y) = \sigma_w(x)^y[1-\sigma_w(x)]^{(1-y)}$$
取对数有：
$$Cost(\sigma_w(x), y) = ylog\sigma_w(x) + (1-y)log(1-\sigma_w(x))$$
上式即为代价函数，通过上式就能求出给定样本为某一类别的概率。求解函数的最大值即为寻找最佳分类的过程。
假定特征之间相互独立，那么样本的概率就是特征概率的乘积，取对数之后就是相加：
$$J(w) = \sum^n_{i = 1}[y^ilog\sigma_w(x^i) + (1 - y^i)log\sigma_w(x^i)]$$

梯度上升算法

对于简单函数

$$f(x) = -x^2 +4x$$
应用梯度上升算法求极值，一步一步逼近极值。数学表达如下：
$$x_{i+1} = x_i + \alpha \frac{\partial f(x_i)}{x_i}$$
其中 $\alpha$即为步长。

import matplotlib.pyplot as plt
def Gradient_Ascent_test():
    def f_partial(x_before):
        '''f(x)的偏导'''
        return -2 * x_before + 4
    x_before =  -1
    x_new = 0
    alpha = 0.01
    presision = 0.00000001
    pression = []
    while abs(x_new - x_before) > presision:
        x_before = x_new
        x_new = x_before + alpha * f_partial(x_before)
        pression.append(x_new)
    return pression
Gradient_Ascent_test()[-1]

1.999999515279857

显然，最后最大值在误差范围内接近真实值2。那么对于$J(w)$函数，同样可以如此获得最大值。接下来推导求系数$w$的公式：

$$w_j = w_j + \alpha\frac{\partial J(w)}{w_j}$$
代入 $J(w)$，以及 $\sigma(z)$即可，接下来求 $J(w)$的偏导:
$$\frac{\partial J(w)}{w_j} = \frac{\partial J(w)}{\partial \sigma(w^Tx)} \cdot \frac{\partial \sigma(w^Tx)}{\partial w^Tx} \cdot \frac{\partial w^Tx}{w_j}$$
其中：
$$\frac{\partial J(w)}{\partial \sigma(w^Tx)} = y \frac{1}{\sigma(w^Tx)} + (y-1)\frac{1}{1 - \sigma(w^Tx)}$$
$$\frac{\partial \sigma(w^Tx)}{\partial w^Tx} = \sigma(w^Tx)(1-\sigma(w^Tx))$$
$$\frac{\partial w^Tx}{w_j} = \frac{\partial J(w_1x_1 + w_2x_2 + \dots + w_nx_n)}{\partial w_j} = x_j$$
综上：
$$\frac{\partial J(w)}{w_j} = (y - \sigma_w(x))x_j$$
代入偏导， $J(w)$的梯度上升公式即为：
$$w_j = w_j + \alpha \sum^n_{i=1}(y^i - \sigma_w(x^i))x_j^i$$

算法测试

import numpy as np
def loadDataSet():
    dataMat = []; labelMat = []
    fr = open('testSet.txt')
    for line in fr.readlines():
        lineArr = line.strip().split()
        dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])])
        labelMat.append(int(lineArr[2]))
    return dataMat,labelMat

def plotDataSet():
    dataMat, labelMat = loadDataSet()
    dataArr = np.array(dataMat)
    n = np.shape(dataMat)[0]
    xcord1 = []; ycord1 = []
    xcord2 = []; ycord2 = []
    for i in range(n):
        if int(labelMat[i]) == 1:
            xcord1.append(dataArr[i,1]); ycord1.append(dataArr[i,2])
        else:
            xcord2.append(dataArr[i,1]); ycord2.append(dataArr[i,2])
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xcord1, ycord1, s = 20, c = 'red', marker = 's',alpha=.5)
    ax.scatter(xcord2, ycord2, s = 20, c = 'blue',alpha=.5)
    plt.title('DataSet')
    plt.xlabel('$x_1$'); plt.ylabel('$x_2$')
    plt.show()

plotDataSet()

def sigmoid(x):
    return 1.0 / (1 + np.exp(-x))
def grad_ascent(data, class_labels):
    data_matrix = np.mat(data)
    label_mat = np.mat(class_labels).T
    m, n = np.shape(data_matrix)
    alpha = 0.001
    times = 500
    weights = np.ones((n, 1))
    for k in range(times):
        z = sigmoid(data_matrix * weights)
        error = label_mat - z
        weights = weights + alpha * data_matrix.T * error
    return weights.getA()

data, label = loadDataSet()
weights = grad_ascent(data, label)
weights

array([[ 4.12414349],
       [ 0.48007329],
       [-0.6168482 ]])

如上，已经算出回归系数$[w_0, w_1, w_2]$

def plotBestFit(weights):
    dataMat,labelMat=loadDataSet()
    dataArr = np.array(dataMat)
    n = np.shape(dataArr)[0] 
    xcord1 = []; ycord1 = []
    xcord2 = []; ycord2 = []
    for i in range(n):
        if int(labelMat[i])== 1:
            xcord1.append(dataArr[i,1]); ycord1.append(dataArr[i,2])
        else:
            xcord2.append(dataArr[i,1]); ycord2.append(dataArr[i,2])
    fig = plt.figure()
    ax = fig.add_subplot(111)
    ax.scatter(xcord1, ycord1, s=30, c='red', marker='s')
    ax.scatter(xcord2, ycord2, s=30, c='green')
    x = np.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()

plotBestFit(weights)

随机梯度上升

import random
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
    m,n = np.shape(dataMatrix)
    weights = np.ones(n)
    for j in range(numIter):                                           
        dataIndex = list(range(m))
        for i in range(m):           
            alpha = 4/(1.0+j+i)+0.01
            randIndex = int(random.uniform(0,len(dataIndex)))
            h = sigmoid(sum(dataMatrix[randIndex]*weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            del(dataIndex[randIndex])
    return weights

dataMat, labelMat = loadDataSet()
weights = stocGradAscent1(np.array(dataMat), labelMat)
plotBestFit(weights)

示例：从疝气病症状预测病马的死亡率

def colicTest():
    fr_train = open('horseColicTraining.txt')
    fr_test = open('horseColicTest.txt')
    training_set = []
    training_labels = []
    for line in fr_train.readlines():
        current = line.strip().split('\t')
        line_arr = []
        for i in range(len(current) - 1):
            line_arr.append(float(current[i]))
        training_set.append(line_arr)
        training_labels.append(float(current[-1]))
    train_weights = stocGradAscent1(np.array(training_set), training_labels, 500)
    error = 0
    test_vec = 0.0
    for line in fr_test.readlines():
        test_vec += 1
        current = line.strip().split('\t')
        line_arr = []
        for i in range(len(current) - 1):
            line_arr.append(float(current[i]))
        if int(classify_vector(np.array(line_arr), train_weights)) != int(current[-1]):
            error += 1
    error_rate = (float(error) / test_vec) * 100
    print("测试集错误率为: %.2f%%" % error_rate)
    return error_rate

def classify_vector(inx, weights):
    prob = sigmoid(sum(inx*weights))
    if prob > 0.5:
        return 1.0
    else:
        return 0.0

def multi_test():
    num_tests = 10
    error_sum = 0.0
    for k in range(num_tests):
        error_sum += colicTest()
    print("平均错误率为", error_sum / float(num_tests), '%')

multi_test()

C:\ProgramData\Anaconda3\lib\site-packages\ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp



测试集错误率为: 40.30%
测试集错误率为: 35.82%
测试集错误率为: 35.82%
测试集错误率为: 29.85%
测试集错误率为: 41.79%
测试集错误率为: 47.76%
测试集错误率为: 28.36%
测试集错误率为: 38.81%
测试集错误率为: 28.36%
测试集错误率为: 20.90%
平均错误率为 34.77611940298507 %