python logistic 回归 AIC logistic回归分析命令

Logistic回归

Sigmod函数：Б(z) = 1/(1+exp(-z)) 具有可以输出0或者1的性质。
Logistic回归：任何大于0.5的数据被分为1类，小于0.5即被归为0类，所以，Logistic回归也可以被看成是一种概率估计。

import numpy as np
import matplotlib.pyplot as pp
%matplotlib inline
z = np.linspace(-60,60,10000)
y = 1/(1+np.exp(-1*z))
pp.plot(z,y)
import numpy as np
import matplotlib.pyplot as pp
%matplotlib inline
z = np.linspace(-60,60,10000)
y = 1/(1+np.exp(-1*z))
pp.plot(z,y)

[<matplotlib.lines.Line2D at 0x14ec0c5f9b0>]

一、基于最优化方法的最佳回归系数确定

1. 梯度上升法

最佳回归系数的确定：
z = w0*x0+w1*x1+w2*x2+...+wn*xn = W^T*X
梯度上升法的迭代公式：：
w:=w+α*(f(w)的对每个w的偏导)

2. 训练算法：使用梯度上升找到最佳参数

# Logistic回归梯度上升优化算法
def loadDataSet():
    dataMat = []
    labelMat = []
    fr = open('data/testSet.txt')
    for line in fr.readlines():
        lineArr = line.strip().split()
        dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])]) # x0 = 1.0
        labelMat.append(int(lineArr[2]))
    fr.close()
    return dataMat, labelMat
# Logistic回归梯度上升优化算法
def loadDataSet():
    dataMat = []
    labelMat = []
    fr = open('data/testSet.txt')
    for line in fr.readlines():
        lineArr = line.strip().split()
        dataMat.append([1.0, float(lineArr[0]), float(lineArr[1])]) # x0 = 1.0
        labelMat.append(int(lineArr[2]))
    fr.close()
    return dataMat, labelMat

def sigmoid(inX):
    return (1.0/(1+np.exp(-inX)))
def sigmoid(inX):
    return (1.0/(1+np.exp(-inX)))

def gradAscent(dataMatIn, classLabels):
    dataMatrix = np.mat(dataMatIn) # 100*3
    labelMat = np.mat(classLabels).transpose() # 1*100
    m, n = np.shape(dataMatrix)
    alpha = 0.001 # step
    maxCycles = 500 # times
    weights = np.ones((n,1)) # 3*1
    for k in range(maxCycles):
        h = sigmoid(dataMatrix*weights) # 100*1
        error = (labelMat - h)
        weights = weights + alpha * dataMatrix.transpose()*error # 按照该差值的方向调整回归系数
    return weights
def gradAscent(dataMatIn, classLabels):
    dataMatrix = np.mat(dataMatIn) # 100*3
    labelMat = np.mat(classLabels).transpose() # 1*100
    m, n = np.shape(dataMatrix)
    alpha = 0.001 # step
    maxCycles = 500 # times
    weights = np.ones((n,1)) # 3*1
    for k in range(maxCycles):
        h = sigmoid(dataMatrix*weights) # 100*1
        error = (labelMat - h)
        weights = weights + alpha * dataMatrix.transpose()*error # 按照该差值的方向调整回归系数
    return weights

dataArr, labelMat = loadDataSet()
gradAscent(dataArr, labelMat)
dataArr, labelMat = loadDataSet()
gradAscent(dataArr, labelMat)

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

3. 分析数据：画出决策边界

# 画出数据集和Logistic回归最佳拟合直线的函数
def plotBestFit(weights):
    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 = pp.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] # w0x0+w1x1+w2x2=0
    ax.plot(x,y)
    pp.xlabel('X1'); pp.ylabel('X2')
    pp.show()
# 画出数据集和Logistic回归最佳拟合直线的函数
def plotBestFit(weights):
    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 = pp.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] # w0x0+w1x1+w2x2=0
    ax.plot(x,y)
    pp.xlabel('X1'); pp.ylabel('X2')
    pp.show()

weights = gradAscent(dataArr, labelMat)
plotBestFit(weights.getA())
weights = gradAscent(dataArr, labelMat)
plotBestFit(weights.getA())

4. 训练算法：随机梯度上升

1. 梯度上升算法每次更新回归系数时都需要遍历整个数据集。
2. 改进方法是一次仅用一个样本点来更新回归系数，即随机梯度上升算法。
3. 前一种成为“批处理”，随机梯度上升算法可以在新样本到来时对分类器进行增量式更新，是“在线学习”算法。

# 随机梯度上升算法
def stocGradAscent0(dataMatrix, classLabels):
    m, n = np.shape(dataMatrix) # 100x3
    alpha = 0.01
    weights = np.ones(n) # 1x3 p.s.np.ones((3,1))为3x1
    for i in range(m):
        h = sigmoid(sum(dataMatrix[i]*weights)) # 标量
        error = classLabels[i] - h
        weights = weights + alpha*error*dataMatrix[i]
    return weights
# 随机梯度上升算法
def stocGradAscent0(dataMatrix, classLabels):
    m, n = np.shape(dataMatrix) # 100x3
    alpha = 0.01
    weights = np.ones(n) # 1x3 p.s.np.ones((3,1))为3x1
    for i in range(m):
        h = sigmoid(sum(dataMatrix[i]*weights)) # 标量
        error = classLabels[i] - h
        weights = weights + alpha*error*dataMatrix[i]
    return weights

dataArr, labelMat = loadDataSet()
dataArr, labelMat = loadDataSet()

weights = stocGradAscent0(np.array(dataArr), labelMat)
weights = stocGradAscent0(np.array(dataArr), labelMat)

plotBestFit(weights)
plotBestFit(weights)

一个判断优化算法优劣的可靠方法是看它是否收敛。

#在整个数据集上运行200次
def stocGradAscent0_200(dataMatrix, classLabels):
    m, n = np.shape(dataMatrix) # 100x3
    alpha = 0.01
    weights = np.ones(n) # 1x3 p.s.np.ones((3,1))为3x1
    weights_all = []
    for times in range(200):
        for i in range(m):
            h = sigmoid(sum(dataMatrix[i]*weights)) # 标量
            error = classLabels[i] - h
            weights = weights + alpha*error*dataMatrix[i]
            #print(weights)
            weights_all.append(weights)
    return weights,weights_all
#在整个数据集上运行200次
def stocGradAscent0_200(dataMatrix, classLabels):
    m, n = np.shape(dataMatrix) # 100x3
    alpha = 0.01
    weights = np.ones(n) # 1x3 p.s.np.ones((3,1))为3x1
    weights_all = []
    for times in range(200):
        for i in range(m):
            h = sigmoid(sum(dataMatrix[i]*weights)) # 标量
            error = classLabels[i] - h
            weights = weights + alpha*error*dataMatrix[i]
            #print(weights)
            weights_all.append(weights)
    return weights,weights_all

dataArr, labelMat = loadDataSet()
weights, weights_all = stocGradAscent0_200(np.array(dataArr), labelMat)
dataArr, labelMat = loadDataSet()
weights, weights_all = stocGradAscent0_200(np.array(dataArr), labelMat)

len(weights_all)
len(weights_all)

20000

x0 = []; x1 = []; x2 = []
for i in range(len(weights_all)):
    x0.append(float(list(weights_all[i])[0]))
    x1.append(float(list(weights_all[i])[1]))
    x2.append(float(list(weights_all[i])[2]))
x0 = []; x1 = []; x2 = []
for i in range(len(weights_all)):
    x0.append(float(list(weights_all[i])[0]))
    x1.append(float(list(weights_all[i])[1]))
    x2.append(float(list(weights_all[i])[2]))

X = np.linspace(0,200,20000,endpoint=True)
fig = pp.subplot(311)
pp.plot(X,x0)
fig = pp.subplot(312)
pp.plot(X,x1)
fig = pp.subplot(313)
pp.plot(X,x2)
X = np.linspace(0,200,20000,endpoint=True)
fig = pp.subplot(311)
pp.plot(X,x0)
fig = pp.subplot(312)
pp.plot(X,x1)
fig = pp.subplot(313)
pp.plot(X,x2)

[<matplotlib.lines.Line2D at 0x14ec15c1eb8>]

与书上的图不太一样
x1只经过了50次迭代就达到了稳定值，但x0,x2需要更多次的迭代。另外，在大的波动停止后，还有一些小的周期性波动。
产生这种现象的原因是存在一些不能正确分类的样本点（数据集并非线性可分），在每次迭代时会引发系数的剧烈改动。
我们期望算法能避免来回波动，从而收敛到某个值。

plotBestFit(weights)
plotBestFit(weights)

# 改进的随机梯度上升算法
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
    m, n = np.shape(dataMatrix)
    weights = np.ones(n)
    weights_all = []
    for j in range(numIter):
        dataIndex = list(range(m))
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.01 # 会随着迭代次数不断减少，当j<<max(i),alpha就不是严格下降的（类似于模拟退火算法）
            randIndex = int(np.random.uniform(0,len(dataIndex)))
            h = sigmoid(sum(dataMatrix[randIndex]*weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            weights_all.append(weights)
            del(dataIndex[randIndex])
    return weights, weights_all
# 改进的随机梯度上升算法
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
    m, n = np.shape(dataMatrix)
    weights = np.ones(n)
    weights_all = []
    for j in range(numIter):
        dataIndex = list(range(m))
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.01 # 会随着迭代次数不断减少，当j<<max(i),alpha就不是严格下降的（类似于模拟退火算法）
            randIndex = int(np.random.uniform(0,len(dataIndex)))
            h = sigmoid(sum(dataMatrix[randIndex]*weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            weights_all.append(weights)
            del(dataIndex[randIndex])
    return weights, weights_all

dataArr, labelMat = loadDataSet()
weights, weights_all = stocGradAscent1(np.array(dataArr), labelMat)
dataArr, labelMat = loadDataSet()
weights, weights_all = stocGradAscent1(np.array(dataArr), labelMat)

x0 = []; x1 = []; x2 = []
for i in range(len(weights_all)):
    x0.append(float(list(weights_all[i])[0]))
    x1.append(float(list(weights_all[i])[1]))
    x2.append(float(list(weights_all[i])[2]))
X = np.linspace(0,4000,len(weights_all),endpoint=True)
fig = pp.subplot(311)
pp.plot(X,x0)
fig = pp.subplot(312)
pp.plot(X,x1)
fig = pp.subplot(313)
pp.plot(X,x2)
x0 = []; x1 = []; x2 = []
for i in range(len(weights_all)):
    x0.append(float(list(weights_all[i])[0]))
    x1.append(float(list(weights_all[i])[1]))
    x2.append(float(list(weights_all[i])[2]))
X = np.linspace(0,4000,len(weights_all),endpoint=True)
fig = pp.subplot(311)
pp.plot(X,x0)
fig = pp.subplot(312)
pp.plot(X,x1)
fig = pp.subplot(313)
pp.plot(X,x2)

[<matplotlib.lines.Line2D at 0x14ec11626d8>]

1. 没有周期性波动，归功于样本随机选择机制
2. 水平轴短了很多，归功于alpha，可以收敛得更快。?

plotBestFit(weights)
plotBestFit(weights)

二、示例：从疝气病症预测病马的死亡率

1. 准备数据：处理数据中的缺失值

1. 使用可用特征的均值来填补缺失值
2. 使用特殊值来填补缺失值，如-1
3. 忽略有缺失值的样本
4. 使用相似样本的均值填补缺失值
5. 使用另外的机器学习算法预测缺失值

本例选择使用实数0来替换缺失值

1. 恰好能适用于Logistic回归，因为在更新时不会影响系数的值weights = weights+alpha*error*dataMatrix[randIndex]
2. sigmoid(0） = 0.5，即对结果不具有任何倾向性。

如果在测试数据集中发现了一条数据的类别标签已经缺失，简单做法是丢弃
这是因为类别标签和特征不同，很难找到合适的值来替换。

2. 测试算法：用Logistic回归进行分类

# Logistic回归分类函数
def classifyVector(inX, weights):
    prob = sigmoid(sum(inX*weights))
    if prob > 0.5: return 1.0
    else: return 0.0
# Logistic回归分类函数
def classifyVector(inX, weights):
    prob = sigmoid(sum(inX*weights))
    if prob > 0.5: return 1.0
    else: return 0.0

def colicTest():
    frTrain = open('data/horseColicTraining.txt')
    frTest = open('data/horseColicTest.txt')
    trainingSet = []; trainingLabels = []
    for line in frTrain.readlines():
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        trainingSet.append(lineArr)
        trainingLabels.append(float(currLine[21]))
    trainWeights,trainWeights_all = stocGradAscent1(np.array(trainingSet), trainingLabels, 500)
    errorCount = 0; numTestVec = 0.0
    for line in frTest.readlines():
        numTestVec += 1.0
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        if int(classifyVector(np.array(lineArr), trainWeights)) != int(currLine[21]):
            errorCount += 1
    errorRate = (float(errorCount)/numTestVec)
    print('the error rate of this test is: %f' % errorRate)
    return errorRate
def colicTest():
    frTrain = open('data/horseColicTraining.txt')
    frTest = open('data/horseColicTest.txt')
    trainingSet = []; trainingLabels = []
    for line in frTrain.readlines():
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        trainingSet.append(lineArr)
        trainingLabels.append(float(currLine[21]))
    trainWeights,trainWeights_all = stocGradAscent1(np.array(trainingSet), trainingLabels, 500)
    errorCount = 0; numTestVec = 0.0
    for line in frTest.readlines():
        numTestVec += 1.0
        currLine = line.strip().split('\t')
        lineArr = []
        for i in range(21):
            lineArr.append(float(currLine[i]))
        if int(classifyVector(np.array(lineArr), trainWeights)) != int(currLine[21]):
            errorCount += 1
    errorRate = (float(errorCount)/numTestVec)
    print('the error rate of this test is: %f' % errorRate)
    return errorRate

def multiTest():
    numTests = 10; errorSum = 0.0
    for k in range(numTests):
        errorSum += colicTest()
    print('after %d iterations the average error rate is: %f' % (numTests, errorSum/float(numTests)))
def multiTest():
    numTests = 10; errorSum = 0.0
    for k in range(numTests):
        errorSum += colicTest()
    print('after %d iterations the average error rate is: %f' % (numTests, errorSum/float(numTests)))

multiTest()
multiTest()

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


the error rate of this test is: 0.328358
the error rate of this test is: 0.373134
the error rate of this test is: 0.313433
the error rate of this test is: 0.358209
the error rate of this test is: 0.313433
the error rate of this test is: 0.343284
the error rate of this test is: 0.402985
the error rate of this test is: 0.402985
the error rate of this test is: 0.417910
the error rate of this test is: 0.492537
after 10 iterations the average error rate is: 0.374627