基於Logistic迴歸的思想，利用梯度上升的方法，求取回歸係數。並且完成對馬生病資料的訓練和預測。

例子來自《Machine Learning in Action》 Peter Harrington

梯度上升

載入資料集

資料集合中有兩類共100個數據點
""" 載入資料集 """
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

sigmoid函式

利用該函式的函式性質，用於分類

""" sigmoid函式 """
def sigmoid(inX):
    return 1.0/(1+exp(-inX))

梯度上升求權重向量

""" 梯度上升 """
def gradAscent(dataMatIn, classLabels):
    dataMatrix = mat(dataMatIn)             #轉換為 NumPy 矩陣
 

    labelMat = mat(classLabels).transpose() #轉換為 NumPy 矩陣，求轉置 （行向量-->列向量）
    m,n = shape(dataMatrix)                 #獲取矩陣的大小
    alpha = 0.001                           #步長
    maxCycles = 500                         #迭代代數
    weights = ones((n,1))                   #權重向量
    for k in range(maxCycles):              
        h = sigmoid(dataMatrix*weights)
        error = (labelMat - h)              # 懲罰度
 

        weights = weights + alpha * dataMatrix.transpose()* error
    return weights

測試

def testGradAscent():
    dataArr,labelMat = logRegres.loadDataSet()
    weights=logRegres.gradAscent(dataArr,labelMat)
    print(weights)

結果

[[ 4.12414349]
 [ 0.48007329]
 [-0.6168482 ]]

視覺化結果

""" 繪製擬合後的直線 """
def plotBestFit(weights):
    import matplotlib.pyplot as plt
    dataMat,labelMat=loadDataSet()
    dataArr = array(dataMat)
    n = shape(dataArr)[0]      #  資料點的個數
    xcord1 = []; ycord1 = []
    xcord2 = []; ycord2 = []
    for i in range(n):         #  根據資料點的型別進行分類
        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 = 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()

測試

注意numpy矩陣轉換為python陣列

"""測試繪製擬合的直線"""
def testPlotBestFit():
    dataArr,labelMat = logRegres.loadDataSet()
    weights =logRegres.gradAscent(dataArr,labelMat)
    logRegres.plotBestFit(weights.getA())    # getA() : matrix --> array

結果

隨機梯度

之前的梯度計算，當資料集很大的時候，計算量會很大，所以採用隨機梯度演算法，即每一次迭代只計算一個點。

隨機梯度演算法_0

""" 隨機梯度上升0 """
def stocGradAscent0(dataMatrix, classLabels):
    m,n = shape(dataMatrix)
    alpha = 0.01
    weights = ones(n)   
    for i in range(m):
        h = sigmoid(sum(dataMatrix[i]*weights)) # 每次只選取一個特徵點進行訓練
        error = classLabels[i] - h
        weights = weights + alpha * error * dataMatrix[i]
    return weights

測試

"""測試隨機梯度上升0"""
def teststocGradAscent0():
    dataArr,labelMat = logRegres.loadDataSet()
    weights =logRegres.stocGradAscent0(array(dataArr),labelMat)
    logRegres.plotBestFit(weights)

結果

由於迭代次數比較少，所以劃分效果不是很理想。

隨機梯度演算法_1

分析之前效果不理想的原因：
1. 由於迭代過程中步長固定，所以在最後收斂的過程中，會週期震盪。
2. 每次的訓練點不是隨機取得，會收到資料週期性的影響。

針對這兩點，做出如下修改：
1. alpha 步長大小隨著迭代的次數而減少
2. 隨機選取資料點進行訓練

""" 隨機梯度上升1 """
def stocGradAscent1(dataMatrix, classLabels, numIter=150):
    m,n = shape(dataMatrix)
    weights = ones(n)   
    for j in range(numIter):
        dataIndex = list(range(m))    # rang 物件無法迭代
        for i in range(m):
            alpha = 4/(1.0+j+i)+0.0001    # 步長會隨著迭代進行而減少，但不會為0。防止波動和停止不前
            randIndex = int(random.uniform(0,len(dataIndex)))  # 隨機選取迭代值，防止週期波動
            h = sigmoid(sum(dataMatrix[randIndex]*weights))
            error = classLabels[randIndex] - h
            weights = weights + alpha * error * dataMatrix[randIndex]
            del(dataIndex[randIndex])
    return weights

測試

"""測試隨機梯度上升1"""
def teststocGradAscent1():
    dataArr,labelMat = logRegres.loadDataSet()
    weights =logRegres.stocGradAscent1(array(dataArr),labelMat)
    logRegres.plotBestFit(weights)

結果

可以看到，這次的劃分效果就很好了。

應用

利用Logistic迴歸來預測病馬的死亡率

訓練

""" 利用迴歸係數和特徵量計算類別"""
def classifyVector(inX, weights):
    prob = sigmoid(sum(inX*weights))
    if prob > 0.5: return 1.0
    else: return 0.0

""" 載入資料 訓練 測試"""
def colicTest():
    # 訓練迴歸係數
    frTrain = open('horseColicTraining.txt'); frTest = open('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 = stocGradAscent1(array(trainingSet), trainingLabels, 500)
    # 測試分類效果
    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(array(lineArr), trainWeights))!= int(currLine[21]):
            errorCount += 1
    errorRate = (float(errorCount)/numTestVec)
    print ("the error rate of this test is: %f" % errorRate)
    return errorRate

測試

"""預測病馬死亡率"""    
def multiTest():
    numTests = 10; errorSum=0.0
    for k in range(numTests):
        errorSum += logRegres.colicTest()
    print ("after %d iterations the average error rate is: %f" % (numTests, errorSum/float(numTests)))

結果

the error rate of this test is: 0.432836
the error rate of this test is: 0.268657
the error rate of this test is: 0.417910
the error rate of this test is: 0.313433
the error rate of this test is: 0.298507
the error rate of this test is: 0.358209
the error rate of this test is: 0.298507
the error rate of this test is: 0.283582
the error rate of this test is: 0.388060
the error rate of this test is: 0.402985
after 10 iterations the average error rate is: 0.346269