1. 问题描述
根据299个训练样本,进行Logistic回归,预测67个测试样本中的疝气病马是否死亡。值得注意的是,特征中有30%的值是缺失的。
2. 数据准备
训练数据在horseColicTraining文件中,有299个样本。测试数据在horseColicTest文件中,有67个样本。每一行为一个样本,每一列为一个特征。
3. 模型原理
logistic回归是一种广义线性回归(generalized linear model),因此与多重线性回归分析有很多相同之处。它们的模型形式基本上相同,都具有 w‘x+b,其中w和b是待求参数,其区别在于他们的因变量不同,多重线性回归直接将w‘x+b作为因变量,即y =w‘x+b,而logistic回归则通过函数L将w‘x+b对应一个隐状态p,p =L(w‘x+b),然后根据p 与1-p的大小决定因变量的值。如果L是logistic函数,就是logistic回归,如果L是多项式函数就是多项式回归。
4. 算法实现
4.1. Logistic回归梯度上升优化算法
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 sigmoid(inX): return 1.0/(1+exp(-inX)) def gradAscent(dataMatIn, classLabels): dataMatrix = mat(dataMatIn) #convert to NumPy matrix labelMat = mat(classLabels).transpose() #convert to NumPy matrix m,n = shape(dataMatrix) alpha = 0.001 maxCycles = 500 weights = ones((n,1)) for k in range(maxCycles): #heavy on matrix operations h = sigmoid(dataMatrix*weights) #matrix mult error = (labelMat - h) #vector subtraction weights = weights + alpha * dataMatrix.transpose()* error #matrix mult return weights
4.2. 画出数据集和Logistics回归最佳拟合直线的函数
import matplotlib.pyplot as plt dataMat,labelMat=loadDataSet() dataArr = array(dataMat) n = 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 = 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()
4.3. 随机梯度上升算法
def stocGradAscent0(dataMatrix, classLabels): m,n = shape(dataMatrix) alpha = 0.01 weights = ones(n) #initialize to all ones for i in range(m): h = sigmoid(sum(dataMatrix[i]*weights)) error = classLabels[i] - h weights = weights + alpha * error * dataMatrix[i] return weights
4.4改进的随机梯度上升算法
def stocGradAscent1(dataMatrix, classLabels, numIter=150): m,n = shape(dataMatrix) weights = ones(n) #initialize to all ones for j in range(numIter): dataIndex = list(range(m)) for i in range(m): alpha = 4/(1.0+j+i)+0.0001 #apha decreases with iteration, does not randIndex = int(random.uniform(0,len(dataIndex)))#go to 0 because of the constant h = sigmoid(sum(dataMatrix[randIndex]*weights)) error = classLabels[randIndex] - h weights = weights + alpha * error * dataMatrix[randIndex] del(dataIndex[randIndex]) return weights
4.5.数据预测
def classifyVector(inX, weights): prob = sigmoid(sum(inX*weights)) if prob > 0.5: return 1.0 else: return 0.0 def colicTest(): 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, 1000) 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 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)))
5. 测试方法及结果
经过测试,平均错误率在32%左右。在有着30%数据缺失的情况下,这个效果并不差。
6. 总结
本文采用书上的相应算法、引入回归分析法,通过拟合来预测疝气病马的死亡率。但由于数据集内容有限及缺失数据过多,有一定的错误率。针对这一情况,应获取无缺失的数据集进行训练。