降维和PCA

去除平均值
计算协方差矩阵
计算协方差矩阵的特征值和特征向量
将特征值从大到小排序
保留最上面的N个特征向量
将数据转换到上述N个特征向量构建的新空间中

# 加载数据的函数
def loadData(filename, delim = ‘\t‘):
    fr = open(filename)
    stringArr = [line.strip().split(delim) for line in fr.readlines()]
    datArr = [map(float,line) for line in stringArr]
    return mat(datArr)
# =================================
# 输入：dataMat：数据集
#       topNfeat：可选参数，需要应用的N个特征，可以指定，不指定的话就会返回全部特征
# 输出：降维之后的数据和重构之后的数据
# =================================
def pca(dataMat, topNfeat=9999999):
    meanVals = mean(dataMat, axis=0)# axis = 0表示计算纵轴
    meanRemoved = dataMat - meanVals #remove mean
    covMat = cov(meanRemoved, rowvar=0)# 计算协方差矩阵
    eigVals,eigVects = linalg.eig(mat(covMat))# 计算特征值（eigenvalue）和特征向量
    eigValInd = argsort(eigVals)            #sort, sort goes smallest to largest
    eigValInd = eigValInd[:-(topNfeat+1):-1]  #cut off unwanted dimensions
    redEigVects = eigVects[:,eigValInd]       #reorganize eig vects largest to smallest
    lowDDataMat = meanRemoved * redEigVects#transform data into new dimensions
    reconMat = (lowDDataMat * redEigVects.T) + meanVals
    return lowDDataMat, reconMat

filename = r‘E:\ml\machinelearninginaction\Ch13\testSet.txt‘
dataMat = loadData(filename)
lowD, reconM = pca(dataMat, 1)

def plotData(dataMat,reconMat):
    fig = plt.figure()
    ax = fig.add_subplot(111)
    # 绘制原始数据
    ax.scatter(dataMat[:, 0].flatten().A[0], dataMat[:,1].flatten().A[0], marker=‘^‘, s = 90)
    # 绘制重构后的数据
    ax.scatter(reconMat[:,0].flatten().A[0], reconMat[:,1].flatten().A[0], marker=‘o‘, s = 10, c=‘red‘)
    plt.show()

lowD, reconM = pca(dataMat, 2)