机器学习Python实现 SVD 分解

# -*- coding: cp936 -*-
'''
Created on Mar 8, 2011

@author: Peter
'''
from numpy import *
from numpy import linalg as la #用到别名

#这里主要结合推荐系统介绍SVD，所以这里的数据都可以看成是用户对物品的一个打分
def loadExData():
    return[[0, 0, 0, 2, 2],
           [0, 0, 0, 3, 3],
           [0, 0, 0, 1, 1],
           [1, 1, 1, 0, 0],
           [2, 2, 2, 0, 0],
           [5, 5, 5, 0, 0],
           [1, 1, 1, 0, 0]]
    
def loadExData2():
    return[[0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 5],
           [0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 3],
           [0, 0, 0, 0, 4, 0, 0, 1, 0, 4, 0],
           [3, 3, 4, 0, 0, 0, 0, 2, 2, 0, 0],
           [5, 4, 5, 0, 0, 0, 0, 5, 5, 0, 0],
           [0, 0, 0, 0, 5, 0, 1, 0, 0, 5, 0],
           [4, 3, 4, 0, 0, 0, 0, 5, 5, 0, 1],
           [0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 4],
           [0, 0, 0, 2, 0, 2, 5, 0, 0, 1, 2],
           [0, 0, 0, 0, 5, 0, 0, 0, 0, 4, 0],
           [1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0]]
    
def ecludSim(inA,inB):
    return 1.0/(1.0 + la.norm(inA - inB))  #计算向量的第二范式,相当于直接计算了欧式距离

def pearsSim(inA,inB):
    if len(inA) < 3 : return 1.0
    return 0.5+0.5*corrcoef(inA, inB, rowvar = 0)[0][1] #corrcoef直接计算皮尔逊相关系数

def cosSim(inA,inB):
    num = float(inA.T*inB)
    denom = la.norm(inA)*la.norm(inB)
    return 0.5+0.5*(num/denom)  #计算余弦相似度

#协同过滤算法
#dataMat 用户数据 user 用户 simMeas 相似度计算方式 item 物品
def standEst(dataMat, user, simMeas, item): 
    n = shape(dataMat)[1] #计算列的数量，物品的数量
    simTotal = 0.0; ratSimTotal = 0.0 
    for j in range(n):
        userRating = dataMat[user,j]
        print(dataMat[user,j])
        if userRating == 0: continue  #如果用户u没有对物品j进行打分，那么这个判断就可以跳过了
        overLap = nonzero(logical_and(dataMat[:,item].A>0, \
                                      dataMat[:,j].A>0))[0]    #找到对物品 j 和item都打过分的用户
        if len(overLap) == 0: similarity = 0
        else: similarity = simMeas(dataMat[overLap,item], dataMat[overLap,j])     #利用相似度计算两个物品之间的相似度
                                   
        print 'the %d and %d similarity is: %f' % (item, j, similarity)
        simTotal += similarity
        ratSimTotal += similarity * userRating  #待推荐物品与用户打过分的物品之间的相似度*用户对物品的打分
    if simTotal == 0: return 0
    else: return ratSimTotal/simTotal


#利用SVD进行分解，但是这里是直接用的库里面的函数
    #如果自己实现一个SVD分解，我想就是和矩阵论里面的求解知识是一样的吧，但是可能在求特征值的过程中会比较痛苦
def svdEst(dataMat, user, simMeas, item):
    n = shape(dataMat)[1]
    simTotal = 0.0; ratSimTotal = 0.0
    U,Sigma,VT = la.svd(dataMat) #直接进行分解
    Sig4 = mat(eye(4)*Sigma[:4]) #arrange Sig4 into a diagonal matrix
    xformedItems = dataMat.T * U[:,:4] * Sig4.I  #create transformed items
    for j in range(n):
        userRating = dataMat[user,j]
        if userRating == 0 or j==item: continue
        similarity = simMeas(xformedItems[item,:].T,\
                             xformedItems[j,:].T)
        print 'the %d and %d similarity is: %f' % (item, j, similarity)
        simTotal += similarity
        ratSimTotal += similarity * userRating
    if simTotal == 0: return 0
    else: return ratSimTotal/simTotal

#真正的推荐函数，后面两个函数就是采用的相似度的计算方法和推荐用的方法
def recommend(dataMat, user, N=3, simMeas=cosSim, estMethod=standEst):
    unratedItems = nonzero(dataMat[user,:].A==0)[1]  #find unrated items  nonzero()[1]返回的是非零值所在的行数，返回的是一个元组   if len(unratedItems) == 0: return 'you rated everything'
    itemScores = []
    for item in unratedItems:
        estimatedScore = estMethod(dataMat, user, simMeas, item)
        itemScores.append((item, estimatedScore))
    return sorted(itemScores, key=lambda jj: jj[1], reverse=True)[:N]


#扩展的例子，利用SVD进行图像的压缩
#将图像打印出来
def printMat(inMat, thresh=0.8):
    for i in range(32):
        for k in range(32):
            if float(inMat[i,k]) > thresh:
                print 1,
            else: print 0,
        print ''



#最后发现重构出来的数据图是差不多的
def imgCompress(numSV=3, thresh=0.8):
    myl = []
    for line in open('0_5.txt').readlines():
        newRow = []
        for i in range(32):
            newRow.append(int(line[i]))
        myl.append(newRow)
    myMat = mat(myl)   #将数据读入了myMat当中
    
    print "****original matrix******"
    printMat(myMat, thresh)
    U,Sigma,VT = la.svd(myMat)
    SigRecon = mat(zeros((numSV, numSV)))   #构建一个3*3的空矩阵
    for k in range(numSV):#construct diagonal matrix from vector
        SigRecon[k,k] = Sigma[k]
    reconMat = U[:,:numSV]*SigRecon*VT[:numSV,:]
    print "****reconstructed matrix using %d singular values******" % numSV
    printMat(reconMat, thresh)