奇异值分解：singular value decomposition ，SVD，用来提取矩阵的重要信息。
利用SVD能够用小得多的数据集来表示原始数据集。实际上是去除了噪声和冗余信息。我们把SVD看成从有噪声的数据中抽取相关特征。
协同过滤：通过将用户和其他用户的数据对比来实现推荐。利用用户对物品的评价来计算相似度。
相似度计算方式有欧氏距离、皮尔逊相关系数、余弦相似度。
相似度计算：
def euclidSim(inA,inB):
return 1.0/(1.0+la.norm(inA-inB))
def pearsSim(inA,inB):
if len(inA)<3:
return 1.0
return 0.5+0.5*corrcoef(inA,inB,rowvar=0)[0][1] #rowvar=0表示每列是一个变量，求列之间的相关系数，corrcoef返回的是2*2的对称方阵,[0][1]才能表示inA和inB的相关系数
def cosSim(inA,inB):
num=float(inA.T*inB)
denom=la.norm(inA)*la.norm(inB)
return 0.5+0.5*(num/denom)
本例中，用户对菜品的评价矩阵，如果行与行之间比较就是基于用户的相似度，列与列之间比较是基于菜品的相似度。
基于菜品相似度的推荐引擎（用的是原始数据集，没有使用SVD）：
def standEst(dataMat,userNum,simMeas,itemNum):
n=shape(dataMat)[1]
simTotal=0.0
ratSimTotal=0.0
for i in range(n):
userRating=dataMat[userNum,i]
if userRating==0:
continue
overLap=nonzero(logical_and(dataMat[:,i].A>0,dataMat[:,itemNum].A>0))[0]
if len(overLap)==0:
similarity=0
else:
similarity=simMeas(dataMat[overLap,i],dataMat[overLap,itemNum])
print('%d和%d的相似性为:%f'%(itemNum,i,similarity))
simTotal+=similarity
ratSimTotal+=similarity*userRating
if simTotal==0:
return 0
else:
return ratSimTotal/simTotal
def recommend(dataMat,userNum,N=3,simMeas=cosSim,estMethod=standEst):
unratedItems=nonzero(dataMat[userNum,:].A==0)[1]
if len(unratedItems)==0:
print('已经评价过所有物品了。')
itemScores=[]
for item in unratedItems:
estimatedScore=estMethod(dataMat,userNum,simMeas,item)
itemScores.append((item,estimatedScore))
return sorted(itemScores,key=lambda i:i[1],reverse=True)[:N]
基于SVD的评分估计：
def svdEst(dataMat, user, simMeas, item):
n = shape(dataMat)[1]
simTotal = 0.0
ratSimTotal = 0.0
U,Sigma,VT = la.svd(dataMat)
# print('sigma:',Sigma[:4])
# print('type:',type(Sigma))
Sig4 = mat(eye(4)*Sigma[:4])
# print('Sig4',Sig4)
xformedItems = dataMat.T * U[:,:4] * Sig4.I #没看懂，n*m m*4 4*4 ==>n*4 11个特征，原数据集转换为11*4的数据集
# print('xformedItems:',xformedItems)
# print('shape:',shape(xformedItems))
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('%d和%d的相似性为:%f'%(item,j,similarity))
simTotal += similarity
ratSimTotal += similarity * userRating
if simTotal == 0:
return 0
else:
return ratSimTotal/simTotal
加载数据集：
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]]
测试：
if __name__=='__main__':
myMat = mat(loadExData2())
result = recommend(myMat, 1, simMeas=euclidSim, estMethod=svdEst)
print(result)
输出：
0和3的相似性为:0.612384 0和5的相似性为:0.554262 0和10的相似性为:0.557208 1和3的相似性为:0.636421 1和5的相似性为:0.569152 1和10的相似性为:0.570929 2和3的相似性为:0.607708 2和5的相似性为:0.550613 2和10的相似性为:0.553660 4和3的相似性为:0.526776 4和5的相似性为:0.504518 4和10的相似性为:0.503709 6和3的相似性为:0.543511 6和5的相似性为:0.455741 6和10的相似性为:0.447819 7和3的相似性为:0.605924 7和5的相似性为:0.553717 7和10的相似性为:0.557372 8和3的相似性为:0.606861 8和5的相似性为:0.552125 8和10的相似性为:0.555494 9和3的相似性为:0.557828 9和5的相似性为:0.513662 9和10的相似性为:0.510451 [(4, 3.328675674700045), (9, 3.3247038080937834), (7, 3.3224884985810177)]
对用户1来说，本来对菜品的评分为：[0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 3]，推荐系统对菜品4的评分是3.33分，对菜品9的评分是3.32分，对菜品7的评分是3.32分，默认只推荐评分前三的菜品。
基于SVD的图像压缩：
def printMat(inMat, thresh=0.8):
for i in range(32):
for k in range(32):
if float(inMat[i, k]) > thresh:
print(1,end="") #输出不换行
else:
print(0,end="")
print('')
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)
print("**********原始数据矩阵**********")
printMat(myMat, thresh)
U, Sigma, VT = la.svd(myMat)
SigRecon = mat(zeros((numSV, numSV)))
for k in range(numSV):
SigRecon[k, k] = Sigma[k]
reconMat= U[:,:numSV]*SigRecon*VT[:numSV,:]
print('*****使用%d个奇异值重构的数据矩阵为*****'%numSV)
printMat(reconMat, thresh)
测试：
if __name__=='__main__':
imgCompress(2)
输出：
**********原始数据矩阵********** 00000000000000110000000000000000 00000000000011111100000000000000 00000000000111111110000000000000 00000000001111111111000000000000 00000000111111111111100000000000 00000001111111111111110000000000 00000000111111111111111000000000 00000000111111100001111100000000 00000001111111000001111100000000 00000011111100000000111100000000 00000011111100000000111110000000 00000011111100000000011110000000 00000011111100000000011110000000 00000001111110000000001111000000 00000011111110000000001111000000 00000011111100000000001111000000 00000001111100000000001111000000 00000011111100000000001111000000 00000001111100000000001111000000 00000001111100000000011111000000 00000000111110000000001111100000 00000000111110000000001111100000 00000000111110000000001111100000 00000000111110000000011111000000 00000000111110000000111111000000 00000000111111000001111110000000 00000000011111111111111110000000 00000000001111111111111110000000 00000000001111111111111110000000 00000000000111111111111000000000 00000000000011111111110000000000 00000000000000111111000000000000 *****使用2个奇异值重构的数据矩阵为***** 00000000000000000000000000000000 00000000000000000000000000000000 00000000000001111100000000000000 00000000000011111111000000000000 00000000000111111111100000000000 00000000001111111111110000000000 00000000001111111111110000000000 00000000011110000000001000000000 00000000111100000000001100000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001110000000 00000000111100000000001100000000 00000000001111111111111000000000 00000000001111111111110000000000 00000000001111111111110000000000 00000000000011111111100000000000 00000000000011111111000000000000 00000000000000000000000000000000
原始图片使用了32*32=1024个0或者1表示，使用SVD方法只需要：32*2*2+2=130个数字即可。