一、基于协同过滤的推荐系统
协同过滤(Collaborative Filtering)的推荐系统的原理是通过将用户和其他用户的数据进行比对来实现推荐的。比对的具体方法就是通过计算两个用户数据之间的相似性,通过相似性的计算来说明两个用户数据之间的相似程度。相似度函数的设计必须满足度量空间的三点要求,即非负性,对称性和三角不等性。常用的相似度的计算方法有:欧式距离法、皮尔逊相关系数法和夹角余弦相似度法。具体的可以参见上一篇文章“协同过滤推荐算法(1) ”。二、面临的问题
在基本的协同过滤的推荐系统中(主要指上面所提到的基本模型中),我们是在整个空间上计算相似度,进而实现推荐的。但是现实中的数据往往并不是那么规整,普遍的现象就是在用户数据中出现很多未评分项,如下面所示的数据:![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1UQTBPRFUxTnprMlAzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
对于这样的稀疏矩阵,我们利用基本的协同过滤推荐算法的效率必将很低。对于这样的稀疏矩阵,我们可以利用SVD对其进行降维,将这样的稀疏矩阵映射到另一个具体的主题空间,SVD降维的原理可以参见博文“SVD奇异值分解”。
三、利用SVD构造主题空间
我们对上面所示的这样一个矩阵进行SVD分解,分解的结果为:1、U矩阵
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1UQTFPVEU1TURrelAzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
(U矩阵,矩阵U主要反应的是用户信息)
2、对角阵S
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1URXdNVEV3TnpVd1AzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
(S矩阵,矩阵S主要反映的是11个奇异值)
3、VT矩阵
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1URXdNakUxTkRZNFAzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
(VT矩阵,矩阵VT主要反映的是物品信息)
4、选取奇异值并映射主题空间
奇异值分解公式为:![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwyeGhkR1Y0TG1OdlpHVmpiMmR6TG1OdmJTOW5hV1l1YkdGMFpYZ19iMnhrUkdGMFlTWnpjR0ZqWlRzOUpuTndZV05sTzFVbE5VTjBhVzFsY3laemNHRmpaVHRUSlRWRGRHbHRaWE1tYzNCaFkyVTdWaVUxUlZRPQ%3D%3D.jpg?w=700&webp=1)
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwyeGhkR1Y0TG1OdlpHVmpiMmR6TG1OdmJTOW5hV1l1YkdGMFpYZ19iMnhrUkdGMFlWOGxOMEl4TVNVMVEzUnBiV1Z6Sm5Od1lXTmxPekV4SlRkRVBWVmZKVGRDTVRFbE5VTjBhVzFsY3laemNHRmpaVHMxSlRkRUpUVkRkR2x0WlhNbWMzQmhZMlU3VTE4bE4wSTFKVFZEZEdsdFpYTW1jM0JoWTJVN05TVTNSQ1UxUTNScGJXVnpKbk53WVdObE8xWmZKVGRDTlNVMVEzUnBiV1Z6Sm5Od1lXTmxPekV4SlRkRUpUVkZWQT09.jpg?w=700&webp=1)
即可得新的主题空间:
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwyeGhkR1Y0TG1OdlpHVmpiMmR6TG1OdmJTOW5hV1l1YkdGMFpYZ19hWFJsYlZSeVlXNXpabTl5YldWa1h5VTNRakV4SlRWRGRHbHRaWE1tYzNCaFkyVTdOU1UzUkQxdmJHUkVZWFJoWHlVM1FqRXhKVFZEZEdsdFpYTW1jM0JoWTJVN01URWxOMFFsTlVWVUpUVkRkR2x0WlhNbWMzQmhZMlU3VlY4bE4wSXhNU1UxUTNScGJXVnpKbk53WVdObE96VWxOMFFsTlVOMGFXMWxjeVp6Y0dGalpUdFRYeVUzUWpVbE5VTjBhVzFsY3laemNHRmpaVHMxSlRkRUpUVkZKVGRDTFRFbE4wUT0%3D.jpg?w=700&webp=1)
四、实验的仿真
我们在这样的数据集上做推荐计算。其中user为2号用户。![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1URXhNVFV6TVRRd1AzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
(相似度的计算)
![机器学习算法(推荐算法)—协同过滤推荐算法(2) 机器学习算法(推荐算法)—协同过滤推荐算法(2)](https://image.shishitao.com:8440/aHR0cHM6Ly93d3cuaXRkYWFuLmNvbS9nby9hSFIwY0RvdkwybHRaeTVpYkc5bkxtTnpaRzR1Ym1WMEx6SXdNVFF3TmpBek1URXhNalE0TkRZNFAzZGhkR1Z5YldGeWF5OHlMM1JsZUhRdllVaFNNR05FYjNaTU1rcHpZakpqZFZrelRtdGlhVFYxV2xoUmRsb3lPWFphTW5oc1RWUnJORTlVUVhoTlJFazlMMlp2Ym5Rdk5XRTJURFZNTWxRdlptOXVkSE5wZW1Vdk5EQXdMMlpwYkd3dlNUQktRbEZyUmtOTlFUMDlMMlJwYzNOdmJIWmxMemN3TDJkeVlYWnBkSGt2VTI5MWRHaEZZWE4w.jpg?w=700&webp=1)
(推荐结果) MATLAB代码 主程序
- %% 主函数
- % 导入数据
- %data = [4,4,0,2,2;4,0,0,3,3;4,0,0,1,1;1,1,1,2,0;2,2,2,0,0;1,1,1,0,0;5,5,5,0,0];
- data = [2,0,0,4,4,0,0,0,0,0,0;0,0,0,0,0,0,0,0,0,0,5;0,0,0,0,0,0,0,1,0,4,0;3,3,4,0,3,0,0,2,2,0,0;5,5,5,0,0,0,0,0,0,0,0;
- 0,0,0,0,0,0,5,0,0,5,0;4,0,4,0,0,0,0,0,0,0,5;0,0,0,0,0,4,0,0,0,0,4;0,0,0,0,0,0,5,0,0,5,0;0,0,0,3,0,0,0,0,4,5,0;
- 1,1,2,1,1,2,1,0,4,5,0];
- % reccomendation
- %[sortScore, sortIndex] = recommend(data, 3, 'cosSim');
- [sortScore, sortIndex] = recommend(data, 2, 'cosSim');
- len = size(sortScore);
- finalRec = [sortIndex, sortScore];
- disp(finalRec);
SVD空间映射的函数
- function [ score ] = SVDEvaluate( data, user, simMeas, item )
- [m,n] = size(data);
- simTotal = 0;
- ratSimTotal = 0;
- % 奇异值分解
- [U S V] = svd(data);
- % 求使得保留90%能量的奇异值
- sizeN = 0;%记录维数
- [m_1,n_1] = size(S);
- a = 0;%求总能量
- for i = 1:m_1
- a = a + S(i,i)*S(i,i);
- end
- b = a*0.9;%能量的90%
- c = 0;
- for i = 1:n_1
- c = c + S(i,i)*S(i,i);
- if c >= b
- sizeN = i;
- break;
- end
- end
- %物品降维后的空间
- itemTransformed = data' * U(:,1:sizeN) * S(1:sizeN,1:sizeN)^(-1);
- for j = 1:n
- userRating = data(user, j);%此用户评价的商品
- if userRating == 0 || j == item%只是找到已评分的商品
- continue;
- end
- vectorA = itemTransformed(item,:);
- vectorB = itemTransformed(j,:);
- switch simMeas
- case {'cosSim'}
- similarity = cosSim(vectorA,vectorB);
- case {'ecludSim'}
- similarity = ecludSim(vectorA,vectorB);
- case {'pearsSim'}
- similarity = pearsSim(vectorA,vectorB);
- end
- disp(['the ', num2str(item), ' and ', num2str(j), ' similarity is ', num2str(similarity)]);
- simTotal = simTotal + similarity;
- ratSimTotal = ratSimTotal + similarity * userRating;
- end
- if simTotal == 0
- score = 0;
- else
- score = ratSimTotal./simTotal;
- end
- end
推荐的函数
- function [ sortScore, sortIndex ] = recommend( data, user, simMeas )
- % 获取data的大小
- [m, n] = size(data);%m为用户,n为商品
- if user > m
- disp('The user is not in the dataBase');
- end
- % 寻找用户user未评分的商品
- unratedItem = zeros(1,n);
- numOfUnrated = 0;
- for j = 1:n
- if data(user, j) == 0
- unratedItem(1,j) = 1;%0表示已经评分,1表示未评分
- numOfUnrated = numOfUnrated + 1;
- end
- end
- if numOfUnrated == 0
- disp('the user has rated all items');
- end
- % 对未评分项打分,已达到推荐的作用
- itemScore = zeros(numOfUnrated,2);
- r = 0;
- for j = 1:n
- if unratedItem(1,j) == 1%找到未评分项
- r = r + 1;
- %score = evaluate(data, user, simMeas, j);
- score = SVDEvaluate(data, user, simMeas, j);
- itemScore(r,1) = j;
- itemScore(r,2) = score;
- end
- end
- %排序,按照分数的高低进行推荐
- [sortScore, sortIndex_1] = sort(itemScore(:,2),'descend');
- [numOfIndex,x] = size(sortIndex_1(:,1));
- sortIndex = zeros(numOfIndex,1);
- for m = 1:numOfIndex
- sortIndex(m,:) = itemScore(sortIndex_1(m,:),1);
- end
- end
相似度的计算与前文一致。