From 3a1bae9fe05646333b6883d906f2907df041bbf4 Mon Sep 17 00:00:00 2001 From: jiangzhonglian Date: Fri, 16 Jun 2017 17:05:13 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0SVD=E7=9A=84=E6=96=87?= =?UTF-8?q?=E6=A1=A3?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/13.利用PCA来简化数据.md | 10 +- docs/14.利用SVD简化数据.md | 132 +++++++++++-------- images/14.SVD/SVD_推荐系统_主题空间案例1.jpg | Bin 0 -> 59437 bytes images/14.SVD/使用SVD简化数据-余弦相似度.png | Bin 64991 -> 0 bytes 4 files changed, 83 insertions(+), 59 deletions(-) create mode 100644 images/14.SVD/SVD_推荐系统_主题空间案例1.jpg delete mode 100644 images/14.SVD/使用SVD简化数据-余弦相似度.png diff --git a/docs/13.利用PCA来简化数据.md b/docs/13.利用PCA来简化数据.md index d0b77f1e..90bfb2bd 100644 --- a/docs/13.利用PCA来简化数据.md +++ b/docs/13.利用PCA来简化数据.md @@ -1,4 +1,4 @@ -# 13) 利用PCA来简化数据 +# 第13章 利用PCA来简化数据 ![利用PCA简化数据_首页](/images/13.PCA/利用PCA简化数据_首页.jpg) @@ -9,7 +9,7 @@ * 人们实时的将显示器上的百万像素转换成为一个三维图像,该图像就给出运动场上球的位置。 * 在这个过程中,人们已经将数据从一百万维降至了三维。这就被称为`降维(dimensionality reduction)` -## 1.降维技术 +## 降维技术 > 数据显示并非大规模特征下的唯一难题,对数据进行简化还有如下一系列的原因: @@ -48,7 +48,7 @@ * 假设数据为多个数据源的混合观察结果,这些数据源之间在统计上是相互独立的,而在PCA中只假设数据是不相关的。 * 同因子分析一样,如果数据源的数目少于观察数据的数目,则可以实现降维过程。 -## 2.主成分分析(PCA) +## 主成分分析(PCA) > PCA的优缺点 @@ -66,7 +66,7 @@ * 例如下图: * ![应用PCA降维](/images/13.PCA/应用PCA降维.png) -## 3.对半导体数据进行降维处理 +## 对半导体数据进行降维处理 ``` 半导体是在一些极为先进的工厂中制造出来的。设备的生命早期有限,并且话费极其巨大。 @@ -79,7 +79,7 @@ 目前该章节处理的方案是:将缺失值NaN(Not a Number缩写),全部用平均值来替代(如果用0来处理的策略就太差劲了)。 ``` -## 4.本章小节 +## 本章小节 ``` 降维技术使得数据变的更易使用,并且它们往往能够去除数据中的噪音,使得其他机器学习任务更加精确。 diff --git a/docs/14.利用SVD简化数据.md b/docs/14.利用SVD简化数据.md index d26e3ece..4f8f81cf 100644 --- a/docs/14.利用SVD简化数据.md +++ b/docs/14.利用SVD简化数据.md @@ -1,85 +1,109 @@ -# 章节内容: -* SVD矩阵分解 -* 推荐引擎 -* 利用SVD提示推荐引擎的性能 +# 第14章 利用SVD简化数据 + -# 1. SVD 的应用:隐性语义索引(LSI)和推荐系统。 -* 奇异值分解(SVD):提取信息的方法。可以把SVD看成是从噪声数据中抽取相关特征。 -* 优点:简化数据,去除噪声,提高算法的结果。 -* 缺点:数据的转换可能难以理解。 -* 使用的数据类型:数值型数据。 +``` +奇异值分解(SVD, Singular Value Decomposition): + 提取信息的一种方法,可以把SVD看成是从噪声数据中抽取相关特征。从生物信息学到金融学,SVD是提出信息的强大工具。 -# 补充:降维 -* 降维(dimensionality reduction) -* 如果样本数据的特征维度很大,会使得难以分析和理解。我们可以通过降维技术减少维度。 -* 降维技术并不是将影响少的特征去掉,而是将样本数据集转换成一个低维度的数据集。 +* 优点:简化数据,去除噪声,提高算法的结果 +* 缺点:数据的转换可能难以理解 +* 使用的数据类型:数值型数据 +``` -# 1.1 隐性语义索引:矩阵 = 文档 + 词语 +## SVD的应用 + +> 信息检索-隐形语义检索(Lstent Semantic Indexing, LSI)或 隐形语义分析(Latent Semantic Analysis, LSA) + +隐性语义索引:矩阵 = 文档 + 词语 * 是最早的SVD应用之一。我们称利用SVD的方法为隐性语义索引(LSI)或隐性语义分析(LSA)。 * ![LSA举例](/images/14.SVD/使用SVD简化数据-LSI举例.png) -# 2. 矩阵分解 -* SVD 是矩阵分解的一种类型,而矩阵分解是将数据矩阵分解为多个独立部分的过程。 -* 矩阵分解可以将原始矩阵表示成新的易于处理的形式,这种新形式时两个或多个矩阵的乘积。(类似代数中的因数分解) +> 推荐引擎 +1. 利用SVD从数据中构建一个主题空间。 +2. 再在该空间下计算其相似度。 + +* ![主题空间案例1](/images/14.SVD/SVD_推荐系统_主题空间案例1.jpg) +* 上图右边标注的为一组共同特征,表示美式BBQ 空间;另一组在上图右边未标注的为日式食品 空间。 + +> 图像压缩 + + + +## SVD矩阵分解 + +> 矩阵分解 + +* 矩阵分解是将数据矩阵分解为多个独立部分的过程。 +* 矩阵分解可以将原始矩阵表示成新的易于处理的形式,这种新形式时两个或多个矩阵的乘积。(类似代数中的因数分解) * 举例:如何将12分解成两个数的乘积?(1,12)、(2,6)、(3,4)都是合理的答案。 -# SVD 时矩阵最常见的分解技术。 -* SVD将原始的数据集矩阵Data分解成三个矩阵U、∑、VT。 -* 举例:如果原始矩阵Data是m行n列,那么U、∑、VT 就分别是m行m列、m行n列和n行n列。 +> SVD 是矩阵分解的一种类型,也是矩阵分解最常见的技术。 + +* SVD 将原始的数据集矩阵Data分解成三个矩阵U、∑、\\(V^T\\) +* 举例:如果原始矩阵 \\(Data_{m*n}\\) 是m行n列, + * \\(U_{m*m}\\) 表示m行m列 + * \\(∑_{m*n}\\) 表示m行n列 + * \\(V^T_{n*n}\\) 表示n行n列。 * ![SVD公式](/images/14.SVD/使用SVD简化数据-SVD公式.png) -* 上述分解中会构建出一个矩阵∑,该矩阵只有对角元素,其他元素均为0。另一个惯例就是,∑的对角元素是从大到小排列的。这些对角元素称为奇异值,它们对应原始矩阵Data的奇异值。 -* 奇异值与特征值是有关系的。这里的奇异值就是矩阵 Data * DateT 特征值的平方根。 +* 上述分解中会构建出一个矩阵∑,该矩阵只有对角元素,其他元素均为0(近似于0)。另一个惯例就是,∑的对角元素是从大到小排列的。这些对角元素称为奇异值,它们对应原始矩阵Data的奇异值。 +* 奇异值与特征值(PCA数据中重要特征)是有关系的。这里的奇异值就是矩阵 \\(Data * Date^T\\) 特征值的平方根。 +* 普遍的事实:在某个奇异值的数目(r个=>奇异值的平方和累加到总值的90%以上)之后,其他的奇异值都置为0(近似于0)。这意味着数据集中仅有r个重要特征,而其余特征则都是噪声或冗余特征。 -* 一个普遍的事实:在某个奇异值的数目(r 个)之后,其他的奇异值都置为0。这意味着数据集中仅有r个重要特征,而其余特征则都是噪声或冗余特征。 +## 推荐引擎 -# 3. 利用Python 实现SVD -* Numpy 有一个称为linalg的线性代数工具箱。 +> 基于协同过滤(collaborative filtering)的推荐引擎 -# 4. 基于协同过滤的推荐引擎 +* 利用Python 实现SVD(Numpy 有一个称为linalg的线性代数工具箱) * 协同过滤:是通过将用户和其他用户的数据进行对比来实现推荐的。 - * 当知道了两个用户或两个物品之间的相似度,我们就可以利用已有的数据来预测未知用户的喜好。 -# 4.1 相似度计算 -* 我们不利用专家所给出的重要属性来描述物品从而计算它们之间的相似度,而是利用用户对它们的意见来计算相似度。 +> 相似度计算 -* 1)欧氏距离:指在m维空间中两个点之间的真实距离,或者向量的自然长度(即改点到原点的距离)。二维或三维中的欧氏距离就是两点之间的实际距离。 -* 相似度= 1/(1+距离) -* 物品对越相似,它们的相似度值就越大。 +* inA, inB 对于的为列向量 +1. 欧氏距离:指在m维空间中两个点之间的真实距离,或者向量的自然长度(即改点到原点的距离)。二维或三维中的欧氏距离就是两点之间的实际距离。 + * 相似度= 1/(1+欧式距离) + * `相似度= 1.0/(1.0 + la.norm(inA - inB))` + * 物品对越相似,它们的相似度值就越大。 +2. 皮尔逊相关系数:度量的是两个向量之间的相似度。 + * 相似度= 0.5 + 0.5*corrcoef() 【皮尔逊相关系数的取值范围从 -1 到 +1,通过函数0.5 + 0.5\*corrcoef()这个函数计算,把值归一化到0到1之间】 + * `相似度= 0.5 + 0.5 * corrcoef(inA, inB, rowvar = 0)[0][1]` + * 相对欧氏距离的优势:它对用户评级的量级并不敏感。 +3. 余弦相似度:计算的是两个向量夹角的余弦值。 + * 余弦值 = (A·B)/(||A||·||B||) 【余弦值的取值范围也在-1到+1之间】 + * 相似度= 0.5 + 0.5*余弦值 + * `相似度= 0.5 + 0.5*( float(inA.T*inB) / la.norm(inA)*la.norm(inB))` + * 如果夹角为90度,则相似度为0;如果两个向量的方向相同,则相似度为1.0。 -* 2)皮尔逊相关系数:度量的是两个向量之间的相似度。 -* 相对欧氏距离的优势:它对用户评级的量级并不敏感。 -* 皮尔逊相关系数的取值范围从 -1 到 +1,通过函数0.5 + 0.5*corrcoef()这个函数计算,把值归一化到0到1之间。 +> 基于物品的相似度和基于用户的相似度:物品比较少则选择物品相似度,用户比较少则选择用户相似度。【矩阵还是小一点好计算】 -* 3)余弦相似度:计算的是两个向量夹角的余弦值。 -* 如果夹角为90度,则相似度为0;如果两个向量的方向相同,则相似度为1.0。 -* 余弦相似度的取值范围也在-1到+1之间。 -* ![SVD公式](/images/14.SVD/使用SVD简化数据-余弦相似度.png) - -# 4.2 基于物品的相似度和基于用户的相似度:物品比较多则选择物品相似度,用户比较多则选择用户相似度。 -* 基于物品的相似度:计算物品之间的距离的方法。 -* 基于物品相似度计算的时间会随物品数量的增加而增加。 +* 基于物品的相似度:计算物品之间的距离。【耗时会随物品数量的增加而增加】 * ![SVD公式](/images/14.SVD/使用SVD简化数据-基于物品相似度.png) -* 基于用户的相似度:计算用户距离的方法。 -* 基于用户的相似度计算的时间则会随着用户数量的增加而增加。 +* 基于用户的相似度:计算用户之间的距离。【耗时会随用户数量的增加而增加】 * ![SVD公式](/images/14.SVD/使用SVD简化数据-基于用户相似度.png) -# 4.3 推荐引擎的评价 -* 最小均方根误差(RMSE):推荐引擎评价的指标,先计算均方误差然后取其平方根。 +> 推荐引擎的评价 +* 采用交叉测试的方法。【就是将某些已知的评分值去掉,然后进行预测,最后计算预测值和真实值之间的差异】 +* 推荐引擎评价的指标: 最小均方根误差(Root mean squared error, RMSE),也称标准误差(Standard error),就是计算均方误差的平均值然后取其平方根。 + * 如果RMSE=1, 表示相差1个星级;如果RMSE=2.5, 表示相差2.5个星级。 -# 5. 示例:(看代码) -# 5.1 推荐未尝过的菜肴 -* 1)寻找用户没有评级的菜肴,即在用户-物品矩阵中的0值; -* 2)在用户没有评级的所有物品中,对每个物品预计一个可能的评级分数。这就是说,我们认为用户可能会对物品的打分(这个就是相似度计算的初衷) -* 3)对这些物品的评分从高到低进行排序,返回前N个物品。 +> 餐馆菜肴推荐引擎-案例 -# 5.2 利用SVD提高推荐效果 +* 推荐系统的工作过程:给定一个用户,系统会为此用户返回N个最好的推荐菜。 +* 实现流程大致如下: + 1. 寻找用户没有评级的菜肴,即在用户-物品矩阵中的0值。 + 2. 在用户没有评级的所有物品中,对每个物品预计一个可能的评级分数。这就是说:我们认为用户可能会对物品的打分(这就是相似度计算的初衷)。 + 3. 对这些物品的评分从高到低进行排序,返回前N个物品。 +* 现在我们来观测代码实现。 + +## 利用SVD提升推荐引擎的性能 + +> 利用SVD提高推荐效果 * 将矩阵降维 -# 5.3 构建推荐引擎面临的挑战 +> 构建推荐引擎面临的挑战 * 1)在大规模的数据集上,SVD分解会降低程序的速度 * 2)存在其他很多规模扩展性的挑战性问题,比如矩阵的表示方法和计算相似度得分消耗资源。 * 3)如何在缺乏数据时给出好的推荐(冷启动问题,解决方案就是将推荐看成是搜索问题) diff --git a/images/14.SVD/SVD_推荐系统_主题空间案例1.jpg b/images/14.SVD/SVD_推荐系统_主题空间案例1.jpg new file mode 100644 index 0000000000000000000000000000000000000000..6ba47e17306f2fd8661c5a791fab07aff6b5191e GIT binary patch literal 59437 zcmb@t1yqz@_dog!1Jd0M0s_)qLra5{bSXV_GjvG}B_Sv%A+6v@H%Nnuq;!KIJ)?jl zdB^YjzQ6abyZ&q4d;jad;XJeFJZH{x_H%0Q&)(;D{&pQ8(^l6~2S6YI(8T-!w;O0RUDp)_-!qcZ$LP;Xwv}Z{8jO(gv>H0p9+u z-ac$%g2I5bik2Ss-x@IV56|@vOE$xtx{v|XWA8=dzc@szMBZ*2 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