术语文档矩阵中的SVD不能给出我想要的值

Question

我试图在一篇名为“LSA简介”的论文中复制一个例子：An introduction to LSA

在示例中，他们具有以下术语 - 文档矩阵：

然后他们应用SVD并获得以下内容：

试图复制这个，我写了以下R代码：

library(lsa); library(tm)

d1 = "Human machine interface for ABC computer applications"
d2 = "A survey of user opinion of computer system response time"
d3 = "The EPS user interface management system"
d4 = "System and human system engineering testing of EPS"
d5 <- "Relation of user perceived response time to error measurement"
d6 <- "The generation of random, binary, ordered trees"
d7 <- "The intersection graph of paths in trees"
d8 <- "Graph minors IV: Widths of trees and well-quasi-ordering"
d9 <- "Graph minors: A survey"

# Words that appear in at least two of the titles
D <- c(d1, d2, d3, d4, d5, d6, d7, d8, d9)

corpus <- Corpus(VectorSource(D))

# Remove Punctuation
corpus <- tm_map(corpus, removePunctuation)

# tolower
corpus <- tm_map(corpus, content_transformer(tolower))

# Stopword Removal
corpus <- tm_map(corpus, function(x) removeWords(x, stopwords("english")))

# term document Matrix
myMatrix <- TermDocumentMatrix(corpus)

# Delete terms that only appear in a document
rowTotals <- apply(myMatrix, 1, sum)
myMatrix.new <- myMatrix[rowTotals > 1, ]

# Correlation Matrix of terms
cor(t(as.matrix(myMatrix.new)))

# lsaSpace <- lsa(myMatrix.new)
# myMatrix.reduced <- lsaSpace$tk %*% diag(lsaSpace$sk) %*% t(lsaSpace$dk)

mySVD <- svd(myMatrix.new)

我得到了相同的术语 - 文档矩阵，并且实际上获得了相同的相关性：

> inspect(myMatrix.new)
<<TermDocumentMatrix (terms: 12, documents: 9)>>
Non-/sparse entries: 28/80
Sparsity           : 74%
Maximal term length: 9
Weighting          : term frequency (tf)

           Docs
Terms       1 2 3 4 5 6 7 8 9
  computer  1 1 0 0 0 0 0 0 0
  eps       0 0 1 1 0 0 0 0 0
  graph     0 0 0 0 0 0 1 1 1
  human     1 0 0 1 0 0 0 0 0
  interface 1 0 1 0 0 0 0 0 0
  minors    0 0 0 0 0 0 0 1 1
  response  0 1 0 0 1 0 0 0 0
  survey    0 1 0 0 0 0 0 0 1
  system    0 1 1 2 0 0 0 0 0
  time      0 1 0 0 1 0 0 0 0
  trees     0 0 0 0 0 1 1 1 0
  user      0 1 1 0 1 0 0 0 0
> cor(as.matrix(t(myMatrix.new)))
            computer        eps      graph      human  interface     minors
computer   1.0000000 -0.2857143 -0.3779645  0.3571429  0.3571429 -0.2857143
eps       -0.2857143  1.0000000 -0.3779645  0.3571429  0.3571429 -0.2857143
graph     -0.3779645 -0.3779645  1.0000000 -0.3779645 -0.3779645  0.7559289
human      0.3571429  0.3571429 -0.3779645  1.0000000  0.3571429 -0.2857143
interface  0.3571429  0.3571429 -0.3779645  0.3571429  1.0000000 -0.2857143
minors    -0.2857143 -0.2857143  0.7559289 -0.2857143 -0.2857143  1.0000000
response   0.3571429 -0.2857143 -0.3779645 -0.2857143 -0.2857143 -0.2857143
survey     0.3571429 -0.2857143  0.1889822 -0.2857143 -0.2857143  0.3571429
system     0.0433555  0.8237545 -0.4588315  0.4335550  0.0433555 -0.3468440
time       0.3571429 -0.2857143 -0.3779645 -0.2857143 -0.2857143 -0.2857143
trees     -0.3779645 -0.3779645  0.5000000 -0.3779645 -0.3779645  0.1889822
user       0.1889822  0.1889822 -0.5000000 -0.3779645  0.1889822 -0.3779645
            response     survey     system       time      trees       user
computer   0.3571429  0.3571429  0.0433555  0.3571429 -0.3779645  0.1889822
eps       -0.2857143 -0.2857143  0.8237545 -0.2857143 -0.3779645  0.1889822
graph     -0.3779645  0.1889822 -0.4588315 -0.3779645  0.5000000 -0.5000000
human     -0.2857143 -0.2857143  0.4335550 -0.2857143 -0.3779645 -0.3779645
interface -0.2857143 -0.2857143  0.0433555 -0.2857143 -0.3779645  0.1889822
minors    -0.2857143  0.3571429 -0.3468440 -0.2857143  0.1889822 -0.3779645
response   1.0000000  0.3571429  0.0433555  1.0000000 -0.3779645  0.7559289
survey     0.3571429  1.0000000  0.0433555  0.3571429 -0.3779645  0.1889822
system     0.0433555  0.0433555  1.0000000  0.0433555 -0.4588315  0.2294157
time       1.0000000  0.3571429  0.0433555  1.0000000 -0.3779645  0.7559289
trees     -0.3779645 -0.3779645 -0.4588315 -0.3779645  1.0000000 -0.5000000
user       0.7559289  0.1889822  0.2294157  0.7559289 -0.5000000  1.0000000

然而，我试图将SVD应用于矩阵，唯一相同的值是特征值，我无法得到他们在论文中得到的东西。

> mySVD
$d
[1] 3.3408838 2.5417010 2.3539435 1.6445323 1.5048316 1.3063820 0.8459031
[8] 0.5601344 0.3636768

$u
             [,1]        [,2]       [,3]          [,4]        [,5]        [,6]
 [1,] -0.24047023 -0.04315195  0.1644291  0.5949618181 -0.10675529 -0.25495513
 [2,] -0.30082816  0.14127047 -0.3303084 -0.1880919179  0.11478462  0.27215528
 [3,] -0.03613585 -0.62278523 -0.2230864 -0.0007000721 -0.06825294  0.11490895
 [4,] -0.22135078  0.11317962 -0.2889582  0.4147507404 -0.10627512 -0.34098332
 [5,] -0.19764540  0.07208778 -0.1350396  0.5522395837  0.28176894  0.49587801
 [6,] -0.03175633 -0.45050892 -0.1411152  0.0087294706 -0.30049511  0.27734340
 [7,] -0.26503747 -0.10715957  0.4259985 -0.0738121922  0.08031938 -0.16967639
 [8,] -0.20591786 -0.27364743  0.1775970  0.0323519366 -0.53715000  0.08094398
 [9,] -0.64448115  0.16730121 -0.3611482 -0.3334616013 -0.15895498 -0.20652259
[10,] -0.26503747 -0.10715957  0.4259985 -0.0738121922  0.08031938 -0.16967639
[11,] -0.01274618 -0.49016179 -0.2311202 -0.0248019985  0.59416952 -0.39212506
[12,] -0.40359886 -0.05707026  0.3378035 -0.0991137295  0.33173372  0.38483192
              [,7]          [,8]        [,9]
 [1,] -0.302240236  0.0623280150 -0.49244436
 [2,]  0.032994110 -0.0189980144  0.16533917
 [3,]  0.159575477 -0.6811254380 -0.23196123
 [4,]  0.522657771 -0.0604501376  0.40667751
 [5,] -0.070423441 -0.0099400372  0.10893027
 [6,]  0.339495286  0.6784178789 -0.18253498
 [7,]  0.282915727 -0.0161465472  0.05387469
 [8,] -0.466897525 -0.0362988295  0.57942611
 [9,] -0.165828575  0.0342720233 -0.27069629
[10,]  0.282915727 -0.0161465472  0.05387469
[11,] -0.288317461  0.2545679452  0.22542407
[12,]  0.002872175 -0.0003905042 -0.01232935

$v
              [,1]        [,2]        [,3]        [,4]        [,5]          [,6]
 [1,] -0.197392802  0.05591352 -0.11026973  0.94978502  0.04567856 -7.659356e-02
 [2,] -0.605990269 -0.16559288  0.49732649  0.02864890 -0.20632728 -2.564752e-01
 [3,] -0.462917508  0.12731206 -0.20760595 -0.04160920  0.37833623  7.243996e-01
 [4,] -0.542114417  0.23175523 -0.56992145 -0.26771404 -0.20560471 -3.688609e-01
 [5,] -0.279469108 -0.10677472  0.50544991 -0.15003543  0.32719441  3.481305e-02
 [6,] -0.003815213 -0.19284794 -0.09818424 -0.01508149  0.39484121 -3.001611e-01
 [7,] -0.014631468 -0.43787488 -0.19295557 -0.01550719  0.34948535 -2.122014e-01
 [8,] -0.024136835 -0.61512190 -0.25290398 -0.01019901  0.14979847  9.743417e-05
 [9,] -0.081957368 -0.52993707 -0.07927315  0.02455491 -0.60199299  3.622190e-01
             [,7]         [,8]        [,9]
 [1,]  0.17731830 -0.014393259  0.06369229
 [2,] -0.43298424  0.049305326 -0.24278290
 [3,] -0.23688970  0.008825502 -0.02407687
 [4,]  0.26479952 -0.019466944  0.08420690
 [5,]  0.67230353 -0.058349563  0.26237588
 [6,] -0.34083983  0.454476523  0.61984719
 [7,] -0.15219472 -0.761527011 -0.01797518
 [8,]  0.24914592  0.449642757 -0.51989050
 [9,]  0.03803419 -0.069637550  0.45350675

我错过了什么吗？

最好的祝福

编辑：

在该示例中假设尺寸减小并且它们删除了较少的特征值。我的问题是我在SVD之后获得的相关性与示例中的相关性不同：

Answer 1

我设法找到了我的错误。当我重建矩阵时，没有正确计算M = U D V'的转置。现在它工作了，对不起，这是我的错误...而且，我在计算文件之间的文件，当我想要的是在术语之间。

我添加了以下几行：

mySVD <- svd(myMatrix.new)

Mp <- mySVD$u[, c(1,2)] %*% diag(mySVD$d)[c(1, 2), c(1, 2)] %*% t(mySVD$v[, c(1, 2)])

rownames(Mp) <- rownames(myMatrix.new)
cor(t(Mp))

Answer 2

只是为了协议，设置你的矩阵myMatrix我能够接近完全重建的例子。唯一的区别（可能的解释？）是图2中的一些相反的符号（例如u[1,1]是-0.22而不是0.22，如图2中的W[1,1]）。相关矩阵是相同的。

但应该提到的是，与论文（第13页）中的索赔相反，使用Spearman相关性，使用（默认）Pearson相关方法获得精确结果。

这里的代码：

> # term document Matrix
> myMatrix <- TermDocumentMatrix(corpus)
> 
> ## reorder rows
> myMatrix <- mm[match(c("human","interface","computer","user","system","response","time","eps","survey","trees","graph","minors"), rownames(mm)), ]
> 
> # Delete terms that only appear in a document
> rowTotals <- apply(myMatrix, 1, sum)
> myMatrix.new <- myMatrix[rowTotals > 1, ]
> 
> mySVD <- svd(myMatrix.new)
> 
> ## Figure 1
> myMatrix.new 
           Docs
Terms       1 2 3 4 5 6 7 8 9
  human     1 0 0 1 0 0 0 0 0
  interface 1 0 1 0 0 0 0 0 0
  computer  1 1 0 0 0 0 0 0 0
  user      0 1 1 0 1 0 0 0 0
  system    0 1 1 2 0 0 0 0 0
  response  0 1 0 0 1 0 0 0 0
  time      0 1 0 0 1 0 0 0 0
  eps       0 0 1 1 0 0 0 0 0
  survey    0 1 0 0 0 0 0 0 1
  trees     0 0 0 0 0 1 1 1 0
  graph     0 0 0 0 0 0 1 1 1
  minors    0 0 0 0 0 0 0 1 1
> 
> ## mySVD Figure 2
> lapply(mySVD,round,2)
$d
[1] 3.34 2.54 2.35 1.64 1.50 1.31 0.85 0.56 0.36

$u
       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9]
 [1,] -0.22 -0.11  0.29 -0.41 -0.11 -0.34 -0.52  0.06  0.41
 [2,] -0.20 -0.07  0.14 -0.55  0.28  0.50  0.07  0.01  0.11
 [3,] -0.24  0.04 -0.16 -0.59 -0.11 -0.25  0.30 -0.06 -0.49
 [4,] -0.40  0.06 -0.34  0.10  0.33  0.38  0.00  0.00 -0.01
 [5,] -0.64 -0.17  0.36  0.33 -0.16 -0.21  0.17 -0.03 -0.27
 [6,] -0.27  0.11 -0.43  0.07  0.08 -0.17 -0.28  0.02  0.05
 [7,] -0.27  0.11 -0.43  0.07  0.08 -0.17 -0.28  0.02  0.05
 [8,] -0.30 -0.14  0.33  0.19  0.11  0.27 -0.03  0.02  0.17
 [9,] -0.21  0.27 -0.18 -0.03 -0.54  0.08  0.47  0.04  0.58
[10,] -0.01  0.49  0.23  0.02  0.59 -0.39  0.29 -0.25  0.23
[11,] -0.04  0.62  0.22  0.00 -0.07  0.11 -0.16  0.68 -0.23
[12,] -0.03  0.45  0.14 -0.01 -0.30  0.28 -0.34 -0.68 -0.18

$v
       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]  [,7]  [,8]  [,9]
 [1,] -0.20 -0.06  0.11 -0.95  0.05 -0.08 -0.18  0.01  0.06
 [2,] -0.61  0.17 -0.50 -0.03 -0.21 -0.26  0.43 -0.05 -0.24
 [3,] -0.46 -0.13  0.21  0.04  0.38  0.72  0.24 -0.01 -0.02
 [4,] -0.54 -0.23  0.57  0.27 -0.21 -0.37 -0.26  0.02  0.08
 [5,] -0.28  0.11 -0.51  0.15  0.33  0.03 -0.67  0.06  0.26
 [6,]  0.00  0.19  0.10  0.02  0.39 -0.30  0.34 -0.45  0.62
 [7,] -0.01  0.44  0.19  0.02  0.35 -0.21  0.15  0.76 -0.02
 [8,] -0.02  0.62  0.25  0.01  0.15  0.00 -0.25 -0.45 -0.52
 [9,] -0.08  0.53  0.08 -0.02 -0.60  0.36 -0.04  0.07  0.45

> 
> Mp <- mySVD$u[, c(1,2)] %*% diag(mySVD$d)[c(1, 2), c(1, 2)] %*% t(mySVD$v[, c(1, 2)])
> rownames(Mp) <- rownames(myMatrix.new)
> 
> ## Figure 3
> round(Mp,2)
           [,1] [,2]  [,3]  [,4] [,5]  [,6]  [,7]  [,8]  [,9]
human      0.16 0.40  0.38  0.47 0.18 -0.05 -0.12 -0.16 -0.09
interface  0.14 0.37  0.33  0.40 0.16 -0.03 -0.07 -0.10 -0.04
computer   0.15 0.51  0.36  0.41 0.24  0.02  0.06  0.09  0.12
user       0.26 0.84  0.61  0.70 0.39  0.03  0.08  0.12  0.19
system     0.45 1.23  1.05  1.27 0.56 -0.07 -0.15 -0.21 -0.05
response   0.16 0.58  0.38  0.42 0.28  0.06  0.13  0.19  0.22
time       0.16 0.58  0.38  0.42 0.28  0.06  0.13  0.19  0.22
eps        0.22 0.55  0.51  0.63 0.24 -0.07 -0.14 -0.20 -0.11
survey     0.10 0.53  0.23  0.21 0.27  0.14  0.31  0.44  0.42
trees     -0.06 0.23 -0.14 -0.27 0.14  0.24  0.55  0.77  0.66
graph     -0.06 0.34 -0.15 -0.30 0.20  0.31  0.69  0.98  0.85
minors    -0.04 0.25 -0.10 -0.21 0.15  0.22  0.50  0.71  0.62
> 
> cor(Mp["human",],Mp["minors",])
[1] -0.83
> 
> cor(Mp["human",],Mp["user",])
[1] 0.94
> 
> ## Figure 4
> corMo <- cor(myMatrix.new)
> corMo[upper.tri(corMo,diag=TRUE)] <- 0
> corMo
      1     2     3     4     5     6    7    8 9
1  0.00  0.00  0.00  0.00  0.00  0.00 0.00 0.00 0
2 -0.19  0.00  0.00  0.00  0.00  0.00 0.00 0.00 0
3  0.00  0.00  0.00  0.00  0.00  0.00 0.00 0.00 0
4  0.00  0.00  0.47  0.00  0.00  0.00 0.00 0.00 0
5 -0.33  0.58  0.00 -0.31  0.00  0.00 0.00 0.00 0
6 -0.17 -0.30 -0.21 -0.16 -0.17  0.00 0.00 0.00 0
7 -0.26 -0.45 -0.32 -0.24 -0.26  0.67 0.00 0.00 0
8 -0.33 -0.58 -0.41 -0.31 -0.33  0.52 0.77 0.00 0
9 -0.33 -0.19 -0.41 -0.31 -0.33 -0.17 0.26 0.56 0
> 
> corMp <- cor(Mp)
> corMp[upper.tri(corMp,diag=TRUE)] <- 0
> corMp
       [,1]  [,2]  [,3]  [,4]  [,5] [,6] [,7] [,8] [,9]
 [1,]  0.00  0.00  0.00  0.00  0.00    0    0    0    0
 [2,]  0.91  0.00  0.00  0.00  0.00    0    0    0    0
 [3,]  1.00  0.91  0.00  0.00  0.00    0    0    0    0
 [4,]  1.00  0.88  1.00  0.00  0.00    0    0    0    0
 [5,]  0.84  0.99  0.84  0.81  0.00    0    0    0    0
 [6,] -0.86 -0.57 -0.86 -0.89 -0.44    0    0    0    0
 [7,] -0.85 -0.56 -0.85 -0.88 -0.44    1    0    0    0
 [8,] -0.85 -0.56 -0.85 -0.88 -0.43    1    1    0    0
 [9,] -0.81 -0.50 -0.81 -0.84 -0.37    1    1    1    0
>