SVD in the document term matrix does not give me the meanings I want

Question

I am trying to reproduce an example in an article entitled "Introduction to LSA": Introduction to LSA

In this example, they have the following term-document matrix:

And then they apply SVD and get the following:

Trying to reproduce this, I wrote the following R code:

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) 

I got the same term-document matrix and actually got the same correlations:

 > 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 

However, I tried to apply SVD to the matrix, and the only values ​​that are equal are eigenvalues, I cannot get what they got in the document.

 > 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 

Did I miss something?

Best wishes

EDIT:

The example assumes that the dimension is reduced, and they remove fewer eigenvalues. My problem is that the correlations that I get after SVD are different from the correlations than in the example:

Answer 1

I managed to find my mistake. When I restored the matrix, the transpose M = UDV 'was not calculated correctly. Now it works, sorry, this was my mistake ... Also, I calculated cor between documents when what I wanted was between terms.

I added the following lines:

 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

Only for the protocol, having tuned into your myMatrix matrix, I was able to almost exactly restore the example. The only difference (perhaps explainable?) Is located on some opposite signs in Figure 2 (for example, u[1,1] is -0.22 instead of 0.22 , as in W[1,1] Figure 2). The correlation matrix is ​​identical.

It should be noted that, contrary to the statement in the article (p. 13) that Spearman correlation is used, the exact result is obtained using the Pearson correlation method (by default).

Here is the code:

 > # 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 >