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There is a software called Brandmap$^1$ which can return a biplot from a matrix. I am trying to run the same result in R but the coordinates are not the same.

First I input a simple matrix into the software. 

  x1 x2 x3
a  6  3  7
b  8  6  7
c  9  4  2

There were several options of data centering and I chose to center the data by subtracting every number in the matrix from its (row mean*column mean/grand mean).

         x1    x2         x3
a -1.076923 -1.00  2.0769231
b -1.288462  0.75  0.5384615
c  2.365385  0.25 -2.6153846

Then I chose column factorization to create a biplot. I guess it means a covariance biplot which the singular values are totally assigned to the right singular vectors.

It showed the coordinates:

     dim1    dim2
x1  -2.81   -0.73
x2  -0.55    1.15
x3   3.36   -0.42

a    1.58   -1.78
b    0.75    2.26
c   -2.33   -0.48

I tried to calculate the same results in R.

> P = matrix(c(6,8,9,3,6,4,7,7,2),nrow=3)
> row.names(P)=c("a","b","c")
> colnames(P)=c("x1","x2","x3")
> P
  x1 x2 x3
a  6  3  7
b  8  6  7
c  9  4  2
> r1 = matrix(rep(1,3))     #row sum
> c1 = matrix(rep(1,3))     #column sum
> r = P%*%r1
> c = t(P)%*%c1
> L = P - r%*%t(c)/sum(P)   #subtract row mean*column mean/grand mean
> L
         x1    x2         x3
a -1.076923 -1.00  2.0769231
b -1.288462  0.75  0.5384615
c  2.365385  0.25 -2.6153846
> S = svd(L)
> S$v%*%diag(S$d)
           [,1]       [,2]         [,3]
[1,]  2.8077724  0.7289408 -8.10596e-17
[2,]  0.5487104 -1.1506159 -8.10596e-17
[3,] -3.3564829  0.4216750 -8.10596e-17
> S$u
           [,1]       [,2]      [,3]
[1,] -0.5420705  0.6105950 0.5773503
[2,] -0.2577555 -0.7747443 0.5773503
[3,]  0.7998260  0.1641494 0.5773503

I found that the values in the right vector are the same but with negative sign and all the values in the left vector are multiplied by -2.918. I am not sure if there is any weighting in the calculation of that software. What kind of adjustment I can try so that I can run the same results in R?

$^1$ Note from @ttnphns: I suppose this software does correspondence analysis, not just arbitrary biplot. CA is very often used in brand research.

ttnphns
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bbbbbliu
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  • I've done (not weighted) biplot with your 2nd (centered) table. I used formulas described [here](http://stats.stackexchange.com/q/141754/3277) - they are in aggreement with Greenacre "Biplots in practice" well known paper and other sources. I gave all the inertia to the columns, like you. Your table has rank 2, so only 2 principal dimensions exist, with eigenvalues `2.16118 .22590` The coordinates (for shortness, I display here only Dimension1 coordinates): For rows: `-.93889 -.44645 .38534`, For columns: `1.62107 .31680 -1.93787`. – ttnphns Sep 28 '16 at 09:14
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    When you look in the thread with formulas I link to, pay attention that biplot is not simply svd(X), it is svd(Z) where Z is a normalized version of X taking number of rows and columns in consideration. After svd, a "back rescaling" of eigenvectors is performed, which gives the standardized coordinates. – ttnphns Sep 28 '16 at 09:19
  • @ttnphns Thank you for your answer! I tried to normalize the matrix by dividing by (r*c)^0.5 which is Z = X/(3*3^0.5) but I got the eigenvalues 1.470096 .475287 and couldn't get the desired results. I need to generate the exact same results of coordinates from that software in R. – bbbbbliu Sep 28 '16 at 14:47
  • You got correct singular values. Eigenvalues are just their squares. You should be able to reproduce my results. As for the results of Brandmap software - I absolutly can't comment it since I don't know its algorithms. It might use other normalizations or even other approaches. Or even have a bug. Consult with the documentation or the developers. – ttnphns Sep 28 '16 at 15:48
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    Hey, possibly I cracked it. I performed, on your initial table1 correspondence analysis (euclidaen model, see my link) with preprocessing = two-way centering + normalizing by the mean mass. I gave full inertia to columns, as you did. I got coordinates which have the cosine similarity with your coordinates from Brandmap equal 1. I.e. results are fully proportional. I tentatively conclude that Brandmap actually does correspondence analysis, only it does normalization which is not same but is equivalent in results with correspondence analysis I described in the link. – ttnphns Sep 28 '16 at 17:13
  • @ttnphns what is the step of a two-way centering + normalizing by mass mean to be done on the matrix? I saw there are some variation of two-way centering and I'm not sure what are their difference. – bbbbbliu Sep 29 '16 at 00:25
  • I'm familiar with the two-way centering variant I've described in my answer of the link (see Correspondence analysis (Euclidean model) section). If you know other variants, please describe them, possibly with sources. – ttnphns Sep 29 '16 at 02:39
  • @ttnphns For example is subtracting from (row mean*column mean/grand mean) one of the two-way centering method? Would you mind showing your steps in details how to calculate your results which are fully proportional? I'd much appreciate it :) Thanks. – bbbbbliu Sep 29 '16 at 05:26

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