# ASReml-R: Storing A inverse as a sparse matrix

December 18, 2010
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(This article was first published on Gregor Gorjanc (gg), and kindly contributed to R-bloggers)

I was testing ASReml-R program (an R package that links propriety ASReml binaries that can be used only with valid licence) this week and had to do some manipulations with the numerator relationship matrix (A). ASReml-R provides a function (asreml.Ainverse) that can create inverse of A directly from the pedigree as this inverse is needed in pedigree based mixed model. Bulding inverse of A directly from a pedigree is a well known result dating back to Henderson in 1970s or so. The funny thing is that it is cheaper to setup inverse of A directly than to setup up first A and then to invert it. In addition, inverse of A is very spare so it is easy/cheap to store it. Documentation for asreml.Ainverse has a tiny example of usage. Since the result of this function is a list with several elements (data.frame with "triplets" for non-zero elements of inverse of A, inbreeding coefficients, ...) example also shows how to create a matrix object in R as shown bellow:
library(package="asreml")## Create test pedigreeped <- data.frame( me=c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10),                  dad=c(0, 0, 0, 1, 1, 2, 4, 5, 7,  9),                  mum=c(0, 0, 0, 1, 1, 2, 6, 6, 8,  9))## Create inverse of A in triplet formtmp <- asreml.Ainverse(pedigree=ped)$ginv## Create a "proper" matrixAInv <- asreml.sparse2mat(x=tmp)## Print AInvAInv So the inverse of A would be:  [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10][1,] 5 0 0 -2 -2 0 0.0 0.0 0.000000 0.000000[2,] 0 3 0 0 0 -2 0.0 0.0 0.000000 0.000000[3,] 0 0 1 0 0 0 0.0 0.0 0.000000 0.000000[4,] -2 0 0 3 0 1 -2.0 0.0 0.000000 0.000000[5,] -2 0 0 0 3 1 0.0 -2.0 0.000000 0.000000[6,] 0 -2 0 1 1 4 -2.0 -2.0 0.000000 0.000000[7,] 0 0 0 -2 0 -2 4.5 0.5 -1.000000 0.000000[8,] 0 0 0 0 -2 -2 0.5 4.5 -1.000000 0.000000[9,] 0 0 0 0 0 0 -1.0 -1.0 4.909091 -2.909091[10,] 0 0 0 0 0 0 0.0 0.0 -2.909091 2.909091 However, this is problematic as it creates a dense matrix - zero values are also stored (you can see them). If we would have 1,000 individuals, such a matrix would consume 7.6 Mb of RAM (= (((1000 * (1000 + 1)) / 2) * 16) / 2^20). This is not a lot, but with 10,000 individuals we would already need 763 Mb of RAM, which can create some bottlenecks. A solution is to create a sparse matrix using the Matrix R package. Luckily we have all the ingredients prepared by asreml.Ainverse function - the triplets. However, the essential R code is a bit daunting and I had to test several options before I figured it out - code from my previous post helped;) ## Load packagelibrary(package="Matrix")## Number of pedigree membersnI <- nrow(ped)## Store inverse of A in sparse formAInv2 <- as(new("dsTMatrix", Dim=c(nI, nI), uplo="L", i=(as.integer(tmp$Row) - 1L),                 j=(as.integer(tmp$Column) - 1L), x=tmp$Ainverse),             "dsCMatrix")## Add row and column names - optionaldimnames(AInv2) <- list(attr(x=tmp, which="rowNames"),                        attr(x=tmp, which="rowNames"))## Print AInvAInv2
And the inverse of A is now:
10 x 10 sparse Matrix of class "dsCMatrix"[[ suppressing 10 column names ‘1’, ‘2’, ‘3’ ... ]]                                       1   5  . . -2 -2  .  .    .    .         .  2   .  3 .  .  . -2  .    .    .         .  3   .  . 1  .  .  .  .    .    .         .  4  -2  . .  3  .  1 -2.0  .    .         .  5  -2  . .  .  3  1  .   -2.0  .         .  6   . -2 .  1  1  4 -2.0 -2.0  .         .  7   .  . . -2  . -2  4.5  0.5 -1.000000  .  8   .  . .  . -2 -2  0.5  4.5 -1.000000  .  9   .  . .  .  .  . -1.0 -1.0  4.909091 -2.90909110  .  . .  .  .  .  .    .   -2.909091  2.909091
you can clearly see the structure and it soon becomes obvious why such a storage is more efficient.

If we want to go back from matrix to triplet form (this might be useful if we want to create a matrix for programs as ASReml) we can use the following code:
## Convert back to triplet form - first the matrixtmp2 <- as(AInv2, "dsTMatrix")## ...                          - now to data.frametmp3 <- data.frame(Row=tmp2@i + 1, Column=tmp2@j + 1, Ainverse=tmp2@x)## Sorttmp3 <- tmp3[order(tmp3$Row, tmp3$Column), ]## Test that we get the same stuffany((tmp[, 3] - tmp3[, 3]) > 0)## ASReml-R specificitiesattr(x=tmp3, which="rowNames") <- rownames(tmp)attr(x=tmp3, which="geneticGroups") <- c(0, 0)