(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 pedigree

ped <- 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 form

tmp <- asreml.Ainverse(pedigree=ped)$ginv

## Create a "proper" matrix

AInv <- asreml.sparse2mat(x=tmp)

## Print AInv

AInv

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 package

library(package="Matrix")

## Number of pedigree members

nI <- nrow(ped)

## Store inverse of A in sparse form

AInv2 <- 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 - optional

dimnames(AInv2) <- list(attr(x=tmp, which="rowNames"),

attr(x=tmp, which="rowNames"))

## Print AInv

AInv2

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 -2.909091

10 . . . . . . . . -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 matrix

tmp2 <- as(AInv2, "dsTMatrix")

## ... - now to data.frame

tmp3 <- data.frame(Row=tmp2@i + 1, Column=tmp2@j + 1, Ainverse=tmp2@x)

## Sort

tmp3 <- tmp3[order(tmp3$Row, tmp3$Column), ]

## Test that we get the same stuff

any((tmp[, 3] - tmp3[, 3]) > 0)

## ASReml-R specificities

attr(x=tmp3, which="rowNames") <- rownames(tmp)

attr(x=tmp3, which="geneticGroups") <- c(0, 0)

To

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