Question

Given a matrix M and smaller matrices with different possible values, I seek to list all possible matrices resulting from superimposing combinations of these small matrices into the matrix M. The small matrices are to be inserted in the locations of M having the same row/column names.

For instance, say have:

 M <- matrix(rep(0, 49), nrow =7, ncol =7)
 rownames(M) <- colnames(M) <-seq(1,7)
 > M
 1 2 3 4 5 6 7
 1 0 0 0 0 0 0 0
 2 0 0 0 0 0 0 0
 3 0 0 0 0 0 0 0
 4 0 0 0 0 0 0 0
 5 0 0 0 0 0 0 0
 6 0 0 0 0 0 0 0
 7 0 0 0 0 0 0 0

# Generate first set of small matrices:
sub_mat_1_1 <- matrix(rep(1, 9), nrow =3, ncol =3)
rownames(sub_mat_1_1) <- colnames(sub_mat_1_1) <- c(2,3,5)
sub_mat_1_2 <- matrix(rep(2, 9), nrow =3, ncol =3)
rownames(sub_mat_1_2) <- colnames(sub_mat_1_2) <- c(2,3,5)
sub_mat_1_3 <- matrix(rep(3, 9), nrow =3, ncol =3)
rownames(sub_mat_1_3) <- colnames(sub_mat_1_3) <- c(2,3,5)
submatrix_1 <- list(sub_mat_1_1, sub_mat_1_2, sub_mat_1_3)

# Generate second set of small matrices:
submatrix_2 <- list()
sub_mat_2_1 <- matrix(rep(1, 4), nrow =2, ncol =2)
rownames(sub_mat_2_1) <- colnames(sub_mat_2_1) <- c(1,6)
sub_mat_2_2 <- matrix(rep(2, 4), nrow =2, ncol =2)
rownames(sub_mat_2_2) <- colnames(sub_mat_2_2) <- c(1,6)
submatrix_2 <- list(sub_mat_2_1, sub_mat_2_2)

# Generate list of small matrices:
submatrices <- list()
submatrices[[1]] <- submatrix_1
submatrices[[2]] <- submatrix_2

[[1]]
[[1]][[1]]
  2 3 5
2 1 1 1
3 1 1 1
5 1 1 1

[[1]][[2]]
  2 3 5
2 2 2 2
3 2 2 2
5 2 2 2

[[1]][[3]]
  2 3 5
2 3 3 3
3 3 3 3
5 3 3 3


[[2]]
[[2]][[1]]
  1 6
1 1 1
6 1 1

[[2]][[2]]
  1 6
1 2 2
6 2 2

Since there are 3 possibilities for the first small matrix set and 2 for the second, I seek to output, without using a for loop, all 6 possible matrices as a list:

[[1]]
  1 2 3 4 5 6 7
1 1 0 0 0 0 1 0
2 0 1 1 0 1 0 0
3 0 1 1 0 1 0 0
4 0 0 0 0 0 0 0
5 0 1 1 0 1 0 0
6 1 0 0 0 0 1 0
7 0 0 0 0 0 0 0

[[2]]
  1 2 3 4 5 6 7
1 1 0 0 0 0 1 0
2 0 2 2 0 2 0 0
3 0 2 2 0 2 0 0
4 0 0 0 0 0 0 0
5 0 2 2 0 2 0 0
6 1 0 0 0 0 1 0
7 0 0 0 0 0 0 0

[[3]]
  1 2 3 4 5 6 7
1 1 0 0 0 0 1 0
2 0 3 3 0 3 0 0
3 0 3 3 0 3 0 0
4 0 0 0 0 0 0 0
5 0 3 3 0 3 0 0
6 1 0 0 0 0 1 0
7 0 0 0 0 0 0 0

[[4]]
  1 2 3 4 5 6 7
1 2 0 0 0 0 2 0
2 0 1 1 0 1 0 0
3 0 1 1 0 1 0 0
4 0 0 0 0 0 0 0
5 0 1 1 0 1 0 0
6 2 0 0 0 0 2 0
7 0 0 0 0 0 0 0

[[5]]
  1 2 3 4 5 6 7
1 2 0 0 0 0 2 0
2 0 2 2 0 2 0 0
3 0 2 2 0 2 0 0
4 0 0 0 0 0 0 0
5 0 2 2 0 2 0 0
6 2 0 0 0 0 2 0
7 0 0 0 0 0 0 0

[[6]]
  1 2 3 4 5 6 7
1 2 0 0 0 0 2 0
2 0 3 3 0 3 0 0
3 0 3 3 0 3 0 0
4 0 0 0 0 0 0 0
5 0 3 3 0 3 0 0
6 2 0 0 0 0 2 0
7 0 0 0 0 0 0 0

In general, I may have n given “lists of small matrices”, each with its own number of matrices. How would I use an apply type function in this context?

Answer 1

This gives the requested output, but I have to agree with the commenters that the problem is not very well posed. How do you define the positions into which the values of the submatrices are inserted? I simply assumed that you wanted them inserted into a particular subset of rows and columns as shown in your requested output …

A function to insert two matrices into specified rows/columns of M (defined globally, probably bad practice)

tmpmatf <- function(m1,m2,rc1=c(2,3,5),rc2=c(1,6)) {
    pos1 <- as.matrix(expand.grid(rc1,rc1))
    pos2 <- as.matrix(expand.grid(rc2,rc2))
    M[pos1] <- m1
    M[pos2] <- m2
    M
}

Now use expand.grid to create a data frame with all combinations of indices from each list of submatrices, and use alply (array-to-list) from plyr to run tmpmatf on each combination:

library(plyr)
alply(expand.grid(seq(length(submatrices[[1]])),
                  seq(length(submatrices[[2]]))),
      1,
      function(x) {
          tmpmatf(submatrices[[1]][[x[[1]]]],submatrices[[2]][[x[[2]]]])
      })

This should work with an arbitrary number of submatrices in each your two lists of submatrices, but if you really have more (>2) lists of submatrices, then you haven’t given us enough information to specify how (e.g.) the third list of submatrices should be pasted into the larger matrix …

Note that the first part of this (using matrix indexing via a two-column matrix) is a lot faster than using a for loop to insert individual elements into the matrix, but the second part (alply) is not actually much (any?) faster than a pair of nested for loops that iterate over all combinations — the latter might be clearer/easier to debug in this case …