How to write a list of lower triangular matrices where the sum of all matches values are equal to one

Question

The main idea is how to generate matrices where the sum of each matches values is 1. That is, I do not have matrices but would like to generate them based on this condition.

I aim to generate a list of lower.tri matrices. The main trick here is how I can generate all these matrices where the sum of every match elements is 1.

For example,

w1 <- c(0,0.7,0.8,0.5,0.2,
        0,0,0.7,0.6,0.3,
        0,0,0,0.9,0.8,
        0,0,0,0,0.3,
        0,0,0,0,0)
w1 <- matrix(w1,5,5)

 w2 <- c(0,0.3,0.2,0.5,0.8,
         0,0,0.3,0.4,0.7,
         0,0,0,0.1,0.2,
         0,0,0,0,0.7,
         0,0,0,0,0)
 w2 <- matrix(w2,5,5)

The main idea is:

w5 <- 1 - (w1+w2+w3+w4). w1 <- 1-w2 , w3 <- 1- (w1+w2). w4 <- 1- (w1+w2+w3).

Here we can see that the sum of each matches values is 1.

For example:

> w1
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.0  0.0  0.0  0.0    0
[2,]  0.7  0.0  0.0  0.0    0
[3,]  0.8  0.7  0.0  0.0    0
[4,]  0.5  0.6  0.9  0.0    0
[5,]  0.2  0.3  0.8  0.3    0
> w2
     [,1] [,2] [,3] [,4] [,5]
[1,]  0.0  0.0  0.0  0.0    0
[2,]  0.3  0.0  0.0  0.0    0
[3,]  0.2  0.3  0.0  0.0    0
[4,]  0.5  0.4  0.1  0.0    0
[5,]  0.8  0.7  0.2  0.7    0

w1[2,1] = 0.7
w2[2,1]= 0.3

Then their sum = 1. The same for all w1[k,j] + w2[k,j].

The question:

How can I generate 5 matrices where the sum of each matches values is 1. I was thinking about lapply however, I really cannot get the idea for this part.

For example, The output should a list of 5 matrices. For example,

> w1
         [,1] [,2] [,3] [,4] [,5]
    [1,]  0.0  0.0  0.0  0.0    0
    [2,]  0.5  0.0  0.0  0.0    0
    [3,]  0.6  0.6  0.0  0.0    0
    [4,]  0.5  0.6  0.7  0.0    0
    [5,]  0.2  0.2  0.2  0.3    0

    > w2
         [,1] [,2] [,3] [,4] [,5]
    [1,]  0.0  0.0  0.0  0.0    0
    [2,]  0.2  0.0  0.0  0.0    0
    [3,]  0.2  0.3  0.0  0.0    0
    [4,]  0.5  0.2  0.1  0.0    0
    [5,]  0.7  0.5  0.2  0.5    0

> w3
         [,1] [,2] [,3] [,4] [,5]
    [1,]  0.0  0.0  0.0  0.0    0
    [2,]  0.1  0.0  0.0  0.0    0
    [3,]  0.2  0.1  0.0  0.0    0
    [4,]  0.3  0.2  0.3  0.0    0
    [5,]  0.1  0.3  0.6  0.2    0

I think this could be a duplicate of https://stackoverflow.com/questions/24845909/generate-n-random-integers-that-sum-to-m-in-r — akrun, Apr 03 '18 at 06:14
I.e. `replicate(5, replicate(10, rand_vect_cont(5, 1)), simplify = FALSE)` and then change it to matrix — akrun, Apr 03 '18 at 06:19

Stéphane Laurent · Accepted Answer · 2018-04-03T09:49:34.487

Assuming you want positive numbers, you can use the Dirichlet distribution to generate a vector which sums to 1. For example, the rdirichlet function in the MCMCpack library. Then you can do:

library(MCMCpack)

nmatrices <- 3 # number of matrices
matrix_size <- 5
matrices <- replicate(nmatrices, matrix(0, ncol=matrix_size, nrow=matrix_size), 
                      simplify=FALSE)
for(i in seq_along(matrices)){
  matrices[[i]][lower.tri(matrices[[i]])] <- rdirichlet(1, rep(1,10))
}

Result:

> matrices
[[1]]
           [,1]       [,2]       [,3]       [,4] [,5]
[1,] 0.00000000 0.00000000 0.00000000 0.00000000    0
[2,] 0.15475899 0.00000000 0.00000000 0.00000000    0
[3,] 0.01094188 0.10190545 0.00000000 0.00000000    0
[4,] 0.18999179 0.08817135 0.06223413 0.00000000    0
[5,] 0.05055661 0.02846215 0.26253043 0.05044721    0

[[2]]
          [,1]       [,2]      [,3]       [,4] [,5]
[1,] 0.0000000 0.00000000 0.0000000 0.00000000    0
[2,] 0.1592983 0.00000000 0.0000000 0.00000000    0
[3,] 0.1277237 0.03365446 0.0000000 0.00000000    0
[4,] 0.0380273 0.07080578 0.0728188 0.00000000    0
[5,] 0.1745902 0.01504430 0.2400594 0.06797775    0

[[3]]
           [,1]       [,2]         [,3]       [,4] [,5]
[1,] 0.00000000 0.00000000 0.0000000000 0.00000000    0
[2,] 0.13537093 0.00000000 0.0000000000 0.00000000    0
[3,] 0.14060566 0.03186886 0.0000000000 0.00000000    0
[4,] 0.27152069 0.23647956 0.0008148619 0.00000000    0
[5,] 0.00176678 0.04141304 0.1052544582 0.03490516    0

EDIT

I didn't understand the question. You can do:

nmatrices <- 3 # number of matrices
matrix_size <- 4
matrices <- replicate(nmatrices, matrix(0, ncol=matrix_size, nrow=matrix_size), 
                      simplify=FALSE)
samples <- rdirichlet(matrix_size*(matrix_size-1)/2, rep(1,nmatrices))
for(m in seq_along(matrices)){
  for(k in seq_len(matrix_size*(matrix_size-1)/2)){
    matrices[[m]][lower.tri(matrices[[m]])][k] <- samples[k, m]
  }
}

Result:

> matrices
[[1]]
          [,1]      [,2]      [,3] [,4]
[1,] 0.0000000 0.0000000 0.0000000    0
[2,] 0.8444768 0.0000000 0.0000000    0
[3,] 0.0509210 0.2392519 0.0000000    0
[4,] 0.5588843 0.1921104 0.2482988    0

[[2]]
           [,1]      [,2]      [,3] [,4]
[1,] 0.00000000 0.0000000 0.0000000    0
[2,] 0.03179142 0.0000000 0.0000000    0
[3,] 0.49615251 0.2485702 0.0000000    0
[4,] 0.03481287 0.1113190 0.7224681    0

[[3]]
          [,1]      [,2]       [,3] [,4]
[1,] 0.0000000 0.0000000 0.00000000    0
[2,] 0.1237318 0.0000000 0.00000000    0
[3,] 0.4529265 0.5121779 0.00000000    0
[4,] 0.4063028 0.6965705 0.02923302    0

@F.AMER Oh sorry, you don't want that *all* entries of *each* matrix sum to 1 ? Then I didn't understand the question. — Stéphane Laurent, Apr 03 '18 at 09:31

How to write a list of lower triangular matrices where the sum of all matches values are equal to one

1 Answers1

EDIT