**Mark van der Loo**, and kindly contributed to R-bloggers)

**update** Henrik Bengtsson commented that **which**(x, arr.ind=TRUE) gives the same result, rendering the blog below academic (thanks for the comment!). So, for academic interest, I’ll leave it. In my defense, I implemented this kind of functionality in C some time ago, so I did not even think about RTFM. (So there’s a lesson in this blog after all )

The well-known **which** function accepts a logical vector and returns the indices where its value equals **TRUE**. Actually, **which** also accepts matrices or multidimensional arrays. Internally, R uses a single index to run through such two- or higher-dimensional structures, in a column-first fashion. This is easy for computers, but for us poor humans it is less readable. The following function gives the multi-index of **TRUE** values into any *d _{1} x d_{2} x … x d_{n}* – dimensional array.

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# A which for multidimensional arrays. # Mark van der Loo 16.09.2011 # # A Array of booleans # returns a sum(A) x length(dim(A)) array of multi-indices where A == TRUE # multi.which <- function(A){ if ( is.vector(A) ) return(which(A)) d <- dim(A) T <- which(A) - 1 nd <- length(d) t( sapply(T, function(t){ I <- integer(nd) I[1] <- t %% d[1] sapply(2:nd, function(j){ I[j] <<- (t %/% prod(d[1:(j-1)])) %% d[j] }) I }) + 1 ) } |

For example. Let’s create a 2x3x2 logical array (2 rows, three columns, and this structure times 2):

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> set.seed(1) > set.seed(1) > (B <- array(sample(c(TRUE,FALSE),12,replace=TRUE),dim=c(2,3,2)) ) , , 1 [,1] [,2] [,3] [1,] TRUE FALSE TRUE [2,] TRUE FALSE FALSE , , 2 [,1] [,2] [,3] [1,] FALSE FALSE TRUE [2,] FALSE TRUE TRUE |

The standard **which** function gives 1-dimensional indices:

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> which(B) [1] 1 2 5 10 11 12 |

If you don’t need to see the result, this is fine. However, sometimes it is convenient to have the multi-index available. For example, the element in the first row of the first column of the first matrix of B equals **TRUE**. That is, element (1,1,1). The **multi.which** function returns all multi-indices where coefficients are **TRUE**:

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> multi.which(B) [,1] [,2] [,3] [1,] 1 1 1 [2,] 2 1 1 [3,] 1 3 1 [4,] 2 2 2 [5,] 1 3 2 [6,] 2 3 2 |

The result is a 2-dimensional array, where each row is a single multi-index. You can check the last row by confirming that the second row of the third column of the second matrix indeed has coefficient **TRUE**. As noted, the function works for any multidimensional array (including vectors and matrices).

So, how does it all work? I will just give the basic equation here, but see this paper for a more thorough description and the inverse relation. Basically, you can regard the multi-index as a positional number system, with the first index running fastest. (Remember, that our decimal notation system is a positional system, but with the first number running slowest).

Denote the single index in a *d _{1} x d_{2} x … x d_{n}* – dimensional array with

*t*. The multi-index

*I*may be written as

where

and the product equals 1 if *j=1*. The symbols *div* and *mod* stand for integer division and remainder upon division. This equation assumes *base 0* indexing, meaning that both the single and multi-indexing start at 0. Since R uses * base 1* indexing, the first and 11th line in **multi.which** first subtract, then add one to correct for this.

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**Mark van der Loo**.

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