iam.RdMaps the elements of an array containing symmetric positive-definite matrices to a matrix with sufficient columns to hold them (called matrix-band format.)
iam(j, k, M, both = FALSE, diag = TRUE)Usually an integer from the set {1:M} giving the row
number of an element.
However, the argument can also be a vector of length M,
for selecting an entire row or column, e.g.,
iam(1:M, 1, M) or iam(1, 1:M, M).
An integer from the set {1:M} giving the column number
of an element.
The number of linear/additive predictors. This is the dimension of each positive-definite symmetric matrix.
Logical. Return both the row and column indices? See below for more details.
Logical. Return the indices for the diagonal elements?
If FALSE then only the strictly upper triangular part
of the matrix elements are used.
Suppose we have \(n\) symmetric positive-definite square
matrices,
each \(M\) by \(M\), and
these are stored in an array of dimension c(n,M,M).
Then these can be more compactly represented by a matrix
of dimension c(n,K) where K is an integer between
M and M*(M+1)/2 inclusive. The mapping between
these two representations is given by this function.
It firstly enumerates by the diagonal elements, followed by
the band immediately above the diagonal, then the band above
that one, etc.
The last element is (1,M).
This function performs the mapping from elements (j,k)
of symmetric positive-definite square matrices to the columns
of another matrix representing such. This is called the
matrix-band format and is used by the VGAM package.
This function has a dual purpose depending on the value of
both. If both = FALSE then the column number
corresponding to the j-k element of the matrix is
returned. If both = TRUE then j and k are
ignored and a list with the following components are returned.
The row indices of the upper triangular part of the
matrix (This may or may not include the diagonal elements,
depending on the argument diagonal).
The column indices of the upper triangular part of the
matrix (This may or may not include the diagonal elements,
depending on the argument diagonal).
This function is used in the weight slot of many
VGAM family functions
(see vglmff-class),
especially those whose \(M\) is determined by the data,
e.g., dirichlet, multinomial.
iam(1, 2, M = 3) # The 4th coln represents elt (1,2) of a 3x3 matrix
#> [1] 4
iam(NULL, NULL, M = 3, both = TRUE) # Return the row & column indices
#> $row.index
#> [1] 1 2 3 1 2 1
#>
#> $col.index
#> [1] 1 2 3 2 3 3
#>
dirichlet()@weight
#> expression({
#> index <- iam(NA, NA, M, both = TRUE, diag = TRUE)
#> wz <- matrix(-trigamma(sumshape), nrow = n, ncol = dimm(M))
#> wz[, 1:M] <- trigamma(shape) + wz[, 1:M]
#> wz <- c(w) * wz * dsh.deta[, index$row] * dsh.deta[, index$col]
#> wz
#> })
M <- 4
temp1 <- iam(NA, NA, M = M, both = TRUE)
mat1 <- matrix(NA, M, M)
mat1[cbind(temp1$row, temp1$col)] = 1:length(temp1$row)
mat1 # More commonly used
#> [,1] [,2] [,3] [,4]
#> [1,] 1 5 8 10
#> [2,] NA 2 6 9
#> [3,] NA NA 3 7
#> [4,] NA NA NA 4
temp2 <- iam(NA, NA, M = M, both = TRUE, diag = FALSE)
mat2 <- matrix(NA, M, M)
mat2[cbind(temp2$row, temp2$col)] = 1:length(temp2$row)
mat2 # Rarely used
#> [,1] [,2] [,3] [,4]
#> [1,] NA 1 4 6
#> [2,] NA NA 2 5
#> [3,] NA NA NA 3
#> [4,] NA NA NA NA