Matrix (version 1.7-6)

indMatrix-class: Index Matrices

Description

The indMatrix class is the class of row and column index matrices, stored as 1-based integer index vectors. A row (column) index matrix is a matrix whose rows (columns) are standard unit vectors. Such matrices are useful when mapping observations to discrete sets of covariate values.

Multiplying a matrix on the left by a row index matrix is equivalent to indexing its rows, i.e., sampling the rows “with replacement”. Analogously, multiplying a matrix on the right by a column index matrix is equivalent to indexing its columns. Indeed, such products are implemented in Matrix as indexing operations; see ‘Details’ below.

A matrix whose rows and columns are standard unit vectors is called a permutation matrix. This special case is designated by the pMatrix class, a direct subclass of indMatrix.

Details

The transpose of an index matrix is an index matrix with identical perm but opposite margin. Hence the transpose of a row index matrix is a column index matrix, and vice versa.

The cross product of a row index matrix R and itself is a diagonal matrix whose diagonal entries are the the number of entries in each column of R.

Given a row index matrix R with perm slot p, a column index matrix C with perm slot q, and a matrix M with conformable dimensions, we have

\(R M\)	=	R %*% M	=	M[p, ]
\(M C\)	=	M %*% C	=	M[, q]
\(C'M\)	=	crossprod(C, M)	=	M[q, ]
\(MR'\)	=	tcrossprod(M, R)	=	M[, p]
\(R'R\)	=	crossprod(R)	=	Diagonal(x=tabulate(p, ncol(R)))
\(CC'\)	=	tcrossprod(C)	=	Diagonal(x=tabulate(q, nrow(C)))

Operations on index matrices that result in index matrices will accordingly return an indMatrix. These include products of two column index matrices and (equivalently) column-indexing of a column index matrix (when dimensions are not dropped). Most other operations on indMatrix treat them as sparse nonzero pattern matrices (i.e., inheriting from virtual class nsparseMatrix). Hence vector-valued subsets of indMatrix, such as those given by diag, are always of type "logical".

See Also

Subclass pMatrix of permutation matrices, a special case of index matrices; virtual class nMatrix of nonzero pattern matrices, and its subclasses.

Examples

p1 <- as(c(2,3,1), "pMatrix")
(sm1 <- as(rep(c(2,3,1), e=3), "indMatrix"))
stopifnot(all(sm1 == p1[rep(1:3, each=3),]))

## row-indexing of a  turns it into an :
class(p1[rep(1:3, each=3),])

set.seed(12) # so we know '10' is in sample
## random index matrix for 30 observations and 10 unique values:
(s10 <- as(sample(10, 30, replace=TRUE),"indMatrix"))

## Sample rows of a numeric matrix :
(mm <- matrix(1:10, nrow=10, ncol=3))
s10 %*% mm

set.seed(27)
IM1 <- as(sample(1:20, 100, replace=TRUE), "indMatrix")
IM2 <- as(sample(1:18, 100, replace=TRUE), "indMatrix")
(c12 <- crossprod(IM1,IM2))
## same as cross-tabulation of the two index vectors:
stopifnot(all(c12 - unclass(table(IM1@perm, IM2@perm)) == 0))

# 3 observations, 4 implied values, first does not occur in sample:
as(2:4, "indMatrix")
# 3 observations, 5 values, first and last do not occur in sample:
as(list(2:4, 5), "indMatrix")

as(sm1, "nMatrix")
s10[1:7, 1:4] # gives an "ngTMatrix" (most economic!)
s10[1:4, ]  # preserves "indMatrix"-class

I1 <- as(c(5:1,6:4,7:3), "indMatrix")
I2 <- as(7:1, "pMatrix")
(I12 <- rbind(I1, I2))
stopifnot(is(I12, "indMatrix"),
          identical(I12, rbind(I1, I2)),
	  colSums(I12) == c(2L,2:4,4:2))