79 lines
2.7 KiB
R
79 lines
2.7 KiB
R
#' Kronecker decomposed Variance Matrix estimation.
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#'
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#' @description Computes the kronecker decomposition factors of the variance
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#' matrix
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#' \deqn{\var{X} = tr(L)tr(R) (L\otimes R).}
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#'
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#' @param X numeric matrix or 3d array.
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#' @param shape in case of \code{X} being a matrix, this specifies the predictor
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#' shape, otherwise ignored.
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#' @param center boolean specifying if \code{X} is centered before computing the
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#' left/right second moments. This is usefull in the case of allready centered
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#' data.
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#'
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#' @returns List containing
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#' \describe{
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#' \item{lhs}{Left Hand Side \eqn{L} of the kronecker decomposed variance matrix}
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#' \item{rhs}{Right Hand Side \eqn{R} of the kronecker decomposed variance matrix}
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#' \item{trace}{Scaling factor \eqn{tr(L)tr(R)} for the variance matrix}
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#' }
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#'
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#' @examples
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#' n <- 503L # nr. of observations
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#' p <- 32L # first predictor dimension
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#' q <- 27L # second predictor dimension
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#' lhs <- 0.5^abs(outer(seq_len(q), seq_len(q), `-`)) # Left Var components
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#' rhs <- 0.5^abs(outer(seq_len(p), seq_len(p), `-`)) # Right Var components
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#' X <- rmvnorm(n, sigma = kronecker(lhs, rhs)) # Multivariate normal data
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#'
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#' # Estimate kronecker decomposed variance matrix
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#' dim(X) # c(n, p * q)
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#' fit <- var.kronecker(X, shape = c(p, q))
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#'
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#' # equivalent
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#' dim(X) <- c(n, p, q)
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#' fit <- var.kronecker(X)
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#'
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#' # Compute complete estimated variance matrix
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#' varX.hat <- fit$trace^-1 * kronecker(fit$lhs, fit$rhs)
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#'
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#' # or its inverse
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#' varXinv.hat <- fit$trace * kronecker(solve(fit$lhs), solve(fit$rhs))
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#'
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var.kronecker <- function(X, shape = dim(X)[-1], center = TRUE) {
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# Get and check predictor dimensions
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n <- nrow(X)
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if (length(dim(X)) == 2L) {
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stopifnot(ncol(X) == prod(shape[1:2]))
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p <- as.integer(shape[1]) # Predictor "height"
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q <- as.integer(shape[2]) # Predictor "width"
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dim(X) <- c(n, p, q)
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} else if (length(dim(X)) == 3L) {
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p <- dim(X)[2]
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q <- dim(X)[3]
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} else {
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stop("'X' must be a matrix or 3-tensor")
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}
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if (isTRUE(center)) {
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# Center X; X[, i, j] <- X[, i, j] - mean(X[, i, j])
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X <- scale(X, scale = FALSE)
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print(range(attr(X, "scaled:center")))
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dim(X) <- c(n, p, q)
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}
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# Calc left/right side of kronecker structures covariance
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# var(X) = var.lhs %x% var.rhs
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var.lhs <- .rowMeans(apply(X, 1, crossprod), q * q, n)
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dim(var.lhs) <- c(q, q)
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var.rhs <- .rowMeans(apply(X, 1, tcrossprod), p * p, n)
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dim(var.rhs) <- c(p, p)
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# Estimate scalling factor tr(var(X)) = tr(var.lhs) tr(var.rhs)
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trace <- sum(X^2) / n
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list(lhs = var.lhs, rhs = var.rhs, trace = trace)
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}
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