```
#' Calculate continuity based on distance matrices
#'
#' The continuity was proposed by Venna and Kaski, as a local quality measure of
#' a low-dimensional representation. The metric focuses on the preservation of
#' local neighborhoods, and compares the neighborhoods of points in the
#' low-dimensional representation to those in the reference data. Hence, the
#' continuity measure indicates to which degree we can trust that the points
#' closest to a given sample in the reference data set are placed close to the
#' sample also in the low-dimensional representation. The \code{kTM} parameter
#' defines the size of the neighborhoods to consider.
#'
#' @references
#' Venna J., Kaski S. (2001). Neighborhood preservation in nonlinear
#' projection methods: An experimental study. In Dorffner G., Bischof H.,
#' Hornik K., editors, Proceedings of ICANN 2001, pp 485–491. Springer,
#' Berlin.
#'
#' @keywords internal
#'
#' @author Charlotte Soneson
#'
#' @return The continuity value.
#'
#' @param distReference N x N matrix or dist object, representing pairwise
#' sample distances based on the reference (high-dimensional) observed values.
#' For each column, samples (rows) will be ranked by the provided distances.
#' @param rankLowDim N x N matrix or dist object, representing pairwise sample
#' distances based on the low-dimensional representation.
#' For each column, samples (rows) will be ranked by the provided distances.
#' @param kTM The number of nearest neighbors (excluding the sample itself).
#'
calcContinuityFromDist <- function(distReference, distLowDim, kTM) {
distReference <- as.matrix(distReference)
distLowDim <- as.matrix(distLowDim)
## Check that matrices have the same dimensions, and that the
## observations are ordered in the same way
stopifnot(ncol(distReference) == ncol(distLowDim))
stopifnot(nrow(distReference) == nrow(distLowDim))
stopifnot(rownames(distReference) == colnames(distReference))
stopifnot(rownames(distReference) == rownames(distLowDim))
stopifnot(rownames(distLowDim) == colnames(distLowDim))
diag(distReference) <- NA
diag(distLowDim) <- NA
rankReference <- apply(distReference, 2, function(w) order(order(w)))
rankLowDim <- apply(distLowDim, 2, function(w) order(order(w)))
diag(rankReference) <- 0
diag(rankLowDim) <- 0
rownames(rankReference) <- rownames(distReference)
rownames(rankLowDim) <- rownames(distLowDim)
calcContinuityFromRank(rankReference = rankReference,
rankLowDim = rankLowDim, kTM = kTM)
}
#' Calculate continuity based on sample rankings
#'
#' The continuity was proposed by Venna and Kaski, as a local quality measure of
#' a low-dimensional representation. The metric focuses on the preservation of
#' local neighborhoods, and compares the neighborhoods of points in the
#' low-dimensional representation to those in the reference data. Hence, the
#' continuity measure indicates to which degree we can trust that the points
#' closest to a given sample in the reference data set are placed close to the
#' sample also in the low-dimensional representation. The \code{kTM} parameter
#' defines the size of the neighborhoods to consider.
#'
#' @references
#' Venna J., Kaski S. (2001). Neighborhood preservation in nonlinear
#' projection methods: An experimental study. In Dorffner G., Bischof H.,
#' Hornik K., editors, Proceedings of ICANN 2001, pp 485–491. Springer,
#' Berlin.
#'
#' @keywords internal
#'
#' @author Charlotte Soneson
#'
#' @return The continuity value.
#'
#' @param rankReference N x N matrix, each row/column corresponding to one
#' sample. The value of entry (i, j) represents the position of sample i in
#' the ranking of all samples with respect to their distance from sample j,
#' based on the reference (high-dimensional) observed values. The sample
#' itself has rank 0.
#' @param rankLowDim N x N matrix, each row/column corresponding to one sample.
#' The value of entry (i, j) represents the position of sample i in the
#' ranking of all samples with respect to their distance from sample j, based
#' on the low-dimensional representation. The sample itself has rank 0.
#' @param kTM The number of nearest neighbors.
#'
calcContinuityFromRank <- function(rankReference, rankLowDim, kTM) {
stopifnot(ncol(rankReference) == ncol(rankLowDim))
stopifnot(nrow(rankReference) == nrow(rankLowDim))
stopifnot(rownames(rankReference) == colnames(rankReference))
stopifnot(rownames(rankReference) == rownames(rankLowDim))
stopifnot(rownames(rankLowDim) == colnames(rankLowDim))
N <- ncol(rankReference)
if (kTM < N/2) {
normConst <- N * kTM * (2 * N - 3 * kTM - 1)
} else {
normConst <- N * (N - kTM) * (N - kTM - 1)
}
1 - 2/(normConst) *
sum(vapply(seq_len(ncol(rankReference)), function(i) {
sum((rankLowDim[, i] - kTM) * (rankReference[, i] <= kTM) *
(rankLowDim[, i] > kTM))
}, NA_real_))
}
```

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