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#' Transformation from SRSF Space
#'
#' This function transforms SRVFs back to the original functional space.
#'
#' @param q Either a numeric vector of a numeric matrix or a numeric array
#' specifying the SRSFs that need to be transformed.
#'
#' - If a vector, it must be of shape \eqn{M} and it is interpreted as a
#' single \eqn{1}-dimensional curve observed on a grid of size \eqn{M}.
#' - If a matrix and `multidimensional == FALSE`, it must be of shape
#' \eqn{M \times N}. In this case, it is interpreted as a sample of \eqn{N}
#' curves observed on a grid of size \eqn{M}, unless \eqn{M = 1} in which case
#' it is interpreted as a single \eqn{1}-dimensional curve observed on a grid
#' of size \eqn{M}.
#' - If a matrix and `multidimensional == TRUE`,it must be of shape
#' \eqn{L \times M} and it is interpreted as a single \eqn{L}-dimensional
#' curve observed on a grid of size \eqn{M}.
#' - If a 3D array, it must be of shape \eqn{L \times M \times N} and it is
#' interpreted as a sample of \eqn{N} \eqn{L}-dimensional curves observed on a
#' grid of size \eqn{M}.
#' @param time A numeric vector of length \eqn{M} specifying the grid on which
#' SRSFs are evaluated.
#' @param f0 Either a numeric value or a numeric vector of or a numeric matrix
#' specifying the initial value of the curves in the original functional
#' space. It must be:
#'
#' - a value if `q` represents a single \eqn{1}-dimensional SRSF.
#' - a vector of length \eqn{L} if `q` represents a single
#' \eqn{L}-dimensional SRSF.
#' - a vector of length \eqn{N} if `q` represents a sample of \eqn{N}
#' \eqn{1}-dimensional SRSFs.
#' - a matrix of shape \eqn{L \times M} if `q` represents a sample of \eqn{N}
#' \eqn{L}-dimensional SRSFs.
#' @param multidimensional A boolean specifying if the curves are
#' multi-dimensional. This is useful when `q` is provided as a matrix to
#' determine whether it is a single multi-dimensional curve or a collection of
#' uni-dimensional curves. Defaults to `FALSE`.
#'
#' @return A numeric array of the same shape as the input `q` storing the
#' transformation of the SRSFs `q` back to the original functional space.
#'
#' @keywords srsf alignment
#' @references Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S.,
#' May 2011. Registration of functional data using fisher-rao metric,
#' arXiv:1103.3817v2.
#' @references Tucker, J. D., Wu, W., Srivastava, A., Generative models for
#' functional data using amplitude and phase separation, Computational
#' Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.
#'
#' @export
#' @examples
#' q <- f_to_srvf(simu_data$f, simu_data$time)
#' f <- srvf_to_f(q, simu_data$time, simu_data$f[1, ])
srvf_to_f <- function(q, time, f0 = 0.0, multidimensional = FALSE) {
dims <- dim(q)
dims0 <- dim(f0)
if (is.null(dims)) { # One uni-dimensional curve
L <- 1
stopifnot(length(f0) == L)
M <- length(q)
N <- 1
integrand <- q * abs(q)
f <- f0 + cumtrapz(time, integrand)
} else {
if (length(dims) == 2) {
if (multidimensional || dims[1] == 1) { # One multi-dimensional curve
L <- dims[1]
stopifnot(is.null(dims0) && length(f0) == L)
M <- dims[2]
N <- 1
norm_q <- sqrt(colSums(q^2))
f <- lapply(1:L, function(l) {
integrand <- q[l, ] * norm_q
f0[l] + cumtrapz(time, integrand)
})
f <- do.call(rbind, f)
} else { # N uni-dimensional curves
L <- 1
M <- dims[1]
N <- dims[2]
stopifnot(is.null(dims0) && length(f0) == N)
f <- lapply(1:N, function(n) {
integrand <- q[, n] * abs(q[, n])
f0[n] + cumtrapz(time, integrand)
})
f <- do.call(cbind, f)
}
} else { # f is 3D array so N multi-dimensional curves and f0 must be (LxN) matrix
stopifnot(!is.null(dims0) && length(dims0) == 2)
L <- dims[1]
stopifnot(dims0[1] == L)
M <- dims[2]
N <- dims[3]
stopifnot(dims0[2] == N)
norm_q <- lapply(1:N, function(n) {
sqrt(colSums(q[, , n]^2))
})
f <- lapply(1:N, function(n) {
res <- lapply(1:L, function(l) {
integrand <- q[l, , n] * norm_q[[n]][l, ]
f0[l, n] + cumtrapz(time, integrand)
})
do.call(rbind, res)
})
f <- do.call(cbind, f)
}
}
f
}
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