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#'@author Verónica Torralba, \email{veronica.torralba@bsc.es}
#'@author Bert Van Schaeybroeck, \email{bertvs@meteo.be}
#'@description Equivalent to function \code{Calibration} but for objects of
#'class \code{s2dv_cube}.
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#'@param exp An object of class \code{s2dv_cube} as returned by \code{CST_Load}
#' function, containing the seasonal hindcast experiment data in the element
#' named \code{$data}. The hindcast is used to calibrate the forecast in case
#' the forecast is provided; if not, the same hindcast will be calibrated
#' instead.
#'@param obs An object of class \code{s2dv_cube} as returned by \code{CST_Load}
#' function, containing the observed data in the element named \code{$data}.
#'@param exp_cor An optional object of class \code{s2dv_cube} as returned by
#' \code{CST_Load} function, containing the seasonal forecast experiment data
#' in the element named \code{$data}. If the forecast is provided, it will be
#' calibrated using the hindcast and observations; if not, the hindcast will be
#' calibrated instead.
#'@param cal.method Is the calibration method used, can be either \code{bias},
#' \code{evmos}, \code{mse_min}, \code{crps_min} or \code{rpc-based}. Default
#' value is \code{mse_min}.
#'@param eval.method Is the sampling method used, can be either \code{in-sample}
#' or \code{leave-one-out}. Default value is the \code{leave-one-out} cross
#' validation. In case the forecast is provided, any chosen eval.method is
#' over-ruled and a third option is used.
#'@param multi.model Is a boolean that is used only for the \code{mse_min}
#' method. If multi-model ensembles or ensembles of different sizes are used,
#' it must be set to \code{TRUE}. By default it is \code{FALSE}. Differences
#' between the two approaches are generally small but may become large when
#' using small ensemble sizes. Using multi.model when the calibration method is
#' \code{bias}, \code{evmos} or \code{crps_min} will not affect the result.
#'@param na.fill Is a boolean that indicates what happens in case calibration is
#' not possible or will yield unreliable results. This happens when three or
#' less forecasts-observation pairs are available to perform the training phase
#' of the calibration. By default \code{na.fill} is set to true such that NA
#' values will be returned. If \code{na.fill} is set to false, the uncorrected
#' data will be returned.
#'@param na.rm Is a boolean that indicates whether to remove the NA values or
#' not. The default value is \code{TRUE}. See Details section for further
#' information about its use and compatibility with \code{na.fill}.
#'@param apply_to Is a character string that indicates whether to apply the
#' calibration to all the forecast (\code{"all"}) or only to those where the
#' correlation between the ensemble mean and the observations is statistically
#' significant (\code{"sign"}). Only useful if \code{cal.method == "rpc-based"}.
#'@param alpha Is a numeric value indicating the significance level for the
#' correlation test. Only useful if \code{cal.method == "rpc-based" &
#' apply_to == "sign"}.
#'@param memb_dim Is a character string indicating the name of the member
#' dimension. By default, it is set to 'member'.
#'@param sdate_dim Is a character string indicating the name of the start date
#' dimension. By default, it is set to 'sdate'.
#'@param ncores Is an integer that indicates the number of cores for parallel
#' computations using multiApply function. The default value is one.
#'@return An object of class \code{s2dv_cube} containing the calibrated
#'forecasts in the element \code{$data} with the same dimensions as the one in
#'the exp object.
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#'@examples
#'# Example 1:
#'mod1 <- 1 : (1 * 3 * 4 * 5 * 6 * 7)
#'dim(mod1) <- c(dataset = 1, member = 3, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'obs1 <- 1 : (1 * 1 * 4 * 5 * 6 * 7)
#'dim(obs1) <- c(dataset = 1, member = 1, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'lon <- seq(0, 30, 5)
#'lat <- seq(0, 25, 5)
#'exp <- list(data = mod1, lat = lat, lon = lon)
#'obs <- list(data = obs1, lat = lat, lon = lon)
#'attr(exp, 'class') <- 's2dv_cube'
#'attr(obs, 'class') <- 's2dv_cube'
#'a <- CST_Calibration(exp = exp, obs = obs, cal.method = "mse_min",
#' eval.method = "in-sample")
#'
#'# Example 2:
#'mod1 <- 1 : (1 * 3 * 4 * 5 * 6 * 7)
#'mod2 <- 1 : (1 * 3 * 1 * 5 * 6 * 7)
#'dim(mod1) <- c(dataset = 1, member = 3, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'dim(mod2) <- c(dataset = 1, member = 3, sdate = 1, ftime = 5, lat = 6, lon = 7)
#'obs1 <- 1 : (1 * 1 * 4 * 5 * 6 * 7)
#'dim(obs1) <- c(dataset = 1, member = 1, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'lon <- seq(0, 30, 5)
#'lat <- seq(0, 25, 5)
#'exp <- list(data = mod1, lat = lat, lon = lon)
#'obs <- list(data = obs1, lat = lat, lon = lon)
#'exp_cor <- list(data = mod2, lat = lat, lon = lon)
#'attr(exp, 'class') <- 's2dv_cube'
#'attr(obs, 'class') <- 's2dv_cube'
#'attr(exp_cor, 'class') <- 's2dv_cube'
#'a <- CST_Calibration(exp = exp, obs = obs, exp_cor = exp_cor,
#' cal.method = "evmos")
CST_Calibration <- function(exp, obs, exp_cor = NULL, cal.method = "mse_min",
eval.method = "leave-one-out", multi.model = FALSE,
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na.fill = TRUE, na.rm = TRUE, apply_to = NULL, alpha = NULL,
memb_dim = 'member', sdate_dim = 'sdate', ncores = 1) {
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if(!missing(multi.model) & !(cal.method == "mse_min")){
warning(paste0("The multi.model parameter is ignored when using the calibration method ", cal.method))
}
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if(is.null(exp_cor)){ #exp will be used to calibrate and will also be calibrated: "calibrate hindcast"
if (!inherits(exp, "s2dv_cube") || !inherits(obs, "s2dv_cube")) {
stop("Parameter 'exp' and 'obs' must be of the class 's2dv_cube', ",
"as output by CSTools::CST_Load.")
}
exp$data <- Calibration(exp = exp$data, obs = obs$data, exp_cor = NULL,
cal.method = cal.method,
eval.method = eval.method,
multi.model = multi.model,
na.fill = na.fill, na.rm = na.rm,
apply_to = apply_to, alpha = alpha,
memb_dim = memb_dim, sdate_dim = sdate_dim,
ncores = ncores)
exp$Datasets <- c(exp$Datasets, obs$Datasets)
exp$source_files <- c(exp$source_files, obs$source_files)
return(exp)
}else{ #if exp_cor is provided, it will be calibrated: "calibrate forecast instead of hindcast"
eval.method = "hindcast-vs-forecast" #if exp_cor is provided, eval.method is overrruled (because if exp_cor is provided, the train data will be all data of "exp" and the evalutaion data will be all data of "exp_cor"; no need for "leave-one-out" or "in-sample")
if (!inherits(exp, "s2dv_cube") || !inherits(obs, "s2dv_cube") || !inherits(exp_cor, "s2dv_cube")) {
stop("Parameter 'exp', 'obs' and 'exp_cor' must be of the class 's2dv_cube', ",
"as output by CSTools::CST_Load.")
}
exp_cor$data <- Calibration(exp = exp$data, obs = obs$data, exp_cor = exp_cor$data,
cal.method = cal.method,
eval.method = eval.method,
multi.model = multi.model,
na.fill = na.fill, na.rm = na.rm,
apply_to = apply_to, alpha = alpha,
memb_dim = memb_dim, sdate_dim = sdate_dim,
ncores = ncores)
exp_cor$Datasets <- c(exp_cor$Datasets, obs$Datasets)
exp_cor$source_files <- c(exp_cor$source_files, exp$source_files, obs$source_files)
return(exp_cor)
}
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#'Forecast Calibration
#'
#'@author Verónica Torralba, \email{veronica.torralba@bsc.es}
#'@author Bert Van Schaeybroeck, \email{bertvs@meteo.be}
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#'@description Five types of member-by-member bias correction can be performed.
#'The \code{"bias"} method corrects the bias only, the \code{"evmos"} method
#'applies a variance inflation technique to ensure the correction of the bias
#'and the correspondence of variance between forecast and observation (Van
#'Schaeybroeck and Vannitsem, 2011). The ensemble calibration methods
#'\code{"mse_min"} and \code{"crps_min"} correct the bias, the overall forecast
#'variance and the ensemble spread as described in Doblas-Reyes et al. (2005)
#'and Van Schaeybroeck and Vannitsem (2015), respectively. While the
#'\code{"mse_min"} method minimizes a constrained mean-squared error using three
#'parameters, the \code{"crps_min"} method features four parameters and
#'minimizes the Continuous Ranked Probability Score (CRPS). The
#'\code{"rpc-based"} method adjusts the forecast variance ensuring that the
#'ratio of predictable components (RPC) is equal to one, as in Eade et al. (2014).
#'@description Both in-sample or our out-of-sample (leave-one-out cross
#'validation) calibration are possible.
#'@references Doblas-Reyes F.J, Hagedorn R, Palmer T.N. The rationale behind the
#'success of multi-model ensembles in seasonal forecasting-II calibration and
#'combination. Tellus A. 2005;57:234-252. doi:10.1111/j.1600-0870.2005.00104.x
#'@references Eade, R., Smith, D., Scaife, A., Wallace, E., Dunstone, N.,
#'Hermanson, L., & Robinson, N. (2014). Do seasonal-to-decadal climate
#'predictions underestimate the predictability of the read world? Geophysical
#'Research Letters, 41(15), 5620-5628. doi: 10.1002/2014GL061146
#'@references Van Schaeybroeck, B., & Vannitsem, S. (2011). Post-processing
#'through linear regression. Nonlinear Processes in Geophysics, 18(2), 147.
#'doi:10.5194/npg-18-147-2011
#'@references Van Schaeybroeck, B., & Vannitsem, S. (2015). Ensemble
#'post-processing using member-by-member approaches: theoretical aspects.
#'Quarterly Journal of the Royal Meteorological Society, 141(688), 807-818.
#'doi:10.1002/qj.2397
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#'
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#'@param exp A multidimensional array with named dimensions (at least 'sdate'
#' and 'member') containing the seasonal hindcast experiment data. The hindcast
#' is used to calibrate the forecast in case the forecast is provided; if not,
#' the same hindcast will be calibrated instead.
#'@param obs A multidimensional array with named dimensions (at least 'sdate')
#' containing the observed data.
#'@param exp_cor An optional multidimensional array with named dimensions (at
#' least 'sdate' and 'member') containing the seasonal forecast experiment data.
#' If the forecast is provided, it will be calibrated using the hindcast and
#' observations; if not, the hindcast will be calibrated instead.
#'@param cal.method Is the calibration method used, can be either \code{bias},
#' \code{evmos}, \code{mse_min}, \code{crps_min} or \code{rpc-based}. Default
#' value is \code{mse_min}.
#'@param eval.method Is the sampling method used, can be either \code{in-sample}
#' or \code{leave-one-out}. Default value is the \code{leave-one-out} cross
#' validation. In case the forecast is provided, any chosen eval.method is
#' over-ruled and a third option is used.
#'@param multi.model Is a boolean that is used only for the \code{mse_min}
#' method. If multi-model ensembles or ensembles of different sizes are used,
#' it must be set to \code{TRUE}. By default it is \code{FALSE}. Differences
#' between the two approaches are generally small but may become large when
#' using small ensemble sizes. Using multi.model when the calibration method is
#' \code{bias}, \code{evmos} or \code{crps_min} will not affect the result.
#'@param na.fill Is a boolean that indicates what happens in case calibration is
#' not possible or will yield unreliable results. This happens when three or
#' less forecasts-observation pairs are available to perform the training phase
#' of the calibration. By default \code{na.fill} is set to true such that NA
#' values will be returned. If \code{na.fill} is set to false, the uncorrected
#' data will be returned.
#'@param na.rm Is a boolean that indicates whether to remove the NA values or
#' not. The default value is \code{TRUE}.
#'@param apply_to Is a character string that indicates whether to apply the
#' calibration to all the forecast (\code{"all"}) or only to those where the
#' correlation between the ensemble mean and the observations is statistically
#' significant (\code{"sign"}). Only useful if \code{cal.method == "rpc-based"}.
#'@param alpha Is a numeric value indicating the significance level for the
#' correlation test. Only useful if \code{cal.method == "rpc-based" &
#' apply_to == "sign"}.
#'@param memb_dim Is a character string indicating the name of the member
#' dimension. By default, it is set to 'member'.
#'@param sdate_dim Is a character string indicating the name of the start date
#' dimension. By default, it is set to 'sdate'.
#'@param ncores Is an integer that indicates the number of cores for parallel
#' computations using multiApply function. The default value is one.
#'@return An array containing the calibrated forecasts with the same dimensions
#'as the \code{exp} array.
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#'
#'@importFrom s2dv InsertDim MeanDims Reorder
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#'@import abind
#'@importFrom ClimProjDiags Subset
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#'
#'@seealso \code{\link{CST_Load}}
#'
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#'@details
#'Both the \code{na.fill} and \code{na.rm} parameters can be used to indicate
#'how the function has to handle the NA values. The \code{na.fill} parameter
#'checks whether there are more than three forecast-observations pairs to
#'perform the computation. In case there are three or less pairs, the
#'computation is not carried out, and the value returned by the function depends
#'on the value of this parameter (either NA if \code{na.fill == TRUE} or the
#'uncorrected value if \code{na.fill == TRUE}). On the other hand, \code{na.rm}
#'is used to indicate the function whether to remove the missing values during
#'the computation of the parameters needed to perform the calibration.
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#'
#'@examples
#'mod1 <- 1 : (1 * 3 * 4 * 5 * 6 * 7)
#'dim(mod1) <- c(dataset = 1, member = 3, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'obs1 <- 1 : (1 * 1 * 4 * 5 * 6 * 7)
#'dim(obs1) <- c(dataset = 1, member = 1, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'a <- Calibration(exp = mod1, obs = obs1)
#'str(a)
#'@export
Calibration <- function(exp, obs, exp_cor=NULL, cal.method = "mse_min",
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multi.model = FALSE, na.fill = TRUE,
na.rm = TRUE, apply_to = NULL, alpha = NULL,
memb_dim = 'member', sdate_dim = 'sdate', ncores = 1) {
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dim.exp <- dim(exp)
amt.dims.exp <- length(dim.exp)
dim.obs <- dim(obs)
amt.dims.obs <- length(dim.obs)
dim.names.exp <- names(dim.exp)
dim.names.obs <- names(dim.obs)
if(!is.null(exp_cor)){
dim.exp_cor <- dim(exp_cor)
amt.dims.exp_cor <- length(dim.exp_cor)
dim.names.exp_cor <- names(dim.exp_cor)
}
if (is.null(memb_dim) || !is.character(memb_dim)) {
stop("Parameter 'memb_dim' should be a character string indicating the",
"name of the dimension where members are stored in 'exp'.")
}
if (length(memb_dim) > 1) {
memb_dim <- memb_dim[1]
warning("Parameter 'memb_dim' has length greater than 1 and only",
" the first element will be used.")
}
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if (is.null(sdate_dim) || !is.character(sdate_dim)) {
stop("Parameter 'sdate_dim' should be a character string indicating the",
"name of the dimension where start dates are stored in 'exp'.")
}
sdate_dim <- sdate_dim[1]
warning("Parameter 'sdate_dim' has length greater than 1 and only",
" the first element will be used.")
}
target.dim.names.exp <- c(memb_dim, sdate_dim)
target.dim.names.obs <- sdate_dim
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if (!all(target.dim.names.exp %in% dim.names.exp)) {
stop("Parameter 'exp' must have the dimensions defined in memb_dim ",
"and sdate_dim.")
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}
if(!is.null(exp_cor)){
if (!all(target.dim.names.exp %in% dim.names.exp_cor)) {
stop("Parameter 'exp_cor' must have the dimensions defined in memb_dim ",
"and sdate_dim.")
}
}
if (!all(c(sdate_dim) %in% dim.names.obs)) {
stop("Parameter 'obs' must have the dimension defined in sdate_dim ",
"parameter.")
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}
if (any(is.na(exp))) {
warning("Parameter 'exp' contains NA values.")
}
if(!is.null(exp_cor)){
if (any(is.na(exp_cor))) {
warning("Parameter 'exp_cor' contains NA values.")
}
}
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if (any(is.na(obs))) {
warning("Parameter 'obs' contains NA values.")
}
if (memb_dim %in% names(dim(obs))) {
obs <- Subset(obs, along = memb_dim, indices = 1, drop = "selected")
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}
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data.set.sufficiently.large.out <-
Apply(data = list(exp = exp, obs = obs),
target_dims = list(exp = target.dim.names.exp, obs = target.dim.names.obs),
ncores = ncores,
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fun = .data.set.sufficiently.large)$output1
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if(!all(data.set.sufficiently.large.out)){
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if(na.fill){
warning("Some forecast data could not be corrected due to data lack",
" and is replaced with NA values")
} else {
warning("Some forecast data could not be corrected due to data lack",
" and is replaced with uncorrected values")
}
}
if (!na.rm %in% c(TRUE,FALSE)) {
stop("Parameter 'na.rm' must be TRUE or FALSE.")
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}
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if (cal.method == 'rpc-based') {
if (is.null(apply_to)) {
apply_to <- 'sign'
warning("'apply_to' cannot be NULL for 'rpc-based' method so it has been set to 'sign', as in Eade et al. (2014).")
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} else if (!apply_to %in% c('all','sign')) {
stop("'apply_to' must be either 'all' or 'sign' when 'rpc-based' method is used.")
}
if (apply_to == 'sign') {
if (is.null(alpha)) {
alpha <- 0.1
warning("'alpha' cannot be NULL for 'rpc-based' method so it has been set to 0.1, as in Eade et al. (2014).")
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} else if (!is.numeric(alpha) | alpha <= 0 | alpha >= 1) {
stop("'alpha' must be a number between 0 and 1.")
}
}
}
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if(is.null(exp_cor)){
calibrated <- Apply(data = list(exp = exp, obs = obs),
cal.method = cal.method,
eval.method = eval.method,
multi.model = multi.model,
na.fill = na.fill, na.rm = na.rm,
apply_to = apply_to, alpha = alpha,
target_dims = list(exp = target.dim.names.exp, obs = target.dim.names.obs),
ncores = ncores, output_dims = target.dim.names.exp,
fun = .cal)$output1
dexes <- match(names(dim(exp)), names(dim(calibrated)))
calibrated <- aperm(calibrated, dexes)
dimnames(calibrated) <- dimnames(exp)[dexes]
}else{
calibrated <- Apply(data = list(exp = exp, obs = obs, exp_cor = exp_cor),
cal.method = cal.method,
eval.method = eval.method,
multi.model = multi.model,
na.fill = na.fill, na.rm = na.rm,
apply_to = apply_to, alpha = alpha,
target_dims = list(exp = target.dim.names.exp, obs = target.dim.names.obs, exp_cor = target.dim.names.exp),
ncores = ncores, output_dims = target.dim.names.exp,
fun = .cal)$output1
dexes <- match(names(dim(exp_cor)), names(dim(calibrated)))
calibrated <- aperm(calibrated, dexes)
dimnames(calibrated) <- dimnames(exp_cor)[dexes]
}
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return(calibrated)
}
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.data.set.sufficiently.large <- function(exp, obs){
amt.min.samples <- 3
amt.good.pts <- sum(!is.na(obs) & !apply(exp, c(2), function(x) all(is.na(x))))
return(amt.good.pts > amt.min.samples)
}
.make.eval.train.dexes <- function(eval.method, amt.points, amt.points_cor){
if(eval.method == "leave-one-out"){
dexes.lst <- lapply(seq(1, amt.points), function(x) return(list(eval.dexes = x,
train.dexes = seq(1, amt.points)[-x])))
} else if (eval.method == "in-sample"){
dexes.lst <- list(list(eval.dexes = seq(1, amt.points),
train.dexes = seq(1, amt.points)))
} else if (eval.method == "hindcast-vs-forecast"){
dexes.lst <- list(list(eval.dexes = seq(1,amt.points_cor),
train.dexes = seq(1, amt.points)))
} else {
stop(paste0("unknown sampling method: ",eval.method))
}
return(dexes.lst)
}
.cal <- function(exp, obs, exp_cor = NULL, cal.method, eval.method, multi.model, na.fill, na.rm, apply_to, alpha) {
if(is.null(exp_cor)){
exp_cor <- exp ## generate a copy of exp so that the same function can run
## when exp_cor is provided and when it's not
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obs <- as.vector(obs)
dims.fc <- dim(exp)
dims.fc_cor <- dim(exp_cor) ## new line
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amt.mbr <- dims.fc[1]
amt.sdate <- dims.fc[2]
amt.sdate_cor <- dims.fc_cor[2] ## new line
var.cor.fc <- NA * exp_cor ## modified line (exp_cor instead of exp);
## in case of exp_cor not provided, at this point exp_cor
## is already the same as exp so no change
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if(!.data.set.sufficiently.large(exp = exp, obs = obs)){
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if(na.fill){
return(var.cor.fc)
} else {
var.cor.fc[] <- exp[]
return(var.cor.fc)
}
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}
eval.train.dexeses <- .make.eval.train.dexes(eval.method, amt.points = amt.sdate,
amt.points_cor = amt.sdate_cor)
amt.resamples <- length(eval.train.dexeses)
for (i.sample in seq(1, amt.resamples)) {
# defining training (tr) and evaluation (ev) subsets
eval.dexes <- eval.train.dexeses[[i.sample]]$eval.dexes
train.dexes <- eval.train.dexeses[[i.sample]]$train.dexes
fc.ev <- exp_cor[ , eval.dexes, drop = FALSE] ## modified line (exp_cor instead of exp)
## fc.ev is used to evaluate (not train; train should be done with exp (hindcast))
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fc.tr <- exp[ , train.dexes]
obs.tr <- obs[train.dexes , drop = FALSE]
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var.cor.fc[ , eval.dexes] <- fc.ev + mean(obs.tr, na.rm = na.rm) - mean(fc.tr, na.rm = na.rm)
} else if(cal.method == "evmos"){ # forecast correction implemented
#calculate ensemble and observational characteristics
quant.obs.fc.tr <- .calc.obs.fc.quant(obs = obs.tr, fc = fc.tr, na.rm = na.rm)
#calculate value for regression parameters
init.par <- c(.calc.evmos.par(quant.obs.fc.tr, na.rm = na.rm))
var.cor.fc[ , eval.dexes] <- .correct.evmos.fc(fc.ev , init.par, na.rm = na.rm)
} else if (cal.method == "mse_min"){
#calculate ensemble and observational characteristics
quant.obs.fc.tr <- .calc.obs.fc.quant(obs = obs.tr, fc = fc.tr, na.rm = na.rm)
#calculate value for regression parameters
init.par <- .calc.mse.min.par(quant.obs.fc.tr, multi.model, na.rm = na.rm)
var.cor.fc[ , eval.dexes] <- .correct.mse.min.fc(fc.ev , init.par, na.rm = na.rm)
} else if (cal.method == "crps_min"){
#calculate ensemble and observational characteristics
quant.obs.fc.tr <- .calc.obs.fc.quant.ext(obs = obs.tr, fc = fc.tr, na.rm = na.rm)
#calculate initial value for regression parameters
init.par <- c(.calc.mse.min.par(quant.obs.fc.tr, na.rm = na.rm), 0.001)
init.par[3] <- sqrt(init.par[3])
#calculate regression parameters on training dataset
optim.tmp <- optim(par = init.par,
fn = .calc.crps.opt,
gr = .calc.crps.grad.opt,
quant.obs.fc = quant.obs.fc.tr,
na.rm = na.rm,
method = "BFGS")
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var.cor.fc[ , eval.dexes] <- .correct.crps.min.fc(fc.ev , mbm.par, na.rm = na.rm)
} else if (cal.method == 'rpc-based') {
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ens_mean.ev <- multiApply::Apply(data = fc.ev, target_dims = names(amt.mbr), fun = mean, na.rm = na.rm)$output1 ## Ensemble mean
ens_mean.tr <- multiApply::Apply(data = fc.tr, target_dims = names(amt.mbr), fun = mean, na.rm = na.rm)$output1 ## Ensemble mean
ens_spread.tr <- multiApply::Apply(data = list(fc.tr, ens_mean.tr), target_dims = names(amt.sdate), fun = "-")$output1 ## Ensemble spread
exp_mean.tr <- mean(fc.tr, na.rm = na.rm) ## Mean (climatology)
var_signal.tr <- var(ens_mean.tr, na.rm = na.rm) ## Ensemble mean variance
var_noise.tr <- var(as.vector(ens_spread.tr), na.rm = na.rm) ## Variance of ensemble members about ensemble mean (= spread)
var_obs.tr <- var(obs.tr, na.rm = na.rm) ## Variance in the observations
r.tr <- cor(x = ens_mean.tr, y = obs.tr, method = 'pearson', use = ifelse(test = isTRUE(na.rm), yes = "pairwise.complete.obs", no = "everything")) ## Correlation between observations and the ensemble mean
if ((apply_to == 'all') || (apply_to == 'sign' && cor.test(ens_mean.tr, obs.tr, method = 'pearson', alternative = 'greater')$p.value < alpha)) {
ens_mean_cal <- (ens_mean.ev - exp_mean.tr) * r.tr * sqrt(var_obs.tr) / sqrt(var_signal.tr) + exp_mean.tr
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var.cor.fc[ , eval.dexes] <- s2dv::Reorder(data = multiApply::Apply(data = list(exp = fc.ev, ens_mean = ens_mean.ev, ens_mean_cal = ens_mean_cal), target_dims = names(amt.sdate), fun = .CalibrationMembersRPC, var_obs = var_obs.tr, var_noise = var_noise.tr, r = r.tr)$output1,
order = names(dims.fc))
dim(var.cor.fc) <- dims.fc
} else { ## no significant -> replacing with observed climatology
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var.cor.fc[ , eval.dexes] <- array(data = mean(obs.tr, na.rm = na.rm), dim = dim(fc.ev))
stop("unknown calibration method: ",cal.method)
.calc.obs.fc.quant <- function(obs, fc, na.rm){ #function to calculate different quantities of a series of ensemble forecasts and corresponding observations
obs.per.ens <- InsertDim(obs, posdim = 1, lendim = amt.mbr)
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fc.ens.av <- apply(fc, c(2), mean, na.rm = na.rm)
cor.obs.fc <- cor(fc.ens.av, obs, use = "complete.obs")
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obs.av <- mean(obs, na.rm = na.rm)
obs.sd <- sd(obs, na.rm = na.rm)
.calc.fc.quant(fc = fc, na.rm = na.rm),
list(
obs.per.ens = obs.per.ens,
cor.obs.fc = cor.obs.fc,
obs.av = obs.av,
obs.sd = obs.sd
)
)
)
}
.calc.obs.fc.quant.ext <- function(obs, fc, na.rm){ #extended function to calculate different quantities of a series of ensemble forecasts and corresponding observations
obs.per.ens <- InsertDim(obs, posdim = 1, lendim = amt.mbr)
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fc.ens.av <- apply(fc, c(2), mean, na.rm = na.rm)
cor.obs.fc <- cor(fc.ens.av, obs, use = "complete.obs")
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obs.av <- mean(obs, na.rm = na.rm)
obs.sd <- sd(obs, na.rm = na.rm)
.calc.fc.quant.ext(fc = fc, na.rm = na.rm),
list(
obs.per.ens = obs.per.ens,
cor.obs.fc = cor.obs.fc,
obs.av = obs.av,
obs.sd = obs.sd
)
)
)
}
.calc.fc.quant <- function(fc, na.rm){ #function to calculate different quantities of a series of ensemble forecasts
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fc.ens.av <- apply(fc, c(2), mean, na.rm = na.rm)
fc.ens.av.av <- mean(fc.ens.av, na.rm = na.rm)
fc.ens.av.sd <- sd(fc.ens.av, na.rm = na.rm)
fc.ens.av.per.ens <- InsertDim(fc.ens.av, posdim = 1, lendim = amt.mbr)
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fc.ens.sd <- apply(fc, c(2), sd, na.rm = na.rm)
fc.ens.var.av.sqrt <- sqrt(mean(fc.ens.sd^2, na.rm = na.rm))
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fc.dev.sd <- sd(fc.dev, na.rm = na.rm)
fc.av <- mean(fc, na.rm = na.rm)
fc.sd <- sd(fc, na.rm = na.rm)
return(
list(
fc.ens.av = fc.ens.av,
fc.ens.av.av = fc.ens.av.av,
fc.ens.av.sd = fc.ens.av.sd,
fc.ens.av.per.ens = fc.ens.av.per.ens,
fc.ens.sd = fc.ens.sd,
fc.ens.var.av.sqrt = fc.ens.var.av.sqrt,
fc.av = fc.av,
fc.sd = fc.sd
)
)
}
.calc.fc.quant.ext <- function(fc, na.rm){ #extended function to calculate different quantities of a series of ensemble forecasts
repmat1.tmp <- InsertDim(fc, posdim = 1, lendim = amt.mbr)
repmat2.tmp <- aperm(repmat1.tmp, c(2, 1, 3))
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spr.abs <- apply(abs(repmat1.tmp - repmat2.tmp), c(3), mean, na.rm = na.rm)
spr.abs.per.ens <- InsertDim(spr.abs, posdim = 1, lendim = amt.mbr)
append(.calc.fc.quant(fc, na.rm = na.rm),
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list(spr.abs = spr.abs, spr.abs.per.ens = spr.abs.per.ens))
#Below are the core or elementary functions to calculate the regression parameters for the different methods
.calc.mse.min.par <- function(quant.obs.fc, multi.model = F, na.rm){
if(multi.model){
par.out[3] <- with(quant.obs.fc, obs.sd * sqrt(1. - cor.obs.fc^2) / fc.ens.var.av.sqrt)
} else {
par.out[3] <- with(quant.obs.fc, obs.sd * sqrt(1. - cor.obs.fc^2) / fc.dev.sd)
}
par.out[2] <- with(quant.obs.fc, abs(cor.obs.fc) * obs.sd / fc.ens.av.sd)
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par.out[1] <- with(quant.obs.fc, obs.av - par.out[2] * fc.ens.av.av, na.rm = na.rm)
.calc.evmos.par <- function(quant.obs.fc, na.rm){
par.out <- rep(NA, 2)
par.out[2] <- with(quant.obs.fc, obs.sd / fc.sd)
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par.out[1] <- with(quant.obs.fc, obs.av - par.out[2] * fc.ens.av.av, na.rm = na.rm)
#Below are the core or elementary functions to calculate the functions necessary for the minimization of crps
.calc.crps.opt <- function(par, quant.obs.fc, na.rm){
return(
with(quant.obs.fc,
mean(abs(obs.per.ens - (par[1] + par[2] * fc.ens.av.per.ens +
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((par[3])^2 + par[4] / spr.abs.per.ens) * fc.dev)), na.rm = na.rm) -
mean(abs((par[3])^2 * spr.abs + par[4]) / 2., na.rm = na.rm)
.calc.crps.grad.opt <- function(par, quant.obs.fc, na.rm){
sgn1 <- with(quant.obs.fc,sign(obs.per.ens -
(par[1] + par[2] * fc.ens.av.per.ens +
((par[3])^2 + par[4] / spr.abs.per.ens) * fc.dev)))
sgn2 <- with(quant.obs.fc, sign((par[3])^2 + par[4] / spr.abs.per.ens))
sgn3 <- with(quant.obs.fc,sign((par[3])^2 * spr.abs + par[4]))
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deriv.par1 <- mean(sgn1, na.rm = na.rm)
deriv.par2 <- with(quant.obs.fc, mean(sgn1 * fc.dev, na.rm = na.rm))
deriv.par3 <- with(quant.obs.fc,
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mean(2* par[3] * sgn1 * sgn2 * fc.ens.av.per.ens, na.rm = na.rm) -
mean(spr.abs * sgn3, na.rm = na.rm) / 2.)
deriv.par4 <- with(quant.obs.fc,
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mean(sgn1 * sgn2 * fc.ens.av.per.ens / spr.abs.per.ens, na.rm = na.rm) -
mean(sgn3, na.rm = na.rm) / 2.)
return(c(deriv.par1, deriv.par2, deriv.par3, deriv.par4))
}
#Below are the core or elementary functions to correct the evaluation set based on the regression parameters
.correct.evmos.fc <- function(fc, par, na.rm){
quant.fc.mp <- .calc.fc.quant(fc = fc, na.rm = na.rm)
return(with(quant.fc.mp, par[1] + par[2] * fc))
}
.correct.mse.min.fc <- function(fc, par, na.rm){
quant.fc.mp <- .calc.fc.quant(fc = fc, na.rm = na.rm)
return(with(quant.fc.mp, par[1] + par[2] * fc.ens.av.per.ens + fc.dev * par[3]))
}
.correct.crps.min.fc <- function(fc, par, na.rm){
quant.fc.mp <- .calc.fc.quant.ext(fc = fc, na.rm = na.rm)
return(with(quant.fc.mp, par[1] + par[2] * fc.ens.av.per.ens + fc.dev * abs((par[3])^2 + par[4] / spr.abs)))
}
# Function to calibrate the individual members with the RPC-based method
.CalibrationMembersRPC <- function(exp, ens_mean, ens_mean_cal, var_obs, var_noise, r){
member_cal <- (exp - ens_mean) * sqrt(var_obs) * sqrt(1 - r^2) / sqrt(var_noise) + ens_mean_cal
return(member_cal)