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#'Forecast Calibration
#'
#'@author Verónica Torralba, \email{veronica.torralba@bsc.es} and Bert Van Schaeybroeck, \email{bertvs@meteo.be}
#'@description The \textit{cal} function applies a variance inflation technique described in Doblas-Reyes et al. (2005) in leave-one-out cross-validation. This bias adjustment method produces calibrated forecasts with equivalent mean and variance to that of the reference dataset, but at the same time preserve reliability.
#'@description This \textit{mbm_cal} function applies a member-by-member ensemble bias correction described in Van Schaeybroeck and Vannitsem (Quart J, 2015). The adjusted forecasts has an optimized CRPS score. This implies the (near) correspondence of 1) forecast mean with observational mean, 2) forecast variability with observational variability, 3) mean squared error with average ensemble variability such that the ensemble is reliable.
#'
#'@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 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.
#'
#'@param exp an object of class \code{s2dv_cube} as returned by \code{CST_Load} function, containing the seasonal forecast experiment data in the element named \code{$data}.
#'@param obs an object of class \code{s2dv_cube} as returned by \code{CST_Load} function, containing the observed data in the element named \code{$data}.
#'@param cal.method is the calibration method used, can be either "bias", "cal" or "mbm_cal". Default value is "bias".
#'@param eval.method is the sampling method used, can be either "in-sample" or "take-one-out". Default value is "take-one-out".
#'@return an object of class \code{s2dv_cube} containing the calibrated forecasts in the element \code{$data} with the same dimensions of the experimental data.
#'
#'@import s2dverification
#'# Example
#'# Creation of sample s2dverification objects. These are not complete
#'# s2dverification objects though. The Load function returns complete objects.
#'mod1 <- 1 : (1 * 3 * 4 * 5 * 6 * 7)
#'dim(mod1) <- c(dataset = 1, member = 3, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'obs1 <- 1 : (1 * 1 * 4 * 5 * 6 * 7)
#'dim(obs1) <- c(dataset = 1, member = 1, sdate = 4, ftime = 5, lat = 6, lon = 7)
#'lon <- seq(0, 30, 5)
#'lat <- seq(0, 25, 5)
#'exp <- list(data = mod1, lat = lat, lon = lon)
#'obs <- list(data = obs1, lat = lat, lon = lon)
#'attr(exp, 'class') <- 's2dv_cube'
#'attr(obs, 'class') <- 's2dv_cube'
#'a <- CST_Calibration(exp = exp, obs = obs, cal.method = "cal", eval.method = "in-sample")
#'str(a)
#'@export
CST_Calibration <- function(exp, obs, ...) {
if (!inherits(exp, "s2dv_cube") || !inherits(exp, "s2dv_cube")) {
stop("Parameter 'exp' and 'obs' must be of the class 's2dv_cube', ",
"as output by CSTools::CST_Load.")
stop("The length of the dimension 'member' in the component 'data' ",
"of the parameter 'obs' must be equal to 1.")
exp$data <- .calibration(exp = exp$data, obs = obs$data, ...)
exp$Datasets <- c(exp$Datasets, obs$Datasets)
exp$source_files <- c(exp$source_files, obs$source_files)
return(exp)
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make.eval.train.dexeses <- function(eval.method, amt.points){
if(amt.points < 10 & eval.method != "in-sample"){
cat("Too few points, so sample method will necessarily be in-sample")
eval.method <- "in-sample"
}
if(eval.method == "take-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 {
stop(paste0("unknown sampling method: ",eval.method))
}
return(dexes.lst)
}
.cal <- function(obs.fc, cal.method = "bias", eval.method = "take-one-out", ...) {
dims.tmp=dim(obs.fc)
amt.mbr <- dims.tmp["member"][] - 1
amt.sdate <- dims.tmp["sdate"][]
pos <- match(c("member","sdate"), names(dims.tmp))
obs.fc <- aperm(obs.fc, pos)
var.obs <- asub(obs.fc, list(1),1)
var.fc <- asub(obs.fc, list(1+seq(1, amt.mbr)),1)
dims.fc <- dim(var.fc)
var.cor.fc <- NA * var.fc
eval.train.dexeses <- make.eval.train.dexeses(eval.method, amt.points = amt.sdate)
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 <- var.fc[ , eval.dexes, drop = FALSE]
fc.tr <- var.fc[ , train.dexes]
obs.tr <- var.obs[train.dexes , drop = FALSE]
#calculate ensemble and observational characteristics
if(cal.method == "bias"){
var.cor.fc[ , eval.dexes] <- fc.ev + mean(obs.tr, na.rm = TRUE) - mean(fc.tr, na.rm = TRUE)
} else if (cal.method == "cal"){
quant.obs.fc.tr <- calc.obs.fc.quant(obs = obs.tr, fc = fc.tr)
#calculate value for regression parameters
init.par <- c(calc.cal.par(quant.obs.fc.tr), 0.)
#correct evaluation subset
var.cor.fc[ , eval.dexes] <- correct.cal.fc(fc.ev , init.par)
} else if (cal.method == "mbm_cal"){
quant.obs.fc.tr <- calc.obs.fc.quant.ext(obs = obs.tr, fc = fc.tr)
#calculate initial value for regression parameters
init.par <- c(calc.cal.par(quant.obs.fc.tr), 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) #, method = "CG"
mbm.par <- optim.tmp$par
#correct evaluation subset
var.cor.fc[ , eval.dexes] <- correct.mbm.fc(fc.ev , mbm.par)
} else {
stop("unknown calibration method: ",cal.method)
}
}
names(dim(var.cor.fc)) <- c("member", "sdate")
return(var.cor.fc)
}
.calibration <- function(exp, obs, ...) {
target.dims <- c("member", "sdate")
if (!all(target.dims %in% names(dim(exp)))) {
stop("Parameter 'exp' must have the dimensions 'member' and 'sdate'.")
}
if (!all(c("sdate") %in% names(dim(obs)))) {
stop("Parameter 'obs' must have the dimension 'sdate'.")
}
if (any(is.na(exp))) {
warning("Parameter 'exp' contains NA values.")
if (any(is.na(obs))) {
warning("Parameter 'obs' contains NA values.")
target_dims_obs <- "sdate"
if ("member" %in% names(dim(obs))) {
target_dims_obs <- c("member", target_dims_obs)
amt.member=dim(exp)["member"]
amt.sdate=dim(exp)["sdate"]
target.dims <- c("member", "sdate")
return.feat <- list(dim = c(amt.member, amt.sdate))
return.feat$name <- c("member", "sdate")
return.feat$dim.name <- list(dimnames(exp)[["member"]],dimnames(exp)[["sdate"]])
ptm <- proc.time()
calibrated <- .apply.obs.fc(obs = obs,
fc = exp,
target.dims = target.dims,
return.feat = return.feat,
FUN = .cal, ...)
return(calibrated)
.apply.obs.fc <- function(obs, fc, target.dims, FUN, return.feat, ...){
dimnames.tmp <- dimnames(fc)
fc.dims.tmp <- dim(fc)
dims.out.tmp <- return.feat$dim
obs.fc <- .combine.obs.fc(obs, fc)
names.dim <- names(dim(obs.fc))
amt.dims <- length(names.dim)
margin.all <- seq(1, amt.dims)
matched.dims <- match(target.dims, names.dim)
margin.to.use <- margin.all[-matched.dims]
arr.out <- apply(X = obs.fc,
MARGIN = margin.to.use,
FUN = FUN,
...)
dims.tmp <- dim(arr.out)
names.dims.tmp <- names(dim(arr.out))
if(prod(return.feat$dim) != dims.tmp[1]){
stop("apply.obs.fc: returned dimensions not as expected: ", prod(return.feat$dim), " and ", dims.tmp[1])
dim(arr.out) <- c(dims.out.tmp, dims.tmp[-1])
names(dim(arr.out)) <- c(return.feat$name, names.dims.tmp[c(-1)])
names.dim[matched.dims] <- return.feat$name
pos <- match(names.dim, names(dim(arr.out)))
pos_inv <- match(names(dim(arr.out)), names.dim)
arr.out <- aperm(arr.out, pos)
for (i.item in seq(1,length(return.feat$name))){
dimnames.tmp[[pos_inv[i.item]]] <- return.feat$dim.name[[i.item]]
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dimnames(arr.out) <- dimnames.tmp
return(arr.out)
}
.combine.obs.fc <- function(obs,fc){
names.dim.tmp <- names(dim(obs))
members.dim <- which(names.dim.tmp == "member")
arr.out <- abind(obs, fc, along = members.dim)
dimnames.tmp <- dimnames(arr.out)
names(dim(arr.out)) <- names.dim.tmp
dimnames(arr.out) <- dimnames.tmp
names(dimnames(arr.out)) <- names.dim.tmp
return(arr.out)
}
calc.obs.fc.quant <- function(obs, fc){
amt.mbr <- dim(fc)[1]
obs.per.ens <- .spr(obs, amt.mbr)
fc.ens.av <- apply(fc, c(2), mean, na.rm = TRUE)
cor.obs.fc <- cor(fc.ens.av, obs, use = "complete.obs")
obs.av <- mean(obs, na.rm = TRUE)
obs.sd <- sd(obs, na.rm = TRUE)
return(
append(
calc.fc.quant(fc = fc),
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){
amt.mbr <- dim(fc)[1]
obs.per.ens <- .spr(obs, amt.mbr)
fc.ens.av <- apply(fc, c(2), mean, na.rm = TRUE)
cor.obs.fc <- cor(fc.ens.av, obs, use = "complete.obs")
obs.av <- mean(obs, na.rm = TRUE)
obs.sd <- sd(obs, na.rm = TRUE)
return(
append(
calc.fc.quant.ext(fc = fc),
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){
amt.mbr <- dim(fc)[1]
fc.ens.av <- apply(fc, c(2), mean, na.rm = TRUE)
fc.ens.av.av <- mean(fc.ens.av, na.rm = TRUE)
fc.ens.av.sd <- sd(fc.ens.av, na.rm = TRUE)
fc.ens.av.per.ens <- .spr(fc.ens.av, amt.mbr)
fc.ens.sd <- apply(fc, c(2), sd, na.rm = TRUE)
fc.ens.sd.av <- sqrt(mean(fc.ens.sd^2,na.rm = TRUE))
fc.dev <- fc - fc.ens.av.per.ens
fc.av <- mean(fc, na.rm = TRUE)
fc.sd <- sd(fc, na.rm = TRUE)
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.sd.av = fc.ens.sd.av,
fc.dev = fc.dev,
fc.av = fc.av,
fc.sd = fc.sd
)
)
}
calc.fc.quant.ext <- function(fc){
amt.mbr <- dim(fc)[1]
fc.ens.av <- apply(fc, c(2), mean, na.rm = TRUE)
fc.ens.av.av <- mean(fc.ens.av, na.rm = TRUE)
fc.ens.av.sd <- sd(fc.ens.av, na.rm = TRUE)
fc.ens.av.per.ens <- .spr(fc.ens.av, amt.mbr)
fc.ens.sd <- apply(fc, c(2), sd, na.rm = TRUE)
fc.ens.sd.av <- sqrt(mean(fc.ens.sd^2, na.rm = TRUE))
fc.dev <- fc - fc.ens.av.per.ens
repmat1.tmp <- .spr(fc, amt.mbr)
repmat2.tmp <- aperm(repmat1.tmp, c(2, 1, 3))
spr.abs <- apply(abs(repmat1.tmp - repmat2.tmp), c(3), mean, na.rm = TRUE)
spr.abs.per.ens <- .spr(spr.abs, amt.mbr)
fc.av <- mean(fc, na.rm = TRUE)
fc.sd <- sd(fc, na.rm = TRUE)
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.sd.av = fc.ens.sd.av,
fc.dev = fc.dev,
spr.abs = spr.abs,
spr.abs.per.ens = spr.abs.per.ens,
fc.av = fc.av,
fc.sd = fc.sd
)
)
}
calc.cal.par <- function(quant.obs.fc){
par.out <- rep(NA, 3)
par.out[3] <- with(quant.obs.fc, obs.sd * sqrt(1 - cor.obs.fc^2) / fc.ens.sd.av)
par.out[2] <- with(quant.obs.fc, cor.obs.fc * obs.sd / fc.ens.av.sd)
par.out[1] <- with(quant.obs.fc, obs.av - par.out[2] * fc.ens.av.av, na.rm = TRUE)
return(par.out)
}
correct.mbm.fc <- function(fc, par){
quant.fc.mp <- calc.fc.quant.ext(fc = fc)
return(with(quant.fc.mp, par[1] + par[2] * fc.ens.av.per.ens + fc.dev * abs((par[3])^2 + par[4] / spr.abs)))
}
correct.cal.fc <- function(fc, par){
quant.fc.mp <- calc.fc.quant(fc = fc)
return(with(quant.fc.mp, par[1] + par[2] * fc.ens.av.per.ens + fc.dev * par[3]))
}
calc.crps <- function(obs, fc){
quant.obs.fc <- calc.obs.fc.quant.ext(obs = obs, fc = fc)
return(with(quant.obs.fc,
mean(apply(abs(obs.per.ens - fc), c(2), mean, na.rm = TRUE) - spr.abs / 2., na.rm = TRUE)))
}
calc.crps.opt <- function(par, quant.obs.fc){
return(
with(quant.obs.fc,
mean(abs(obs.per.ens - (par[1] + par[2] * fc.ens.av.per.ens +
((par[3])^2 + par[4] / spr.abs.per.ens) * fc.dev)), na.rm = TRUE) -
mean(abs((par[3])^2 * spr.abs + par[4]) / 2., na.rm = TRUE)
)
)
}
calc.crps.grad.opt <- function(par, quant.obs.fc){
attach(quant.obs.fc)
sgn1 <- 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 <- sign((par[3])^2 + par[4] / spr.abs.per.ens)
sgn3 <- sign((par[3])^2 * spr.abs + par[4])
deriv.par1 <- mean(sgn1, na.rm = TRUE)
deriv.par2 <- mean(sgn1 * fc.dev, na.rm = TRUE)
deriv.par3 <- mean(2* par[3] * sgn1 * sgn2 * fc.ens.av.per.ens, na.rm = TRUE) -
mean(spr.abs * sgn3, na.rm = TRUE) / 2.
deriv.par4 <- mean(sgn1 * sgn2 * fc.ens.av.per.ens / spr.abs.per.ens, na.rm = TRUE) -
mean(sgn3, na.rm = TRUE) / 2.
return(c(deriv.par1, deriv.par2, deriv.par3, deriv.par4))
}
.spr <- function(x, amt.spr, dim = 1) {
if(is.vector(x)){
amt.dims <- 1
if(dim == 2){
arr.out <- array(rep(x, amt.spr), c(length(x), amt.spr))
} else if(dim == 1){
arr.out <- t(array(rep(x, amt.spr), c(length(x), amt.spr)))
} else {
stop(paste0("error in .spr: amt.dims = ",amt.dims," while dim = ",dim))
}
} else if(is.array(x)) {
amt.dims <- length(dim(x))
if(dim > amt.dims + 1){
stop(paste0("error in .spr: amt.dims = ",amt.dims," while dim = ",dim))
}
arr.out <- array(rep(as.vector(x), amt.spr), c(dim(x), amt.spr))
if(dim != amt.dims + 1){
amt.dims.out <- amt.dims + 1
dims.tmp <- seq(1, amt.dims.out)
dims.tmp[seq(dim, amt.dims.out)] <- c(amt.dims.out, seq(dim,amt.dims.out-1))
arr.out <- aperm(arr.out, dims.tmp)
}
} else {
stop("x is not array nor vector but is ", class(x))