CST_Calibration.R

#'Forecast Calibration 
#'
#'@author Verónica Torralba, \email{veronica.torralba@bsc.es} and Bert Van Schaeybroeck, \email{bertvs@meteo.be}
#'@description Three types bias correction can be implemented. The first \code{"bias"} method simply subtracts the mean bias.
#'@description The \code{"cal"} calibration method 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 The \code{"mbm_cal"} calibration method applies a member-by-member ensemble bias correction described in Van Schaeybroeck and Vannitsem (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 \code{"bias"}, \code{"cal"} or \code{"mbm_cal"}. Default value is \code{"bias"}.
#'@param eval.method is the sampling method used, can be either \code{"in-sample"} or \code{"take-one-out"}. Default value is the \code{"take-one-out"} cross validation.
#'@param ... other parameters to be passed on to the calibration procedure
#'@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
#'@import abind
#'
#'@seealso \code{\link{CST_Load}}
#'
#'# 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, cal.method = "bias", eval.method = "take-one-out", ...) {
  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.")
  }

  if (dim(obs$data)["member"] != 1) {
    stop("The length of the dimension 'member' in the component 'data' ",
         "of the parameter 'obs' must be equal to 1.")
  }

  exp$data <- .calibration.wrap(exp = exp$data, obs = obs$data, cal.method = cal.method, eval.method = eval.method, ...)
  exp$Datasets <- c(exp$Datasets, obs$Datasets)
  exp$source_files <- c(exp$source_files, obs$source_files)
  return(exp)
  
}

.make.eval.train.dexes <- 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, eval.method, ...) {
  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.dexes(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 = "BFGS")
      
      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.wrap <- function(exp, obs, cal.method, eval.method, ...) {
  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,
                      cal.method = cal.method,
                      eval.method = eval.method,
                      ...)
  print(proc.time()-ptm)
                      
  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]]
  }
  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){
  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]))
  deriv.par1 <- mean(sgn1, na.rm = TRUE)
  deriv.par2 <- with(quant.obs.fc, mean(sgn1 * fc.dev, na.rm = TRUE))
  deriv.par3 <- with(quant.obs.fc, 
    mean(2* par[3] * sgn1 * sgn2 * fc.ens.av.per.ens, na.rm = TRUE) -
    mean(spr.abs * sgn3, na.rm = TRUE) / 2.)
  deriv.par4 <- with(quant.obs.fc,
    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))
  }
  return(arr.out)
}