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```
# FairSprErr.R Spread to Error Ratio
#
# Copyright (C) 2016 MeteoSwiss
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
#
#' Fair Spread to Error Ratio
#'
#' Compute the spread to error ratio (\code{SPR}) for probabilistic forecasts -
#' not unlike the functions in SpecsVerification. \code{SPR > 1} indicates
#' overdispersion (underconfidence), whereas \code{SPR < 1} indicates
#' overconfidence in the forecasts.
#'
#' @param ens n x k matrix of n forecasts for k ensemble members
#' @param obs vector with n verifying observations
#'
#' @details Here we define the spread-error rate as the square root of the ratio
#' of mean ensemble variance to the mean squared error of the ensemble mean
#' with the verifying observations. We inflate the intra ensemble sample
#' variance to account for the finite ensemble size as in Weigel (2011).
#'
#'
#' @references Weigel, A.P. (2012). Ensemble forecasts. Forecast Verification: A
#' Practitioner's Guide in Atmospheric Science, Second Edition, 141-166.
#'
#' @seealso \code{\link{veriApply}}, \code{\link{FairSprErr}}
#'
#' @examples
#' tm <- toymodel()
#' FairSprErr(tm$fcst, tm$obs)
#'
#' ## compute spread to error ratio using veriApply
#' veriApply('FairSprErr', fcst=tm$fcst, obs=tm$obs)
#'
#' ## compare with 'unfair' spread to error ratio
#' veriApply("EnsSprErr", fcst=tm$fcst, obs=tm$obs)
#'
#' @export
FairSprErr <- function(ens, obs){
stopifnot(is.matrix(ens), is.vector(obs), nrow(ens) == length(obs))
xmask <- apply(!is.na(ens), 1, any) & !is.na(obs)
nens <- ncol(ens)
spread <- mean(apply(ens[xmask,,drop=F], 1, sd, na.rm=T)**2, na.rm=T)
error <- mean((obs - rowMeans(ens))**2, na.rm=T)
return(sqrt((nens + 1) / nens * spread/error))
}
```

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