#'Compute the ratio between the ensemble spread and RMSE
#'
#'Compute the ratio between the spread of the members around the 
#'ensemble mean in experimental data and the RMSE between the ensemble mean of 
#'experimental and observational data. The p-value and/or the statistical 
#'significance is provided by a two-sided Fisher's test.
#'
#'@param exp A named numeric array of experimental data with at least two 
#'  dimensions 'memb_dim' and 'time_dim'.
#'@param obs A named numeric array of observational data with at least two 
#'  dimensions 'memb_dim' and 'time_dim'. It should have the same dimensions as
#'  parameter 'exp' except along 'dat_dim' and 'memb_dim'.
#'@param dat_dim A character string indicating the name of dataset (nobs/nexp) 
#'  dimension. The default value is NULL (no dataset).
#'@param memb_dim A character string indicating the name of the member 
#'  dimension. It must be one dimension in 'exp' and 'obs'. The default value 
#'  is 'member'.
#'@param time_dim A character string indicating the name of dimension along  
#'  which the ratio is computed. The default value is 'sdate'.
#'@param pval A logical value indicating whether to compute the p-value 
#'  of the test Ho : SD/RMSE = 1 or not. The default value is TRUE.
#'@param sign A logical value indicating whether to retrieve the statistical
#'  significance of the test Ho: ACC = 0 based on 'alpha'. The default value is
#'  FALSE.
#'@param alpha A numeric indicating the significance level for the statistical
#'  significance test. The default value is 0.05. 
#'@param na.rm A logical value indicating whether to remove NA values. The 
#'  default value is FALSE.
#'@param ncores An integer indicating the number of cores to use for parallel 
#'  computation. The default value is NULL.
#'
#'@return A list of two arrays with dimensions c(nexp, nobs, the rest of 
#'  dimensions of 'exp' and 'obs' except memb_dim and time_dim), which nexp is
#'  the length of dat_dim of 'exp' and nobs is the length of dat_dim of 'obs'. 
#'  If dat_dim is NULL, nexp and nobs are omitted. \cr
#'\item{$ratio}{
#'  The ratio of the ensemble spread and RMSE.
#'}
#'\item{$p_val}{
#'  The p-value of the two-sided Fisher's test with Ho: Spread/RMSE = 1. Only 
#'  present if \code{pval = TRUE}.
#'}
#'
#'@examples
#'exp <- array(rnorm(30), dim = c(lat = 2, sdate = 3, member = 5))
#'obs <- array(rnorm(30), dim = c(lat = 2, sdate = 3))
#'sprerr1 <- SprErr(exp, obs)
#'sprerr2 <- SprErr(exp, obs, pval = FALSE, sign = TRUE)
#'sprerr3 <- SprErr(exp, obs, pval = TRUE, sign = TRUE)
#'
#'@import multiApply
#'@export
SprErr <- function(exp, obs, dat_dim = NULL, memb_dim = 'member', 
                   time_dim = 'sdate', pval = TRUE, sign = FALSE, 
                   alpha = 0.05, na.rm = FALSE, ncores = NULL) {
  
  # Check inputs 
  ## exp and obs (1)
  if (is.null(exp) | is.null(obs)) {
    stop("Parameter 'exp' and 'obs' cannot be NULL.")
  }
  if (!is.numeric(exp) | !is.numeric(obs)) {
    stop("Parameter 'exp' and 'obs' must be a numeric array.")
  }
  if (is.null(dim(exp)) | is.null(dim(obs))) {
    stop(paste0("Parameter 'exp' and 'obs' must be array with as least two ",
                "dimensions memb_dim and time_dim."))
  }
  if (any(is.null(names(dim(exp))))| any(nchar(names(dim(exp))) == 0) |
      any(is.null(names(dim(obs))))| any(nchar(names(dim(obs))) == 0)) {
    stop("Parameter 'exp' and 'obs' must have dimension names.")
  }
  ## dat_dim
  if (!is.null(dat_dim)) {
    if (!is.character(dat_dim) | length(dat_dim) > 1) {
      stop("Parameter 'dat_dim' must be a character string.")
    }
    if (!dat_dim %in% names(dim(exp)) | !dat_dim %in% names(dim(obs))) {
      stop("Parameter 'dat_dim' is not found in 'exp' or 'obs' dimension.")
    }
  }
  ## memb_dim
  if (!is.character(memb_dim) | length(memb_dim) > 1) {
    stop("Parameter 'memb_dim' must be a character string.")
  }
  if (!memb_dim %in% names(dim(exp))) {
    stop("Parameter 'memb_dim' is not found in 'exp' dimensions. ",
         "'exp' must have the member dimension to compute the spread.")
  }
  # Add [member = 1] 
  if (memb_dim %in% names(dim(exp)) & !memb_dim %in% names(dim(obs))) {
    dim(obs) <- c(dim(obs), 1)
    names(dim(obs))[length(dim(obs))] <- memb_dim
  }
  if (!memb_dim %in% names(dim(exp)) & memb_dim %in% names(dim(obs))) { ## check no longer needed?
    dim(exp) <- c(dim(exp), 1)
    names(dim(exp))[length(dim(exp))] <- memb_dim
  }
  ## time_dim
  if (!is.character(time_dim) | length(time_dim) > 1) {
    stop("Parameter 'time_dim' must be a character string.")
  }
  if (!time_dim %in% names(dim(exp)) | !time_dim %in% names(dim(obs))) {
    stop("Parameter 'time_dim' is not found in 'exp' or 'obs' dimension.")
  }
  ## exp and obs (2)
  name_exp <- sort(names(dim(exp)))
  name_obs <- sort(names(dim(obs)))
  if (!is.null(dat_dim)) {
    name_exp <- name_exp[-which(name_exp == dat_dim)]
    name_obs <- name_obs[-which(name_obs == dat_dim)]
  }
  name_exp <- name_exp[-which(name_exp == memb_dim)]
  name_obs <- name_obs[-which(name_obs == memb_dim)]
  if (!identical(dim(exp)[name_exp], dim(obs)[name_obs])) {
    stop(paste0("Parameter 'exp' and 'obs' must have same length of ",
                "all the dimensions except 'dat_dim' and 'memb_dim'."))
  }
  ## pval
  if (!is.logical(pval) | length(pval) > 1) {
    stop("Parameter 'pval' must be one logical value.")
  }
  ## sign
  if (!is.logical(sign) | length(sign) > 1) {
    stop("Parameter 'sign' must be one logical value.")
  }
  # alpha
  if (!is.numeric(alpha) | any(alpha < 0) | any(alpha > 1) | length(alpha) > 1) {
    stop("Parameter 'alpha' must be a numeric number between 0 and 1.")
  }
  # na.rm
  if (!na.rm %in% c(TRUE, FALSE)) {
    stop("Parameter 'na.rm' must be TRUE or FALSE")
  }
  ## ncores
  if (!is.null(ncores)) {
    if (!is.numeric(ncores) | ncores %% 1 != 0 | ncores <= 0 |
        length(ncores) > 1) {
      stop("Parameter 'ncores' must be a positive integer.")
    }
  } 
  
  
  ###############################
  # Calculate RatioSDRMS
  
  # If dat_dim = NULL, insert dat dim
  remove_dat_dim <- FALSE
  if (is.null(dat_dim)) {
    dat_dim <- 'dataset'
    exp <- InsertDim(exp, posdim = 1, lendim = 1, name = 'dataset')
    obs <- InsertDim(obs, posdim = 1, lendim = 1, name = 'dataset')
    remove_dat_dim <- TRUE
  }
  
  res <- Apply(list(exp, obs), 
               target_dims = list(c(dat_dim, memb_dim, time_dim), 
                                  c(dat_dim, memb_dim, time_dim)),
               pval = pval,  
               sign = sign,
               alpha = alpha,
               na.rm = na.rm,
               fun = .SprErr, 
               ncores = ncores)
  
  if (remove_dat_dim) {
    if (length(dim(res[[1]])) > 2) {
      res <- lapply(res, Subset, c('nexp', 'nobs'), list(1, 1), drop = 'selected')
    } else {
      res <- lapply(res, as.vector)
    }
  }
  
  return(res)
}

.SprErr <- function(exp, obs, pval = TRUE, sign = FALSE, alpha = 0.05, na.rm = FALSE) {
  
  # exp: [dat_exp, member, sdate]
  # obs: [dat_obs, member, sdate]
  nexp <- dim(exp)[1]
  nobs <- dim(obs)[1]
  
  # ensemble mean
  ens_exp <- MeanDims(exp, 2, na.rm = na.rm) # [dat, sdate]
  ens_obs <- MeanDims(obs, 2, na.rm = na.rm)
  
  # Create empty arrays
  ratio <- array(dim = c(nexp = as.numeric(nexp), nobs = as.numeric(nobs)))  # [nexp, nobs]
  p.val <- array(dim = c(nexp = as.numeric(nexp), nobs = as.numeric(nobs)))  # [nexp, nobs]
  
  for (jexp in 1:nexp) {
    for (jobs in 1:nobs) {
      
      # spread and error
      spread <- sqrt(mean(apply(exp[jexp,,], 2, var, na.rm = na.rm), na.rm = na.rm))
      error <- sqrt(mean((ens_obs - ens_exp[jexp,])^2, na.rm = na.rm))
      ratio[jexp, jobs] <- spread/error
      
      # effective sample size
      enospr <- sum(Eno(apply(exp[jexp,,], 2, var, na.rm = na.rm), names(dim(exp))[3]))
      enodif <- .Eno((ens_exp[jexp, ] - ens_obs[jobs, ])^2, na.action = na.pass)
      if (pval | sign) {
        f_statistic <- (enospr * spread^2 / (enospr - 1)) / (enodif * error^2 / (enodif - 1))
        if (!is.na(f_statistic) & !is.na(enospr) & !is.na(enodif) & any(enospr > 2) & enodif > 2) {
          p.val[jexp, jobs] <- pf(f_statistic, enospr - 1, enodif - 1)
          p.val[jexp, jobs] <- 2 * min(p.val[jexp, jobs], 1 - p.val[jexp, jobs])
        } else {
          p.val[jexp, jobs] <- NA
        }
      }
    }
  }
  
  res <- list(ratio = ratio)
  if (pval) {res$p.val <- p.val}
  if (sign) {res$sign <- p.val <= alpha}
  
  return(res)
}