gevdfit.R 21.4 KB
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# This file contains the functions:
# - gev.d.fit, gev.d.init for fitting
# - gev.d.diag for diagnostic plots
# - gev.d.params, gev.d.rl for calculation of parameters/ return levels
# and the documentation of the example data

#### gev.d.fit ####

#' @title Maximum-likelihood Fitting of the duration dependent GEV Distribution
#' @description Modified \code{\link[ismev]{gev.fit}} function for Maximum-likelihood fitting 
#' for the duration dependent generalized extreme 
#' value distribution, following Koutsoyiannis et al. (1988), including generalized linear 
#' modelling of each parameter.
#' @param xdat A vector containing maxima for different durations. 
#' This can be obtained from \code{\link{IDF.agg}}.
#' @param ds A vector of aggregation levels corresponding to the maxima in xdat.
#' @param ydat A matrix of covariates for generalized linear modelling of the parameters 
#' (or NULL (the default) for stationary fitting). The number of rows should be the same as the 
#' length of xdat.
#' @param  mul,sigl,shl,thetal,etal Numeric vectors of integers, giving the columns of ydat that contain
#'  covariates for generalized linear modelling of the parameters (or NULL (the default) 
#'  if the corresponding parameter is stationary).
#'  Parameters are: modified location, scale_0, shape, duration offset, duration exponent repectively.
#' @param mulink,siglink,shlink,thetalink,etalink Inverse link functions for generalized linear 
#' modelling of the parameters.
#' @param muinit,siginit,shinit,thetainit,etainit initial values as numeric of length 
#' equal to total number of parameters 
#' used to model the parameters. If NULL (the default) is given, initial parameters are obtained 
#' internally by fitting the GEV seperately for each duration and applying a linear model to optain the 
#' duration dependency of the location and shape parameter.
#' @param show Logical; if TRUE (the default), print details of the fit.
#' @param method The optimization method used in \code{\link{optim}}.
#' @param maxit The maximum number of iterations.
#' @param ... Other control parameters for the optimization.
#' @return A list containing the following components. 
#' A subset of these components are printed after the fit. 
#' If show is TRUE, then assuming that successful convergence is indicated, 
#' the components nllh, mle and se are always printed. 
#' \item{nllh}{single numeric giving the negative log-likelihood value.} 
#' \item{mle}{numeric vector giving the MLE's for the modified location, scale_0, shape, 
#' duration offset and duration exponent, resp.} 
#' \item{se}{numeric vector giving the standard errors for the MLE's (in the same order).}
#' \item{trans}{An logical indicator for a non-stationary fit.}
#' \item{model}{A list with components mul, sigl, shl, thetal and etal.}
#' \item{link}{A character vector giving inverse link functions.}
#' \item{conv}{The convergence code, taken from the list returned by \code{\link{optim}}. 
#' A zero indicates successful convergence.}
#' \item{data}{data is standardized to standart Gumbel.} 
#' \item{cov}{The covariance matrix.} 
#' @seealso \code{\link{dgev.d}}, \code{\link{IDF.agg}}, \code{\link{gev.fit}}, \code{\link{optim}}
#' @export
#' @importFrom stats optim 
#' 
#' @examples 
#' # sampled random data from d-gev with covariates
#' # GEV parameters:
#' # mu = 4 + 0.2*cov1 +0.5*cov2
#' # sigma = 2+0.5*cov1
#' # xi = 0.5
#' # theta = 0
#' # eta = 0.5
#' 
#' data('example',package ='IDF')
#' 
#' gev.d.fit(xdat=example$dat,ds = example$d,ydat=as.matrix(example[,c('cov1','cov2')])
#' ,mul=c(1,2),sigl=1)

gev.d.fit<-
  function(xdat, ds, ydat = NULL, mul = NULL, sigl = NULL, shl = NULL, thetal = NULL, etal = NULL, 
           mulink = identity, siglink = identity, shlink = identity, thetalink = identity, etalink = identity,  
           muinit = NULL, siginit = NULL, shinit = NULL, thetainit = NULL, etainit = NULL,
           show = TRUE, method = "Nelder-Mead", maxit = 10000, ...)
  {
    #
    # obtains mles etc for d-gev distn
    #
    
    # test for NA values:
    if(any(is.na(xdat))) stop('xdat contains NA values. NA values need to be removed first.')
    
    z <- list()
    # number of parameters (betas) to estimate for each parameter: 
    npmu <- length(mul) + 1
    npsc <- length(sigl) + 1
    npsh <- length(shl) + 1
    npth <- length(thetal) + 1
    npet <- length(etal) + 1
    z$trans <- FALSE  # indicates if fit is non-stationary
    
    # calculate initial values for mu.d, sigma_0, xi, eta using IDF.init:  (thetainit=0)
    init.vals <- gev.d.init(xdat,ds,ifelse(is.null(thetainit),0,thetainit[1]))
    
    # generate covariates matrices: 
    if (is.null(mul)) {
      mumat <- as.matrix(rep(1, length(xdat)))
      if (is.null(muinit)) 
        muinit <- init.vals$mu
    }else {
      z$trans <- TRUE
      mumat <- cbind(rep(1, length(xdat)), ydat[, mul])
      if (is.null(muinit)) 
        muinit <- c(init.vals$mu, rep(0, length(mul)))
    }
    if (is.null(sigl)) {
      sigmat <- as.matrix(rep(1, length(xdat)))
      if (is.null(siginit)) 
        siginit <- init.vals$sigma
    }else {
      z$trans <- TRUE
      sigmat <- cbind(rep(1, length(xdat)), ydat[, sigl])
      if (is.null(siginit)) 
        siginit <- c(init.vals$sigma, rep(0, length(sigl)))
    }
    if (is.null(shl)) {
      shmat <- as.matrix(rep(1, length(xdat)))
      if (is.null(shinit)) 
        shinit <- init.vals$xi 
    }else {
      z$trans <- TRUE
      shmat <- cbind(rep(1, length(xdat)), ydat[, shl])
      if (is.null(shinit)) 
        shinit <- c(init.vals$xi, rep(0, length(shl)))
    }
    if (is.null(thetal)) {
      thmat <- as.matrix(rep(1, length(xdat)))
      if (is.null(thetainit))  
        thetainit <- 0
    }else {
      z$trans <- TRUE
      thmat <- cbind(rep(1, length(xdat)), ydat[, thetal])
      if (is.null(thetainit))  
        thetainit <- c(0, rep(0, length(thetal)))
    }
    if (is.null(etal)) {
      etmat <- as.matrix(rep(1, length(xdat)))
      if (is.null(etainit)) 
        etainit <- init.vals$eta
    }else {
      z$trans <- TRUE
      etmat <- cbind(rep(1, length(xdat)), ydat[, etal])
      if (is.null(etainit)) 
        etainit <- c(init.vals$eta, rep(0, length(etal)))
    }
    
    z$model <- list(mul, sigl, shl, thetal, etal)
    z$link <- deparse(substitute(c(mulink, siglink, shlink, thetalink, etalink)))
    init <- c(muinit, siginit, shinit, thetainit, etainit)
    
    # function to calculate neg log-likelihood:
    gev.lik <- function(a) {
      # computes neg log lik of d-gev model
      mu <- mulink(mumat %*% (a[1:npmu]))
      sigma <- siglink(sigmat %*% (a[seq(npmu + 1, length = npsc)]))
      xi <- shlink(shmat %*% (a[seq(npmu + npsc + 1, length = npsh)]))
      theta <- thetalink(thmat %*% (a[seq(npmu + npsc + npsh + 1, length = npth)]))
      eta <- etalink(etmat %*% (a[seq(npmu + npsc + npsh + npth + 1, length = npet)]))
      
      ds.t <- ds+theta
      sigma.d <- sigma/(ds.t^eta)
      y <- xdat/sigma.d - mu
      y <- 1 + xi * y
      
      if(any(eta <= 0) ||any(theta <= -0.01) || any(sigma.d <= 0) || any(y <= 0)) return(10^6)
      sum(log(sigma.d)) + sum(y^(-1/xi)) + sum(log(y) * (1/xi + 1))
    }
    
    #####################################################################################
    # derivations of nll after d-gev-parameters (for boosting):
    
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    # get parameters from covariates (and a vector containing predictors)
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    # mu <- mulink(mumat %*% (a[1:npmu]))
    # sigma <- siglink(sigmat %*% (a[seq(npmu + 1, length = npsc)]))
    # xi <- shlink(shmat %*% (a[seq(npmu + npsc + 1, length = npsh)]))
    # theta <- thetalink(thmat %*% (a[seq(npmu + npsc + npsh + 1, length = npth)]))
    # eta <- etalink(etmat %*% (a[seq(npmu + npsc + npsh + npth + 1, length = npet)]))
    # 
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    # xsd <- xdat*(ds+theta)^eta/sigma
    # y <- 1 + xi * (xsd - mu)
    # 
    # nll <- log(sigma.d) + y^(-1/xi) + log(y) * (1/xi + 1)
    # dnll.mu <- y^(-1/xi-1)-(1+xi)/y
    # dnll.sigma <- xsd*y^(-1/xi-1)/sigma -(1+xi)*xsd/sigma/y+1/sigma
    # dnll.xi <- y^(-1/xi)*(log(y)/xi^2-(xsd-mu)/(xi*y))-log(y)/xi^2+(1/xi+1)*(xsd-mu)/y
    # dnll.theta <-  (-eta*xsd*y^(-1/xi-1)+eta*(1+xi)*xsd/y-eta)/(ds+theta)
    # dnll.eta <- (-xsd*y^(-1/xi-1)+(1+xi)*xsd/y-1)*log(ds+theta)
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    #####################################################################################
    
    
    # finding minimum of log-likelihood:
    x <- optim(init, gev.lik, hessian = TRUE, method = method,
               control = list(maxit = maxit, ...))
    
    # saving output parameters:
    z$conv <- x$convergence
    mut <- mulink(mumat %*% (x$par[1:npmu]))
    sc0 <- siglink(sigmat %*% (x$par[seq(npmu + 1, length = npsc)]))
    xi <- shlink(shmat %*% (x$par[seq(npmu + npsc + 1, length = npsh)]))
    theta <- thetalink(thmat %*% (x$par[seq(npmu + npsc + npsh + 1, length = npth)]))
    eta <- etalink(etmat %*% (x$par[seq(npmu + npsc + npsh + npth + 1, length = npet)]))
    z$nllh <- x$value
    # normalize data to standart gumbel:
    sc.d <- sc0/((ds+theta)^eta)
    z$data <-  - log(as.vector((1 + xi * (xdat/sc.d-mut))^(-1/xi))) 
    z$mle <- x$par
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    test <- try(              # catch error 
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    z$cov <- solve(x$hessian) # invert hessian to get estimation on var-covar-matrix
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    ,silent = TRUE )
    if("try-error" %in% class(test)){
      warning("Hessian could not be inverted. NAs were produced.")
      z$cov <- matrix(NA,length(z$mle),length(z$mle))
        }
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    z$se <- sqrt(diag(z$cov)) # sqrt(digonal entries) = standart error of mle's 
    z$vals <- cbind(mut, sc0, xi, theta, eta)
    colnames(z$vals) <- c('mut','sigma0','xi','theta','eta')
    z$ds <- ds
    if(show) {
      if(z$trans) # for nonstationary fit
        print(z[c(2, 3, 4)]) # print model, link, conv
      else print(z[4]) # for stationary fit print only conv
      if(!z$conv) # if fit converged 
        print(z[c(5, 7, 9)]) # print nll, mle, se
    }
    class( z) <- "gev.d.fit"
    invisible(z)
}


#### gev.d.init ####

# function to get initial values for gev.d.fit:
# obtain initial values 
# by fitting every duration seperately

# possible ways to improve:
# take given initial values into accout, if there are any
# xi -> mean vs. median ... how do we improve that?
# mu_tilde -> is not very good for small sample sizes yet
# improved inital value for eta, by fitting both mu~d and sigma~d in log-log scale

#' @title get initial values for gev.d.fit
#' @description obtain initial values by fitting every duration seperately
#' @param xdat vector of maxima for differnt durations
#' @param ds vector of durations belonging to maxima in xdat
#' @param thetainit initial parameter for theta
#' @return list of initail values for mu_tilde, sigma_0, xi, eta
#' @importFrom stats lm 
#' @importFrom ismev gev.fit
#' @keywords internal 

gev.d.init <- function(xdat,ds,thetainit){
  
  durs <- unique(ds)
  mles <- matrix(NA, nrow=length(durs), ncol= 3)
  for(i in 1:length(durs)){
    mles[i,] <- gev.fit(xdat[ds==durs[i]],show = FALSE)$mle
  }
  # get values for sig0 and eta (also mu_0) from linear model in log-log scale
  lmsig <- lm(log(mles[,2])~log(durs+thetainit))
  lmmu <- lm(log(mles[,1])~log(durs+thetainit))
  
  # sig0 <- exp Intercept
  siginit <- exp(lmsig$coefficients[[1]])
  # eta <- mean of negativ slopes 
  etainit <- mean(c(-lmsig$coefficients[[2]],-lmmu$coefficients[[2]]))
  # mean of mu_d/sig_d 
  # could try:
  # mu0/sig0 is also an estimate but needs to be weighted in mean
  muinit <- mean(c(mles[,1]/mles[,2])) #exp(lmmu$coefficients[[1]])/exp(lmsig$coefficients[[1]])
  # mean of shape parameters 
  shinit <- mean(mles[,3])
  
  return(list(mu=muinit,sigma=siginit,xi=shinit,eta=etainit))
}


#### gev.d.diag ####

#' Diagnostic Plots for d-gev Models
#'
#' @description  Produces diagnostic plots for d-gev models using 
#' the output of the function \code{\link{gev.d.fit}}. Values for different durations can be plotted in 
#' different colors of with different symbols.
#' @param fit object returned by \code{\link{gev.d.fit}}
#' @param subset an optional vector specifying a subset of observations to be used in the plot
#' @param cols optional either one value or vector of same length as \code{unique(durations)} to
#' specify the colors of plotting points. 
#' The default uses the \code{rainbow} function.
#' @param pch optional either one value or vector of same length as \code{unique(durations)} containing
#' integers or symbols to specify the plotting points.
#' @param which string containing 'both', 'pp' or 'qq' to specify, which plots should be produced.
#' @param mfrow vector specifying layout of plots. If both plots should be produced seperately,
#' set to \code{c(1,1)}.
#' @param legend logical indicating if legends should be plotted
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#' @param title character vector of length 2, giving the titles for the pp- and the qq-plot
#' @param emp.lab,mod.lab character string containing names for empirical and model axis
#' @param ... additional parameters passed on to the plotting function 
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#'
#' @export
#' @importFrom graphics plot abline par title
#' @importFrom grDevices rainbow
#'
#' @examples
#' data('example',package ='IDF')
#' 
#' fit <- gev.d.fit(xdat=example$dat,ds = example$d,ydat=as.matrix(example[,c('cov1','cov2')])
#'                  ,mul=c(1,2),sigl=1)
#' # diagnostic plots for complete data                
#' gev.d.diag(fit)    
#' # diagnostic plots for subset of data (e.g. one station)            
#' gev.d.diag(fit,subset = example$cov1==1)
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gev.d.diag <- function(fit,subset=NULL,cols=NULL,pch=NULL,which='both',mfrow=c(1,2),legend=TRUE,
                       title=c('Residual Probability Plot','Residual Quantile Plot'),
                       emp.lab='Empirical',mod.lab='Model',...){
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  # check parameter:
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  if(!is.element(which,c('both','pp','qq'))) stop("Parameter 'which'= ",which,
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                                                 " but only 'both','pp' or 'qq' are allowed.")
  # subset data
  df <- data.frame(data=fit$data,ds=fit$ds)
  if(!is.null(subset))df <- df[subset,]
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  # get single durations
  durs <- sort(unique(df$ds))
  # rescale durations to assign colors
  df$cval <- sapply(df$ds,function(d){which(durs==d)})

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  # sort data 
  df <- df[order(df$data),]
  
  # plotting position
  n <- length(df$data)
  px <- (1:n)/(n + 1)

  # create plots:
  if(which=='both') par(mfrow=mfrow) # 2 subplots
  # colors and symbols
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  if(is.null(cols))cols <- rainbow(length(durs))
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  if(is.null(pch))pch <- df$cval
  
  if(which=='both'|which=='pp'){
    # pp
    plot(px, exp( - exp( - df$data)), xlab =
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           emp.lab, ylab = mod.lab,col=cols[df$cval],pch=pch,...)
    abline(0, 1, col = 1,lwd=1)
    title(title[1])
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    if(legend){legend('bottomright',legend = round(durs,digits = 2),pch=pch,
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                      col = cols[1:length(durs)],title = 'Durations',ncol = 2)}
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  }
  if(which=='both'|which=='qq'){
    # qq
    plot( - log( - log(px)), df$data, ylab =
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            emp.lab, xlab = mod.lab,col=cols[df$cval],pch=pch,...)
    abline(0, 1, col = 1,lwd=1)
    title(title[2])
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    if(legend){legend('bottomright',legend = round(durs,digits = 2),pch=pch,
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                      col = cols[1:length(durs)],title = 'Durations',ncol = 2)}
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  }
  if(which=='both') par(mfrow=c(1,1)) # reset par
}

#### gev.d.params ####

#' Calculate gev(d) parameters from \code{gev.d.fit} output
#'
#' @description function to calculate mut, sigma0, xi, theta, eta 
#' (modified location, scale, shape, duration offset, duration exponent) 
#' from results of \code{\link{gev.d.fit}} with covariates
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#' @param fit fit object returned by \code{gev.d.fit} or \code{gev.fit}
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#' @param ydat A matrix containing the covariates in the same order as used in \code{gev.d.fit}.
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#' @seealso \code{\link{dgev.d}}
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#' @return data.frame containing mu_tilde, sigma0, xi, theta, eta (or mu, sigma, xi for gev.fit objects)
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#' @export
#' 
#' @examples
#' data('example',package = 'IDF')
#' fit <- gev.d.fit(example$dat,example$d,ydat = as.matrix(example[,c("cov1","cov2")])
#'                  ,mul = c(1,2),sigl = 1)
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#' gev.d.params(fit = fit,ydat = cbind(c(0.9,1),c(0.5,1)))


gev.d.params <- function(fit,ydat){
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  if(!class(fit)%in%c("gev.d.fit","gev.fit"))stop("'fit' must be an object returned by 'gev.d.fit' or 'gev.fit'.")
  if(!fit$trans){warning('No glm for parameters. Max. likelihood estimates are returned.')
    return(fit$mle)}
  if(!is.matrix(ydat))stop("'ydat' must be of class matrix.")
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  n.par <- max(sapply(fit$model,function(x){return(ifelse(is.null(x),0,max(x)))}))
  if(n.par>ncol(ydat))stop("Covariates-Matrix 'ydat' has ",ncol(ydat), " columns, but ", n.par," are required.")
  
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  #ydat <- rbind(0,ydat) # no error in case ncols=1
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  # number of parameters
  npmu <- length(fit$model[[1]]) + 1
  npsc <- length(fit$model[[2]]) + 1
  npsh <- length(fit$model[[3]]) + 1
  if(class(fit)=="gev.d.fit"){npth <- length(fit$model[[4]]) + 1}
  if(class(fit)=="gev.d.fit"){npet <- length(fit$model[[5]]) + 1}
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  # link functions
  mulink <- eval(parse(text=fit$link))[[1]]
  siglink <- eval(parse(text=fit$link))[[2]]
  shlink <- eval(parse(text=fit$link))[[3]]
  if(class(fit)=="gev.d.fit"){thetalink <- eval(parse(text=fit$link))[[4]]}
  if(class(fit)=="gev.d.fit"){etalink <- eval(parse(text=fit$link))[[5]]}
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  # covariates matrices
  mumat <- cbind(rep(1, dim(ydat)[1]), matrix(ydat[, fit$model[[1]]],dim(ydat)[1],npmu-1))
  sigmat <- cbind(rep(1, dim(ydat)[1]), matrix(ydat[, fit$model[[2]]],dim(ydat)[1],npsc-1))
  shmat <- cbind(rep(1, dim(ydat)[1]), matrix(ydat[, fit$model[[3]]],dim(ydat)[1],npsh-1))
  if(class(fit)=="gev.d.fit"){thmat <- cbind(rep(1, dim(ydat)[1]), matrix(ydat[, fit$model[[4]]],dim(ydat)[1],npth-1))}
  if(class(fit)=="gev.d.fit"){etmat <- cbind(rep(1, dim(ydat)[1]), matrix(ydat[, fit$model[[5]]],dim(ydat)[1],npet-1))}
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  # calculate parameters
  mut <- mulink(mumat %*% (fit$mle[1:npmu]))
  sc0 <- siglink(sigmat %*% (fit$mle[seq(npmu + 1, length = npsc)]))
  xi <- shlink(shmat %*% (fit$mle[seq(npmu + npsc + 1, length = npsh)]))
  if(class(fit)=="gev.d.fit"){theta <- thetalink(thmat %*% (fit$mle[seq(npmu + npsc + npsh + 1, length = npth)]))}
  if(class(fit)=="gev.d.fit"){eta <- etalink(etmat %*% (fit$mle[seq(npmu + npsc + npsh + npth + 1, length = npet)]))}
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  if(class(fit)=="gev.d.fit"){return(data.frame(mut=mut,sig0=sc0,xi=xi,theta=theta,eta=eta))
  }else{return(data.frame(mu=mut,sig=sc0,xi=xi))}
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}


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#### gev.d.rl ####

#' Calculate (spatial) Returnlevel 
#'
#' @description calculate (spatial) Returnlevel for chosen duration and return period 
#' from \code{\link{gev.d.fit}} parameters
#' @param params list of parameters mu_tilde, sigma0, xi, theta, eta (single values and/or compatible matrices)
#' as obtained from \code{\link{gev.d.fit}} or \code{\link{gev.d.params}}
#' @param p.d numeric vector of length 2 containing one value the for exeedance probability p = 1-1/RP 
#' and one value for the duration at which to calculate the return level
#'
#' @return one return level value or matrix with return levels (depending on input to params)
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#' unit: e.g. mm/h 
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#' @export
#' 
#' @examples 
#' data('example',package = 'IDF')
#' fit <- gev.d.fit(example$dat,example$d,ydat = as.matrix(example[,c("cov1","cov2")])
#'                  ,mul = c(1,2),sigl = 1)
#' ### calculate rl on grid:
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#' # covariates values
#' cov1 <- rep(seq(-1,1,0.1),11)
#' cov2 <- rep(seq(0,1,0.1),each=21)
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#' # calculate parameters of d-gev on grid, based on output of gev.d.fit
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#' par <- gev.d.params(fit = fit,ydat = cbind(cov1,cov2))
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#' # calculate 100 year (p=0.99) rl for a duration of d=24 hours
#' rl <- gev.d.rl(params = par,p.d = c(0.99,24))
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#' dim(rl) <- c(21,11)
#' # rl map:
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#' image(x=seq(-1,1,0.1),y=seq(0,1,0.1),z=rl,xlab = 'cov1', ylab = 'cov2')
gev.d.rl <- function(params,p.d){
  sigma.d <- params[[2]]/((p.d[2]+params[[4]])^params[[5]])
  mu <- params[[1]]*sigma.d
  yt <- -1/log(p.d[1])
  rl <- mu+sigma.d/params[[3]]*(yt^params[[3]]-1)
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  return(rl)
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}


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#### gev.d2stdgumbel ####

#' Transform data to standart gumbel
#'
#' @param xdat A vector containing maxima for different durations.
#' @param ds A vector of aggregation levels corresponding to the maxima in xdat.
#' @param params list of parameters mu_tilde, sigma0, xi, theta, eta 
#' as obtained from \code{\link{gev.d.params}}
#'
#' @return Vector containing transformed data.
#' @export
#'
#' @examples 
#' data('example',package = 'IDF')
#' # fit subset
#' ind <- sample(1:10, length(example$dat), replace=TRUE)
#' fit.subs <- gev.d.fit(example$dat[ind!=1],example$d[ind!=1]
#'                       ,ydat = as.matrix(example[ind!=1,c("cov1","cov2")])
#'                       ,mul = c(1,2),sigl = 1)
#' # calculate parameters for unfitted values
#' par <- gev.d.params(fit = fit.subs
#'                       ,ydat = as.matrix(example[ind==1,c("cov1","cov2")]))
#' # transform unfitted values to standart gumbel
#' sg.data <- gev.d2stdgumbel(xdat = example$dat[ind==1]
#'                       ,ds = example$d[ind==1],params = par)
#' # check unfitted values agains standart gumbel
#' gev.d.diag(data.frame(data=sg.data,ds=example$d[ind==1]),pch=20)
gev.d2stdgumbel <- function(xdat,ds,params){
  sc.d <- params$sig0/((ds+params$theta)^params$eta)
  sg.data <-  - log(as.vector((1 + params$xi * (xdat/sc.d-params$mut))^(-1/params$xi))) 
  return(sg.data)
}



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#### example data ####

#' Sampled data for duration dependent GEV
#'
#' A dataset containing:
#' \itemize{
#'   \item \code{$xdat}: 'annual' maxima values
#'   \item \code{$ds}: corresponding durations
#'   \item \code{$cov1}, \code{$cov2}: covariates}
#' GEV parameters:
#' \itemize{
#'   \item mu = 4 + 0.2*cov1 +0.5*cov2
#'   \item sigma = 2+0.5*cov1
#'   \item xi = 0.5
#'   \item theta = 0
#'   \item eta = 0.5}
#'
#' @docType data
#' @keywords datasets
#' @name example
#' @usage data('example',package ='IDF')
NULL