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#' @export
#'
#' @title MLE Fitting of Generalised Pareto Distribution (GPD)
#'
#' @description Maximum likelihood estimation for fitting the GPD with
#' parameters scale \code{sigmau} and shape \code{xi} to the threshold
#' exceedances, conditional on being above a threshold \code{u}. Unconditional
#' likelihood fitting also provided when the probability \code{phiu} of being
#' above the threshold \code{u} is given.
#'
#' @param x vector of sample data
#' @param u scalar threshold
#' @param sigmau scalar scale parameter (positive)
#' @param xi scalar shape parameter
#' @param pvector vector of initial values of GPD parameters (\code{sigmau}, \code{xi}) or \code{NULL}
#' @param phiu probability of being above threshold \eqn{[0, 1]} or \code{NULL}, see Details
#' @param std.err logical, should standard errors be calculated
#' @param method optimisation method (see \code{\link[stats:optim]{optim}})
#' @param control optimisation control list (see \code{\link[stats:optim]{optim}})
#' @param finitelik logical, should log-likelihood return finite value for invalid parameters
#' @param log logical, if \code{TRUE} then log-likelihood rather than likelihood is output
#' @param ... optional inputs passed to \code{\link[stats:optim]{optim}}
#'
#' @details The GPD is fitted to the exceedances of the threshold \code{u} using
#' maximum likelihood estimation. The estimated parameters,
#' variance-covariance matrix and their standard errors are automatically
#' output.
#'
#' The log-likelihood and negative log-likelihood are also provided for wider
#' usage, e.g. constructing your own extreme value mixture model or profile
#' likelihood functions. The
#' parameter vector \code{pvector} must be specified in the negative
#' log-likelihood \code{\link[evmix:fgpd]{nlgpd}}.
#'
#' Log-likelihood calculations are carried out in
#' \code{\link[evmix:fgpd]{lgpd}}, which takes parameters as inputs in the
#' same form as distribution functions. The negative log-likelihood is a
#' wrapper for \code{\link[evmix:fgpd]{lgpd}}, designed towards making it
#' useable for optimisation (e.g. parameters are given a vector as first
#' input).
#'
#' The default value for the tail fraction \code{phiu} in the fitting function
#' \code{\link[evmix:fgpd]{fgpd}} is \code{NULL}, in which case the MLE is calculated
#' using the sample proportion of exceedances. In this case the standard error for \code{phiu} is
#' estimated and output as \code{se.phiu}, otherwise it is set to \code{NA}. Consistent with the
#' \code{\link[evd:fpot]{evd}} library the missing values (\code{NA} and
#' \code{NaN}) are assumed to be below the threshold in calculating the tail fraction.
#'
#' Otherwise, in the fitting function \code{\link[evmix:fgpd]{fgpd}} the tail
#' fraction \code{phiu} can be specified as any value over \eqn{(0, 1]}, i.e.
#' excludes \eqn{\phi_u=0}, leading to the unconditional log-likelihood being
#' used for estimation. In this case the standard error will be output as \code{NA}.
#'
#' In the log-likelihood functions \code{\link[evmix:fgpd]{lgpd}} and
#' \code{\link[evmix:fgpd]{nlgpd}} the tail fraction \code{phiu} cannot be
#' \code{NULL} but can be over the range \eqn{[0, 1]}, i.e. which includes
#' \eqn{\phi_u=0}.
#'
#' The value of \code{phiu} does not effect the GPD parameter estimates, only
#' the value of the likelihood, as:
#'
#' \deqn{L(\sigma_u, \xi; u, \phi_u) = (\phi_u ^ {n_u}) L(\sigma_u, \xi; u,
#' \phi_u=1)}
#'
#' where the GPD has scale \eqn{\sigma_u} and shape \eqn{\xi}, the threshold
#' is \eqn{u} and \eqn{nu} is the number of exceedances. A non-unit value for
#' \code{phiu} simply scales the likelihood and shifts the log-likelihood,
#' thus the GPD parameter estimates are invariant to \code{phiu}.
#'
#' The default optimisation algorithm is "BFGS", which requires a finite
#' negative log-likelihood function evaluation \code{finitelik=TRUE}. For
#' invalid parameters, a zero likelihood is replaced with \code{exp(-1e6)}.
#' The "BFGS" optimisation algorithms require finite values for likelihood, so
#' any user input for \code{finitelik} will be overridden and set to
#' \code{finitelik=TRUE} if either of these optimisation methods is chosen.
#'
#' It will display a warning for non-zero convergence result comes from
#' \code{\link[stats:optim]{optim}} function call.
#'
#' If the hessian is of reduced rank then the variance covariance (from
#' inverse hessian) and standard error of parameters cannot be calculated,
#' then by default \code{std.err=TRUE} and the function will stop. If you want
#' the parameter estimates even if the hessian is of reduced rank (e.g. in a
#' simulation study) then set \code{std.err=FALSE}.
#'
#' @return \code{\link[evmix:fgpd]{lgpd}} gives (log-)likelihood and
#' \code{\link[evmix:fgpd]{nlgpd}} gives the negative log-likelihood.
#' \code{\link[evmix:fgpd]{fgpd}} returns a simple list with the following
#' elements
#'
#' \tabular{ll}{
#' \code{call}: \tab \code{optim} call\cr
#' \code{x}: \tab data vector \code{x}\cr
#' \code{init}: \tab \code{pvector}\cr
#' \code{optim}: \tab complete \code{optim} output\cr
#' \code{mle}: \tab vector of MLE of parameters\cr
#' \code{cov}: \tab variance-covariance matrix of MLE of parameters\cr
#' \code{se}: \tab vector of standard errors of MLE of parameters\cr
#' \code{rate}: \tab \code{phiu} to be consistent with \code{\link[evd:fpot]{evd}}\cr
#' \code{nllh}: \tab minimum negative log-likelihood\cr
#' \code{n}: \tab total sample size\cr
#' \code{u}: \tab threshold\cr
#' \code{sigmau}: \tab MLE of GPD scale\cr
#' \code{xi}: \tab MLE of GPD shape\cr
#' \code{phiu}: \tab MLE of tail fraction\cr
#' \code{se.phiu}: \tab standard error of MLE of tail fraction (parameterised approach using sample proportion)\cr
#' }
#'
#' The output list has some duplicate entries and repeats some of the inputs to both
#' provide similar items to those from \code{\link[evd:fpot]{fpot}} and increase usability.
#'
#' @note Unlike all the distribution functions for the GPD, the MLE fitting only
#' permits single scalar values for each parameter, \code{phiu} and threshold
#' \code{u}.
#'
#' When \code{pvector=NULL} then the initial values are calculated, type
#' \code{fgpd} to see the default formulae used. The GPD fitting is not very
#' sensitive to the initial values, so you will rarely have to give
#' alternatives. Avoid setting the starting value for the shape parameter to
#' \code{xi=0} as depending on the optimisation method it may be get stuck.
#'
#' Default values for the threshold \code{u=0} and tail fraction
#' \code{phiu=NULL} are given in the fitting \code{\link[evmix:fgpd]{fpgd}},
#' in which case the MLE assumes that excesses over the threshold are given,
#' rather than exceedances.
#'
#' The usual default of \code{phiu=1} is given in the likelihood functions
#' \code{\link[evmix:fgpd]{lpgd}} and \code{\link[evmix:fgpd]{nlpgd}}.
#'
#' The \code{\link[evmix:fgpd]{lgpd}} also has the usual defaults for the
#' other parameters, but \code{\link[evmix:fgpd]{nlgpd}} has no defaults.
#'
#' Infinite sample values are dropped in fitting function
#' \code{\link[evmix:fgpd]{fpgd}}, but missing values are used to estimate
#' \code{phiu} as described above. But in likelihood functions
#' \code{\link[evmix:fgpd]{lpgd}} and \code{\link[evmix:fgpd]{nlpgd}} both
#' infinite and missing values are ignored.
#'
#' Error checking of the inputs is carried out and will either stop or give
#' warning message as appropriate.
#'
#' @references
#'
#' \url{http://en.wikipedia.org/wiki/Generalized_Pareto_distribution}
#'
#' Hu Y. and Scarrott, C.J. (2018). evmix: An R Package for Extreme Value Mixture Modeling,
#' Threshold Estimation and Boundary Corrected Kernel Density Estimation. Journal of
#' Statistical Software 84(5), 1-27. doi: 10.18637/jss.v084.i05.
#'
#' @author Yang Hu and Carl Scarrott \email{carl.scarrott@@canterbury.ac.nz}
#'
#' @section Acknowledgments: Based on the \code{\link[ismev:gpd.fit]{gpd.fit}} and
#' \code{\link[evd:fpot]{fpot}} functions in the
#' \code{\link[ismev:gpd.fit]{ismev}} and
#' \code{\link[evd:fpot]{evd}} packages for which their author's contributions are gratefully acknowledged.
#' They are designed to have similar syntax and functionality to simplify the transition for users of these packages.
#'
#' @seealso \code{\link[evd:gpd]{dgpd}}, \code{\link[evd:fpot]{fpot}} and
#' \code{\link[MASS:fitdistr]{fitdistr}}
#'
#' @aliases fgpd lgpd nlgpd
#' @family gpd
#' @family fgpd
#'
#' @examples
#' set.seed(1)
#' par(mfrow = c(2, 1))
#'
#' # GPD is conditional model for threshold exceedances
#' # so tail fraction phiu not relevant when only have exceedances
#' x = rgpd(1000, u = 10, sigmau = 5, xi = 0.2)
#' xx = seq(0, 100, 0.1)
#' hist(x, breaks = 100, freq = FALSE, xlim = c(0, 100))
#' lines(xx, dgpd(xx, u = 10, sigmau = 5, xi = 0.2))
#' fit = fgpd(x, u = 10)
#' lines(xx, dgpd(xx, u = fit$u, sigmau = fit$sigmau, xi = fit$xi), col="red")
#'
#' # but tail fraction phiu is needed for conditional modelling of population tail
#' x = rnorm(10000)
#' xx = seq(-4, 4, 0.01)
#' hist(x, breaks = 200, freq = FALSE, xlim = c(0, 4))
#' lines(xx, dnorm(xx), lwd = 2)
#' fit = fgpd(x, u = 1)
#' lines(xx, dgpd(xx, u = fit$u, sigmau = fit$sigmau, xi = fit$xi, phiu = fit$phiu),
#' col = "red", lwd = 2)
#' legend("topright", c("True Density","Fitted Density"), col=c("black", "red"), lty = 1)
#'
# maximum likelihood fitting for GPD
fgpd <- function(x, u = 0, phiu = NULL, pvector = NULL, std.err = TRUE,
method = "BFGS", control = list(maxit = 10000), finitelik = TRUE, ...) {
call <- match.call()
np = 2 # maximum number of parameters
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.param(u)
check.prob(phiu, allownull = TRUE) # don't use check.phiu as TRUE only valid for mixture models
check.nparam(pvector, nparam = np, allownull = TRUE)
check.logic(std.err)
check.optim(method)
check.control(control)
check.logic(finitelik)
if (any(is.infinite(x))) warning("infinite cases have been removed")
x = x[!is.infinite(x)] # ignore infinite cases only (all mixture models also ignore missing)
if (any(is.na(x)))
warning("missing values are treated as below threshold when estimating tail fraction")
check.quant(x, allowna = TRUE)
n = length(x)
if ((method == "L-BFGS-B") | (method == "BFGS")) finitelik = TRUE
# assume NA or NaN are below threshold consistent with evd library
# hence use which() to ignore these
xu = x[which(x > u)]
nu = length(xu)
if (nu < 1)
stop("no elements of x are above threshold")
# set default values if pvector is NULL
if (is.null(pvector)) {
yu = xu - u
pvector[1] = sqrt(6 * var(yu)) / pi
pvector[2] = 0.1
}
if (is.null(phiu)) {
phiu = nu / n
se.phiu = sqrt(phiu * (1 - phiu) / n)
} else {
if (phiu == 0) stop("tail probability must be in (0, 1]")
se.phiu = NA
}
# check initial parameter vector and try different shape
nllh = nlgpd(pvector, xu, u, phiu)
if (is.infinite(nllh)) {
pvector[2] = 0.1
nllh = nlgpd(pvector, xu, u, phiu)
}
if (is.infinite(nllh)) stop("initial parameter values are invalid")
fit = optim(par = as.vector(pvector), fn = nlgpd, x = xu, u = u, phiu = phiu,
finitelik = finitelik, control = control, method = method, hessian = TRUE, ...)
conv = TRUE
if ((fit$convergence != 0) | any(fit$par == pvector) | (abs(fit$value) >= 1e6)) {
conv = FALSE
warning("check convergence")
}
if (conv & std.err) {
qrhess = qr(fit$hessian)
if (qrhess$rank != ncol(qrhess$qr)) {
warning("observed information matrix is singular")
se = NULL
invhess = NULL
} else {
invhess = solve(qrhess)
vars = diag(invhess)
if (any(vars <= 0)) {
warning("observed information matrix is singular")
invhess = NULL
se = NULL
} else {
se = sqrt(vars)
}
}
} else {
invhess = NULL
se = NULL
}
list(call = call, x = as.vector(x), init = as.vector(pvector), optim = fit,
conv = conv, cov = invhess, mle = fit$par, se = se, rate = phiu, nllh = fit$value,
n = n, u = u, sigmau = fit$par[1], xi = fit$par[2], phiu = phiu, se.phiu = se.phiu)
}
#' @export
#' @aliases fgpd lgpd nlgpd
#' @rdname fgpd
# log-likelihood function for GPD
lgpd <- function(x, u = 0, sigmau = 1, xi = 0, phiu = 1, log = TRUE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.param(u)
check.param(sigmau)
check.param(xi)
check.prob(phiu) # don't use check.phiu as TRUE only valid for mixture models
check.logic(log)
check.inputn(c(length(u), length(sigmau), length(xi), length(phiu)), allowscalar = TRUE)
if (any(!is.finite(x))) {
warning("non-finite cases have been removed")
x[!is.finite(x)] = NA # ignore missing and infinite cases
}
# assume NA or NaN are below threshold consistent with evd library
# hence use which() to ignore these
xu = x[which(x > u)]
nu = length(xu)
yu = (xu - u) / sigmau # used when shape is zero
syu = 1 + xi * yu # used when shape non-zero
if ((min(syu) <= 0) | (sigmau <= 0) | (phiu <= 0) | (phiu > 1)) { # phiu = 1 is conditional likelihood
l = -Inf
} else {
if (abs(xi) < 1e-6) {
l = - nu * log(sigmau) - sum(yu) + nu * log(phiu)
} else {
l = - nu * log(sigmau) - (1/xi + 1) * sum(log(syu)) + nu * log(phiu)
}
}
if (!log) l = exp(l)
l
}
#' @export
#' @aliases fgpd lgpd nlgpd
#' @rdname fgpd
# negative log-likelihood function for GPD
# (wrapper for likelihood, inputs and checks designed for optimisation)
nlgpd <- function(pvector, x, u = 0, phiu = 1, finitelik = FALSE) {
np = 2 # maximum number of parameters
# Check properties of inputs
check.nparam(pvector, nparam = np) # must be provided
check.param(u)
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.prob(phiu) # don't use check.phiu as TRUE only valid for mixture models
check.logic(finitelik)
sigmau = pvector[1]
xi = pvector[2]
nllh = -lgpd(x, u, sigmau, xi, phiu)
if (finitelik & is.infinite(nllh)) {
nllh = sign(nllh) * 1e6
}
nllh
}
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