# Taken from the now archived R package POT
#Descripton of the last available version:
#Package: POT
#Version: 1.1-3
#Date: 2012-10-30
#Title: Generalized Pareto Distribution and Peaks Over Threshold
#Author: Mathieu Ribatet <mathieu.ribatet@math.univ-montp2.fr>
# Maintainer: Mathieu Ribatet <mathieu.ribatet@math.univ-montp2.fr>
# Depends: R (>= 1.8.0)
#Description: Some functions useful to perform a Peak Over Threshold
#analysis in univariate and bivariate cases. A user's guide is
# available.
# License: GPL (>= 2)
# URL: http://r-forge.r-project.org/projects/pot/
# Repository: CRAN
# Repository/R-Forge/Project: pot
# Repository/R-Forge/Revision: 492
# Repository/R-Forge/DateTimeStamp: 2012-10-30 14:21:03
# Date/Publication: 2012-11-06 09:49:26
# Packaged: 2012-10-30 15:22:42 UTC; rforge
## In this file, several functions to estimates the GPD parameters
## are available:
## 1) Moments Estimator
## 2) Unbiased Probability Weighted Moment (PWMU) Estimator
## 3) Biased Probability Weighted Moment (PWMB) Estimator
## 4) Maximum Likelihood Estimator
## 5) Pickands' Estimator
## 6) Minimum Density Power Divergence Estimator
## 7) Method of Medians Estimator
## 8) Likelihood Moment Estimator
## 9) Maximum Goodness-of-Fit Estimator
## A generic function for estimate the GPD parameters
fitgpd <- function(data, threshold, est = "mle", ...){
threshold.call <- deparse(threshold)
if (!(est %in% c("moments", "pwmb", "pwmu", "mle", "pickands",
"mdpd", "med", "lme", "mgf", "mple")))
stop("Unknown estimator. Please check the ``est'' argument.")
fitted <- switch(est, 'moments' = gpdmoments(data, threshold),
'pwmb' = gpdpwmb(data, threshold, ...),
'pwmu' = gpdpwmu(data, threshold),
'mle' = gpdmle(data, threshold, ...),
'pickands' = gpdpickands(data, threshold),
'mdpd' = gpdmdpd(data, threshold, ...),
'med' = gpdmed(data, threshold, ...),
'lme' = gpdlme(data, threshold, ...),
'mgf' = gpdmgf(data, threshold, ...),
'mple' = gpdmple(data, threshold, ...)
)
fitted$threshold.call <- threshold.call
class(fitted) <- c("uvpot","pot")
return(fitted)
}
##Maximum penalized likelihood estimator
gpdmple <- function(x, threshold, start, ..., std.err.type =
"observed", corr = FALSE, method = "BFGS",
warn.inf = TRUE, alpha = 1, lambda = 1){
if (all(c("observed", "expected", "none") != std.err.type))
stop("``std.err.type'' must be one of 'observed', 'expected' or 'none'")
nlpot <- function(scale, shape) {
ans <- -.C("gpdlik", exceed, nat, threshold, scale,
shape, dns = double(1), PACKAGE = "simPop")$dns
if ((shape > 0) & (shape <1))
ans <- lambda * ((1 / (1 - shape) - 1)^alpha) + ans
if (shape >= 1)
ans <- 1e6
return(ans)
}
nn <- length(x)
threshold <- rep(threshold, length.out = nn)
high <- (x > threshold) & !is.na(x)
threshold <- as.double(threshold[high])
exceed <- as.double(x[high])
nat <- length(exceed)
if(!nat) stop("no data above threshold")
pat <- nat/nn
param <- c("scale", "shape")
if(missing(start)) {
start <- list(scale = 0, shape = 0)
start$scale <- mean(exceed) - min(threshold)
start <- start[!(param %in% names(list(...)))]
}
if(!is.list(start))
stop("`start' must be a named list")
if(!length(start))
stop("there are no parameters left to maximize over")
nm <- names(start)
l <- length(nm)
f <- formals(nlpot)
names(f) <- param
m <- match(nm, param)
if(any(is.na(m)))
stop("`start' specifies unknown arguments")
formals(nlpot) <- c(f[m], f[-m])
nllh <- function(p, ...) nlpot(p, ...)
if(l > 1)
body(nllh) <- parse(text = paste("nlpot(", paste("p[",1:l,
"]", collapse = ", "), ", ...)"))
fixed.param <- list(...)[names(list(...)) %in% param]
if(any(!(param %in% c(nm,names(fixed.param)))))
stop("unspecified parameters")
start.arg <- c(list(p = unlist(start)), fixed.param)
if( warn.inf && do.call("nllh", start.arg) == 1e6 )
warning("negative log-likelihood is infinite at starting values")
opt <- optim(start, nllh, hessian = TRUE, ..., method = method)
if ((opt$convergence != 0) || (opt$value == 1e6)) {
warning("optimization may not have succeeded")
if(opt$convergence == 1) opt$convergence <- "iteration limit reached"
}
else opt$convergence <- "successful"
if (std.err.type != "none"){
tol <- .Machine$double.eps^0.5
if(std.err.type == "observed") {
var.cov <- qr(opt$hessian, tol = tol)
if(var.cov$rank != ncol(var.cov$qr)){
warning("observed information matrix is singular; passing std.err.type to ``expected''")
obs.fish <- FALSE
return()
}
if (std.err.type == "observed"){
var.cov <- solve(var.cov, tol = tol)
std.err <- diag(var.cov)
if(any(std.err <= 0)){
warning("observed information matrix is singular; passing std.err.type to ``expected''")
std.err.type <- "expected"
return()
}
std.err <- sqrt(std.err)
if(corr) {
.mat <- diag(1/std.err, nrow = length(std.err))
corr.mat <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm))
diag(corr.mat) <- rep(1, length(std.err))
}
else {
corr.mat <- NULL
}
}
}
if (std.err.type == "expected"){
shape <- opt$par[2]
scale <- opt$par[1]
a22 <- 2/((1+shape)*(1+2*shape))
a12 <- 1/(scale*(1+shape)*(1+2*shape))
a11 <- 1/((scale^2)*(1+2*shape))
##Expected Matix of Information of Fisher
expFisher <- nat * matrix(c(a11,a12,a12,a22),nrow=2)
expFisher <- qr(expFisher, tol = tol)
var.cov <- solve(expFisher, tol = tol)
std.err <- sqrt(diag(var.cov))
if(corr) {
.mat <- diag(1/std.err, nrow = length(std.err))
corr.mat <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm))
diag(corr.mat) <- rep(1, length(std.err))
}
else
corr.mat <- NULL
}
colnames(var.cov) <- nm
rownames(var.cov) <- nm
names(std.err) <- nm
}
else{
std.err <- std.err.type <- corr.mat <- NULL
var.cov <- NULL
}
param <- c(opt$par, unlist(fixed.param))
scale <- param["scale"]
var.thresh <- !all(threshold == threshold[1])
if (!var.thresh)
threshold <- threshold[1]
list(fitted.values = opt$par, std.err = std.err, std.err.type = std.err.type,
var.cov = var.cov, fixed = unlist(fixed.param), param = param,
deviance = 2*opt$value, corr = corr.mat, convergence = opt$convergence,
counts = opt$counts, message = opt$message, threshold = threshold,
nat = nat, pat = pat, data = x, exceed = exceed, scale = scale,
var.thresh = var.thresh, est = "MPLE", logLik = -opt$value,
opt.value = opt$value)
}
## Maximum goodness-of-fit estimator
gpdmgf <- function(x, threshold, start, stat, ...,
method = "BFGS", warn.inf = TRUE){
nn <- length(x)
high <- (x > threshold) & !is.na(x)
exceed <- as.double(x[high])
nat <- length(exceed)
if (!nat)
stop("no data above threshold")
if (!(stat %in% c("KS","CM","AD","ADR","ADL","AD2R","AD2L",
"AD2")))
stop("`stat' must be one of 'KS','CM','AD','ADR','ADL','AD2R','AD2L', 'AD2'.")
pat <- nat/nn
param <- c("scale", "shape")
excess <- exceed - threshold
excess <- sort(excess)
if(missing(start)) {
start <- list(scale = 0, shape = 0)
start$scale <- mean(exceed) - min(threshold)
start <- start[!(param %in% names(list(...)))]
}
if(!is.list(start))
stop("`start' must be a named list")
if(!length(start))
stop("there are no parameters left to maximize over")
if (stat == "KS")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else
1 / 2 / nat + max(abs(pgpd(excess, 0, scale, shape) -
ppoints(nat)))
}
if (stat == "CM")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else
1/12/nat + sum((pgpd(excess, 0, scale, shape) -
ppoints(nat))^2)
}
if (stat == "AD")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if ((pgpd(max(excess), 0, scale, shape) >= 1) ||
(pgpd(min(excess), 0, scale, shape) <= 0))
1e6
else
-nat - mean((2*1:nat - 1) *
(log(pgpd(excess, 0, scale, shape)) +
log(1 - pgpd(rev(excess), 0, scale, shape))))
}
}
if (stat == "ADR")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if (pgpd(max(excess), 0, scale, shape) >= 1)
1e6
else
nat / 2 - 2 * sum(pgpd(excess, 0, scale, shape)) -
mean((2 * 1:nat -1)*log(1 - pgpd(rev(excess), 0, scale, shape)))
}
}
if (stat == "ADL")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if (pgpd(min(excess), 0, scale, shape) <= 0)
1e6
else
- 3 * nat / 2 + 2 * sum(pgpd(excess, 0, scale, shape)) -
mean((2 * 1:nat -1)*log(pgpd(excess, 0, scale, shape)))
}
}
if (stat == "AD2R")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if (pgpd(max(excess), 0, scale, shape) >= 1)
1e6
else
2 * sum(log(1 - pgpd(excess, 0, scale, shape))) +
mean((2* 1:nat - 1) / (1 - pgpd(rev(excess), 0, scale, shape)))
}
}
if (stat == "AD2L")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if (pgpd(min(excess), 0, scale, shape) <= 0)
1e6
else
2 * sum(log(pgpd(excess, 0, scale, shape))) +
mean((2 * 1:nat - 1) / pgpd(excess, 0, scale, shape))
}
}
if (stat == "AD2")
fun <- function(scale, shape){
if (scale <= 0)
1e6
else{
if ((pgpd(max(excess), 0, scale, shape) >= 1) ||
(pgpd(min(excess), 0, scale, shape) <= 0))
1e6
else
2 * sum(log(pgpd(excess, 0, scale, shape)) +
log(1 - pgpd(excess, 0, scale, shape))) +
mean((2 * 1:nat - 1) / pgpd(excess, 0, scale, shape) +
(2 * 1:nat - 1) /
(1 - pgpd(rev(excess), 0, scale, shape)))
}
}
nm <- names(start)
l <- length(nm)
f <- formals(fun)
names(f) <- param
m <- match(nm, param)
if(any(is.na(m)))
stop("`start' specifies unknown arguments")
formals(fun) <- c(f[m], f[-m])
mgf <- function(p, ...) fun(p, ...)
if(l > 1)
body(mgf) <- parse(text = paste("fun(", paste("p[",1:l,
"]", collapse = ", "), ", ...)"))
fixed.param <- list(...)[names(list(...)) %in% param]
if(any(!(param %in% c(nm,names(fixed.param)))))
stop("unspecified parameters")
start.arg <- c(list(p = unlist(start)), fixed.param)
if( warn.inf && do.call("mgf", start.arg) == 1e6 )
warning("Maximum goodness-of-fit function is infinite at starting values")
opt <- optim(start, mgf, hessian = TRUE, ..., method = method)
if ((opt$convergence != 0) || (opt$value == 1e6)) {
warning("optimization may not have succeeded")
if(opt$convergence == 1) opt$convergence <- "iteration limit reached"
}
else opt$convergence <- "successful"
tol <- .Machine$double.eps^0.5
param <- c(opt$par, unlist(fixed.param))
scale <- param["scale"]
var.thresh <- !all(threshold == threshold[1])
if (!var.thresh)
threshold <- threshold[1]
std.err <- std.err.type <- corr.mat <- NULL
var.cov <- NULL
list(fitted.values = opt$par, std.err = std.err, std.err.type = std.err.type,
var.cov = var.cov, fixed = unlist(fixed.param), param = param,
corr = corr.mat, convergence = opt$convergence, counts = opt$counts,
message = opt$message, threshold = threshold, nat = nat, pat = pat,
data = x, exceed = exceed, scale = scale, var.thresh = var.thresh,
est = "MGF", opt.value = opt$value, stat = stat)
}
##Likelihood moment estimation
gpdlme <- function(x, threshold, r = -.5, start, ...,
method = "BFGS"){
nn <- length(x)
high <- (x > threshold) & !is.na(x)
exceed <- as.double(x[high])
nat <- length(exceed)
if (!nat)
stop("no data above threshold")
pat <- nat/nn
excess <- exceed - threshold
fun <- function(x){
if (x >= 1/max(excess))
return(1e6)
p <- r / mean(log(1 - x * excess))
abs(mean((1 - x * excess)^p) - 1 / (1 - r))
}
if (missing(start))
start <- list(x = -1)
opt <- optim(start, fun, hessian = TRUE, ..., method = method)
if (opt$convergence != 0){
warning("optimization may not have succeeded")
if(opt$convergence == 1) opt$convergence <- "iteration limit reached"
}
else opt$convergence <- "successful"
counts <- opt$counts
b <- opt$par
zero <- opt$value
shape <- mean(log(1 - b*excess))
scale <- - shape / b
param <- c(scale, shape)
names(param) <- c("scale", "shape")
a11 <- scale^2 * (2 + ((r - shape)^2 + 2 * shape) /
(1 - 2 * r))
a12 <- scale * (1 + (r^2 + shape^2 + shape) /
(1 - 2 * r))
a22 <- (1 - r) * (1 + (2*shape^2 - 2 * shape + r) /
(1 - 2 * r))
var.cov <- matrix(c(a11, a12, a12, a22), 2) / nat
colnames(var.cov) <- c("scale", "shape")
rownames(var.cov) <- c("scale", "shape")
std.err <- sqrt(diag(var.cov))
.mat <- diag(1/std.err, nrow = length(std.err))
corr <- structure(.mat %*% var.cov %*% .mat)
diag(corr) <- rep(1, length(std.err))
colnames(corr) <- c("scale", "shape")
rownames(corr) <- c("scale", "shape")
if (shape < -0.5)
message <- "Assymptotic theory assumptions\nfor standard error may not be fullfilled !"
else message <- NULL
var.thresh <- FALSE
return(list(fitted.values = param, std.err = std.err, std.err.type = "expected",
var.cov = var.cov, param = param, message = message, data = x,
threshold = threshold, corr = corr, convergence = opt$convergence,
counts = counts, nat = nat, pat = pat, exceed = exceed, scale = scale,
var.thresh = var.thresh, est = "LME", opt.value = opt$value))
}
##Pickand's Estimator
gpdpickands <- function(data, threshold){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'pickands'. Taking only the first value !!!")
threshold <- threshold[1]
}
exceed <- data[data>threshold]
nat <- length( exceed )
pat <- nat / length( data )
excess <- sort(exceed - threshold)
n <- length(excess)
xn.2 <- excess[ceiling(n/2)]
x3n.4 <- excess[ceiling(.75*n)]
d <- xn.2^2 / (2 * xn.2 - x3n.4)
shape <- -log(xn.2 / (x3n.4 - xn.2) ) / log(2)
scale <- -shape * d
if ( (max(excess) * shape) > -scale)
message <- "Estimates are valid"
else
message <- "Estimates are not valid"
estim <- param <- c(scale = scale, shape = shape)
std.err <- var.cov <- corr <- NULL
convergence <- counts <- NA
var.thresh <- FALSE
return(list(fitted.values = estim, std.err = std.err, var.cov = var.cov,
param = param, message = message, threshold = threshold,
nat = nat, pat = pat, convergence = convergence,
corr = corr, counts = counts, exceed = exceed,
scale = scale, var.thresh = var.thresh, est = "pickands"))
}
## Moments Estimator
gpdmoments <- function(data, threshold){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'moments'. Taking only the first value !!!")
threshold <- threshold[1]
}
exceed <- data[data>threshold]
nat <- length( exceed )
pat <- nat / length( data )
if ( nat == 0 )
stop("None observation above the specified threshold !!!")
exceed <- sort(exceed)
loc <- threshold
## Evaluate the excess above the threshold
exces <- exceed - loc
m <- mean(exces)
v <- var(exces)
scale <- m / 2 * ( m^2 / v +1 )
shape <- - ( m^2 / v -1 ) / 2
estim <- param <- c(scale = scale, shape = shape)
convergence <- counts <- NA
a11 <- 2*scale^2 * ( 1 - 6*shape + 12*shape^2)
a12 <- - scale * (1-2*shape) * (1-4*shape+12*shape^2)
a21 <- a12
a22 <- (1-2*shape)^2 * (1-shape+6*shape^2)
var.cov <- (1 - shape)^2 / ( (1-2*shape)*(1-3*shape)*(1-4*shape)*nat ) *
matrix(c(a11,a21,a12,a22),2)
colnames(var.cov) <- c('scale','shape')
rownames(var.cov) <- c('scale','shape')
std.err <- sqrt( diag(var.cov) )
.mat <- diag(1/std.err, nrow = length(std.err))
corr <- structure(.mat %*% var.cov %*% .mat)
diag(corr) <- rep(1, length(std.err))
colnames(corr) <- c('scale','shape')
rownames(corr) <- c('scale','shape')
if ( shape > 0.25 ) message <- 'Assymptotic theory assumptions
for standard error may not be fullfilled !'
else message <- NULL
var.thresh <- FALSE
return(list(fitted.values = estim, std.err = std.err, var.cov = var.cov,
param = param, message = message, threshold = threshold,
nat = nat, pat = pat, convergence = convergence,
corr= corr, counts = counts, exceed = exceed,
scale=scale, var.thresh = var.thresh, est = "moments"))
}
##PWMB Estimator
gpdpwmb <- function(data, threshold, a=0.35, b=0, hybrid = FALSE){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'pwmb'. Taking only the first value !!!")
threshold <- threshold[1]
}
exceed <- data[data>threshold]
nat <- length( exceed )
pat <- nat / length( data )
if ( nat == 0 )
stop("None observation above the specified threshold !!!")
exceed <- sort(exceed)
loc <- threshold
excess <- exceed - loc
m <- mean(excess)
n <- length(excess)
p <- (1:n - a) / (n + b)
t <- sum((1-p)*excess)/n
shape <- - m / (m- 2*t ) + 2
scale <- 2 * m * t / (m - 2*t )
est <- 'PWMB'
if (hybrid)
if ( (max(excess) >= (-scale / shape)) & (shape < 0) ){
shape <- -scale / max(excess)
est <- 'PWMB Hybrid'
}
estim <- c(scale = scale, shape = shape)
param <- c(scale = scale, shape = shape)
convergence <- NA
counts <- NA
a11 <- scale^2 * (7-18*shape+11*shape^2-2*shape^3)
a12 <- - scale * (2-shape) * (2-6*shape+7*shape^2-2*shape^3)
a21 <- a12
a22 <- (1-shape) * (2 -shape)^2 * (1-shape+2*shape^2)
var.cov <- 1 / ( (1-2*shape) * (3-2*shape)*nat ) *
matrix(c(a11,a21,a12,a22),2)
colnames(var.cov) <- c('scale','shape')
rownames(var.cov) <- c('scale','shape')
std.err <- sqrt( diag(var.cov) )
.mat <- diag(1/std.err, nrow = length(std.err))
corr <- structure(.mat %*% var.cov %*% .mat)
diag(corr) <- rep(1, length(std.err))
colnames(corr) <- c('scale','shape')
rownames(corr) <- c('scale','shape')
if ( shape > 0.5 )
message <- "Assymptotic theory assumptions for standard error may not be fullfilled !"
else message <- NULL
var.thresh <- FALSE
return(list(fitted.values = estim, std.err = std.err, var.cov = var.cov,
param = param, message = message, threshold = threshold,
corr = corr, convergence = convergence, counts = counts,
nat = nat, pat = pat, exceed = exceed,
scale=scale, var.thresh = var.thresh, est = est))
}
## PWMU Estimator
## First, we need a function which computes the samples L-moments
samlmu <- function (x, nmom = 4, sort.data = TRUE)
{
xok <- x[!is.na(x)]
n <- length(xok)
if (nmom <= 0) return(numeric(0))
if (nmom <= 2) rnames <- paste("l", 1:nmom, sep = "_")
else rnames <- c("l_1", "l_2", paste("t", 3:nmom, sep = "_"))
lmom <- rep(NA, nmom)
names(lmom) <- rnames
if (n == 0) return(lmom)
if (sort.data == TRUE) xok <- sort(xok)
nmom.actual <- min(nmom, n)
lmom <- .C("samlmu", as.double(xok), as.integer(nmom.actual),
as.integer(n), lmom = double(nmom.actual),
PACKAGE = "simPop")$lmom
names(lmom) <- rnames
return(lmom)
}
gpdpwmu <- function(data,threshold, hybrid = FALSE){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'pwmu'. Taking only the first value !!!")
threshold <- threshold[1]
}
exceed <- data[data>threshold]
if ( length(exceed) == 0 )
stop("None observation above the specified threshold !!!")
exceed <- sort(exceed)
nat <- length( exceed )
pat <- nat / length( data )
loc <- threshold
excess <- exceed - loc
lmoments <- samlmu(excess, nmom=2, sort.data = FALSE)
shape <- - lmoments[1]/lmoments[2] + 2
scale <- (1 - shape)*lmoments[1]
names(shape) <- NULL
names(scale) <- NULL
est <- "PWMU"
if (hybrid)
if ( (excess[nat] >= (-scale / shape)) & (shape < 0) ){
shape <- -scale / excess[nat]
est <- 'PWMU Hybrid'
}
estim <- param <- c(scale = scale, shape = shape)
convergence <- counts <- NA
a11 <- scale^2 * (7-18*shape+11*shape^2-2*shape^3)
a12 <- - scale * (2-shape) * (2-6*shape+7*shape^2-2*shape^3)
a21 <- a12
a22 <- (1-shape) * (2 -shape)^2 * (1-shape+2*shape^2)
var.cov <- 1 / ( (1-2*shape) * (3-2*shape)*nat ) * matrix(c(a11,a21,a12,a22),2)
colnames(var.cov) <- c('scale','shape')
rownames(var.cov) <- c('scale','shape')
std.err <- sqrt( diag(var.cov) )
.mat <- diag(1/std.err, nrow = length(std.err))
corr <- structure(.mat %*% var.cov %*% .mat)
diag(corr) <- rep(1, length(std.err))
colnames(corr) <- c('scale','shape')
rownames(corr) <- c('scale','shape')
if ( shape > 0.5 ) message <- "Assymptotic theory assumptions
for standard error may not be fullfilled !"
else message <- NULL
var.thresh <- FALSE
return(list(fitted.values = estim, std.err = std.err, var.cov = var.cov,
param = param, message = message, threshold = threshold,
corr = corr, convergence = convergence, counts = counts,
nat = nat, pat = pat, exceed = exceed,
scale=scale, var.thresh = var.thresh, est = est))
}
##MDPD estimators for the GPD.
gpdmdpd <- function(x, threshold, a, start, ...,
method = "BFGS", warn.inf = TRUE){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'mdpd'. Taking only the first value !!!")
threshold <- threshold[1]
}
if (missing(a))
a <- .1
nn <- length(x)
threshold <- rep(threshold, length.out = nn)
high <- (x > threshold) & !is.na(x)
threshold <- as.double(threshold[high])
exceed <- as.double(x[high])
nat <- length(exceed)
excess <- exceed - threshold
if(!nat) stop("no data above threshold")
pat <- nat/nn
if(missing(start)) {
start <- list(scale = 0, shape = 0.01)
start$scale <- mean(exceed) - min(threshold)
}
start <- c(scale = start$scale, shape = start$shape)
pddf <- function(param){
## Evaluates the (P)ower (D)ensity (D)ivergence (F)unction which is
## criterion function of the MDPDE
scale <- param[1]
shape <- param[2]
if ( ((max(excess) < (-scale / shape)) && (shape < 0)) ||
(shape > 0) ){
y <- pmax(0, 1 + shape * excess / scale)^
((-1/shape - 1) * a)
c1 <- 1 / (scale^a * (1 + a + a * shape))
c2 <- (1 + 1/a ) / scale^a
div <- c1 - c2 * mean(y)
}
else
div <- 1e6
return(div)
}
opt <- optim(start, pddf, hessian = TRUE, ..., method = method)
if ((opt$convergence != 0) || (opt$value == 1e6)) {
warning("optimization may not have succeeded")
if(opt$convergence == 1) opt$convergence <- "iteration limit reached"
}
else opt$convergence <- "successful"
shape <- opt$par[2]
scale <- opt$par[1]
param <- c(scale, shape)
names(param) <- c("scale", "shape")
std.err <- std.err.type <- var.cov <- corr <- NULL
var.thresh <- FALSE
list(fitted.values = opt$par, std.err = std.err, std.err.type = std.err.type,
var.cov = var.cov, fixed = NULL, param = param,
deviance = NULL, corr = corr, convergence = opt$convergence,
counts = opt$counts, message = opt$message, threshold = threshold,
nat = nat, pat = pat, data = x, exceed = exceed,
scale = scale, var.thresh = var.thresh, est = "MDPD",
opt.value = opt$value)
}
## This function comes from the evd package. The gpdmle function
## corresponds to the fpot function. Nevertheless, it was sligthly modified
## to simplify it. So, this function is a ligther version of fpot.
## So, I'm very gratefull to Alec Stephenson.
gpdmle <- function(x, threshold, start, ...,
std.err.type = "observed", corr = FALSE,
method = "BFGS", warn.inf = TRUE){
if (all(c("observed", "expected", "none") != std.err.type))
stop("``std.err.type'' must be one of 'observed', 'expected' or 'none'")
nlpot <- function(scale, shape) {
-.C("gpdlik", exceed, nat, threshold, scale,
shape, dns = double(1), PACKAGE = "simPop")$dns
}
nn <- length(x)
threshold <- rep(threshold, length.out = nn)
high <- (x > threshold) & !is.na(x)
threshold <- as.double(threshold[high])
exceed <- as.double(x[high])
nat <- length(exceed)
if(!nat) stop("no data above threshold")
pat <- nat/nn
param <- c("scale", "shape")
if(missing(start)) {
start <- list(scale = 0, shape = 0)
start$scale <- mean(exceed) - min(threshold)
start <- start[!(param %in% names(list(...)))]
}
if(!is.list(start))
stop("`start' must be a named list")
if(!length(start))
stop("there are no parameters left to maximize over")
nm <- names(start)
l <- length(nm)
f <- formals(nlpot)
names(f) <- param
m <- match(nm, param)
if(any(is.na(m)))
stop("`start' specifies unknown arguments")
formals(nlpot) <- c(f[m], f[-m])
nllh <- function(p, ...) nlpot(p, ...)
if(l > 1)
body(nllh) <- parse(text = paste("nlpot(", paste("p[",1:l,
"]", collapse = ", "), ", ...)"))
fixed.param <- list(...)[names(list(...)) %in% param]
if(any(!(param %in% c(nm,names(fixed.param)))))
stop("unspecified parameters")
start.arg <- c(list(p = unlist(start)), fixed.param)
if( warn.inf && do.call("nllh", start.arg) == 1e6 )
warning("negative log-likelihood is infinite at starting values")
opt <- optim(start, nllh, hessian = TRUE, ..., method = method)
if ((opt$convergence != 0) || (opt$value == 1e6)) {
warning("optimization may not have succeeded")
if(opt$convergence == 1) opt$convergence <- "iteration limit reached"
}
else opt$convergence <- "successful"
if (std.err.type != "none"){
tol <- .Machine$double.eps^0.5
if(std.err.type == "observed") {
var.cov <- qr(opt$hessian, tol = tol)
if(var.cov$rank != ncol(var.cov$qr)){
warning("observed information matrix is singular; passing std.err.type to ``expected''")
obs.fish <- FALSE
return()
}
if (std.err.type == "observed"){
var.cov <- try(solve(var.cov, tol = tol), silent = TRUE)
if(!is.matrix(var.cov)){
warning("observed information matrix is singular; passing std.err.type to ''none''")
std.err.type <- "expected"
return()
}
else{
std.err <- diag(var.cov)
if(any(std.err <= 0)){
warning("observed information matrix is singular; passing std.err.type to ``expected''")
std.err.type <- "expected"
return()
}
std.err <- sqrt(std.err)
if(corr) {
.mat <- diag(1/std.err, nrow = length(std.err))
corr.mat <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm))
diag(corr.mat) <- rep(1, length(std.err))
}
else {
corr.mat <- NULL
}
}
}
}
if (std.err.type == "expected"){
shape <- opt$par[2]
scale <- opt$par[1]
a22 <- 2/((1+shape)*(1+2*shape))
a12 <- 1/(scale*(1+shape)*(1+2*shape))
a11 <- 1/((scale^2)*(1+2*shape))
##Expected Matix of Information of Fisher
expFisher <- nat * matrix(c(a11,a12,a12,a22),nrow=2)
expFisher <- qr(expFisher, tol = tol)
var.cov <- solve(expFisher, tol = tol)
std.err <- sqrt(diag(var.cov))
if(corr) {
.mat <- diag(1/std.err, nrow = length(std.err))
corr.mat <- structure(.mat %*% var.cov %*% .mat, dimnames = list(nm,nm))
diag(corr.mat) <- rep(1, length(std.err))
}
else
corr.mat <- NULL
}
colnames(var.cov) <- nm
rownames(var.cov) <- nm
names(std.err) <- nm
}
else{
std.err <- std.err.type <- corr.mat <- NULL
var.cov <- NULL
}
param <- c(opt$par, unlist(fixed.param))
scale <- param["scale"]
var.thresh <- !all(threshold == threshold[1])
if (!var.thresh)
threshold <- threshold[1]
list(fitted.values = opt$par, std.err = std.err, std.err.type = std.err.type,
var.cov = var.cov, fixed = unlist(fixed.param), param = param,
deviance = 2*opt$value, corr = corr.mat, convergence = opt$convergence,
counts = opt$counts, message = opt$message, threshold = threshold,
nat = nat, pat = pat, data = x, exceed = exceed, scale = scale,
var.thresh = var.thresh, est = "MLE", logLik = -opt$value,
opt.value = opt$value, hessian = opt$hessian)
}
##Medians estimation for the GPD ( Peng, L. and Welsh, A. (2002) )
gpdmed <- function(x, threshold, start, tol = 10^-3, maxit = 500,
show.trace = FALSE){
if ( length(unique(threshold)) != 1){
warning("Threshold must be a single numeric value for est = 'med'. Taking only the first value !!!")
threshold <- threshold[1]
}
nn <- length(x)
threshold <- rep(threshold, length.out = nn)
high <- (x > threshold) & !is.na(x)
threshold <- as.double(threshold[high])
exceed <- as.double(x[high])
nat <- length(exceed)
excess <- exceed - threshold
if(!nat) stop("no data above threshold")
pat <- nat/nn
if(missing(start)) {
start <- list(scale = 0, shape = 0.1)
start["scale"] <- mean(exceed) - min(threshold)
}
start <- c(scale = start$scale, shape = start$shape)
iter <- 1
trace <- round(start, 3)
##Definition of a function to solve
f <- function(x, y){
-log(x)/y - (1+y)/y^2 * (1 - x^y) + log(x + .5)/y +
(1+y)/y^2 * (1 - (x+.5)^y)
}
while (iter < maxit){
##If we have a non feasible point, we move back to feasible region
if ( (start[2] < 0) & (max(excess) >= (-start[1] / start[2])))
start[2] <- -start[1] / max(excess) + .1
r1 <- start[2] * median(excess) / (2^start[2] - 1) - start[1]
a <- log( 1 + start[2] * excess / start[1] ) / start[2]^2
b <- (1 + start[2]) * excess / (start[1]*start[2] +
start[2]^2 * excess)
if (start[2] <= -1)
y1 <- .5
else{
opt <- uniroot(f, c(10^-12, .5), y = start[2])
y1 <- opt$root
}
r2 <- median(a - b) + log(y1)/start[2] + (1 + start[2]) /
start[2]^2 * (1 - y1^start[2])
next.point <- c(r1, r2) + start
if (sqrt(sum( (next.point - start)^2) ) < tol)
break
trace <- rbind(trace, next.point)
iter <- iter + 1
start <- next.point
}
if(iter == maxit) opt$convergence <- "iteration limit reached"
else opt$convergence <- "successful"
opt$counts <- iter - 1
names(opt$counts) <- "function"
shape <- start[2]
scale <- start[1]
param <- c(scale = scale, shape = shape)
names(param) <- c("scale", "shape")
std.err <- std.err.type <- var.cov <- corr <- NULL
var.thresh <- FALSE
if (show.trace){
if (iter >= 2)
rownames(trace) <- c("Init. Val.", 1:(iter-1))
print(round(trace, 3))
}
list(fitted.values = param, std.err = std.err, std.err.type = std.err.type,
var.cov = var.cov, fixed = NULL, param = param,
deviance = NULL, corr = corr, convergence = opt$convergence,
counts = opt$counts, message = opt$message, threshold = threshold,
nat = nat, pat = pat, data = x, exceed = exceed,
scale = scale, var.thresh = var.thresh, est = "MEDIANS",
opt.value = opt$f.root)
}
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