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#' @export
#'
#' @title MLE Fitting of Kernel Density Estimate for Bulk and GPD for Both Tails
#' Extreme Value Mixture Model
#'
#' @description Maximum likelihood estimation for fitting the extreme value
#' mixture model with kernel density estimate for bulk distribution between thresholds and conditional
#' GPDs beyond thresholds. With options for profile likelihood estimation for both thresholds and
#' fixed threshold approach.
#'
#' @inheritParams fgng
#' @inheritParams fkdengpd
#' @inheritParams fkden
#' @inheritParams kden
#' @inheritParams fnormgpd
#' @inheritParams fgpd
#'
#' @details The extreme value mixture model with kernel density estimate for bulk and
#' GPD for both tails is
#' fitted to the entire dataset using maximum likelihood estimation. The estimated
#' parameters, variance-covariance matrix and their standard errors are automatically
#' output.
#'
#' See help for \code{\link[evmix:fnormgpd]{fnormgpd}} and \code{\link[evmix:fgkg]{fgkg}}
#' for details, type \code{help fnormgpd} and \code{help fgkg}.
#' Only the different features are outlined below for brevity.
#'
#' The full parameter vector is
#' (\code{lambda}, \code{ul}, \code{sigmaul}, \code{xil}, \code{ur}, \code{sigmaur}, \code{xir})
#' if thresholds are also estimated and
#' (\code{lambda}, \code{sigmaul}, \code{xil}, \code{sigmaur}, \code{xir})
#' for profile likelihood or fixed threshold approach.
#'
#' Cross-validation likelihood is used for KDE, but standard likelihood is used
#' for GPD components. See help for \code{\link[evmix:fkden]{fkden}} for details,
#' type \code{help fkden}.
#'
#' The alternate bandwidth definitions are discussed in the
#' \code{\link[evmix:kernels]{kernels}}, with the \code{lambda} as the default
#' used in the likelihood fitting. The \code{bw} specification is the same as
#' used in the \code{\link[stats:density]{density}} function.
#'
#' The possible kernels are also defined in \code{\link[evmix:kernels]{kernels}}
#' with the \code{"gaussian"} as the default choice.
#'
#' The tail fractions \code{phiul} and \code{phiur} are treated separately to the other parameters,
#' to allow for all their representations. In the fitting functions
#' \code{\link[evmix:fgkg]{fgkg}} and
#' \code{\link[evmix:fgkg]{proflugkg}} they are logical:
#' \itemize{
#' \item default values \code{phiul=TRUE} and \code{phiur=TRUE} - tail fractions specified by
#' KDE distribution and survivior functions respectively and
#' standard error is output as \code{NA}.
#' \item \code{phiul=FALSE} and \code{phiur=FALSE} - treated as extra parameters estimated using
#' the MLE which is the sample proportion beyond the thresholds and
#' standard error is output.
#' }
#' In the likelihood functions \code{\link[evmix:fgkg]{lgkg}},
#' \code{\link[evmix:fgkg]{nlgkg}} and \code{\link[evmix:fgkg]{nlugkg}}
#' it can be logical or numeric:
#' \itemize{
#' \item logical - same as for fitting functions with default values \code{phiul=TRUE} and \code{phiur=TRUE}.
#' \item numeric - any value over range \eqn{(0, 1)}. Notice that the tail
#' fraction probability cannot be 0 or 1 otherwise there would be no
#' contribution from either tail or bulk components respectively. Also,
#' \code{phiul+phiur<1} as bulk must contribute.
#' }
#'
#' If the profile likelihood approach is used, then a grid search over all combinations of both thresholds
#' is carried out. The combinations which lead to less than 5 in any datapoints beyond the thresholds are not considered.
#'
#' @section Warning:
#' See important warnings about cross-validation likelihood estimation in
#' \code{\link[evmix:fkden]{fkden}}, type \code{help fkden}.
#'
#' @return Log-likelihood is given by \code{\link[evmix:fgkg]{lgkg}} and it's
#' wrappers for negative log-likelihood from \code{\link[evmix:fgkg]{nlgkg}}
#' and \code{\link[evmix:fgkg]{nlugkg}}. Profile likelihood for both
#' thresholds given by \code{\link[evmix:fgkg]{proflugkg}}. Fitting function
#' \code{\link[evmix:fgkg]{fgkg}} 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{fixedu}: \tab fixed thresholds, logical\cr
#' \code{ulseq}: \tab lower threshold vector for profile likelihood or scalar for fixed threshold\cr
#' \code{urseq}: \tab upper threshold vector for profile likelihood or scalar for fixed threshold\cr
#' \code{nllhuseq}: \tab profile negative log-likelihood at each threshold pair in (ulseq, urseq)\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{lambda}: \tab MLE of lambda (kernel half-width)\cr
#' \code{ul}: \tab lower threshold (fixed or MLE)\cr
#' \code{sigmaul}: \tab MLE of lower tail GPD scale\cr
#' \code{xil}: \tab MLE of lower tail GPD shape\cr
#' \code{phiul}: \tab MLE of lower tail fraction (bulk model or parameterised approach)\cr
#' \code{se.phiul}: \tab standard error of MLE of lower tail fraction\cr
#' \code{ur}: \tab upper threshold (fixed or MLE)\cr
#' \code{sigmaur}: \tab MLE of upper tail GPD scale\cr
#' \code{xir}: \tab MLE of upper tail GPD shape\cr
#' \code{phiur}: \tab MLE of upper tail fraction (bulk model or parameterised approach)\cr
#' \code{se.phiur}: \tab standard error of MLE of upper tail fraction\cr
#' \code{bw}: \tab MLE of bw (kernel standard deviations)\cr
#' \code{kernel}: \tab kernel name\cr
#' }
#'
#' @note The data and kernel centres are both vectors. Infinite and missing sample values
#' (and kernel centres) are dropped.
#'
#' When \code{pvector=NULL} then the initial values are:
#' \itemize{
#' \item normal reference rule for bandwidth, using the \code{\link[stats:bandwidth]{bw.nrd0}} function, which is
#' consistent with the \code{\link[stats:density]{density}} function. At least two kernel
#' centres must be provided as the variance needs to be estimated.
#' \item lower threshold 10\% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
#' \item upper threshold 90\% quantile (not relevant for profile likelihood for threshold or fixed threshold approaches);
#' \item MLE of GPD parameters beyond thresholds.
#' }
#'
#' @references
#' \url{http://www.math.canterbury.ac.nz/~c.scarrott/evmix}
#'
#' \url{http://en.wikipedia.org/wiki/Kernel_density_estimation}
#'
#' \url{http://en.wikipedia.org/wiki/Cross-validation_(statistics)}
#'
#' \url{http://en.wikipedia.org/wiki/Generalized_Pareto_distribution}
#'
#' Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value
#' threshold estimation and uncertainty quantification. REVSTAT - Statistical
#' Journal 10(1), 33-59. Available from \url{http://www.ine.pt/revstat/pdf/rs120102.pdf}
#'
#' Hu, Y. (2013). Extreme value mixture modelling: An R package and simulation study.
#' MSc (Hons) thesis, University of Canterbury, New Zealand.
#' \url{http://ir.canterbury.ac.nz/simple-search?query=extreme&submit=Go}
#'
#' Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of
#' density estimates. Biometrika 71(2), 353-360.
#'
#' Duin, R.P.W. (1976). On the choice of smoothing parameters for Parzen estimators of
#' probability density functions. IEEE Transactions on Computers C25(11), 1175-1179.
#'
#' MacDonald, A., Scarrott, C.J., Lee, D., Darlow, B., Reale, M. and Russell, G. (2011).
#' A flexible extreme value mixture model. Computational Statistics and Data Analysis
#' 55(6), 2137-2157.
#'
#' Wand, M. and Jones, M.C. (1995). Kernel Smoothing. Chapman && Hall.
#'
#' @author Yang Hu and Carl Scarrott \email{carl.scarrott@@canterbury.ac.nz}
#'
#' @section Acknowledgments: See Acknowledgments in
#' \code{\link[evmix:fnormgpd]{fnormgpd}}, type \code{help fnormgpd}. Based on code
#' by Anna MacDonald produced for MATLAB.
#'
#' @seealso \code{\link[evmix:kernels]{kernels}}, \code{\link[evmix:kfun]{kfun}},
#' \code{\link[stats:density]{density}}, \code{\link[stats:bandwidth]{bw.nrd0}}
#' and \code{\link[ks:kde]{dkde}} in \code{\link[ks:kde]{ks}} package.
#' \code{\link[evmix:fgpd]{fgpd}} and \code{\link[evmix:gpd]{gpd}}.
#'
#' @aliases fgkg lgkg nlgkg proflugkg nlugkg
#' @family kden
#' @family kdengpd
#' @family gkg
#' @family gkgcon
#' @family fgkg
#'
#' @examples
#' \dontrun{
#' set.seed(1)
#' par(mfrow = c(2, 1))
#'
#' x = rnorm(1000)
#' xx = seq(-4, 4, 0.01)
#' y = dnorm(xx)
#'
#' # Bulk model based tail fraction
#' fit = fgkg(x)
#' hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
#' lines(xx, y)
#' with(fit, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
#' ur, sigmaur, xir, phiur), col="red"))
#' abline(v = c(fit$ul, fit$ur), col = "red")
#'
#' # Parameterised tail fraction
#' fit2 = fgkg(x, phiul = FALSE, phiur = FALSE)
#' with(fit2, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
#' ur, sigmaur, xir, phiur), col="blue"))
#' abline(v = c(fit2$ul, fit2$ur), col = "blue")
#' legend("topright", c("True Density","Bulk Tail Fraction","Parameterised Tail Fraction"),
#' col=c("black", "red", "blue"), lty = 1)
#'
#' # Profile likelihood for initial value of threshold and fixed threshold approach
#' fitu = fgkg(x, ulseq = seq(-2, -0.2, length = 10),
#' urseq = seq(0.2, 2, length = 10))
#' fitfix = fgkg(x, ulseq = seq(-2, -0.2, length = 10),
#' urseq = seq(0.2, 2, length = 10), fixedu = TRUE)
#'
#' hist(x, breaks = 100, freq = FALSE, xlim = c(-4, 4))
#' lines(xx, y)
#' with(fit, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
#' ur, sigmaur, xir, phiur), col="red"))
#' abline(v = c(fit$ul, fit$ur), col = "red")
#' with(fitu, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
#' ur, sigmaur, xir, phiur), col="purple"))
#' abline(v = c(fitu$ul, fitu$ur), col = "purple")
#' with(fitfix, lines(xx, dgkg(xx, x, lambda, ul, sigmaul, xil, phiul,
#' ur, sigmaur, xir, phiur), col="darkgreen"))
#' abline(v = c(fitfix$ul, fitfix$ur), col = "darkgreen")
#' legend("topright", c("True Density","Default initial value (90% quantile)",
#' "Prof. lik. for initial value", "Prof. lik. for fixed threshold"),
#' col=c("black", "red", "purple", "darkgreen"), lty = 1)
#' }
#'
# maximum likelihood fitting for kernel density estimate for bulk with GPD for both tails
fgkg <- function(x, phiul = TRUE, phiur = TRUE, ulseq = NULL, urseq = NULL, fixedu = FALSE,
pvector = NULL, kernel = "gaussian", add.jitter = FALSE, factor = 0.1, amount = NULL,
std.err = TRUE, method = "BFGS", control = list(maxit = 10000), finitelik = TRUE, ...) {
call <- match.call()
np = 7 # maximum number of parameters
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.logic(phiul)
check.logic(phiur)
check.param(ulseq, allowvec = TRUE, allownull = TRUE)
check.param(urseq, allowvec = TRUE, allownull = TRUE)
check.logic(fixedu)
check.logic(std.err)
check.optim(method)
check.control(control)
check.logic(finitelik)
check.kernel(kernel)
check.posparam(factor)
check.posparam(amount, allownull = TRUE)
check.logic(add.jitter)
if (any(!is.finite(x))) {
warning("non-finite cases have been removed")
x = x[is.finite(x)] # ignore missing and infinite cases
}
check.quant(x)
n = length(x)
if (add.jitter) x = jitter(x, factor, amount)
xuniq = unique(x)
if (length(xuniq) < (0.95*n))
warning("data may be rounded, as more than 5% are ties, so bandwidth could be biased to zero")
if ((method == "L-BFGS-B") | (method == "BFGS")) finitelik = TRUE
# useq must be specified if threshold is fixed
if (fixedu & (is.null(ulseq) | is.null(urseq)))
stop("for fixed threshold approach, ulseq and urseq must be specified (as scalar or vector)")
# Check if profile likelihood or fixed threshold is being used
# and determine initial values for parameters in each case
if (is.null(ulseq) | is.null(ulseq)) { # not profile or fixed
check.nparam(pvector, nparam = np, allownull = TRUE)
if (is.null(pvector)) {
if (n == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (n < 10) {
stop("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
pvector[1] = klambda(bw.nrd0(x), kernel)
pvector[2] = as.vector(quantile(x, 0.1))
initfgpd = fgpd(-x, -pvector[2], std.err = FALSE)
pvector[3] = initfgpd$sigmau
pvector[4] = initfgpd$xi
pvector[5] = as.vector(quantile(x, 0.9))
initfgpd = fgpd(x, pvector[5], std.err = FALSE)
pvector[6] = initfgpd$sigmau
pvector[7] = initfgpd$xi
}
} else { # profile or fixed
check.nparam(pvector, nparam = np - 2, allownull = TRUE)
# profile likelihood for threshold or scalar given
if ((length(ulseq) != 1) | (length(urseq) != 1)) {
# remove thresholds with less than 5 excesses
ulseq = ulseq[sapply(ulseq, FUN = function(u, x) sum(x < u) > 5, x = x)]
check.param(ulseq, allowvec = TRUE)
urseq = urseq[sapply(urseq, FUN = function(u, x) sum(x > u) > 5, x = x)]
check.param(urseq, allowvec = TRUE)
ulrseq = expand.grid(ulseq, urseq)
# remove those where ulseq >= urseq
if (any(ulrseq[1] >= ulrseq[2])) {
warning("lower thresholds above or equal to upper threshold are ignored")
ulrseq = ulrseq[ulrseq[1] < ulrseq[2],]
}
nllhu = apply(ulrseq, 1, proflugkg, pvector = pvector, x = x,
phiul = phiul, phiur = phiur, kernel = kernel,
method = method, control = control, finitelik = finitelik, ...)
if (all(!is.finite(nllhu))) stop("thresholds are all invalid")
ul = ulrseq[which.min(nllhu), 1]
ur = ulrseq[which.min(nllhu), 2]
} else {
if (ulseq >= urseq) stop("lower threshold cannot be above or equal to upper threshold")
ul = ulseq
ur = urseq
}
if (fixedu) { # threshold fixed
if (is.null(pvector)) {
if (n == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (n < 10) {
stop("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
pvector[1] = klambda(bw.nrd0(x), kernel)
initfgpd = fgpd(-x, -ul, std.err = FALSE)
pvector[2] = initfgpd$sigmau
pvector[3] = initfgpd$xi
initfgpd = fgpd(x, ur, std.err = FALSE)
pvector[4] = initfgpd$sigmau
pvector[5] = initfgpd$xi
}
} else { # threshold as initial value in usual MLE
if (is.null(pvector)) {
if (n == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (n < 10) {
stop("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
pvector[1] = klambda(bw.nrd0(x), kernel)
pvector[2] = ul
initfgpd = fgpd(-x, -pvector[2], std.err = FALSE)
pvector[3] = initfgpd$sigmau
pvector[4] = initfgpd$xi
pvector[5] = ur
initfgpd = fgpd(x, pvector[5], std.err = FALSE)
pvector[6] = initfgpd$sigmau
pvector[7] = initfgpd$xi
} else {
pvector[7] = pvector[5] # shift upper tail GPD scale and shape to add in ur
pvector[6] = pvector[4]
pvector[5] = ur
pvector[4] = pvector[3] # shift lower tail GPD scale and shape to add in ul
pvector[3] = pvector[2]
pvector[2] = ul
}
}
}
if (fixedu) { # fixed threshold (separable) likelihood
nllh = nlugkg(pvector, ul, ur, x, phiul, phiur, kernel = kernel)
if (is.infinite(nllh)) {
pvector[c(3, 5)] = 0.1
nllh = nlugkg(pvector, ul, ur, x, phiul, phiur, kernel = kernel)
}
if (is.infinite(nllh)) stop("initial parameter values are invalid")
fit = optim(par = as.vector(pvector), fn = nlugkg, ul = ul, ur = ur, x = x,
phiul = phiul, phiur = phiur, kernel = kernel,
finitelik = finitelik, method = method, control = control, hessian = TRUE, ...)
lambda = fit$par[1]
sigmaul = fit$par[2]
xil = fit$par[3]
sigmaur = fit$par[4]
xir = fit$par[5]
} else { # complete (non-separable) likelihood
nllh = nlgkg(pvector, x, phiul, phiur, kernel = kernel)
if (is.infinite(nllh)) {
pvector[c(4, 7)] = 0.1
nllh = nlgkg(pvector, x, phiul, phiur, kernel = kernel)
}
if (is.infinite(nllh)) stop("initial parameter values are invalid")
fit = optim(par = as.vector(pvector), fn = nlgkg, x = x, phiul = phiul, phiur = phiur, kernel = kernel,
finitelik = finitelik, method = method, control = control, hessian = TRUE, ...)
lambda = fit$par[1]
ul = fit$par[2]
sigmaul = fit$par[3]
xil = fit$par[4]
ur = fit$par[5]
sigmaur = fit$par[6]
xir = fit$par[7]
}
bw = kbw(fit$par[1], kernel)
conv = TRUE
if ((fit$convergence != 0) | any(fit$par == pvector) | (abs(fit$value) >= 1e6)) {
conv = FALSE
warning("check convergence")
}
pul = pkdenx(ul, x, lambda, kernel)
if (phiul) {
phiul = pul
se.phiul = NA
} else {
phiul = mean(x < ul, na.rm = TRUE)
se.phiul = sqrt(phiul * (1 - phiul) / n)
}
pur = pkdenx(ur, x, lambda, kernel)
if (phiur) {
phiur = 1 - pur
se.phiur = NA
} else {
phiur = mean(x > ur, na.rm = TRUE)
se.phiur = sqrt(phiur * (1 - phiur) / n)
}
if (std.err) {
qrhess = qr(fit$hessian)
if (qrhess$rank != ncol(qrhess$qr)) {
warning("observed information matrix is singular")
se = NULL
invhess = NULL
} else {
invhess = solve(qrhess)
vars = diag(invhess)
if (any(vars <= 0)) {
warning("observed information matrix is singular")
invhess = NULL
se = NULL
} else {
se = sqrt(vars)
}
}
} else {
invhess = NULL
se = NULL
}
if (!exists("nllhu")) nllhu = NULL
list(call = call, x = as.vector(x),
init = as.vector(pvector), fixedu = fixedu, ulseq = ulseq, urseq = urseq, nllhuseq = nllhu,
optim = fit, conv = conv, cov = invhess, mle = fit$par, se = se, ratel = phiul, rater = phiur,
nllh = fit$value, n = n, lambda = lambda,
ul = ul, sigmaul = sigmaul, xil = xil, phiul = phiul, se.phiul = se.phiul,
ur = ur, sigmaur = sigmaur, xir = xir, phiur = phiur, se.phiur = se.phiur, bw = bw, kernel = kernel)
}
#' @export
#' @aliases fgkg lgkg nlgkg proflugkg nlugkg
#' @rdname fgkg
# log-likelihood function for kernel density estimate for bulk with GPD for both tails
# cross-validation for KDE component
lgkg <- function(x, lambda = NULL,
ul = 0, sigmaul = 1, xil = 0, phiul = TRUE,
ur = 0, sigmaur = 1, xir = 0, phiur = TRUE, bw = NULL, kernel = "gaussian", log = TRUE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.param(lambda, allownull = TRUE)
check.param(bw, allownull = TRUE)
check.param(ul)
check.param(sigmaul)
check.param(xil)
check.param(ur)
check.param(sigmaur)
check.param(xir)
check.phiu(phiul, allowfalse = TRUE)
check.phiu(phiur, allowfalse = TRUE)
check.logic(log)
check.kernel(kernel)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
if (any(!is.finite(x))) {
warning("non-finite cases have been removed")
x = x[is.finite(x)] # ignore missing and infinite cases
}
check.quant(x)
n = length(x)
if (is.null(lambda) & is.null(bw)) {
if (n == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (n < 10) {
warning("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
bw = bw.nrd0(x)
}
if (is.null(lambda)) lambda = klambda(bw, kernel, lambda)
check.inputn(c(length(lambda), length(ul), length(sigmaul), length(xil), length(phiul),
length(ur), length(sigmaur), length(xir), length(phiur)), allowscalar = TRUE)
# assume NA or NaN are irrelevant as entire lower tail is now modelled
# inconsistent with evd library definition
# hence use which() to ignore these
xur = x[which(x > ur)]
nur = length(xur)
xul = x[which(x < ul)]
nul = length(xul)
xb = x[which((x >= ul) & (x <= ur))]
nb = length(xb)
if ((lambda <= 0) | (sigmaul <= 0) | (ul <= min(x)) | (ul >= max(x))
| (sigmaur <= 0) | (ur <= min(x)) | (ur >= max(x))| (ur <= ul)) {
l = -Inf
} else {
if (is.logical(phiul)) {
pul = pkdenx(ul, x, lambda, kernel)
if (phiul) {
phiul = pul
} else {
phiul = nul / n
}
}
if (is.logical(phiur)) {
pur = pkdenx(ur, x, lambda, kernel)
if (phiur) {
phiur = 1 - pur
} else {
phiur = nur / n
}
}
phib = (1 - phiul - phiur) / (pur - pul)
syul = 1 + xil * (ul - xul) / sigmaul
syur = 1 + xir * (xur - ur) / sigmaur
if ((min(syul) <= 0) | (phiul <= 0) | (phiul >= 1) |
(min(syur) <= 0) | (phiur <= 0) | (phiur >= 1) | ((phiul + phiur) > 1) |
(pul <= 0) | (pul >= 1) | (pur <= 0) | (pur >= 1) |
(phib < .Machine$double.eps)) {
l = -Inf
} else {
l = lgpd(-xul, -ul, sigmaul, xil, phiul)
l = l + lgpd(xur, ur, sigmaur, xir, phiur)
l = l + lkden(xb, lambda, kernel = kernel, extracentres = c(xul, xur)) + nb*log(phib)
}
}
if (!log) l = exp(l)
l
}
#' @export
#' @aliases fgkg lgkg nlgkg proflugkg nlugkg
#' @rdname fgkg
# negative log-likelihood function for kernel density estimate for bulk with GPD for both tails
# cross-validation for KDE component
# (wrapper for likelihood, inputs and checks designed for optimisation)
nlgkg <- function(pvector, x, phiul = TRUE, phiur = TRUE, kernel = "gaussian", finitelik = FALSE) {
np = 7 # maximum number of parameters
# Check properties of inputs
check.nparam(pvector, nparam = np)
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.phiu(phiul, allowfalse = TRUE)
check.phiu(phiur, allowfalse = TRUE)
check.logic(finitelik)
check.kernel(kernel)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
lambda = pvector[1]
ul = pvector[2]
sigmaul = pvector[3]
xil = pvector[4]
ur = pvector[5]
sigmaur = pvector[6]
xir = pvector[7]
nllh = -lgkg(x, lambda, ul, sigmaul, xil, phiul, ur, sigmaur, xir, phiur, kernel = kernel)
if (finitelik & is.infinite(nllh)) {
nllh = sign(nllh) * 1e6
}
nllh
}
#' @export
#' @aliases fgkg lgkg nlgkg proflugkg nlugkg
#' @rdname fgkg
# profile negative log-likelihood function for given threshold for
# kernel density estimate for bulk with GPD for both tails
# designed for apply to loop over vector of thresholds (hence c(ul, ur) vector is first input)
# cross-validation for KDE component
proflugkg <- function(ulr, pvector, x, phiul = TRUE, phiur = TRUE, kernel = "gaussian",
method = "BFGS", control = list(maxit = 10000), finitelik = TRUE, ...) {
np = 7 # maximum number of parameters
# Check properties of inputs
check.nparam(pvector, nparam = np - 2, allownull = TRUE)
check.param(ulr, allowvec = TRUE)
check.nparam(ulr, nparam = 2)
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.phiu(phiul, allowfalse = TRUE)
check.phiu(phiur, allowfalse = TRUE)
check.optim(method)
check.control(control)
check.logic(finitelik)
check.kernel(kernel)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
if (any(!is.finite(x))) {
warning("non-finite cases have been removed")
x = x[is.finite(x)] # ignore missing and infinite cases
}
check.quant(x)
n = length(x)
ul = ulr[1]
ur = ulr[2]
if (ul >= ur) stop("lower threshold cannot be above or equal to upper threshold")
# check initial values for other parameters, try usual alternative
if (!is.null(pvector)) {
nllh = nlugkg(pvector, ul, ur, x, phiul, phiur, kernel = kernel)
if (is.infinite(nllh)) pvector = NULL
}
if (is.null(pvector)) {
if (n == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (n < 10) {
stop("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
pvector[1] = klambda(bw.nrd0(x), kernel)
initfgpd = fgpd(-x, -ul, std.err = FALSE)
pvector[2] = initfgpd$sigmau
pvector[3] = initfgpd$xi
initfgpd = fgpd(x, ur, std.err = FALSE)
pvector[4] = initfgpd$sigmau
pvector[5] = initfgpd$xi
nllh = nlugkg(pvector, ul, ur, x, phiul, phiur, kernel = kernel)
}
if (is.infinite(nllh)) {
pvector[c(3, 5)] = 0.1
nllh = nlugkg(pvector, ul, ur, x, phiul, phiur, kernel = kernel)
}
# if still invalid then output cleanly
if (is.infinite(nllh)) {
warning(paste("initial parameter values for thresholds ul =", ul, "and ur =", ur,"are invalid"))
fit = list(par = rep(NA, np), value = Inf, counts = 0, convergence = NA,
message = "initial values invalid", hessian = rep(NA, np))
} else {
fit = optim(par = as.vector(pvector), fn = nlugkg, ul = ul, ur = ur, x = x,
phiul = phiul, phiur = phiur, kernel = kernel,
finitelik = finitelik, method = method, control = control, hessian = TRUE, ...)
}
if (finitelik & is.infinite(fit$value)) {
fit$value = sign(fit$value) * 1e6
}
fit$value
}
#' @export
#' @aliases fgkg lgkg nlgkg proflugkg nlugkg
#' @rdname fgkg
# negative log-likelihood function for kernel density estimate for bulk with GPD for both tails
# (wrapper for likelihood, designed for threshold to be fixed and other parameters optimised)
# cross-validation for KDE component
nlugkg <- function(pvector, ul, ur, x, phiul = TRUE, phiur = TRUE, kernel = "gaussian", finitelik = FALSE) {
np = 7 # maximum number of parameters
# Check properties of inputs
check.nparam(pvector, nparam = np - 2)
check.param(ul)
check.param(ur)
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.phiu(phiul, allowfalse = TRUE)
check.phiu(phiur, allowfalse = TRUE)
check.logic(finitelik)
check.kernel(kernel)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
if (ul >= ur) stop("lower threshold cannot be above or equal to upper threshold")
lambda = pvector[1]
sigmaul = pvector[2]
xil = pvector[3]
sigmaur = pvector[4]
xir = pvector[5]
nllh = -lgkg(x, lambda, ul, sigmaul, xil, phiul, ur, sigmaur, xir, phiur, kernel = kernel)
if (finitelik & is.infinite(nllh)) {
nllh = sign(nllh) * 1e6
}
nllh
}
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