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R/fkden.r
In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation
Defines functions
fkden
lkden
nlkden
Documented in
fkden
lkden
nlkden
#' @export
#'
#' @title Cross-validation MLE Fitting of Kernel Density Estimator, With Variety of Kernels
#'
#' @description Maximum (cross-validation) likelihood estimation for fitting kernel density estimator
#' for a variety of possible kernels, by treating it as a mixture model.
#'
#' @param linit initial value for bandwidth (as kernel half-width) or \code{NULL}
#' @param bwinit initial value for bandwidth (as kernel standard deviations) or \code{NULL}
#' @param extracentres extra kernel centres used in KDE,
#' but likelihood contribution not evaluated, or \code{NULL}
#' @param add.jitter logical, whether jitter is needed for rounded kernel centres
#' @param factor see \code{\link[base:jitter]{jitter}}
#' @param amount see \code{\link[base:jitter]{jitter}}
#' @inheritParams fnormgpd
#' @inheritParams kden
#' @inheritParams fgpd
#'
#' @details The kernel density estimator (KDE) with one of possible kernels is
#' fitted to the entire dataset using maximum (cross-validation) likelihood estimation.
#' The estimated bandwidth, variance and standard error are automatically output.
#'
#' The alternate bandwidth definitions are discussed in the
#' \code{\link[evmix:kernels]{kernels}}, with the \code{lambda} used here but
#' \code{bw} also output. 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}} help
#' documentation with the \code{"gaussian"} as the default choice.
#'
#' Missing values (\code{NA} and \code{NaN}) are assumed to be invalid data so are ignored.
#'
#' Cross-validation likelihood is used for kernel density component, obtained by
#' leaving each point out in turn and evaluating the KDE at the point left out:
#' \deqn{L(\lambda)\prod_{i=1}^{n} \hat{f}_{-i}(x_i)}
#' where
#' \deqn{\hat{f}_{-i}(x_i) = \frac{1}{(n-1)\lambda} \sum_{j=1: j\ne i}^{n} K(\frac{x_i - x_j}{\lambda})}
#' is the KDE obtained when the \eqn{i}th datapoint is dropped out and then
#' evaluated at that dropped datapoint at \eqn{x_i}.
#'
#' Normally for likelihood estimation of the bandwidth the kernel centres and
#' the data where the likelihood is evaluated are the same. However, when using
#' KDE for extreme value mixture modelling the likelihood only those data in the
#' bulk of the distribution should contribute to the likelihood, but all the
#' data (including those beyond the threshold) should contribute to the density
#' estimate. The \code{extracentres} option allows the use to specify extra
#' kernel centres used in estimating the density, but not evaluated in the
#' likelihood. Suppose the first \code{nb} data are below the threshold, followed
#' by \code{nu} exceedances of the threshold, so \eqn{i = 1,\ldots,nb, nb+1, \ldots, nb+nu}.
#' The cross-validation likelihood using the extra kernel centres is then:
#' \deqn{L(\lambda)\prod_{i=1}^{nb} \hat{f}_{-i}(x_i)}
#' where
#' \deqn{\hat{f}_{-i}(x_i) = \frac{1}{(nb+nu-1)\lambda} \sum_{j=1: j\ne i}^{nb+nu} K(\frac{x_i - x_j}{\lambda})}
#' which shows that the complete set of data is used in evaluating the KDE, but only those
#' below the threshold contribute to the cross-validation likelihood. The default is to
#' use the existing data, so \code{extracentres=NULL}.
#'
#' The following functions are provided:
#' \itemize{
#' \item \code{\link[evmix:fkden]{fkden}} - maximum (cross-validation) likelihood fitting with all the above options;
#' \item \code{\link[evmix:fkden]{lkden}} - cross-validation log-likelihood;
#' \item \code{\link[evmix:fkden]{nlkden}} - negative cross-validation log-likelihood;
#' }
#' The log-likelihood functions are provided for wider usage, e.g. constructing
#' profile likelihood functions.
#'
#' The log-likelihood and negative log-likelihood are also provided for wider
#' usage, e.g. constructing your own extreme value
#' mixture models or profile likelihood functions. The parameter
#' \code{lambda} must be specified in the negative log-likelihood
#' \code{\link[evmix:fkden]{nlkden}}.
#'
#' Log-likelihood calculations are carried out in
#' \code{\link[evmix:fkden]{lkden}}, which takes bandwidths as inputs in
#' the same form as distribution functions. The negative log-likelihood is a
#' wrapper for \code{\link[evmix:fkden]{lkden}}, designed towards making
#' it useable for optimisation (e.g. \code{lambda} given as first input).
#'
#' Defaults values for the bandwidth \code{linit} and \code{lambda} are given in the fitting
#' \code{\link[evmix:fkden]{fkden}} and cross-validation likelihood functions
#' \code{\link[evmix:fkden]{lkden}}. The bandwidth \code{linit} must be specified in
#' the negative log-likelihood function \code{\link[evmix:fkden]{nlkden}}.
#'
#' Missing values (\code{NA} and \code{NaN}) are assumed to be invalid data so are ignored,
#' which is inconsistent with the \code{\link[evd:fpot]{evd}} library which assumes the
#' missing values are below the threshold.
#'
#' The function \code{\link[evmix:fkden]{lkden}} carries out the calculations
#' for the log-likelihood directly, which can be exponentiated to give actual
#' likelihood using (\code{log=FALSE}).
#'
#' 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 or for common indicators of lack
#' of convergence (e.g. estimated bandwidth equal to initial value).
#'
#' 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}.
#'
#' @section Warning:
#' Two important practical issues arise with MLE for the kernel bandwidth:
#' 1) Cross-validation likelihood is needed for the KDE bandwidth parameter
#' as the usual likelihood degenerates, so that the MLE \eqn{\hat{\lambda} \rightarrow 0} as
#' \eqn{n \rightarrow \infty}, thus giving a negative bias towards a small bandwidth.
#' Leave one out cross-validation essentially ensures that some smoothing between the kernel centres
#' is required (i.e. a non-zero bandwidth), otherwise the resultant density estimates would always
#' be zero if the bandwidth was zero.
#'
#' This problem occassionally rears its ugly head for data which has been heavily rounded,
#' as even when using cross-validation the density can be non-zero even if the bandwidth is zero.
#' To overcome this issue an option to add a small jitter should be added to the data
#' (\code{x} only) has been included in the fitting inputs, using the
#' \code{\link[base:jitter]{jitter}} function, to remove the ties. The default options red in the
#' \code{\link[base:jitter]{jitter}} are specified above, but the user can override these.
#' Notice the default scaling \code{factor=0.1}, which is a tenth of the default value in the
#' \code{\link[base:jitter]{jitter}}
#' function itself.
#'
#' A warning message is given if the data appear to be rounded
#' (i.e. more than 5% of data are tied). If the estimated bandwidth is too small, then
#' data rounding is the likely culprit. Only use the jittering when the MLE of
#' the bandwidth is far too small.
#'
#' 2) For heavy tailed populations the bandwidth is positively biased, giving oversmoothing
#' (see example). The bias is due to the distance between the upper (or lower) order statistics not
#' necessarily decaying to zero as the sample size tends to infinity. Essentially, as the distance
#' between the two largest (or smallest) sample datapoints does not decay to zero, some smoothing between
#' them is required (i.e. bandwidth cannot be zero). One solution to this problem is to trim
#' the data at a suitable threshold to remove the problematic tail from the inference for the bandwidth,
#' using either the \code{\link[evmix:fkdengpd]{fkdengpd}} function for a single heavy tail
#' or the \code{\link[evmix:fgkg]{fgkg}} function
#' if both tails are heavy. See MacDonald et al (2013).
#'
#' @return Log-likelihood is given by \code{\link[evmix:fkden]{lkden}} and it's
#' wrappers for negative log-likelihood from \code{\link[evmix:fkden]{nlkden}}.
#' Fitting function \code{\link[evmix:fkden]{fkden}} returns a simple list with the
#' following elements
#'
#' \tabular{ll}{
#' \code{call}: \tab \code{optim} call\cr
#' \code{x}: \tab (jittered) data vector \code{x}\cr
#' \code{kerncentres}: \tab actual kernel centres used \code{x}\cr
#' \code{init}: \tab \code{linit} for lambda\cr
#' \code{optim}: \tab complete \code{optim} output\cr
#' \code{mle}: \tab vector of MLE of bandwidth\cr
#' \code{cov}: \tab variance of MLE of bandwidth\cr
#' \code{se}: \tab standard error of MLE of bandwidth\cr
#' \code{nllh}: \tab minimum negative cross-validation log-likelihood\cr
#' \code{n}: \tab total sample size\cr
#' \code{lambda}: \tab MLE of lambda (kernel half-width)\cr
#' \code{bw}: \tab MLE of bw (kernel standard deviations)\cr
#' \code{kernel}: \tab kernel name\cr
#' }
#'
#' @note When \code{linit=NULL} then the initial value for the \code{lambda}
#' bandwidth is calculated
#' using \code{\link[stats:bandwidth]{bw.nrd0}} function and transformed using
#' \code{\link[evmix:kfun]{klambda}} function.
#'
#' The extra kernel centres \code{extracentres} can either be a vector of data or \code{NULL}.
#'
#' Invalid parameter ranges will give \code{0} for likelihood, \code{log(0)=-Inf} for
#' log-likelihood and \code{-log(0)=Inf} for negative log-likelihood.
#'
#' Infinite and missing sample values are dropped.
#'
#' 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/Kernel_density_estimation}
#'
#' \url{http://en.wikipedia.org/wiki/Cross-validation_(statistics)}
#'
#' 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. 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.
#'
#' 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.
#'
#' MacDonald, A., C. J. Scarrott, and D. S. Lee (2011). Boundary correction, consistency
#' and robustness of kernel densities using extreme value theory. Submitted.
#' Available from: \url{http://www.math.canterbury.ac.nz/~c.scarrott}.
#'
#' 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[base:jitter]{jitter}}, \code{\link[stats:density]{density}} and
#' \code{\link[stats:bandwidth]{bw.nrd0}}
#'
#' @aliases fkden lkden nlkden
#' @family kden
#' @family kdengpd
#' @family bckden
#' @family fkden
#'
#' @examples
#' \dontrun{
#' set.seed(1)
#' par(mfrow = c(1, 1))
#'
#' nk=50
#' x = rnorm(nk)
#' xx = seq(-5, 5, 0.01)
#' fit = fkden(x)
#' hist(x, nk/5, freq = FALSE, xlim = c(-5, 5), ylim = c(0,0.6))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$lambda)*0.05)
#' lines(xx,dnorm(xx), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
#' lines(density(x), lty = 2, lwd = 2, col = "green")
#' lines(density(x, bw = fit$bw), lwd = 2, lty = 2, col = "blue")
#' legend("topright", c("True Density", "KDE fitted evmix",
#' "KDE Using density, default bandwidth", "KDE Using density, c-v likelihood bandwidth"),
#' lty = c(1, 1, 2, 2), lwd = c(1, 2, 2, 2), col = c("black", "red", "green", "blue"))
#'
#' par(mfrow = c(2, 1))
#'
#' # bandwidth is biased towards oversmoothing for heavy tails
#' nk=100
#' x = rt(nk, df = 2)
#' xx = seq(-8, 8, 0.01)
#' fit = fkden(x)
#' hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$lambda)*0.05)
#' lines(xx,dt(xx , df = 2), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
#' legend("topright", c("True Density", "KDE fitted evmix, c-v likelihood bandwidth"),
#' lty = c(1, 1), lwd = c(1, 2), col = c("black", "red"))
#'
#' # remove heavy tails from cv-likelihood evaluation, but still include them in KDE within likelihood
#' # often gives better bandwidth (see MacDonald et al (2011) for justification)
#' nk=100
#' x = rt(nk, df = 2)
#' xx = seq(-8, 8, 0.01)
#' fit2 = fkden(x[(x > -4) & (x < 4)], extracentres = x[(x <= -4) | (x >= 4)])
#' hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit2$lambda)*0.05)
#' lines(xx,dt(xx , df = 2), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit2$lambda), lwd = 2, col = "red")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "blue")
#' legend("topright", c("True Density", "KDE fitted evmix, tails removed",
#' "KDE fitted evmix, tails included"),
#' lty = c(1, 1, 1), lwd = c(1, 2, 2), col = c("black", "red", "blue"))
#' }
#'
# maximum cross-validation likelihood fitting for kernel density estimator
fkden <- function(x, linit = NULL, bwinit = NULL, kernel = "gaussian", extracentres = NULL,
add.jitter = FALSE, factor = 0.1, amount = NULL,
std.err = TRUE, method = "BFGS", control = list(maxit = 10000), finitelik = TRUE, ...) {
call <- match.call()
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.kbw(linit, bwinit, allownull = TRUE)
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 (!is.null(extracentres)) {
if (any(!is.finite(extracentres))) warning("non-finite extracentres have been removed")
extracentres = extracentres[is.finite(extracentres)]
check.quant(extracentres)
}
if ((method == "L-BFGS-B") | (method == "BFGS")) finitelik = TRUE
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 (is.null(linit) & is.null(bwinit)) {
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")
}
bwinit = bw.nrd0(x)
}
linit = klambda(bwinit, kernel, linit)
nllh = nlkden(linit, x = x, kernel = kernel, extracentres = extracentres)
if (is.infinite(nllh)) stop("initial bandwidth is invalid")
fit = optim(par = linit, fn = nlkden, x = x, kernel = kernel, extracentres = extracentres,
finitelik = finitelik, method = method, control = control, hessian = TRUE, ...)
conv = TRUE
if ((fit$convergence != 0) | any(fit$par == linit) | (abs(fit$value) >= 1e6)) {
conv = FALSE
warning("check convergence")
}
lambda = fit$par[1]
if (conv & std.err) {
se = sqrt(1/fit$hessian)
} else {
se = NULL
}
list(call = call, x = as.vector(x), kerncentres = c(x, extracentres),
init = as.vector(linit), optim = fit, conv = conv, cov = 1/fit$hessian, mle = fit$par,
se = se, nllh = fit$value, n = n, lambda = lambda, bw = kbw(lambda, kernel), kernel = kernel)
}
#' @export
#' @aliases fkden lkden nlkden
#' @rdname fkden
# cross-validation log-likelihood function for kernel density estimator
# will not stop evaluation unless it has to
lkden <- function(x, lambda = NULL, bw = NULL, kernel = "gaussian",
extracentres = NULL, log = TRUE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.param(lambda, allownull = TRUE)
check.param(bw, allownull = TRUE)
check.kernel(kernel)
check.logic(log)
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(extracentres)) {
if (any(!is.finite(extracentres))) warning("non-finite extracentres have been removed")
extracentres = extracentres[is.finite(extracentres)]
}
kerncentres = c(x, extracentres)
check.quant(kerncentres)
nk = length(kerncentres)
if (is.null(lambda) & is.null(bw)) {
if (nk == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (nk < 10) {
warning("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
bw = bw.nrd0(kerncentres)
}
if (is.null(lambda)) lambda = klambda(bw, kernel, lambda)
kdenxi <- function(i, kerncentres, lambda, kernel)
mean(kdz((x[i] - kerncentres[-i])/lambda, kernel))/lambda
if (lambda <= 0) {
l = -Inf
} else {
# evaluate likelihood for all points
# then correct for one left out to give CV-likelihood
cvl = sapply(1:n, kdenxi, kerncentres = kerncentres, lambda = lambda, kernel = kernel)
l = sum(log(cvl))
}
if (!log) l = exp(l)
l
}
#' @export
#' @aliases fkden lkden nlkden
#' @rdname fkden
# negative cross-validation log-likelihood function for kernel density estimator
# (wrapper for likelihood, inputs and checks designed for optimisation)
nlkden <- function(lambda, x, bw = NULL, kernel = "gaussian", extracentres = NULL,
finitelik = FALSE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.param(lambda, allownull = TRUE)
check.param(bw, allownull = TRUE)
check.kernel(kernel)
check.logic(finitelik)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
nllh = -lkden(x, lambda = lambda, bw = bw, kernel = kernel, extracentres = extracentres)
if (finitelik & is.infinite(nllh)) {
nllh = sign(nllh) * 1e6
}
nllh
}
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Defines functions fkden lkden nlkden
Documented in fkden lkden nlkden
#' @export
#'
#' @title Cross-validation MLE Fitting of Kernel Density Estimator, With Variety of Kernels
#'
#' @description Maximum (cross-validation) likelihood estimation for fitting kernel density estimator
#' for a variety of possible kernels, by treating it as a mixture model.
#'
#' @param linit initial value for bandwidth (as kernel half-width) or \code{NULL}
#' @param bwinit initial value for bandwidth (as kernel standard deviations) or \code{NULL}
#' @param extracentres extra kernel centres used in KDE,
#' but likelihood contribution not evaluated, or \code{NULL}
#' @param add.jitter logical, whether jitter is needed for rounded kernel centres
#' @param factor see \code{\link[base:jitter]{jitter}}
#' @param amount see \code{\link[base:jitter]{jitter}}
#' @inheritParams fnormgpd
#' @inheritParams kden
#' @inheritParams fgpd
#'
#' @details The kernel density estimator (KDE) with one of possible kernels is
#' fitted to the entire dataset using maximum (cross-validation) likelihood estimation.
#' The estimated bandwidth, variance and standard error are automatically output.
#'
#' The alternate bandwidth definitions are discussed in the
#' \code{\link[evmix:kernels]{kernels}}, with the \code{lambda} used here but
#' \code{bw} also output. 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}} help
#' documentation with the \code{"gaussian"} as the default choice.
#'
#' Missing values (\code{NA} and \code{NaN}) are assumed to be invalid data so are ignored.
#'
#' Cross-validation likelihood is used for kernel density component, obtained by
#' leaving each point out in turn and evaluating the KDE at the point left out:
#' \deqn{L(\lambda)\prod_{i=1}^{n} \hat{f}_{-i}(x_i)}
#' where
#' \deqn{\hat{f}_{-i}(x_i) = \frac{1}{(n-1)\lambda} \sum_{j=1: j\ne i}^{n} K(\frac{x_i - x_j}{\lambda})}
#' is the KDE obtained when the \eqn{i}th datapoint is dropped out and then
#' evaluated at that dropped datapoint at \eqn{x_i}.
#'
#' Normally for likelihood estimation of the bandwidth the kernel centres and
#' the data where the likelihood is evaluated are the same. However, when using
#' KDE for extreme value mixture modelling the likelihood only those data in the
#' bulk of the distribution should contribute to the likelihood, but all the
#' data (including those beyond the threshold) should contribute to the density
#' estimate. The \code{extracentres} option allows the use to specify extra
#' kernel centres used in estimating the density, but not evaluated in the
#' likelihood. Suppose the first \code{nb} data are below the threshold, followed
#' by \code{nu} exceedances of the threshold, so \eqn{i = 1,\ldots,nb, nb+1, \ldots, nb+nu}.
#' The cross-validation likelihood using the extra kernel centres is then:
#' \deqn{L(\lambda)\prod_{i=1}^{nb} \hat{f}_{-i}(x_i)}
#' where
#' \deqn{\hat{f}_{-i}(x_i) = \frac{1}{(nb+nu-1)\lambda} \sum_{j=1: j\ne i}^{nb+nu} K(\frac{x_i - x_j}{\lambda})}
#' which shows that the complete set of data is used in evaluating the KDE, but only those
#' below the threshold contribute to the cross-validation likelihood. The default is to
#' use the existing data, so \code{extracentres=NULL}.
#'
#' The following functions are provided:
#' \itemize{
#' \item \code{\link[evmix:fkden]{fkden}} - maximum (cross-validation) likelihood fitting with all the above options;
#' \item \code{\link[evmix:fkden]{lkden}} - cross-validation log-likelihood;
#' \item \code{\link[evmix:fkden]{nlkden}} - negative cross-validation log-likelihood;
#' }
#' The log-likelihood functions are provided for wider usage, e.g. constructing
#' profile likelihood functions.
#'
#' The log-likelihood and negative log-likelihood are also provided for wider
#' usage, e.g. constructing your own extreme value
#' mixture models or profile likelihood functions. The parameter
#' \code{lambda} must be specified in the negative log-likelihood
#' \code{\link[evmix:fkden]{nlkden}}.
#'
#' Log-likelihood calculations are carried out in
#' \code{\link[evmix:fkden]{lkden}}, which takes bandwidths as inputs in
#' the same form as distribution functions. The negative log-likelihood is a
#' wrapper for \code{\link[evmix:fkden]{lkden}}, designed towards making
#' it useable for optimisation (e.g. \code{lambda} given as first input).
#'
#' Defaults values for the bandwidth \code{linit} and \code{lambda} are given in the fitting
#' \code{\link[evmix:fkden]{fkden}} and cross-validation likelihood functions
#' \code{\link[evmix:fkden]{lkden}}. The bandwidth \code{linit} must be specified in
#' the negative log-likelihood function \code{\link[evmix:fkden]{nlkden}}.
#'
#' Missing values (\code{NA} and \code{NaN}) are assumed to be invalid data so are ignored,
#' which is inconsistent with the \code{\link[evd:fpot]{evd}} library which assumes the
#' missing values are below the threshold.
#'
#' The function \code{\link[evmix:fkden]{lkden}} carries out the calculations
#' for the log-likelihood directly, which can be exponentiated to give actual
#' likelihood using (\code{log=FALSE}).
#'
#' 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 or for common indicators of lack
#' of convergence (e.g. estimated bandwidth equal to initial value).
#'
#' 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}.
#'
#' @section Warning:
#' Two important practical issues arise with MLE for the kernel bandwidth:
#' 1) Cross-validation likelihood is needed for the KDE bandwidth parameter
#' as the usual likelihood degenerates, so that the MLE \eqn{\hat{\lambda} \rightarrow 0} as
#' \eqn{n \rightarrow \infty}, thus giving a negative bias towards a small bandwidth.
#' Leave one out cross-validation essentially ensures that some smoothing between the kernel centres
#' is required (i.e. a non-zero bandwidth), otherwise the resultant density estimates would always
#' be zero if the bandwidth was zero.
#'
#' This problem occassionally rears its ugly head for data which has been heavily rounded,
#' as even when using cross-validation the density can be non-zero even if the bandwidth is zero.
#' To overcome this issue an option to add a small jitter should be added to the data
#' (\code{x} only) has been included in the fitting inputs, using the
#' \code{\link[base:jitter]{jitter}} function, to remove the ties. The default options red in the
#' \code{\link[base:jitter]{jitter}} are specified above, but the user can override these.
#' Notice the default scaling \code{factor=0.1}, which is a tenth of the default value in the
#' \code{\link[base:jitter]{jitter}}
#' function itself.
#'
#' A warning message is given if the data appear to be rounded
#' (i.e. more than 5% of data are tied). If the estimated bandwidth is too small, then
#' data rounding is the likely culprit. Only use the jittering when the MLE of
#' the bandwidth is far too small.
#'
#' 2) For heavy tailed populations the bandwidth is positively biased, giving oversmoothing
#' (see example). The bias is due to the distance between the upper (or lower) order statistics not
#' necessarily decaying to zero as the sample size tends to infinity. Essentially, as the distance
#' between the two largest (or smallest) sample datapoints does not decay to zero, some smoothing between
#' them is required (i.e. bandwidth cannot be zero). One solution to this problem is to trim
#' the data at a suitable threshold to remove the problematic tail from the inference for the bandwidth,
#' using either the \code{\link[evmix:fkdengpd]{fkdengpd}} function for a single heavy tail
#' or the \code{\link[evmix:fgkg]{fgkg}} function
#' if both tails are heavy. See MacDonald et al (2013).
#'
#' @return Log-likelihood is given by \code{\link[evmix:fkden]{lkden}} and it's
#' wrappers for negative log-likelihood from \code{\link[evmix:fkden]{nlkden}}.
#' Fitting function \code{\link[evmix:fkden]{fkden}} returns a simple list with the
#' following elements
#'
#' \tabular{ll}{
#' \code{call}: \tab \code{optim} call\cr
#' \code{x}: \tab (jittered) data vector \code{x}\cr
#' \code{kerncentres}: \tab actual kernel centres used \code{x}\cr
#' \code{init}: \tab \code{linit} for lambda\cr
#' \code{optim}: \tab complete \code{optim} output\cr
#' \code{mle}: \tab vector of MLE of bandwidth\cr
#' \code{cov}: \tab variance of MLE of bandwidth\cr
#' \code{se}: \tab standard error of MLE of bandwidth\cr
#' \code{nllh}: \tab minimum negative cross-validation log-likelihood\cr
#' \code{n}: \tab total sample size\cr
#' \code{lambda}: \tab MLE of lambda (kernel half-width)\cr
#' \code{bw}: \tab MLE of bw (kernel standard deviations)\cr
#' \code{kernel}: \tab kernel name\cr
#' }
#'
#' @note When \code{linit=NULL} then the initial value for the \code{lambda}
#' bandwidth is calculated
#' using \code{\link[stats:bandwidth]{bw.nrd0}} function and transformed using
#' \code{\link[evmix:kfun]{klambda}} function.
#'
#' The extra kernel centres \code{extracentres} can either be a vector of data or \code{NULL}.
#'
#' Invalid parameter ranges will give \code{0} for likelihood, \code{log(0)=-Inf} for
#' log-likelihood and \code{-log(0)=Inf} for negative log-likelihood.
#'
#' Infinite and missing sample values are dropped.
#'
#' 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/Kernel_density_estimation}
#'
#' \url{http://en.wikipedia.org/wiki/Cross-validation_(statistics)}
#'
#' 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. 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.
#'
#' 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.
#'
#' MacDonald, A., C. J. Scarrott, and D. S. Lee (2011). Boundary correction, consistency
#' and robustness of kernel densities using extreme value theory. Submitted.
#' Available from: \url{http://www.math.canterbury.ac.nz/~c.scarrott}.
#'
#' 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[base:jitter]{jitter}}, \code{\link[stats:density]{density}} and
#' \code{\link[stats:bandwidth]{bw.nrd0}}
#'
#' @aliases fkden lkden nlkden
#' @family kden
#' @family kdengpd
#' @family bckden
#' @family fkden
#'
#' @examples
#' \dontrun{
#' set.seed(1)
#' par(mfrow = c(1, 1))
#'
#' nk=50
#' x = rnorm(nk)
#' xx = seq(-5, 5, 0.01)
#' fit = fkden(x)
#' hist(x, nk/5, freq = FALSE, xlim = c(-5, 5), ylim = c(0,0.6))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$lambda)*0.05)
#' lines(xx,dnorm(xx), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
#' lines(density(x), lty = 2, lwd = 2, col = "green")
#' lines(density(x, bw = fit$bw), lwd = 2, lty = 2, col = "blue")
#' legend("topright", c("True Density", "KDE fitted evmix",
#' "KDE Using density, default bandwidth", "KDE Using density, c-v likelihood bandwidth"),
#' lty = c(1, 1, 2, 2), lwd = c(1, 2, 2, 2), col = c("black", "red", "green", "blue"))
#'
#' par(mfrow = c(2, 1))
#'
#' # bandwidth is biased towards oversmoothing for heavy tails
#' nk=100
#' x = rt(nk, df = 2)
#' xx = seq(-8, 8, 0.01)
#' fit = fkden(x)
#' hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$lambda)*0.05)
#' lines(xx,dt(xx , df = 2), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
#' legend("topright", c("True Density", "KDE fitted evmix, c-v likelihood bandwidth"),
#' lty = c(1, 1), lwd = c(1, 2), col = c("black", "red"))
#'
#' # remove heavy tails from cv-likelihood evaluation, but still include them in KDE within likelihood
#' # often gives better bandwidth (see MacDonald et al (2011) for justification)
#' nk=100
#' x = rt(nk, df = 2)
#' xx = seq(-8, 8, 0.01)
#' fit2 = fkden(x[(x > -4) & (x < 4)], extracentres = x[(x <= -4) | (x >= 4)])
#' hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10))
#' rug(x)
#' for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit2$lambda)*0.05)
#' lines(xx,dt(xx , df = 2), col = "black")
#' lines(xx, dkden(xx, x, lambda = fit2$lambda), lwd = 2, col = "red")
#' lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "blue")
#' legend("topright", c("True Density", "KDE fitted evmix, tails removed",
#' "KDE fitted evmix, tails included"),
#' lty = c(1, 1, 1), lwd = c(1, 2, 2), col = c("black", "red", "blue"))
#' }
#'
# maximum cross-validation likelihood fitting for kernel density estimator
fkden <- function(x, linit = NULL, bwinit = NULL, kernel = "gaussian", extracentres = NULL,
add.jitter = FALSE, factor = 0.1, amount = NULL,
std.err = TRUE, method = "BFGS", control = list(maxit = 10000), finitelik = TRUE, ...) {
call <- match.call()
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.kbw(linit, bwinit, allownull = TRUE)
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 (!is.null(extracentres)) {
if (any(!is.finite(extracentres))) warning("non-finite extracentres have been removed")
extracentres = extracentres[is.finite(extracentres)]
check.quant(extracentres)
}
if ((method == "L-BFGS-B") | (method == "BFGS")) finitelik = TRUE
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 (is.null(linit) & is.null(bwinit)) {
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")
}
bwinit = bw.nrd0(x)
}
linit = klambda(bwinit, kernel, linit)
nllh = nlkden(linit, x = x, kernel = kernel, extracentres = extracentres)
if (is.infinite(nllh)) stop("initial bandwidth is invalid")
fit = optim(par = linit, fn = nlkden, x = x, kernel = kernel, extracentres = extracentres,
finitelik = finitelik, method = method, control = control, hessian = TRUE, ...)
conv = TRUE
if ((fit$convergence != 0) | any(fit$par == linit) | (abs(fit$value) >= 1e6)) {
conv = FALSE
warning("check convergence")
}
lambda = fit$par[1]
if (conv & std.err) {
se = sqrt(1/fit$hessian)
} else {
se = NULL
}
list(call = call, x = as.vector(x), kerncentres = c(x, extracentres),
init = as.vector(linit), optim = fit, conv = conv, cov = 1/fit$hessian, mle = fit$par,
se = se, nllh = fit$value, n = n, lambda = lambda, bw = kbw(lambda, kernel), kernel = kernel)
}
#' @export
#' @aliases fkden lkden nlkden
#' @rdname fkden
# cross-validation log-likelihood function for kernel density estimator
# will not stop evaluation unless it has to
lkden <- function(x, lambda = NULL, bw = NULL, kernel = "gaussian",
extracentres = NULL, log = TRUE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.param(lambda, allownull = TRUE)
check.param(bw, allownull = TRUE)
check.kernel(kernel)
check.logic(log)
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(extracentres)) {
if (any(!is.finite(extracentres))) warning("non-finite extracentres have been removed")
extracentres = extracentres[is.finite(extracentres)]
}
kerncentres = c(x, extracentres)
check.quant(kerncentres)
nk = length(kerncentres)
if (is.null(lambda) & is.null(bw)) {
if (nk == 1) {
stop("Automated bandwidth estimation requires 2 or more kernel centres")
} else if (nk < 10) {
warning("Automated bandwidth estimation unreliable with less than 10 kernel centres")
}
bw = bw.nrd0(kerncentres)
}
if (is.null(lambda)) lambda = klambda(bw, kernel, lambda)
kdenxi <- function(i, kerncentres, lambda, kernel)
mean(kdz((x[i] - kerncentres[-i])/lambda, kernel))/lambda
if (lambda <= 0) {
l = -Inf
} else {
# evaluate likelihood for all points
# then correct for one left out to give CV-likelihood
cvl = sapply(1:n, kdenxi, kerncentres = kerncentres, lambda = lambda, kernel = kernel)
l = sum(log(cvl))
}
if (!log) l = exp(l)
l
}
#' @export
#' @aliases fkden lkden nlkden
#' @rdname fkden
# negative cross-validation log-likelihood function for kernel density estimator
# (wrapper for likelihood, inputs and checks designed for optimisation)
nlkden <- function(lambda, x, bw = NULL, kernel = "gaussian", extracentres = NULL,
finitelik = FALSE) {
# Check properties of inputs
check.quant(x, allowna = TRUE, allowinf = TRUE)
check.quant(extracentres, allownull = TRUE, allowna = TRUE, allowinf = TRUE)
check.param(lambda, allownull = TRUE)
check.param(bw, allownull = TRUE)
check.kernel(kernel)
check.logic(finitelik)
kernel = ifelse(kernel == "rectangular", "uniform", kernel)
kernel = ifelse(kernel == "normal", "gaussian", kernel)
nllh = -lkden(x, lambda = lambda, bw = bw, kernel = kernel, extracentres = extracentres)
if (finitelik & is.infinite(nllh)) {
nllh = sign(nllh) * 1e6
}
nllh
}
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