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R/mqdedist.R
In BMT: The BMT Distribution
#' @title Minimum Quantile Distance Fit of Univariate Distributions.
#' @description Fit of univariate distributions for non-censored data using
#' minimum quantile distance estimation (mqde), which can also be called
#' maximum quantile goodness-of-fit estimation.
#' @rdname mqdedist
#' @name mqdedist
#'
#' @details The \code{mqdedist} function carries out the minimum quantile
#' distance estimation numerically, by minimization of a distance between
#' observed and theoretical quantiles.
#'
#' The optimization process is the same as
#' \code{\link{mledist}}, see the 'details' section of that
#' function.
#'
#' Optionally, a vector of \code{weights} can be used in the fitting process.
#' By default (when \code{weigths=NULL}), ordinary mqde is carried out,
#' otherwise the specified weights are used to compute a weighted distance.
#'
#' We believe this function should be added to the package
#' \code{\link{fitdistrplus}}. Until it is accepted and incorporated into that
#' package, it will remain in the package \code{\link{BMT}}. This function is
#' internally called in \code{\link{BMTfit.mqde}}.
#'
#' @param data A numeric vector with the observed values for non-censored data.
#' @param distr A character string \code{"name"} naming a distribution for which
#' the corresponding quantile function \code{qname} and the corresponding
#' density distribution \code{dname} must be classically defined.
#' @param probs A numeric vector of the probabilities for which the minimum
#' quantile distance estimation is done. \eqn{p[k] = (k - 0.5) / n} (default).
#' @param qtype The quantile type used by the R \code{\link{quantile}} function
#' to compute the empirical quantiles. Type 5 (default), i.e. \eqn{x[k]} is
#' both the \eqn{k}th order statistic and the type 5 sample quantile of
#' \eqn{p[k] = (k - 0.5) / n}.
#' @param dist The distance measure between observed and theoretical quantiles
#' to be used. This must be one of "euclidean" (default), "maximum", or
#' "manhattan". Any unambiguous substring can be given.
#' @param start A named list giving the initial values of parameters of the
#' named distribution or a function of data computing initial values and
#' returning a named list. This argument may be omitted (default) for some
#' distributions for which reasonable starting values are computed (see the
#' 'details' section of \code{\link{mledist}}).
#' @param fix.arg An optional named list giving the values of fixed parameters
#' of the named distribution or a function of data computing (fixed) parameter
#' values and returning a named list. Parameters with fixed value are thus NOT
#' estimated.
#' @param optim.method \code{"default"} (see details) or optimization method to pass to
#' \code{\link{optim}}.
#' @param lower Left bounds on the parameters for the \code{"L-BFGS-B"} method
#' (see \code{\link{optim}}) or the \code{\link{constrOptim}} function (as an
#' equivalent linear constraint).
#' @param upper Right bounds on the parameters for the \code{"L-BFGS-B"} method
#' (see \code{\link{optim}}) or the \code{\link{constrOptim}} function (as an
#' equivalent linear constraint).
#' @param custom.optim A function carrying the optimization (see details).
#' @param weights An optional vector of weights to be used in the fitting
#' process. Should be \code{NULL} or a numeric vector with strictly positive
#' numbers. If non-\code{NULL}, weighted mqde is used, otherwise ordinary
#' mqde.
#' @param silent A logical to remove or show warnings when bootstraping.
#' @param gradient A function to return the gradient of the optimization
#' objective function for the \code{"BFGS"}, \code{"CG"} and \code{"L-BFGS-B"}
#' methods. If it is \code{NULL}, a finite-difference approximation will be
#' used, see \code{\link{optim}}.
#' @param \dots Further arguments passed to the \code{\link{optim}},
#' \code{\link{constrOptim}} or \code{custom.optim} function.
#'
#' @return \code{mqdedist} returns a list with following components,
#'
#' \item{estimate}{ the parameter estimates.}
#'
#' \item{convergence}{ an integer code for the convergence of
#' \code{\link{optim}} defined as below or defined by the user in the
#' user-supplied optimization function.
#'
#' \code{0} indicates successful convergence.
#'
#' \code{1} indicates that the iteration limit of \code{\link{optim}} has been
#' reached.
#'
#' \code{10} indicates degeneracy of the Nealder-Mead simplex.
#'
#' \code{100} indicates that \code{\link{optim}} encountered an internal
#' error. }
#'
#' \item{value}{the value of the optimization objective function at the solution found.}
#'
#' \item{hessian}{ a symmetric matrix computed by \code{\link{optim}} as an
#' estimate of the Hessian at the solution found or computed in the
#' user-supplied optimization function. }
#'
#' \item{probs}{ the probability vector on which observed and theoretical quantiles were calculated. }
#'
#' \item{dist}{ the name of the distance between observed and theoretical quantiles used. }
#'
#' \item{optim.function }{ the name of the optimization function used. }
#'
#' \item{fix.arg}{ the named list giving the values of parameters of the named
#' distribution that must kept fixed rather than estimated by maximum
#' likelihood or \code{NULL} if there are no such parameters. }
#'
#' \item{loglik}{ the log-likelihood. }
#'
#' \item{optim.method}{when \code{\link{optim}} is used, the name of the
#' algorithm used, \code{NULL} otherwise.}
#'
#' \item{fix.arg.fun}{the function used to set the value of \code{fix.arg} or
#' \code{NULL}.}
#'
#' \item{weights}{the vector of weigths used in the estimation process or
#' \code{NULL}.}
#'
#' \item{counts}{A two-element integer vector giving the number of calls to
#' the log-likelihood function and its gradient respectively. This excludes
#' those calls needed to compute the Hessian, if requested, and any calls to
#' log-likelihood function to compute a finite-difference approximation to the
#' gradient. \code{counts} is returned by \code{\link{optim}} or the
#' user-supplied optimization function, or set to \code{NULL}.}
#'
#' \item{optim.message}{A character string giving any additional information
#' returned by the optimizer, or \code{NULL}. To understand exactly the
#' message, see the source code.}
#'
#' @references LaRiccia, V. N. (1982). \emph{Asymptotic Properties of Weighted
#' $L^2$ Quantile Distance Estimators}. The Annals of Statistics, 10(2),
#' 621-624.
#'
#' Torres-Jimenez, C. J. (2017, September), \emph{Comparison of estimation methods
#' for the BMT distribution}. ArXiv e-prints.
#'
#' @seealso \code{\link{mpsedist}}, \code{\link{mledist}},
#' \code{\link{mmedist}}, \code{\link{qmedist}},
#' \code{\link{mgedist}}, \code{\link{optim}},
#' \code{\link{constrOptim}}, and \code{\link{quantile}}.
#'
#' @author Camilo Jose Torres-Jimenez [aut,cre] \email{cjtorresj@unal.edu.co}
#'
#' @source Based on the function mledist of the R package: fitdistrplus
#'
#' Delignette-Muller ML and Dutang C (2015), \emph{fitdistrplus: An R Package
#' for Fitting Distributions}. Journal of Statistical Software, 64(4), 1-34.
#'
#' Functions \code{checkparam} and \code{start.arg.default} are needed and
#' were copied from the same package (fitdistrplus version: 1.0-9).
#'
#' @examples
#' # (1) basic fit of a normal distribution
#' set.seed(1234)
#' x1 <- rnorm(n = 100)
#' mean(x1); sd(x1)
#' mqde1 <- mqdedist(x1, "norm")
#' mqde1$estimate
#'
#' # (2) defining your own distribution functions, here for the Gumbel
#' # distribution for other distributions, see the CRAN task view dedicated
#' # to probability distributions
#' dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
#' pgumbel <- function(q, a, b) exp(-exp((a-q)/b))
#' qgumbel <- function(p, a, b) a-b*log(-log(p))
#' mqde1 <- mqdedist(x1, "gumbel", start = list(a = 10, b = 5))
#' mqde1$estimate
#'
#' # (3) fit a discrete distribution (Poisson)
#' set.seed(1234)
#' x2 <- rpois(n = 30, lambda = 2)
#' mqde2 <- mqdedist(x2, "pois")
#' mqde2$estimate
#'
#' # (4) fit a finite-support distribution (beta)
#' set.seed(1234)
#' x3 <- rbeta(n = 100, shape1 = 5, shape2 = 10)
#' mqde3 <- mqdedist(x3, "beta")
#' mqde3$estimate
#'
#' # (5) fit frequency distributions on USArrests dataset.
#' x4 <- USArrests$Assault
#' mqde4pois <- mqdedist(x4, "pois")
#' mqde4pois$estimate
#' mqde4nbinom <- mqdedist(x4, "nbinom")
#' mqde4nbinom$estimate
#'
#' # (6) weighted fit of a normal distribution
#' set.seed(1234)
#' w1 <- runif(100)
#' weighted.mean(x1, w1)
#' mqde1 <- mqdedist(x1, "norm", weights = w1)
#' mqde1$estimate
#'
#' @keywords distribution
###################
#' @rdname mqdedist
#' @export mqdedist
mqdedist <- function (data, distr, probs = (1:length(data)-0.5)/length(data), qtype = 5,
dist = "euclidean", start = NULL, fix.arg = NULL, optim.method = "default",
lower = -Inf, upper = Inf, custom.optim = NULL, weights = NULL,
silent = TRUE, gradient = NULL, ...)
{
if (!is.character(distr))
stop("distr must be a character string naming a distribution")
else distname <- distr
qdistname <- paste("q", distname, sep = "")
if (!exists(qdistname, mode = "function"))
stop(paste("The ", qdistname, " function must be defined"))
ddistname <- paste("d", distname, sep = "")
if (!exists(ddistname, mode = "function"))
stop(paste("The ", ddistname, " function must be defined"))
if (!(is.vector(probs) & is.numeric(probs)) | anyNA(probs) | any(probs <= 0 | probs >= 1))
stop("probs must be a numeric vector with all elements greater than zero and less than one")
probs <- unique(sort(probs))
if (qtype < 0 || qtype > 9)
stop("wrong type for the R quantile function")
if (is.null(custom.optim))
optim.method <- match.arg(optim.method, c("default", "Nelder-Mead", "BFGS", "CG",
"L-BFGS-B", "SANN", "Brent"))
int.dist <- pmatch(dist, c("euclidean", "maximum", "manhattan"))
if (is.na(int.dist))
stop("invalid distance measure to be used")
if (int.dist == -1)
stop("ambiguous distance measure to be used")
start.arg <- start
if (is.vector(start.arg))
start.arg <- as.list(start.arg)
my3dots <- list(...)
if (!is.null(weights)) {
if (any(weights <= 0))
stop("weights should be a vector of numbers greater than 0")
if (length(weights) != NROW(probs))
stop("weights should be a vector with a length equal to the the probabilities probs")
warning("weights are not taken into account in the default initial values")
}
if (is.vector(data)) {
cens <- FALSE
if (!(is.numeric(data) & length(data) > 1))
stop("data must be a numeric vector of length greater than 1 for non-censored data")
# \n or a dataframe with two columns named left and right and more than one line for censored data")
}
else {
stop("Minimum quantile distance estimation is not yet available for censored data.")
# cens <- TRUE
# censdata <- data
# if (!(is.vector(censdata$left) & is.vector(censdata$right) &
# length(censdata[, 1]) > 1))
# stop("data must be a numeric vector of length greater than 1 for non censored data\n or a dataframe with two columns named left and right and more than one line for censored data")
# pdistname <- paste("p", distname, sep = "")
# if (!exists(pdistname, mode = "function"))
# stop(paste("The ", pdistname, " function must be defined to apply maximum likelihood to censored data"))
}
# if (cens) {
# lcens <- censdata[is.na(censdata$left), ]$right
# if (any(is.na(lcens)))
# stop("An observation cannot be both right and left censored, coded with two NA values")
# rcens <- censdata[is.na(censdata$right), ]$left
# ncens <- censdata[censdata$left == censdata$right &
# !is.na(censdata$left) & !is.na(censdata$right),
# ]$left
# icens <- censdata[censdata$left != censdata$right &
# !is.na(censdata$left) & !is.na(censdata$right),
# ]
# data <- c(rcens, lcens, ncens, (icens$left + icens$right)/2)
# }
argqdistname <- names(formals(qdistname))
chfixstt <- checkparam(start.arg = start.arg, fix.arg = fix.arg,
argdistname = argqdistname, errtxt = NULL,
data10 = head(data, 10), distname = distname)
if (!chfixstt$ok)
stop(chfixstt$txt)
if (is.function(chfixstt$start.arg))
vstart <- unlist(chfixstt$start.arg(data))
else vstart <- unlist(chfixstt$start.arg)
if (is.function(fix.arg)) {
fix.arg.fun <- fix.arg
fix.arg <- fix.arg(data)
}
else fix.arg.fun <- NULL
if(qtype == 0)
s <- data
else
s <- quantile(data, probs = probs, type = qtype, names = FALSE)
# if (!cens) {
if(is.null(weights)){
if (int.dist == 1) # euclidean
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum((obs - theoq)^2)
}
else if (int.dist == 2) # maximum
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
max(abs(obs - theoq))
}
else if (int.dist == 3) # manhattan
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(abs(obs - theoq))
}
}
else{
if (int.dist == 1) # euclidean
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(weights*(obs - theoq)^2)
}
else if (int.dist == 2) # maximum
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
max(weights*abs(obs - theoq))
}
else if (int.dist == 3) # manhattan
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(weights*abs(obs - theoq))
}
}
# }
# else stop("Minimum quantile distance estimation is not yet available for censored data.")
owarn <- getOption("warn")
if (is.null(custom.optim)) {
hasbound <- any(is.finite(lower) | is.finite(upper))
if (optim.method == "default") {
meth <- ifelse(length(vstart) > 1, "Nelder-Mead", "BFGS")
}
else meth <- optim.method
if (meth == "BFGS" && hasbound && is.null(gradient)) {
meth <- "L-BFGS-B"
txt1 <- "The BFGS method cannot be used with bounds without provided the gradient."
txt2 <- "The method is changed to L-BFGS-B."
warning(paste(txt1, txt2))
}
options(warn = ifelse(silent, -1, 0))
if (hasbound) {
if (!is.null(gradient)) {
opt.fun <- "constrOptim"
}
else {
if (meth == "Nelder-Mead")
opt.fun <- "constrOptim"
else if (meth %in% c("L-BFGS-B", "Brent"))
opt.fun <- "optim"
else {
txt1 <- paste("The method", meth, "cannot be used by constrOptim() nor optim() without gradient and bounds.")
txt2 <- "Only optimization methods L-BFGS-B, Brent and Nelder-Mead can be used in such case."
stop(paste(txt1, txt2))
}
}
if (opt.fun == "constrOptim") {
npar <- length(vstart)
lower <- as.double(rep_len(lower, npar))
upper <- as.double(rep_len(upper, npar))
haslow <- is.finite(lower)
Mat <- diag(npar)[haslow, ]
hasupp <- is.finite(upper)
Mat <- rbind(Mat, -diag(npar)[hasupp, ])
colnames(Mat) <- names(vstart)
rownames(Mat) <- paste0("constr", 1:NROW(Mat))
Bnd <- c(lower[is.finite(lower)], -upper[is.finite(upper)])
names(Bnd) <- paste0("constr", 1:length(Bnd))
initconstr <- Mat %*% vstart - Bnd
if (any(initconstr < 0))
stop("Starting values must be in the feasible region.")
opttryerror <- try(opt <- constrOptim(theta = vstart,
f = fnobj, ui = Mat, ci = Bnd, grad = gradient,
fix.arg = fix.arg, obs = s, qdistnam = qdistname,
probs = probs, weights = weights,
hessian = !is.null(gradient), method = meth,
...), silent = TRUE)
if (!inherits(opttryerror, "try-error"))
if (length(opt$counts) == 1)
opt$counts <- c(opt$counts, NA)
}
else {
opttryerror <- try(opt <- optim(par = vstart,
fn = fnobj, fix.arg = fix.arg, obs = s,
gr = gradient, qdistnam = qdistname,
probs = probs, weights = weights, hessian = TRUE,
method = meth, lower = lower, upper = upper,
...), silent = TRUE)
}
}
else {
opt.fun <- "optim"
opttryerror <- try(opt <- optim(par = vstart, fn = fnobj,
fix.arg = fix.arg, obs = s, gr = gradient,
qdistnam = qdistname, probs = probs, weights = weights,
hessian = TRUE, method = meth,
lower = lower, upper = upper, ...), silent = TRUE)
}
options(warn = owarn)
if (inherits(opttryerror, "try-error")) {
warnings("The function optim encountered an error and stopped.")
if (getOption("show.error.messages"))
print(attr(opttryerror, "condition"))
return(list(estimate = rep(NA, length(vstart)),
convergence = 100, loglik = NA, hessian = NA))
}
if (opt$convergence > 0) {
warnings("The function optim failed to converge, with the error code ",
opt$convergence)
}
if (is.null(names(opt$par)))
names(opt$par) <- names(vstart)
res <- list(estimate = opt$par, convergence = opt$convergence, value = opt$value,
hessian = opt$hessian, probs = probs, dist = dist,
optim.function = opt.fun, fix.arg = fix.arg,
loglik = .loglik(opt$par, fix.arg, data, ddistname),
optim.method = meth, fix.arg.fun = fix.arg.fun, weights = weights,
counts = opt$counts, optim.message = opt$message)
}
else {
options(warn = ifelse(silent, -1, 0))
# if (!cens)
opttryerror <- try(opt <- custom.optim(fn = fnobj,
fix.arg = fix.arg, obs = s, qdistnam = qdistname,
probs = probs, weights = weights,
par = vstart, ...), silent = TRUE)
# else stop("Maximum goodness-of-fit estimation is not yet available for censored data.")
options(warn = owarn)
if (inherits(opttryerror, "try-error")) {
warnings("The customized optimization function encountered an error and stopped.")
if (getOption("show.error.messages"))
print(attr(opttryerror, "condition"))
return(list(estimate = rep(NA, length(vstart)),
convergence = 100, value = NA, hessian = NA))
}
if (opt$convergence > 0) {
warnings("The customized optimization function failed to converge, with the error code ",
opt$convergence)
}
if (is.null(names(opt$par)))
names(opt$par) <- names(vstart)
res <- list(estimate = opt$par, convergence = opt$convergence, value = opt$value,
probs = probs, dist = dist, hessian = opt$hessian,
optim.function = custom.optim, fix.arg = fix.arg,
loglik = .loglik(opt$par, fix.arg, data, ddistname),
optim.method = NULL, fix.arg.fun = fix.arg.fun, weights = weights,
counts = opt$counts, optim.message = opt$message)
}
return(res)
}
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#' @title Minimum Quantile Distance Fit of Univariate Distributions.
#' @description Fit of univariate distributions for non-censored data using
#' minimum quantile distance estimation (mqde), which can also be called
#' maximum quantile goodness-of-fit estimation.
#' @rdname mqdedist
#' @name mqdedist
#'
#' @details The \code{mqdedist} function carries out the minimum quantile
#' distance estimation numerically, by minimization of a distance between
#' observed and theoretical quantiles.
#'
#' The optimization process is the same as
#' \code{\link{mledist}}, see the 'details' section of that
#' function.
#'
#' Optionally, a vector of \code{weights} can be used in the fitting process.
#' By default (when \code{weigths=NULL}), ordinary mqde is carried out,
#' otherwise the specified weights are used to compute a weighted distance.
#'
#' We believe this function should be added to the package
#' \code{\link{fitdistrplus}}. Until it is accepted and incorporated into that
#' package, it will remain in the package \code{\link{BMT}}. This function is
#' internally called in \code{\link{BMTfit.mqde}}.
#'
#' @param data A numeric vector with the observed values for non-censored data.
#' @param distr A character string \code{"name"} naming a distribution for which
#' the corresponding quantile function \code{qname} and the corresponding
#' density distribution \code{dname} must be classically defined.
#' @param probs A numeric vector of the probabilities for which the minimum
#' quantile distance estimation is done. \eqn{p[k] = (k - 0.5) / n} (default).
#' @param qtype The quantile type used by the R \code{\link{quantile}} function
#' to compute the empirical quantiles. Type 5 (default), i.e. \eqn{x[k]} is
#' both the \eqn{k}th order statistic and the type 5 sample quantile of
#' \eqn{p[k] = (k - 0.5) / n}.
#' @param dist The distance measure between observed and theoretical quantiles
#' to be used. This must be one of "euclidean" (default), "maximum", or
#' "manhattan". Any unambiguous substring can be given.
#' @param start A named list giving the initial values of parameters of the
#' named distribution or a function of data computing initial values and
#' returning a named list. This argument may be omitted (default) for some
#' distributions for which reasonable starting values are computed (see the
#' 'details' section of \code{\link{mledist}}).
#' @param fix.arg An optional named list giving the values of fixed parameters
#' of the named distribution or a function of data computing (fixed) parameter
#' values and returning a named list. Parameters with fixed value are thus NOT
#' estimated.
#' @param optim.method \code{"default"} (see details) or optimization method to pass to
#' \code{\link{optim}}.
#' @param lower Left bounds on the parameters for the \code{"L-BFGS-B"} method
#' (see \code{\link{optim}}) or the \code{\link{constrOptim}} function (as an
#' equivalent linear constraint).
#' @param upper Right bounds on the parameters for the \code{"L-BFGS-B"} method
#' (see \code{\link{optim}}) or the \code{\link{constrOptim}} function (as an
#' equivalent linear constraint).
#' @param custom.optim A function carrying the optimization (see details).
#' @param weights An optional vector of weights to be used in the fitting
#' process. Should be \code{NULL} or a numeric vector with strictly positive
#' numbers. If non-\code{NULL}, weighted mqde is used, otherwise ordinary
#' mqde.
#' @param silent A logical to remove or show warnings when bootstraping.
#' @param gradient A function to return the gradient of the optimization
#' objective function for the \code{"BFGS"}, \code{"CG"} and \code{"L-BFGS-B"}
#' methods. If it is \code{NULL}, a finite-difference approximation will be
#' used, see \code{\link{optim}}.
#' @param \dots Further arguments passed to the \code{\link{optim}},
#' \code{\link{constrOptim}} or \code{custom.optim} function.
#'
#' @return \code{mqdedist} returns a list with following components,
#'
#' \item{estimate}{ the parameter estimates.}
#'
#' \item{convergence}{ an integer code for the convergence of
#' \code{\link{optim}} defined as below or defined by the user in the
#' user-supplied optimization function.
#'
#' \code{0} indicates successful convergence.
#'
#' \code{1} indicates that the iteration limit of \code{\link{optim}} has been
#' reached.
#'
#' \code{10} indicates degeneracy of the Nealder-Mead simplex.
#'
#' \code{100} indicates that \code{\link{optim}} encountered an internal
#' error. }
#'
#' \item{value}{the value of the optimization objective function at the solution found.}
#'
#' \item{hessian}{ a symmetric matrix computed by \code{\link{optim}} as an
#' estimate of the Hessian at the solution found or computed in the
#' user-supplied optimization function. }
#'
#' \item{probs}{ the probability vector on which observed and theoretical quantiles were calculated. }
#'
#' \item{dist}{ the name of the distance between observed and theoretical quantiles used. }
#'
#' \item{optim.function }{ the name of the optimization function used. }
#'
#' \item{fix.arg}{ the named list giving the values of parameters of the named
#' distribution that must kept fixed rather than estimated by maximum
#' likelihood or \code{NULL} if there are no such parameters. }
#'
#' \item{loglik}{ the log-likelihood. }
#'
#' \item{optim.method}{when \code{\link{optim}} is used, the name of the
#' algorithm used, \code{NULL} otherwise.}
#'
#' \item{fix.arg.fun}{the function used to set the value of \code{fix.arg} or
#' \code{NULL}.}
#'
#' \item{weights}{the vector of weigths used in the estimation process or
#' \code{NULL}.}
#'
#' \item{counts}{A two-element integer vector giving the number of calls to
#' the log-likelihood function and its gradient respectively. This excludes
#' those calls needed to compute the Hessian, if requested, and any calls to
#' log-likelihood function to compute a finite-difference approximation to the
#' gradient. \code{counts} is returned by \code{\link{optim}} or the
#' user-supplied optimization function, or set to \code{NULL}.}
#'
#' \item{optim.message}{A character string giving any additional information
#' returned by the optimizer, or \code{NULL}. To understand exactly the
#' message, see the source code.}
#'
#' @references LaRiccia, V. N. (1982). \emph{Asymptotic Properties of Weighted
#' $L^2$ Quantile Distance Estimators}. The Annals of Statistics, 10(2),
#' 621-624.
#'
#' Torres-Jimenez, C. J. (2017, September), \emph{Comparison of estimation methods
#' for the BMT distribution}. ArXiv e-prints.
#'
#' @seealso \code{\link{mpsedist}}, \code{\link{mledist}},
#' \code{\link{mmedist}}, \code{\link{qmedist}},
#' \code{\link{mgedist}}, \code{\link{optim}},
#' \code{\link{constrOptim}}, and \code{\link{quantile}}.
#'
#' @author Camilo Jose Torres-Jimenez [aut,cre] \email{cjtorresj@unal.edu.co}
#'
#' @source Based on the function mledist of the R package: fitdistrplus
#'
#' Delignette-Muller ML and Dutang C (2015), \emph{fitdistrplus: An R Package
#' for Fitting Distributions}. Journal of Statistical Software, 64(4), 1-34.
#'
#' Functions \code{checkparam} and \code{start.arg.default} are needed and
#' were copied from the same package (fitdistrplus version: 1.0-9).
#'
#' @examples
#' # (1) basic fit of a normal distribution
#' set.seed(1234)
#' x1 <- rnorm(n = 100)
#' mean(x1); sd(x1)
#' mqde1 <- mqdedist(x1, "norm")
#' mqde1$estimate
#'
#' # (2) defining your own distribution functions, here for the Gumbel
#' # distribution for other distributions, see the CRAN task view dedicated
#' # to probability distributions
#' dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
#' pgumbel <- function(q, a, b) exp(-exp((a-q)/b))
#' qgumbel <- function(p, a, b) a-b*log(-log(p))
#' mqde1 <- mqdedist(x1, "gumbel", start = list(a = 10, b = 5))
#' mqde1$estimate
#'
#' # (3) fit a discrete distribution (Poisson)
#' set.seed(1234)
#' x2 <- rpois(n = 30, lambda = 2)
#' mqde2 <- mqdedist(x2, "pois")
#' mqde2$estimate
#'
#' # (4) fit a finite-support distribution (beta)
#' set.seed(1234)
#' x3 <- rbeta(n = 100, shape1 = 5, shape2 = 10)
#' mqde3 <- mqdedist(x3, "beta")
#' mqde3$estimate
#'
#' # (5) fit frequency distributions on USArrests dataset.
#' x4 <- USArrests$Assault
#' mqde4pois <- mqdedist(x4, "pois")
#' mqde4pois$estimate
#' mqde4nbinom <- mqdedist(x4, "nbinom")
#' mqde4nbinom$estimate
#'
#' # (6) weighted fit of a normal distribution
#' set.seed(1234)
#' w1 <- runif(100)
#' weighted.mean(x1, w1)
#' mqde1 <- mqdedist(x1, "norm", weights = w1)
#' mqde1$estimate
#'
#' @keywords distribution
###################
#' @rdname mqdedist
#' @export mqdedist
mqdedist <- function (data, distr, probs = (1:length(data)-0.5)/length(data), qtype = 5,
dist = "euclidean", start = NULL, fix.arg = NULL, optim.method = "default",
lower = -Inf, upper = Inf, custom.optim = NULL, weights = NULL,
silent = TRUE, gradient = NULL, ...)
{
if (!is.character(distr))
stop("distr must be a character string naming a distribution")
else distname <- distr
qdistname <- paste("q", distname, sep = "")
if (!exists(qdistname, mode = "function"))
stop(paste("The ", qdistname, " function must be defined"))
ddistname <- paste("d", distname, sep = "")
if (!exists(ddistname, mode = "function"))
stop(paste("The ", ddistname, " function must be defined"))
if (!(is.vector(probs) & is.numeric(probs)) | anyNA(probs) | any(probs <= 0 | probs >= 1))
stop("probs must be a numeric vector with all elements greater than zero and less than one")
probs <- unique(sort(probs))
if (qtype < 0 || qtype > 9)
stop("wrong type for the R quantile function")
if (is.null(custom.optim))
optim.method <- match.arg(optim.method, c("default", "Nelder-Mead", "BFGS", "CG",
"L-BFGS-B", "SANN", "Brent"))
int.dist <- pmatch(dist, c("euclidean", "maximum", "manhattan"))
if (is.na(int.dist))
stop("invalid distance measure to be used")
if (int.dist == -1)
stop("ambiguous distance measure to be used")
start.arg <- start
if (is.vector(start.arg))
start.arg <- as.list(start.arg)
my3dots <- list(...)
if (!is.null(weights)) {
if (any(weights <= 0))
stop("weights should be a vector of numbers greater than 0")
if (length(weights) != NROW(probs))
stop("weights should be a vector with a length equal to the the probabilities probs")
warning("weights are not taken into account in the default initial values")
}
if (is.vector(data)) {
cens <- FALSE
if (!(is.numeric(data) & length(data) > 1))
stop("data must be a numeric vector of length greater than 1 for non-censored data")
# \n or a dataframe with two columns named left and right and more than one line for censored data")
}
else {
stop("Minimum quantile distance estimation is not yet available for censored data.")
# cens <- TRUE
# censdata <- data
# if (!(is.vector(censdata$left) & is.vector(censdata$right) &
# length(censdata[, 1]) > 1))
# stop("data must be a numeric vector of length greater than 1 for non censored data\n or a dataframe with two columns named left and right and more than one line for censored data")
# pdistname <- paste("p", distname, sep = "")
# if (!exists(pdistname, mode = "function"))
# stop(paste("The ", pdistname, " function must be defined to apply maximum likelihood to censored data"))
}
# if (cens) {
# lcens <- censdata[is.na(censdata$left), ]$right
# if (any(is.na(lcens)))
# stop("An observation cannot be both right and left censored, coded with two NA values")
# rcens <- censdata[is.na(censdata$right), ]$left
# ncens <- censdata[censdata$left == censdata$right &
# !is.na(censdata$left) & !is.na(censdata$right),
# ]$left
# icens <- censdata[censdata$left != censdata$right &
# !is.na(censdata$left) & !is.na(censdata$right),
# ]
# data <- c(rcens, lcens, ncens, (icens$left + icens$right)/2)
# }
argqdistname <- names(formals(qdistname))
chfixstt <- checkparam(start.arg = start.arg, fix.arg = fix.arg,
argdistname = argqdistname, errtxt = NULL,
data10 = head(data, 10), distname = distname)
if (!chfixstt$ok)
stop(chfixstt$txt)
if (is.function(chfixstt$start.arg))
vstart <- unlist(chfixstt$start.arg(data))
else vstart <- unlist(chfixstt$start.arg)
if (is.function(fix.arg)) {
fix.arg.fun <- fix.arg
fix.arg <- fix.arg(data)
}
else fix.arg.fun <- NULL
if(qtype == 0)
s <- data
else
s <- quantile(data, probs = probs, type = qtype, names = FALSE)
# if (!cens) {
if(is.null(weights)){
if (int.dist == 1) # euclidean
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum((obs - theoq)^2)
}
else if (int.dist == 2) # maximum
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
max(abs(obs - theoq))
}
else if (int.dist == 3) # manhattan
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(abs(obs - theoq))
}
}
else{
if (int.dist == 1) # euclidean
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(weights*(obs - theoq)^2)
}
else if (int.dist == 2) # maximum
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
max(weights*abs(obs - theoq))
}
else if (int.dist == 3) # manhattan
fnobj <- function(par, fix.arg, obs, qdistnam, probs, weights) {
theoq <- do.call(qdistnam, c(list(p = probs), as.list(par), as.list(fix.arg)))
sum(weights*abs(obs - theoq))
}
}
# }
# else stop("Minimum quantile distance estimation is not yet available for censored data.")
owarn <- getOption("warn")
if (is.null(custom.optim)) {
hasbound <- any(is.finite(lower) | is.finite(upper))
if (optim.method == "default") {
meth <- ifelse(length(vstart) > 1, "Nelder-Mead", "BFGS")
}
else meth <- optim.method
if (meth == "BFGS" && hasbound && is.null(gradient)) {
meth <- "L-BFGS-B"
txt1 <- "The BFGS method cannot be used with bounds without provided the gradient."
txt2 <- "The method is changed to L-BFGS-B."
warning(paste(txt1, txt2))
}
options(warn = ifelse(silent, -1, 0))
if (hasbound) {
if (!is.null(gradient)) {
opt.fun <- "constrOptim"
}
else {
if (meth == "Nelder-Mead")
opt.fun <- "constrOptim"
else if (meth %in% c("L-BFGS-B", "Brent"))
opt.fun <- "optim"
else {
txt1 <- paste("The method", meth, "cannot be used by constrOptim() nor optim() without gradient and bounds.")
txt2 <- "Only optimization methods L-BFGS-B, Brent and Nelder-Mead can be used in such case."
stop(paste(txt1, txt2))
}
}
if (opt.fun == "constrOptim") {
npar <- length(vstart)
lower <- as.double(rep_len(lower, npar))
upper <- as.double(rep_len(upper, npar))
haslow <- is.finite(lower)
Mat <- diag(npar)[haslow, ]
hasupp <- is.finite(upper)
Mat <- rbind(Mat, -diag(npar)[hasupp, ])
colnames(Mat) <- names(vstart)
rownames(Mat) <- paste0("constr", 1:NROW(Mat))
Bnd <- c(lower[is.finite(lower)], -upper[is.finite(upper)])
names(Bnd) <- paste0("constr", 1:length(Bnd))
initconstr <- Mat %*% vstart - Bnd
if (any(initconstr < 0))
stop("Starting values must be in the feasible region.")
opttryerror <- try(opt <- constrOptim(theta = vstart,
f = fnobj, ui = Mat, ci = Bnd, grad = gradient,
fix.arg = fix.arg, obs = s, qdistnam = qdistname,
probs = probs, weights = weights,
hessian = !is.null(gradient), method = meth,
...), silent = TRUE)
if (!inherits(opttryerror, "try-error"))
if (length(opt$counts) == 1)
opt$counts <- c(opt$counts, NA)
}
else {
opttryerror <- try(opt <- optim(par = vstart,
fn = fnobj, fix.arg = fix.arg, obs = s,
gr = gradient, qdistnam = qdistname,
probs = probs, weights = weights, hessian = TRUE,
method = meth, lower = lower, upper = upper,
...), silent = TRUE)
}
}
else {
opt.fun <- "optim"
opttryerror <- try(opt <- optim(par = vstart, fn = fnobj,
fix.arg = fix.arg, obs = s, gr = gradient,
qdistnam = qdistname, probs = probs, weights = weights,
hessian = TRUE, method = meth,
lower = lower, upper = upper, ...), silent = TRUE)
}
options(warn = owarn)
if (inherits(opttryerror, "try-error")) {
warnings("The function optim encountered an error and stopped.")
if (getOption("show.error.messages"))
print(attr(opttryerror, "condition"))
return(list(estimate = rep(NA, length(vstart)),
convergence = 100, loglik = NA, hessian = NA))
}
if (opt$convergence > 0) {
warnings("The function optim failed to converge, with the error code ",
opt$convergence)
}
if (is.null(names(opt$par)))
names(opt$par) <- names(vstart)
res <- list(estimate = opt$par, convergence = opt$convergence, value = opt$value,
hessian = opt$hessian, probs = probs, dist = dist,
optim.function = opt.fun, fix.arg = fix.arg,
loglik = .loglik(opt$par, fix.arg, data, ddistname),
optim.method = meth, fix.arg.fun = fix.arg.fun, weights = weights,
counts = opt$counts, optim.message = opt$message)
}
else {
options(warn = ifelse(silent, -1, 0))
# if (!cens)
opttryerror <- try(opt <- custom.optim(fn = fnobj,
fix.arg = fix.arg, obs = s, qdistnam = qdistname,
probs = probs, weights = weights,
par = vstart, ...), silent = TRUE)
# else stop("Maximum goodness-of-fit estimation is not yet available for censored data.")
options(warn = owarn)
if (inherits(opttryerror, "try-error")) {
warnings("The customized optimization function encountered an error and stopped.")
if (getOption("show.error.messages"))
print(attr(opttryerror, "condition"))
return(list(estimate = rep(NA, length(vstart)),
convergence = 100, value = NA, hessian = NA))
}
if (opt$convergence > 0) {
warnings("The customized optimization function failed to converge, with the error code ",
opt$convergence)
}
if (is.null(names(opt$par)))
names(opt$par) <- names(vstart)
res <- list(estimate = opt$par, convergence = opt$convergence, value = opt$value,
probs = probs, dist = dist, hessian = opt$hessian,
optim.function = custom.optim, fix.arg = fix.arg,
loglik = .loglik(opt$par, fix.arg, data, ddistname),
optim.method = NULL, fix.arg.fun = fix.arg.fun, weights = weights,
counts = opt$counts, optim.message = opt$message)
}
return(res)
}
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