disfitqua: Fit a Distribution using Minimization of Available Quantiles

Description

This function fits a distribution to available quantiles (or irregular quantiles) through $n$-dimensional minimization using the optim function. Objective function forms are either root mean-square error (RMSE) or mean absolute deviation (MAD), and the objective functions are expected to result in slightly different estimates of distribution parameters. The RMSE form ($\sigma_{\mathrm{RMSE}}$) is defined as $$\sigma_{\mathrm{RMSE}} = \biggl[ \frac{1}{m}\,\sum_{i=1}^m \bigl[x_o(f_i) - \hat{x}(f_i)\bigr]^2\biggr]^{1/2}\mbox{,}$$ where $m$ is the length of the vector of $o$bserved quantiles $x_o(f_i)$ for nonexceedance probability $f_i$ for $i \in 1, 2, \cdots, m$, and $\hat{x}(f_i)$ for $i \in 1, 2, \cdots, m$ are quantile estimates based on the “current” iteration of the parameters for the selected distribution having $n$ parameters for $n \le m$. Similarly, the MAD form ($\sigma_{\mathrm{MAD}}$) is defined as $$\sigma_{\mathrm{MAD}} = \frac{1}{m}\,\sum_{i=1}^m \mid x_o(f_i) - \hat{x}(f_i) \mid \mbox{.}$$

The disfitqua function is not intended to be an implementation of the method of percentiles but rather is intended for circumstances in which the available quantiles are restricted to either the left or right tails of the distribution. It is evident that a form of the method of percentiles however could be pursued by disfitqua when the length of $x(f)$ is equal to the number of distribution parameters ($n = m$). The situation of $n < m$ however is thought to be the most common application.

The right-tail restriction is the general case in flood-peak hydrology in which the median and select quantiles greater than the median can be available from empirical studies (e.g. Asquith and Roussel, 2009) or rainfall-runoff models. The available quantiles suit engineering needs and thus left-tail quantiles simply are not available. This circumstance might appear quite unusual to users from most statistical disciplines but quantile estimates can exist from regional study of observed data. The Examples section provides further motivation and discussion.

Usage

disfitqua(x, f, objfun=c("rmse", "mad"),
                init.lmr=NULL, init.para=NULL, type=NA,
                ptransf=  function(t) return(t),
                pretransf=function(t) return(t), verbose=FALSE, ... )

Value

An R

list is returned, and this list contains at least the following items:

type: The type of distribution in character format (see dist.list).
para: The parameters of the distribution.
source: Attribute specifying source of the parameters---“disfitqua”.
init.para: A vector of the initial parameters actually passed to the optim function to serve only as a reminder.
disfitqua: The returned list from the optim function. This list contains a repeat of the parameters, the value of the objective function ($\sigma_{\mathrm{RMSE}}$ or $\sigma_{\mathrm{MAD}}$), the interation count, and convergence status.

Arguments

x: The quantiles $x_o(f)$ for the nonexceedance probabilities in f.
f: The nonexceedance probabilities $f$ of the quantiles $x_o(f)$ in x.
objfun: The form of the objective function as previously described.
init.lmr: Optional initial values for the L-moments from which the initial starting parameters for the optimization will be determined. The optimizations by this function are not performed on the L-moments during the optimization. The form of init.lmr is that of an L-moment object from the lmomco package (e.g. lmoms).
init.para: Optional initial values for the parameters used for starting values for the optim function. If this argument is not set nor is init.lmr, then unrigorous estimates of the mean $\lambda_1$ and L-scale $\lambda_2$ are made from the available quantiles, higher L-moment ratios $\tau_r$ for $r \ge 3$ are set to zero, and the L-moments converted to the initial parameters.
type: The distribution type specified by the abbreviations listed under dist.list.
ptransf: An optional parameter transformation function (see Examples) that is useful to guide the optimization run. For example, suppose the first parameter of a three parameter distribution resides in the positive domain, then
ptransf(t) = function(t) c(log(t[1]), t[2], t[3]).
pretransf: An optional parameter retransformation function (see Examples) that is useful to guide the optimization run. For example, suppose the first parameter of a three parameter distribution resides in the positive domain, then
pretransf(t) = function(t) c(exp(t[1]), t[2], t[3]).
verbose: A logical switch on the verbosity of output.
...: Additional arguments to pass to the optim function.

Author

W.H. Asquith

References