tbea (version 1.5.0)

findParams: Function for estimation of probability density function parameters through quadratic optimization

Description

Function for estimation of probability density function parameters through quadratic optimization

Usage

findParams(q, p, output = "complete", pdfunction, params, initVals = NULL)

Value

Either a list with the complete output of convergence, squared errors and parameter values, or just a vector of parameter values. Depends on the value of output. Warnings may be triggered by the function optim since the optimization is a heuristic process, whenever a given iteration results in an invalid value for a given combination of parameters, the optim function tries another combination of values but inform the user about the problem through a warning. In general these can be safely disregarded.

Arguments

q

A numeric vector of observed quantiles, might come from a HPD from a previous study (along with a median), or from other sources of prior information. See Details.

p

A numeric vector of percentiles.

output

One of two possible values: "complete" and "parameters". For the latter the complete output of the optim function is returned with information on convergence and squared errors (that might be useless for simple cases) or just the parameters.

pdfunction

A character vector (of length one) with the name of the PDF function of interest. Technically this argument supports any PDF function of the form pDIST (e.g., pnorm, ppois, pexp).

params

A character vector with the name of the parameter(s) to optimize in the probability density function. These should match the parameter names of the respective PDF function, e.g., "lambda" in the function ppois

initVals

A numeric vector with default value NULL. It allows the user to provide initial values, althought this is discouraged in most cases.

Author

Main code by Gustavo A. Ballen with important contributions in expression call structure and vectorized design by Klaus Schliep ().

Details

This function comes handy whenever we have some values of uncertainty, (e.g., confidence intervals, HPDs, biostratigraphic age constrains) and want to express it in the form of a probability density function of the form \(P(x;\theta)\). As we have some values (the quantiles) already and their corresponding percentiles, all we need is a way to approximate the parameters \(\theta\) that produce the same combination of quantiles for the given percentiles under a given PDF. This is carried out through optimization of a quadratic error function. This is accomplished through the function optim. For instance, if the estimated age of a fossil is Lutetian, in the Eocene (41.2 to 47.8 Ma), and we want to model such uncertainty through a normal distribution, we could assume that these age boundaries are the quantiles for percentiles 0.025 and 0.975 respectively, and add a thir pair with the midpoint corresponding to the percentile 0.5. This is all the information needed in order to estimate the parameters mean and sd in the functiono pnorm.

Examples

# Find the best parameters for a standard normal density that fit the observed quantiles
# -1.644854, 0, and 1.644854, providing full output for the calculations in the form of
# a list
findParams(q = c(-1.959964, 0.000000, 1.959964),
           p = c(0.025, 0.50, 0.975),
           output = "complete",
           pdfunction = "pnorm",
           params = c("mean", "sd"))

# Given that we have prior on the age of a fossil to be 1 - 10 Ma and that we want to
# model it with a lognormal distribution, fin the parameters of the PDF that best reflect
# the uncertainty in question (i.e., the parameters  for which the observed quantiles are
# 1, 5.5, and 10, assuming that we want the midpoint to reflect the mean of the PDF.
findParams(q = c(1, 5.5, 10),
           p = c(0.025,  0.50, 0.975),
           output = "complete",
           pdfunction = "plnorm",
           params = c("meanlog", "sdlog"))