README.md

In dmi3kno/qpd: Tools for Quantile-Parametrized Distribiutions

qpd

The goal of qpd is to provide essential functions and tools for using quantile and quantile-parameterized distributions in Bayesian analysis in R.

Installation

You can install the development version of qpd from GitHub with:

# install.packages("remotes")
remotes::install_github("dmi3kno/qpd")

Quantile distributions and quantile-parameterized distributions

Most people consider cumulative distribution function to be the fundamental way of defining a distribution (there’s a reason it is often simply called a distribution function). Some even believe that PDF is the only correct way to define a distribution. But equally plausible way to define a distribution is via the inverse cumulative distribution function, also know as the quantile function. Distributions defined by non-analytically-invertible quantile function are called quantile distributions (Gilchrist 2000; Parzen 2004; Perepolkin, Goodrich, and Sahlin 2021b).

Quantile-parameterized distributions (often also defined by a non-invertible quantile function) use special kind of parameterization: a vector of cumulative probabilities p and a vector of corresponding quantiles q (e.g. p = {0.1, 0.5, 0.9}, q = {4, 9, 17} is a valid parameterization) (Keelin and Powley 2011; Hadlock 2017).

Examples

The following example shows how to solve a common problem.

library(qpd)

Probability grids

When working with quantile distributions it is often convenient to use regularly spaced grids of probability values, because quantile values take probabilities (also referred to as depths) p and output the quantile values q given the parameters. qpd provides a couple of convenience functions for creating regularly spaced probability grids. make_pgrid() creates “a smooth” S-shaped grid, denser toward the tails and sparser in the middle. make_tgrid() creates linear (equispaced) with several tiers. Certain part of the grid (defined by tail) argument is also split into linear grid, forming an S-shaped piecewise-linear grid. Curvature of the grid is controlled by parameters.

p_grd <- make_pgrid()
plot(p_grd, type="l", main="P-grid using beta QF")
t_grd <- make_tgrid()
plot(t_grd, type="l", main="Tiered grid (3 tiers)")

Quantile distributions

The package contains standard distribution functions for the following quantile distributions: GLD, g-and-h, g-and-k, Govindarajulu, Wakeby.

Generalized Lambda Distribution (GLD)

q <- qpd::qgld(p_grd, 1, 1, 6, 3)
plot(q~p_grd, type="l", main="GLD QF")
plot(pgld(q, 1,1,6,3)~q, type="l", main="GLD CDF (approx iQF)")
plot(fgld(p_grd, 1, 1, 6, 3)~q, type="l", main="GLD QDF")
plot(dqgld(p_grd, 1, 1, 6, 3)~q, type="l", main="GLD DQF")

Generalized g-and-h distribution

q <- qpd::qgnh(p_grd,  3, 1, 0.8, 2, 0.5)
plot(q~p_grd, type="l", main="g-and-h QF")
plot(pgnh(q, 3, 1, 0.8, 2, 0.5)~q, type="l", main="g-and-h CDF (approx iQF)")
plot(fgnh(p_grd, 3, 1, 0.8, 2, 0.5)~q, type="l", main="g-and-h QDF")
plot(dqgnh(p_grd, 3, 1, 0.8, 2, 0.5)~q, type="l", main="g-and-h DQF")

Govindarajulu distribution

q <- qpd::qgovindarajulu(p_grd,  4, 2.2)
plot(q~p_grd, type="l", main="Govindarajulu QF")
plot(pgovindarajulu(q, 4, 2.2)~q, type="l", main="Govindarajulu CDF (approx iQF)")
plot(fgovindarajulu(p_grd, 4, 2.2)~q, type="l", main="Govindarajulu QDF")
plot((qpd::dqgovindarajulu(p_grd, 4, 2.2))~q, type="l", main="Govindarajulu DQF")

Wakeby distribution

q <- qpd::qwakeby(p_grd,  5, 3, 0.1, 0.2, 0)
plot(q~p_grd, type="l", main="Wakeby QF")
plot(pwakeby(q, 5, 3, 0.1, 0.2, 0)~q, type="l", main="Wakeby CDF (approx iQF)")
plot(fwakeby(p_grd, 5, 3, 0.1, 0.2, 0)~q, type="l", main="Wakeby QDF")
plot((qpd::dqwakeby(p_grd, 5, 3, 0.1, 0.2, 0))~q, type="l", main="Wakeby DQF")

Quantile-parameterized distributions

The package implements standard distribution functions for the following quantile-parametrized distributions: Myerson, SkewNormal (special case of Myerson distribution), J-QPD and metalog distribution.

Myerson distribution

Myerson is a Normal-based probability distribution, therefore its quantile function builds on the Normal quantile function.

q <- qpd::qMyerson(p_grd,  4,9,17, alpha=0.1)
plot(q~p_grd, type="l", main="Myerson QF")
plot(pMyerson(q, 4,9,17, alpha=0.1)~q, type="l", main="Myerson CDF")
plot(qpd::dMyerson(q, 4,9,17, alpha=0.1)~q, type="l", main="Myerson PDF")

Skew-Normal distribution

Quantile-parametrized Skew-Normal is a special case of Myerson distribution

q <- qpd::qSkewNorm (p_grd,  4,17, s=-0.3)
plot(q~p_grd, type="l", main="Skew-Normal QF")
plot(pSkewNorm(q, 4,17, s=-0.3)~q, type="l", main="Skew-Normal CDF")
plot(qpd::dSkewNorm(q,4,17, s=-0.3)~q, type="l", main="Skew-Normal PDF")

Metalog distribution

Fitting metalog distribution (Keelin 2016) a metalog distribution requires calculation of a coefficients which can be passed to all functions. It is a good idea to check whether the resulting metalog is valid.

as <- qpd::fit_metalog(p=c(0.1, 0.5, 0.9, 0.99), q=c(4,9,17, 22), bl=0)
is_metalog_valid(as, bl=0)
#> [1] TRUE
q <- qpd::qmetalog(p_grd,  as, bl=0)
plot(q~p_grd, type="l", main="Metalog (4-terms) QF")
plot(pmetalog(q, as, bl=0)~q, type="l", main="Metalog (4-terms) CDF (approx iQF)")
plot(fmetalog(p_grd, as, bl=0)~q, type="l", main="Metalog (4-terms) QDF")
plot((qpd::dqmetalog(p_grd, as, bl=0))~q, type="l", main="Metalog (4-terms) DQF")

Other functionality

qpd can also fit (using conditional beta distributions) and sample from Dirichlet and Connor-Mosimann distributions (Perepolkin, Goodrich, and Sahlin 2021a). There are also functions for fitting Chebyshev polynomials to arbitrary functions and automatic proxy root-finding for validation of a user-defined quantile density function (Perepolkin, Goodrich, and Sahlin 2021b). The package also has quantile versions of density (DQF and QDF) for some standard distributions, including normal, exponential, Rayleigh, and generalized exponential. There’s also fixed-seed HDR pseudo-random number generator.

References

Gilchrist, Warren. 2000. *Statistical Modelling with Quantile Functions*. Boca Raton: Chapman & Hall/CRC.

Hadlock, Christopher Campbell. 2017. “Quantile-Parameterized Methods for Quantifying Uncertainty in Decision Analysis.” Thesis, Austin, TX: University of Texas. .

Keelin, Thomas W. 2016. “The Metalog Distributions.” *Decision Analysis* 13 (4): 243–77. .

Keelin, Thomas W., and Bradford W. Powley. 2011. “Quantile-Parameterized Distributions.” *Decision Analysis* 8 (3): 206–19. .

Parzen, Emanuel. 2004. “Quantile Probability and Statistical Data Modeling.” *Statistical Science* 19 (4): 652–62. .

Perepolkin, Dmytro, Benjamin Goodrich, and Ullrika Sahlin. 2021a. “Hybrid Elicitation and Indirect Bayesian Inference with Quantile-Parametrized Likelihood.” OSF Preprints. .

———. 2021b. “The Tenets of Indirect Inference in Bayesian Models.” Preprint. https://osf.io/enzgs: Open Science Framework. .

dmi3kno/qpd documentation built on Sept. 29, 2024, 6:39 p.m.