quantile.dtn: Interpolated quantile distribution with exponential tails and Nadaraya-Watson quantile distribution

Description

dquantile gives a probability density function (pdf) by combining step-interpolation of probability densities for specified tau-quantiles (quant) with exponential lower/upper tails (Quiñonero-Candela, 2006; Cannon, 2011). Point mass (e.g., as might occur when using censored QRNN models) can be defined by setting lower to the left censoring point. pquantile gives the cumulative distribution function (cdf); the integrate function is used for values outside the range of quant. The inverse cdf is given by qquantile; the uniroot function is used for values outside the range of tau. rquantile is used for generating random variates.

Alternative formulations (without left censoring) based on the Nadaraya-Watson estimator [p,q,r]quantile.nw are also provided (Passow and Donner, 2020).

Note: these functions have not been extensively tested or optimized and should be considered experimental.

Usage

dquantile(x, tau, quant, lower=-Inf)
pquantile(q, tau, quant, lower=-Inf, ...)
pquantile.nw(q, tau, quant, h=0.001, ...)
qquantile(p, tau, quant, lower=-Inf,
          tol=.Machine$double.eps^0.25, maxiter=1000,
          range.mult=1.1, max.error=100, ...)
qquantile.nw(p, tau, quant, h=0.001)
rquantile(n, tau, quant, lower=-Inf,
          tol=.Machine$double.eps^0.25, maxiter=1000,
          range.mult=1.1, max.error=100, ...)
rquantile.nw(n, tau, quant, h=0.001)

Value

dquantile gives the pdf, pquantile gives the cdf, qquantile gives the inverse cdf (or quantile function), and rquantile generates random deviates.

Arguments

x, q: vector of quantiles.
p: vector of cumulative probabilities.
n: number of random samples.
tau: ordered vector of cumulative probabilities associated with quant argument.
quant: ordered vector of quantiles associated with tau argument.
lower: left censoring point.
tol: tolerance passed to uniroot.
h: bandwidth for Nadaraya-Watson kernel.
maxiter: maximum number of iterations passed to uniroot.
range.mult: values of lower and upper in uniroot are initialized to
quant[1]-range.mult*diff(range(quant)) and
quant[length(quant)]+range.mult*diff(range(quant)) respectively; range.mult is squared, lower and upper are recalculated, and uniroot is rerun if the current values lead to an exception.
max.error: maximum number of uniroot errors allowed before termination.
...: additional arguments passed to integrate or uniroot.

References

Cannon, A.J., 2011. Quantile regression neural networks: implementation in R and application to precipitation downscaling. Computers & Geosciences, 37: 1277-1284. doi:10.1016/j.cageo.2010.07.005

Passow, C., R.V. Donner, 2020. Regression-based distribution mapping for bias correction of climate model outputs using linear quantile regression. Stochastic Environmental Research and Risk Assessment, 34:87-102. doi:10.1007/s00477-019-01750-7

Quiñonero-Candela, J., C. Rasmussen, F. Sinz, O. Bousquet, B. Scholkopf, 2006. Evaluating Predictive Uncertainty Challenge. Lecture Notes in Artificial Intelligence, 3944: 1-27.

Examples

Run this code

## Normal distribution
tau <- c(0.01, seq(0.05, 0.95, by=0.05), 0.99)
quant <- qnorm(tau)

x <- seq(-3, 3, length=500)
plot(x, dnorm(x), type="l", col="red", lty=2, lwd=2,
     main="pdf")
lines(x, dquantile(x, tau, quant), col="blue")

q <- seq(-3, 3, length=20)
plot(q, pnorm(q), type="b", col="red", lty=2, lwd=2,
     main="cdf")
lines(q, pquantile(q, tau, quant),
      col="blue")

abline(v=1.96, lty=2)
abline(h=pnorm(1.96), lty=2)
abline(h=pquantile(1.96, tau, quant), lty=3)
abline(h=pquantile.nw(1.96, tau, quant, h=0.01), lty=3)

p <- c(0.001, 0.01, 0.025, seq(0.05, 0.95, by=0.05),
       0.975, 0.99, 0.999)
plot(p, qnorm(p), type="b", col="red", lty=2, lwd=2,
     main="inverse cdf")
lines(p, qquantile(p, tau, quant), col="blue")

## Distribution with point mass at zero
tau.0 <- c(0.3, 0.5, 0.7, 0.8, 0.9)
quant.0 <- c(0, 5, 7, 15, 20)

r.0 <- rquantile(500, tau=tau.0, quant=quant.0, lower=0)
x.0 <- seq(0, 40, by=0.5)
d.0 <- dquantile(x.0, tau=tau.0, quant=quant.0, lower=0)
p.0 <- pquantile(x.0, tau=tau.0, quant=quant.0, lower=0)
q.0 <- qquantile(p.0, tau=tau.0, quant=quant.0, lower=0)

par(mfrow=c(2, 2))
plot(r.0, pch=20, main="random")
plot(x.0, d.0, type="b", col="red", main="pdf")
plot(x.0, p.0, type="b", col="blue", ylim=c(0, 1),
     main="cdf")
plot(p.0, q.0, type="b", col="green", xlim=c(0, 1),
     main="inverse cdf")