quantile.dtn: Interpolated quantile distribution with exponential tails and...

In qrnn: Quantile Regression Neural Network

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)

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.

Value

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

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.

See Also

integrate, uniroot, qrnn.predict

Examples

## 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")

qrnn documentation built on May 29, 2024, 1:27 a.m.