evmix (version 1.0)

dwm: Dynamically Weighted Mixture Model

Description

Density, cumulative distribution function, quantile function and random number generation for the dynamically weighted mixture model. The parameters are the Weibull shape wshape and scale wscale, Cauchy location cmu, Cauchy scale ctau, GPD scale sigmau, shape xi and initial value for the quantile qinit.

Usage

ddwm(x, wshape = 1, wscale = 1, cmu = 1, ctau = 1,
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, log = FALSE)

  pdwm(q, wshape = 1, wscale = 1, cmu = 1, ctau = 1,
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, lower.tail = TRUE)

  qdwm(p, wshape = 1, wscale = 1, cmu = 1, ctau = 1,
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0, lower.tail = TRUE, qinit = NULL)

  rdwm(n = 1, wshape = 1, wscale = 1, cmu = 1, ctau = 1,
    sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale * gamma(1 + 1/wshape))^2),
    xi = 0)

Arguments

cmu

Cauchy location

ctau

Cauchy scale

qinit

vector of initial values for the quantile estimate

x

quantile

wshape

Weibull shape (non-negative)

wscale

Weibull scale (non-negative)

sigmau

scale parameter (non-negative)

xi

shape parameter

log

logical, if TRUE then log density

q

quantile

lower.tail

logical, if FALSE then upper tail probabilities

p

cumulative probability

n

sample size (non-negative integer)

Value

• ddwm gives the density, pdwm gives the cumulative distribution function, qdwm gives the quantile function and rdwm gives a random sample.

Details

The dynamic weighted mixture model combines a Weibull for the bulk model with GPD for the tail model. However, unlike all the other mixture models the GPD is defined over the entire range of support rather than as a conditional model above some threshold. A transition function is used to apply weights to transition between the bulk and GPD for the upper tail, thus providing the dynamically weighted mixture. They use a Cauchy cumulative distribution function for the transition function. The density function is then a dynamically weighted mixture given by: $$f(x) = {[1 - p(x)] h(x) + p(x) g(x)}/r$$ where $h(x)$ and $g(x)$ are the Weibull and unscaled GPD density functions respectively (i.e. pweibull(x, wshape, wscale) and pgpd(x, u, sigmau, xi)). The Cauchy cumulative distribution function used to provide the transition is defined by $p(x)$ (i.e. pcauchy(x, cmu, ctau. The normalisation constant $r$ ensures a proper density. The quantile function is not available in closed form, so has to be solved numerically. The argument qinit is the initial quantile estimate which is used for numerical optimisation and should be set to a reasonable guess. When the qinit is NULL, the initial quantile value is given by the midpoint between the Weibull and GPD quantiles. As with the other inputs qinit is also vectorised, but R does not permit vectors combining NULL and numeric entries.

References

http://en.wikipedia.org/wiki/Weibull_distribution http://en.wikipedia.org/wiki/Generalized_Pareto_distribution Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value threshold estimation and uncertainty quantification. REVSTAT - Statistical Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf Frigessi, A., Haug, O. and Rue, H. (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5 (3), 219-235

See Also

weibullgpd, gpd and dweibull

Examples

par(mfrow = c(2, 2))
xx = seq(0.001, 5, 0.01)
f = ddwm(xx, wshape = 2, wscale = 1/gamma(1.5), cmu = 1, ctau = 1, sigmau = 1, xi = 0.5)
plot(xx, f, ylim = c(0, 1), xlim = c(0, 5), type = 'l', lwd = 2,
  ylab = "density", main = "Plot example in Frigessi et al. (2002)")
lines(xx, dgpd(xx, xi = 1, sigmau = 0.5), col = "red", lty = 2, lwd = 2)
lines(xx, dweibull(xx, shape = 2, scale = 1/gamma(1.5)), col = "blue", lty = 2, lwd = 2)
legend('topright', c('DWM', 'Weibull', 'GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)

# three tail behaviours
plot(xx, pdwm(xx, xi = 0), type = "l")
lines(xx, pdwm(xx, xi = 0.3), col = "red")
lines(xx, pdwm(xx, xi = -0.3), col = "blue")
legend("bottomright", paste("xi =",c(0, 0.3, -0.3)), col=c("black", "red", "blue"), lty = 1)

x = rdwm(10000, wshape = 2, wscale = 1/gamma(1.5), cmu = 1, ctau = 1, sigmau = 1, xi = 0.1)
xx = seq(0, 15, 0.01)
hist(x, freq = FALSE, breaks = 100)
lines(xx,ddwm(xx, wshape = 2, wscale = 1/gamma(1.5), cmu = 1, ctau = 1, sigmau = 1, xi = 0.1),
  lwd = 2, col = 'black')

plot(xx, pdwm(xx, wshape = 2, wscale = 1/gamma(1.5), cmu = 1, ctau = 1, sigmau = 1, xi = 0.1),
 xlim = c(0, 15), type = 'l', lwd = 2,
  xlab = "x", ylab = "F(x)")
lines(xx, pgpd(xx, sigmau = 1, xi = 0.1), col = "red", lty = 2, lwd = 2)
lines(xx, pweibull(xx, shape = 2, scale = 1/gamma(1.5)), col = "blue", lty = 2, lwd = 2)
legend('bottomright', c('DWM', 'Weibull', 'GPD'),
      col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)