| itmnormgpd {evmix} | R Documentation |
Normal Bulk and GPD Tail Interval Transition Mixture Model
Description
Density, cumulative distribution function, quantile function and
random number generation for the normal bulk and GPD tail
interval transition mixture model. The
parameters are the normal mean nmean and standard deviation nsd,
threshold u, interval half-width epsilon, GPD scale
sigmau and shape xi.
Usage
ditmnormgpd(x, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
nmean, nsd), sigmau = nsd, xi = 0, log = FALSE)
pitmnormgpd(q, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
nmean, nsd), sigmau = nsd, xi = 0, lower.tail = TRUE)
qitmnormgpd(p, nmean = 0, nsd = 1, epsilon = nsd, u = qnorm(0.9,
nmean, nsd), sigmau = nsd, xi = 0, lower.tail = TRUE)
ritmnormgpd(n = 1, nmean = 0, nsd = 1, epsilon = nsd,
u = qnorm(0.9, nmean, nsd), sigmau = nsd, xi = 0)
Arguments
x |
quantiles |
nmean |
normal mean |
nsd |
normal standard deviation (positive) |
epsilon |
interval half-width |
u |
threshold |
sigmau |
scale parameter (positive) |
xi |
shape parameter |
log |
logical, if TRUE then log density |
q |
quantiles |
lower.tail |
logical, if FALSE then upper tail probabilities |
p |
cumulative probabilities |
n |
sample size (positive integer) |
Details
The interval transition mixture model combines a normal for
the bulk model with GPD for the tail model, with a smooth transition
over the interval (u-epsilon, u+epsilon). The mixing function warps
the normal to map from (u-epsilon, u) to (u-epsilon, u+epsilon) and
warps the GPD from (u, u+epsilon) to (u-epsilon, u+epsilon).
The cumulative distribution function is defined by
F(x)=\kappa(H_t(q(x)) + G(p(x)))
where H_t(x) and G(x) are the truncated normal and
conditional GPD cumulative distribution functions
(i.e. pnorm(x, nmean, nsd) and
pgpd(x, u, sigmau, xi)) respectively. The truncated
normal is not renormalised to be proper, so H_t(x) contrubutes
pnorm(u, nmean, nsd) to the cdf for all x\geq (u + \epsilon).
The normalisation constant \kappa ensures a proper density, given by
1/(1+pnorm(u, nmean, nsd)) where 1 is from GPD component and
latter is contribution from normal component.
The mixing functions q(x) and p(x) suggested by Holden and Haug (2013)
have been implemented. These are symmetric about the threshold u. So for
computational convenience only q(x;u) has been implemented as
qmix
for a given u, with the complementary mixing function is then defined as
p(x;u)=-q(-x;-u).
A minor adaptation of the mixing function has been applied. For the mixture model to
function correctly q(x)>=u for all x\ge u+\epsilon, as then the bulk model will contribute
the constant H_t(u)=H(u) for all x above the interval. Holden and Haug (2013) define
q(x)=x-\epsilon for all x\ge u. For more straightforward and interpretable
computational implementation the mixing function has been set to the threshold
q(x)=u for all x\ge u, so the cdf/pdf of the normal model can be used
directly. We do not have to define cdf/pdf for the non-proper truncated normal
seperately. As such q'(x)=0 for all x\ge u in
qmixxprime, which also makes it clearer that
normal does not contribute to the tail above the interval and vice-versa.
The quantile function within the transition interval is not available in closed form, so has to be solved numerically. Outside of the interval, the quantile are obtained from the normal and GPD components directly.
Value
ditmnormgpd gives the density,
pitmnormgpd gives the cumulative distribution function,
qitmnormgpd gives the quantile function and
ritmnormgpd gives a random sample.
Note
All inputs are vectorised except log and lower.tail.
The main inputs (x, p or q) and parameters must be either
a scalar or a vector. If vectors are provided they must all be of the same length,
and the function will be evaluated for each element of vector. In the case of
ritmnormgpd any input vector must be of length n.
Default values are provided for all inputs, except for the fundamentals
x, q and p. The default sample size for
ritmnormgpd is 1.
Missing (NA) and Not-a-Number (NaN) values in x,
p and q are passed through as is and infinite values are set to
NA. None of these are not permitted for the parameters.
Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.
Author(s)
Alfadino Akbar and Carl Scarrott carl.scarrott@canterbury.ac.nz
References
http://en.wikipedia.org/wiki/Normal_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
Holden, L. and Haug, O. (2013). A mixture model for unsupervised tail estimation. arxiv:0902.4137
See Also
Other itmnormgpd: fitmgng,
fitmnormgpd, itmgng
Other normgpd: fgng, fhpd,
fitmnormgpd, flognormgpd,
fnormgpdcon, fnormgpd,
gngcon, gng,
hpdcon, hpd,
lognormgpdcon, lognormgpd,
normgpdcon, normgpd
Examples
## Not run:
set.seed(1)
par(mfrow = c(2, 2))
xx = seq(-4, 5, 0.01)
u = 1.5
epsilon = 0.4
kappa = 1/(1 + pnorm(u, 0, 1))
f = ditmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, f, ylim = c(0, 1), xlim = c(-4, 5), type = 'l', lwd = 2, xlab = "x", ylab = "density")
lines(xx, kappa * dgpd(xx, u, sigmau = 1, xi = 0.5), col = "red", lty = 2, lwd = 2)
lines(xx, kappa * dnorm(xx, 0, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topright', c('Normal-GPD ITM', 'kappa*Normal', 'kappa*GPD'),
col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)
# cdf contributions
F = pitmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
plot(xx, F, ylim = c(0, 1), xlim = c(-4, 5), type = 'l', lwd = 2, xlab = "x", ylab = "cdf")
lines(xx[xx > u], kappa * (pnorm(u, 0, 1) + pgpd(xx[xx > u], u, sigmau = 1, xi = 0.5)),
col = "red", lty = 2, lwd = 2)
lines(xx[xx <= u], kappa * pnorm(xx[xx <= u], 0, 1), col = "blue", lty = 2, lwd = 2)
abline(v = u + epsilon * seq(-1, 1), lty = c(2, 1, 2))
legend('topleft', c('Normal-GPD ITM', 'kappa*Normal', 'kappa*GPD'),
col = c("black", "blue", "red"), lty = c(1, 2, 2), lwd = 2)
# simulated data density histogram and overlay true density
x = ritmnormgpd(10000, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5)
hist(x, freq = FALSE, breaks = seq(-4, 1000, 0.1), xlim = c(-4, 5))
lines(xx, ditmnormgpd(xx, nmean = 0, nsd = 1, epsilon, u, sigmau = 1, xi = 0.5),
lwd = 2, col = 'black')
## End(Not run)