theta2fit: Fit time series extremes

Description

Appropriate marginal transforms are done before the fit using standard procedures, before the dependence model is fitted to the data. Then the measure of dependence $\theta(x,m)$ is derived using a method described in Eastoe and Tawn (2012).

Usage

theta2fit(ts, lapl = FALSE, nlag = 1, R = 1000,
          u.mar = 0, u.dep, probs = seq(u.dep, 0.9999, length.out = 30),
          method.mar = c("mle","mom","pwm"), method = c("prop", "MCi"),
          silent = FALSE,
          R.boot = 0, block.length = m * 5, levels = c(.025,.975))

Value

List containing:

depfit: an object of class 'stepfit'
probs: probs
levels: probs transformed to original scale of ts
theta: a matrix with proportion or Monte Carlo estimates of $\theta(x,m)$. Rows correspond to values in probs, columns are point estimates and bootstrap quantiles

Arguments

ts: numeric vector; time series to be fitted.
lapl: logical; is ts on the Laplace scale already? The default (FALSE) assumes unknown marginal distribution.
nlag: integer; number of lags to be considered when modelling the dependence in time.
R: integer; the number of samples used for estimating $\theta(x,m)$.
u.mar: marginal threshold; used when transforming the time series to Laplace scale if lapl is FALSE; not used otherwise.
u.dep: dependence threshold; level above which the dependence is modelled. u.dep can be lower than u.mar.
probs: vector of probabilities; the values of $x$ for which to evaluate $\theta(x,m)$.
method.mar: a character string defining the method used to estimate the marginal GPD; either "mle" for maximum likelihood of "mom" for method of moments or "pwm" for probability weighted moments methods. Defaults to "mle".
method: a character string defining the method used to estimate the dependence measure; either "prop" for proportions or "MCi" for Monte Carlo integration (see Details).
silent: logical (FALSE); verbosity.
R.boot: integer; the number of samples used for the block bootstrap for the confidence intervals.
block.length: integer; the block length used for the block-bootstrapped confidence intervals.
levels: vector of probabilities; the quantiles of the bootstrap distribution of the extremal measure to be computed.

Details

The standard procedure (method="prop") to estimating probabilities from a Heffernan-Tawn fit best illustrated in the bivariate context ($Y\mid X>u$):

1. sample $X$ from an exponential distribution above $v \ge u$,

2. sample $Z$ (residuals) from their empirical distribution,

3. compute $Y$ using the relation $Y = \alpha\times X + X^\beta\times Z$,

4. estimate $Pr(X > v_x, Y > v_y)$ by calculating the proportion $p$ of $Y$ samples above $v_y$ and multiply $p$ with the marginal survival distribution evaluated at $v_x$.

With method="MCi" a Monte Carlo integration approach is used, where the survivor distribution of $Z$ is evaluated at pseudo-residuals of the form $$\frac{v_y - \alpha\times X}{X^\beta},$$ where $X$ is sampled from an exponential distribution above $v_x$. Taking the mean of these survival probabilities, we get the Monte Carlo equivalent of $p$ in the proportion approach.

Examples

Run this code

## generate data from an AR(1)
## with Gaussian marginal distribution
n   <- 10000
dep <- 0.5
ar    <- numeric(n)
ar[1] <- rnorm(1)
for(i in 2:n)
  ar[i] <- rnorm(1, mean=dep*ar[i-1], sd=1-dep^2)
plot(ar, type="l")
plot(density(ar))
grid <- seq(-3,3,0.01)
lines(grid, dnorm(grid), col="blue")

## rescale the margin (focus on dependence)
ar <- qlapl(pnorm(ar))

## fit the data
fit <- theta2fit(ts=ar, u.mar=0.95, u.dep=0.98)

## plot theta(x,1)
plot(fit)
abline(h=1, lty="dotted")