depmeasures: Estimate dependence measures

Description

Appropriate marginal transforms are done before the fit using standard procedures, before the dependence model is fitted to the data. Then the posterior distribution of a measure of dependence is derived. thetafit gives posterior samples for the extremal index $\theta(x,m)$ and chifit does the same for the coefficient of extremal dependence $\chi_m(x)$.

Usage

thetafit(ts, lapl = FALSE, nlag = 1,
      R = 1000, S = 500,
      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,
      fit = TRUE, prev.fit=bayesfit(), par = bayesparams(),
      submodel = c("fom", "none"), levels=c(.025,.975))
chifit(ts, lapl = FALSE, nlag = 1,
      R = 1000, S = 500,
      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,
      fit = TRUE, prev.fit=bayesfit(), par = bayesparams(),
      submodel = c("fom", "none"), levels=c(.025,.975))

Value

An object of class 'depmeasure', containing a subset of:

bayesfit: An object of class 'bayesfit'
theta: An array with dimensions m $\times$ length(probs) $\times$ (2+length(levels)), with the last dimension listing the posterior mean and median, and the level posterior quantiles
distr: An array with dimensions m $\times$ length(probs) $\times$ S; posterior samples of theta
chi: An array with dimensions m $\times$ length(probs) $\times$ (2+length(levels)), with the last dimension listing the posterior mean and median, and the level posterior quantiles
probs: probs
levels: probs transformed to original scale of ts

Arguments

ts: a vector, the time series for which to estimate the extremal index $\theta(x,m)$ or the coefficient of extremal dependence $\chi_m(x)$, with $x$ a probability level and $m$ a run-length (see details).
lapl: logical; TRUE indicates that ts has a marginal Laplace distribution. If FALSE (default), method.mar is used to transform the marginal distribution of ts to Laplace.
nlag: the run-length; an integer larger or equal to 1.
R: the number of samples per MCMC iteration drawn from the sampled posterior distributions; used for the estimation of the dependence measure.
S: the number of posterior distributions sampled to be used for the estimation of the dependence measure.
u.mar: probability; threshold used for marginal transformation if lapl is FALSE. Not used otherwise.
u.dep: probability; threshold used for the extremal dependence model.
probs: vector of probabilities; the values of $x$ for which to evaluate $\theta(x,m)$ or $\chi_m(x)$.
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.
fit: logical; TRUE means that the dependence model must be fitted and the values in par are used. Otherwise the result from a previous call to depfit.
prev.fit: an object of class 'bayesfit'. Needed if fit is FALSE. Typically returned by a previous call to depfit.
par: an object of class 'bayesparams' to be used for the fit of dependence model.
submodel: a character string, either "fom" for first order Markov or "none" for no specification.
levels: vector of probabilites; the quantiles of the posterior distribution of the extremal measure to be computed.

Details

The sub-asymptotic extremal index is defined as $$\theta(x,m) = Pr(X_1 < x,\ldots,X_m < x | X_0 > x),$$ whose limit as $x$ and $m$ go to $\infty$appropriately is the extremal index $\theta$. The extremal index can be interpreted as the inverse of the asymptotic mean cluster size (see thetaruns).

The sub-asymptotic coefficient of extremal dependence is $$\chi_m(x) = Pr(X_m > x | X_0 > x),$$ whose limit $\chi$ defines asymptotic dependence ($\chi > 0$) or asymptotic independence ($\chi = 0$).

Both types of extremal dependence measures can be estimated either using a

* proportion method (method == "prop"), sampling from the conditional probability given $X_0 > x$ and counting the proportion of sampled points falling in the region of interest, or

* Monte Carlo integration (method == "MCi"), sampling replicates from the marginal exponential tail distribution and evaluating the conditional tail distribution in these replicates, then taking their mean as an approximation of the integral.

submodel == "fom" imposes a first order Markov structure to the model, namely a geometrical decrease in $\alpha$ and a constant $\beta$ across lags, i.e. $\alpha_j = \alpha^j$ and $\beta_j = \beta$, $j=1,\ldots,m$.

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
params <- bayesparams()
params$maxit <- 100 # bigger numbers would be
params$burn  <- 10  # more sensible...
params$thin  <- 4
theta <- thetafit(ts=ar, R=500, S=100, u.mar=0.95, u.dep=0.98,
                  probs = c(0.98, 0.999), par=params)
## or, same thing in two steps to control fit output before computing theta:
fit <- depfit(ts=ar, u.mar=0.95, u.dep=0.98, par=params)
plot(fit)
theta <- thetafit(ts=ar, R=500, S=100, u.mar=0.95, u.dep=0.98,
                  probs = c(0.98, 0.999), fit=FALSE, prev.fit=fit)