depmeasures {tsxtreme}

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))

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 \inftyappropriately 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.

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

See Also

depfit, theta2fit, thetaruns

Examples

## 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)