mev (version 2.1)

extcoef: Estimators of the extremal coefficient

Description

These functions estimate the extremal coefficient using an approximate sample from the Frechet distribution.

Usage

extcoef(
  xdat,
  coord = NULL,
  thresh = NULL,
  estimator = c("schlather", "smith", "fmado"),
  margtrans = c("emp", "gev", "none"),
  ties.method = "random",
  prob = 0,
  plot = TRUE,
  ...
)

Value

an invisible list with vectors dist if coord is non-null or else a matrix of pairwise indices ind,

extcoef and the supplied estimator, fmado and binned. If estimator == "schlather", an additional matrix with 2 columns containing the binned distance binned with the h and the binned extremal coefficient.

Arguments

xdat

an n by D matrix of unit Frechet observations

coord

an optional d by D matrix of location coordinates

thresh

threshold parameter (default is to keep all data if prob = 0).

estimator

string indicating which estimator to compute, one of smith, schlather or fmado.

margtrans

string indicating which method to use to transform the margins to unit Frechet scale, either emp for nonparametric transformation via rank transform, gev for fit of generalized extreme value distribution to marginals, or none

ties.method

method for handling of ties in rank transformation

prob

probability of not exceeding threshold thresh

plot

logical; should cloud or matrix of pairwise empirical estimates be plotted? Default to TRUE.

...

additional parameters passed to the function, currently ignored.

Details

The Smith estimator: suppose \(Z(x)\) is simple max-stable vector (i.e., with unit Frechet marginals). Then \(1/Z\) is unit exponential and \(1/\max(Z(s_1), Z(s_2))\) is exponential with rate \(\theta = \max\{Z(s_1), Z(s_2)\}\). The extremal index for the pair can therefore be calculated using the reciprocal mean.

The Schlather and Tawn estimator: the likelihood of the naive estimator for a pair of two sites \(A\) is $$ \mathrm{card}\left\{ j: \max_{i \in A} X_i^{(j)}\bar{X}_i)>z \right\} \log(\theta_A) - \theta_A \sum_{j=1}^n \left[ \max \left\{z, \max_{i \in A} (X_i^{(j)}\bar{X}_i)\right\}\right]^{-1},$$ where \(\bar{X}_i = n^{-1} \sum_{j=1}^n 1/X_i^{(j)}\) is the harmonic mean and \(z\) is a threshold on the unit Frechet scale. The search for the maximum likelihood estimate for every pair \(A\) is restricted to the interval \([1,3]\). A binned version of the extremal coefficient cloud is also returned. The Schlather estimator is not self-consistent. The Schlather and Tawn estimator includes as special case the Smith estimator if we do not censor the data (p = 0) and do not standardize observations by their harmonic mean.

The F-madogram estimator is a non-parametric estimate based on a stationary process \(Z\); the extremal coefficient satisfies $$\theta(h)=\frac{1+2\nu(h)}{1-2\nu(h)},$$ where $$\nu(h) = \frac{1}{2} \mathsf{E}[|F(Z(s+h)-F(Z(s))|]$$ The implementation only uses complete pairs to calculate the relative ranks.

All estimators are coded in plain R and computations are not optimized. The estimation time can therefore be large for large data sets. If there are no missing observations, the routine fmadogram from the SpatialExtremes package should be preferred as it is noticeably faster.

The data will typically consist of max-stable vectors or block maxima. Both of the Smith and the Schlather--Tawn estimators require unit Frechet margins; the margins will be standardized to the unit Frechet scale, either parametrically or nonparametrically unless margtrans="none". If margtrans = "gev", a parametric GEV model is fitted to each column of dat using maximum likelihood estimation and transformed back using the probability integral transform. If method = "emp", using the empirical distribution function. The latter is the default, as it is appreciably faster.

References

Schlather, M. and J. Tawn (2003). A dependence measure for multivariate and spatial extremes, Biometrika, 90(1), pp.139--156.

Cooley, D., P. Naveau and P. Poncet (2006). Variograms for spatial max-stable random fields, In: Bertail P., Soulier P., Doukhan P. (eds) Dependence in Probability and Statistics. Lecture Notes in Statistics, vol. 187. Springer, New York, NY

R. J. Erhardt, R. L. Smith (2012), Approximate Bayesian computing for spatial extremes, Computational Statistics and Data Analysis, 56, pp.1468--1481.

Examples

if (FALSE) {
coord <- 10 * cbind(runif(50), runif(50))
di <- as.matrix(dist(coord))
dat <- rmev(
  n = 1000,
  d = 100,
  param = 3,
  sigma = exp(-di / 2),
  model = 'xstud'
)
res <- extcoef(xdat = dat, coord = coord)
# Extremal Student extremal coefficient function

XT.extcoeffun <- function(h, nu, corrfun, ...) {
  if (!is.function(corrfun)) {
    stop('Invalid function \"corrfun\".')
  }
  h <- unique(as.vector(h))
  rhoh <- sapply(h, corrfun, ...)
  cbind(
    h = h,
    extcoef = 2 * pt(sqrt((nu + 1) * (1 - rhoh) / (1 + rhoh)), nu + 1)
  )
}
#This time, only one graph with theoretical extremal coef
plot(res$dist, res$extcoef, ylim = c(1, 2), pch = 20)
abline(v = 2, col = 'gray')
extcoefxt <- XT.extcoeffun(
  seq(0, 10, by = 0.1),
  nu = 3,
  corrfun = function(x) {
    exp(-x / 2)
  }
)
lines(
  extcoefxt[, 'h'],
  extcoefxt[, 'extcoef'],
  type = 'l',
  col = 'blue',
  lwd = 2
)
# Brown--Resnick extremal coefficient function
BR.extcoeffun <- function(h, vario, ...) {
  if (!is.function(vario)) {
    stop('Invalid function \"vario\".')
  }
  h <- unique(as.vector(h))
  gammah <- sapply(h, vario, ...)
  cbind(h = h, extcoef = 2 * pnorm(sqrt(gammah / 4)))
}
extcoefbr <- BR.extcoeffun(
  seq(0, 20, by = 0.25),
  vario = function(x) {
    2 * x^0.7
  }
)
lines(
  extcoefbr[, 'h'],
  extcoefbr[, 'extcoef'],
  type = 'l',
  col = 'orange',
  lwd = 2
)

coord <- 10 * cbind(runif(20), runif(20))
di <- as.matrix(dist(coord))
dat <- rmev(
  n = 1000,
  d = 20,
  param = 3,
  sigma = exp(-di / 2),
  model = 'xstud'
)
res <- extcoef(
  xdat = dat,
  coord = coord,
  estimator = "smith"
)
}