chi: Measures of extremal dependence

In texmex: Statistical Modelling of Extreme Values

Measures of extremal dependence

Description

Compute measures of extremal dependence for 2 variables.

Usage

chi(data, nq = 100, qlim = NULL, alpha = 0.05, trunc = TRUE)

## S3 method for class 'chi'
summary(object, ...)

## S3 method for class 'summary.chi'
print(x, digits=3, ...)

## S3 method for class 'chi'
print(x, ...)

## S3 method for class 'chi'
plot(x, show=c("Chi"=TRUE,"ChiBar"=TRUE), lty=1,
cilty=2, col=1, spcases=TRUE, cicol=1, xlim=c(0, 1), ylimChi =
c(-1, 1), ylimChiBar = c(-1, 1), mainChi = "Chi", mainChiBar =
"Chi Bar", xlab = "Quantile", ylabChi =
expression(chi(u)), ylabChiBar = expression(bar(chi)(u)),
ask, ...)

## S3 method for class 'chi'
ggplot(data=NULL, mapping, xlab = "Quantile", 
ylab=c("ChiBar" = expression(bar(chi)(u)), "Chi" = expression(chi(u))),
main=c("ChiBar" = "Chi Bar",       "Chi" = "Chi"),
xlim = c(0, 1), ylim =list("Chi" = c(-1, 1),"ChiBar" = c(-1, 1)),
ptcol="blue",fill="orange",show=c("ChiBar"=TRUE,"Chi"=TRUE),
spcases = TRUE,plot., ..., environment)

Arguments

data	A matrix containing 2 numeric columns.
nq	The number of quantiles at which to evaluate the dependence measures.
qlim	The minimum and maximum quantiles at which to do the evaluation.
alpha	The size of the confidence interval to be used. Defaults to alpha = 0.05.
trunc	Logical flag indicating whether the estimates should be truncated at their theoretical bounds. Defaults to trunc = TRUE.
x, object	An object of class chi.
digits	Number of digits for printing.
show	Logical, of length 2, names "Chi" and "ChiBar". Defaults to c("Chi" = TRUE, "ChiBar" = TRUE).
lty, cilty, col, cicol	Line types and colours for the the estimated quantities and their confidence intervals.
xlim, ylimChi, ylimChiBar	Limits for the axes.
mainChi, mainChiBar	Main titles for the plots.
xlab, ylabChi, ylabChiBar	Axis labels for the plots.
mapping, ylab, main, ylim, ptcol, fill, environment	Arguments to ggplot methods.
spcases	Whether or not to plot special cases of perfect (positive and negative) dependence and indpenendence. Defaults to FALSE.
plot.	whether to plot to active graphics device.
ask	Whether or not to ask before reusing the graphics device.
...	Further arguments to be passed to methods.

Details

Computes the functions chi and chi-bar described by Coles, Heffernan and Tawn (1999). The limiting values of these functions as the quantile approaches 1 give an empirical measure of the type and strength of tail dependendce exhibited by the data.

A limiting value of ChiBar equal to 1 indicates Asymptotic Dependence, in which case the limiting value of Chi gives a measure of the strength of dependence in this class. A limiting value of ChiBar of less than 1 indicates Asymptotic Independence in which case Chi is irrelevant and the limiting value of ChiBar gives a measure of the strength of dependence.

The plot and ggplot methods show the ChiBar and Chi functions. In the case of the confidence interval for ChiBar excluding the value 1 for all of the largest quantiles, the plot of the Chi function is shown in grey.

Value

An object of class chi containing the following.

chi	Values of chi and their estimated upper and lower confidence limits.
chibar	Values of chibar and their estimated upper and lower confidence limits.
quantile	The quantiles at which chi and chi-bar were evaluated.
chiulb, chibarulb	Upper and lower bounds for chi and chi-bar.

Note

When the data contain ties, the values of chi and chibar are calculated by assigning distinct ranks to tied values using the rank function with argument ties.method = "first". This results in the values of chi and chibar being sensitive to the order in which the tied values appear in the data.

The code is a fairly simple reorganization of code written by Janet E. Heffernan and Alec Stephenson and which appears in the chiplot function in the evd package.

Author(s)

Janet E. Heffernan, Alec Stephenson, Harry Southworth

References

S. Coles, J. E. Heffernan and J. A. Tawn, Dependence measures for extreme values analyses, Extremes, 2, 339 – 365, 1999.

A. G. Stephenson. evd: Extreme Value Distributions, R News, 2, 2002.

See Also

MCS, rank

Examples

D <- liver[liver$dose == "D",]
chiD <- chi(D[, 5:6])
par(mfrow=c(1,2))
ggplot(chiD)

A <- liver[liver$dose == "A",]
chiA <- chi(A[, 5:6])
# here the limiting value of chi bar(u) lies away from one so the chi plot is
# not relevant and is plotted in grey
ggplot(chiA) 

texmex documentation built on June 22, 2024, 12:26 p.m.