chi: Measures of extremal dependence
In texmex: Threshold exceedences and multivariate extremes

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Compute measures of extremal dependence for 2 variables.

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

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

## 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)),#"Chi(u)"
    ylabChiBar = expression(bar(chi)(u)), #"Chi Bar(u)"
    ask, ...)

`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.
`spcases`	Whether or not to plot special cases of perfect (positive and negative) dependence and indpenendence. Defaults to `FALSE`.
`ask`	Whether or not to ask before reusing the graphics device.
`...`	Further arguments to be passed to methods.

Computes the functions chi and chi-bar described by Coles, Heffernan and Tawn. 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 the limiting value of ChiBar gives a measure of the strength of dependence.

The plot method shows the ChiBar and Chi functions. In the case of the confidence interval for ChiBar excluding the value 1 for all of the largest values of Quantile, the plot of the Chi function is shown in grey.

An object of class chi containing the following.

`chi`	Values of chi and their esitmated upper and lower confidence limits.
`chibar`	Values of chibar and their esitmated 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.

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.

Janet E. Heffernan, Alec Stephenson, Harry Southworth

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

MCS, rank

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