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

## Description

Compute measures of extremal dependence for 2 variables.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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, ...) ```

## 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. `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.

## Details

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.

## Value

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.

## 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.

`MCS`, `rank`
 ```1 2 3 4 5 6 7 8 9``` ```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) ```