chiplot {evd}

Dependence Measure Plots

Description

Plots of estimates of the dependence measures chi and chi-bar for bivariate data.

Usage

chiplot(data, nq = 100, qlim = NULL, which = 1:2, conf = 0.95, trunc =
    TRUE, spcases = FALSE, lty = 1, cilty = 2, col = 1, cicol = 1,
    xlim = c(0,1), ylim1 = c(-1,1), ylim2 = c(-1,1), main1 = "Chi Plot",
    main2 = "Chi Bar Plot", xlab = "Quantile", ylab1 = "Chi", ylab2 =
    "Chi Bar", ask = nb.fig < length(which) && dev.interactive(), ...)

Arguments

data

A matrix or data frame with two columns. Rows (observations) with missing values are stripped from the data before any computations are performed.

nq

The number of quantiles at which the measures are evaluated.

qlim

The limits of the quantiles at which the measures are evaluated (see Details).

which

If only one plot is required, specify 1 for chi and 2 for chi-bar.

conf

The confidence coefficient of the plotted confidence intervals.

trunc

Logical; truncate the estimates at their theoretical upper and lower bounds?

spcases

If TRUE, plots greyed lines corresponding to the special cases of perfect positive/negative dependence and exact independence.

lty, cilty

Line types for the estimates of the measures and for the confidence intervals respectively. Use zero to supress.

col, cicol

Colour types for the estimates of the measures and for the confidence intervals respectively.

xlim, xlab

Limits and labels for the x-axis; they apply to both plots.

ylim1

Limits for the y-axis of the chi plot. If this is NULL (the default) the upper limit is one, and the lower limit is the minimum of zero and the smallest plotted value.

ylim2

Limits for the y-axis of the chi-bar plot.

main1, main2

The plot titles for the chi and chi-bar plots respectively.

ylab1, ylab2

The y-axis labels for the chi and chi-bar plots respectively.

ask

Logical; if TRUE, the user is asked before each plot.

...

Other arguments to be passed to matplot.

Details

These measures are explained in full detail in Coles, Heffernan and Tawn (1999). A brief treatment is also given in Section 8.4 of Coles(2001). A short summary is given as follows. We assume that the data are iid random vectors with common bivariate distribution function G, and we define the random vector (X,Y) to be distributed according to G.

The chi plot is a plot of q against empirical estimates of

\chi(q) = 2 - \log(\Pr(F_X(X) < q, F_Y(Y) < q)) / \log(q)

where F_X and F_Y are the marginal distribution functions, and where q is in the interval (0,1). The quantity \chi(q) is bounded by

2 - \log(2u - 1)/\log(u) \leq \chi(q) \leq 1

where the lower bound is interpreted as -Inf for q \leq 1/2 and zero for q = 1. These bounds are reflected in the corresponding estimates.

The chi bar plot is a plot of q against empirical estimates of

\bar{\chi}(q) = 2\log(1-q)/\log(\Pr(F_X(X) > q, F_Y(Y) > q)) - 1

where F_X and F_Y are the marginal distribution functions, and where q is in the interval (0,1). The quantity \bar{\chi}(q) is bounded by -1 \leq \bar{\chi}(q) \leq 1 and these bounds are reflected in the corresponding estimates.

Note that the empirical estimators for \chi(q) and \bar{\chi}(q) are undefined near q=0 and q=1. By default the function takes the limits of q so that the plots depicts all values at which the estimators are defined. This can be overridden by the argument qlim, which must represent a subset of the default values (and these can be determined using the component quantile of the invisibly returned list; see Value).

The confidence intervals within the plot assume that observations are independent, and that the marginal distributions are estimated exactly. The intervals are constructed using the delta method; this may lead to poor interval estimates near q=0 and q=1.

The function \chi(q) can be interpreted as a quantile dependent measure of dependence. In particular, the sign of \chi(q) determines whether the variables are positively or negatively associated at quantile level q. By definition, variables are said to be asymptotically independent when \chi(1) (defined in the limit) is zero. For independent variables, \chi(q) = 0 for all q in (0,1). For perfectly dependent variables, \chi(q) = 1 for all q in (0,1). For bivariate extreme value distributions, \chi(q) = 2(1-A(1/2)) for all q in (0,1), where A is the dependence function, as defined in abvevd. If a bivariate threshold model is to be fitted (using fbvpot), this plot can therefore act as a threshold identification plot, since e.g. the use of 95% marginal quantiles as threshold values implies that \chi(q) should be approximately constant above q = 0.95.

The function \bar{\chi}(q) can again be interpreted as a quantile dependent measure of dependence; it is most useful within the class of asymptotically independent variables. For asymptotically dependent variables (i.e. those for which \chi(1) < 1), we have \bar{\chi}(1) = 1, where \bar{\chi}(1) is again defined in the limit. For asymptotically independent variables, \bar{\chi}(1) provides a measure that increases with dependence strength. For independent variables \bar{\chi}(q) = 0 for all q in (0,1), and hence \bar{\chi}(1) = 0.

Value

A list with components quantile, chi (if 1 is in which) and chibar (if 2 is in which) is invisibly returned. The components quantile and chi contain those objects that were passed to the formal arguments x and y of matplot in order to create the chi plot. The components quantile and chibar contain those objects that were passed to the formal arguments x and y of matplot in order to create the chi-bar plot.

Author(s)

Jan Heffernan and Alec Stephenson

References

Coles, S. G., Heffernan, J. and Tawn, J. A. (1999) Dependence measures for extreme value analyses. Extremes, 2, 339–365.

Coles, S. G. (2001) An Introduction to Statistical Modelling of Extreme Values, London: Springer–Verlag.

See Also

fbvevd, fbvpot, matplot

Examples

par(mfrow = c(1,2))
smdat1 <- rbvevd(1000, dep = 0.6, model = "log")
smdat2 <- rbvevd(1000, dep = 1, model = "log")
chiplot(smdat1)
chiplot(smdat2)