Dependence Measure Plots
Plots of estimates of the dependence measures chi and chi-bar for bivariate data.
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(), ...)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 |
conf |
The confidence coefficient of the plotted confidence intervals. |
trunc |
Logical; truncate the estimates at their theoretical upper and lower bounds? |
spcases |
If |
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 |
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 |
... |
Other arguments to be passed to |
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) <= chi(q) <= 1
where the lower bound is interpreted as -Inf for
q <= 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
chibar(q) = 2log(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 chibar(q) is bounded by -1 <= chibar(q) <= 1 and these bounds are reflected in the corresponding estimates.
Note that the empirical estimators for chi(q) and
chibar(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 chibar(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 chibar(1) = 1, where chibar(1) is again defined in the limit. For asymptotically independent variables, chibar(1) provides a measure that increases with dependence strength. For independent variables chibar(q) = 0 for all q in (0,1), and hence chibar(1) = 0.
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.
Jan Heffernan and Alec Stephenson
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.
par(mfrow = c(1,2)) smdat1 <- rbvevd(1000, dep = 0.6, model = "log") smdat2 <- rbvevd(1000, dep = 1, model = "log") chiplot(smdat1) chiplot(smdat2)
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