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(), ...)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.
A matrix or data frame with two columns. Rows (observations) with missing values are stripped from the data before any computations are performed.
The number of quantiles at which the measures are evaluated.
The limits of the quantiles at which the measures are evaluated (see Details).
If only one plot is required, specify 1
for chi and 2 for chi-bar.
The confidence coefficient of the plotted confidence intervals.
Logical; truncate the estimates at their theoretical upper and lower bounds?
If TRUE, plots greyed lines corresponding
to the special cases of perfect positive/negative dependence
and exact independence.
Line types for the estimates of the measures and for the confidence intervals respectively. Use zero to supress.
Colour types for the estimates of the measures and for the confidence intervals respectively.
Limits and labels for the x-axis; they apply to both plots.
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.
Limits for the y-axis of the chi-bar plot.
The plot titles for the chi and chi-bar plots respectively.
The y-axis labels for the chi and chi-bar plots respectively.
Logical; if TRUE, the user is asked before
each plot.
Other arguments to be passed to matplot.
Jan Heffernan and Alec Stephenson
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 \(\bar{\chi}(1) = 0\).
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.
fbvevd, fbvpot,
matplot
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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