BiCopChiPlot | R Documentation |
This function creates a chi-plot of given bivariate copula data.
BiCopChiPlot(u1, u2, PLOT = TRUE, mode = "NULL", ...)
u1 , u2 |
Data vectors of equal length with values in |
PLOT |
Logical; whether the results are plotted. If |
mode |
Character; whether a general, lower or upper chi-plot is
calculated. Possible values are |
... |
Additional plot arguments. |
For observations u_{i,j},\ i=1,...,N,\ j=1,2,
the chi-plot is based on the following two quantities: the
chi-statistics
\chi_i = \frac{\hat{F}_{1,2}(u_{i,1},u_{i,2})
- \hat{F}_{1}(u_{i,1})\hat{F}_{2}(u_{i,2})}{
\sqrt{\hat{F}_{1}(u_{i,1})(1-\hat{F}_{1}(u_{i,1}))
\hat{F}_{2}(u_{i,2})(1-\hat{F}_{2}(u_{i,2}))}},
and the lambda-statistics
\lambda_i = 4 sgn\left( \tilde{F}_{1}(u_{i,1}),\tilde{F}_{2}(u_{i,2}) \right)
\cdot \max\left( \tilde{F}_{1}(u_{i,1})^2,\tilde{F}_{2}(u_{i,2})^2 \right),
where \hat{F}_{1}
, \hat{F}_{2}
and
\hat{F}_{1,2}
are the empirical distribution functions
of the uniform random variables U_1
and U_2
and of
(U_1,U_2)
, respectively. Further,
\tilde{F}_{1}=\hat{F}_{1}-0.5
and
\tilde{F}_{2}=\hat{F}_{2}-0.5
.
These quantities only depend on the ranks of the data and are scaled to the
interval [0,1]
. \lambda_i
measures a distance of a data point
\left(u_{i,1},u_{i,2}\right)
to the center of the
bivariate data set, while \chi_i
corresponds to a correlation
coefficient between dichotomized values of U_1
and U_2
. Under
independence it holds that \chi_i \sim
\mathcal{N}(0,\frac{1}{N})
and \lambda_i \sim
\mathcal{U}[-1,1]
asymptotically, i.e., values of
\chi_i
close to zero indicate independence—corresponding to
F_{1, 2}=F_{1}F_{2}
.
When plotting these quantities, the pairs of \left(\lambda_i, \chi_i
\right)
will tend to be located above zero for
positively dependent margins and vice versa for negatively dependent
margins. Control bounds around zero indicate whether there is significant
dependence present.
If mode = "lower"
or "upper"
, the above quantities are
calculated only for those u_{i,1}
's and u_{i,2}
's which are
smaller/larger than the respective means of
u1
=(u_{1,1},...,u_{N,1})
and
u2
=(u_{1,2},...,u_{N,2})
.
lambda |
Lambda-statistics (x-axis). |
chi |
Chi-statistics (y-axis). |
control.bounds |
A 2-dimensional vector of bounds
|
Natalia Belgorodski, Ulf Schepsmeier
Abberger, K. (2004). A simple graphical method to explore tail-dependence in stock-return pairs. Discussion Paper, University of Konstanz, Germany.
Genest, C. and A. C. Favre (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12 (4), 347-368.
BiCopMetaContour()
, BiCopKPlot()
,
BiCopLambda()
## chi-plots for bivariate Gaussian copula data
# simulate copula data
fam <- 1
tau <- 0.5
par <- BiCopTau2Par(fam, tau)
cop <- BiCop(fam, par)
set.seed(123)
dat <- BiCopSim(500, cop)
# create chi-plots
op <- par(mfrow = c(1, 3))
BiCopChiPlot(dat[,1], dat[,2], xlim = c(-1,1), ylim = c(-1,1),
main="General chi-plot")
BiCopChiPlot(dat[,1], dat[,2], mode = "lower", xlim = c(-1,1),
ylim = c(-1,1), main = "Lower chi-plot")
BiCopChiPlot(dat[,1], dat[,2], mode = "upper", xlim = c(-1,1),
ylim = c(-1,1), main = "Upper chi-plot")
par(op)
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