psepolar: Pseudo-Polar Representation of Bivariate Data

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

Pseudo-Polar Representation of Bivariate Data

Description

Kiriliouk et al. (2016, pp. 358–360) describe a pseudo-polar representation of bivariate data as a means to explore right-tail extremal dependency between the variables. Let (X_i, Y_i) (real values) or (U_i, V_i) (as probabilities) for i = 1, \ldots, n be a bivariate sample of size n. When such data are transformed into a “unit-Pareto” scale by

\widehat{X}^\star_i = n/(n+1-R_{X,i}) \mbox{\ and\ } \widehat{Y}^\star_i = n/(n+1-R_{Y,i})\mbox{,}

where R is rank(), then letting each component sum or pseudo-polar radius be defined as

\widehat{S}_i = \widehat{X}^\star_i + \widehat{Y}^\star_i\mbox{,}

and each respective pseudo-polar angle be defined as

\widehat{W}_i = \widehat{X}^\star_i / (\widehat{X}^\star_i + \widehat{Y}^\star_i) = \widehat{X}^\star_i / \widehat{S}_i\mbox{,}

a pseudo-polar representation is available for study.

A scatter plot of \widehat{W}_i (horizontal) versus \widehat{S}_i (vertical) will depict a pseudo-polar plot of the data. Kiriliouk et al. (2016) approach the pseudo-polar concept as a means to study extremal dependency in the sense of what are the contributions of the X and Y to their sum conditional on the sum being large. The largeness of \widehat{S}_i is assessed by its empirical cumulative distribution function and a threshold S_f stemming from f as a nonexceedance probability f \in [0,1].

A density plot of the \widehat{W}_i is a representation of extremal dependence. If the density plot shows low density for pseudo-polar angles away from 0 and 1 or bimodality on the edges then weak extremal dependency is present. If the density is substantial and uniform away from the the angles 0 and 1 or if the density peaks near \widehat{W} \approx 0.5 then extremal dependency is strong.

Usage

psepolar(u, v=NULL, f=0.90, ...)

Arguments

u	Nonexceedance probability u in the X direction (actually the ranks are used so this can be a real-value argument as well);
v	Nonexceedance probability v in the Y direction (actually the ranks are used so this can be a real-value argument as well) and if NULL then u is treated as a two column R data.frame;
f	The nonexceedance probability of the distal \widehat{S} to flag in Shat_ge_Sf column of the output; and
...	Additional arguments to pass to the dat2bernqua() function of the lmomco package.

Value

An R data.frame is returned in the table element and the S_f is in the Sf element.

U	An echo of the u input;
V	An echo of the v input;
Xstar	The \widehat{X}^\star_i (Kiriliouk et al., 2016, eq. 17.8, p. 359);
Ystar	The \widehat{Y}^\star_i (Kiriliouk et al., 2016, eq. 17.8, p. 359);
FXhat1	The F_{X,i} = 1 - 1/X^\star_i, which is the inverse of Kiriliouk et al. (2016, eq. 17.1, p. 354);
FYhat1	The F_{Y,i} = 1 - 1/Y^\star_i, which is the inverse of Kiriliouk et al. (2016, eq. 17.1, p. 354);
FXhat3	The F_{3,X,i} = (R_{X,i} - 0.5)/n corresponding to the “3” alternative identified by Kiriliouk et al. (2016, p. 365);
FYhat3	The F_{3,Y,i} = (R_{Y,i} - 0.5)/n corresponding to the “3” alternative identified by Kiriliouk et al. (2016, p. 365);
What	The \widehat{W}_i (Kiriliouk et al., 2016, eq. 17.9, p. 359);
Shat	The \widehat{S}_i (Kiriliouk et al., 2016, eq. 17.9, p. 359); and
Shat_ge_Sf	A logical on whether the \widehat{S}_i are larger than S_f.

Note

The default of f=0.90 means that the upper 90th percentile of the component sum will be identified in the output. This percentile is computed by the Bernstein empirical distribution function provided by the lmomco package through the dat2bernqua() function. Suggested arguments for ... are poly.type="Bernstein" and bound.type="Carv" though the former is redundant because it is the default of dat2bernqua().

Author(s)

William Asquith [e-mail]

References

Kiriliouk, Anna, Segers, Johan, Warchoł, Michał, 2016, Nonparameteric estimation of extremal dependence: in Extreme Value Modeling and Risk Analysis, D.K. Dey and Jun Yan eds., Boca Raton, FL, CRC Press, ISBN 978–1–4987–0129–7.

See Also

spectralmeas, stabtaildepf

Examples

## Not run: 
pse <- psepolar(simCOP(n=799, cop=PARETOcop, para=4.3,graphics=FALSE),bound.type="Carv")
pse <- pse$table # The Pareto copula has right-tail extreme dependency
plot(1/(1-pse$U), 1/(1-pse$V), col=pse$Shat_ge_Sf+1, lwd=0.8, cex=0.5, log="xy", pch=16)
plot(pse$What, pse$Shat, log="y", col=pse$Shat_ge_Sf+1, lwd=0.8, cex=0.5, pch=16)
plot(density(pse$What[pse$Shat_ge_Sf]), pch=16, xlim=c(0,1)) # then try the
# non-right tail extremal copula PSP as cop=PSP in the above psepolar() call.
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.