psepolar: 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, ...)

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 $S_f$.

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.

Author

William Asquith william.asquith@ttu.edu

References