copBasic-package: Basic Theoretical Copula, Empirical Copula, and Various Utility Functions
Description
The copBasic package is oriented around bivariate copula theory and mathematical operations closely follow the standard texts of Nelsen (2006) and Joe (2014) as well as select other references. Another recommended text is Salvadori et al. (2007) and is cited herein, but about half of that excellent book concerns univariate applications. The primal objective of copBasic is to provide a basic application programming interface (API) to numerous results shown by authoritative texts on copulas. It is intended that the package will help other copula students in self study, potential course work, and applied circumstances.
Notes on copulas that are supported. The author has focused on pedagogical aspects of copulas, and this package is a diary of sorts. Originally, the author did not implement many copulas in the copBasic in order to deliberately avoid redundancy to that support such as it exists on the R CRAN. Though as time has progressed, other copulas have been added occasionally based on needs of the user community, need to show some specific concept in the general theory, or test algorithms. For example, the Clayton copula (CLcop) is a late arriving addition to the copBasic package (c.2017), which was added to assist a specific user.
Helpful Navigation of Copulas Implemented in the copBasic Package
Some entry points to the copulas implemented are listed in the Table of Copulas:
| Name | Symbol | Function | Concept |
| Lower-bounds copula | \(\mathbf{W}(u,v)\) | W | copula |
| Independence copula | \(\mathbf{\Pi}(u,v)\) | P | copula |
| Upper-bounds copula | \(\mathbf{M}(u,v)\) | M | copula |
| Fréchet Family copula | \(\mathbf{FF}(u,v)\) | FRECHETcop | copula |
| Ali--Mikhail--Haq copula | \(\mathbf{AMH}(u,v)\) | AMHcop | copula |
| Clayton copula | \(\mathbf{CL}(u,v)\) | CLcop | copula |
| Copula of uniform circle | \(\mathbf{CIRC}(u,v)\) | CIRCcop | copula |
| Farlie--Gumbel--Morgenstern (generalized) | \(\mathbf{FGM}(u,v)\) | FGMcop | copula |
| Galambos copula | \(\mathbf{GL}(u,v)\) | GLcop | copula |
| Gumbel--Hougaard copula | \(\mathbf{GH}(u,v)\) | GHcop | copula |
| Hüsler--Reiss copula | \(\mathbf{HR}(u,v)\) | HRcop | copula |
| Joe B5 copula | \(\mathbf{B5}(u,v)\) | JOcopB5 | copula |
| Nelsen eq.4-2-12 copula | \(\mathbf{N4212cop}(u,v)\) | N4212cop | copula |
| Ordinal Sums by Copula | \(\mathbf{C}_\mathcal{J}(u,v)\) | ORDSUMcop | copula |
| Pareto copula | \(\mathbf{PA}(u,v)\) | PLcop | copula |
| Plackett copula | \(\mathbf{PL}(u,v)\) | PLcop | copula |
| PSP copula | \(\mathbf{PSP}(u,v)\) | PSP | copula |
| Raftery copula | \(\mathbf{RF}(u,v)\) | RFcop | copula |
| Rayleigh copula | \(\mathbf{RAY}(u,v)\) | RAYcop | copula |
| g-EV copula (Gaussian extreme value) | \(\mathbf{gEV}(u,v)\) | gEVcop | copula |
| t-EV copula (t-distribution extreme value) | \(\mathbf{tEV}(u,v)\) | tEVcop | copula |
The language and vocabulary of copulas is formidable. The author (Asquith) has often emphasized “vocabulary” words in italics, which is used extensively and usually near the opening of function-by-function documentation to identify vocabulary words, such as survival copula (see surCOP). This syntax tries to mimic and accentuate the word usage in Nelsen (2006) and Joe (2014).
The italics then are used to draw connections between concepts. In conjunction with the summary of functions in copBasic-package, the extensive cross referencing to functions and expansive keyword indexing should be beneficial. The author had no experience with copulas prior to a chance happening upon Nelsen (2006) in c.2008. The copBasic package is a personal tour de force in self-guided learning. Hopefully, this package and user's manual will be helpful to others.
A few comments on notation herein are needed. A bold math typeface is used to represent a copula such as \(\mathbf{\Pi}\) (see P) for the independence copula. The syntax \(\mathcal{R}\times\mathcal{R} \equiv \mathcal{R}^2\) denotes the orthogonal domain of two real numbers, and \([0,1]\times [0,1]\) \(\equiv\) \(\mathcal{I}\times\mathcal{I} \equiv \mathcal{I}^2\) denotes the orthogonal domain on the unit square of probabilities. Limits of integration \([0,1]\) or \([0,1]^2\) involving copulas are thus shown as \(\mathcal{I}\) and \(\mathcal{I}^2\), respectively.
The random variables \(X\) and \(Y\) respectively denote the horizontal and vertical directions in \(\mathcal{R}^2\). Their probabilistic counterparts are uniformly distributed random variables on \([0,1]\), are respectively denoted as \(U\) and \(V\), and necessarily also are the respective directions in \(\mathcal{I}^2\) (\(U\) denotes the horizontal, \(V\) denotes the vertical). Often realizations of these random variables are respectively \(x\) and \(y\) for \(X\) and \(Y\) and \(u\) and \(v\) for \(U\) and \(V\).
There is an obvious difference between nonexceedance probability \(F\) and its complement, which is exceedance probability defined as \(1-F\). Both \(u\) and \(v\) herein are in nonexceedance probability. Arguments to many functions herein are u \(= u\) and v \(= v\) and are almost exclusively nonexceedance but there are instances for which the probability arguments are u \(= 1 - u = u'\) and v \(= 1 - v = v'\).
Helpful Navigation of the copBasic Package
Some other entry points into the package are listed in the following table:
| Name | Symbol | Function | Concept |
| Copula | \(\mathbf{C}(u,v)\) | COP | copula theory |
| Survival copula | \(\hat{\mathbf{C}}(u',v')\) | surCOP | copula theory |
| Joint survival function | \(\overline{\mathbf{C}}(u,v)\) | surfuncCOP | copula theory |
| Co-copula | \(\mathbf{C}^\star(u',v')\) | coCOP | copula theory |
| Dual of a copula | \(\tilde{\mathbf{C}}(u,v)\) | duCOP | copula theory |
| Primary copula diagonal | \(\delta(t)\) | diagCOP | copula theory |
| Secondary copula diagonal | \(\delta^\star(t)\) | diagCOP | copula theory |
| Inverse copula diagonal | \(\delta^{(-1)}(f)\) | diagCOPatf | copula theory |
| Joint probability | \({-}{-}\) | jointCOP | copula theory |
| Bivariate L-moments | \(\delta^{[\ldots]}_{k;\mathbf{C}}\) | bilmoms and lcomCOP | bivariate moments |
| Bivariate L-comoments | \(\tau^{[\ldots]}_{k;\mathbf{C}}\) | bilmoms and lcomCOP | bivariate moments |
| Blomqvist Beta | \(\beta_\mathbf{C}\) | blomCOP | bivariate association |
| Gini Gamma | \(\gamma_\mathbf{C}\) | giniCOP | bivariate association |
| Hoeffding Phi | \(\Phi_\mathbf{C}\) | hoefCOP | bivariate association |
| Nu-Skew | \(\nu_\mathbf{C}\) | nuskewCOP | bivariate moments |
| Nu-Star (skew) | \(\nu^\star_\mathbf{C}\) | nustarCOP | bivariate moments |
| Lp distance to independence | \(\Phi_\mathbf{C} \rightarrow L_p\) | LpCOP | bivariate association |
| Permutation-Mu | \(\mu_{\infty\mathbf{C}}^\mathrm{permsym}\) | LzCOPpermsym | permutation asymmetry |
| Kendall Tau | \(\tau_\mathbf{C}\) | tauCOP | bivariate association |
| Kendall Measure | \(K_\mathbf{C}(z)\) | kmeasCOP | copula theory |
| Kendall Function | \(F_K(z)\) | kfuncCOP | copula theory |
| Inverse Kendall Function | \(F_K^{(-1)}(z)\) | kfuncCOPinv | copula theory |
| An L-moment of \(F_K(z)\) | \(\lambda_r(F_K)\) | kfuncCOPlmom | L-moment theory |
| L-moments of \(F_K(z)\) | \(\lambda_r(F_K)\) | kfuncCOPlmoms | L-moment theory |
| Semi-correlations (negatives) | \(\rho_N^{-}(a)\) | semicorCOP | bivariate tail association |
| Semi-correlations (positives) | \(\rho_N^{+}(a)\) | semicorCOP | bivariate tail association |
| Spearman Footrule | \(\psi_\mathbf{C}\) | footCOP | bivariate association |
| Spearman Rho | \(\rho_\mathbf{C}\) | rhoCOP | bivariate association |
| Schweizer--Wolff Sigma | \(\sigma_\mathbf{C}\) | wolfCOP | bivariate association |
| Density of a copula | \(c(u,v)\) | densityCOP | copula density |
| Density visualization | \({-}{-}\) | densityCOPplot | copula density |
| Empirical copula | \(\mathbf{C}_n(u,v)\) | EMPIRcop | copula |
| Empirical simulation | \({-}{-}\) | EMPIRsim | copula simulation |
| Empirical simulation | \({-}{-}\) | EMPIRsimv | copula simulation |
| Empirical copulatic surface | \({-}{-}\) | EMPIRgrid | copulatic surface |
| Parametric copulatic surface | \({-}{-}\) | gridCOP | copulatic surface |
| Parametric simulation | \({-}{-}\) | simCOP or rCOP | copula simulation |
| Parametric simulation | \({-}{-}\) | simCOPmicro | copula simulation |
| Maximum likelihood | \(\mathcal{L}(\Theta_d)\) | mleCOP | copula fitting |
| Akaike information criterion | \(\mathrm{AIC}_\mathbf{C}\) | aicCOP | goodness-of-fit |
| Bayesian information criterion | \(\mathrm{BIC}_\mathbf{C}\) | bicCOP | goodness-of-fit |
| Root mean square error | \(\mathrm{RMSE}_\mathbf{C}\) | rmseCOP | goodness-of-fit |
| Another goodness-of-fit | \(T_n\) | statTn | goodness-of-fit |
Several of the functions listed above are measures of “bivariate association.” Two of the measures (Kendall Tau, tauCOP; Spearman Rho, rhoCOP) are widely known. R provides native support for their sample estimation of course, but each function can be used to call the cor() function in R for parallelism to the other measures of this package. The other measures (Blomqvist Beta, Gini Gamma, Hoeffding Phi, Schweizer--Wolff Sigma, Spearman Footrule) support sample estimation by specially formed calls to their respective functions: blomCOP, giniCOP, hoefCOP, wolfCOP, and footCOP. Gini Gamma (giniCOP) documentation (also joeskewCOP) shows extensive use of theoretical and sample computations for these and other functions.
Concerning goodness-of-fit and although not quite the same as copula properties (such as “correlation”) per se as the coefficients aforementioned in the prior paragraph, three goodness-of-fit metrics of a copula compared to the empirical copula, which are all based the mean square error (MSE), are aicCOP, bicCOP, and rmseCOP. This triad of functions is useful for making decisions on whether a copula is more favorable than another to a given dataset. However, because they are genetically related by using MSE and if these are used for copula fitting by minimization, the fits will be identical. A statement of “not quite the same” is made because the previously described copula properties are generally defined as types of deviations from other copulas (such as P). Another goodness-of-fit statistic is statTn, which is based on magnitude summation of fitted copula difference from the empirical copula. These four (aicCOP, bicCOP, rmseCOP, and statTn) collectively are relative simple and readily understood measures. These bulk sample statistics are useful, but generally thought to not capture the nuances of tail behavior (semicorCOP and taildepCOP might be useful).
Bivariate skewness measures are supported in the functions joeskewCOP (nuskewCOP and nustarCOP) and uvlmoms (uvskew). Extensive discussion and example computations of bivariate skewness are provided in the joeskewCOP documentation. Lastly, so-called bivariate L-moments and bivariate L-comoments of a copula are directly computable in bilmoms (though that function using Monte Carlo integration is deprecated) and lcomCOP (direct numerical integration). The lcomCOP function is the theoretical counterpart to the sample L-comoments provided in the lmomco package.
Bivariate random simulation methods by several functions are identified in the previous table. The copBasic package explicitly uses only conditional simulation also known as the conditional distribution method for random variate generation following Nelsen (2006, pp. 40--41) (see also simCOPmicro, simCOP). The numerical derivatives (derCOP and derCOP2) and their inversions (derCOPinv and derCOPinv2) represent the foundation of the conditional simulation. There are other methods in the literature and available in other R packages, and a comparison of some methods is made in the Examples section of the Gumbel--Hougaard copula (GHcop).
Several functions in copBasic make the distinction between \(V\) with respect to (wrt) \(U\) and \(U\) wrt \(V\), and a guide for the nomenclature involving wrt distinctions is listed in the following table:
| Name | Symbol | Function | Concept |
| Copula inversion | \(V\) wrt \(U\) | COPinv | copula operator |
| Copula inversion | \(U\) wrt \(V\) | COPinv2 | copula operator |
| Copula derivative | \(\delta \mathbf{C}/\delta u\) | derCOP | copula operator |
| Copula derivative | \(\delta \mathbf{C}/\delta v\) | derCOP2 | copula operator |
| Copula derivative inversion | \(V\) wrt \(U\) | derCOPinv | copula operator |
| Copula derivative inversion | \(U\) wrt \(V\) | derCOPinv2 | copula operator |
| Joint curves | \(t \mapsto \mathbf{C}(u=U, v)\) | joint.curvesCOP | copula theory |
| Joint curves | \(t \mapsto \mathbf{C}(u, v=V)\) | joint.curvesCOP2 | copula theory |
| Level curves | \(t \mapsto \mathbf{C}(u=U, v)\) | level.curvesCOP | copula theory |
| Level curves | \(t \mapsto \mathbf{C}(u, v=V)\) | level.curvesCOP2 | copula theory |
| Level set | \(V\) wrt \(U\) | level.setCOP | copula theory |
| Level set | \(U\) wrt \(V\) | level.setCOP2 | copula theory |
| Median regression | \(V\) wrt \(U\) | med.regressCOP | copula theory |
| Median regression | \(U\) wrt \(V\) | med.regressCOP2 | copula theory |
| Quantile regression | \(V\) wrt \(U\) | qua.regressCOP | copula theory |
| Quantile regression | \(U\) wrt \(V\) | qua.regressCOP2 | copula theory |
| Copula section | \(t \mapsto \mathbf{C}(t,a)\) | sectionCOP | copula theory |
| Copula section | \(t \mapsto \mathbf{C}(a,t)\) | sectionCOP | copula theory |
The previous two tables do not include all of the myriad of special functions to support similar operations on empirical copulas. All empirical copula operators and utilities are prepended with EMPIR in the function name. An additional note concerning package nomenclature is that an appended “2” to a function name indicates \(U\) wrt \(V\) (e.g. EMPIRgridderinv2 for an inversion of the partial derivatives \(\delta \mathbf{C}/\delta v\) across the grid of the empirical copula).
Some additional functions to compute often salient features or characteristics of copulas or bivariate data, including functions for bivariate inference or goodness-of-fit, are listed in the following table:
| Name | Symbol | Function | Concept |
| Left-tail decreasing | \(V\) wrt \(U\) | isCOP.LTD | bivariate association |
| Left-tail decreasing | \(U\) wrt \(V\) | isCOP.LTD | bivariate association |
| Right-tail increasing | \(V\) wrt \(U\) | isCOP.RTI | bivariate association |
| Right-tail increasing | \(U\) wrt \(V\) | isCOP.RTI | bivariate association |
| Pseudo-polar representation | \((\widehat{S},\widehat{W})\) | psepolar | extremal dependency |
| Tail concentration function | \(q_\mathbf{C}(t)\) | tailconCOP | bivariate tail association |
| Tail (lower) dependency | \(\lambda^L_\mathbf{C}\) | taildepCOP | bivariate tail association |
| Tail (upper) dependency | \(\lambda^U_\mathbf{C}\) | taildepCOP | bivariate tail association |
| Tail (lower) order | \(\kappa^L_\mathbf{C}\) | tailordCOP | bivariate tail association |
| Tail (upper) order | \(\kappa^U_\mathbf{C}\) | tailordCOP | bivariate tail association |
| Neg'ly quadrant dependency | NQD | isCOP.PQD | bivariate association |
| Pos'ly quadrant dependency | PQD | isCOP.PQD | bivariate association |
| Permutation symmetry | \(\mathrm{permsym}\) | isCOP.permsym | copula symmetry |
| Radial symmetry | \(\mathrm{radsym}\) | isCOP.radsym | copula symmetry |
| Skewness (Joe, 2014) | \(\eta(p; \psi)\) | uvskew | bivariate skewness |
| Kullback--Leibler Divergence | \(\mathrm{KL}(f \mid g)\) | kullCOP | bivariate inference |
| KL sample size | \(n_{f\!g}\) | kullCOP | bivariate inference |
| The Vuong Procedure | \({-}{-}\) | vuongCOP | bivariate inference |
| Spectral measure | \(H(w)\) | spectralmeas | extremal dependency inference |
| Stable tail dependence | \(\widehat{l}(x,y)\) | stabtaildepf | extremal dependency inference |
| L-comoments (samp. distr.) | \({-}{-}\) | lcomCOPpv | experimental bivariate inference |
The Table of Probabilities that follows lists important relations between various joint probability concepts, the copula, nonexceedance probabilities \(u\) and \(v\), and exceedance probabilities \(u'\) and \(v'\). A compact summary of these probability relations has obvious usefulness. The notation \([\ldots, \ldots]\) is to read as \([\ldots \mathrm{\ and\ } \ldots]\), and the \([\ldots \mid \ldots]\) is to be read as \([\ldots \mathrm{\ given\ } \ldots]\).
| Probability | and | Symbol Convention |
| \(\mathrm{Pr}[\,U \le u, V \le v\,]\) | \(=\) | \(\mathbf{C}(u,v)\) --- The copula, COP |
| \(\mathrm{Pr}[\,U > u, V > v\,]\) | \(=\) | \(\hat{\mathbf{C}}(u',v')\) --- The survival copula, surCOP |
| \(\mathrm{Pr}[\,U \le u, V > v\,]\) | \(=\) | \(u - \mathbf{C}(u,v')\) |
| \(\mathrm{Pr}[\,U > u, V \le v\,]\) | \(=\) | \(v - \mathbf{C}(u',v)\) |
| \(\mathrm{Pr}[\,U \le u \mid V \le v\,]\) | \(=\) | \(\mathbf{C}(u,v)/v\) |
| \(\mathrm{Pr}[\,V \le v \mid U \le u\,]\) | \(=\) | \(\mathbf{C}(u,v)/u\) |
| \(\mathrm{Pr}[\,U \le u \mid V > v\,]\) | \(=\) | \(\bigl(u - \mathbf{C}(u,v)\bigr)/(1 - v)\) |
| \(\mathrm{Pr}[\,V \le v \mid U > u\,]\) | \(=\) | \(\bigl(v - \mathbf{C}(u,v)\bigr)/(1 - u)\) |
| \(\mathrm{Pr}[\,U > u \mid V > v\,]\) | \(=\) | \(\hat{\mathbf{C}}(u',v')/u' = \overline{\mathbf{C}}(u,v)/(1-u)\) |
| \(\mathrm{Pr}[\,V > v \mid U > u\,]\) | \(=\) | \(\hat{\mathbf{C}}(u',v')/v' = \overline{\mathbf{C}}(u,v)/(1-v)\) |
| \(\mathrm{Pr}[\,V \le v \mid U = u\,]\) | \(=\) | \(\delta \mathbf{C}(u,v)/\delta u\) --- Partial derivative, derCOP |
| \(\mathrm{Pr}[\,U \le u \mid V = v\,]\) | \(=\) | \(\delta \mathbf{C}(u,v)/\delta v\) --- Partial derivative, derCOP2 |
| \(\mathrm{Pr}[\,U > u \mathrm{\ or\ } V > v\,]\) | \(=\) | \(\mathbf{C}^\star(u',v') = 1 - \mathbf{C}(u',v')\) --- The co-copula, coCOP |
| \(\mathrm{Pr}[\,U \le u \mathrm{\ or\ } V \le v\,]\) | \(=\) | \(\tilde{\mathbf{C}}(u,v) = u + v - \mathbf{C}(u,v)\) --- The dual of a copula, duCOP |
| \(E[\,U \mid V = v\,]\) | \(=\) | \(\int_0^1 (1 - \delta \mathbf{C}(u,v)/\delta v)\mathrm{d}u\) --- Expectation of U given V, EuvCOP |
| \(E[\,V \mid U = u\,]\) | \(=\) | \(\int_0^1 (1 - \delta \mathbf{C}(u,v)/\delta u)\mathrm{d}v\) --- Expectation of V given U, EvuCOP |
The function jointCOP has considerable demonstration in its Note section of the joint and and joint or relations shown through simulation and counting scenarios. Also there is a demonstration in the Note section of function duCOP on application of the concepts of joint and conditions, joint or conditions, and importantly joint mutually exclusive or conditions.
Copula Construction Methods
Permutation asymmetry can be added to a copula by breveCOP. One, two, or more copulas can be “composited,” “combined,” or “multiplied” in interesting ways to create highly unique bivariate relations and as a result, complex dependence structures can be formed. The package provides three main functions for copula composition: composite1COP composites a single copula with two compositing parameters, composite2COP composites two copulas with two compositing parameters, and composite3COP composites two copulas with four compositing parameters. Also two copulas can be combined through a weighted convex combination using convex2COP with a single weighting parameter, and even \(N\) number of copulas can be combined by weights using convexCOP. So-called “gluing” two copula by a parameter is provided by glueCOP. Multiplication of two copulas to form a third is supported by prod2COP. All eight functions for compositing, combining, or multipling copulas are compatible with joint probability simulation (simCOP), measures of association (e.g. \(\rho_\mathbf{C}\)), and presumably all other copula operations using copBasic features. Finally, ordinal sums of copula are provided by ORDSUMcop and ORDSUWcop as particularly interesting methods of combining copulas.
| No. of copulas | Combining Parameters | Function | Concept |
| 1 | \(\beta\) | breveCOP | adding permuation asymmetry |
| 1 | \(\alpha, \beta\) | composite1COP | copula combination |
| 2 | \(\alpha, \beta\) | composite2COP | copula combination |
| 2 | \(\alpha, \beta, \kappa, \gamma\) | composite3COP | copula combination |
| 2 | \(\alpha, (1-\alpha)\) | convex2COP | weighted copula combination |
| \(N\) | \(\omega_{i \in N}\) | convexCOP | weighted copula combination |
| 2 | \(\gamma\) | glueCOP | gluing of coupla |
| 2 | \(\bigl(\mathbf{C}_1 \ast \mathbf{C}_2 \bigr)\) | prod2COP | copula multiplication |
| \(N\) | \(\mathbf{C}_{\mathcal{J}i}\) for \(\mathcal{J}_{i \in N}\) partitions | ORDSUMcop | M-ordinal sums of copulas |
| \(N\) | \(\mathbf{C}_{\mathcal{J}i}\) for \(\mathcal{J}_{i \in N}\) partitions | ORDSUWcop | W-ordinal sums of copulas |
Useful Copula Relations by Visualization
There are a myriad of relations amongst variables computable through copulas, and these were listed in the Table of Probabilities earlier in this documentation. There is a script located in the inst/doc directory of the copBasic sources titled CopulaRelations_BaseFigure_inR.txt. This script demonstrates, using the PSP copula, relations between the copula (COP), survival copula (surCOP), joint survival function of a copula (surfuncCOP), co-copula (coCOP), and dual of a copula function (duCOP). The script performs simulation and manual counts observations meeting various criteria in order to compute their empirical probabilities. The script produces a base figure, which after extending in editing software, is suitable for educational description and is provided at the end of this documentation.
A Review of “Return Periods” using Copulas
Risk analyses of natural hazards are commonly expressed as annual return periods \(T\) in years, which are defined for a nonexceedance probability \(q\) as \(T = 1/(1-q)\). In bivariate analysis, there immediately emerge two types of return periods representing \(T_{q;\,\mathrm{coop}}\) and \(T_{q;\,\mathrm{dual}}\) conditions between nonexceedances of the two hazard sources (random variables) \(U\) and \(V\). It is usual in many applications for \(T\) to be expressed equivalently as a probability \(q\) in common for both variables.
Incidentally, the \(\mathrm{Pr}[\,U > u \mid V > v\,]\) and \(\mathrm{Pr}[\,V > v \mid U > u\,]\) probabilities also are useful for conditional return period computations following Salvadori et al. (2007, pp. 159--160) but are not further considered here. Also the \(F_K(w)\) (Kendall Function or Kendall Measure of a copula) is the core tool for secondary return period computations (see kfuncCOP).
Let the copula \(\mathbf{C}(u,v; \Theta)\) for nonexceedances \(u\) and \(v\) be set for some copula family (formula) by a parameter vector \(\Theta\). The copula family and parameters define the joint coupling (loosely meant the dependency/correlation) between hazards \(U\) and \(V\). If “failure” occurs if either or both hazards \(U\) and \(V\) are at probability \(q\) threshold (\(u = v = 1 - 1/T = q\)) for \(T\)-year return period, then the real return period of failure is defined using either the copula \(\mathbf{C}(q,q; \Theta)\) or the co-copula \(\mathbf{C}^\star(q',q'; \Theta)\) for exceedance probability \(q' = 1 - q\) is
$$T_{q;\,\mathrm{coop}} = \frac{1}{1 - \mathbf{C}(q, q; \Theta)} = \frac{1}{\mathbf{C}^\star(1-q, 1-q; \Theta)}\mbox{\ and}$$ $$T_{q;\,\mathrm{coop}} \equiv \frac{1}{\mathrm{cooperative\ risk}}\mbox{.}$$
Or in words, the hazard sources collaborate or cooperate to cause failure. If failure occurs, however, if and only if both hazards \(U\) and \(V\) occur simultaneously (the hazards must “dually work together” or be “conjunctive”), then the real return period is defined using either the dual of a copula (function) \(\tilde{\mathbf{C}}(q,q; \Theta)\), the joint survival function \(\overline{\mathbf{C}}(q,q;\Theta)\), or survival copula \(\hat{\mathbf{C}}(q',q'; \Theta)\) as
$$T_{q;\,\mathrm{dual}} = \frac{1}{1 - \tilde{\mathbf{C}}(q,q; \Theta)} = \frac{1}{\overline{\mathbf{C}}(q,q;\Theta)} = \frac{1}{\hat{\mathbf{C}}(q',q';\Theta)} \mbox{\ and}$$ $$T_{q;\,\mathrm{dual}} \equiv \frac{1}{\mathrm{complement\ of\ dual\ protection}}\mbox{.}$$
Numerical demonstration is informative. Salvadori et al. (2007, p. 151) show for a Gumbel--Hougaard copula (GHcop) having \(\Theta =\) 3.055 and \(T =\) 1,000 years (\(q = 0.999\)) that \(T_{q;\,\mathrm{coop}} = 797.1\) years and that \(T_{q;\,\mathrm{dual}}\) = 1,341.4 years, which means that average return periods between “failures” are
$$T_{q;\,\mathrm{coop}} \le T \le T_{q;\,\mathrm{dual}}\mbox{\ and thus}$$
$$797.1 \le T \le 1314.4\mbox{\ years.}$$
With the following code, these bounding return-period values are readily computed and verified using the prob2T() function from the lmomco package along with copBasic functions COP (generic functional interface to a copula) and duCOP (dual of a copula):
q <- T2prob(1000)
lmomco::prob2T( COP(q,q, cop=GHcop, para=3.055)) # 797.110
lmomco::prob2T(duCOP(q,q, cop=GHcop, para=3.055)) # 1341.438
An early source (in 2005) by some of those authors cited on p. 151 of Salvadori et al. (2007; their citation “[67]”) shows \(T_{q;\,\mathrm{dual}} = 798\) years---a rounding error seems to have been committed. Finally just for reference, a Gumbel--Hougaard copula having \(\Theta = 3.055\) corresponds to an analytical Kendall Tau (see GHcop) of \(\tau \approx 0.673\), which can be verified through numerical integration available from tauCOP as:
tauCOP(cop=GHcop, para=3.055, brute=TRUE) # 0.6726542
Thus, a “better understanding of the statistical characteristics of [multiple hazard sources] requires the study of their joint distribution” (Salvadori et al., 2007, p. 150).
Interaction of copBasic to Copulas in Other Packages
Originally, the copBasic package was not intended to be a port of the numerous bivariate copulas or over re-implementation other bivariate copulas available in R though as the package passed its 10th year in 2018, the original intent changed. It is useful to point out a demonstration showing an implementation of the Gaussian copula from the copula package, which is shown in the Note section of med.regressCOP in a circumstance of ordinary least squares linear regression compared to median regression of a copula as well as prediction limits of both regressions. Another demonstration in context of maximum pseudo-log-likelihood estimation of copula parameters is seen in the Note section mleCOP, and also see “API to the copula package” or “package copula (comparison to)” entries in the Index of this user manual.
Arguments
Author
William Asquith william.asquith@ttu.edu
References
Cherubini, U., Luciano, E., and Vecchiato, W., 2004, Copula methods in finance: Hoboken, NJ, Wiley, 293 p.
Hernández-Maldonado, V., Díaz-Viera, M., and Erdely, A., 2012, A joint stochastic simulation method using the Bernstein copula as a flexible tool for modeling nonlinear dependence structures between petrophysical properties: Journal of Petroleum Science and Engineering, v. 90--91, pp. 112--123.
Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.
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