Package 'copBasic'

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:

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:

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:

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:

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 $[\ldots \mathrm{\ given\ } \ldots]$.

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.

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.

Author(s)

William Asquith [email protected]

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.

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in nature—An approach using copulas: Dordrecht, Netherlands, Springer, Water Science and Technology Library 56, 292 p.

Examples

## Not run: 
# Nelsen (2006, p. 75, exer. 3.15b) provides for a nice test of copBasic features.
"mcdurv" <- function(u,v, theta) {
   ifelse(u > theta & u < 1-theta & v > theta & v < 1 - theta,
             return(M(u,v) - theta), # Upper bounds copula with a shift
             return(W(u,v)))         # Lower bounds copula
}
"MCDURV" <- function(u,v, para=NULL) {
   if(is.null(para))         stop("need theta")
   if(para < 0 | para > 0.5) stop("theta ! in [0, 1/2]")
   return(asCOP(u, v, f=mcdurv, para))
}
"afunc" <- function(t) { # a sample size = 1,000 hard wired
   return(cov(simCOP(n=1000, cop=MCDURV, para=t, ploton=FALSE, points=FALSE))[1,2])
}
set.seed(6234) # setup covariance based on parameter "t" and the "root" parameter
print(uniroot(afunc, c(0, 0.5))) # "t" by simulation = 0.1023742
# Nelsen reports that if theta appox. 0.103 then covariance of U and V is zero.
# So, one will have mutually completely dependent uncorrelated uniform variables!

# Let us check some familiar measures of association:
rhoCOP( cop=MCDURV, para=0.1023742) # Spearman Rho = 0.005854481 (near zero)
tauCOP( cop=MCDURV, para=0.1023742) # Kendall Tau  = 0.2648521
wolfCOP(cop=MCDURV, para=0.1023742) # S & W Sigma  = 0.4690174 (less familiar)
D <- simCOP(n=1000, cop=MCDURV, para=0.1023742) # Plot mimics Nelsen (2006, fig. 3.11)
# Lastly, open research problem. L-comoments (matrices) measure high dimension of
# variable comovements (see lmomco package)---"method of L-comoments" for estimation?
lmomco::lcomoms2(simCOP(n=1000, cop=MCDURV, para=0),   nmom=5) # Perfect neg. corr.
lmomco::lcomoms2(simCOP(n=1000, cop=MCDURV, para=0.1023742), nmom=5)
lmomco::lcomoms2(simCOP(n=1000, cop=MCDURV, para=0.5), nmom=5) # Perfect pos. corr.
# T2 (L-correlation), T3 (L-coskew), T4 (L-cokurtosis), and T5 matrices result. For
# Theta = 0 or 0.5 see the matrix symmetry with a sign change for L-coskew and T5 on
# the off diagonals (offdiags). See unities for T2. See near zero for offdiag terms
# in T2 near zero. But then see that T4 off diagonals are quite different from those
# for Theta 0.1024 relative to 0 or 0.5. As a result, T4 has captured a unique
# property of U vs V.
## End(Not run)

Description

Compute the Akaike information criterion (AIC) $\mathrm{AIC}_\mathbf{C}$ (Chen and Guo, 2019, p. 29), which is computed using mean square error $\mathrm{MSE}_\mathbf{C}$ as

$\mathrm{MSE}_\mathbf{C} = \frac{1}{n}\sum_{i=1}^n \bigl(\mathbf{C}_n(u_i,v_i) - \mathbf{C}_{\Theta_m}(u_i, v_i)\bigr)^2\mbox{ and}$

$\mathrm{AIC}_\mathbf{C} = 2m + n\log(\mathrm{MSE}_\mathbf{C})\mbox{,}$

where $\mathbf{C}_n(u_i,v_i)$ is the empirical copula (empirical joint probability) for the $i$th observation, $\mathbf{C}_{\Theta_m}(u_i, v_i)$ is the fitted copula having $m$ parameters in $\Theta$. The $\mathbf{C}_n(u_i,v_i)$ comes from EMPIRcop. The $\mathrm{AIC}_\mathbf{C}$ is in effect saying that the best copula will have its joint probabilities plotting on a 1:1 line with the empirical joint probabilities, which is an $\mathrm{AIC}_\mathbf{C} = -\infty$. From the $\mathrm{MSE}_\mathbf{C}$ shown above, the root mean square error rmseCOP and Bayesian information criterion (BIC) bicCOP can be computed. These goodness-of-fits can assist in deciding one copula favorability over another, and another goodness-of-fit using the absolute differences between $\mathbf{C}_n(u,v)$ and $\mathbf{C}_{\Theta_m}(u, v)$ is found under statTn.

Usage

aicCOP(u, v=NULL, cop=NULL, para=NULL, m=NA, ...)

Arguments

Value

The value for $\mathrm{AIC}_\mathbf{C}$ is returned.

Author(s)

W.H. Asquith

References

Chen, Lu, and Guo, Shenglian, 2019, Copulas and its application in hydrology and water resources: Springer Nature, Singapore, ISBN 978–981–13–0574–0.

See Also

EMPIRcop, bicCOP, rmseCOP

Examples

## Not run: 
S <- simCOP(80, cop=GHcop, para=5) # Simulate some probabilities, but we
# must then treat these as data and recompute empirical probabilities.
U <- lmomco::pp(S$U, sort=FALSE); V <- lmomco::pp(S$V, sort=FALSE)
# The parent distribution is Gumbel-Hougaard extreme value copula.
# But in practical application we do not know that, but say we speculate
# that perhaps the Galambos extreme value might be the parent. Then maximum
# likelihood is used to fit the single parameter.
pGL <- mleCOP(U,V, cop=GLcop, interval=c(0,20))$par

aics <- c(aicCOP(U,V, cop=GLcop, para=pGL), aicCOP(U,V, cop=P), aicCOP(U,V, cop=PSP))
names(aics) <- c("GLcop", "P", "PSP")
print(aics) # We will see that the first AIC is the smallest as the
# Galambos has the nearest overall behavior than the P and PSP copulas.
## End(Not run)

Description

The Ali–Mikhail–Haq copula (Nelsen, 2006, p. 92–93, 172) is

$\mathbf{C}_{\Theta}(u,v) = \mathbf{AMH}(u,v) = \frac{uv}{1 - \Theta(1-u)(1-v)}\mbox{,}$

where $\Theta \in [-1,+1)$, where the right boundary, $\Theta = 1$, can sometimes be considered valid according to Mächler (2014). The copula $\Theta \rightarrow 0$ becomes the independence copula ($\mathbf{\Pi}(u,v)$; P), and the parameter $\Theta$ is readily computed from a Kendall Tau (tauCOP) by

$\tau_\mathbf{C} = \frac{3\Theta - 2}{3\Theta} - \frac{2(1-\Theta)^2\log(1-\Theta)}{3\Theta^2}\mbox{,}$

and by Spearman Rho (rhoCOP), through Mächler (2014), by

$\rho_\mathbf{C} = \sum_{k=1}^\infty \frac{3\Theta^k}{{k + 2 \choose 2}^2}\mbox{.}$

The support of $\tau_\mathbf{C}$ is $[(5 - 8\log(2))/3, 1/3]$ $\approx$ $[-0.1817258, 0.3333333]$ and the $\rho_\mathbf{C}$ is $[33 - 48\log(2), 4\pi^2 - 39]$ $\approx$ $[-0.2710647, 0.4784176]$, which shows that this copula has a limited range of dependency. The infinite summation is easier to work with than Nelsen (2006, p. 172) definition of

$\rho_\mathbf{C} = \frac{12(1+\Theta)}{\Theta^2}\mathrm{dilog}(1-\Theta)- \frac{24(1-\Theta)}{\Theta^2}\log(1-\Theta)- \frac{3(\Theta+12)}{\Theta}\mbox{,}$

where the $\mathrm{dilog(x)}$ is the dilogarithm function defined by

$\mathrm{dilog}(x) = \int_1^x \frac{\log(t)}{1-t}\,\mathrm{d}t\mbox{.}$

The integral version has more nuances with approaches toward $\Theta = 0$ and $\Theta = 1$ than the infinite sum version.

Usage

AMHcop(u, v, para=NULL, rho=NULL, tau=NULL, fit=c("rho", "tau"), ...)

Arguments

Value

Value(s) for the copula are returned. Otherwise if tau is given, then the $\Theta$ is computed and a list having

and if para=NULL and tau=NULL, then the values within u and v are used to compute Kendall Tau and then compute the parameter, and these are returned in the aforementioned list.

Note

Mächler (2014) reports on accurate computation of $\tau_\mathbf{C}$ and $\rho_\mathbf{C}$ for this copula for conditions of $\Theta \rightarrow 0$ and in particular derives the following equation, which does not have $\Theta$ in the denominator:

$\rho_\mathbf{C} = \sum_{k=1}^{\infty} \frac{3\Theta^k}{{k+2 \choose 2}^2}\mbox{.}$

The copula package provides a Taylor series expansion for $\tau_\mathbf{C}$ for small $\Theta$ in the copula::tauAMH(). This is demonstrated here between the implementation of $\tau = 0$ for parameter estimation in the copBasic package to that in the more sophisticated implementation in the copula package.

  copula::tauAMH(AMHcop(tau=0)$para) # theta = -2.313076e-07

It is seen that the numerical approaches yield quite similar results for small $\tau_\mathbf{C}$, and finally, a comparison to the $\rho_\mathbf{C}$ is informative:

  rhoCOP(AMHcop, para=1E-9)    # 3.333333e-10 (two nested integrations)
  copula:::.rhoAmhCopula(1E-9) # 3.333333e-10 (cutoff based)
  theta <- seq(-1,1, by=.0001)
  RHOa <- sapply(theta, function(t) rhoCOP(AMHcop, para=t))
  RHOb <- sapply(theta, function(t) copula:::.rhoAmhCopula(t))
  plot(10^theta, RHOa-RHOb, type="l", col=2)

The plot shows that the apparent differences are less than 1 part in 100 million—The copBasic computation is radically slower though, but rhoCOP was designed for generality of copula family.

Author(s)

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

Mächler, Martin, 2014, Spearman's Rho for the AMH copula—A beautiful formula: copula package vignette, accessed on April 7, 2018, at https://CRAN.R-project.org/package=copula under the vignette rhoAMH-dilog.pdf.

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Pranesh, Kumar, 2010, Probability distributions and estimation of Ali–Mikhail–Haq copula: Applied Mathematical Sciences, v. 4, no. 14, p. 657–666.

See Also

P

Examples

## Not run: 
t <- 0.9 # The Theta of the copula and we will compute Spearman Rho.
di <- integrate(function(t) log(t)/(1-t), lower=1, upper=(1-t))$value
A <- di*(1+t) - 2*log(1-t) + 2*t*log(1-t) - 3*t # Nelsen (2007, p. 172)
rho <- 12*A/t^2 - 3    # 0.4070369
rhoCOP(AMHcop, para=t) # 0.4070369
sum(sapply(1:100, function(k) 3*t^k/choose(k+2, 2)^2)) # Machler (2014)
# 0.4070369 (see Note section, very many tens of terms are needed) 
## End(Not run)

## Not run: 
  layout(matrix(1:2, byrow=TRUE)) # Note: Kendall Tau is same on reversal.
  s <- 2; set.seed(s); nsim <- 10000
  UVn <- simCOP(nsim, cop=AMHcop, para=c(-0.9, "FALSE" ), col=4)
  mtext("Normal definition [default]") # '2nd' parameter could be skipped
  set.seed(s) # seed used to keep Rho/Tau inside attainable limits
  UVr <- simCOP(nsim, cop=AMHcop, para=c(-0.9, "TRUE"),   col=2)
  mtext("Reversed definition")
  AMHcop(UVn[,1], UVn[,2], fit="rho")$rho # -0.2581653
  AMHcop(UVr[,1], UVr[,2], fit="rho")$rho # -0.2570689
  rhoCOP(cop=AMHcop, para=-0.9)           # -0.2483124
  AMHcop(UVn[,1], UVn[,2], fit="tau")$tau # -0.1731904
  AMHcop(UVr[,1], UVr[,2], fit="tau")$tau # -0.1724820
  tauCOP(cop=AMHcop, para=-0.9)           # -0.1663313 
## End(Not run)

Description

This function is intended to document and then to extend a simple API to end users to aid in implementation of other copulas for use within the copBasic package. There is no need or requirement to use asCOP for almost all users. However, for the mathematical definition of some copulas, the asCOP function might help considerably. This is because there is a need for special treatment of $u$ and $v$ vectors of probability as each interacts with the vectorization implicit in R. The special treatment is needed because many copulas are based on the operators such as min() and max(). When numerical integration used by the integrate() function in R in some copula operators, such as tauCOP for the Kendall Tau of a copula, special accommodation is needed related to the inherent vectorization in R and how integrate() works.

Basically, the problem is that one can not strictly rely in all circumstances on what R does in terms of value recycling when $u$ and $v$ are of unequal lengths. The source code is straightforward. Simply put, if lengths of $u$ and $v$ are unity, then there is no concern, and even if the length of $u$ (say) is unity and $v$ is 21, then recycling of $u$ would often be okay. The real danger is when $u$ and $v$ have unequal lengths and those lengths are each greater than unity—the R treatment can not be universally relied upon with the various numerics herein involving optimization and nested numerical integration.

The example shows how a formula definition of a copula that is not a copula already implemented by copBasic is set into a function deltacop and then used inside another function UsersCop that will be the official copula that is compatible with a host of functions in copBasic. The use of asCOP provides the length check necessary on $u$ and $v$, and the argument ... provides optional parameter support should the user's formula require more settings.

Usage

asCOP(u, v, f=NULL, ...)

Arguments

Value

The value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

COP

Examples

## Not run: 
# Concerning Nelsen (2006, exer. 3.7, pp. 64--65)
"trianglecop" <- function(u,v, para=NULL, ...) {
   # If para is set, then the triangle is rotated 90d clockwise.
   if(! is.null(para) && para == 1) { t <- u; u <- v; v <- t }
   if(length(u) > 1 | length(v) > 1) stop("only scalars for this function")
   v2<-v/2; if(0   <= u    &  u   <= v2 & v2 <= 1/2) { return(u    )
   } else   if(0   <= v2   & v2   <  u  &  u < 1-v2) { return(v2   )
   } else   if(1/2 <= 1-v2 & 1-v2 <= u  &  u <= 1  ) { return(u+v-1)
   } else { stop("should not be here in logic") }
}
"UsersCop" <- function(u,v, ...) { asCOP(u,v, f=trianglecop, ...) }
n=20000; UV <- simCOP(n=n, cop=UsersCop)
# The a-d elements of the problem now follow:
# (a) Pr[V = 1 - |2*U -1|] = 1 and Cov(U,V) = 0; so that two random variables
# can be uncorrelated but each is perfectly predictable from the other
mean(UV$V - (1 - abs(2*UV$U -1))) # near zero; Nelsen says == 0
cov(UV$U, UV$V)                   # near zero; Nelsen says == 0

# (b) Cop(m,n) = Cop(n,m); so that two random variables can be identically
# distributed, uncorrelated, and not exchangeable
EMPIRcop(0.95,0.17, para=UV) # = A
EMPIRcop(0.17,0.95, para=UV) # = B; then A != B

# (c) Pr[V - U > 0] = 2/3; so that two random variables can be identically
# distributed, but their difference need not be symmetric about zero
tmp <- (UV$V - UV$U) > 0
length(tmp[tmp == TRUE])/n # about 2/3; Nelsen says == 2/3
# the prior two lines yield about 1/2 for independence copula P()

# (d) Pr[X + Y > 0] = 2/3; so that uniform random variables on (-1,1) can each
# be symmetric about zero, but their sum need not be.
tmp <- ((2*UV$V - 1) + (2*UV$U - 1)) > 0
length(tmp[tmp == TRUE])/n # about 2/3; Nelsen says == 2/3 
## End(Not run)

## Not run: 
# Concerning Nelsen (2006, exam. 3.10, p. 73)
"shufflecop" <- # assume scalar arguments for u and v
function(u,v, para, ...) {
   m <- para$mixer; subcop <- para$subcop
   if(is.na(m) | m <= 0 | m >= 1) stop("m ! in [0,1]")
   if(u <= m) { return(    subcop(1-m+u, v, para=para$para) -
                           subcop(1-m,   v, para=para$para))
   } else {     return(v - subcop(1-m,   v, para=para$para) +
                           subcop(u-m,   v, para=para$para))
}
}
"UsersCop" <- function(u,v, para=NULL) {
   asCOP(u,v, f=shufflecop, para=para)
}
n <- 1000; u <- runif(n)
para <- list(mixer=runif(1), subcop=W, para=20)
v <- sapply(1:n, function(i) {
      simCOPmicro(u[i], cop=UsersCop, para=para) } )
plot(data.frame(U=u, V=v), pch=17, col=rgb(1,0,1,1),
     xlab="U, NONEXCEEDANCE PROBABILTY", ylab="V, NONEXCEEDANCE PROBABILITY")
mtext("Shuffle Copula Nelsen (2006, exam. 3.10, p. 73)")

# Concerning Nelsen (2006, exam. 5.14, p. 195)
"deltacop" <- function(u,v, ...) { min(c(u,v,(u^2+v^2)/2))     }
"UsersCop" <- function(u,v, ...) { asCOP(u,v, f=deltacop, ...) }
isCOP.PQD(cop=UsersCop) # TRUE + Rho=0.288 and Tau=0.333 as Nelsen says
isCOP.LTD(cop=UsersCop, wrtV=TRUE) # FALSE as Nelsen says
isCOP.RTI(cop=UsersCop, wrtV=TRUE) # FALSE as Nelsen says 
## End(Not run)

Description

Compute the Bayesian information criterion (BIC) $\mathrm{BIC}_\mathbf{C}$ (Chen and Guo, 2019, p. 29), which is computed using mean square error $\mathrm{MSE}_\mathbf{C}$ as

$\mathrm{MSE}_\mathbf{C} = \frac{1}{n}\sum_{i=1}^n \bigl(\mathbf{C}_n(u_i,v_i) - \mathbf{C}_{\Theta_m}(u_i, v_i)\bigr)^2\mbox{ and}$

$\mathrm{BIC}_\mathbf{C} = m\log(n) + n\log(\mathrm{MSE}_\mathbf{C})\mbox{,}$

where $\mathbf{C}_n(u_i,v_i)$ is the empirical copula (empirical joint probability) for the $i$th observation, $\mathbf{C}_{\Theta_m}(u_i, v_i)$ is the fitted copula having $m$ parameters in $\Theta$. The $\mathbf{C}_n(u_i,v_i)$ comes from EMPIRcop. The $\mathrm{BIC}_\mathbf{C}$ is in effect saying that the best copula will have its joint probabilities plotting on a 1:1 line with the empirical joint probabilities, which is an $\mathrm{BIC}_\mathbf{C} = -\infty$. From the $\mathrm{MSE}_\mathbf{C}$ shown above, the root mean square error rmseCOP and Akaike information criterion (AIC) aicCOP can be computed. These goodness-of-fits can assist in deciding on one copula favorability over another, and another goodness-of-fit using the absolute differences between $\mathbf{C}_n(u,v)$ and $\mathbf{C}_{\Theta_m}(u, v)$ is found under statTn.

Usage

bicCOP(u, v=NULL, cop=NULL, para=NULL, m=NA, ...)

Arguments