duCOP: The Dual of a Copula Function

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

The Dual of a Copula Function

Description

Compute the dual of a copula (function) from a copula (Nelsen, 2006, pp. 33–34), which is defined as

\mathrm{Pr}[U \le v \mathrm{\ or\ } V \le v] = \tilde{\mathbf{C}}(u,v) = u + v - \mathbf{C}(u,v)\mbox{,}

where \tilde{\mathbf{C}}(u,v) is the dual of a copula and u and v are nonexceedance probabilities. The dual of a copula is the expression for the probability that either U \le u or V \le v, which is unlike the co-copula (function) (see coCOP) that provides \mathrm{Pr}[U > u \mathrm{\ or\ } V > v]. The dual of a copula is a function and not in itself a copula. The dual of the survival copula (surCOP) is the co-copula (function) (coCOP). Some rules of copulas mean that

\hat{\mathbf{C}}(u',v') + \tilde{\mathbf{C}}(u,v) = 1\mbox{,}

where \hat{\mathbf{C}}(u',v') is the survival copula in terms of exceedance probabilities u' and v' or in copBasic code that the functions surCOP + duCOP equal unity.

The function duCOP gives “protection” against simultaneous (concurrent or dual) risk by failure if and only if failure is caused (defined) by both hazard sources U and V being by themselves responsible for failure. Expressing this in terms of an annual probability of occurrence (q), one has

q = 1 - \mathrm{Pr}[U \le v \mathrm{\ or\ } V \le v] = 1 - \tilde{\mathbf{C}}(u,v)\mbox{\ or}

in R code q <- 1 - duCOP(u,v). So, as a mnemonic: A dual of a copula is the probabililty of nonexceedance if the hazard sources must dual (concur, link, pair, twin, twain) between each other to cause failure. An informative graphic is shown within copBasic-package.

Usage

duCOP(u, v, cop=NULL, para=NULL, ...)

Arguments

u	Nonexceedance probability u in the X direction;
v	Nonexceedance probability v in the Y direction;
cop	A copula function;
para	Vector of parameters or other data structure, if needed, to pass to the copula; and
...	Additional arguments to pass (such as parameters, if needed, for the copula in the form of a list.

Value

Value(s) for the dual of a copula are returned.

Note

There can be confusion in the interpretation and implemenation of the or condition of joint probability provided by \tilde{\mathbf{C}}(u,v). Two types of or's seemingly exist depending on one's concept of the meaning of “or.” To start, there is the “either or both” conceptualization (joint or) that encompasses either “event” (say a loss) of importance for random variables U and V as well as the joint and conditions where both variables simultaneously are generating an event of importance.

Let us continue by performing a massive simulation for the \mathbf{PSP}(u,v) copula (PSP) and set an either event standard on the margins as 10 percent for an arbitrary starting point. The \mathbf{PSP} has positive association with lower tail dependency, and the example here considers the left tail as the risk tail.

  Event <- 0.1; nn <- 100000; set.seed(9238)
  UV <- simCOP(n=nn, cop=PSP, graphics=FALSE) # 1E5 realizations

Next, let us step through counting and then make theoretical comparisons using copula theory. The joint and condition as nonexceedances is

  ANDs <- length(UV$U[UV$U <= Event & UV$V <= Event]) / nn
  ANDt <- COP(Event, Event, cop=PSP)
  message(   "Joint AND by simulation = ", round(ANDs, digits=5),
           "\n    Joint AND by theory = ", round(ANDt, digits=5))
  # ANDs = 0.05348 and ANDt = 0.05263 (numerical congruence)

where it is obvious that the simulations and theory estimate about the same joint and condition. Now, the joint or condition as nonexceedances is

  ORs <- length(UV$U[UV$U <= Event | UV$V <= Event]) / nn
  ORt <- duCOP(Event, Event, cop=PSP)
  message(   "Joint OR by simulation = ", round(ORs, digits=5),
           "\n    Joint OR by theory = ", round(ORt, digits=5))
  # ORs = 0.14779 and ORt = 0.14737 (numerical congruence)

where it is obvious that the simulations and theory estimate about the same joint or condition. Finally, the joint mutually exclusive or condition as nonexceedances is

  eORs <- length((UV$U[(UV$U <= Event | UV$V <= Event) &
                     ! (UV$U <= Event & UV$V <= Event)])) / nn
  eORt <- ORt - ANDt # theoretical computation
  message(   "Joint exclusive OR by simulation = ", round(eORs, digits=5),
           "\n    Joint exclusive OR by theory = ", round(eORt, digits=5))
  # eORs = 0.09431 and eORt = 0.09474 (numerical congruence)

where it is obvious that the simulations and theory estimate about the same joint mutually exclusive or condition, and where it is shown that the prior two theoretical joint probabilities can be subtracted from each to yield the mutually exclusive or condition.

Let us then play out a scenario in which it is judged that of the events causing damage that the simultaneous occurrance is worse but that engineering against about 5 percent of events not occurring at the same time represents the most funding available. Using numerical methods, it is possible to combine \tilde{\mathbf{C}} and \mathbf{C} and assume equal marginal risk in U and V as the following list shows:

  "designf" <- function(t) { # a one-off function just for this example
     duCOP(t, t, cop=PSP) - COP(t, t, cop=PSP) - 5/100 # 5 percent
  }
  dThres <- uniroot(designf, c(.Machine$double.eps,0.5))$root

where the uniroot function performs the optimization and the .Machine$double.eps value is used because the \mathbf{PSP} is NaN for zero probability. (It is unity for unity marginal probabilities.)

The design threshold on the margins then is dThres \approx 0.05135. In other words, the designThres is the marginal probability that results in about 5 percent of events not occurring at the same time. Then considering the simulated sample and counting the nonexceedances by code one achieves:

  Damage       <- length( UV$U[ UV$U <= dThres | UV$V <= dThres ])
  SimDamage    <- length( UV$U[ UV$U <= dThres & UV$V <= dThres ])
  NonSimDamage <- length((UV$U[(UV$U <= dThres | UV$V <= dThres) &
                             ! (UV$U <= dThres & UV$V <= dThres)]) )
  message(  "                 Damaging Events (sim.) = ", Damage,
          "\n    Simultaneous damaging events (sim.) = ", SimDamage,
          "\n Nonsimultaneous damaging events (sim.) = ", NonSimDamage)

but also the theoretical expectations are readily computed using copula theory:

  tDamage       <- as.integer(duCOP(dThres, dThres, cop=PSP) * nn)
  tSimDamage    <- as.integer(  COP(dThres, dThres, cop=PSP) * nn)
  tNonSimDamage <- tDamage - tSimDamage
  message(  "                 Damaging Events (theory) = ", tDamage,
          "\n    Simultaneous damaging events (theory) = ", tSimDamage,
          "\n Nonsimultaneous damaging events (theory) = ", tNonSimDamage)

The counts from the former listing are 7,670; 2,669; and 5,001, whereas the respective counts from the later listing are 7,635; 2,635; and 5,000. Numerical congruency in the counts thus exists.

Author(s)

W.H. Asquith

References

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

See Also

COP, coCOP, surCOP, jointCOP, joint.curvesCOP

Examples

u <- runif(1); t <- runif(1)
duCOP(cop=W,u,t)    # joint or probability for perfect negative dependence
duCOP(cop=P,u,t)    # joint or probability for perfect        independence
duCOP(cop=M,u,t)    # joint or probability for perfect positive dependence
duCOP(cop=PSP,u,t)  # joint or probability for some positive    dependence

# Next demonstrate COP + duCOP = unity.
"MOcop.formula" <- function(u,v, para=para, ...) {
   alpha <- para[1]; beta <- para[2]; return(min(v*u^(1-alpha), u*v^(1-beta)))
}
"MOcop" <- function(u,v, ...) { asCOP(u,v, f=MOcop.formula, ...) }

u <- 0.2; v <- 0.75; ab <- c(1.5, 0.3)
surCOP(1-u,1-v, cop=MOcop, para=ab) + duCOP(u,v, cop=MOcop, para=ab) # UNITY

# See extended code listings and discussion in the Note section

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