# The Co-Copula Function Compute the **co-copula (function)** from a copula (Nelsen, 2006, pp. 33--34), which is defined as [REMOVE_ME]$$ \mathrm{Pr}[U > u \mathrm{\ or\ } V > v] = \mathbf{C}^{\star}(u',v') = 1 - \mathbf{C}(u',v')\mbox{,} [REMOVE_ME_2]$$ where $\mathbf{C}^{\star}(u',v')$ is the co-copula and $u'$ and $v'$ are exceedance probabilities and are equivalent to $1-u$ and $1-v$ respectively. The co-copula is the expression for the probability that either $U > u$ or $V > v$ when the arguments to $\mathbf{C}^{\star}(u',v')$ are exceedance probabilities, which is unlike the **dual of a copula (function)** (see `duCOP`) that provides $\mathrm{Pr}[U \le u \mathrm{\ or\ } V \le v]$. The co-copula is a function and not in itself a copula. Some rules of copulas mean that $\mathbf{C}(u,v) + \mathbf{C}^{\star}(u',v') \equiv 1$ or in `copBasic` syntax that the functions `COP(u,v)` + `coCOP(u,v)` equal unity if the `exceedance` argument to `coCOP` is set to `FALSE`. The function `coCOP` gives risk against failure if failure is defined as either hazard source $U$ or $V$ occuring by themselves or if both occurred at the same time. Expressing this in terms of an annual probability of occurrence ($q$), one has [REMOVE_ME]$$ q = 1 - \mathrm{Pr}[U > u \mathrm{\ or\ } V > v] = \mathbf{C}^{\star}(u',v') \mbox{\ or} [REMOVE_ME_2]$$ in code `q <- coCOP(u,v, exceedance=FALSE, ...)`. So, in yet other words and as a mnemonic: **A co-copula is the probabililty of exceedance if the hazard sources collaborate or cooperate to cause failure.** Also, $q$ can be computed by `q <-``coCOP(1 - u, 1 - v,` `exceedance=TRUE, ...)`. utf8 ## Description Compute the **co-copula (function)** from a copula (Nelsen, 2006, pp. 33--34), which is defined as $$ \mathrm{Pr}[U > u \mathrm{\ or\ } V > v] = \mathbf{C}^{\star}(u',v') = 1 - \mathbf{C}(u',v')\mbox{,} $$ where $\mathbf{C}^{\star}(u',v')$ is the co-copula and $u'$ and $v'$ are exceedance probabilities and are equivalent to $1-u$ and $1-v$ respectively. The co-copula is the expression for the probability that either $U > u$ or $V > v$ when the arguments to $\mathbf{C}^{\star}(u',v')$ are exceedance probabilities, which is unlike the **dual of a copula (function)** (see `duCOP`) that provides $\mathrm{Pr}[U \le u \mathrm{\ or\ } V \le v]$. The co-copula is a function and not in itself a copula. Some rules of copulas mean that $\mathbf{C}(u,v) + \mathbf{C}^{\star}(u',v') \equiv 1$ or in `copBasic` syntax that the functions `COP(u,v)` + `coCOP(u,v)` equal unity if the `exceedance` argument to `coCOP` is set to `FALSE`. The function `coCOP` gives risk against failure if failure is defined as either hazard source $U$ or $V$ occuring by themselves or if both occurred at the same time. Expressing this in terms of an annual probability of occurrence ($q$), one has $$ q = 1 - \mathrm{Pr}[U > u \mathrm{\ or\ } V > v] = \mathbf{C}^{\star}(u',v') \mbox{\ or} $$ in code `q <- coCOP(u,v, exceedance=FALSE, ...)`. So, in yet other words and as a mnemonic: **A co-copula is the probabililty of exceedance if the hazard sources collaborate or cooperate to cause failure.** Also, $q$ can be computed by `q <-``coCOP(1 - u, 1 - v,` `exceedance=TRUE, ...)`. ```r coCOP(u, v, cop=NULL, para=NULL, exceedance=TRUE, ...) ``` ## Arguments - `u`: Exceedance probability ($u' = 1-u$) in the $X$ direction; - `v`: Exceedance probability ($v' = 1-v$) in the $Y$ direction; - `cop`: A copula function; - `para`: Vector of parameters or other data structure, if needed, to pass to the copula; - `exceedance`: A logical controlling the probability direction. Are `u` and `v` values given really $u'$ and $v'$, respectively? If `FALSE`, then the complements of the two are made internally and the nonexceedances can thus be passed; and - `...`: Additional arguments to pass to the copula. ## Returns The value(s) for the co-copula are returned. ## Note The author (Asquith) finds the use of exceedance probabilities delicate in regards to Nelsen's notation. The `coCOP` function and `surCOP` have the `exceedance` argument to serve as a reminder that the co-copula as defined in the literature uses **exceedance probabilities** as its arguments, although the arguments as code `u` and `v` do not mimic the overline nomenclature ($\,\overline{\cdots}\,$) of the exceedance (survival) probabilities. ## References Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. ## Author(s) W.H. Asquith ## See Also `COP`, `surCOP`, `duCOP` ## Examples ```r u <- 1 - runif(1); v <- 1 - runif(1) # as exceedance, in order to reinforce the # change to exceedance instead of nonexceedance that otherwise dominates this package message("Exceedance probabilities u' and v' are ", u, " and ", v) coCOP(u,v,cop=PLACKETTcop, para=10) # Positive association Plackett # computation using manual manipulation to nonexceedance probability 1 - COP(cop=PSP,(1-u),(1-v)) # computation using internal manipulation to nonexceedance probability coCOP(cop=PSP, u, v) # Next demonstrate COP + coCOP = unity. "MOcop.formula" <- function(u,v, para=para, ...) { # Marshall-Olkin copula 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) COP(u,v, cop=MOcop, para=ab) + coCOP(1-u,1-v, cop=MOcop, para=ab) # UNITY ```

coCOP function