# The Product of Two Copulas Perform copula multiplication (so-called \ast-product or **Markov Product**) (Darsow and others, 1992) is a continuous analog of matrix multiplication and yields another copula: [REMOVE_ME]$$ \bigl(\mathbf{C}_1 \ast \mathbf{C}_2 \bigr)(u,v) = \mathbf{C}_3(u,v) = \int_\mathcal{I} \frac{\delta \mathbf{C}_1(u, t)}{\delta v} \frac{\delta \mathbf{C}_2(t, v)}{\delta u}\,\mathrm{d}t\mbox{,} [REMOVE_ME_2]$$ for copulas $\mathbf{C}_1(u, v)$ and $\mathbf{C}_2(u, v)$ are copulas whose $\ast$-product yields copula $\mathbf{C}_3(u, v)$ in terms of partial derivatives (`derCOP` and `derCOP2`) of the other two. Nelsen (2006, p. 245) lists several identities of the $\ast$-product involving the product ($\mathbf{\Pi}$; `P`), lower bound ($\mathbf{W}$; `W`), and upper bound ($\mathbf{M}$; `M`) copulas: [REMOVE_ME]$$ \mathbf{\Pi} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{\Pi} = \mathbf{\Pi}\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \mathbf{M} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{M} = \mathbf{M}\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \bigl(\mathbf{W} \ast \mathbf{C}\bigr)(u,v) = v - \mathbf{C}(1-u, v)\mbox{\ and\ } \bigl(\mathbf{C} \ast \mathbf{W}\bigr)(u,v) = u - \mathbf{C}(u, 1-v)\mbox{, and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \mathbf{W} \ast \mathbf{W} = \mathbf{M}\mbox{ and } \mathbf{W} \ast \mathbf{C} \ast \mathbf{W} = \hat{\mathbf{C}}\mbox{,} [REMOVE_ME_2]$$ where $\hat{\mathbf{C}}$ is the **survival copula** (`surCOP`). The $\ast$-product is associative: [REMOVE_ME]$$ \mathbf{A} \ast (\mathbf{B} \ast \mathbf{C}) = (\mathbf{A} \ast \mathbf{B}) \ast \mathbf{C}\mbox{,} [REMOVE_ME_2]$$ but $\ast$-product is not commutative (order independent). Nelsen (2006, p. 245) reports that if we view \ast as a binary operation on the set of copulas, then\mathbf{Pi} is the null element, and \mathbf{M} is the identity. Copula mulitiplication is closely linked to **Markov Processes** (Nelsen, 2006, pp. 244--248). For other descriptions and computations of copula combination are possible using the `copBasic` package, see `convexCOP`, `convex2COP`, `composite1COP`, `composite2COP`, `composite3COP`, `glueCOP`, and `convexCOP`. utf8 ## Description Perform copula multiplication (so-called \ast-product or **Markov Product**) (Darsow and others, 1992) is a continuous analog of matrix multiplication and yields another copula: $$ \bigl(\mathbf{C}_1 \ast \mathbf{C}_2 \bigr)(u,v) = \mathbf{C}_3(u,v) = \int_\mathcal{I} \frac{\delta \mathbf{C}_1(u, t)}{\delta v} \frac{\delta \mathbf{C}_2(t, v)}{\delta u}\,\mathrm{d}t\mbox{,} $$ for copulas $\mathbf{C}_1(u, v)$ and $\mathbf{C}_2(u, v)$ are copulas whose $\ast$-product yields copula $\mathbf{C}_3(u, v)$ in terms of partial derivatives (`derCOP` and `derCOP2`) of the other two. Nelsen (2006, p. 245) lists several identities of the $\ast$-product involving the product ($\mathbf{\Pi}$; `P`), lower bound ($\mathbf{W}$; `W`), and upper bound ($\mathbf{M}$; `M`) copulas: $$ \mathbf{\Pi} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{\Pi} = \mathbf{\Pi}\mbox{,} $$ $$ \mathbf{M} \ast \mathbf{C} = \mathbf{C} \ast \mathbf{M} = \mathbf{M}\mbox{,} $$ $$ \bigl(\mathbf{W} \ast \mathbf{C}\bigr)(u,v) = v - \mathbf{C}(1-u, v)\mbox{\ and\ } \bigl(\mathbf{C} \ast \mathbf{W}\bigr)(u,v) = u - \mathbf{C}(u, 1-v)\mbox{, and} $$ $$ \mathbf{W} \ast \mathbf{W} = \mathbf{M}\mbox{ and } \mathbf{W} \ast \mathbf{C} \ast \mathbf{W} = \hat{\mathbf{C}}\mbox{,} $$ where $\hat{\mathbf{C}}$ is the **survival copula** (`surCOP`). The $\ast$-product is associative: $$ \mathbf{A} \ast (\mathbf{B} \ast \mathbf{C}) = (\mathbf{A} \ast \mathbf{B}) \ast \mathbf{C}\mbox{,} $$ but $\ast$-product is not commutative (order independent). Nelsen (2006, p. 245) reports that if we view \ast as a binary operation on the set of copulas, then\mathbf{Pi} is the null element, and \mathbf{M} is the identity. Copula mulitiplication is closely linked to **Markov Processes** (Nelsen, 2006, pp. 244--248). For other descriptions and computations of copula combination are possible using the `copBasic` package, see `convexCOP`, `convex2COP`, `composite1COP`, `composite2COP`, `composite3COP`, `glueCOP`, and `convexCOP`. ```r prod2COP(u,v, cop1=NULL, para1=NULL, cop2=NULL, para2=NULL, para=NULL, pinterval=NULL, ...) ``` ## Arguments - `u`: Nonexceedance probability $u$ in the $X$ direction; - `v`: Nonexceedance probability $v$ in the $Y$ direction; - `cop1`: The $\mathbf{C}_1(u,v; \Theta_1)$ copula function with vectorization as in `asCOP`; - `para1`: Vector of parameters or other data structures for $\Theta_1$, if needed, to pass to copula $\mathbf{C}_1(u,v; \Theta_1)$; - `cop2`: The $\mathbf{C}_2(u,v; \Theta_2)$ copula function with vectorization as in `asCOP`; - `para2`: Vector of parameters or other data structures for $\Theta_2$, if needed, to pass to copula $\mathbf{C}_2(u,v; \Theta_2)$; - `para`: An `list` that can take the place of the `cop1`, `para1`, `cop2`, and `para2` arguments. These four will be populated from same named elements of the `list`, and if the other four arguments were specified through the function interface, these are silently ignored; - `pinterval`: An optional interval for the above integral. The default is $\mathcal{I} = [0,1]$ but the option of the user to replace exact end points with small numbers is possible (**e.g.** `interval=``c(lo, 1-lo)` for say `lo=.Machine$double.eps`). This interval is uniquely picked up for the interval in the above definition of `prod2COP`. The `pinterval` can also be set within the `para` and the function will pick it up from there; and - `...`: Additional arguments to pass to the copulas. ## Returns Value(s) for the copula are returned. ## Note The **Farlie--Gumbel--Morgenstern copula** ($\mathbf{FGM}(u,v; \Theta)$; `FGMcop`) is $$ \mathbf{FGM}(u,v; \Theta) = uv[1+\Theta(1-u)(1-v)]\mbox{,} $$ where $-1 \le \Theta \le 1$. Nelsen (2006, exer. 6.12, p. 249) asserts that for $\mathbf{FGM}_{(\Theta = \alpha)}$ and $\mathbf{FGM}_{(\Theta = \beta)}$ with $\ast$-product as $\mathbf{FGM}_\alpha \ast \mathbf{FGM}_\beta$ that a closed-form solution exists and is $$ \mathbf{FGM}_\alpha \ast \mathbf{FGM}_\beta = \mathbf{FGM}_{(\alpha\beta) / 3}\mbox{.} $$ This assertion is numerically true as readily verified using the `prod2COP` function: ``` u <- c(0.41, 0.87); v <- c(0.13,0.35); A <- -0.532; B <- 0.235 FGMcop( u,v, para= A*B / 3) # 0.0521598638574___ 0.3034277347150___ prod2COP(u,v, cop1=FGMcop, para1=A, cop2=FGMcop, para2=B) # 0.0521598638312605 0.3034277344807909 ``` ## References Darsow, W.F., Nguyen, B., and Olsen, E.T., 1992, Copulas and Markov processes: Illinois Journal of Mathematics, v. 26, pp. 600--624, tools:::Rd_expr_doi("10.1215/IJM/1255987328") . Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. ## Author(s) W.H. Asquith ## See Also `COP`, `composite1COP`, `composite2COP`, `composite3COP`, `convexCOP`, `convex2COP`, `glueCOP` ## Examples ```r ## Not run: # Product P * N4212 ---> P (by identity) u <- c(0.41, 0.87); v <- c(0.13, 0.35) prod2COP(u,v, cop1=P, cop2=N4212cop, para1=NA, para2=2.12) # 0.0533 and 0.3045 COP(u,v, cop=P) # 0.0533 and 0.3045 ## End(Not run) ## Not run: para <- list(cop1=PLcop, para1=0.19, cop2=PLcop, para2=34.5) UV <- simCOP(n=1000, cop=prod2COP, para=para, resamv01=FALSE, showresamv01=FALSE) # This is large simulation run (with a lot of numerical operations) is expected # at least for the Placketts and chosen parameters to trigger one or more NAs # from derCOPinv(). The simCOP() function simply continues on with ignoring the # solution or lack thereof for certain combinations, and simCOP() will report how # many of the simulated values for sample of size n were computed. For example, # for one n=1000, some 965 simulated values were returned. The defaults require # that NAs, empty simulations, remain intact. We can try resampling: UV <- simCOP(n=1000, cop=prod2COP, para=para, resamv01=TRUE, showresamv01=TRUE) rhoCOP(cop=prod2COP, para=para) # -0.4271195 (theoretical) rhoCOP(para=UV, as.sample=TRUE) # -0.4274703 # ## End(Not run) ## Not run: para <- list(cop1=PLcop, para1=0.19, cop2=PLcop, para2=34.5) # The prod2COP() might be one of the more sensitive to NAs in simulation because # of the two partial numerical derivatives involved. para$pinterval <- c(0.4, 0.6) # totally inappropriate interval for the integral # for the prod2COP() definition. Because the ... are used so extensively, we have # the "pinterval" for this function so that interval itself can be passed also. UV <- simCOP(n=1000, cop=prod2COP, para=para, resamv01=TRUE, showresamv01=TRUE, pinterval=c(0, 1 )) UV <- simCOP(n=1000, cop=prod2COP, para=para, resamv01=TRUE, showresamv01=TRUE, pinterval=c(0.4, 0.6)) # ## End(Not run) ```

prod2COP function