# Gluing Two Copulas The **gluing copula** technique (Erdely, 2017, p. 71), given two bivariate copulas $\mathbf{C}_A$ and $\mathbf{C}_B$ and a fixed value $0 \le \gamma \le 1$ is [REMOVE_ME]$$ \mathbf{C}_{\gamma}(u,v) = \gamma\cdot\mathbf{C}_A(u/\gamma, v) [REMOVE_ME_2]$$ for $0 \le u \le \gamma$ and [REMOVE_ME]$$ \mathbf{C}_{\gamma}(u,v) = (1-\gamma)\cdot\mathbf{C}_B((u-\gamma)\,/\,(1-\gamma), v) [REMOVE_ME_2]$$ for $\gamma \le u \le 1$ and $\gamma$ represents the **gluing point** in $u$ (horizontal axis). The logic is simply the rescaling of $\mathbf{C}_A$ to $[0,\gamma] \times [0,1]$ and $\mathbf{C}_B$ to $[\gamma,1] \times [0,1]$. Copula gluing is potentially useful in circumstances for which regression is non-monotone. utf8 ## Description The **gluing copula** technique (Erdely, 2017, p. 71), given two bivariate copulas $\mathbf{C}_A$ and $\mathbf{C}_B$ and a fixed value $0 \le \gamma \le 1$ is $$ \mathbf{C}_{\gamma}(u,v) = \gamma\cdot\mathbf{C}_A(u/\gamma, v) $$ for $0 \le u \le \gamma$ and $$ \mathbf{C}_{\gamma}(u,v) = (1-\gamma)\cdot\mathbf{C}_B((u-\gamma)\,/\,(1-\gamma), v) $$ for $\gamma \le u \le 1$ and $\gamma$ represents the **gluing point** in $u$ (horizontal axis). The logic is simply the rescaling of $\mathbf{C}_A$ to $[0,\gamma] \times [0,1]$ and $\mathbf{C}_B$ to $[\gamma,1] \times [0,1]$. Copula gluing is potentially useful in circumstances for which regression is non-monotone. ```r glueCOP(u, v, para=NULL, ...) ``` ## Arguments - `u`: Nonexceedance probability $u$ in the $X$ direction; - `v`: Nonexceedance probability $v$ in the $Y$ direction; - `para`: A special parameter `list` (see Note ) with a mandatory element of `glue` parameter $\gamma$; and - `...`: Additional arguments to pass to the copulas. ## Returns Value(s) for the copula are returned. ## Note The following descriptions list in detail the structure and content of the `para` argument: - **`glue`**: --- The $\gamma$ gluing parameter; - **`cop1`**: --- Function of the first copula $\mathbf{A}$; - **`cop2`**: --- Function of the second copula $\mathbf{B}$; - **`para1`**: --- Vector of parameters $\Theta_\mathbf{A}$ for $\mathbf{A}$; and - **`para2`**: --- Vector of parameters $\Theta_\mathbf{B}$ for $\mathbf{B}$. ## References Erdely, A., 2017, Copula-based piecewise regression (chap. 5) **in** Copulas and dependence models with applications---Contributions in honor of Roger B. Nelsen, **eds.** Flores, U.M., Amo Artero, E., Durante, F., , J.F.: Springer, Cham, Switzerland, ISBN 978--3--319--64220--9, tools:::Rd_expr_doi("10.1007/978-3-319-64221-5") . ## Author(s) W.H. Asquith ## See Also `COP`, `breveCOP`, `composite1COP`, `composite2COP`, `composite3COP`, `convexCOP` ## Examples ```r ## Not run: para <- list(cop1=PLACKETTcop, para1=.2, cop2=GLcop, para2=1.2, glue=0.6) densityCOPplot(cop=glueCOP, para=para) # ## End(Not run) ## Not run: # Concerning Nelsen (2006, exam. 3.3, pp. 59-61) # Concerning Erdely (2017, exam. 5.1, p. 71) # Concerning Erdely (2017, exam. 5.2, p. 75) # Nelsen's example is a triangle with vertex at [G, 1]. # Erdley's example permits the construction using glueCOP from M and W. "coptri" <- function(u,v, para=NA, ...) { p <- para[1]; r <- 1 - (1-p)*v if(length(u) > 1 | length(v) > 1) stop("only scalars for this function") if(0 <= u & u <= p*v & p*v <= p) { return(u) } else if( 0 <= p*v & p*v < u & u < r) { return(p*v) } else if( p <= r & r <= u & u <= 1 ) { return(u+v-1) } else { stop("should not be here in logic") } } "UsersCop" <- function(u,v, ...) { asCOP(u,v, f=coptri, ...) } # Demonstrate Nelsen's triangular copula (black dots ) UV <- simCOP(cop=UsersCop, para=0.35, cex=0.5, pch=16) # Add Erdley's gluing of M() and W() copula (red circles) para <- list(cop1=M, cop2=W, para1=NA, para2=NA, glue=0.35) UV <- simCOP(cop=glueCOP, para=para, col=2, ploton=FALSE) # We see in the plot that the triangular copulas are the same. # For G = 0.5, Erdley shows Spearman Rho = 2*G-1 = 0, but # Schweizer-Wolff = G^2 + (G-1)^2 = 0.5, let us check these: para <- list(cop1=M, cop2=W, para1=NA, para2=NA, glue=0.5) rhoCOP( cop=glueCOP, para=para) # -2.181726e-17 wolfCOP(cop=glueCOP, para=para) # 0.4999953 # So, rhoCOP() indicates independence, but wolfCOP() indicates # dependence at the minimum value possible for a triangular copula. ## End(Not run) ```

glueCOP function