# Bernstein Copula Approximation Bernstein copula approximation from the empirical subcopula of given bivariate data. ```r Bcopula(mat.xy, m, both.cont = FALSE, tolimit = 1e-05) ``` ## Arguments - `mat.xy`: 2-column matrix with bivariate observations of a random vector $(X,Y)$. - `m`: integer value of approximation order, where $m = 2,...,n$ with $n$ equal to sample size. A recommended value for $m$ would be the minimum between $\sqrt{n}$ and $50$. - `both.cont`: logical value, if TRUE then (X,Y) are considered (both) as continuos random variables, and jittering will be applied to repeated values (if any). - `tolimit`: tolerance limit in numerical approximation of the inverse of the first partial derivatives of the estimated Bernstein copula. ## Details Each of the random variables $X$ and $Y$ may be of any kind (discrete, continuous, or mixed). NA values are not allowed. ## Returns A list containing the following components: - **copula**: bivariate Bernstein Copula function (BC) of order $m$ - **du**: bivariate function $\partial BC(u,v)/\partial u$ - **dv**: bivariate function $\partial BC(u,v)/\partial v$ - **du.inv**: inverse of `du` with respect to v, given u and alpha (numerical approx) - **dv.inv**: inverse of `dv` with respect to u, given v and alpha (numerical approx) - **density**: bivariate Bernstein copula density function of order $m$ - **bilinearCopula**: bivariate function of bilinear approximation of copula - **bilinearSubcopula**: $(m+1)\times (m+1)$ matrix with empirical subcopula values - **sample.size**: sample size of bivariate observations - **order**: approximation order $m$ used - **both.cont**: logical value, TRUE if both variables considered as continuous - **tolimit**: tolerance limit in numerical approximation of `du.inv` and `dv.inv` - **subcopemObject**: list object with the output from `subcopem` if `both.cont = FALSE` or from `subcopemc` if `both.cont = TRUE` ## References Erdely, A. (2017) **A subcopula based dependence measure**. Kybernetika 53(2), 231-243. DOI: 10.14736/kyb-2017-2-0231 Nelsen, R.B. (2006) **An Introduction to Copulas**. Springer, New York. Sancetta, A., Satchell, S. (2004) **The Bernstein copula and its applications to modeling and approximations of multivariate distributions.** Econometric Theory 20, 535-562. DOI: 10.1017/S026646660420305X ## Author(s) Arturo Erdely [https://sites.google.com/site/arturoerdely](https://sites.google.com/site/arturoerdely) ## Note If both $X$ and $Y$ are continuous random variables it is faster and better to set `both.cont = TRUE`. ## Acknowledgement Development of this code was partially supported by Programa UNAM DGAPA PAPIIT through project IN115817. ## See Also `subcopem`, `subcopemc` ## Examples ```r ## (X,Y) continuous random variables with copula FGM(param = 1) # Theoretical formulas FGMcopula <- function(u, v) u*v*(1 + (1 - u)*(1 - v)) dFGM.du <- function(u, v) (2*u - 1)*(v^2) + 2*v*(1 - u) dFGM.dv <- function(u, v) (2*v - 1)*(u^2) + 2*u*(1 - v) A1 <- function(u) 2*(1 - u) A2 <- function(u, z) sqrt(A1(u)^2 - 4*(A1(u) - 1)*z) dFGM.du.inv <- function(u, z) 2*z/(A1(u) + A2(u, z)) FGMdensity <- function(u, v) 2*(1 - u - v + 2*u*v) # Simulating FGM observations n <- 3000 U <- runif(n) Z <- runif(n) V <- mapply(dFGM.du.inv, U, Z) # Applying Bcopula to FGM simulated values B <- Bcopula(cbind(U, V), 50, TRUE) str(B) # Comparing theoretical values versus Bernstein and Bilinear approximations u <- 0.70; v <- 0.55 FGMcopula(u, v); B[["copula"]](u, v); B[["bilinearCopula"]](u, v) dFGM.du(u, v); B[["du"]](u, v) dFGM.dv(u, v); B[["dv"]](u, v) dFGM.du.inv(u, 0.8); B[["du.inv"]](u, 0.8) FGMdensity(u, v); B[["density"]](u, v) ```

Bcopula function