# Bivariate Empirical Subcopula Calculation of bivariate empirical subcopula matrix, induced partitions, standardized bivariate sample, and dependence measures for a given bivariate sample. ```r subcopem(mat.xy, display = FALSE) ``` ## Arguments - `mat.xy`: 2-column matrix with bivariate observations of a random vector $(X,Y)$. - `display`: logical value indicating if graphs and dependence measures should be displayed. ## 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: - **depMon**: monotone standardized supremum distance in $[-1,1].$ - **depMonNonSTD**: monotone non-standardized supremum distance $[min,value,max].$ - **depSup**: standardized supremum distance in $[0,1].$ - **depSupNonSTD**: non-standardized supremum distance $[min,value,max].$ - **matrix**: matrix with empirical subcopula values. - **part1**: vector with partition induced by first variable $X$. - **part2**: vector with partition induced by second variable $Y$. - **sample.size**: numeric value of sample size. - **std.sample**: 2-column matrix with the standardized bivariate sample. - **sample**: 2-column matrix with the original bivariate sample of $(X,Y)$. If `display = TRUE` then the values of `depMon`, `depMonNonSTD`, `depSup`, and `depSupNonSTD` will be displayed, and the following graphs will be generated: marginal histograms of $X$ and $Y$, scatterplots of the original and the standardized bivariate sample, contour and image bivariate graphs of the empirical subcopula. ## References Durante, F. and Sempi, C. (2016) **Principles of Copula Theory**. Taylor and Francis Group, Boca Raton. 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. ## 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 use `subcopemc`. ## See Also `subcopemc` ## Examples ```r ## Example 1: Discrete-discrete Poisson positive dependence n <- 1000 # sample size X <- rpois(n, 5) # Poisson(parameter = 5) p <- 2 # another parameter Y <- mapply(rpois, rep(1, n), 1 + p*X) # creating dependence XY <- cbind(X, Y) # 2-column matrix with bivariate sample cor(XY, method = "pearson")[1, 2] # Pearson's correlation cor(XY, method = "spearman")[1, 2] # Spearman's correlation cor(XY, method = "kendall")[1, 2] # Kendall's correlation SC <- subcopem(XY, display = TRUE) str(SC) ## Example 2: Continuous-discrete non-monotone dependence n <- 1000 # sample size X <- rnorm(n) # Normal(0,1) Y <- 2*(X > 1) - 1*(X > -1) # Discrete({-1, 0, 1}) XY <- cbind(X, Y) # 2-column matrix with bivariate sample cor(XY, method = "pearson")[1, 2] # Pearson's correlation cor(XY, method = "spearman")[1, 2] # Spearman's correlation cor(XY, method = "kendall")[1, 2] # Kendall's correlation SC <- subcopem(XY, display = TRUE) str(SC) ```

subcopem function