# Bivariate Empirical Sucopula of Given Approximation Order Calculation of bivariate empirical subcopula matrix of given approximation order, induced partitions, standardized bivariate sample, and dependence measures for a given continuous bivariate sample. ```r subcopemc(mat.xy, m = nrow(mat.xy), display = FALSE) ``` ## Arguments - `mat.xy`: 2-column matrix with bivariate observations of a continuous random vector $(X,Y)$. - `m`: integer value of approximation order, where $m = 2,...,n$ with $n$ equal to sample size. - `display`: logical value indicating if graphs and dependence values should be displayed. ## Details Both random variables $X$ and $Y$ must be continuous, and therefore repeated values in the sample are not expected. If found, `jitter` will be applied to break ties. 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. - **order**: numeric value of approximation order. - **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 approximation order $m > 2000$ calculation may take more than 2 minutes. Usually $m = 50$ would be enough for an acceptable approximation. ## See Also `subcopem` ## Examples ```r ## Example 1: Independent Normal and Gamma n <- 300 # sample size X <- rnorm(n) # Normal(0,1) Y <- rgamma(n, 2, 3) # Gamma(2,3) 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 <- subcopemc(XY,, display = TRUE) str(SC) ## Approximation of order m = 15 SCm15 <- subcopemc(XY, 15, display = TRUE) str(SCm15) ## Example 2: Non-monotone dependence n <- 300 # sample size Theta <- runif(n, 0, 2*pi) # Uniform circular distribution X <- cos(Theta) Y <- sin(Theta) 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 <- subcopemc(XY,, display = TRUE) str(SC) ## Approximation of order m = 15 SCm15 <- subcopemc(XY, 15, display = TRUE) str(SCm15) ```

subcopemc function