# Density evaluation, sampling, and parameter estimation of the bivariate wrapped Cauchy distribution Computation of the density of a bivariate wrapped Cauchy: [REMOVE_ME]$$ f(\theta_1, \theta_2)=c(\xi_1,\xi_2,\rho)\{c_0(\xi_1,\xi_2,\rho)-c_1(\xi_1,\xi_2,\rho) \cos (\theta_1-\mu_1)-c_2(\xi_1,\xi_2,\rho)\cos (\theta_2-\mu_2)-\\c_3(\xi_1,\xi_2,\rho) \cos (\theta_1-\mu_1) \cos (\theta_2-\mu_2)-c_4(\xi_1,\xi_2,\rho) \sin (\theta_1-\mu_1) \sin (\theta_2-\mu_2)\}^{-1}. [REMOVE_ME_2]$$ Simulation of samples from a bivariate wrapped Cauchy. Maximum likelihood and method of moments estimation of the parameters $(\mu_1, \mu_2, \xi_1, \xi_2, \rho)$. ```r d_bwc(x, mu, xi) r_bwc(n, mu, xi) fit_bwc_mm(x, hom = FALSE, indep = FALSE) fit_bwc_mle( x, start = NULL, lower = c(-pi, -pi, 0, 0, -1 + 0.001), upper = c(pi, pi, 1 - 0.001, 1 - 0.001, 1 - 0.001), hom = FALSE, indep = FALSE, ... ) ``` ## Arguments - `x`: matrix of size `c(nx, 2)` with the angles on which the density is evaluated. - `mu`: circular means of the density, a vector of length `2`. - `xi`: a vector of length `3` with the marginal concentrations $(\xi_1, \xi_2)$, and the dependence parameter $\rho$. - `n`: sample size. - `hom`: assume a homogeneous distribution with equal marginal concentrations? Defaults to `FALSE`. - `indep`: set the dependence parameter to zero? Defaults to `FALSE`. - `start`: a vector of length `5` with the initial values for the maximum likelihood optimizer. If `NULL` (default), the method of moments estimates are employed. - `lower, upper`: vectors of length `5` with the bounds for the likelihood optimizer. Default to `c(-pi, -pi, 0, 0, -1 + 1e-3)` and `c(pi, pi, 1 - 1e-3, 1 - 1e-3, 1 - 1e-3)`. - `...`: further parameters passed to `mleOptimWrapper`. ## Returns * `d_bwc`: a vector of length `nx` with the density evaluated at `x`. * `r_bwc`: a matrix of size `c(n, 2)` with the random sample. * `fit_mme_bwc, fit_mle_bwc`: a list with the parameters $(\mu_1, \mu_2, \xi_1, \xi_2, \rho)$ and the object `opt` containing the optimization summary. ## Description Computation of the density of a bivariate wrapped Cauchy: $$ f(\theta_1, \theta_2)=c(\xi_1,\xi_2,\rho)\{c_0(\xi_1,\xi_2,\rho)-c_1(\xi_1,\xi_2,\rho) \cos (\theta_1-\mu_1)-c_2(\xi_1,\xi_2,\rho)\cos (\theta_2-\mu_2)-\\c_3(\xi_1,\xi_2,\rho) \cos (\theta_1-\mu_1) \cos (\theta_2-\mu_2)-c_4(\xi_1,\xi_2,\rho) \sin (\theta_1-\mu_1) \sin (\theta_2-\mu_2)\}^{-1}. $$ Simulation of samples from a bivariate wrapped Cauchy. Maximum likelihood and method of moments estimation of the parameters $(\mu_1, \mu_2, \xi_1, \xi_2, \rho)$. ## Examples ```r ## Density evaluation mu <- c(0, 0) xi <- c(0.3, 0.5, 0.4) nth <- 50 th <- seq(-pi, pi, l = nth) x <- as.matrix(expand.grid(th, th)) d <- d_bwc(x = x, mu = mu, xi = xi) filled.contour(th, th, matrix(d, nth, nth), col = viridisLite::viridis(20), levels = seq(0, max(d), l = 20)) ## Sampling and estimation n <- 100 samp <- r_bwc(n = n, mu = mu, xi = xi) (param_mm <- fit_bwc_mm(samp)$par) (param_mle <- fit_bwc_mle(samp)$par) ``` ## References Kato, S. and Pewsey, A. (2015). A Möbius transformation-induced distribution on the torus. **Biometrika**, 102(2):359--370. tools:::Rd_expr_doi("10.1093/biomet/asv003") ## Author(s) The original code for `r_bwc` was supplied by Arthur Pewsey.

bwc function