# Density evaluation, sampling, and parameter estimation of the bivariate sine von Mises distribution Computation of the density and normalizing constant $T(\kappa_1, \kappa_2, \lambda)$ of the bivariate sine von Mises [REMOVE_ME]$$ f(\theta_1, \theta_2)= T(\kappa_1, \kappa_2, \lambda)\exp\{\kappa_1 \cos(\theta_1-\mu_1) +\kappa_2 \cos(\theta_2-\mu_2) +\lambda \sin(\theta_1-\mu_1) \sin(\theta_2-\mu_2)\}. [REMOVE_ME_2]$$ Simulation of samples from a bivariate sine von Mises. Maximum likelihood and method of moments estimation of the parameters $(\mu_1, \mu_2, \kappa_1, \kappa_2, \lambda)$. ```r d_bvm(x, mu, kappa, log_const = NULL) const_bvm(kappa, M = 25, MC = 10000) r_bvm(n, mu, kappa) fit_bvm_mm( x, lower = c(0, 0, -30), upper = c(30, 30, 30), start = NULL, M = 25, hom = FALSE, indep = FALSE, ... ) fit_bvm_mle( x, start = NULL, M = 25, lower = c(-pi, -pi, 0, 0, -30), upper = c(pi, pi, 30, 30, 30), 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`. - `kappa`: vector of length `3` with the concentrations $(\kappa_1, \kappa_2)$ and the dependence parameter $\lambda$ of the density. - `log_const`: logarithm of the normalizing constant. Computed internally if `NULL` (default). - `M`: truncation of the series expansion for computing the normalizing constant. Defaults to `25`. - `MC`: Monte Carlo replicates for computing the normalizing constant when there is no series expansion. Defaults to `1e4`. - `n`: sample size. - `lower, upper`: vectors of length `5` with the bounds for the likelihood optimizer. Default to `c(-pi, -pi, 0, 0, -30)` and `c(pi, pi, 30, 30, 30)`. - `start`: a vector of length `5` with the initial values for the maximum likelihood optimizer. The first two entries are disregarded in `fit_bvm_mm`. If `NULL` (default), the starting values are taken from the estimation of marginal von Mises in `fit_bvm_mm`. In `fit_bvm_mle`, the method of moments estimates are employed. - `hom`: assume a homogeneous distribution with equal marginal concentrations? Defaults to `FALSE`. - `indep`: set the dependence parameter to zero? Defaults to `FALSE`. - `...`: further parameters passed to `mleOptimWrapper`. ## Returns * `d_bvm`: a vector of length `nx` with the density evaluated at `x`. * `const_bvm`: the value of the normalizing constant $T(\kappa_1, \kappa_2, \lambda)$. * `r_bvm`: a matrix of size `c(n, 2)` with the random sample. * `fit_mme_bvm, fit_mle_bvm`: a list with the estimated parameters $(\mu_1, \mu_2, \kappa_1, \kappa_2, \lambda)$ and the object `opt` containing the optimization summary. ## Description Computation of the density and normalizing constant $T(\kappa_1, \kappa_2, \lambda)$ of the bivariate sine von Mises $$ f(\theta_1, \theta_2)= T(\kappa_1, \kappa_2, \lambda)\exp\{\kappa_1 \cos(\theta_1-\mu_1) +\kappa_2 \cos(\theta_2-\mu_2) +\lambda \sin(\theta_1-\mu_1) \sin(\theta_2-\mu_2)\}. $$ Simulation of samples from a bivariate sine von Mises. Maximum likelihood and method of moments estimation of the parameters $(\mu_1, \mu_2, \kappa_1, \kappa_2, \lambda)$. ## Examples ```r ## Density evaluation mu <- c(0, 0) kappa <- 3:1 nth <- 50 th <- seq(-pi, pi, l = nth) x <- as.matrix(expand.grid(th, th)) const <- const_bvm(kappa = kappa) d <- d_bvm(x = x, mu = mu, kappa = kappa, log_const = log(const)) filled.contour(th, th, matrix(d, nth, nth), col = viridisLite::viridis(31), levels = seq(0, max(d), l = 30)) ## Sampling and estimation n <- 100 samp <- r_bvm(n = n, mu = mu, kappa = kappa) (param_mm <- fit_bvm_mm(samp)$par) (param_mle <- fit_bvm_mle(samp)$par) ``` ## References Mardia, K. V., Hughes, G., Taylor, C. C., and Singh, H. (2008). A multivariate von Mises with applications to bioinformatics. **Canadian Journal of Statistics**, 36(1):99--109. tools:::Rd_expr_doi("10.1002/cjs.5550360110") Singh, H., Hnizdo, V., and Demchuk, E. (2002). Probabilistic model for two dependent circular variables. **Biometrika**, 89(3):719--723. tools:::Rd_expr_doi("10.1093/biomet/89.3.719")

bvm function