# Connected component of the toroidal density ridge of a bivariate sine von Mises, bivariate wrapped Cauchy, and bivariate wrapped normal Computation of the connected component of the density ridge of in a given set of points or, if not specified, in a regular grid on $[-\pi, \pi)$. ```r ridge_bvm(mu, kappa, eval_points, subint_1, subint_2) ridge_bwc(mu, xi, eval_points, subint_1, subint_2) ridge_bwn(mu, Sigma, kmax = 2, eval_points, subint_1, subint_2) ``` ## Arguments - `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. - `eval_points`: evaluation points for the ridge. - `subint_1`: number of points for $\theta_1$. - `subint_2`: number of points for $\theta_2$ at each $\theta_1$. - `xi`: a vector of length `3` with the marginal concentrations $(\xi_1, \xi_2)$, and the dependence parameter $\rho$. - `Sigma`: covariance matrix of size `c(2, 2)`. - `kmax`: integer number up to truncate the wrapped normal series in `-kmax:kmax`. Defaults to `2`. ## Returns A matrix of size `c(subint_1, 2)` containing the points of the connected component of the ridge. ## Examples ```r # Bivariate von Mises mu <- c(0, 0) kappa <- c(0.3, 0.5, 0.4) nth <- 100 th <- seq(-pi, pi, l = nth) x <- as.matrix(expand.grid(th, th)) d <- d_bvm(x = x, mu = mu, kappa = kappa) image(th, th, matrix(d, nth, nth), col = viridisLite::viridis(20)) ridge <- ridge_bvm(mu = mu, kappa = kappa, subint_1 = 5e2, subint_2 = 5e2) points(ridge) # Bivariate wrapped Cauchy mu <- c(0, 0) xi <- c(0.3, 0.6, 0.25) nth <- 100 th <- seq(-pi, pi, l = nth) x <- as.matrix(expand.grid(th, th)) d <- d_bwc(x = x, mu = mu, xi = xi) image(th, th, matrix(d, nth, nth), col = viridisLite::viridis(20)) ridge <- ridge_bwc(mu = mu, xi = xi, subint_1 = 5e2, subint_2 = 5e2) points(ridge) # Bivariate wrapped normal mu <- c(0, 0) Sigma <- matrix(c(10, 3, 3, 5), nrow = 2) nth <- 100 th <- seq(-pi, pi, l = nth) x <- as.matrix(expand.grid(th, th)) d <- d_bwn(x = x, mu = mu, Sigma = Sigma) image(th, th, matrix(d, nth, nth), col = viridisLite::viridis(20)) ridge <- ridge_bwn(mu = mu, Sigma = Sigma, subint_1 = 5e2, subint_2 = 5e2) points(ridge) ``` ## References Ozertem, U. and Erdogmus, D. (2011). Locally defined principal curves and surfaces. **Journal of Machine Learning Research**, 12(34):1249--1286. tools:::Rd_expr_doi("10.6083/M4ZG6Q60")

ridge_distr function