# Index a ridge curve, creating the Smoothed and Indexed Estimated Ridge (SIER) Indexing of an unordered collection of points defining the estimated density ridge curve. The indexing is done by a multidimensional scaling map to the real line, while the smoothing is done by local polynomial regression for polyspherical-on-scalar regression. ```r index_ridge(endpoints, X, d, l_index = 1000, f_index = 2, probs_scores = seq(0, 1, l = 101), verbose = FALSE, type_bwd = c("1se", "min")[1], p = 0, ...) ``` ## Arguments - `endpoints`: a matrix of size `c(nx, sum(d) + r)` with the end points of the ridge algorithm to be indexed. - `X`: a matrix of size `c(n, sum(d) + r)` with the sample. - `d`: vector of size `r` with dimensions. - `l_index`: length of the grid index used for finding projections. Defaults to `1e3`. - `f_index`: factor with the range of the grid for finding ridge projections. Defaults to `2`, which is twice the range of MDS indexes. - `probs_scores`: probabilities for indexing the ridge on the `probs_scores`-quantiles of the scores. Defaults to `seq(0, 1, l = 101)`. - `verbose`: show diagnostic plots? Defaults to `FALSE`. - `type_bwd`: type of cross-validation bandwidth for Nadaraya--Watson, either `"min"` (minimizer of the cross-validation loss) or `"1se"` (the "one standard error rule", smoother). Defaults to `"1se"`. - `p`: degree of local fit, either `0` or `1`. Defaults to `0`. - `...`: further arguments passed to `bw_cv_kre_polysph`. ## Returns A list with the following fields: - **scores_X**: a vector of size `n` with the SIER scores for `X`. - **projs_X**: a matrix of size `c(n, sum(d) + r)` with the projections of `X` onto the SIER. - **ord_X**: a vector of size `n` with the ordering of `X` induced by the SIER scores. - **scores_grid**: a vector of size `length(probs_scores)` with score quantiles associated to the probabilities `probs_scores`. - **ridge_grid**: a vector of size `length(probs_scores)` with the SIER evaluated at `scores_grid`. - **mds_index**: a vector of size `nx` with the multidimensional scaling indexes. - **ridge_fun**: a function that parametrizes the SIER. - **h**: bandwidth used for the local polynomial regression. - **probs_scores**: object `probs_scores`. ## Details Indexing is designed to work with collection of ridge points that admit a linear ordering, so that mapping to the real line by multidimensional scaling is adequate. The indexing will not work properly if the ridge points define a closed curve. ## Examples ```r ## Test on (S^1)^2 # Sample set.seed(132121) r <- 2 d <- rep(1, r) n <- 200 ind_dj <- comp_ind_dj(d = d) Th <- matrix(runif(n = n * (r - 1), min = -pi / 2, max = pi / 2), nrow = n, ncol = r - 1) Th[, r - 1] <- sort(Th[, r - 1]) Th <- cbind(Th, sdetorus::toPiInt( pi + Th[, r - 1] + runif(n = n, min = -pi / 4, max = pi / 4))) X <- angles_to_torus(Th) col_X_alp <- viridis::viridis(n, alpha = 0.25) col_X <- viridis::viridis(n) # Euler h_rid <- rep(0.75, r) h_eu <- h_rid^2 N <- 200 eps <- 1e-6 Y <- euler_ridge(x = X, X = X, d = d, h = h_rid, h_euler = h_eu, N = N, eps = eps, keep_paths = TRUE) # Visualization i <- N # Between 1 and N plot(rbind(torus_to_angles(Y$paths[, , 1])), col = col_X_alp, pch = 19, axes = FALSE, xlim = c(-pi, pi), ylim = c(-pi, pi), xlab = "", ylab = "") points(rbind(torus_to_angles(Y$paths[, , i])), col = col_X, pch = 16, cex = 0.75) sdetorus::torusAxis(1:2) for (k in seq_len(nrow(Y$paths))) { xy <- torus_to_angles(t(Y$paths[k, , ])) sdetorus::linesTorus(x = xy[, 1], y = xy[, 2], col = col_X_alp[k]) } # SIER ind_rid <- index_ridge(endpoints = Y$ridge_y, X = X, d = d, probs_scores = seq(0, 1, l = 50)) xy <- torus_to_angles(ind_rid$ridge_grid) sdetorus::linesTorus(x = xy[, 1], y = xy[, 2], col = 2, lwd = 2) points(torus_to_angles(ind_rid$ridge_fun(quantile(ind_rid$scores_grid))), col = 4, pch = 19) # Scores plot(density(ind_rid$scores_X), type = "l", xlab = "Scores", ylab = "Density", main = "Scores density") for (i in 1:n) rug(ind_rid$scores_X[i], col = col_X[i]) ```

index_ridge function