# Density Estimation Based on Distribution Element Trees The function `det.query` evaluates probability densities at the query points `x` based on a distribution element tree (DET). The latter is calculable with `det.construct` based on available data. ```r det.query(det, x, cores = 1) ``` ## Arguments - `det`: distribution element tree object resulting from `det.construct` based on data with `d` components or dimensions. - `x`: matrix containing `n` query points (columns) with `d` components or dimensions (rows). - `cores`: for large query-point sets, `cores > 1` allows for parallel tree query using the indicated number of cores. `cores = Inf` allocates half of the available cores (see `detectCores`). The default is `cores = 1` corresponding to serial tree query. ## Returns A vector containing the probability density at the query points `x` is returned. ## Examples ```r ## 1d example require(stats); require(graphics) # DET generation based on Gaussian/uniform data det <- det.construct(t(c(rnorm(1e5,2,3),runif(1e5)-3))) # density evaluation based on DET at equidistant query points x <- t(seq(-10,14,0.01)); p <- det.query(det, x) # compare DET estimate (black) against Gaussian/uniform reference (red) plot(x, p, type = "l", col = "black") lines(x, (dnorm(x,2,3)+dunif(x+3))/2, col = "red") ## 2d example require(stats); require(graphics) # mean and covariance of Gaussian, data generation mu <- c(3,5); C <- matrix(c(4.0,-2.28,-2.28,1.44), nrow = 2) A <- eigen(C); B <- diag(A$values); A <- A$vectors x <- matrix(rnorm(2e4), nrow = 2) x <- t(A %*% (sqrt(B) %*% x) + mu %*% t(rep(1,ncol(x)))) # bounds and resolution of x1-x2 query grid lb <- c(-5,0); ub <- c(11,10); np <- c(320,200) x1 <- lb[1] + (ub[1]-lb[1])*((1:np[1])-0.5)/np[1] x2 <- lb[2] + (ub[2]-lb[2])*((1:np[2])-0.5)/np[2] xp <- rbind(rep(x1,np[2]), rep(x2,each = np[1])) # grid points # plotting split.screen(c(2, 2)); screen(1) plot(x, type = "p", pch = ".", asp = 1, main = "data") # DET estimator det <- det.construct(t(x)) yd <- matrix(det.query(det, xp), nrow = np[1]) screen(2) image(list(x = x1, y = x2, z = yd), asp = 1, col = grDevices::gray((100:0)/100), main = "det") # Gaussian density for comparison yr <- yr <- exp(-1/2 * colSums( (t(solve(C)) %*% (xp - mu%*%t(rep(1,ncol(xp))))) * (xp - mu%*%t(rep(1,ncol(xp))))) ) / sqrt((2*pi)^2*det(C)) yr <- matrix(yr, nrow = np[1]) screen(3) image(list(x = x1, y = x2, z = yr), asp = 1, col = grDevices::gray((100:0)/100), main = "reference") ```

det.query function