knnDE: k Nearest Neighbors Density Estimator over a Grid of Points

Description

Given a point cloud X ($n$ points), The function knnDE computes the k Nearest Neighbors Density Estimator over a grid of points. For each $x \in R^d$, the knn Density Estimator is defined by $$ p_X(x)=\frac{k}{n \; v_d \; r_k^d(x)}, $$ where $v_n$ is the volume of the Euclidean $d$ dimensional unit ball and $r_k^d(x)$ is the Euclidean distance from point x to its $k$'th closest neighbor.

Usage

knnDE(X, Grid, k)

Value

The function knnDE returns a vector of length $m$ (the number of points in the grid) containing the value of the knn Density Estimator for each point in the grid.

Arguments

X: an $n$ by $d$ matrix of coordinates of points used in the density estimation process, where $n$ is the number of points and $d$ is the dimension.
Grid: an $m$ by $d$ matrix of coordinates, where $m$ is the number of points in the grid.
k: number: the smoothing paramter of the k Nearest Neighbors Density Estimator.

Author

Fabrizio Lecci

Examples

Run this code

## Generate Data from the unit circle
n <- 300
X <- circleUnif(n)

## Construct a grid of points over which we evaluate the function
by <- 0.065
Xseq <- seq(-1.6, 1.6, by = by)
Yseq <- seq(-1.7, 1.7, by = by)
Grid <- expand.grid(Xseq, Yseq)

## kernel density estimator
k <- 50
KNN <- knnDE(X, Grid, k)