# knnDE: k Nearest Neighbors Density Estimator over a Grid of Points In TDA: Statistical Tools for Topological Data Analysis

## 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) = 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

 1 knnDE(X, Grid, k) 

## 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.

## 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.

## Author(s)

Fabrizio Lecci

kde, kernelDist, distFct, dtm

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 ## 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) 

### Example output




TDA documentation built on March 30, 2021, 5:10 p.m.