## Kernel Density Estimator for High Dimensions

version 1.0.0.0 (4.85 KB) by
fast multivariate kernel density estimation for high dimensions

Updated 21 Jul 2016

Fast adaptive kernel density estimation in high dimensions in one m-file.
Provides optimal accuracy/speed trade-off, controlled via a parameter "gam";
To increase speed for "big data" applications, use small "gam";
Typically gam=n^(1/2), where "n" is the number of points. '

USAGE: [pdf,X1,X2]=akde(X,grid,gam)
INPUTS:
X - data as a 'n' by 'd' vector;
grid - 'm' points of dimension 'd' over which pdf is computed;
default provided only for 2-dimensional data;
see example below on how to construct it in higher dimensions;
gam - (optional) cost/accuracy trade-off parameter, where gam<n;
default value is gam=ceil(n^(1/2)); larger values
may result in better accuracy, but reduce speed;
to speedup the code, use smaller "gam";

OUTPUT:
pdf - the value of the estimated density at 'grid'
X1,X2 - default grid (used only for 2 dimensional data)
see example on how to construct grid on higher dimensions

EXAMPLE IN 2 DIMENSIONS:
L=chol([1,-0.999;-0.999,1],'lower');L1=chol([1,0.999;0.999,1],'lower');
data=[(L1*randn(10^3,2)')';(L*randn(10^3,2)')'*2;rand(10^4,2)*5-2.5];
[pdf,X1,X2]=akde(data);pdf=reshape(pdf,size(X1));contour(X1,X2,pdf,20)

EXAMPLE IN 3 DIMENSIONS:
data=[randn(10^3,3);randn(10^3,3)/2+2]; % three dimensional data
[n,d]=size(data); ng=100; % total grid points = ng^d
MAX=max(data,[],1); MIN=min(data,[],1); scaling=MAX-MIN;
% create meshgrid in 3-dimensions
[X1,X2,X3]=meshgrid(MIN(1):scaling(1)/(ng-1):MAX(1),...
MIN(2):scaling(2)/(ng-1):MAX(2),MIN(3):scaling(3)/(ng-1):MAX(3));
grid=reshape([X1(:),X2(:),X3(:)],ng^d,d); % create points for plotting
pdf=reshape(pdf,size(X1)); % reshape pdf for use with meshgrid
for iso=[0.005:0.005:0.015] % isosurfaces with pdf = 0.005,0.01,0.015
isosurface(X1,X2,X3,pdf,iso),view(3),alpha(.3),box on,hold on,colormap cool
end

Reference:
Kernel density estimation via diffusion
Z. I. Botev, J. F. Grotowski, and D. P. Kroese (2010)
Annals of Statistics, Volume 38, Number 5, pages 2916-2957.

### Cite As

Zdravko Botev (2022). Kernel Density Estimator for High Dimensions (https://www.mathworks.com/matlabcentral/fileexchange/58312-kernel-density-estimator-for-high-dimensions), MATLAB Central File Exchange. Retrieved .

##### MATLAB Release Compatibility
Created with R2016a
Compatible with any release
##### Platform Compatibility
Windows macOS Linux