Code covered by the BSD License

### Highlights fromkernel density estimation

from kernel density estimation by Zdravko Botev
fast and accurate state-of-the-art bivariate kernel density estimator

[bandwidth,density,X,Y]=kde2d(data,n,MIN_XY,MAX_XY)
```function [bandwidth,density,X,Y]=kde2d(data,n,MIN_XY,MAX_XY)
% fast and accurate state-of-the-art
% bivariate kernel density estimator
% with diagonal bandwidth matrix.
% The kernel is assumed to be Gaussian.
% The two bandwidth parameters are
% chosen optimally without ever
% using/assuming a parametric model for the data or any "rules of thumb".
% Unlike many other procedures, this one
% is immune to accuracy failures in the estimation of
% multimodal densities with widely separated modes (see examples).
% INPUTS: data - an N by 2 array with continuous data
%            n - size of the n by n grid over which the density is computed
%                n has to be a power of 2, otherwise n=2^ceil(log2(n));
%                the default value is 2^8;
% MIN_XY,MAX_XY- limits of the bounding box over which the density is computed;
%                the format is:
%                MIN_XY=[lower_Xlim,lower_Ylim]
%                MAX_XY=[upper_Xlim,upper_Ylim].
%                The dafault limits are computed as:
%                MAX=max(data,[],1); MIN=min(data,[],1); Range=MAX-MIN;
%                MAX_XY=MAX+Range/4; MIN_XY=MIN-Range/4;
% OUTPUT: bandwidth - a row vector with the two optimal
%                     bandwidths for a bivaroate Gaussian kernel;
%                     the format is:
%                     bandwidth=[bandwidth_X, bandwidth_Y];
%          density  - an n by n matrix containing the density values over the n by n grid;
%                     density is not computed unless the function is asked for such an output;
%              X,Y  - the meshgrid over which the variable "density" has been computed;
%                     the intended usage is as follows:
%                     surf(X,Y,density)
% Example (simple Gaussian mixture)
% clear all
%   % generate a Gaussian mixture with distant modes
%   data=[randn(500,2);
%       randn(500,1)+3.5, randn(500,1);];
%   % call the routine
%     [bandwidth,density,X,Y]=kde2d(data);
%   % plot the data and the density estimate
%     contour3(X,Y,density,50), hold on
%     plot(data(:,1),data(:,2),'r.','MarkerSize',5)
%
% Example (Gaussian mixture with distant modes):
%
% clear all
%  % generate a Gaussian mixture with distant modes
%  data=[randn(100,1), randn(100,1)/4;
%      randn(100,1)+18, randn(100,1);
%      randn(100,1)+15, randn(100,1)/2-18;];
%  % call the routine
%    [bandwidth,density,X,Y]=kde2d(data);
%  % plot the data and the density estimate
%  surf(X,Y,density,'LineStyle','none'), view([0,60])
%  colormap hot, hold on, alpha(.8)
%  set(gca, 'color', 'blue');
%  plot(data(:,1),data(:,2),'w.','MarkerSize',5)
%
% Example (Sinusoidal density):
%
% clear all
%   X=rand(1000,1); Y=sin(X*10*pi)+randn(size(X))/3; data=[X,Y];
%  % apply routine
%  [bandwidth,density,X,Y]=kde2d(data);
%  % plot the data and the density estimate
%  surf(X,Y,density,'LineStyle','none'), view([0,70])
%  colormap hot, hold on, alpha(.8)
%  set(gca, 'color', 'blue');
%  plot(data(:,1),data(:,2),'w.','MarkerSize',5)
%
% Notes: If you have a more accurate density estimator
%        (as measured by which routine attains the smallest
%         L_2 distance between the estimate and the true density) or you have
%        problems running this code, please email me at botev@maths.uq.edu.au

%  Reference: Z. I. Botev, J. F. Grotowski and D. P. Kroese
%             "KERNEL DENSITY ESTIMATION VIA DIFFUSION" ,Submitted to the
%             Annals of Statistics, 2009
global N A2 I
if nargin<2
n=2^8;
end
n=2^ceil(log2(n)); % round up n to the next power of 2;
N=size(data,1);
if nargin<3
MAX=max(data,[],1); MIN=min(data,[],1); Range=MAX-MIN;
MAX_XY=MAX+Range/4; MIN_XY=MIN-Range/4;
end
scaling=MAX_XY-MIN_XY;
if N<=size(data,2)
error('data has to be an N by 2 array where each row represents a two dimensional observation')
end
transformed_data=(data-repmat(MIN_XY,N,1))./repmat(scaling,N,1);
%bin the data uniformly using regular grid;
initial_data=ndhist(transformed_data,n);
% discrete cosine transform of initial data
a= dct2d(initial_data);
% now compute the optimal bandwidth^2
I=(0:n-1).^2; A2=a.^2;

t_star=fzero( @(t)(t-evolve(t)),[0,0.1]);

p_02=func([0,2],t_star);p_20=func([2,0],t_star); p_11=func([1,1],t_star);
t_y=(p_02^(3/4)/(4*pi*N*p_20^(3/4)*(p_11+sqrt(p_20*p_02))))^(1/3);
t_x=(p_20^(3/4)/(4*pi*N*p_02^(3/4)*(p_11+sqrt(p_20*p_02))))^(1/3);
% smooth the discrete cosine transform of initial data using t_star
a_t=exp(-(0:n-1)'.^2*pi^2*t_x/2)*exp(-(0:n-1).^2*pi^2*t_y/2).*a;
% now apply the inverse discrete cosine transform
if nargout>1
density=idct2d(a_t)*(numel(a_t)/prod(scaling));
[X,Y]=meshgrid(MIN_XY(1):scaling(1)/(n-1):MAX_XY(1),MIN_XY(2):scaling(2)/(n-1):MAX_XY(2));
end
bandwidth=sqrt([t_x,t_y]).*scaling;
end
%#######################################
function  [out,time]=evolve(t)
global N
Sum_func = func([0,2],t) + func([2,0],t) + 2*func([1,1],t);
time=(2*pi*N*Sum_func)^(-1/3);
out=(t-time)/time;
end
%#######################################
function out=func(s,t)
global N
if sum(s)<=4
Sum_func=func([s(1)+1,s(2)],t)+func([s(1),s(2)+1],t); const=(1+1/2^(sum(s)+1))/3;
time=(-2*const*K(s(1))*K(s(2))/N/Sum_func)^(1/(2+sum(s)));
out=psi(s,time);
else
out=psi(s,t);
end

end
%#######################################
function out=psi(s,Time)
global I A2
% s is a vector
w=exp(-I*pi^2*Time).*[1,.5*ones(1,length(I)-1)];
wx=w.*(I.^s(1));
wy=w.*(I.^s(2));
out=(-1)^sum(s)*(wy*A2*wx')*pi^(2*sum(s));
end
%#######################################
function out=K(s)
out=(-1)^s*prod((1:2:2*s-1))/sqrt(2*pi);
end
%#######################################
function data=dct2d(data)
% computes the 2 dimensional discrete cosine transform of data
% data is an nd cube
[nrows,ncols]= size(data);
if nrows~=ncols
error('data is not a square array!')
end
% Compute weights to multiply DFT coefficients
w = [1;2*(exp(-i*(1:nrows-1)*pi/(2*nrows))).'];
weight=w(:,ones(1,ncols));
data=dct1d(dct1d(data)')';
function transform1d=dct1d(x)

% Re-order the elements of the columns of x
x = [ x(1:2:end,:); x(end:-2:2,:) ];

% Multiply FFT by weights:
transform1d = real(weight.* fft(x));
end
end
%#######################################
function data = idct2d(data)
% computes the 2 dimensional inverse discrete cosine transform
[nrows,ncols]=size(data);
% Compute wieghts
w = exp(i*(0:nrows-1)*pi/(2*nrows)).';
weights=w(:,ones(1,ncols));
data=idct1d(idct1d(data)');
function out=idct1d(x)
y = real(ifft(weights.*x));
out = zeros(nrows,ncols);
out(1:2:nrows,:) = y(1:nrows/2,:);
out(2:2:nrows,:) = y(nrows:-1:nrows/2+1,:);
end
end
%#######################################
function binned_data=ndhist(data,M)
% this function computes the histogram
% of an n-dimensional data set;
% 'data' is nrows by n columns
% M is the number of bins used in each dimension
% so that 'binned_data' is a hypercube with
% size length equal to M;
[nrows,ncols]=size(data);
bins=zeros(nrows,ncols);
for i=1:ncols
[dum,bins(:,i)] = histc(data(:,i),[0:1/M:1],1);
bins(:,i) = min(bins(:,i),M);
end
% Combine the  vectors of 1D bin counts into a grid of nD bin
% counts.
binned_data = accumarray(bins(all(bins>0,2),:),1/nrows,M(ones(1,ncols)));
end

```