Mauch MIREX 2010

function [bandwidth,density,xmesh]=kde(data,n,MIN,MAX)

% Reliable and extremely fast kernel density estimator for 1 dimensional data;

%        Gaussian kernel is assumed and the bandwidth is chosen automatically;

%

% INPUTS:

%     data    - a vector of data from which the density estimate is constructed;

%   MIN, MAX  - defines the interval [MIN,MAX] on which the density estimate is constructed;

%               the default values of MIN and MAX are: 

%               MIN=min(data)-Range/100 and MAX=max(data)+Range/100, where Range=max(data)-min(data);   

%          n  - the number of mesh points used in the uniform discretization of the

%               interval [MIN, MAX]; n has to be a power of two; if n is not a power of two, then

%               n is rounded up to the next power of two, i.e., n is set to n=2^ceil(log2(n));

%               the default value of n is n=2^12;

% OUTPUTS:

%   bandwidth - the optimal bandwidth (Gaussian kernel assumed);

%     density - column vector of length 'n' with the values of the density

%               estimate at the grid points;

%     xmesh   - the grid over which the density estimate is computed;

%  Reference: Botev, Z. I.,

%             "A Novel Nonparametric Density Estimator",Technical Report,The University of Queensland

%             http://espace.library.uq.edu.au/view.php?pid=UQ:12535

%

%  Example:

%    data=randn(1000,1); 

%    [bandwidth,density,xmesh]=kde(data,2^12,min(data)-1,max(data)+1);

%    plot(xmesh,density)

data=data(:); %make data a column vector

if nargin<2 % if n is not supplied switch to the default

    n=2^12;

end

n=2^ceil(log2(n)); % round up n to the next power of 2;

if nargin<4 %define the default  interval [MIN,MAX]

    minimum=min(data); maximum=max(data);

    Range=maximum-minimum;

    MIN=minimum-Range/10; MAX=maximum+Range/10;

end

% set up the grid over which the density estimate is computed;

R=MAX-MIN; dx=R/(n-1); xmesh=MIN+[0:dx:R]; N=length(data);

%bin the data uniformly using the grid define above;

initial_data=histc(data,xmesh)/N;

a=dct1d(initial_data); % discrete cosine transform of initial data

% now compute the optimal bandwidth^2 using the GCE method

t_star=gce(a,n,N);

% smooth the discrete cosine transform of initial data using t_star

a_t=a.*exp(-[0:n-1]'.^2*pi^2*t_star/2);

% now apply the inverse discrete cosine transform

  if nargout>1

    density=idct1d(a_t)/R;

  end

bandwidth=sqrt(t_star)*R;

end

function t_star=gce(a,n,N)

a=a(2:end)/2;

I=[1:n-1]'.^2;

a2=a.^2;

Var_a=zeros(n-1,1);

Var_a(1:n/2-1)=(1/2+1/2*a(2:2:n-1)-a2(1:n/2-1))/N;

t_star=fzero(@mise,[0,1]);

NORM=2*pi^4*sum(I.^2.*a2.*exp(-I*pi^2*t_star));

%NORM=.5*pi^4*sum([1:n-1]'.^4.*a_t(2:end).^2)/R^5;

t_star=[2*N*sqrt(pi)*NORM]^(-2/5);

    function  out=mise(t)

        out=sum((a2+Var_a).*(1-exp(-I*pi^2*t/2)).^2./I)+...

            sqrt(t/pi)/N*(pi^2/2)-sum(Var_a./I);

    end

end

function data=dct1d(data)

% computes the discrete cosine transform of the column vector data

[nrows,ncols]= size(data);

% Compute weights to multiply DFT coefficients

weight = [1;2*(exp(-i*(1:nrows-1)*pi/(2*nrows))).'];

% Re-order the elements of the columns of x

data = [ data(1:2:end,:); data(end:-2:2,:) ];

% Multiply FFT by weights:

data= real(weight.* fft(data));

end

function out = idct1d(data)

% computes the inverse discrete cosine transform

[nrows,ncols]=size(data);

% Compute weights

weights = nrows*exp(i*(0:nrows-1)*pi/(2*nrows)).';

% Compute x tilde using equation (5.93) in Jain

data = real(ifft(weights.*data));

% Re-order elements of each column according to equations (5.93) and

% (5.94) in Jain

out = zeros(nrows,1);

out(1:2:nrows) = data(1:nrows/2);

out(2:2:nrows) = data(nrows:-1:nrows/2+1);

%   Reference: 

%      A. K. Jain, "Fundamentals of Digital Image

%      Processing", pp. 150-153.

end