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Locally Adaptive Kernel Density Estimation

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Locally Adaptive Kernel Density Estimation

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02 Jul 2012 (Updated )

Kernel density estimation with bandwidths locally adapted to data.

sshist(x,N)
function [optN, optD, edges, C, N] = sshist(x,N)
% [optN, edges, C, N] = sshist(x,N)
%
% Function `sshist' returns the optimal number of bins in a histogram
% used for density estimation.
% Optimization principle is to minimize expected L2 loss function between 
% the histogram and an unknown underlying density function.
% An assumption made is merely that samples are drawn from the density
% independently each other.
%
% The optimal binwidth D* is obtained as a minimizer of the formula, 
% (2K-V) / D^2,
% where K and V are mean and variance of sample counts across bins with width D.
% Optimal number of bins is given as (max(x) - min(x)) / D*.
%
% For more information, visit 
% http://2000.jukuin.keio.ac.jp/shimazaki/res/histogram.html
%
% Original paper:
% Hideaki Shimazaki and Shigeru Shinomoto
% A method for selecting the bin size of a time histogram
% Neural Computation 19(6), 1503-1527, 2007
% http://dx.doi.org/10.1162/neco.2007.19.6.1503
%
% Example usage:
% optN = sshist(x); hist(x,optN);
%
% Input argument
% x:    Sample data vector.
% N (optinal):
%       A vector that specifies the number of bins to be examined. 
%       The optimal number of bins is selected from the elements of N.  
%       Default value is N = 2:100.
%       * Do not search binwidths smaller than a sampling resolution of data.
%
% Output argument
% optN: Optimal number of bins.
% N:    Bin numbers examined.
% C:    Cost function of N.
%
% See also SSKERNEL
%
% Copyright (c) 2009 2010, Hideaki Shimazaki All rights reserved.
% http://2000.jukuin.keio.ac.jp/shimazaki

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Parameters Setting
x = reshape(x,1,numel(x));
x_min = min(x);
x_max = max(x);

if nargin < 2
    buf = abs(diff(sort(x)));
    dx = min(buf(logical(buf ~= 0)));
    N_MIN = 2;              % Minimum number of bins (integer)
                            % N_MIN must be more than 1 (N_MIN > 1).            
    N_MAX = min(floor((x_max - x_min)/(2*dx)),500);
                            % Maximum number of bins (integer)
    N = N_MIN:N_MAX;        % # of Bins
end
    
SN = 30;                    % # of partitioning positions for shift average
D = (x_max - x_min) ./ N;   % Bin Size Vector

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Computation of the Cost Function
Cs = zeros(length(N),SN);
for i = 1: length(N)

       shift = linspace(0,D(i),SN);
       for p = 1 : SN
               edges = linspace(x_min+shift(p)-D(i)/2,...
                        x_max+shift(p)-D(i)/2,N(i)+1);   % Bin edges

               ki = histc(x,edges);               % Count # of events in bins
               ki = ki(1:end-1);

               k = mean(ki);                      % Mean of event count
               v = sum( (ki-k).^2 )/N(i);         % Variance of event count

               Cs(i,p) = ( 2*k - v ) / D(i)^2;    % The Cost Function
       end

end
C = mean(Cs,2);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Optimal Bin Size Selectioin
[Cmin idx] = min(C);
optN = N(idx);                          % Optimal number of bins
optD = D(idx);                         % *Optimal binwidth

[Cminp idxp] = min(Cs(idx,:));
shift = linspace(0,D(idx),SN);
edges = linspace(x_min+shift(idxp)-D(idx)/2,...
                        x_max+shift(idxp)-D(idx)/2,N(idx)+1);
%edges = linspace(x_min,x_max,N(idx));  % Optimal segmentation

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