diff _misc/probability/kde.m @ 8:b5b38998ef3b

added all that other stuff

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/_misc/probability/kde.m	Fri Apr 11 15:54:25 2014 +0100
@@ -0,0 +1,98 @@
+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
+