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Zoltan Szabo committed 5810f95

EASI real/complex ICA method, adaptive (k-d) partitioning based Shannon entropy estimation: added; see 'estimate_ICA_EASI.m', 'estimate_ICA.m', 'estimate_complex_ICA.m', 'HShannon_KDP_initialization.m', 'HShannon_KDP_estimation.m'.

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  • Parent commits 06f4ae4
  • Tags release-0.31

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Files changed (18)

-v0.30 (Jan 25, 2012):
+v0.31 (Feb 9, 2013):
+-EASI (equivariant adaptive separation via independence) real/complex ICA method: added; see 'estimate_ICA_EASI.m', 'estimate_ICA.m', 'estimate_complex_ICA.m'.
+-Adaptive (k-d) partitioning based Shannon entropy estimation: added; see 'HShannon_KDP_initialization.m', 'HShannon_KDP_estimation.m'.
+
+v0.30 (Jan 25, 2013):
 -Upper tail dependence via conditional Spearman's rho: added; see 'ASpearman_U_initialization.m', 'ASpearman_U_estimation.m'.
 -Multivariate conditional version of Spearman's rho weighting the lower tail: added; see 'ASpearman_ut_initialization.m', 'ASpearman_ut_estimation.m'.
 
-v0.29 (Jan 13, 2012):
+v0.29 (Jan 13, 2013):
 -Lower tail dependence via conditional Spearman's rho: added; see 'ASpearman_L_initialization.m', 'ASpearman_L_estimation.m'.
 -Multivariate conditional version of Spearman's rho weighting the lower tail: added; see 'ASpearman_lt_initialization.m', 'ASpearman_lt_estimation.m'.
 -Notes on lower- and upper orthant dependence: added (doc: see 'Spearman1' and 'Spearman2').
 
-v0.28 (Jan 2, 2012):
+v0.28 (Jan 2, 2013):
 -Multivariate extension of Blomqvist's beta (medial correlation coefficient): added; see 'ABlomqvist_initialization.m', 'ABlomqvist_estimation.m'.
 -Average pairwise Spearman's rho: added; see 'ASpearman4_initialization.m', 'ASpearman4_estimation.m'.
 -Definition of the multivariate measure of concordance and -independence measure: added (doc).
 
 **Download** the latest release: 
 
-- code: [zip](https://bitbucket.org/szzoli/ite/downloads/ITE-0.30_code.zip), [tar.bz2](https://bitbucket.org/szzoli/ite/downloads/ITE-0.30_code.tar.bz2), 
-- [documentation (pdf)](https://bitbucket.org/szzoli/ite/downloads/ITE-0.30_documentation.pdf).
+- code: [zip](https://bitbucket.org/szzoli/ite/downloads/ITE-0.31_code.zip), [tar.bz2](https://bitbucket.org/szzoli/ite/downloads/ITE-0.31_code.tar.bz2), 
+- [documentation (pdf)](https://bitbucket.org/szzoli/ite/downloads/ITE-0.31_documentation.pdf).
 
 

code/H_I_D_A_C/base_estimators/HShannon_KDP_estimation.m

+function [H] = HShannon_KDP_estimation(Y,co)
+%Estimates the Shannon differential entropy (H) of Y using k-d partitioning.
+%
+%We use the naming convention 'H<name>_estimation' to ease embedding new entropy estimation methods.
+%
+%INPUT:
+%   Y: Y(:,t) is the t^th sample.
+%  co: entropy estimator object.
+%
+%REFERENCE:
+%   Dan Stowell and Mark D. Plumbley. Fast multidimensional entropy estimation by k-d partitioning. IEEE Signal Processing Letters 16 (6), 537-540, 2009.
+%
+%Copyright (C) 2012 Zoltan Szabo ("http://nipg.inf.elte.hu/szzoli", "szzoli (at) cs (dot) elte (dot) hu")
+%
+%This file is part of the ITE (Information Theoretical Estimators) Matlab/Octave toolbox.
+%
+%ITE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by
+%the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
+%
+%This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
+%MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.
+%
+%You should have received a copy of the GNU General Public License along with ITE. If not, see <http://www.gnu.org/licenses/>.
+
+%co.mult:OK.
+
+%H estimation:
+     H =  kdpee(Y.');
+ 

code/H_I_D_A_C/base_estimators/HShannon_KDP_initialization.m

+function [co] = HShannon_KDP_initialization(mult)
+%Initialization of the k-d partitioning based Shannon differential entropy estimator.
+%
+%Note:
+%   1)The estimator is treated as a cost object (co).
+%   2)We use the naming convention 'H<name>_initialization' to ease embedding new entropy estimation methods.
+%
+%INPUT:
+%   mult: is a multiplicative constant relevant (needed) in the estimation; '=1' means yes, '=0' no.
+%OUTPUT:
+%   co: cost object (structure).
+%
+%Copyright (C) 2012 Zoltan Szabo ("http://nipg.inf.elte.hu/szzoli", "szzoli (at) cs (dot) elte (dot) hu")
+%
+%This file is part of the ITE (Information Theoretical Estimators) Matlab/Octave toolbox.
+%
+%ITE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by
+%the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
+%
+%This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
+%MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.
+%
+%You should have received a copy of the GNU General Public License along with ITE. If not, see <http://www.gnu.org/licenses/>.
+
+%mandatory fields:
+    co.name = 'Shannon_KDP';
+    co.mult = mult;
+   

code/IPA/demos/estimate_ICA.m

 %INPUT:
 %   x: x(:,t) is the t^th sample.
 %   ICA_method: the method to perform ICA with. ('fastICA')
-%   dim_reduction: dim(x) = size(x,1) >= dim_reduction; if '>' holds, perform dimension reduction to dimension dim_reduction, too.
+%   dim_reduction: <=dim(x)[= size(x,1)]; in case of '<', dimension reduction is also carried out.
 %OUTPUT:
 %   e_hat: estimated ICA elements; e_hat(:,t) is the t^th sample.
 %   W_hat: estimated ICA demixing matrix.
                                      'approach', 'symm', 'g', 'tanh','displayMode', 'off','verbose','off',  ...
                                      'stabilization','on','maxNumIterations',3000); 
         W_hat = W_ICA * W_whiten;
+    case 'EASI'
+        [e_hat,W_hat] = estimate_ICA_EASI(x,dim_reduction);
     otherwise
         error('ICA method=?');
 end
 
-disp('ICA estimation: ready.');
+disp('ICA estimation: ready.');

code/IPA/demos/estimate_complex_ICA.m

 %
 %INPUT:
 %   x: x(:,t) is the t^th sample.
-%   ICA_method: the method to perform ICA with. ('fastICA')
+%   ICA_method: the method to perform ICA with.
 %OUTPUT:
 %   e_hat: estimated ICA elements; e_hat(:,t) is the t^th sample.
 %   W_hat: estimated ICA demixing matrix.
             defl = 1;%1:deflation, 0:no deflation.
             [e_hat,W_ICA] = cfastica_public(x,defl);
             W_hat = W_ICA * W_whiten;
+    case 'EASI'
+        [e_hat,W_hat] = estimate_ICA_EASI(x,size(x,1));
     otherwise
         error('Complex ICA method=?');
 end

code/ITE_install.m

 		compile_Hoeffding_term1 = 1; %1=compile, 0=do not compile; not necessary, but can be more ecomical in terms of memory used + accelerate computations; the package also contains the purely Matlab/Octave 'Hoeffding_term1.m'
 		compile_Edgeworth_t1_t2_t3 = 1; %1=compile, 0=do not compile; not necessary, but can accelerate computations; the ITE package also contains the purely Matlab/Octave 'Edgeworth_t1_t2_t3.m'
 		compile_CDSS = 1; %1=compile, 0=do not compile; not necessary, but can speed up the computations; the ITE package also contains the purely Matlab/Octave 'compute_CDSS.m'
+        compile_KDP = 1; %1=compile, 0=do not compile  
     %download:
 		download_ARfit = 1;%1=download+extract,0=do not download
 	delete_Ncut_in_Octave = 0; %1=yes, 0=no
         build; %compile the knn package (top.cpp)
         disp('knn (top.cpp) compilation: ready.');
     end
-	
+    
+%compile KDP, if needed:    
+    if compile_KDP	%if needed
+        cd(strcat(ITE_code_dir,'/shared/embedded/KDP/mat_oct'));
+        disp('KDP (kdpee.c, kdpeemex.c) compilation: started.');
+        mexme;
+        disp('KDP (kdpee.c, kdpeemex.c) compilation: ready.');        
+    end
+    
 %change back to the stored working directory:
     cd(pd);
     
 			Y = rand(3,100); Q = rand(3,200);
 			knn(Q,Y,1);
 			disp('knn quick test: successful.');
-		end
+        end
+    %KDP:
+        if compile_KDP %verify only in case of compilation
+            Y = rand(1,1000);
+            kdpee(Y.');
+            disp('KDP quick test: successful.');
+        end

code/shared/embedded/KDP/COPYING

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+<http://www.gnu.org/philosophy/why-not-lgpl.html>.

code/shared/embedded/KDP/INSTALL

+
+ "kdpee", k-d partitioning entropy estimator.
+ (c) 2008-2009 Dan Stowell and Queen Mary University of London.
+ All rights reserved.
+
+
+INSTALLING FOR MATLAB / OCTAVE
+==============================
+
+Run the following from within matlab / octave:
+
+  cd ..../path/to/kdpee/mat_oct
+  mexme
+
+The "mexme" script is a very simple M-file that calls "mex" -
+look at the source of "mexme" and tweak it if you like.
+
+
+Basic usage in matlab/octave might look something like this:
+
+  addpath('..../path/to/kdpee/mat_oct');
+  a = [1,6; 2,7; 3,8; 4,9; 5,0; 1,6; 2,7; 3,8; 4,9; 5,0];
+  kdpee(a)           % (result is 3.0723)
+
+
+
+INSTALLING FOR PYTHON (NUMPY)
+=============================
+Tested on python 2.5 and 2.6, on Mac systems.
+
+REQUIREMENTS: 
+You'll need to have python as well as the numpy module installed,
+and you'll need a C compiler (e.g. gcc, I used gcc v4.0) too.
+
+To install this module, first edit the file "python/setup.py". Find the 
+line which specifies a value for "include_dirs", and edit it to match
+the location of numpy/arrayobject.h on your system.
+
+Then run "setup.py" from a command-line as follows:
+
+  cd ..../path/to/kdpee/python
+  python setup.py build
+  python setup.py install
+
+You might need to be the super-user (e.g. use sudo) for the install step.
+
+
+Basic usage in python might look something like this:
+
+  from numpy import *
+  from kdpee import *
+  # entropy of a 2D distribution with 10 data points:
+  kdpee(array( [[1,2,3,4,5,1,2,3,4,5], [6,7,8,9,0,6,7,8,9,0]]))
+  # (result is 3.072320520444161)
+

code/shared/embedded/KDP/NEWS

+
+Version 1.1.1, 2011-07-20
+=========================
+* Better compatibility of 'mexme' across GCC versions - thanks to Anh Cat Le Ngo
+
+Version 1.1, 2009-05-19
+=======================
+* add python (numpy) bindings
+* rearrange file structure of package more cleanly (gnu coding standards style)
+
+
+Version 1.0
+===========
+* kdpee C code, with matlab/octave bindings
+
+

code/shared/embedded/KDP/README

+
+ "kdpee", k-d partitioning entropy estimator.
+ (c) 2008-2009 Dan Stowell and Queen Mary University of London.
+ All rights reserved.
+
+This package contains the entropy estimator described in 
+
+    D. Stowell and M. D. Plumbley
+    Fast multidimensional entropy estimation by k-d partitioning.
+    IEEE Signal Processing Letters 16 (6), 537--540, June 2009.
+    http://dx.doi.org/10.1109/LSP.2009.2017346
+
+The estimator is written in C and so should be easily usable in any C/C++
+software, as well as in other languages that can call upon C.
+
+This package also contains wrappers which allow the code to be used from
+Matlab, GNU Octave, or Python (numpy).
+
+
+INSTALLATION:
+
+For installation instructions, please see the file named INSTALL.
+
+
+CONTENTS:
+
+The folders in this package contain:
+
+ * src     - the kdpee source code
+ * kdpee   - the kdpee header file
+ * mat_oct - the wrapper for matlab / gnu octave
+ * python  - the wrapper for python (numpy)
+
+
+LICENSE:
+
+This package is distributed under the terms of the GNU General Public 
+License, version 3 or (at your option) any later version. For the text
+of the license please see the file named COPYING.
+

code/shared/embedded/KDP/VERSION

+1.1.1

code/shared/embedded/KDP/kdpee/kdpee.h

+/**
+    This file is part of "kdpee", kd-partitioning entropy estimator.
+    (c) 2008 Dan Stowell and Queen Mary University of London
+    All rights reserved.
+
+    kdpee is free software: you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 of the License, or
+    (at your option) any later version.
+
+    kdpee is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with kdpee.  If not, see <http://www.gnu.org/licenses/>.
+*/
+
+// idempotence:
+#ifndef __KDPEE_HEADER__
+#define __KDPEE_HEADER__
+
+// Can be made to use single- or double-precision calculation
+#ifdef KDPEE_PREC_SINGLE
+	typedef  float floatval;
+#else
+	typedef double floatval;
+#endif
+
+/** 
+* kdpee() is the entry function.
+* 
+* Note that the return value, and many of the arguments, use EITHER type 
+* double OR float, depending on whether KDPEE_PREC_SINGLE was defined at
+* compile time.
+* 
+* dimrefs is an array of pointers to your data dimension arrays
+* n is the number of data (i.e. num values in each array in dimrefs)
+* d is number of dimensions (i.e. size of dimrefs itself)
+* mins and maxs specify the data range, arrays of size d
+* zcut (typically 1.96) is the threshold for branch stopping
+* keys should point to an array of indices which we can shuffle during our algo; 
+*    on init it should contain the values (1..n)
+*/
+floatval kdpee(const floatval **dimrefs, const int n, const int d, floatval *mins, 
+								floatval *maxs, const floatval zcut, int *keys);
+
+// idempotence:
+#endif

code/shared/embedded/KDP/mat_oct/kdpee.m

+function h = kdpee (arr, mins, maxs, zcut)
+% function h = kdpee (arr)
+% function h = kdpee (arr, mins, maxs)
+% function h = kdpee (arr, mins, maxs, zcut)
+% 
+% Entropy estimator using k-d partitioning. Returns a value in nats.
+%
+%   "arr" is a matrix of data points, 1 row per datum.
+%
+% The algorithm can run on data with known OR unknown support.
+%   (Support is assumed to be extent of data in the latter case.)
+% If the support is known, pass it in as vectors "mins" and "maxs".
+% If the support is not known simply don't supply "mins" or "maxs".
+%
+%   kdpee is free software: you can redistribute it and/or modify
+%   it under the terms of the GNU General Public License as published by
+%   the Free Software Foundation, either version 3 of the License, or
+%   (at your option) any later version.
+
+
+if (nargin < 4)
+	zcut = 1.96;
+end
+
+n = size(arr, 1);    % num data points
+d = size(arr, 2);    % dimensionality
+
+%if (n < (2^d))
+%	fprintf(1, 'kdpee() warning: n < 2^d, meaning you might not have enough data to make an entropy estimate for the dimensionality\n');
+%end
+
+if(nargin < 3)
+    % Take limits from data, should have tendency to underestimate support
+    %           but seems to work well
+	mins = min(arr, [], 1);
+	maxs = max(arr, [], 1);
+end
+
+
+h = kdpeemex(arr, mins, maxs, zcut);

code/shared/embedded/KDP/mat_oct/mexme.m

+if exist('OCTAVE_VERSION', 'builtin')    % standard test for octave as opposed to matlab
+	mex                    -I../kdpee  private/kdpeemex.c ../src/kdpee.c
+else
+	if isunix
+		mex CFLAGS='-std=gnu99 -D_GNU_SOURCE' -outdir private   -I../kdpee  private/kdpeemex.c ../src/kdpee.c
+	else ispc
+		mex -outdir private   -I../kdpee  private/kdpeemex.c ../src/kdpee.c
+	end
+end
+

code/shared/embedded/KDP/mat_oct/private/kdpeemex.c

+/**
+    This file is part of "kdpee", kd-partitioning entropy estimator.
+    (c) 2008 Dan Stowell and Queen Mary University of London
+    All rights reserved.
+
+    kdpee is free software: you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 of the License, or
+    (at your option) any later version.
+
+    kdpee is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with kdpee.  If not, see <http://www.gnu.org/licenses/>.
+*/
+
+#include "math.h"
+#include "mex.h"
+#include "kdpee.h"
+
+////////////////////////////////////////////////////////////////////////
+// MATLAB ENTRY FUNCTION:
+void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[])
+{
+	const mxArray *xData, *minData, *maxData;
+	double *xValues, *mins, *maxs;
+	int numdims, numdatums;
+	double zcut;
+
+	int *keys;
+	const double **dimrefs;
+	int i;
+	double result;
+
+	if(nrhs < 4){
+		mexErrMsgTxt("kdpee error: 4 arguments required by the C implementation.\n");
+	}
+
+	// Get info about our input matrix
+	xData = prhs[0];
+	xValues = mxGetPr(xData); // indexed by whichdim * numdims + whichdatum
+	numdims = mxGetN(xData);
+	numdatums = mxGetM(xData);
+	
+	// mins and maxes
+	minData = prhs[1];
+	mins = mxGetPr(minData);
+	maxData = prhs[2];
+	maxs = mxGetPr(maxData);
+	if( !(mxGetN(minData)==numdims ||  mxGetN(minData)==numdims)
+				|| !(mxGetN(maxData)==numdims ||  mxGetN(maxData)==numdims) ){
+    	mexErrMsgTxt("kdpee error: mins and maxs must have same dimension as the # dims in the dataset.\n");
+	}
+	
+	zcut = mxGetScalar(prhs[3]);
+	
+	// Now we allocate space for the keys that we'll need to shuffle around
+	keys = mxMalloc(numdatums * sizeof(int));
+	// Here we create a set of pointers, one to each row of the input data
+	dimrefs = mxMalloc(numdims * sizeof(double*));
+	for(i = 0; i < numdims; ++i){
+		dimrefs[i] = xValues + (i * numdatums);
+	}
+	
+	// That's the matlab-specific stuff done, now let's call our calc and return it
+	result = kdpee(dimrefs, numdatums, numdims, mins, maxs, zcut, keys);
+	plhs[0] = mxCreateDoubleScalar(result);
+	
+	mxFree(keys);
+	mxFree(dimrefs);
+	
+	return;
+}

code/shared/embedded/KDP/src/kdpee.c

+/**
+    This file is part of "kdpee", kd-partitioning entropy estimator.
+    (c) 2009 Dan Stowell and Queen Mary University of London
+    All rights reserved.
+
+    kdpee is free software: you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 of the License, or
+    (at your option) any later version.
+
+    kdpee is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with kdpee.  If not, see <http://www.gnu.org/licenses/>.
+*/
+
+#include "../kdpee/kdpee.h"
+#include <math.h>
+//#include <stdbool.h>
+typedef int bool;
+
+//natural logarithm of 2, used in Log2.
+const double ln_2 =  0.693147181;
+
+//Calculates log2 of a number. You need this function in case of using Microsoft (Microsoft does not provide log2...), otherwise you can use the available log2 function.
+double Log2( double n )  
+{  
+    return log(n) / ln_2; // log(n)/log(2) is log2(n)
+}
+
+#ifndef false
+	#define false 0
+	#define true 1
+#endif
+
+////////////////////////////////////////////////////////////////////////////////
+// Declarations of not-particularly-public functions:
+// minindex and maxindex are the INCLUSIVE indices of the partition we're 
+//  interested in
+floatval kdpee_recurse(const floatval **dimrefs, const int n, const int d, floatval *mins, 
+									floatval *maxs, const floatval zcut, int *keys,
+									bool mayTerminate, int curlev, const floatval n_rec,
+									int minindex, int maxindex, int minlev
+									);
+floatval kdpee_hoareMedian(const floatval *oneRow, int *keys, int minindex, int maxindex);
+floatval kdpee_hoareFind(const floatval *oneRow, int *keys, int minindex, int maxindex, int findThis);
+void   kdpee_hoarePartition(const floatval *oneRow, int *keys, int minindex, int maxindex, 
+											int l0, int r0, floatval fulcrum, int* l, int* r);
+
+
+////////////////////////////////////////////////////////////////////////////////
+
+floatval kdpee(const floatval **dimrefs, const int n, const int d, floatval *mins, 
+								floatval *maxs, const floatval zcut, int *keys){
+	//int minlev = ceil(0.5 * log2(n)); // Min partitioning level
+    int minlev = ceil(0.5 * Log2(n)); // Min partitioning level (Microsoft does not provide log2...)
+	int i;
+    floatval result;
+    
+	for(i = 0; i < n; ++i){
+		keys[i] = i; // initialise keys
+	}
+	result = kdpee_recurse(dimrefs, n, d, mins, maxs, zcut, keys,
+						false, 0, 1./n, 0, n-1, minlev);
+	return result;
+}
+
+floatval kdpee_recurse(const floatval **dimrefs, const int n, const int d, floatval *mins, 
+									floatval *maxs, const floatval zcut, int *keys,
+									bool mayTerminate, int curlev, const floatval n_rec,
+									int minindex, int maxindex, int minlev
+									){
+	
+	int dimno = curlev % d; // The dimension along which we're intending to split
+	int thesize = 1+maxindex-minindex; // No of points in this subset
+	
+	// As well as returning the median, this PARTITIONS the data (keys) in-place
+	floatval median = kdpee_hoareMedian(dimrefs[dimno], keys, minindex, maxindex);
+	floatval zscore;
+    
+	if(curlev == minlev){
+		mayTerminate = true; // We have passed the lower termination depth
+	}
+	
+
+	if(mayTerminate){
+		zscore = (sqrt(thesize) * (median+median-mins[dimno]-maxs[dimno]) 
+						/ 
+				(maxs[dimno]-mins[dimno]));
+		if(zscore<0.){
+			zscore = 0. - zscore;
+		}
+	}
+
+	if(thesize==1 || (mayTerminate && (zscore < zcut))){
+		// allowed to terminate, and z-score doesn't disprove uniformity, 
+		// so let's calc the negsurprisal!
+		floatval frac = thesize * n_rec;
+		floatval volume = maxs[0] - mins[0];
+		int i;
+		for(i=1; i<d; ++i){
+			volume *= maxs[i] - mins[i];
+		}
+		
+		if(volume==0.){
+			return 0.;
+		}else{
+			return log(volume / frac) * frac;
+		}
+	}else{
+		// We need to partition and recurse
+		floatval oldextremum;
+		floatval left,right;
+        
+		int newmaxindex, newminindex;
+		if((thesize & 1) == 0){ // even # points
+			newmaxindex = minindex + thesize/2 - 1;
+			newminindex = minindex + thesize/2;
+		}else{ // odd # points
+			newmaxindex = maxindex - (thesize+1)/2;
+			newminindex = minindex + (thesize-1)/2;
+		}
+		
+		// Remember the outer extremum, replace with median, then recurse
+		oldextremum = maxs[dimno];
+		maxs[dimno] = median;
+		left = kdpee_recurse(dimrefs, n, d, mins, 
+								maxs, zcut, keys,
+								mayTerminate, curlev+1, n_rec,
+								minindex, newmaxindex, minlev
+								);
+		// put the extremum back in place
+		maxs[dimno] = oldextremum;
+		
+		// Remember the outer extremum, replace with median, then recurse
+		oldextremum = mins[dimno];
+		mins[dimno] = median;
+		right = kdpee_recurse(dimrefs, n, d, mins, 
+								maxs, zcut, keys,
+								mayTerminate, curlev+1, n_rec,
+								newminindex, maxindex, minlev
+								);
+		// put the extremum back in place
+		mins[dimno] = oldextremum;
+		
+		return left + right;
+	}
+}
+
+// by rearranging the keys between keys[minindex] & keys[maxindex] (inclusive),
+//   find the median value in that section of oneRow
+floatval kdpee_hoareMedian(const floatval *oneRow, int *keys, int minindex, int maxindex){
+	int num = 1 + maxindex - minindex;
+	
+	if((num & 1) == 1){ // odd
+		return kdpee_hoareFind(oneRow, keys, minindex, maxindex, (num - 1) / 2);
+	}else{ // even
+		return (
+			kdpee_hoareFind(oneRow, keys, minindex, maxindex, (num / 2))
+				+
+			kdpee_hoareFind(oneRow, keys, minindex, maxindex, (num / 2) - 1)
+				) * 0.5;
+	}
+}
+
+floatval kdpee_hoareFind(const floatval *oneRow, int *keys, int minindex, int maxindex, int findThis){
+	int l, r, i, j;
+	i = minindex;
+	j = maxindex;
+	findThis += minindex; // offset so that we're actually in the desired section
+	while(i < j){
+		kdpee_hoarePartition(oneRow, keys, minindex, maxindex, i, j, oneRow[keys[findThis]], &l, &r);
+		if(r < findThis){
+			// kth smallest is in right split
+			i = l;
+		}
+		if(findThis < l){
+			// kth smallest is in left split
+			j = r;
+		}
+	}
+	
+	// The desired element is now in desired position, so return its value
+	return oneRow[keys[findThis]];
+}
+
+void kdpee_hoarePartition(const floatval *oneRow, int *keys, int minindex, int maxindex, 
+											int l0, int r0, floatval fulcrum, int* l, int* r){
+	int tmp;
+	*l = l0;
+	*r = r0;
+	while(*l <= *r){
+		// left_scan
+		while(*l <= maxindex && oneRow[keys[*l]] < fulcrum){
+			++*l;
+		}
+		// right_scan
+		while(*r >= minindex && oneRow[keys[*r]] > fulcrum){
+			--*r;
+		}
+		// check and exchange (keys)
+		if(*l <= *r){
+			tmp = keys[*l];
+			keys[*l] = keys[*r];
+			keys[*r] = tmp;
+			// then
+			++*l;
+			--*r;
+		}
+	}
+}

code/shared/estimate_ICA_EASI.m

+function [e_hat,W_hat] = estimate_ICA_EASI(x,dim_reduction)
+%ICA estimation using the EASI (equivariant adaptive separation via independence) method. The function copes with real/complex observations.
+%
+%INPUT:
+%   x: x(:,t) is the t^th observation sample.
+%  dim_reduction: <=dim(x)[= size(x,1)]; in case of '<', dimension reduction is also carried out.
+%OUTPUT:
+%   e_hat: estimated ICA elements; e_hat(:,t) is the t^th sample.
+%   W_hat: estimated ICA demixing matrix.
+%
+%REFERENCE: 
+%   Jean-Francois Cardoso and Beate Hvam Laheld. Equivariant adaptive source separation. IEEE Transactions on Signal Processing, 44:3017-3030, 1996.
+%
+%Copyright (C) 2012 Zoltan Szabo ("http://nipg.inf.elte.hu/szzoli", "szzoli (at) cs (dot) elte (dot) hu")
+%
+%This file is part of the ITE (Information Theoretical Estimators) Matlab/Octave toolbox.
+%
+%ITE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by
+%the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
+%
+%This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
+%MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.
+%
+%You should have received a copy of the GNU General Public License along with ITE. If not, see <http://www.gnu.org/licenses/>.
+
+%initialization:
+    [d,num_of_samples] = size(x); %dimension, number of samples
+    %demixing matrix (W_hat):
+        if isreal(x) %real case
+            W_hat = randn(dim_reduction,d);
+        else %complex case
+            W_hat = randn(dim_reduction,d) + i * randn(dim_reduction,d);
+        end
+    %learning rates:
+        learning_rates = 2./[1:num_of_samples];%2/t    
+        %learning_rates = 0.01*ones(num_of_samples,1); %constant learning rate
+    I = eye(dim_reduction);%identity matrix
+    
+%demixing matrix estimation (W_hat):
+    for t = 1 : num_of_samples
+        learning_rate = learning_rates(t);
+        y = W_hat * x(:,t); %actual observation
+        %H (normalized EASI):
+            diady = y * y';
+            %some g(y) possibilities:
+                gy = diag(diady).*y; 
+                %gy =  y .* diag(0.1*diady).*diag(diady);
+                %gy	=  y .* sqrt(diag(diady));
+                %gy	=  y .* log(diag(diady));
+                %gy = - y .* (diag(diady)<0.9);
+            gyy = gy * y';
+            H = (diady - I) / (1 + learning_rate * (y'*y)) + (gyy - gyy') / ( 1 + learning_rate * abs(y' * gy));
+        W_hat = W_hat - learning_rate * H * W_hat;
+    end
+    
+%source estimation (e_hat):
+    e_hat = W_hat * x;