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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 16:20:06 +0100
parents f0afece42f48
children
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 % SFtest perfomes a Spectral F-Test on PSDs.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Daniele Nicolodi <nicolodi@science.unitn.it>
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3 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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4 % DESCRIPTION: SFtest performes a Spectral F-Test on two input PSD objects.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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5 % The null hypothesis H0 (the two PSDs belong to the same
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Daniele Nicolodi <nicolodi@science.unitn.it>
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6 % statistical distribution) is rejected at the confidence
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Daniele Nicolodi <nicolodi@science.unitn.it>
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7 % level for the alternative hypotheis H1 (the two PSDs belong
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Daniele Nicolodi <nicolodi@science.unitn.it>
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8 % to different statistical distributions) if the test
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Daniele Nicolodi <nicolodi@science.unitn.it>
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9 % statistic falls in the critical region.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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10 % SFtest uses utils.math.Ftest which does the test at each
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Daniele Nicolodi <nicolodi@science.unitn.it>
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11 % frequency bin.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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12 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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13 % VERSION: $Id: SFtest.m,v 1.3 2011/03/07 17:38:05 congedo Exp $
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Daniele Nicolodi <nicolodi@science.unitn.it>
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14 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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15 % HISTORY: 18-02-2011 G. Congedo
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Daniele Nicolodi <nicolodi@science.unitn.it>
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16 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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17 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Daniele Nicolodi <nicolodi@science.unitn.it>
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18
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Daniele Nicolodi <nicolodi@science.unitn.it>
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19 function test = SFtest(X,Y,alpha,showPlots)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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20
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Daniele Nicolodi <nicolodi@science.unitn.it>
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21 % Assume a two-tailed test
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Daniele Nicolodi <nicolodi@science.unitn.it>
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22 twoTailed = 1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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23
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Daniele Nicolodi <nicolodi@science.unitn.it>
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24 % Extract the degree of freedom
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Daniele Nicolodi <nicolodi@science.unitn.it>
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25 dofX = X.getdof.y;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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26 dofY = Y.getdof.y;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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27
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Daniele Nicolodi <nicolodi@science.unitn.it>
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28 % Ratio of the two spectra: this test statistic is F-distributed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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29 F = X/Y;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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30 F.setYunits('');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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31 F.setName('F statistic');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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32
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Daniele Nicolodi <nicolodi@science.unitn.it>
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33 n = numel(F.y);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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34
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Daniele Nicolodi <nicolodi@science.unitn.it>
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35 % Interquartile range
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Daniele Nicolodi <nicolodi@science.unitn.it>
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36 Fy = sort(F.y);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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37 m = median(Fy);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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38 q1 = median(Fy(Fy<=m));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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39 q3 = median(Fy(Fy>=m));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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40 iqr = q3 - q1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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41
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Daniele Nicolodi <nicolodi@science.unitn.it>
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42 % Freedman–Diaconis rule
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Daniele Nicolodi <nicolodi@science.unitn.it>
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43 binSz = 2*iqr*n^(-1/3);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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44 nBin = round((max(Fy)-min(Fy))/binSz);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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45
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Daniele Nicolodi <nicolodi@science.unitn.it>
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46 % Sample PDF
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Daniele Nicolodi <nicolodi@science.unitn.it>
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47 samplePDF = hist(F,plist('N',nBin,'norm',1));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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48 samplePDF.setName('sample PDF');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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49
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Daniele Nicolodi <nicolodi@science.unitn.it>
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50 % Sample PDF moments
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 % sampleK = utils.math.Kurt(samplePDF.x);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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52 % sampleS = utils.math.Skew(samplePDF.x);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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53
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 % Theor PDF
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Daniele Nicolodi <nicolodi@science.unitn.it>
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55 theorPDF = copy(samplePDF,1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 theorPDF.setY(Fpdf(samplePDF.x,dofX,dofY));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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57 theorPDF.setName('theor. PDF');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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58
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 % Theor PDF moments
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 % theorK = utils.math.Kurt(samplePDF.x);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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61 % theorS = utils.math.Skew(samplePDF.x);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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62
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Daniele Nicolodi <nicolodi@science.unitn.it>
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63 % Plot PDFs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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64 if showPlots
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Daniele Nicolodi <nicolodi@science.unitn.it>
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65 iplot(samplePDF,theorPDF);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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66 set(get(gca,'YLabel'),'String','PDF')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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67 set(get(gca,'XLabel'),'String','F statistic')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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68 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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69
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Daniele Nicolodi <nicolodi@science.unitn.it>
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70 % Perform test on the hypothesis that both distributions should be
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Daniele Nicolodi <nicolodi@science.unitn.it>
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71 % chi2-distributed or, equivalently, that the ratio of the two spectra
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Daniele Nicolodi <nicolodi@science.unitn.it>
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72 % should be F-distributed. The comparison is actually done in chi2 sense.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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73 sampleHIST = hist(F,plist('N',nBin));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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74 ix = sampleHIST.y>=5;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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75 C = sum(sampleHIST.y)*mean(diff(sampleHIST.x));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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76 NN = sampleHIST.y(ix);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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77 nn = C*theorPDF.y(ix);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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78 r = (NN-nn).^2./nn;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 chi2 = sum(r);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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80 dof = numel(NN) - 1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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81 Y2 = dof + sqrt(2*dof/(2*dof+sum(1./nn)))*(chi2-dof);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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82 critValue(1) = utils.math.Chi2inv(alpha/2,dof);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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83 critValue(2) = utils.math.Chi2inv(1-alpha/2,dof);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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84
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Daniele Nicolodi <nicolodi@science.unitn.it>
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85 % Perform the test on PDFs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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86 Chi2test = Y2<critValue(1) | Y2>critValue(2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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87
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 % % Sample CDF
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Daniele Nicolodi <nicolodi@science.unitn.it>
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89 % samplePDF = hist(F,plist('N',1000));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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90 % sampleCDF = copy(samplePDF,1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 % sampleCDF.setY(cumsum(samplePDF.y)/sum(samplePDF.y));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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92 % sampleCDF.setName('sample CDF');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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93 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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94 % % Theor CDF
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Daniele Nicolodi <nicolodi@science.unitn.it>
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95 % theorCDF = copy(samplePDF,1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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96 % theorCDF.setY(Fcdf(samplePDF.x,dofX,dofY));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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97 % theorCDF.setName('theoretical CDF');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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98 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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99 % % Plot CDFs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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100 % iplot(sampleCDF,theorCDF);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 % set(get(gca,'YLabel'),'String','CDF')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102 % set(get(gca,'XLabel'),'String','F statistic')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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103
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Daniele Nicolodi <nicolodi@science.unitn.it>
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104 % Critical values
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Daniele Nicolodi <nicolodi@science.unitn.it>
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105 [test,critValue,pValue] = utils.math.Ftest(F.y,dofX,dofY,alpha,twoTailed);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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106
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Daniele Nicolodi <nicolodi@science.unitn.it>
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107 % Build AOs for critical values
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Daniele Nicolodi <nicolodi@science.unitn.it>
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108 % if twoTailed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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109 critValueLB = ao(plist('xvals',X.x,'yvals',repmat(critValue(1),size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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110 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','critical values'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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111 critValueUB = ao(plist('xvals',X.x,'yvals',repmat(critValue(2),size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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112 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','critical values'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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113 % else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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114 % critValue = ao(plist('xvals',X.x,'yvals',repmat(critValue,size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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115 % 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','critical values'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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116 % end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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117
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Daniele Nicolodi <nicolodi@science.unitn.it>
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118 % % Build AOs for p-values and confidence levels
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Daniele Nicolodi <nicolodi@science.unitn.it>
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119 % pValue = ao(plist('xvals',X.x,'yvals',pValue,...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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120 % 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','p-values'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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121 % % pValue.setDy(pValueDy);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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122 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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123 % if twoTailed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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124 % confLevelUB = ao(plist('xvals',X.x,'yvals',repmat(1-alpha/2/numel(F.y),size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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125 % 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','confidence level'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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126 % confLevelLB = ao(plist('xvals',X.x,'yvals',repmat(alpha/2/numel(F.y),size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 % 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','confidence level'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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128 % else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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129 % confLevel = ao(plist('xvals',X.x,'yvals',repmat(1-alpha/n,size(X.x)),...
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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130 % 'type','fsdata','fs',X.fs,'xunits',X.xunits,'name','confidence level'));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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131 % end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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132
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Daniele Nicolodi <nicolodi@science.unitn.it>
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133 % Rejection index
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Daniele Nicolodi <nicolodi@science.unitn.it>
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134 ix = find(test);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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135 if ~isempty(ix)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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136 H1 = F.setXY(plist('x',F.x(ix),'y',F.y(ix)));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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137 H1.setDy([]);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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138 H1.setName('H0 rejected');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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139 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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140
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Daniele Nicolodi <nicolodi@science.unitn.it>
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141 % Compute the number of sigmal corresponding to the confidence level
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Daniele Nicolodi <nicolodi@science.unitn.it>
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142 % Ns = erfinv(1-alpha)*sqrt(2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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143
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Daniele Nicolodi <nicolodi@science.unitn.it>
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144 % Make the test: H0 rejected?
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Daniele Nicolodi <nicolodi@science.unitn.it>
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145 % if twoTailed
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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146 % % test = any( (F.y-Ns.*F.dy)>critValueUB.y | (F.y+Ns.*F.dy)<critValueLB.y );
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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147 % test = any( F.y>critValueUB.y | F.y<critValueLB.y );
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Daniele Nicolodi <nicolodi@science.unitn.it>
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148 % else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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149 % % test = any( (F.y-Ns.*F.dy)>critValue.y );
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Daniele Nicolodi <nicolodi@science.unitn.it>
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150 % test = any( F.y>critValue.y );
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Daniele Nicolodi <nicolodi@science.unitn.it>
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151 % end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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152
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Daniele Nicolodi <nicolodi@science.unitn.it>
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153 % Perform the F-test on spectra
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Daniele Nicolodi <nicolodi@science.unitn.it>
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154 Ftest = any(test);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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155
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Daniele Nicolodi <nicolodi@science.unitn.it>
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156 % Plot results
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Daniele Nicolodi <nicolodi@science.unitn.it>
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157 pl = plist('yscales',{'All', 'log'},'autoerrors',0);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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158 % if twoTailed
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Daniele Nicolodi <nicolodi@science.unitn.it>
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159 if showPlots
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Daniele Nicolodi <nicolodi@science.unitn.it>
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160 if Ftest
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Daniele Nicolodi <nicolodi@science.unitn.it>
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161 [hfig, hax, hli] = iplot(F,critValueLB,critValueUB,H1,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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162 set(hli(4),'linestyle','none','marker','s','MarkerSize',10,'MarkerEdgeColor','r');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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163 else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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164 [hfig, hax, hli] = iplot(F,critValueLB,critValueUB,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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165 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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166 set(hli(2:3),'color','r');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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167 % h2 = iplot(pValue,confLevelLB,confLevelUB,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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168 % else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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169 % if Ftest
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Daniele Nicolodi <nicolodi@science.unitn.it>
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170 % [hfig, hax, hli] = iplot(F,critValue,H1,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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171 % set(hli(3),'linestyle','none','marker','s','MarkerSize',10,'MarkerEdgeColor','r');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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172 % else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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173 % [hfig, hax, hli] = iplot(F,critValue,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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174 % end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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175 % set(hli(2),'color','r');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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176 % % h2 = iplot(pValue,confLevel,pl);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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177 % end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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178 set(get(gca,'YLabel'),'String','F statistic')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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179 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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180
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Daniele Nicolodi <nicolodi@science.unitn.it>
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181 % Output final result
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Daniele Nicolodi <nicolodi@science.unitn.it>
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182 test = any(Chi2test | Ftest);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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183
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Daniele Nicolodi <nicolodi@science.unitn.it>
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184 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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185