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Fix. Default password should be [] not an empty string
author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 07 Dec 2011 17:29:47 +0100
parents f0afece42f48
children
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 % KSTEST perform the Kolmogorov - Smirnov statistical hypothesis test
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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:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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5 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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6 % Kolmogorov - Smirnov test is typically used to assess if a sample comes
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Daniele Nicolodi <nicolodi@science.unitn.it>
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7 % from a specific distribution or if two data samples came from the same
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Daniele Nicolodi <nicolodi@science.unitn.it>
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8 % distribution. The test statistics is d_K = max|S(x) - K(x)| where S(x)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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9 % and K(x) are cumulative distribution functions of the two inputs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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10 % respectively.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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11 % In the case of the test on a single data series:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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12 % - null hypothesis is that the data are a realizations of a random variable
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Daniele Nicolodi <nicolodi@science.unitn.it>
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13 % which is distributed according to the given probability distribution
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Daniele Nicolodi <nicolodi@science.unitn.it>
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14 % In the case of the test on two data series:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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15 % - null hypothesis is that the two data series are realizations of the same random variable
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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 % CALL:
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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 % H = utils.math.kstest(y1, y2, alpha, distparams)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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20 % [H] = utils.math.kstest(y1, y2, alpha, distparams)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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21 % [H, KSstatistic] = utils.math.kstest(y1, y2, alpha, distparams)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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22 % [H, KSstatistic, criticalValue] = utils.math.kstest(y1, y2, alpha, distparams)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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23 % [H, KSstatistic, criticalValue] = utils.math.kstest(y1, y2, alpha, distparams, shapeparam)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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24 % [H, KSstatistic, criticalValue, pValue] = utils.math.kstest(y1, y2, alpha, distparams, shapeparam, criticalValue)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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25 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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26 % INPUT:
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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 % - Y1 are the data we want to test against Y2.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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29 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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30 % - Y2 can be a theoretical distribution or a second set of data. In case
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Daniele Nicolodi <nicolodi@science.unitn.it>
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31 % of theoretical distribution, Y2 should be a string with the corresponding
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Daniele Nicolodi <nicolodi@science.unitn.it>
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32 % distribution name. Permitted values are:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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33 % - 'NormDist' Nomal distribution
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Daniele Nicolodi <nicolodi@science.unitn.it>
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34 % - 'Chi2Dist' Chi square distribution
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Daniele Nicolodi <nicolodi@science.unitn.it>
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35 % - 'FDist' F distribution
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Daniele Nicolodi <nicolodi@science.unitn.it>
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36 % - 'GammaDist' Gamma distribution
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Daniele Nicolodi <nicolodi@science.unitn.it>
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37 % If Y2 is left empty a normal distribution is assumed.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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38 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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39 % - ALPHA is the desired significance level (default = 0.05). It represents
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Daniele Nicolodi <nicolodi@science.unitn.it>
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40 % the probability of rejecting the null hypothesis when it is true.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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41 % Rejecting the null hypothesis, H0, when it is true is called a Type I
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Daniele Nicolodi <nicolodi@science.unitn.it>
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42 % Error. Therefore, if the null hypothesis is true , the level of the test,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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43 % is the probability of a type I error.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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44 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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45 % - DISTPARAMS are the parameters of the chosen theoretical distribution.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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46 % You should not assign this input if Y2 are experimental data. In general
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Daniele Nicolodi <nicolodi@science.unitn.it>
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47 % DISTPARAMS is a vector containing the following distribution parameters:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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48 % - In case of 'NormDist', DISTPARAMS is a vector containing mean and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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49 % standard deviation of the normal distribution [mean sigma]. Default [0 1]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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50 % - In case of 'Chi2Dist' , DISTPARAMS is a number containing containing
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 % the degrees of freedom of the chi square distribution [dof]. Default [2]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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52 % - In case of 'FDist', DISTPARAMS is a vector containing the two degrees
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Daniele Nicolodi <nicolodi@science.unitn.it>
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53 % of freedom of the F distribution [dof1 dof2]. Default [2 2]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 % - In case of 'GammaDist', DISTPARAMS is a vector containing the shape
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Daniele Nicolodi <nicolodi@science.unitn.it>
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55 % and scale parameters [k, theta]. Default [2 2]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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57 % - SHAPEPARAM In the case of comparison of a data series with a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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58 % theoretical distribution and the data series is composed of correlated
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 % elements. K can be adjusted with a shape parameter in order to recover
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 % test fairness [3]. In such a case the test is performed for K' = Phi * K.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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61 % Phi is the corresponding Shape parameter. The shape parameter depends on
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Daniele Nicolodi <nicolodi@science.unitn.it>
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62 % the correlations and on the significance value. It does not depend on
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Daniele Nicolodi <nicolodi@science.unitn.it>
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63 % data length. Default [1]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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64 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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65 % - CRITICALVALUE In case the critical value for the test is available from
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Daniele Nicolodi <nicolodi@science.unitn.it>
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66 % external calculations, e.g. Monte Carlo simulation, the vale can be input
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Daniele Nicolodi <nicolodi@science.unitn.it>
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67 % to the method
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Daniele Nicolodi <nicolodi@science.unitn.it>
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68 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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69 % OUTPUT:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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70 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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71 % - H indicates the result of the hypothesis test:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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72 % H = false => Do not reject the null hypothesis at significance level ALPHA.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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73 % H = true => Reject the null hypothesis at significance level ALPHA.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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74 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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75 % - TEST STATISTIC is the value of d_K = max|S(x) - K(x)|.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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76 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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77 % - CRITICAL VALUE is the value of the test statistics corresponding to the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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78 % significance level. CRITICAL VALUE is depending on K, where K is the data length of Y1 if
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 % Y2 is a theoretical distribution, otherwise if Y1 and Y2 are two data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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80 % samples K = n1*n2/(n1 + n2) where n1 and n2 are data length of Y1 and Y2
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Daniele Nicolodi <nicolodi@science.unitn.it>
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81 % respectively. In the case of comparison of a data series with a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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82 % theoretical distribution and the data series is composed of correlated
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Daniele Nicolodi <nicolodi@science.unitn.it>
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83 % elements. K can be adjusted with a shape parameter in order to recover
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Daniele Nicolodi <nicolodi@science.unitn.it>
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84 % test fairness [3]. In such a case the test is performed for K' = Phi * K.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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85 % If TEST STATISTIC > CRITICAL VALUE the null hypothesis is rejected.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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86 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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87 % - P VALUE is the probability value associated to the test statistic.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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89 % Luigi Ferraioli 17-02-2011
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Daniele Nicolodi <nicolodi@science.unitn.it>
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90 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 % REFERENCES:
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Daniele Nicolodi <nicolodi@science.unitn.it>
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92 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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93 % [1] Massey, F.J., (1951) "The Kolmogorov-Smirnov Test for Goodness of
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Daniele Nicolodi <nicolodi@science.unitn.it>
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94 % Fit", Journal of the American Statistical Association, 46(253):68-78.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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95 % [2] Miller, L.H., (1956) "Table of Percentage Points of Kolmogorov
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Daniele Nicolodi <nicolodi@science.unitn.it>
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96 % Statistics", Journal of the American Statistical Association,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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97 % 51(273):111-121.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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98 % [3] Ferraioli L. et al, to be published.
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Daniele Nicolodi <nicolodi@science.unitn.it>
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99 %
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Daniele Nicolodi <nicolodi@science.unitn.it>
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100 % % $Id: kstest.m,v 1.8 2011/07/14 07:09:29 mauro Exp $
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102 function [H, KSstatistic, criticalValue, pValue] = kstest(y1, y2, alpha, varargin)
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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 % check inputs
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Daniele Nicolodi <nicolodi@science.unitn.it>
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105 if isempty(y2)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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106 y2 = 'normdist'; % set normal distribution as default
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Daniele Nicolodi <nicolodi@science.unitn.it>
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107 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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108 if isempty(alpha)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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109 alpha = 0.05;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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110 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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111
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Daniele Nicolodi <nicolodi@science.unitn.it>
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112 if nargin > 3
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Daniele Nicolodi <nicolodi@science.unitn.it>
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113 dof = varargin{1};
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Daniele Nicolodi <nicolodi@science.unitn.it>
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114 elseif nargin <= 3 && ischar(y2)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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115 switch lower(y2) % assign dof
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Daniele Nicolodi <nicolodi@science.unitn.it>
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116 case 'fdist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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117 dof = [2 2];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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118 case 'normdist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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119 dof = [0 1];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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120 case 'chi2dist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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121 dof = [2];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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122 case 'gammadist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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123 dof = [2 2];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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124 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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125 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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126
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 shp = 1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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128 if nargin > 4
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Daniele Nicolodi <nicolodi@science.unitn.it>
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129 shp = varargin{2};
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Daniele Nicolodi <nicolodi@science.unitn.it>
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130 if isempty(shp)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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131 shp = 1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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132 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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133 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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134
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Daniele Nicolodi <nicolodi@science.unitn.it>
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135 if nargin > 5
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Daniele Nicolodi <nicolodi@science.unitn.it>
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136 criticalValue = varargin{3};
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Daniele Nicolodi <nicolodi@science.unitn.it>
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137 else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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138 criticalValue = [];
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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 n1 = length(y1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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142
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Daniele Nicolodi <nicolodi@science.unitn.it>
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143 % get empirical distribution for input data
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Daniele Nicolodi <nicolodi@science.unitn.it>
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144 [CD1,x1] = utils.math.ecdf(y1);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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145
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Daniele Nicolodi <nicolodi@science.unitn.it>
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146 % check if we have a second dataset or a theoretical distribution as second
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Daniele Nicolodi <nicolodi@science.unitn.it>
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147 % input
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Daniele Nicolodi <nicolodi@science.unitn.it>
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148 if ischar(y2)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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149 % switch between theoretical distributions
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Daniele Nicolodi <nicolodi@science.unitn.it>
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150 switch lower(y2)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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151 case 'fdist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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152 CD2 = utils.math.Fcdf(x1, dof(1), dof(2));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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153 case 'normdist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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154 CD2 = utils.math.Normcdf(x1, dof(1), dof(2));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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155 case 'chi2dist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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156 CD2 = utils.math.Chi2cdf(x1, dof(1));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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157 case 'gammadist'
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Daniele Nicolodi <nicolodi@science.unitn.it>
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158 CD2 = gammainc(x./dof(2), dof(1));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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159 otherwise
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Daniele Nicolodi <nicolodi@science.unitn.it>
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160 error('??? Unrecognized distribution type')
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Daniele Nicolodi <nicolodi@science.unitn.it>
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161 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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162 n2 = [];
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Daniele Nicolodi <nicolodi@science.unitn.it>
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163 n1 = shp*n1;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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164 % calculate empirical distribution for second input dataset
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Daniele Nicolodi <nicolodi@science.unitn.it>
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165 else
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Daniele Nicolodi <nicolodi@science.unitn.it>
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166 [eCD2, ex2] = utils.math.ecdf(y2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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167 CD2 = interp1(ex2, eCD2, x1, 'linear');
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Daniele Nicolodi <nicolodi@science.unitn.it>
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168 n2 = length(y2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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169 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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170
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Daniele Nicolodi <nicolodi@science.unitn.it>
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171 KSstatistic = max(abs(CD1 - CD2));
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Daniele Nicolodi <nicolodi@science.unitn.it>
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172
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Daniele Nicolodi <nicolodi@science.unitn.it>
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173 if isempty(criticalValue)
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Daniele Nicolodi <nicolodi@science.unitn.it>
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174 criticalValue = utils.math.SKcriticalvalues(n1, n2, alpha/2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
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175 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
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176
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Daniele Nicolodi <nicolodi@science.unitn.it>
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177 % "H = 0" implies that we "Do not reject the null hypothesis at the
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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178 % significance level of alpha," and "H = 1" implies that we "Reject null
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Daniele Nicolodi <nicolodi@science.unitn.it>
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179 % hypothesis at significance level of alpha."
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Daniele Nicolodi <nicolodi@science.unitn.it>
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180 H = (KSstatistic > criticalValue);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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181
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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182 if nargout > 3
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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183 pValue = utils.math.KSpValue(KSstatistic, n1, n2);
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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184 end
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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185
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Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
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186 end