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author | Daniele Nicolodi <nicolodi@science.unitn.it> |
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date | Mon, 05 Dec 2011 16:20:06 +0100 |
parents | f0afece42f48 |
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% KSTEST perform the Kolmogorov - Smirnov statistical hypothesis test %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % DESCRIPTION: % % Kolmogorov - Smirnov test is typically used to assess if a sample comes % from a specific distribution or if two data samples came from the same % distribution. The test statistics is d_K = max|S(x) - K(x)| where S(x) % and K(x) are cumulative distribution functions of the two inputs % respectively. % In the case of the test on a single data series: % - null hypothesis is that the data are a realizations of a random variable % which is distributed according to the given probability distribution % In the case of the test on two data series: % - null hypothesis is that the two data series are realizations of the same random variable % % CALL: % % H = utils.math.kstest(y1, y2, alpha, distparams) % [H] = utils.math.kstest(y1, y2, alpha, distparams) % [H, KSstatistic] = utils.math.kstest(y1, y2, alpha, distparams) % [H, KSstatistic, criticalValue] = utils.math.kstest(y1, y2, alpha, distparams) % [H, KSstatistic, criticalValue] = utils.math.kstest(y1, y2, alpha, distparams, shapeparam) % [H, KSstatistic, criticalValue, pValue] = utils.math.kstest(y1, y2, alpha, distparams, shapeparam, criticalValue) % % INPUT: % % - Y1 are the data we want to test against Y2. % % - Y2 can be a theoretical distribution or a second set of data. In case % of theoretical distribution, Y2 should be a string with the corresponding % distribution name. Permitted values are: % - 'NormDist' Nomal distribution % - 'Chi2Dist' Chi square distribution % - 'FDist' F distribution % - 'GammaDist' Gamma distribution % If Y2 is left empty a normal distribution is assumed. % % - ALPHA is the desired significance level (default = 0.05). It represents % the probability of rejecting the null hypothesis when it is true. % Rejecting the null hypothesis, H0, when it is true is called a Type I % Error. Therefore, if the null hypothesis is true , the level of the test, % is the probability of a type I error. % % - DISTPARAMS are the parameters of the chosen theoretical distribution. % You should not assign this input if Y2 are experimental data. In general % DISTPARAMS is a vector containing the following distribution parameters: % - In case of 'NormDist', DISTPARAMS is a vector containing mean and % standard deviation of the normal distribution [mean sigma]. Default [0 1] % - In case of 'Chi2Dist' , DISTPARAMS is a number containing containing % the degrees of freedom of the chi square distribution [dof]. Default [2] % - In case of 'FDist', DISTPARAMS is a vector containing the two degrees % of freedom of the F distribution [dof1 dof2]. Default [2 2] % - In case of 'GammaDist', DISTPARAMS is a vector containing the shape % and scale parameters [k, theta]. Default [2 2] % % - SHAPEPARAM In the case of comparison of a data series with a % theoretical distribution and the data series is composed of correlated % elements. K can be adjusted with a shape parameter in order to recover % test fairness [3]. In such a case the test is performed for K' = Phi * K. % Phi is the corresponding Shape parameter. The shape parameter depends on % the correlations and on the significance value. It does not depend on % data length. Default [1] % % - CRITICALVALUE In case the critical value for the test is available from % external calculations, e.g. Monte Carlo simulation, the vale can be input % to the method % % OUTPUT: % % - H indicates the result of the hypothesis test: % H = false => Do not reject the null hypothesis at significance level ALPHA. % H = true => Reject the null hypothesis at significance level ALPHA. % % - TEST STATISTIC is the value of d_K = max|S(x) - K(x)|. % % - CRITICAL VALUE is the value of the test statistics corresponding to the % significance level. CRITICAL VALUE is depending on K, where K is the data length of Y1 if % Y2 is a theoretical distribution, otherwise if Y1 and Y2 are two data % samples K = n1*n2/(n1 + n2) where n1 and n2 are data length of Y1 and Y2 % respectively. In the case of comparison of a data series with a % theoretical distribution and the data series is composed of correlated % elements. K can be adjusted with a shape parameter in order to recover % test fairness [3]. In such a case the test is performed for K' = Phi * K. % If TEST STATISTIC > CRITICAL VALUE the null hypothesis is rejected. % % - P VALUE is the probability value associated to the test statistic. % % Luigi Ferraioli 17-02-2011 % % REFERENCES: % % [1] Massey, F.J., (1951) "The Kolmogorov-Smirnov Test for Goodness of % Fit", Journal of the American Statistical Association, 46(253):68-78. % [2] Miller, L.H., (1956) "Table of Percentage Points of Kolmogorov % Statistics", Journal of the American Statistical Association, % 51(273):111-121. % [3] Ferraioli L. et al, to be published. % % % $Id: kstest.m,v 1.8 2011/07/14 07:09:29 mauro Exp $ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [H, KSstatistic, criticalValue, pValue] = kstest(y1, y2, alpha, varargin) % check inputs if isempty(y2) y2 = 'normdist'; % set normal distribution as default end if isempty(alpha) alpha = 0.05; end if nargin > 3 dof = varargin{1}; elseif nargin <= 3 && ischar(y2) switch lower(y2) % assign dof case 'fdist' dof = [2 2]; case 'normdist' dof = [0 1]; case 'chi2dist' dof = [2]; case 'gammadist' dof = [2 2]; end end shp = 1; if nargin > 4 shp = varargin{2}; if isempty(shp) shp = 1; end end if nargin > 5 criticalValue = varargin{3}; else criticalValue = []; end n1 = length(y1); % get empirical distribution for input data [CD1,x1] = utils.math.ecdf(y1); % check if we have a second dataset or a theoretical distribution as second % input if ischar(y2) % switch between theoretical distributions switch lower(y2) case 'fdist' CD2 = utils.math.Fcdf(x1, dof(1), dof(2)); case 'normdist' CD2 = utils.math.Normcdf(x1, dof(1), dof(2)); case 'chi2dist' CD2 = utils.math.Chi2cdf(x1, dof(1)); case 'gammadist' CD2 = gammainc(x./dof(2), dof(1)); otherwise error('??? Unrecognized distribution type') end n2 = []; n1 = shp*n1; % calculate empirical distribution for second input dataset else [eCD2, ex2] = utils.math.ecdf(y2); CD2 = interp1(ex2, eCD2, x1, 'linear'); n2 = length(y2); end KSstatistic = max(abs(CD1 - CD2)); if isempty(criticalValue) criticalValue = utils.math.SKcriticalvalues(n1, n2, alpha/2); end % "H = 0" implies that we "Do not reject the null hypothesis at the % significance level of alpha," and "H = 1" implies that we "Reject null % hypothesis at significance level of alpha." H = (KSstatistic > criticalValue); if nargout > 3 pValue = utils.math.KSpValue(KSstatistic, n1, n2); end end