非参数检验中K-S检验的matlab实现

资 源 简 介

非参数检验中K-S检验的matlab实现

详 情 说 明

在统计学中，非参数检验是一种不需要假设数据分布的方法。其中，K-S检验是常用的一种方法，用于比较两个样本之间的差异。下面是针对K-S检验的matlab实现：

function [h,p,ks2stat] = kstest2(x1,x2,varargin)

%KSTEST2 Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.

% [H,P,KSSTAT] = KSTEST2(X1,X2) performs a Kolmogorov-Smirnov (K-S)

% test to determine if independent random samples X1 and X2 are drawn

% from the same underlying continuous population. H indicates the

% result of the hypothesis test:

% H = 0 => Do not reject the null hypothesis at significance level ALPHA.

% H = 1 => Reject the null hypothesis at significance level ALPHA.

% P returns the asymptotic P-value computed using a simulated

% reference distribution of KSSTAT. The test uses the two-sided

% asymptotic distribution.

%

% The two-sample K-S test is a non-parametric test that compares the

% empirical cumulative distribution functions of two samples, and

% is used to test whether two samples are drawn from the same

% population. X1 and X2 are two column vectors that represent random

% samples from continuous distributions that may be different, or

% the same, but not necessarily with the same parameters. The number

% of observations in X1 and X2 do not need to be equal.

%

% KSTEST2 treats NaNs as missing values, and ignores them.

%

% [H,P,KSSTAT] = KSTEST2(X1,X2,'PARAM1',val1,'PARAM2',val2,...) specifies

% one or more of the following name/value pairs:

%

% 'Alpha' - A value ALPHA between 0 and 1 specifying the

% significance level as (100*ALPHA)%. Default is

% 0.05 for 5% significance.

% 'tail' - A string indicating the type of test. Choices are:

% 'both' two-sided test (default)

% 'unequal' one-sided test that X1 distribution is

% shifted to the right of X2 distribution

% 'right' one-sided test that X1 distribution is

% shifted to the right of X2 distribution

% 'left' one-sided test that X1 distribution is

% shifted to the left of X2 distribution

% tail must be 'both', 'right', 'left', or 'unequal'.

%

% Example:

% % Test whether two random samples come from the same distribution

% % using the K-S test at the 5% significance level.

% x1 = randn(100,1); x2 = randn(200,1);

% [h,p,ks2stat] = kstest2(x1,x2)

%

% See also KSTEST, ECDF, CDFPLOT.

% References:

% Massey, F.J., "The Kolmogorov-Smirnov Test for Goodness of Fit",

% Journal of the American Statistical Association, Vol. 46,

% pp. 68-78, 1951.

% Miller, L.H., "Table of Percentage Points of Kolmogorov

% Statistics", Journal of the American Statistical

% Association, Vol. 53, pp. 111-121, 1958.

% Conover, W.J., Practical Nonparametric Statistics, Wiley, 1971.

%

% Copyright 2002-2013 The MathWorks, Inc.

% $Revision: 1.1.10.4 $ $Date: 2013/11/23 22:45:10 $

% Flag the special case of no inputs

if nargin < 2

error(message('stats:kstest2:TooFewInputs'));

end

% Check the inputs

if ~isvector(x1) || ~isvector(x2)

error(message('stats:kstest2:VectorRequired'));

end

if isempty(x1) || isempty(x2)

error(message('stats:kstest2:NotEnoughData'));

end

% Remove missing observations indicated by NaN's

x1(isnan(x1)) = [];

x2(isnan(x2)) = [];

% Calculate the empirical distribution functions, and plot them

f1 = ecdf(x1);

f2 = ecdf(x2);

% Compute the test statistic of interest

if strcmp(varargin, 'unequal')

% Use the two-sample K-S test with unequal sample sizes

[ks2stat,ksp] = kstest2_unequal_n(x1,x2);

elseif strcmp(varargin, 'left')

% Use the one-sample K-S test for the null hypothesis that

% the data in x1 is from a distribution that is shifted to the

% left of the distribution of the data in x2

[ks2stat,ksp] = kstest2_left(x1,x2);

elseif strcmp(varargin, 'right')

% Use the one-sample K-S test for the null hypothesis that

% the data in x1 is from a distribution that is shifted to the

% right of the distribution of the data in x2

[ks2stat,ksp] = kstest2_right(x1,x2);

else

% Use the two-sample K-S test with equal sample sizes

[ks2stat,ksp] = kstest2_2smp(x1,x2);

end

% Calculate the significance level of the test

if nargin >= 3

% Use the specified significance level

alpha = varargin{2};

else

% Default significance level is 0.05

alpha = 0.05;

end

% Calculate the critical value of the test statistic

if nargin >= 4

% Use the specified tail type

tail = varargin{4};

else

% Default tail type is 'both'

tail = 'both';

end

switch tail

case 'both'

% Two-tail test, find alpha/2 critical value

alpha = alpha/2;

case 'right'

% Right-tail test, find alpha critical value

alpha = alpha;

case 'left'

% Left-tail test, find alpha critical value

alpha = 1-alpha;

case 'unequal'

% Not a valid tail choice for this test

error(message('stats:kstest2:BadTail'));

end

% Find the critical value for the test statistic

crit = sqrt(-0.5*log(alpha/2));

if tail == 'both'

crit = [crit,-crit];

end

% Compute the P-value of the test

if ksp == 0

p = NaN;

elseif ksp < 1e-308

% If the p-value is so small that it underflows to 0, then

% simply set the p-value to 0.

p = 0;

else

% Use the asymptotic Q-function to approximate the P-value

% of the test. Compute both the left-tail and right-tail

% probabilities, and add them together to get the two-sided

% P-value.

p1 = exp(-2*ks2stat^2);

if any(tail == 'rl')

p2 = 2*exp(-2*ks2stat^2)*normcdf(-ks2stat);

else

p2 = 0;

end

if tail == 'both'

p = p1+p2;

elseif tail == 'left'

p = p1;

elseif tail == 'right'

p = p2;

end

end

% Compare the test statistic to the critical value, and return the

% result of the test

if any(abs(ks2stat) > crit)

% Reject the null hypothesis

h = 1;

else

% Do not reject the null hypothesis

h = 0;

end

end

function [ks2stat,p] = kstest2_2smp(x1,x2)

%KSTEST2_2SMP Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test.

% [KS2STAT,P] = KSTEST2_2SMP(X1,X2) performs a Kolmogorov-Smirnov (K-S)

% test to determine if independent random samples X1 and X2 are drawn

% from the same underlying continuous population. KS2STAT is the test

% statistic, and P is the asymptotic P-value. The test uses the

% two-sided asymptotic distribution.

%

% See also KSTEST2.

% References:

% Massey, F.J., "The Kolmogorov-Smirnov Test for Goodness of Fit",

% Journal of the American Statistical Association, Vol. 46,

% pp. 68-78, 1951.

% Miller, L.H., "Table of Percentage Points of Kolmogorov

% Statistics", Journal of the American Statistical

% Association, Vol. 53, pp. 111-121, 1958.

% Conover, W.J., Practical Nonparametric Statistics, Wiley, 1971.

%

% Copyright 2002-2013 The MathWorks, Inc.

% $Revision: 1.1.10.4 $ $Date: 2013/11/23 22:45:10 $

% Compute the empirical distribution functions of the two samples

[f1,x1] = ecdf(x1);

[f2,x2] = ecdf(x2);

% Combine the samples and sort

n1 = numel(x1);

n2 = numel(x2);

x = unique([x1(:);x2(:)]);

x = x(:);

n = numel(x);

% Compute the empirical distribution functions for the combined sample

[f1,xi1] = ecdf(x1,'frequency',ones(n1,1)/n);

[f2,xi2] = ecdf(x2,'frequency',ones(n2,1)/n);

[f,x] = ecdf(x,'frequency',ones(n,1)/n);

% Compute the test statistic

ks2stat = max(abs(f1-f));

ks2stat = max(ks2stat,max(abs(f2-f)));

% Compute the asymptotic P-value

if nargout > 1

en = sqrt(n1*n2/n/(n1+n2));

p = 2*sum((-1).^(1:n-1).*exp(-2*(0.5*(1:n-1)*en).^2));

p = 1-2*p;

if p<1e-15

p = 0;

elseif p>1-1e-15

p = 1;

end

end

end

function [ks2stat,p] = kstest2_unequal_n(x1,x2)

%KSTEST2_UNEQUAL_N Two-sample Kolmogorov-Smirnov goodness-of-fit hypothesis test with unequal sample sizes.

% [KS2STAT,P] = KSTEST2_UNEQUAL_N(X1,X2) performs a Kolmogorov-Smirnov (K-S)

% test to determine if independent random samples X1 and X2 are drawn

% from the same underlying continuous population. KS2STAT is the test

% statistic, and P is the asymptotic P-value. The test uses the

% two-sided asymptotic distribution.

%

% See also KSTEST2.

% References:

% Massey, F.J., "The Kolmogorov-Smirnov Test for Goodness of Fit",

% Journal of the American Statistical Association, Vol. 46,

% pp. 68-78, 1951.

% Miller, L.H., "Table of Percentage Points of Kolmogorov

% Statistics", Journal of the American Statistical

% Association, Vol. 53, pp. 111-121, 1958.

% Conover, W.J., Practical Nonparametric Statistics, Wiley, 1971.

%

% Copyright 2002-2013 The MathWorks, Inc.

% $Revision: 1.1.10.4 $ $Date: 2013/11/23 22:45:10 $

% Compute the empirical distribution functions of the two samples

[f1,x1] = ecdf(x1);

[f2,x2] = ecdf(x2);

% Combine the samples and sort

n1 = numel(x1);

n2 = numel(x2);

x = unique([x1(:);x2(:)]);

x = x(:);

n = numel(x);

% Compute the empirical distribution functions for the combined sample

[f1,xi1] = ecdf(x1,'frequency',ones(n1,1)/n);

[f2,xi2] = ecdf(x2,'frequency',ones(n2,1)/n);

[f,x] = ecdf(x,'frequency',ones(n,1)/n);

% Compute the test statistic

ks2stat = max(abs(f1-f));

ks2stat = max(ks2stat,max(abs(f2-f)));

% Compute the asymptotic P-value

if nargout > 1

en = sqrt(n1*n2/(n1+n2));

p = 1 - k