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/*************************************************************************
Copyright (c) Sergey Bochkanov (ALGLIB project).
>>> SOURCE LICENSE >>>
This program 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 (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program 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.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#ifndef _statistics_pkg_h
#define _statistics_pkg_h
#include "ap.h"
#include "alglibinternal.h"
#include "linalg.h"
#include "specialfunctions.h"
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (DATATYPES)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS C++ INTERFACE
//
/////////////////////////////////////////////////////////////////////////
namespace alglib
{
/*************************************************************************
Calculation of the distribution moments: mean, variance, skewness, kurtosis.
INPUT PARAMETERS:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
OUTPUT PARAMETERS
Mean - mean.
Variance- variance.
Skewness- skewness (if variance<>0; zero otherwise).
Kurtosis- kurtosis (if variance<>0; zero otherwise).
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void samplemoments(const real_1d_array &x, const ae_int_t n, double &mean, double &variance, double &skewness, double &kurtosis);
void samplemoments(const real_1d_array &x, double &mean, double &variance, double &skewness, double &kurtosis);
/*************************************************************************
Calculation of the mean.
INPUT PARAMETERS:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
NOTE:
This function return result which calculated by 'SampleMoments' function
and stored at 'Mean' variable.
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplemean(const real_1d_array &x, const ae_int_t n);
double samplemean(const real_1d_array &x);
/*************************************************************************
Calculation of the variance.
INPUT PARAMETERS:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
NOTE:
This function return result which calculated by 'SampleMoments' function
and stored at 'Variance' variable.
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplevariance(const real_1d_array &x, const ae_int_t n);
double samplevariance(const real_1d_array &x);
/*************************************************************************
Calculation of the skewness.
INPUT PARAMETERS:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
NOTE:
This function return result which calculated by 'SampleMoments' function
and stored at 'Skewness' variable.
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double sampleskewness(const real_1d_array &x, const ae_int_t n);
double sampleskewness(const real_1d_array &x);
/*************************************************************************
Calculation of the kurtosis.
INPUT PARAMETERS:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
NOTE:
This function return result which calculated by 'SampleMoments' function
and stored at 'Kurtosis' variable.
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplekurtosis(const real_1d_array &x, const ae_int_t n);
double samplekurtosis(const real_1d_array &x);
/*************************************************************************
ADev
Input parameters:
X - sample
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
Output parameters:
ADev- ADev
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void sampleadev(const real_1d_array &x, const ae_int_t n, double &adev);
void sampleadev(const real_1d_array &x, double &adev);
/*************************************************************************
Median calculation.
Input parameters:
X - sample (array indexes: [0..N-1])
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
Output parameters:
Median
-- ALGLIB --
Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void samplemedian(const real_1d_array &x, const ae_int_t n, double &median);
void samplemedian(const real_1d_array &x, double &median);
/*************************************************************************
Percentile calculation.
Input parameters:
X - sample (array indexes: [0..N-1])
N - N>=0, sample size:
* if given, only leading N elements of X are processed
* if not given, automatically determined from size of X
P - percentile (0<=P<=1)
Output parameters:
V - percentile
-- ALGLIB --
Copyright 01.03.2008 by Bochkanov Sergey
*************************************************************************/
void samplepercentile(const real_1d_array &x, const ae_int_t n, const double p, double &v);
void samplepercentile(const real_1d_array &x, const double p, double &v);
/*************************************************************************
2-sample covariance
Input parameters:
X - sample 1 (array indexes: [0..N-1])
Y - sample 2 (array indexes: [0..N-1])
N - N>=0, sample size:
* if given, only N leading elements of X/Y are processed
* if not given, automatically determined from input sizes
Result:
covariance (zero for N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double cov2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n);
double cov2(const real_1d_array &x, const real_1d_array &y);
/*************************************************************************
Pearson product-moment correlation coefficient
Input parameters:
X - sample 1 (array indexes: [0..N-1])
Y - sample 2 (array indexes: [0..N-1])
N - N>=0, sample size:
* if given, only N leading elements of X/Y are processed
* if not given, automatically determined from input sizes
Result:
Pearson product-moment correlation coefficient
(zero for N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double pearsoncorr2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n);
double pearsoncorr2(const real_1d_array &x, const real_1d_array &y);
/*************************************************************************
Spearman's rank correlation coefficient
Input parameters:
X - sample 1 (array indexes: [0..N-1])
Y - sample 2 (array indexes: [0..N-1])
N - N>=0, sample size:
* if given, only N leading elements of X/Y are processed
* if not given, automatically determined from input sizes
Result:
Spearman's rank correlation coefficient
(zero for N=0 or N=1)
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double spearmancorr2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n);
double spearmancorr2(const real_1d_array &x, const real_1d_array &y);
/*************************************************************************
Covariance matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with covariance matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X are used
* if not given, automatically determined from input size
M - M>0, number of variables:
* if given, only leading M columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M,M], covariance matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void covm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void smp_covm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void covm(const real_2d_array &x, real_2d_array &c);
void smp_covm(const real_2d_array &x, real_2d_array &c);
/*************************************************************************
Pearson product-moment correlation matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with correlation matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X are used
* if not given, automatically determined from input size
M - M>0, number of variables:
* if given, only leading M columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M,M], correlation matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void smp_pearsoncorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void pearsoncorrm(const real_2d_array &x, real_2d_array &c);
void smp_pearsoncorrm(const real_2d_array &x, real_2d_array &c);
/*************************************************************************
Spearman's rank correlation matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with correlation matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X are used
* if not given, automatically determined from input size
M - M>0, number of variables:
* if given, only leading M columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M,M], correlation matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void spearmancorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void smp_spearmancorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c);
void spearmancorrm(const real_2d_array &x, real_2d_array &c);
void smp_spearmancorrm(const real_2d_array &x, real_2d_array &c);
/*************************************************************************
Cross-covariance matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with covariance matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M1], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
Y - array[N,M2], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X/Y are used
* if not given, automatically determined from input sizes
M1 - M1>0, number of variables in X:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
M2 - M2>0, number of variables in Y:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M1,M2], cross-covariance matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void covm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void smp_covm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void covm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
void smp_covm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
/*************************************************************************
Pearson product-moment cross-correlation matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with correlation matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M1], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
Y - array[N,M2], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X/Y are used
* if not given, automatically determined from input sizes
M1 - M1>0, number of variables in X:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
M2 - M2>0, number of variables in Y:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void smp_pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
void smp_pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
/*************************************************************************
Spearman's rank cross-correlation matrix
SMP EDITION OF ALGLIB:
! This function can utilize multicore capabilities of your system. In
! order to do this you have to call version with "smp_" prefix, which
! indicates that multicore code will be used.
!
! This note is given for users of SMP edition; if you use GPL edition,
! or commercial edition of ALGLIB without SMP support, you still will
! be able to call smp-version of this function, but all computations
! will be done serially.
!
! We recommend you to carefully read ALGLIB Reference Manual, section
! called 'SMP support', before using parallel version of this function.
!
! You should remember that starting/stopping worker thread always have
! non-zero cost. Although multicore version is pretty efficient on
! large problems, we do not recommend you to use it on small problems -
! with correlation matrices smaller than 128*128.
INPUT PARAMETERS:
X - array[N,M1], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
Y - array[N,M2], sample matrix:
* J-th column corresponds to J-th variable
* I-th row corresponds to I-th observation
N - N>=0, number of observations:
* if given, only leading N rows of X/Y are used
* if not given, automatically determined from input sizes
M1 - M1>0, number of variables in X:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
M2 - M2>0, number of variables in Y:
* if given, only leading M1 columns of X are used
* if not given, automatically determined from input size
OUTPUT PARAMETERS:
C - array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)
-- ALGLIB --
Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void spearmancorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void smp_spearmancorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c);
void spearmancorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
void smp_spearmancorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c);
/*************************************************************************
*************************************************************************/
void rankdata(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures);
void smp_rankdata(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures);
void rankdata(real_2d_array &xy);
void smp_rankdata(real_2d_array &xy);
/*************************************************************************
*************************************************************************/
void rankdatacentered(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures);
void smp_rankdatacentered(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures);
void rankdatacentered(real_2d_array &xy);
void smp_rankdatacentered(real_2d_array &xy);
/*************************************************************************
Obsolete function, we recommend to use PearsonCorr2().
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double pearsoncorrelation(const real_1d_array &x, const real_1d_array &y, const ae_int_t n);
/*************************************************************************
Obsolete function, we recommend to use SpearmanCorr2().
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double spearmanrankcorrelation(const real_1d_array &x, const real_1d_array &y, const ae_int_t n);
/*************************************************************************
Pearson's correlation coefficient significance test
This test checks hypotheses about whether X and Y are samples of two
continuous distributions having zero correlation or whether their
correlation is non-zero.
The following tests are performed:
* two-tailed test (null hypothesis - X and Y have zero correlation)
* left-tailed test (null hypothesis - the correlation coefficient is
greater than or equal to 0)
* right-tailed test (null hypothesis - the correlation coefficient is
less than or equal to 0).
Requirements:
* the number of elements in each sample is not less than 5
* normality of distributions of X and Y.
Input parameters:
R - Pearson's correlation coefficient for X and Y
N - number of elements in samples, N>=5.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrelationsignificance(const double r, const ae_int_t n, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Spearman's rank correlation coefficient significance test
This test checks hypotheses about whether X and Y are samples of two
continuous distributions having zero correlation or whether their
correlation is non-zero.
The following tests are performed:
* two-tailed test (null hypothesis - X and Y have zero correlation)
* left-tailed test (null hypothesis - the correlation coefficient is
greater than or equal to 0)
* right-tailed test (null hypothesis - the correlation coefficient is
less than or equal to 0).
Requirements:
* the number of elements in each sample is not less than 5.
The test is non-parametric and doesn't require distributions X and Y to be
normal.
Input parameters:
R - Spearman's rank correlation coefficient for X and Y
N - number of elements in samples, N>=5.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void spearmanrankcorrelationsignificance(const double r, const ae_int_t n, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Jarque-Bera test
This test checks hypotheses about the fact that a given sample X is a
sample of normal random variable.
Requirements:
* the number of elements in the sample is not less than 5.
Input parameters:
X - sample. Array whose index goes from 0 to N-1.
N - size of the sample. N>=5
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
Accuracy of the approximation used (5<=N<=1951):
p-value relative error (5<=N<=1951)
[1, 0.1] < 1%
[0.1, 0.01] < 2%
[0.01, 0.001] < 6%
[0.001, 0] wasn't measured
For N>1951 accuracy wasn't measured but it shouldn't be sharply different
from table values.
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void jarqueberatest(const real_1d_array &x, const ae_int_t n, double &p);
/*************************************************************************
Mann-Whitney U-test
This test checks hypotheses about whether X and Y are samples of two
continuous distributions of the same shape and same median or whether
their medians are different.
The following tests are performed:
* two-tailed test (null hypothesis - the medians are equal)
* left-tailed test (null hypothesis - the median of the first sample
is greater than or equal to the median of the second sample)
* right-tailed test (null hypothesis - the median of the first sample
is less than or equal to the median of the second sample).
Requirements:
* the samples are independent
* X and Y are continuous distributions (or discrete distributions well-
approximating continuous distributions)
* distributions of X and Y have the same shape. The only possible
difference is their position (i.e. the value of the median)
* the number of elements in each sample is not less than 5
* the scale of measurement should be ordinal, interval or ratio (i.e.
the test could not be applied to nominal variables).
The test is non-parametric and doesn't require distributions to be normal.
Input parameters:
X - sample 1. Array whose index goes from 0 to N-1.
N - size of the sample. N>=5
Y - sample 2. Array whose index goes from 0 to M-1.
M - size of the sample. M>=5
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
To calculate p-values, special approximation is used. This method lets us
calculate p-values with satisfactory accuracy in interval [0.0001, 1].
There is no approximation outside the [0.0001, 1] interval. Therefore, if
the significance level outlies this interval, the test returns 0.0001.
Relative precision of approximation of p-value:
N M Max.err. Rms.err.
5..10 N..10 1.4e-02 6.0e-04
5..10 N..100 2.2e-02 5.3e-06
10..15 N..15 1.0e-02 3.2e-04
10..15 N..100 1.0e-02 2.2e-05
15..100 N..100 6.1e-03 2.7e-06
For N,M>100 accuracy checks weren't put into practice, but taking into
account characteristics of asymptotic approximation used, precision should
not be sharply different from the values for interval [5, 100].
-- ALGLIB --
Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void mannwhitneyutest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Sign test
This test checks three hypotheses about the median of the given sample.
The following tests are performed:
* two-tailed test (null hypothesis - the median is equal to the given
value)
* left-tailed test (null hypothesis - the median is greater than or
equal to the given value)
* right-tailed test (null hypothesis - the median is less than or
equal to the given value)
Requirements:
* the scale of measurement should be ordinal, interval or ratio (i.e.
the test could not be applied to nominal variables).
The test is non-parametric and doesn't require distribution X to be normal
Input parameters:
X - sample. Array whose index goes from 0 to N-1.
N - size of the sample.
Median - assumed median value.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
While calculating p-values high-precision binomial distribution
approximation is used, so significance levels have about 15 exact digits.
-- ALGLIB --
Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void onesamplesigntest(const real_1d_array &x, const ae_int_t n, const double median, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
One-sample t-test
This test checks three hypotheses about the mean of the given sample. The
following tests are performed:
* two-tailed test (null hypothesis - the mean is equal to the given
value)
* left-tailed test (null hypothesis - the mean is greater than or
equal to the given value)
* right-tailed test (null hypothesis - the mean is less than or equal
to the given value).
The test is based on the assumption that a given sample has a normal
distribution and an unknown dispersion. If the distribution sharply
differs from normal, the test will work incorrectly.
INPUT PARAMETERS:
X - sample. Array whose index goes from 0 to N-1.
N - size of sample, N>=0
Mean - assumed value of the mean.
OUTPUT PARAMETERS:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
NOTE: this function correctly handles degenerate cases:
* when N=0, all p-values are set to 1.0
* when variance of X[] is exactly zero, p-values are set
to 1.0 or 0.0, depending on difference between sample mean and
value of mean being tested.
-- ALGLIB --
Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void studentttest1(const real_1d_array &x, const ae_int_t n, const double mean, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Two-sample pooled test
This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
* two-tailed test (null hypothesis - the means are equal)
* left-tailed test (null hypothesis - the mean of the first sample is
greater than or equal to the mean of the second sample)
* right-tailed test (null hypothesis - the mean of the first sample is
less than or equal to the mean of the second sample).
Test is based on the following assumptions:
* given samples have normal distributions
* dispersions are equal
* samples are independent.
Input parameters:
X - sample 1. Array whose index goes from 0 to N-1.
N - size of sample.
Y - sample 2. Array whose index goes from 0 to M-1.
M - size of sample.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
NOTE: this function correctly handles degenerate cases:
* when N=0 or M=0, all p-values are set to 1.0
* when both samples has exactly zero variance, p-values are set
to 1.0 or 0.0, depending on difference between means.
-- ALGLIB --
Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void studentttest2(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Two-sample unpooled test
This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
* two-tailed test (null hypothesis - the means are equal)
* left-tailed test (null hypothesis - the mean of the first sample is
greater than or equal to the mean of the second sample)
* right-tailed test (null hypothesis - the mean of the first sample is
less than or equal to the mean of the second sample).
Test is based on the following assumptions:
* given samples have normal distributions
* samples are independent.
Equality of variances is NOT required.
Input parameters:
X - sample 1. Array whose index goes from 0 to N-1.
N - size of the sample.
Y - sample 2. Array whose index goes from 0 to M-1.
M - size of the sample.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
NOTE: this function correctly handles degenerate cases:
* when N=0 or M=0, all p-values are set to 1.0
* when both samples has zero variance, p-values are set
to 1.0 or 0.0, depending on difference between means.
* when only one sample has zero variance, test reduces to 1-sample
version.
-- ALGLIB --
Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void unequalvariancettest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Two-sample F-test
This test checks three hypotheses about dispersions of the given samples.
The following tests are performed:
* two-tailed test (null hypothesis - the dispersions are equal)
* left-tailed test (null hypothesis - the dispersion of the first
sample is greater than or equal to the dispersion of the second
sample).
* right-tailed test (null hypothesis - the dispersion of the first
sample is less than or equal to the dispersion of the second sample)
The test is based on the following assumptions:
* the given samples have normal distributions
* the samples are independent.
Input parameters:
X - sample 1. Array whose index goes from 0 to N-1.
N - sample size.
Y - sample 2. Array whose index goes from 0 to M-1.
M - sample size.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
-- ALGLIB --
Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void ftest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
One-sample chi-square test
This test checks three hypotheses about the dispersion of the given sample
The following tests are performed:
* two-tailed test (null hypothesis - the dispersion equals the given
number)
* left-tailed test (null hypothesis - the dispersion is greater than
or equal to the given number)
* right-tailed test (null hypothesis - dispersion is less than or
equal to the given number).
Test is based on the following assumptions:
* the given sample has a normal distribution.
Input parameters:
X - sample 1. Array whose index goes from 0 to N-1.
N - size of the sample.
Variance - dispersion value to compare with.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
-- ALGLIB --
Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void onesamplevariancetest(const real_1d_array &x, const ae_int_t n, const double variance, double &bothtails, double &lefttail, double &righttail);
/*************************************************************************
Wilcoxon signed-rank test
This test checks three hypotheses about the median of the given sample.
The following tests are performed:
* two-tailed test (null hypothesis - the median is equal to the given
value)
* left-tailed test (null hypothesis - the median is greater than or
equal to the given value)
* right-tailed test (null hypothesis - the median is less than or
equal to the given value)
Requirements:
* the scale of measurement should be ordinal, interval or ratio (i.e.
the test could not be applied to nominal variables).
* the distribution should be continuous and symmetric relative to its
median.
* number of distinct values in the X array should be greater than 4
The test is non-parametric and doesn't require distribution X to be normal
Input parameters:
X - sample. Array whose index goes from 0 to N-1.
N - size of the sample.
Median - assumed median value.
Output parameters:
BothTails - p-value for two-tailed test.
If BothTails is less than the given significance level
the null hypothesis is rejected.
LeftTail - p-value for left-tailed test.
If LeftTail is less than the given significance level,
the null hypothesis is rejected.
RightTail - p-value for right-tailed test.
If RightTail is less than the given significance level
the null hypothesis is rejected.
To calculate p-values, special approximation is used. This method lets us
calculate p-values with two decimal places in interval [0.0001, 1].
"Two decimal places" does not sound very impressive, but in practice the
relative error of less than 1% is enough to make a decision.
There is no approximation outside the [0.0001, 1] interval. Therefore, if
the significance level outlies this interval, the test returns 0.0001.
-- ALGLIB --
Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void wilcoxonsignedranktest(const real_1d_array &x, const ae_int_t n, const double e, double &bothtails, double &lefttail, double &righttail);
}
/////////////////////////////////////////////////////////////////////////
//
// THIS SECTION CONTAINS COMPUTATIONAL CORE DECLARATIONS (FUNCTIONS)
//
/////////////////////////////////////////////////////////////////////////
namespace alglib_impl
{
void samplemoments(/* Real */ ae_vector* x,
ae_int_t n,
double* mean,
double* variance,
double* skewness,
double* kurtosis,
ae_state *_state);
double samplemean(/* Real */ ae_vector* x,
ae_int_t n,
ae_state *_state);
double samplevariance(/* Real */ ae_vector* x,
ae_int_t n,
ae_state *_state);
double sampleskewness(/* Real */ ae_vector* x,
ae_int_t n,
ae_state *_state);
double samplekurtosis(/* Real */ ae_vector* x,
ae_int_t n,
ae_state *_state);
void sampleadev(/* Real */ ae_vector* x,
ae_int_t n,
double* adev,
ae_state *_state);
void samplemedian(/* Real */ ae_vector* x,
ae_int_t n,
double* median,
ae_state *_state);
void samplepercentile(/* Real */ ae_vector* x,
ae_int_t n,
double p,
double* v,
ae_state *_state);
double cov2(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
double pearsoncorr2(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
double spearmancorr2(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
void covm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_covm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c, ae_state *_state);
void pearsoncorrm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_pearsoncorrm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c, ae_state *_state);
void spearmancorrm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_spearmancorrm(/* Real */ ae_matrix* x,
ae_int_t n,
ae_int_t m,
/* Real */ ae_matrix* c, ae_state *_state);
void covm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_covm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c, ae_state *_state);
void pearsoncorrm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_pearsoncorrm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c, ae_state *_state);
void spearmancorrm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c,
ae_state *_state);
void _pexec_spearmancorrm2(/* Real */ ae_matrix* x,
/* Real */ ae_matrix* y,
ae_int_t n,
ae_int_t m1,
ae_int_t m2,
/* Real */ ae_matrix* c, ae_state *_state);
void rankdata(/* Real */ ae_matrix* xy,
ae_int_t npoints,
ae_int_t nfeatures,
ae_state *_state);
void _pexec_rankdata(/* Real */ ae_matrix* xy,
ae_int_t npoints,
ae_int_t nfeatures, ae_state *_state);
void rankdatacentered(/* Real */ ae_matrix* xy,
ae_int_t npoints,
ae_int_t nfeatures,
ae_state *_state);
void _pexec_rankdatacentered(/* Real */ ae_matrix* xy,
ae_int_t npoints,
ae_int_t nfeatures, ae_state *_state);
double pearsoncorrelation(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
double spearmanrankcorrelation(/* Real */ ae_vector* x,
/* Real */ ae_vector* y,
ae_int_t n,
ae_state *_state);
void pearsoncorrelationsignificance(double r,
ae_int_t n,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void spearmanrankcorrelationsignificance(double r,
ae_int_t n,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void jarqueberatest(/* Real */ ae_vector* x,
ae_int_t n,
double* p,
ae_state *_state);
void mannwhitneyutest(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void onesamplesigntest(/* Real */ ae_vector* x,
ae_int_t n,
double median,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void studentttest1(/* Real */ ae_vector* x,
ae_int_t n,
double mean,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void studentttest2(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void unequalvariancettest(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void ftest(/* Real */ ae_vector* x,
ae_int_t n,
/* Real */ ae_vector* y,
ae_int_t m,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void onesamplevariancetest(/* Real */ ae_vector* x,
ae_int_t n,
double variance,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
void wilcoxonsignedranktest(/* Real */ ae_vector* x,
ae_int_t n,
double e,
double* bothtails,
double* lefttail,
double* righttail,
ae_state *_state);
}
#endif
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