weka.core
Class Statistics

java.lang.Object
  weka.core.Statistics

public class Statistics

extends java.lang.Object

Class implementing some distributions, tests, etc. The code is mostly adapted from the CERN Jet Java libraries: Copyright © 2001 University of Waikato Copyright © 1999 CERN - European Organization for Nuclear Research. Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose is hereby granted without fee, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation. CERN and the University of Waikato make no representations about the suitability of this software for any purpose. It is provided "as is" without expressed or implied warranty.

Version:

$Revision: 1.7 $

Author:

peter.gedeck@pharma.Novartis.com, wolfgang.hoschek@cern.ch, Eibe Frank (eibe@cs.waikato.ac.nz), Richard Kirkby (rkirkby@cs.waikato.ac.nz)

Field Summary

protected static double	big
protected static double	biginv
protected static double	LOGPI
protected static double	MACHEP Some constants
protected static double	MAXGAM
protected static double	MAXLOG
protected static double	MINLOG
protected static double[]	P0 COEFFICIENTS FOR METHOD normalInverse() *
protected static double[]	P1
protected static double[]	P2
protected static double[]	Q0
protected static double[]	Q1
protected static double[]	Q2
protected static double	SQRTH
protected static double	SQTPI

Constructor Summary

Statistics()

Method Summary

static double	binomialStandardError(double p, int n) Computes standard error for observed values of a binomial random variable.
static double	chiSquaredProbability(double x, double v) Returns chi-squared probability for given value and degrees of freedom.
(package private) static double	errorFunction(double x) Returns the error function of the normal distribution.
(package private) static double	errorFunctionComplemented(double a) Returns the complementary Error function of the normal distribution.
static double	FProbability(double F, int df1, int df2) Computes probability of F-ratio.
(package private) static double	gamma(double x) Returns the Gamma function of the argument.
static double	incompleteBeta(double aa, double bb, double xx) Returns the Incomplete Beta Function evaluated from zero to xx.
(package private) static double	incompleteBetaFraction1(double a, double b, double x) Continued fraction expansion #1 for incomplete beta integral.
(package private) static double	incompleteBetaFraction2(double a, double b, double x) Continued fraction expansion #2 for incomplete beta integral.
(package private) static double	incompleteGamma(double a, double x) Returns the Incomplete Gamma function.
(package private) static double	incompleteGammaComplement(double a, double x) Returns the Complemented Incomplete Gamma function.
static double	lnGamma(double x) Returns natural logarithm of gamma function.
static void	main(java.lang.String[] ops) Main method for testing this class.
static double	normalInverse(double y0) Returns the value, x, for which the area under the Normal (Gaussian) probability density function (integrated from minus infinity to x) is equal to the argument y (assumes mean is zero, variance is one).
static double	normalProbability(double a) Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x (assumes mean is zero, variance is one).
(package private) static double	p1evl(double x, double[] coef, int N) Evaluates the given polynomial of degree N at x.
(package private) static double	polevl(double x, double[] coef, int N) Evaluates the given polynomial of degree N at x.
(package private) static double	powerSeries(double a, double b, double x) Power series for incomplete beta integral.
(package private) static double	stirlingFormula(double x) Returns the Gamma function computed by Stirling's formula.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

MACHEP

protected static final double MACHEP

Some constants

See Also:

Constant Field Values

MAXLOG

protected static final double MAXLOG

See Also:

Constant Field Values

MINLOG

protected static final double MINLOG

See Also:

Constant Field Values

MAXGAM

protected static final double MAXGAM

See Also:

Constant Field Values

SQTPI

protected static final double SQTPI

See Also:

Constant Field Values

SQRTH

protected static final double SQRTH

See Also:

Constant Field Values

LOGPI

protected static final double LOGPI

See Also:

Constant Field Values

big

protected static final double big

See Also:

Constant Field Values

biginv

protected static final double biginv

See Also:

Constant Field Values

P0

protected static final double[] P0

COEFFICIENTS FOR METHOD normalInverse() *

Q0

protected static final double[] Q0

P1

protected static final double[] P1

Q1

protected static final double[] Q1

P2

protected static final double[] P2

Q2

protected static final double[] Q2

Constructor Detail

Statistics

public Statistics()

Method Detail

binomialStandardError

public static double binomialStandardError(double p,
                                           int n)

Computes standard error for observed values of a binomial random variable.

Parameters:

p - the probability of success

n - the size of the sample

Returns:

the standard error

chiSquaredProbability

public static double chiSquaredProbability(double x,
                                           double v)

Returns chi-squared probability for given value and degrees of freedom. (The probability that the chi-squared variate will be greater than x for the given degrees of freedom.)

Parameters:

x - the value

Returns:

the chi-squared probability

FProbability

public static double FProbability(double F,
                                  int df1,
                                  int df2)

Computes probability of F-ratio.

Parameters:

F - the F-ratio

df1 - the first number of degrees of freedom

df2 - the second number of degrees of freedom

Returns:

the probability of the F-ratio.

normalProbability

public static double normalProbability(double a)

Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x (assumes mean is zero, variance is one).

                            x
                             -
                   1        | |          2
  normal(x)  = ---------    |    exp( - t /2 ) dt
               sqrt(2pi)  | |
                           -
                          -inf.

             =  ( 1 + erf(z) ) / 2
             =  erfc(z) / 2
 

where z = x/sqrt(2). Computation is via the functions errorFunction and errorFunctionComplement.

Parameters:

a - the z-value

Returns:

the probability of the z value according to the normal pdf

normalInverse

public static double normalInverse(double y0)

Returns the value, x, for which the area under the Normal (Gaussian) probability density function (integrated from minus infinity to x) is equal to the argument y (assumes mean is zero, variance is one).

For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). There are two rational functions P/Q, one for 0 < y < exp(-32) and the other for y up to exp(-2). For larger arguments, w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

Parameters:

y0 - the area under the normal pdf

Returns:

the z-value

lnGamma

public static double lnGamma(double x)

Returns natural logarithm of gamma function.

Parameters:

x - the value

Returns:

natural logarithm of gamma function

errorFunction

static double errorFunction(double x)

Returns the error function of the normal distribution. The integral is

                           x 
                            -
                 2         | |          2
   erf(x)  =  --------     |    exp( - t  ) dt.
              sqrt(pi)   | |
                          -
                           0
 

Implementation: For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise erf(x) = 1 - erfc(x).

Code adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C).

errorFunctionComplemented

static double errorFunctionComplemented(double a)

Returns the complementary Error function of the normal distribution.

  1 - erf(x) =

                           inf. 
                             -
                  2         | |          2
   erfc(x)  =  --------     |    exp( - t  ) dt
               sqrt(pi)   | |
                           -
                            x
 

Implementation: For small x, erfc(x) = 1 - erf(x); otherwise rational approximations are computed.

Code adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C).

Parameters:

a - the argument to the function.

p1evl

static double p1evl(double x,
                    double[] coef,
                    int N)

Evaluates the given polynomial of degree N at x. Evaluates polynomial when coefficient of N is 1.0. Otherwise same as polevl().

                     2          N
 y  =  C  + C x + C x  +...+ C x
        0    1     2          N

 Coefficients are stored in reverse order:

 coef[0] = C  , ..., coef[N] = C  .
            N                   0
 

The function p1evl() assumes that coef[N] = 1.0 and is omitted from the array. Its calling arguments are otherwise the same as polevl().

In the interest of speed, there are no checks for out of bounds arithmetic.

Parameters:

x - argument to the polynomial.

coef - the coefficients of the polynomial.

N - the degree of the polynomial.

polevl

static double polevl(double x,
                     double[] coef,
                     int N)

Evaluates the given polynomial of degree N at x.

                     2          N
 y  =  C  + C x + C x  +...+ C x
        0    1     2          N

 Coefficients are stored in reverse order:

 coef[0] = C  , ..., coef[N] = C  .
            N                   0
 

In the interest of speed, there are no checks for out of bounds arithmetic.

Parameters:

x - argument to the polynomial.

coef - the coefficients of the polynomial.

N - the degree of the polynomial.

incompleteGamma

static double incompleteGamma(double a,
                              double x)

Returns the Incomplete Gamma function.

Parameters:

a - the parameter of the gamma distribution.

x - the integration end point.

incompleteGammaComplement

static double incompleteGammaComplement(double a,
                                        double x)

Returns the Complemented Incomplete Gamma function.

Parameters:

a - the parameter of the gamma distribution.

x - the integration start point.

gamma

static double gamma(double x)

Returns the Gamma function of the argument.

stirlingFormula

static double stirlingFormula(double x)

Returns the Gamma function computed by Stirling's formula. The polynomial STIR is valid for 33 <= x <= 172.

incompleteBeta

public static double incompleteBeta(double aa,
                                    double bb,
                                    double xx)

Returns the Incomplete Beta Function evaluated from zero to xx.

Parameters:

aa - the alpha parameter of the beta distribution.

bb - the beta parameter of the beta distribution.

xx - the integration end point.

incompleteBetaFraction1

static double incompleteBetaFraction1(double a,
                                      double b,
                                      double x)

Continued fraction expansion #1 for incomplete beta integral.

incompleteBetaFraction2

static double incompleteBetaFraction2(double a,
                                      double b,
                                      double x)

Continued fraction expansion #2 for incomplete beta integral.

powerSeries

static double powerSeries(double a,
                          double b,
                          double x)

Power series for incomplete beta integral. Use when b*x is small and x not too close to 1.

main

public static void main(java.lang.String[] ops)

Main method for testing this class.