Package moa.core

Class Statistics


  • public class Statistics
    extends 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: 5616 $
    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

      Fields 
      Modifier and Type Field Description
      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

      Constructors 
      Constructor Description
      Statistics()  
    • Method Summary

      All Methods Static Methods Concrete Methods 
      Modifier and Type Method Description
      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.
      static double errorFunction​(double x)
      Returns the error function of the normal distribution.
      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.
      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.
      static double incompleteBetaFraction1​(double a, double b, double x)
      Continued fraction expansion #1 for incomplete beta integral.
      static double incompleteBetaFraction2​(double a, double b, double x)
      Continued fraction expansion #2 for incomplete beta integral.
      static double incompleteGamma​(double a, double x)
      Returns the Incomplete Gamma function.
      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 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).
      static double p1evl​(double x, double[] coef, int N)
      Evaluates the given polynomial of degree N at x.
      static double polevl​(double x, double[] coef, int N)
      Evaluates the given polynomial of degree N at x.
      static double powerSeries​(double a, double b, double x)
      Power series for incomplete beta integral.
      static double stirlingFormula​(double x)
      Returns the Gamma function computed by Stirling's formula.
    • 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
        v - the number of degrees of freedom
        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

        public 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).

        Parameters:
        x - the argument to the function.
      • errorFunctionComplemented

        public 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

        public 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

        public 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

        public 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

        public 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

        public static double gamma​(double x)
        Returns the Gamma function of the argument.
      • stirlingFormula

        public 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

        public static double incompleteBetaFraction1​(double a,
                                                     double b,
                                                     double x)
        Continued fraction expansion #1 for incomplete beta integral.
      • incompleteBetaFraction2

        public static double incompleteBetaFraction2​(double a,
                                                     double b,
                                                     double x)
        Continued fraction expansion #2 for incomplete beta integral.
      • powerSeries

        public 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.