Class Probability

java.lang.Object
math.Constants
math.Probability
public class Probability extends Constants
Custom tailored numerical integration of certain probability distributions.

Implementation: Some code taken and adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C). Most Cephes code (missing from the 2D Graph Package) directly ported.

Version:
0.91, 08-Dec-99
Author:
peter.gedeck@pharma.Novartis.com, wolfgang.hoschek@cern.ch

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    protected static final double[]
    P0
    COEFFICIENTS FOR METHOD normalInverse() *
    protected static final double[]
    P1
     
    protected static final double[]
    P2
     
    protected static final double[]
    Q0
     
    protected static final double[]
    Q1
     
    protected static final double[]
    Q2
     

    Fields inherited from class Constants

    big, biginv, LOGPI, MACHEP, MAXGAM, MAXLOG, MINLOG, SQRTH, SQTPI

  • Constructor Summary

    Constructors
    Modifier
    Constructor
    Description
    protected
    Probability()
    Makes this class non instantiable, but still let's others inherit from it.

  • Method Summary

    Modifier and Type
    Method
    Description
    static double
    beta(double a, double b, double x)
    Returns the area from zero to x under the beta density function.
    static double
    betaComplemented(double a, double b, double x)
    Returns the area under the right hand tail (from x to infinity) of the beta density function.
    static double
    binomial(int k, int n, double p)
    Returns the sum of the terms 0 through k of the Binomial probability density.
    static double
    binomialComplemented(int k, int n, double p)
    Returns the sum of the terms k+1 through n of the Binomial probability density.
    static double
    chiSquare(double v, double x)
    Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom.
    static double
    chiSquareComplemented(double v, double x)
    Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom.
    static double
    errorFunction(double x)
    Returns the error function of the normal distribution; formerly named erf.
    static double
    errorFunctionComplemented(double a)
    Returns the complementary Error function of the normal distribution; formerly named erfc.
    static double
    gamma(double a, double b, double x)
    Returns the integral from zero to x of the gamma probability density function.
    static double
    gammaComplemented(double a, double b, double x)
    Returns the integral from x to infinity of the gamma probability density function:
    static double
    negativeBinomial(int k, int n, double p)
    Returns the sum of the terms 0 through k of the Negative Binomial Distribution.
    static double
    negativeBinomialComplemented(int k, int n, double p)
    Returns the sum of the terms k+1 to infinity of the Negative Binomial distribution.
    static double
    normal(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
    normal(double mean, double variance, double x)
    Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x.
    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); formerly named ndtri.
    static double
    poisson(int k, double mean)
    Returns the sum of the first k terms of the Poisson distribution.
    static double
    poissonComplemented(int k, double mean)
    Returns the sum of the terms k+1 to Infinity of the Poisson distribution.
    static double
    studentT(double k, double t)
    Returns the integral from minus infinity to t of the Student-t distribution with k > 0 degrees of freedom.
    static double
    studentTInverse(double alpha, int size)
    Returns the value, t, for which the area under the Student-t probability density function (integrated from minus infinity to t) is equal to 1-alpha/2.

    Methods inherited from class Object

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

  • Field Details

    • 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 Details

    • Probability

      protected Probability()
      Makes this class non instantiable, but still let's others inherit from it.

  • Method Details

    • beta

      public static double beta(double a, double b, double x)
      Returns the area from zero to x under the beta density function. 
                               x
           -             -
          | (a+b)       | |  a-1      b-1
P(x)  =  ----------     |   t    (1-t)    dt
          -     -     | |
         | (a) | (b)   -
                        0
      This function is identical to the incomplete beta integral function Gamma.incompleteBeta(a, b, x). The complemented function is 1 - P(1-x) = Gamma.incompleteBeta( b, a, x );

    • betaComplemented

      public static double betaComplemented(double a, double b, double x)
      Returns the area under the right hand tail (from x to infinity) of the beta density function. This function is identical to the incomplete beta integral function Gamma.incompleteBeta(b, a, x).

    • binomial

      public static double binomial(int k, int n, double p)
      Returns the sum of the terms 0 through k of the Binomial probability density. 
        k
  --  ( n )   j      n-j
  >   (   )   p     (1-p)
  --  ( j )
 j=0
      The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula

      y = binomial( k, n, p ) = Gamma.incompleteBeta( n-k, k+1, 1-p ).

      All arguments must be positive,

      Parameters:
      k - end term.
      n - the number of trials.
      p - the probability of success (must be in (0.0,1.0)).

    • binomialComplemented

      public static double binomialComplemented(int k, int n, double p)
      Returns the sum of the terms k+1 through n of the Binomial probability density. 
        n
  --  ( n )   j      n-j
  >   (   )   p     (1-p)
  --  ( j )
 j=k+1
      The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula

      y = binomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n-k, p ).

      All arguments must be positive,

      Parameters:
      k - end term.
      n - the number of trials.
      p - the probability of success (must be in (0.0,1.0)).

    • chiSquare

      public static double chiSquare(double v, double x) throws ArithmeticException
      Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom. 
                                       inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x
      where x is the Chi-square variable.

      The incomplete gamma integral is used, according to the formula

      y = chiSquare( v, x ) = incompleteGamma( v/2.0, x/2.0 ).

      The arguments must both be positive.

      Parameters:
      v - degrees of freedom.
      x - integration end point.
      Throws:
      ArithmeticException

    • chiSquareComplemented

      public static double chiSquareComplemented(double v, double x) throws ArithmeticException
      Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom. 
                                       inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x
      where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chiSquareComplemented( v, x ) = incompleteGammaComplement( v/2.0, x/2.0 ). The arguments must both be positive.
      Parameters:
      v - degrees of freedom.
      Throws:
      ArithmeticException

    • errorFunction

      public static double errorFunction(double x) throws ArithmeticException
      Returns the error function of the normal distribution; formerly named erf. 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.
      Throws:
      ArithmeticException

    • errorFunctionComplemented

      public static double errorFunctionComplemented(double a) throws ArithmeticException
      Returns the complementary Error function of the normal distribution; formerly named erfc. 
       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.
      Throws:
      ArithmeticException

    • gamma

      public static double gamma(double a, double b, double x)
      Returns the integral from zero to x of the gamma probability density function. 
                     x
       b       -
      a       | |   b-1  -at
y =  -----    |    t    e    dt
      -     | |
     | (b)   -
              0
      The incomplete gamma integral is used, according to the relation y = Gamma.incompleteGamma( b, a*x ).
      Parameters:
      a - the paramater a (alpha) of the gamma distribution.
      b - the paramater b (beta, lambda) of the gamma distribution.
      x - integration end point.

    • gammaComplemented

      public static double gammaComplemented(double a, double b, double x)
      Returns the integral from x to infinity of the gamma probability density function: 
                    inf.
       b       -
      a       | |   b-1  -at
y =  -----    |    t    e    dt
      -     | |
     | (b)   -
              x
      The incomplete gamma integral is used, according to the relation

      y = Gamma.incompleteGammaComplement( b, a*x ).

      Parameters:
      a - the paramater a (alpha) of the gamma distribution.
      b - the paramater b (beta, lambda) of the gamma distribution.
      x - integration end point.

    • negativeBinomial

      public static double negativeBinomial(int k, int n, double p)
      Returns the sum of the terms 0 through k of the Negative Binomial Distribution. 
        k
  --  ( n+j-1 )   n      j
  >   (       )   p     (1-p)
  --  (   j   )
 j=0
      In a sequence of Bernoulli trials, this is the probability that k or fewer failures precede the n-th success.

      The terms are not computed individually; instead the incomplete beta integral is employed, according to the formula

      y = negativeBinomial( k, n, p ) = Gamma.incompleteBeta( n, k+1, p ). All arguments must be positive,

      Parameters:
      k - end term.
      n - the number of trials.
      p - the probability of success (must be in (0.0,1.0)).

    • negativeBinomialComplemented

      public static double negativeBinomialComplemented(int k, int n, double p)
      Returns the sum of the terms k+1 to infinity of the Negative Binomial distribution. 
        inf
  --  ( n+j-1 )   n      j
  >   (       )   p     (1-p)
  --  (   j   )
 j=k+1
      The terms are not computed individually; instead the incomplete beta integral is employed, according to the formula

      y = negativeBinomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n, 1-p ). All arguments must be positive,

      Parameters:
      k - end term.
      n - the number of trials.
      p - the probability of success (must be in (0.0,1.0)).

    • normal

      public static double normal(double a) throws ArithmeticException
      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.
      Throws:
      ArithmeticException

    • normal

      public static double normal(double mean, double variance, double x) throws ArithmeticException
      Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x. 
                                 x
                            -
                  1        | |                 2
 normal(x)  = ---------    |    exp( - (t-mean) / 2v ) dt
              sqrt(2pi*v)| |
                          -
                         -inf.
      where v = variance. Computation is via the functions errorFunction.
      Parameters:
      mean - the mean of the normal distribution.
      variance - the variance of the normal distribution.
      x - the integration limit.
      Throws:
      ArithmeticException

    • normalInverse

      public static double normalInverse(double y0) throws ArithmeticException
      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); formerly named ndtri.

      For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * getLog(y) ); then the approximation is x = z - getLog(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)).

      Throws:
      ArithmeticException

    • poisson

      public static double poisson(int k, double mean) throws ArithmeticException
      Returns the sum of the first k terms of the Poisson distribution. 
        k         j
  --   -m  m
  >   e    --
  --       j!
 j=0
      The terms are not summed directly; instead the incomplete gamma integral is employed, according to the relation

      y = poisson( k, m ) = Gamma.incompleteGammaComplement( k+1, m ). The arguments must both be positive.

      Parameters:
      k - number of terms.
      mean - the mean of the poisson distribution.
      Throws:
      ArithmeticException

    • poissonComplemented

      public static double poissonComplemented(int k, double mean) throws ArithmeticException
      Returns the sum of the terms k+1 to Infinity of the Poisson distribution. 
       inf.       j
  --   -m  m
  >   e    --
  --       j!
 j=k+1
      The terms are not summed directly; instead the incomplete gamma integral is employed, according to the formula

      y = poissonComplemented( k, m ) = Gamma.incompleteGamma( k+1, m ). The arguments must both be positive.

      Parameters:
      k - start term.
      mean - the mean of the poisson distribution.
      Throws:
      ArithmeticException

    • studentT

      public static double studentT(double k, double t) throws ArithmeticException
      Returns the integral from minus infinity to t of the Student-t distribution with k > 0 degrees of freedom. 
                                           t
                                     -
                                    | |
             -                      |         2   -(k+1)/2
            | ( (k+1)/2 )           |  (     x   )
      ----------------------        |  ( 1 + --- )        dx
                    -               |  (      k  )
      sqrt( k pi ) | ( k/2 )        |
                                  | |
                                   -
                                  -inf.
      Relation to incomplete beta integral:

      1 - studentT(k,t) = 0.5 * Gamma.incompleteBeta( k/2, 1/2, z ) where z = k/(k + t**2).

      Since the function is symmetric about t=0, the area under the right tail of the density is found by calling the function with -t instead of t.

      Parameters:
      k - degrees of freedom.
      t - integration end point.
      Throws:
      ArithmeticException

    • studentTInverse

      public static double studentTInverse(double alpha, int size)
      Returns the value, t, for which the area under the Student-t probability density function (integrated from minus infinity to t) is equal to 1-alpha/2. The value returned corresponds to usual Student t-distribution lookup table for talpha[size].

      The function uses the studentT function to determine the return value iteratively.

      Parameters:
      alpha - probability
      size - size of data set