Probability
cern.jet.stat

Class Probability



  • public class Probabilityextends 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.
    • Method Summary

      Methods 
      Modifier and TypeMethod and Description
      static doublebeta(double a, double b, double x)
      Returns the area from zero to x under the beta density function.
      static doublebetaComplemented(double a, double b, double x)
      Returns the area under the right hand tail (from x to infinity) of the beta density function.
      static doublebinomial(int k, int n, double p)
      Returns the sum of the terms 0 through k of the Binomial probability density.
      static doublebinomialComplemented(int k, int n, double p)
      Returns the sum of the terms k+1 through n of the Binomial probability density.
      static doublechiSquare(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 doublechiSquareComplemented(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 doubleerrorFunction(double x)
      Returns the error function of the normal distribution; formerly named erf.
      static doubleerrorFunctionComplemented(double a)
      Returns the complementary Error function of the normal distribution; formerly named erfc.
      static doublegamma(double a, double b, double x)
      Returns the integral from zero to x of the gamma probability density function.
      static doublegammaComplemented(double a, double b, double x)
      Returns the integral from x to infinity of the gamma probability density function:
      static doublenegativeBinomial(int k, int n, double p)
      Returns the sum of the terms 0 through k of the Negative Binomial Distribution.
      static doublenegativeBinomialComplemented(int k, int n, double p)
      Returns the sum of the terms k+1 to infinity of the Negative Binomial distribution.
      static doublenormal(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 doublenormal(double mean, double variance, double x)
      Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x.
      static doublenormalInverse(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 doublepoisson(int k, double mean)
      Returns the sum of the first k terms of the Poisson distribution.
      static doublepoissonComplemented(int k, double mean)
      Returns the sum of the terms k+1 to Infinity of the Poisson distribution.
      static doublestudentT(double k, double t)
      Returns the integral from minus infinity to t of the Student-t distribution with k > 0 degrees of freedom.
      static doublestudentTInverse(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.
    • Method Detail

      • 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:
        a - 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 * 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)).

        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

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