org.apache.mahout.math.jet.stat
Class Probability

java.lang.Object
  extended by org.apache.mahout.math.jet.math.Constants
      extended by org.apache.mahout.math.jet.stat.Probability

public class Probability
extends Constants

Partially deprecated until unit tests are in place. Until this time, this class/interface is unsupported.


Field Summary
 
Fields inherited from class org.apache.mahout.math.jet.math.Constants
BIG, BIG_INVERSE, LOGPI, MACHEP, MAXGAM, MAXLOG, MINLOG, SQRTH, SQTPI
 
Constructor Summary
protected Probability()
          Makes this class non instantiable, but still let's others inherit from it.
 
Method Summary
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)
          Deprecated. 
static double binomial(int k, int n, double p)
          Deprecated. 
static double binomialComplemented(int k, int n, double p)
          Deprecated. 
static double chiSquare(double v, double x)
          Deprecated. 
static double chiSquareComplemented(double v, double x)
          Deprecated. 
static double errorFunction(double x)
          Deprecated. 
static double errorFunctionComplemented(double a)
          Deprecated. 
static double gamma(double alpha, double beta, double x)
          Returns the integral from zero to x of the gamma probability density function.
static double gammaComplemented(double alpha, double beta, 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)
          Deprecated. 
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)
          Deprecated. 
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)
          Deprecated. 
static double studentT(double k, double t)
          Deprecated. 
static double studentTInverse(double alpha, int size)
          Deprecated. 
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

Probability

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

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

@Deprecated
public static double betaComplemented(double a,
                                                 double b,
                                                 double x)
Deprecated. 

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

@Deprecated
public static double binomial(int k,
                                         int n,
                                         double p)
Deprecated. 

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

@Deprecated
public static double binomialComplemented(int k,
                                                     int n,
                                                     double p)
Deprecated. 

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

@Deprecated
public static double chiSquare(double v,
                                          double x)
Deprecated. 

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.

chiSquareComplemented

@Deprecated
public static double chiSquareComplemented(double v,
                                                      double x)
Deprecated. 

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.

errorFunction

@Deprecated
public static double errorFunction(double x)
Deprecated. 

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.

errorFunctionComplemented

@Deprecated
public static double errorFunctionComplemented(double a)
Deprecated. 

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.

gamma

public static double gamma(double alpha,
                           double beta,
                           double x)
Returns the integral from zero to x of the gamma probability density function.

          alpha     - x
       beta        |     alpha-1  -beta t
 y =  ---------    |    t         e        dt
       -           |
      | (alpha)   -  0
 
The incomplete gamma integral is used, according to the relation y = Gamma.incompleteGamma( alpha, beta*x ). See http://en.wikipedia.org/wiki/Gamma_distribution#Probability_density_function

Parameters:
alpha - the shape parameter of the gamma distribution.
beta - the rate parameter of the gamma distribution.
x - integration end point.

gammaComplemented

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

y = Gamma.incompleteGammaComplement( b, a*x ). TODO this method is inconsistent with gamma(alpha, beta, x)

Parameters:
alpha - the shape parameter of the gamma distribution.
beta - the rate parameter 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

@Deprecated
public static double negativeBinomialComplemented(int k,
                                                             int n,
                                                             double p)
Deprecated. 

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)
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.

Computed using method 26.2.17 from Abramovitz and Stegun (see http://www.math.sfu.ca/~cbm/aands/page_932.htm and http://en.wikipedia.org/wiki/Normal_distribution#Numerical_approximations_of_the_normal_cdf


normal

public 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.
                            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.

normalInverse

@Deprecated
public static double normalInverse(double y0)
Deprecated. 

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)).


poisson

public static double poisson(int k,
                             double mean)
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.

poissonComplemented

@Deprecated
public static double poissonComplemented(int k,
                                                    double mean)
Deprecated. 

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.

studentT

@Deprecated
public static double studentT(double k,
                                         double t)
Deprecated. 

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.

studentTInverse

@Deprecated
public static double studentTInverse(double alpha,
                                                int size)
Deprecated. 

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


Copyright © 2008-2010 The Apache Software Foundation. All Rights Reserved.