WSJT-X/boost/libs/math/doc/distributions/geometric.qbk

[section:geometric_dist Geometric Distribution]

``#include <boost/math/distributions/geometric.hpp>``

   namespace boost{ namespace math{ 
   
   template <class RealType = double, 
             class ``__Policy``   = ``__policy_class`` >
   class geometric_distribution;
   
   typedef geometric_distribution<> geometric;
   
   template <class RealType, class ``__Policy``>
   class geometric_distribution
   {
   public:
      typedef RealType value_type;
      typedef Policy   policy_type;
      // Constructor from success_fraction:
      geometric_distribution(RealType p);
      
      // Parameter accessors:
      RealType success_fraction() const;
      RealType successes() const;
     
      // Bounds on success fraction:
      static RealType find_lower_bound_on_p(
         RealType trials, 
         RealType successes,
         RealType probability); // alpha
      static RealType find_upper_bound_on_p(
         RealType trials, 
         RealType successes,
         RealType probability); // alpha
         
      // Estimate min/max number of trials:
      static RealType find_minimum_number_of_trials(
         RealType k,     // Number of failures.
         RealType p,     // Success fraction.
         RealType probability); // Probability threshold alpha.
      static RealType find_maximum_number_of_trials(
         RealType k,     // Number of failures.
         RealType p,     // Success fraction.
         RealType probability); // Probability threshold alpha.
   };
   
   }} // namespaces
   
The class type `geometric_distribution` represents a
[@http://en.wikipedia.org/wiki/geometric_distribution geometric distribution]:
it is used when there are exactly two mutually exclusive outcomes of a
[@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trial]:
these outcomes are labelled "success" and "failure".

For [@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials]
each with success fraction /p/, the geometric distribution gives
the probability of observing /k/ trials (failures, events, occurrences, or arrivals)
before the first success. 

[note For this implementation, the set of trials *includes zero*
(unlike another definition where the set of trials starts at one, sometimes named /shifted/).]
The geometric distribution assumes that success_fraction /p/ is fixed for all /k/ trials.

The probability that there are /k/ failures before the first success is

__spaces Pr(Y=/k/) = (1-/p/)[super /k/]/p/

For example, when throwing a 6-face dice the success probability /p/ = 1/6 = 0.1666[recur][space].
Throwing repeatedly until a /three/ appears,
the probability distribution of the number of times /not-a-three/ is thrown
is geometric. 

Geometric distribution has the Probability Density Function PDF:

__spaces (1-/p/)[super /k/]/p/

The following graph illustrates how the PDF and CDF vary for three examples
of the success fraction /p/, 
(when considering the geometric distribution as a continuous function),

[graph geometric_pdf_2]

[graph geometric_cdf_2]

and as discrete. 

[graph geometric_pdf_discrete]

[graph geometric_cdf_discrete]


[h4 Related Distributions]

The geometric distribution is a special case of
the __negative_binomial_distrib with successes parameter /r/ = 1,
so only one first and only success is required : thus by definition
__spaces `geometric(p) == negative_binomial(1, p)`

  negative_binomial_distribution(RealType r, RealType success_fraction);
  negative_binomial nb(1, success_fraction);
  geometric g(success_fraction);
  ASSERT(pdf(nb, 1) == pdf(g, 1));

This implementation uses real numbers for the computation throughout
(because it uses the *real-valued* power and exponential functions).
So to obtain a conventional strictly-discrete geometric distribution
you must ensure that an integer value is provided for the number of trials 
(random variable) /k/,
and take integer values (floor or ceil functions) from functions that return 
a number of successes.

[discrete_quantile_warning geometric]
   
[h4 Member Functions]

[h5 Constructor]

   geometric_distribution(RealType p);

Constructor: /p/ or success_fraction is the probability of success of a single trial.

Requires: `0 <= p <= 1`.

[h5 Accessors]

   RealType success_fraction() const; // successes / trials (0 <= p <= 1)
   
Returns the success_fraction parameter /p/ from which this distribution was constructed.
   
   RealType successes() const; // required successes always one,
   // included for compatibility with negative binomial distribution
   // with successes r == 1.
   
Returns unity.

The following functions are equivalent to those provided for the negative binomial,
with successes = 1, but are provided here for completeness.

The best method of calculation for the following functions is disputed:
see __binomial_distrib and __negative_binomial_distrib for more discussion. 

[h5 Lower Bound on success_fraction Parameter ['p]]

      static RealType find_lower_bound_on_p(
        RealType failures, 
        RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
      
Returns a *lower bound* on the success fraction:

[variablelist
[[failures][The total number of failures before the 1st success.]]
[[alpha][The largest acceptable probability that the true value of
         the success fraction is [*less than] the value returned.]]
]

For example, if you observe /k/ failures from /n/ trials
the best estimate for the success fraction is simply 1/['n], but if you
want to be 95% sure that the true value is [*greater than] some value, 
['p[sub min]], then:

   p``[sub min]`` = geometric_distribution<RealType>::
      find_lower_bound_on_p(failures, 0.05);
                       
[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.]
      
This function uses the Clopper-Pearson method of computing the lower bound on the
success fraction, whilst many texts refer to this method as giving an "exact" 
result in practice it produces an interval that guarantees ['at least] the
coverage required, and may produce pessimistic estimates for some combinations
of /failures/ and /successes/.  See:

[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].

[h5 Upper Bound on success_fraction Parameter p]

   static RealType find_upper_bound_on_p(
      RealType trials, 
      RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
      
Returns an *upper bound* on the success fraction:

[variablelist
[[trials][The total number of trials conducted.]]
[[alpha][The largest acceptable probability that the true value of
         the success fraction is [*greater than] the value returned.]]
]

For example, if you observe /k/ successes from /n/ trials the
best estimate for the success fraction is simply ['k/n], but if you
want to be 95% sure that the true value is [*less than] some value, 
['p[sub max]], then:

   p``[sub max]`` = geometric_distribution<RealType>::find_upper_bound_on_p(
                       k, 0.05);

[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.]

This function uses the Clopper-Pearson method of computing the lower bound on the
success fraction, whilst many texts refer to this method as giving an "exact" 
result in practice it produces an interval that guarantees ['at least] the
coverage required, and may produce pessimistic estimates for some combinations
of /failures/ and /successes/.  See:

[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].

[h5 Estimating Number of Trials to Ensure at Least a Certain Number of Failures]

   static RealType find_minimum_number_of_trials(
      RealType k,     // number of failures.
      RealType p,     // success fraction.
      RealType alpha); // probability threshold (0.05 equivalent to 95%).
      
This functions estimates the number of trials required to achieve a certain
probability that [*more than ['k] failures will be observed].

[variablelist
[[k][The target number of failures to be observed.]]
[[p][The probability of ['success] for each trial.]]
[[alpha][The maximum acceptable ['risk] that only ['k] failures or fewer will be observed.]]
]

For example:
   
   geometric_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
      
Returns the smallest number of trials we must conduct to be 95% (1-0.05) sure
of seeing 10 failures that occur with frequency one half.
   
[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.]

This function uses numeric inversion of the geometric distribution
to obtain the result: another interpretation of the result is that it finds
the number of trials (failures) that will lead to an /alpha/ probability
of observing /k/ failures or fewer.

[h5 Estimating Number of Trials to Ensure a Maximum Number of Failures or Less]

   static RealType find_maximum_number_of_trials(
      RealType k,     // number of failures.
      RealType p,     // success fraction.
      RealType alpha); // probability threshold (0.05 equivalent to 95%).
      
This functions estimates the maximum number of trials we can conduct and achieve
a certain probability that [*k failures or fewer will be observed].

[variablelist
[[k][The maximum number of failures to be observed.]]
[[p][The probability of ['success] for each trial.]]
[[alpha][The maximum acceptable ['risk] that more than ['k] failures will be observed.]]
]

For example:
   
   geometric_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05);
      
Returns the largest number of trials we can conduct and still be 95% sure
of seeing no failures that occur with frequency one in one million.
   
This function uses numeric inversion of the geometric distribution
to obtain the result: another interpretation of the result, is that it finds
the number of trials that will lead to an /alpha/ probability
of observing more than k failures.

[h4 Non-member Accessors]

All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
that are generic to all distributions are supported: __usual_accessors.

However it's worth taking a moment to define what these actually mean in 
the context of this distribution:

[table Meaning of the non-member accessors.
[[Function][Meaning]]
[[__pdf]
   [The probability of obtaining [*exactly k failures] from /k/ trials
   with success fraction p.  For example:

``pdf(geometric(p), k)``]]
[[__cdf]
   [The probability of obtaining [*k failures or fewer] from /k/ trials
   with success fraction p and success on the last trial.  For example:

``cdf(geometric(p), k)``]]
[[__ccdf]
   [The probability of obtaining [*more than k failures] from /k/ trials
   with success fraction p and success on the last trial.  For example:
   
``cdf(complement(geometric(p), k))``]]
[[__quantile]
   [The [*greatest] number of failures /k/ expected to be observed from /k/ trials
   with success fraction /p/, at probability /P/.  Note that the value returned
   is a real-number, and not an integer.  Depending on the use case you may
   want to take either the floor or ceiling of the real result.  For example:
``quantile(geometric(p), P)``]]
[[__quantile_c]
   [The [*smallest] number of failures /k/ expected to be observed from /k/ trials
   with success fraction /p/, at probability /P/.  Note that the value returned
   is a real-number, and not an integer.  Depending on the use case you may
   want to take either the floor or ceiling of the real result. For example:
   ``quantile(complement(geometric(p), P))``]]
]

[h4 Accuracy]

This distribution is implemented using the pow and exp functions, so most results
are accurate within a few epsilon for the RealType.
For extreme values of `double` /p/, for example 0.9999999999,
accuracy can fall significantly, for example to 10 decimal digits (from 16).

[h4 Implementation]

In the following table, /p/ is the probability that any one trial will
be successful (the success fraction), /k/ is the number of failures,
/p/ is the probability and /q = 1-p/,
/x/ is the given probability to estimate 
the expected number of failures using the quantile.

[table
[[Function][Implementation Notes]]
[[pdf][pdf =  p * pow(q, k)]]
[[cdf][cdf = 1 - q[super k=1]]]
[[cdf complement][exp(log1p(-p) * (k+1))]]
[[quantile][k = log1p(-x) / log1p(-p) -1]]
[[quantile from the complement][k = log(x) / log1p(-p) -1]]
[[mean][(1-p)/p]]
[[variance][(1-p)/p[sup2]]]
[[mode][0]]
[[skewness][(2-p)/[sqrt]q]]
[[kurtosis][9+p[sup2]/q]]
[[kurtosis excess][6 +p[sup2]/q]]
[[parameter estimation member functions][See __negative_binomial_distrib]]
[[`find_lower_bound_on_p`][See __negative_binomial_distrib]]
[[`find_upper_bound_on_p`][See __negative_binomial_distrib]]
[[`find_minimum_number_of_trials`][See __negative_binomial_distrib]]
[[`find_maximum_number_of_trials`][See __negative_binomial_distrib]]
]

[endsect][/section:geometric_dist geometric]

[/ geometric.qbk
  Copyright 2010 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]