Squashed 'third_party/boostorg/math/' content from commit 0e9549f
Change-Id: I7c2a13cb6a5beea4a471341510d8364cedd71613
git-subtree-dir: third_party/boostorg/math
git-subtree-split: 0e9549ff2f854e6edafaf4627d65026f2f533a18
diff --git a/doc/distributions/binomial.qbk b/doc/distributions/binomial.qbk
new file mode 100644
index 0000000..eae3e83
--- /dev/null
+++ b/doc/distributions/binomial.qbk
@@ -0,0 +1,404 @@
+[section:binomial_dist Binomial Distribution]
+
+``#include <boost/math/distributions/binomial.hpp>``
+
+ namespace boost{ namespace math{
+
+ template <class RealType = double,
+ class ``__Policy`` = ``__policy_class`` >
+ class binomial_distribution;
+
+ typedef binomial_distribution<> binomial;
+
+ template <class RealType, class ``__Policy``>
+ class binomial_distribution
+ {
+ public:
+ typedef RealType value_type;
+ typedef Policy policy_type;
+
+ static const ``['unspecified-type]`` clopper_pearson_exact_interval;
+ static const ``['unspecified-type]`` jeffreys_prior_interval;
+
+ // construct:
+ binomial_distribution(RealType n, RealType p);
+
+ // parameter access::
+ RealType success_fraction() const;
+ RealType trials() const;
+
+ // Bounds on success fraction:
+ static RealType find_lower_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType probability,
+ ``['unspecified-type]`` method = clopper_pearson_exact_interval);
+ static RealType find_upper_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType probability,
+ ``['unspecified-type]`` method = clopper_pearson_exact_interval);
+
+ // estimate min/max number of trials:
+ static RealType find_minimum_number_of_trials(
+ RealType k, // number of events
+ RealType p, // success fraction
+ RealType alpha); // risk level
+
+ static RealType find_maximum_number_of_trials(
+ RealType k, // number of events
+ RealType p, // success fraction
+ RealType alpha); // risk level
+ };
+
+ }} // namespaces
+
+The class type `binomial_distribution` represents a
+[@http://mathworld.wolfram.com/BinomialDistribution.html binomial distribution]:
+it is used when there are exactly two mutually
+exclusive outcomes of a trial. These outcomes are labelled
+"success" and "failure". The
+__binomial_distrib is used to obtain
+the probability of observing k successes in N trials, with the
+probability of success on a single trial denoted by p. The
+binomial distribution assumes that p is fixed for all trials.
+
+[note The random variable for the binomial distribution is the number of successes,
+(the number of trials is a fixed property of the distribution)
+whereas for the negative binomial,
+the random variable is the number of trials, for a fixed number of successes.]
+
+The PDF for the binomial distribution is given by:
+
+[equation binomial_ref2]
+
+The following two graphs illustrate how the PDF changes depending
+upon the distributions parameters, first we'll keep the success
+fraction /p/ fixed at 0.5, and vary the sample size:
+
+[graph binomial_pdf_1]
+
+Alternatively, we can keep the sample size fixed at N=20 and
+vary the success fraction /p/:
+
+[graph binomial_pdf_2]
+
+[discrete_quantile_warning Binomial]
+
+[h4 Member Functions]
+
+[h5 Construct]
+
+ binomial_distribution(RealType n, RealType p);
+
+Constructor: /n/ is the total number of trials, /p/ is the
+probability of success of a single trial.
+
+Requires `0 <= p <= 1`, and `n >= 0`, otherwise calls __domain_error.
+
+[h5 Accessors]
+
+ RealType success_fraction() const;
+
+Returns the parameter /p/ from which this distribution was constructed.
+
+ RealType trials() const;
+
+Returns the parameter /n/ from which this distribution was constructed.
+
+[h5 Lower Bound on the Success Fraction]
+
+ static RealType find_lower_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType alpha,
+ ``['unspecified-type]`` method = clopper_pearson_exact_interval);
+
+Returns a lower bound on the success fraction:
+
+[variablelist
+[[trials][The total number of trials conducted.]]
+[[successes][The number of successes that occurred.]]
+[[alpha][The largest acceptable probability that the true value of
+ the success fraction is [*less than] the value returned.]]
+[[method][An optional parameter that specifies the method to be used
+ to compute the interval (See below).]]
+]
+
+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 [*greater than] some value,
+['p[sub min]], then:
+
+ p``[sub min]`` = binomial_distribution<RealType>::find_lower_bound_on_p(
+ n, k, 0.05);
+
+[link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.]
+
+There are currently two possible values available for the /method/
+optional parameter: /clopper_pearson_exact_interval/
+or /jeffreys_prior_interval/. These constants are both members of
+class template `binomial_distribution`, so usage is for example:
+
+ p = binomial_distribution<RealType>::find_lower_bound_on_p(
+ n, k, 0.05, binomial_distribution<RealType>::jeffreys_prior_interval);
+
+The default method if this parameter is not specified is the Clopper Pearson
+"exact" interval. This produces an interval that guarantees at least
+`100(1-alpha)%` coverage, but which is known to be overly conservative,
+sometimes producing intervals with much greater than the requested coverage.
+
+The alternative calculation method produces a non-informative
+Jeffreys Prior interval. It produces `100(1-alpha)%` coverage only
+['in the average case], though is typically very close to the requested
+coverage level. It is one of the main methods of calculation recommended
+in the review by Brown, Cai and DasGupta.
+
+Please note that the "textbook" calculation method using
+a normal approximation (the Wald interval) is deliberately
+not provided: it is known to produce consistently poor results,
+even when the sample size is surprisingly large.
+Refer to Brown, Cai and DasGupta for a full explanation. Many other methods
+of calculation are available, and may be more appropriate for specific
+situations. Unfortunately there appears to be no consensus amongst
+statisticians as to which is "best": refer to the discussion at the end of
+Brown, Cai and DasGupta for examples.
+
+The two methods provided here were chosen principally because they
+can be used for both one and two sided intervals.
+See also:
+
+Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001),
+Interval Estimation for a Binomial Proportion,
+Statistical Science, Vol. 16, No. 2, 101-133.
+
+T. Tony Cai (2005),
+One-sided confidence intervals in discrete distributions,
+Journal of Statistical Planning and Inference 131, 63-88.
+
+Agresti, A. and Coull, B. A. (1998). Approximate is better than
+"exact" for interval estimation of binomial proportions. Amer.
+Statist. 52 119-126.
+
+Clopper, C. J. and Pearson, E. S. (1934). The use of confidence
+or fiducial limits illustrated in the case of the binomial.
+Biometrika 26 404-413.
+
+[h5 Upper Bound on the Success Fraction]
+
+ static RealType find_upper_bound_on_p(
+ RealType trials,
+ RealType successes,
+ RealType alpha,
+ ``['unspecified-type]`` method = clopper_pearson_exact_interval);
+
+Returns an upper bound on the success fraction:
+
+[variablelist
+[[trials][The total number of trials conducted.]]
+[[successes][The number of successes that occurred.]]
+[[alpha][The largest acceptable probability that the true value of
+ the success fraction is [*greater than] the value returned.]]
+[[method][An optional parameter that specifies the method to be used
+ to compute the interval. Refer to the documentation for
+ `find_upper_bound_on_p` above for the meaning of the
+ method options.]]
+]
+
+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]`` = binomial_distribution<RealType>::find_upper_bound_on_p(
+ n, k, 0.05);
+
+[link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.]
+
+[note
+In order to obtain a two sided bound on the success fraction, you
+call both `find_lower_bound_on_p` *and* `find_upper_bound_on_p`
+each with the same arguments.
+
+If the desired risk level
+that the true success fraction lies outside the bounds is [alpha],
+then you pass [alpha]/2 to these functions.
+
+So for example a two sided 95% confidence interval would be obtained
+by passing [alpha] = 0.025 to each of the functions.
+
+[link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.]
+]
+
+
+[h5 Estimating the Number of Trials Required for a Certain Number of Successes]
+
+ static RealType find_minimum_number_of_trials(
+ RealType k, // number of events
+ RealType p, // success fraction
+ RealType alpha); // probability threshold
+
+This function estimates the minimum number of trials required to ensure that
+more than k events is observed with a level of risk /alpha/ that k or
+fewer events occur.
+
+[variablelist
+[[k][The number of success observed.]]
+[[p][The probability of success for each trial.]]
+[[alpha][The maximum acceptable probability that k events or fewer will be observed.]]
+]
+
+For example:
+
+ binomial_distribution<RealType>::find_number_of_trials(10, 0.5, 0.05);
+
+Returns the smallest number of trials we must conduct to be 95% sure
+of seeing 10 events that occur with frequency one half.
+
+[h5 Estimating the Maximum Number of Trials to Ensure no more than a Certain Number of Successes]
+
+ static RealType find_maximum_number_of_trials(
+ RealType k, // number of events
+ RealType p, // success fraction
+ RealType alpha); // probability threshold
+
+This function estimates the maximum number of trials we can conduct
+to ensure that k successes or fewer are observed, with a risk /alpha/
+that more than k occur.
+
+[variablelist
+[[k][The number of success observed.]]
+[[p][The probability of success for each trial.]]
+[[alpha][The maximum acceptable probability that more than k events will be observed.]]
+]
+
+For example:
+
+ binomial_distribution<RealType>::find_maximum_number_of_trials(0, 1e-6, 0.05);
+
+Returns the largest number of trials we can conduct and still be 95% certain
+of not observing any events that occur with one in a million frequency.
+This is typically used in failure analysis.
+
+[link math_toolkit.stat_tut.weg.binom_eg.binom_size_eg See Worked Example.]
+
+[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.
+
+The domain for the random variable /k/ is `0 <= k <= N`, otherwise a
+__domain_error is returned.
+
+It's worth taking a moment to define what these accessors actually mean in
+the context of this distribution:
+
+[table Meaning of the non-member accessors
+[[Function][Meaning]]
+[[__pdf]
+ [The probability of obtaining [*exactly k successes] from n trials
+ with success fraction p. For example:
+
+`pdf(binomial(n, p), k)`]]
+[[__cdf]
+ [The probability of obtaining [*k successes or fewer] from n trials
+ with success fraction p. For example:
+
+`cdf(binomial(n, p), k)`]]
+[[__ccdf]
+ [The probability of obtaining [*more than k successes] from n trials
+ with success fraction p. For example:
+
+`cdf(complement(binomial(n, p), k))`]]
+[[__quantile]
+ [The [*greatest] number of successes that may be observed from n 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 result. For example:
+
+`quantile(binomial(n, p), P)`]]
+[[__quantile_c]
+ [The [*smallest] number of successes that may be observed from n 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 result. For example:
+
+`quantile(complement(binomial(n, p), P))`]]
+]
+
+[h4 Examples]
+
+Various [link math_toolkit.stat_tut.weg.binom_eg worked examples]
+are available illustrating the use of the binomial distribution.
+
+[h4 Accuracy]
+
+This distribution is implemented using the
+incomplete beta functions __ibeta and __ibetac,
+please refer to these functions for information on accuracy.
+
+[h4 Implementation]
+
+In the following table /p/ is the probability that one trial will
+be successful (the success fraction), /n/ is the number of trials,
+/k/ is the number of successes, /p/ is the probability and /q = 1-p/.
+
+[table
+[[Function][Implementation Notes]]
+[[pdf][Implementation is in terms of __ibeta_derivative: if [sub n]C[sub k ] is the binomial
+ coefficient of a and b, then we have:
+
+[equation binomial_ref1]
+
+Which can be evaluated as `ibeta_derivative(k+1, n-k+1, p) / (n+1)`
+
+The function __ibeta_derivative is used here, since it has already
+ been optimised for the lowest possible error - indeed this is really
+ just a thin wrapper around part of the internals of the incomplete
+ beta function.
+
+There are also various special cases: refer to the code for details.
+ ]]
+[[cdf][Using the relation:
+
+``
+p = I[sub 1-p](n - k, k + 1)
+ = 1 - I[sub p](k + 1, n - k)
+ = __ibetac(k + 1, n - k, p)``
+
+There are also various special cases: refer to the code for details.
+]]
+[[cdf complement][Using the relation: q = __ibeta(k + 1, n - k, p)
+
+There are also various special cases: refer to the code for details. ]]
+[[quantile][Since the cdf is non-linear in variate /k/ none of the inverse
+ incomplete beta functions can be used here. Instead the quantile
+ is found numerically using a derivative free method
+ (__root_finding_TOMS748).]]
+[[quantile from the complement][Found numerically as above.]]
+[[mean][ `p * n` ]]
+[[variance][ `p * n * (1-p)` ]]
+[[mode][`floor(p * (n + 1))`]]
+[[skewness][`(1 - 2 * p) / sqrt(n * p * (1 - p))`]]
+[[kurtosis][`3 - (6 / n) + (1 / (n * p * (1 - p)))`]]
+[[kurtosis excess][`(1 - 6 * p * q) / (n * p * q)`]]
+[[parameter estimation][The member functions `find_upper_bound_on_p`
+ `find_lower_bound_on_p` and `find_number_of_trials` are
+ implemented in terms of the inverse incomplete beta functions
+ __ibetac_inv, __ibeta_inv, and __ibetac_invb respectively]]
+]
+
+[h4 References]
+
+* [@http://mathworld.wolfram.com/BinomialDistribution.html Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource].
+* [@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia binomial distribution].
+* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm NIST Explorary Data Analysis].
+
+[endsect] [/section:binomial_dist Binomial]
+
+[/ binomial.qbk
+ Copyright 2006 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).
+]