mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-18 01:52:05 -05:00
374 lines
14 KiB
Plaintext
374 lines
14 KiB
Plaintext
[section:negative_binomial_dist Negative Binomial Distribution]
|
|
|
|
``#include <boost/math/distributions/negative_binomial.hpp>``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class RealType = double,
|
|
class ``__Policy`` = ``__policy_class`` >
|
|
class negative_binomial_distribution;
|
|
|
|
typedef negative_binomial_distribution<> negative_binomial;
|
|
|
|
template <class RealType, class ``__Policy``>
|
|
class negative_binomial_distribution
|
|
{
|
|
public:
|
|
typedef RealType value_type;
|
|
typedef Policy policy_type;
|
|
// Constructor from successes and success_fraction:
|
|
negative_binomial_distribution(RealType r, 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 `negative_binomial_distribution` represents a
|
|
[@http://en.wikipedia.org/wiki/Negative_binomial_distribution negative_binomial 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 k + r Bernoulli trials each with success fraction p, the
|
|
negative_binomial distribution gives the probability of observing
|
|
k failures and r successes with success on the last trial.
|
|
The negative_binomial distribution
|
|
assumes that success_fraction p is fixed for all (k + r) trials.
|
|
|
|
[note The random variable for the negative binomial distribution is the number of trials,
|
|
(the number of successes is a fixed property of the distribution)
|
|
whereas for the binomial,
|
|
the random variable is the number of successes, for a fixed number of trials.]
|
|
|
|
It has the PDF:
|
|
|
|
[equation neg_binomial_ref]
|
|
|
|
The following graph illustrate how the PDF varies as the success fraction
|
|
/p/ changes:
|
|
|
|
[graph negative_binomial_pdf_1]
|
|
|
|
Alternatively, this graph shows how the shape of the PDF varies as
|
|
the number of successes changes:
|
|
|
|
[graph negative_binomial_pdf_2]
|
|
|
|
[h4 Related Distributions]
|
|
|
|
The name negative binomial distribution is reserved by some to the
|
|
case where the successes parameter r is an integer.
|
|
This integer version is also called the
|
|
[@http://mathworld.wolfram.com/PascalDistribution.html Pascal distribution].
|
|
|
|
This implementation uses real numbers for the computation throughout
|
|
(because it uses the *real-valued* incomplete beta function family of functions).
|
|
This real-valued version is also called the Polya Distribution.
|
|
|
|
The Poisson distribution is a generalization of the Pascal distribution,
|
|
where the success parameter r is an integer: to obtain the Pascal
|
|
distribution you must ensure that an integer value is provided for r,
|
|
and take integer values (floor or ceiling) from functions that return
|
|
a number of successes.
|
|
|
|
For large values of r (successes), the negative binomial distribution
|
|
converges to the Poisson distribution.
|
|
|
|
The geometric distribution is a special case
|
|
where the successes parameter r = 1,
|
|
so only a first and only success is required.
|
|
geometric(p) = negative_binomial(1, p).
|
|
|
|
The Poisson distribution is a special case for large successes
|
|
|
|
poisson([lambda]) = lim [sub r [rarr] [infin]] [space] negative_binomial(r, r / ([lambda] + r)))
|
|
|
|
[discrete_quantile_warning Negative Binomial]
|
|
|
|
[h4 Member Functions]
|
|
|
|
[h5 Construct]
|
|
|
|
negative_binomial_distribution(RealType r, RealType p);
|
|
|
|
Constructor: /r/ is the total number of successes, /p/ is the
|
|
probability of success of a single trial.
|
|
|
|
Requires: `r > 0` and `0 <= p <= 1`.
|
|
|
|
[h5 Accessors]
|
|
|
|
RealType success_fraction() const; // successes / trials (0 <= p <= 1)
|
|
|
|
Returns the parameter /p/ from which this distribution was constructed.
|
|
|
|
RealType successes() const; // required successes (r > 0)
|
|
|
|
Returns the parameter /r/ from which this distribution was constructed.
|
|
|
|
The best method of calculation for the following functions is disputed:
|
|
see __binomial_distrib for more discussion.
|
|
|
|
[h5 Lower Bound on Parameter p]
|
|
|
|
static RealType find_lower_bound_on_p(
|
|
RealType failures,
|
|
RealType successes,
|
|
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 ['r]th success.]]
|
|
[[successes][The number of successes required.]]
|
|
[[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 and /r/ successes from /n/ = k + r trials
|
|
the best estimate for the success fraction is simply ['r/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]`` = negative_binomial_distribution<RealType>::find_lower_bound_on_p(
|
|
failures, successes, 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 Parameter p]
|
|
|
|
static RealType find_upper_bound_on_p(
|
|
RealType trials,
|
|
RealType successes,
|
|
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.]]
|
|
[[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.]]
|
|
]
|
|
|
|
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]`` = negative_binomial_distribution<RealType>::find_upper_bound_on_p(
|
|
r, 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:
|
|
|
|
negative_binomial_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
|
|
|
|
Returns the smallest number of trials we must conduct to be 95% 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 negative binomial distribution
|
|
to obtain the result: another interpretation of the result, is that it finds
|
|
the number of trials (success+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:
|
|
|
|
negative_binomial_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 negative binomial distribution
|
|
to obtain the result: another interpretation of the result, is that it finds
|
|
the number of trials (success+failures) 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+r trials
|
|
with success fraction p. For example:
|
|
|
|
``pdf(negative_binomial(r, p), k)``]]
|
|
[[__cdf]
|
|
[The probability of obtaining [*k failures or fewer] from k+r trials
|
|
with success fraction p and success on the last trial. For example:
|
|
|
|
``cdf(negative_binomial(r, p), k)``]]
|
|
[[__ccdf]
|
|
[The probability of obtaining [*more than k failures] from k+r trials
|
|
with success fraction p and success on the last trial. For example:
|
|
|
|
``cdf(complement(negative_binomial(r, p), k))``]]
|
|
[[__quantile]
|
|
[The [*greatest] number of failures k expected to be observed from k+r 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(negative_binomial(r, p), P)``]]
|
|
[[__quantile_c]
|
|
[The [*smallest] number of failures k expected to be observed from k+r 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(negative_binomial(r, p), P))``]]
|
|
]
|
|
|
|
[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 any one trial will
|
|
be successful (the success fraction), /r/ is the number of successes,
|
|
/k/ is the number of failures, /p/ is the probability and /q = 1-p/.
|
|
|
|
[table
|
|
[[Function][Implementation Notes]]
|
|
[[pdf][pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) * pow((1-p), k)
|
|
|
|
Implementation is in terms of __ibeta_derivative:
|
|
|
|
(p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), p)
|
|
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.
|
|
]]
|
|
[[cdf][Using the relation:
|
|
|
|
cdf = I[sub p](r, k+1) = ibeta(r, k+1, p)
|
|
|
|
= ibeta(r, static_cast<RealType>(k+1), p)]]
|
|
[[cdf complement][Using the relation:
|
|
|
|
1 - cdf = I[sub p](k+1, r)
|
|
|
|
= ibetac(r, static_cast<RealType>(k+1), p)
|
|
]]
|
|
[[quantile][ibeta_invb(r, p, P) - 1]]
|
|
[[quantile from the complement][ibetac_invb(r, p, Q) -1)]]
|
|
[[mean][ `r(1-p)/p` ]]
|
|
[[variance][ `r (1-p) / p * p` ]]
|
|
[[mode][`floor((r-1) * (1 - p)/p)`]]
|
|
[[skewness][`(2 - p) / sqrt(r * (1 - p))`]]
|
|
[[kurtosis][`6 / r + (p * p) / r * (1 - p )`]]
|
|
[[kurtosis excess][`6 / r + (p * p) / r * (1 - p ) -3`]]
|
|
[[parameter estimation member functions][]]
|
|
[[`find_lower_bound_on_p`][ibeta_inv(successes, failures + 1, alpha)]]
|
|
[[`find_upper_bound_on_p`][ibetac_inv(successes, failures, alpha) plus see comments in code.]]
|
|
[[`find_minimum_number_of_trials`][ibeta_inva(k + 1, p, alpha)]]
|
|
[[`find_maximum_number_of_trials`][ibetac_inva(k + 1, p, alpha)]]
|
|
]
|
|
|
|
Implementation notes:
|
|
|
|
* The real concept type (that deliberately lacks the Lanczos approximation),
|
|
was found to take several minutes to evaluate some extreme test values,
|
|
so the test has been disabled for this type.
|
|
|
|
* Much greater speed, and perhaps greater accuracy,
|
|
might be achieved for extreme values by using a normal approximation.
|
|
This is NOT been tested or implemented.
|
|
|
|
[endsect][/section:negative_binomial_dist Negative Binomial]
|
|
|
|
[/ negative_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).
|
|
]
|
|
|