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374 lines
14 KiB
Plaintext
374 lines
14 KiB
Plaintext
[section:negative_binomial_dist Negative Binomial Distribution]
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``#include <boost/math/distributions/negative_binomial.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class negative_binomial_distribution;
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typedef negative_binomial_distribution<> negative_binomial;
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template <class RealType, class ``__Policy``>
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class negative_binomial_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor from successes and success_fraction:
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negative_binomial_distribution(RealType r, RealType p);
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// Parameter accessors:
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RealType success_fraction() const;
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RealType successes() const;
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// Bounds on success fraction:
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static RealType find_lower_bound_on_p(
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RealType trials,
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RealType successes,
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RealType probability); // alpha
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static RealType find_upper_bound_on_p(
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RealType trials,
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RealType successes,
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RealType probability); // alpha
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// Estimate min/max number of trials:
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static RealType find_minimum_number_of_trials(
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RealType k, // Number of failures.
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RealType p, // Success fraction.
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RealType probability); // Probability threshold alpha.
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static RealType find_maximum_number_of_trials(
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RealType k, // Number of failures.
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RealType p, // Success fraction.
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RealType probability); // Probability threshold alpha.
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};
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}} // namespaces
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The class type `negative_binomial_distribution` represents a
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[@http://en.wikipedia.org/wiki/Negative_binomial_distribution negative_binomial distribution]:
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it is used when there are exactly two mutually exclusive outcomes of a
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[@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trial]:
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these outcomes are labelled "success" and "failure".
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For k + r Bernoulli trials each with success fraction p, the
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negative_binomial distribution gives the probability of observing
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k failures and r successes with success on the last trial.
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The negative_binomial distribution
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assumes that success_fraction p is fixed for all (k + r) trials.
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[note The random variable for the negative binomial distribution is the number of trials,
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(the number of successes is a fixed property of the distribution)
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whereas for the binomial,
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the random variable is the number of successes, for a fixed number of trials.]
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It has the PDF:
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[equation neg_binomial_ref]
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The following graph illustrate how the PDF varies as the success fraction
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/p/ changes:
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[graph negative_binomial_pdf_1]
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Alternatively, this graph shows how the shape of the PDF varies as
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the number of successes changes:
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[graph negative_binomial_pdf_2]
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[h4 Related Distributions]
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The name negative binomial distribution is reserved by some to the
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case where the successes parameter r is an integer.
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This integer version is also called the
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[@http://mathworld.wolfram.com/PascalDistribution.html Pascal distribution].
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This implementation uses real numbers for the computation throughout
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(because it uses the *real-valued* incomplete beta function family of functions).
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This real-valued version is also called the Polya Distribution.
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The Poisson distribution is a generalization of the Pascal distribution,
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where the success parameter r is an integer: to obtain the Pascal
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distribution you must ensure that an integer value is provided for r,
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and take integer values (floor or ceiling) from functions that return
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a number of successes.
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For large values of r (successes), the negative binomial distribution
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converges to the Poisson distribution.
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The geometric distribution is a special case
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where the successes parameter r = 1,
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so only a first and only success is required.
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geometric(p) = negative_binomial(1, p).
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The Poisson distribution is a special case for large successes
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poisson([lambda]) = lim [sub r [rarr] [infin]] [space] negative_binomial(r, r / ([lambda] + r)))
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[discrete_quantile_warning Negative Binomial]
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[h4 Member Functions]
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[h5 Construct]
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negative_binomial_distribution(RealType r, RealType p);
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Constructor: /r/ is the total number of successes, /p/ is the
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probability of success of a single trial.
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Requires: `r > 0` and `0 <= p <= 1`.
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[h5 Accessors]
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RealType success_fraction() const; // successes / trials (0 <= p <= 1)
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Returns the parameter /p/ from which this distribution was constructed.
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RealType successes() const; // required successes (r > 0)
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Returns the parameter /r/ from which this distribution was constructed.
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The best method of calculation for the following functions is disputed:
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see __binomial_distrib for more discussion.
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[h5 Lower Bound on Parameter p]
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static RealType find_lower_bound_on_p(
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RealType failures,
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RealType successes,
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RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
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Returns a *lower bound* on the success fraction:
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[variablelist
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[[failures][The total number of failures before the ['r]th success.]]
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[[successes][The number of successes required.]]
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[[alpha][The largest acceptable probability that the true value of
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the success fraction is [*less than] the value returned.]]
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]
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For example, if you observe /k/ failures and /r/ successes from /n/ = k + r trials
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the best estimate for the success fraction is simply ['r/n], but if you
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want to be 95% sure that the true value is [*greater than] some value,
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['p[sub min]], then:
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p``[sub min]`` = negative_binomial_distribution<RealType>::find_lower_bound_on_p(
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failures, successes, 0.05);
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.]
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This function uses the Clopper-Pearson method of computing the lower bound on the
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success fraction, whilst many texts refer to this method as giving an "exact"
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result in practice it produces an interval that guarantees ['at least] the
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coverage required, and may produce pessimistic estimates for some combinations
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of /failures/ and /successes/. See:
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[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
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Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].
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[h5 Upper Bound on Parameter p]
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static RealType find_upper_bound_on_p(
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RealType trials,
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RealType successes,
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RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
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Returns an *upper bound* on the success fraction:
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[variablelist
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[[trials][The total number of trials conducted.]]
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[[successes][The number of successes that occurred.]]
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[[alpha][The largest acceptable probability that the true value of
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the success fraction is [*greater than] the value returned.]]
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]
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For example, if you observe /k/ successes from /n/ trials the
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best estimate for the success fraction is simply ['k/n], but if you
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want to be 95% sure that the true value is [*less than] some value,
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['p[sub max]], then:
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p``[sub max]`` = negative_binomial_distribution<RealType>::find_upper_bound_on_p(
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r, k, 0.05);
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.]
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This function uses the Clopper-Pearson method of computing the lower bound on the
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success fraction, whilst many texts refer to this method as giving an "exact"
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result in practice it produces an interval that guarantees ['at least] the
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coverage required, and may produce pessimistic estimates for some combinations
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of /failures/ and /successes/. See:
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[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
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Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].
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[h5 Estimating Number of Trials to Ensure at Least a Certain Number of Failures]
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static RealType find_minimum_number_of_trials(
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RealType k, // number of failures.
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RealType p, // success fraction.
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RealType alpha); // probability threshold (0.05 equivalent to 95%).
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This functions estimates the number of trials required to achieve a certain
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probability that [*more than k failures will be observed].
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[variablelist
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[[k][The target number of failures to be observed.]]
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[[p][The probability of ['success] for each trial.]]
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[[alpha][The maximum acceptable risk that only k failures or fewer will be observed.]]
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]
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For example:
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negative_binomial_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
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Returns the smallest number of trials we must conduct to be 95% sure
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of seeing 10 failures that occur with frequency one half.
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.]
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This function uses numeric inversion of the negative binomial distribution
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to obtain the result: another interpretation of the result, is that it finds
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the number of trials (success+failures) that will lead to an /alpha/ probability
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of observing k failures or fewer.
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[h5 Estimating Number of Trials to Ensure a Maximum Number of Failures or Less]
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static RealType find_maximum_number_of_trials(
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RealType k, // number of failures.
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RealType p, // success fraction.
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RealType alpha); // probability threshold (0.05 equivalent to 95%).
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This functions estimates the maximum number of trials we can conduct and achieve
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a certain probability that [*k failures or fewer will be observed].
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[variablelist
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[[k][The maximum number of failures to be observed.]]
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[[p][The probability of ['success] for each trial.]]
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[[alpha][The maximum acceptable ['risk] that more than k failures will be observed.]]
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]
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For example:
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negative_binomial_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05);
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Returns the largest number of trials we can conduct and still be 95% sure
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of seeing no failures that occur with frequency one in one million.
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This function uses numeric inversion of the negative binomial distribution
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to obtain the result: another interpretation of the result, is that it finds
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the number of trials (success+failures) that will lead to an /alpha/ probability
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of observing more than k failures.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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However it's worth taking a moment to define what these actually mean in
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the context of this distribution:
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[table Meaning of the non-member accessors.
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[[Function][Meaning]]
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[[__pdf]
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[The probability of obtaining [*exactly k failures] from k+r trials
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with success fraction p. For example:
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``pdf(negative_binomial(r, p), k)``]]
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[[__cdf]
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[The probability of obtaining [*k failures or fewer] from k+r trials
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with success fraction p and success on the last trial. For example:
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``cdf(negative_binomial(r, p), k)``]]
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[[__ccdf]
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[The probability of obtaining [*more than k failures] from k+r trials
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with success fraction p and success on the last trial. For example:
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``cdf(complement(negative_binomial(r, p), k))``]]
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[[__quantile]
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[The [*greatest] number of failures k expected to be observed from k+r trials
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with success fraction p, at probability P. Note that the value returned
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is a real-number, and not an integer. Depending on the use case you may
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want to take either the floor or ceiling of the real result. For example:
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``quantile(negative_binomial(r, p), P)``]]
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[[__quantile_c]
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[The [*smallest] number of failures k expected to be observed from k+r trials
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with success fraction p, at probability P. Note that the value returned
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is a real-number, and not an integer. Depending on the use case you may
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want to take either the floor or ceiling of the real result. For example:
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``quantile(complement(negative_binomial(r, p), P))``]]
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]
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[h4 Accuracy]
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This distribution is implemented using the
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incomplete beta functions __ibeta and __ibetac:
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please refer to these functions for information on accuracy.
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[h4 Implementation]
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In the following table, /p/ is the probability that any one trial will
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be successful (the success fraction), /r/ is the number of successes,
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/k/ is the number of failures, /p/ is the probability and /q = 1-p/.
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[table
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[[Function][Implementation Notes]]
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[[pdf][pdf = exp(lgamma(r + k) - lgamma(r) - lgamma(k+1)) * pow(p, r) * pow((1-p), k)
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Implementation is in terms of __ibeta_derivative:
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(p/(r + k)) * ibeta_derivative(r, static_cast<RealType>(k+1), p)
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The function __ibeta_derivative is used here, since it has already
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been optimised for the lowest possible error - indeed this is really
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just a thin wrapper around part of the internals of the incomplete
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beta function.
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]]
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[[cdf][Using the relation:
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cdf = I[sub p](r, k+1) = ibeta(r, k+1, p)
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= ibeta(r, static_cast<RealType>(k+1), p)]]
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[[cdf complement][Using the relation:
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1 - cdf = I[sub p](k+1, r)
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= ibetac(r, static_cast<RealType>(k+1), p)
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]]
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[[quantile][ibeta_invb(r, p, P) - 1]]
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[[quantile from the complement][ibetac_invb(r, p, Q) -1)]]
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[[mean][ `r(1-p)/p` ]]
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[[variance][ `r (1-p) / p * p` ]]
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[[mode][`floor((r-1) * (1 - p)/p)`]]
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[[skewness][`(2 - p) / sqrt(r * (1 - p))`]]
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[[kurtosis][`6 / r + (p * p) / r * (1 - p )`]]
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[[kurtosis excess][`6 / r + (p * p) / r * (1 - p ) -3`]]
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[[parameter estimation member functions][]]
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[[`find_lower_bound_on_p`][ibeta_inv(successes, failures + 1, alpha)]]
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[[`find_upper_bound_on_p`][ibetac_inv(successes, failures, alpha) plus see comments in code.]]
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[[`find_minimum_number_of_trials`][ibeta_inva(k + 1, p, alpha)]]
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[[`find_maximum_number_of_trials`][ibetac_inva(k + 1, p, alpha)]]
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]
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Implementation notes:
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* The real concept type (that deliberately lacks the Lanczos approximation),
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was found to take several minutes to evaluate some extreme test values,
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so the test has been disabled for this type.
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* Much greater speed, and perhaps greater accuracy,
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might be achieved for extreme values by using a normal approximation.
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This is NOT been tested or implemented.
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[endsect][/section:negative_binomial_dist Negative Binomial]
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[/ negative_binomial.qbk
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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