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162 lines
5.7 KiB
Plaintext
162 lines
5.7 KiB
Plaintext
[section:chi_squared_dist Chi Squared Distribution]
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``#include <boost/math/distributions/chi_squared.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 chi_squared_distribution;
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typedef chi_squared_distribution<> chi_squared;
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template <class RealType, class ``__Policy``>
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class chi_squared_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:
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chi_squared_distribution(RealType i);
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// Accessor to parameter:
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RealType degrees_of_freedom()const;
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// Parameter estimation:
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static RealType find_degrees_of_freedom(
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RealType difference_from_mean,
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RealType alpha,
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RealType beta,
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RealType sd,
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RealType hint = 100);
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};
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}} // namespaces
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The Chi-Squared distribution is one of the most widely used distributions
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in statistical tests. If [chi][sub i][space] are [nu][space]
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independent, normally distributed
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random variables with means [mu][sub i][space] and variances [sigma][sub i][super 2],
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then the random variable:
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[equation chi_squ_ref1]
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is distributed according to the Chi-Squared distribution.
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The Chi-Squared distribution is a special case of the gamma distribution
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and has a single parameter [nu][space] that specifies the number of degrees of
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freedom. The following graph illustrates how the distribution changes
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for different values of [nu]:
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[graph chi_squared_pdf]
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[h4 Member Functions]
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chi_squared_distribution(RealType v);
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Constructs a Chi-Squared distribution with /v/ degrees of freedom.
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Requires v > 0, otherwise calls __domain_error.
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RealType degrees_of_freedom()const;
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Returns the parameter /v/ from which this object was constructed.
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static RealType find_degrees_of_freedom(
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RealType difference_from_variance,
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RealType alpha,
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RealType beta,
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RealType variance,
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RealType hint = 100);
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Estimates the sample size required to detect a difference from a nominal
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variance in a Chi-Squared test for equal standard deviations.
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[variablelist
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[[difference_from_variance][The difference from the assumed nominal variance
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that is to be detected: Note that the sign of this value is critical, see below.]]
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[[alpha][The maximum acceptable risk of rejecting the null hypothesis when it is
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in fact true.]]
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[[beta][The maximum acceptable risk of falsely failing to reject the null hypothesis.]]
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[[variance][The nominal variance being tested against.]]
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[[hint][An optional hint on where to start looking for a result: the current sample
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size would be a good choice.]]
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]
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Note that this calculation works with /variances/ and not /standard deviations/.
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The sign of the parameter /difference_from_variance/ is important: the Chi
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Squared distribution is asymmetric, and the caller must decide in advance
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whether they are testing for a variance greater than a nominal value (positive
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/difference_from_variance/) or testing for a variance less than a nominal value
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(negative /difference_from_variance/). If the latter, then obviously it is
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a requirement that `variance + difference_from_variance > 0`, since no sample
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can have a negative variance!
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This procedure uses the method in Diamond, W. J. (1989).
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Practical Experiment Designs, Van-Nostrand Reinhold, New York.
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See also section on Sample sizes required in
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[@http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm the NIST Engineering Statistics Handbook, Section 7.2.3.2].
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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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(We have followed the usual restriction of the mode to degrees of freedom >= 2,
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but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0,
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and provide an extended definition that would avoid a discontinuity in the mode
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as alternative code in a comment).
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The domain of the random variable is \[0, +[infin]\].
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[h4 Examples]
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Various [link math_toolkit.stat_tut.weg.cs_eg worked examples]
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are available illustrating the use of the Chi Squared Distribution.
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[h4 Accuracy]
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The Chi-Squared distribution is implemented in terms of the
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[link math_toolkit.sf_gamma.igamma incomplete gamma functions]:
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please refer to the accuracy data for those functions.
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[h4 Implementation]
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In the following table /v/ is the number of degrees of freedom of the distribution,
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/x/ is the random variate, /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][Using the relation: pdf = __gamma_p_derivative(v / 2, x / 2) / 2 ]]
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[[cdf][Using the relation: p = __gamma_p(v / 2, x / 2) ]]
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[[cdf complement][Using the relation: q = __gamma_q(v / 2, x / 2) ]]
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[[quantile][Using the relation: x = 2 * __gamma_p_inv(v / 2, p) ]]
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[[quantile from the complement][Using the relation: x = 2 * __gamma_q_inv(v / 2, p) ]]
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[[mean][v]]
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[[variance][2v]]
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[[mode][v - 2 (if v >= 2)]]
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[[skewness][2 * sqrt(2 / v) == sqrt(8 / v)]]
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[[kurtosis][3 + 12 / v]]
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[[kurtosis excess][12 / v]]
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]
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[h4 References]
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* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm NIST Exploratory Data Analysis]
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* [@http://en.wikipedia.org/wiki/Chi-square_distribution Chi-square distribution]
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* [@http://mathworld.wolfram.com/Chi-SquaredDistribution.html Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.]
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[endsect][/section:chi_squared_dist Chi Squared]
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[/ chi_squared.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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