| [section:chi_squared_dist Chi Squared Distribution] |
| |
| ``#include <boost/math/distributions/chi_squared.hpp>`` |
| |
| namespace boost{ namespace math{ |
| |
| template <class RealType = double, |
| class ``__Policy`` = ``__policy_class`` > |
| class chi_squared_distribution; |
| |
| typedef chi_squared_distribution<> chi_squared; |
| |
| template <class RealType, class ``__Policy``> |
| class chi_squared_distribution |
| { |
| public: |
| typedef RealType value_type; |
| typedef Policy policy_type; |
| |
| // Constructor: |
| chi_squared_distribution(RealType i); |
| |
| // Accessor to parameter: |
| RealType degrees_of_freedom()const; |
| |
| // Parameter estimation: |
| static RealType find_degrees_of_freedom( |
| RealType difference_from_mean, |
| RealType alpha, |
| RealType beta, |
| RealType sd, |
| RealType hint = 100); |
| }; |
| |
| }} // namespaces |
| |
| The Chi-Squared distribution is one of the most widely used distributions |
| in statistical tests. If [chi][sub i][space] are [nu][space] |
| independent, normally distributed |
| random variables with means [mu][sub i][space] and variances [sigma][sub i][super 2], |
| then the random variable: |
| |
| [equation chi_squ_ref1] |
| |
| is distributed according to the Chi-Squared distribution. |
| |
| The Chi-Squared distribution is a special case of the gamma distribution |
| and has a single parameter [nu][space] that specifies the number of degrees of |
| freedom. The following graph illustrates how the distribution changes |
| for different values of [nu]: |
| |
| [graph chi_squared_pdf] |
| |
| [h4 Member Functions] |
| |
| chi_squared_distribution(RealType v); |
| |
| Constructs a Chi-Squared distribution with /v/ degrees of freedom. |
| |
| Requires v > 0, otherwise calls __domain_error. |
| |
| RealType degrees_of_freedom()const; |
| |
| Returns the parameter /v/ from which this object was constructed. |
| |
| static RealType find_degrees_of_freedom( |
| RealType difference_from_variance, |
| RealType alpha, |
| RealType beta, |
| RealType variance, |
| RealType hint = 100); |
| |
| Estimates the sample size required to detect a difference from a nominal |
| variance in a Chi-Squared test for equal standard deviations. |
| |
| [variablelist |
| [[difference_from_variance][The difference from the assumed nominal variance |
| that is to be detected: Note that the sign of this value is critical, see below.]] |
| [[alpha][The maximum acceptable risk of rejecting the null hypothesis when it is |
| in fact true.]] |
| [[beta][The maximum acceptable risk of falsely failing to reject the null hypothesis.]] |
| [[variance][The nominal variance being tested against.]] |
| [[hint][An optional hint on where to start looking for a result: the current sample |
| size would be a good choice.]] |
| ] |
| |
| Note that this calculation works with /variances/ and not /standard deviations/. |
| |
| The sign of the parameter /difference_from_variance/ is important: the Chi |
| Squared distribution is asymmetric, and the caller must decide in advance |
| whether they are testing for a variance greater than a nominal value (positive |
| /difference_from_variance/) or testing for a variance less than a nominal value |
| (negative /difference_from_variance/). If the latter, then obviously it is |
| a requirement that `variance + difference_from_variance > 0`, since no sample |
| can have a negative variance! |
| |
| This procedure uses the method in Diamond, W. J. (1989). |
| Practical Experiment Designs, Van-Nostrand Reinhold, New York. |
| |
| See also section on Sample sizes required in |
| [@http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm the NIST Engineering Statistics Handbook, Section 7.2.3.2]. |
| |
| [h4 Non-member Accessors] |
| |
| All the [link math_toolkit.dist.dist_ref.nmp usual non-member accessor functions] |
| that are generic to all distributions are supported: __usual_accessors. |
| |
| (We have followed the usual restriction of the mode to degrees of freedom >= 2, |
| but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0, |
| and provide an extended definition that would avoid a discontinuity in the mode |
| as alternative code in a comment). |
| |
| The domain of the random variable is \[0, +[infin]\]. |
| |
| [h4 Examples] |
| |
| Various [link math_toolkit.dist.stat_tut.weg.cs_eg worked examples] |
| are available illustrating the use of the Chi Squared Distribution. |
| |
| [h4 Accuracy] |
| |
| The Chi-Squared distribution is implemented in terms of the |
| [link math_toolkit.special.sf_gamma.igamma incomplete gamma functions]: |
| please refer to the accuracy data for those functions. |
| |
| [h4 Implementation] |
| |
| In the following table /v/ is the number of degrees of freedom of the distribution, |
| /x/ is the random variate, /p/ is the probability, and /q = 1-p/. |
| |
| [table |
| [[Function][Implementation Notes]] |
| [[pdf][Using the relation: pdf = __gamma_p_derivative(v / 2, x / 2) / 2 ]] |
| [[cdf][Using the relation: p = __gamma_p(v / 2, x / 2) ]] |
| [[cdf complement][Using the relation: q = __gamma_q(v / 2, x / 2) ]] |
| [[quantile][Using the relation: x = 2 * __gamma_p_inv(v / 2, p) ]] |
| [[quantile from the complement][Using the relation: x = 2 * __gamma_q_inv(v / 2, p) ]] |
| [[mean][v]] |
| [[variance][2v]] |
| [[mode][v - 2 (if v >= 2)]] |
| [[skewness][2 * sqrt(2 / v) == sqrt(8 / v)]] |
| [[kurtosis][3 + 12 / v]] |
| [[kurtosis excess][12 / v]] |
| ] |
| |
| [h4 References] |
| |
| * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm NIST Exploratory Data Analysis] |
| * [@http://en.wikipedia.org/wiki/Chi-square_distribution Chi-square distribution] |
| * [@http://mathworld.wolfram.com/Chi-SquaredDistribution.html Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.] |
| |
| |
| [endsect][/section:chi_squared_dist Chi Squared] |
| |
| [/ chi_squared.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). |
| ] |
| |