mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-18 01:52:05 -05:00
191 lines
5.2 KiB
Plaintext
191 lines
5.2 KiB
Plaintext
[section:f_dist F Distribution]
|
|
|
|
``#include <boost/math/distributions/fisher_f.hpp>``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class RealType = double,
|
|
class ``__Policy`` = ``__policy_class`` >
|
|
class fisher_f_distribution;
|
|
|
|
typedef fisher_f_distribution<> fisher_f;
|
|
|
|
template <class RealType, class ``__Policy``>
|
|
class fisher_f_distribution
|
|
{
|
|
public:
|
|
typedef RealType value_type;
|
|
|
|
// Construct:
|
|
fisher_f_distribution(const RealType& i, const RealType& j);
|
|
|
|
// Accessors:
|
|
RealType degrees_of_freedom1()const;
|
|
RealType degrees_of_freedom2()const;
|
|
};
|
|
|
|
}} //namespaces
|
|
|
|
The F distribution is a continuous distribution that arises when testing
|
|
whether two samples have the same variance. If [chi][super 2][sub m][space] and
|
|
[chi][super 2][sub n][space] are independent variates each distributed as
|
|
Chi-Squared with /m/ and /n/ degrees of freedom, then the test statistic:
|
|
|
|
F[sub n,m][space] = ([chi][super 2][sub n][space] / n) / ([chi][super 2][sub m][space] / m)
|
|
|
|
Is distributed over the range \[0, [infin]\] with an F distribution, and
|
|
has the PDF:
|
|
|
|
[equation fisher_pdf]
|
|
|
|
The following graph illustrates how the PDF varies depending on the
|
|
two degrees of freedom parameters.
|
|
|
|
[graph fisher_f_pdf]
|
|
|
|
|
|
[h4 Member Functions]
|
|
|
|
fisher_f_distribution(const RealType& df1, const RealType& df2);
|
|
|
|
Constructs an F-distribution with numerator degrees of freedom /df1/
|
|
and denominator degrees of freedom /df2/.
|
|
|
|
Requires that /df1/ and /df2/ are both greater than zero, otherwise __domain_error
|
|
is called.
|
|
|
|
RealType degrees_of_freedom1()const;
|
|
|
|
Returns the numerator degrees of freedom parameter of the distribution.
|
|
|
|
RealType degrees_of_freedom2()const;
|
|
|
|
Returns the denominator degrees of freedom parameter of the distribution.
|
|
|
|
[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 of the random variable is \[0, +[infin]\].
|
|
|
|
[h4 Examples]
|
|
|
|
Various [link math_toolkit.stat_tut.weg.f_eg worked examples] are
|
|
available illustrating the use of the F Distribution.
|
|
|
|
[h4 Accuracy]
|
|
|
|
The normal distribution is implemented in terms of the
|
|
[link math_toolkit.sf_beta.ibeta_function incomplete beta function]
|
|
and its [link math_toolkit.sf_beta.ibeta_inv_function inverses],
|
|
refer to those functions for accuracy data.
|
|
|
|
[h4 Implementation]
|
|
|
|
In the following table /v1/ and /v2/ are the first and second
|
|
degrees of freedom parameters of the distribution,
|
|
/x/ is the random variate, /p/ is the probability, and /q = 1-p/.
|
|
|
|
[table
|
|
[[Function][Implementation Notes]]
|
|
[[pdf][The usual form of the PDF is given by:
|
|
|
|
[equation fisher_pdf]
|
|
|
|
However, that form is hard to evaluate directly without incurring problems with
|
|
either accuracy or numeric overflow.
|
|
|
|
Direct differentiation of the CDF expressed in terms of the incomplete beta function
|
|
|
|
led to the following two formulas:
|
|
|
|
f[sub v1,v2](x) = y * __ibeta_derivative(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
|
|
|
|
with y = (v2 * v1) \/ ((v2 + v1 * x) * (v2 + v1 * x))
|
|
|
|
and
|
|
|
|
f[sub v1,v2](x) = y * __ibeta_derivative(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
|
|
|
|
with y = (z * v1 - x * v1 * v1) \/ z[super 2]
|
|
|
|
and z = v2 + v1 * x
|
|
|
|
The first of these is used for v1 * x > v2, otherwise the second is used.
|
|
|
|
The aim is to keep the /x/ argument to __ibeta_derivative away from 1 to avoid
|
|
rounding error. ]]
|
|
[[cdf][Using the relations:
|
|
|
|
p = __ibeta(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
|
|
|
|
and
|
|
|
|
p = __ibetac(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
|
|
|
|
The first is used for v1 * x > v2, otherwise the second is used.
|
|
|
|
The aim is to keep the /x/ argument to __ibeta well away from 1 to
|
|
avoid rounding error. ]]
|
|
|
|
[[cdf complement][Using the relations:
|
|
|
|
p = __ibetac(v1 \/ 2, v2 \/ 2, v1 * x \/ (v2 + v1 * x))
|
|
|
|
and
|
|
|
|
p = __ibeta(v2 \/ 2, v1 \/ 2, v2 \/ (v2 + v1 * x))
|
|
|
|
The first is used for v1 * x < v2, otherwise the second is used.
|
|
|
|
The aim is to keep the /x/ argument to __ibeta well away from 1 to
|
|
avoid rounding error. ]]
|
|
[[quantile][Using the relation:
|
|
|
|
x = v2 * a \/ (v1 * b)
|
|
|
|
where:
|
|
|
|
a = __ibeta_inv(v1 \/ 2, v2 \/ 2, p)
|
|
|
|
and
|
|
|
|
b = 1 - a
|
|
|
|
Quantities /a/ and /b/ are both computed by __ibeta_inv without the
|
|
subtraction implied above.]]
|
|
[[quantile
|
|
|
|
from the complement][Using the relation:
|
|
|
|
x = v2 * a \/ (v1 * b)
|
|
|
|
where
|
|
|
|
a = __ibetac_inv(v1 \/ 2, v2 \/ 2, p)
|
|
|
|
and
|
|
|
|
b = 1 - a
|
|
|
|
Quantities /a/ and /b/ are both computed by __ibetac_inv without the
|
|
subtraction implied above.]]
|
|
[[mean][v2 \/ (v2 - 2)]]
|
|
[[variance][2 * v2[super 2 ] * (v1 + v2 - 2) \/ (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))]]
|
|
[[mode][v2 * (v1 - 2) \/ (v1 * (v2 + 2))]]
|
|
[[skewness][2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) \/ (v1 * (v2 + v1 - 2))) \/ (v2 - 6)]]
|
|
[[kurtosis and kurtosis excess]
|
|
[Refer to, [@http://mathworld.wolfram.com/F-Distribution.html
|
|
Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource.] ]]
|
|
]
|
|
|
|
[endsect][/section:f_dist F distribution]
|
|
|
|
[/ fisher.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).
|
|
]
|