F-Distribution¶
Table of contents
Density Function¶
The density function of the F distribution:
where \(\mathcal{B}(a,b)\) denotes the Beta function.
Methods for scalar input, as well as for vector/matrix input, are listed below.
Scalar Input¶
-
template<typename T1, typename T2, typename T3>
constexpr common_return_t<T1, T2, T3> df(const T1 x, const T2 df1_par, const T3 df2_par, const bool log_form = false) noexcept¶ Density function of the F-distribution.
Example:
stats::df(1.5,10.0,12.0,false);
- Parameters
x – a real-valued input.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.
- Returns
the density function evaluated at
x
.
Vector/Matrix Input¶
STL Containers¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline std::vector<rT> df(const std::vector<eT> &x, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Density function of the F-distribution.
Example:
std::vector<double> x = {0.3, 0.5, 0.9}; stats::df(x,3.0,2.0,false);
- Parameters
x – a standard vector.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.
- Returns
a vector of density function values corresponding to the elements of
x
.
Armadillo¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline ArmaMat<rT> df(const ArmaMat<eT> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Density function of the F-distribution.
Example:
arma::mat X = { {0.2, 0.7, 0.1}, {0.9, -0.3, 1.3} }; stats::df(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.
- Returns
a matrix of density function values corresponding to the elements of
X
.
Blaze¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor>
inline BlazeMat<rT, To> df(const BlazeMat<eT, To> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Density function of the F-distribution.
Example:
stats::df(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.
- Returns
a matrix of density function values corresponding to the elements of
X
.
Eigen¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic>
inline EigenMat<rT, iTr, iTc> df(const EigenMat<eT, iTr, iTc> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Density function of the F-distribution.
Example:
stats::df(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-density or the true form.
- Returns
a matrix of density function values corresponding to the elements of
X
.
Cumulative Distribution Function¶
The cumulative distribution function of the F distribution:
where \(I_x (a,b)\) denotes the regularized incomplete Beta function.
Methods for scalar input, as well as for vector/matrix input, are listed below.
Scalar Input¶
-
template<typename T1, typename T2, typename T3>
constexpr common_return_t<T1, T2, T3> pf(const T1 x, const T2 df1_par, const T3 df2_par, const bool log_form = false) noexcept¶ Distribution function of the F-distribution.
Example:
stats::pf(1.5,10.0,12.0,false);
- Parameters
x – a real-valued input.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.
- Returns
the cumulative distribution function evaluated at
x
.
Vector/Matrix Input¶
STL Containers¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline std::vector<rT> pf(const std::vector<eT> &x, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Distribution function of the Beta distribution.
Example:
std::vector<double> x = {0.3, 0.5, 0.9}; stats::pf(x,3.0,2.0,false);
- Parameters
x – a standard vector.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.
- Returns
a vector of CDF values corresponding to the elements of
x
.
Armadillo¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline ArmaMat<rT> pf(const ArmaMat<eT> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Distribution function of the Beta distribution.
Example:
arma::mat X = { {0.2, 0.7, 0.1}, {0.9, -0.3, 1.3} }; stats::pf(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.
- Returns
a matrix of CDF values corresponding to the elements of
X
.
Blaze¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor>
inline BlazeMat<rT, To> pf(const BlazeMat<eT, To> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Distribution function of the Beta distribution.
Example:
stats::pf(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.
- Returns
a matrix of CDF values corresponding to the elements of
X
.
Eigen¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic>
inline EigenMat<rT, iTr, iTc> pf(const EigenMat<eT, iTr, iTc> &X, const T1 df1_par, const T2 df2_par, const bool log_form = false)¶ Distribution function of the Beta distribution.
Example:
stats::pf(X,3.0,2.0,false);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
log_form – return the log-probability or the true form.
- Returns
a matrix of CDF values corresponding to the elements of
X
.
Quantile Function¶
The quantile function of the F distribution:
Methods for scalar input, as well as for vector/matrix input, are listed below.
Scalar Input¶
-
template<typename T1, typename T2, typename T3>
constexpr common_return_t<T1, T2, T3> qf(const T1 p, const T2 df1_par, const T3 df2_par) noexcept¶ Quantile function of the F-distribution.
Example:
stats::qf(0.5,10.0,12.0);
- Parameters
p – a real-valued input.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
- Returns
the quantile function evaluated at
p
.
Vector/Matrix Input¶
STL Containers¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline std::vector<rT> qf(const std::vector<eT> &x, const T1 df1_par, const T2 df2_par)¶ Quantile function of the F-distribution.
Example:
std::vector<double> x = {0.3, 0.5, 0.9}; stats::qf(x,3.0,2.0);
- Parameters
x – a standard vector.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
- Returns
a vector of quantile values corresponding to the elements of
x
.
Armadillo¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>>
inline ArmaMat<rT> qf(const ArmaMat<eT> &X, const T1 df1_par, const T2 df2_par)¶ Quantile function of the F-distribution.
Example:
arma::mat X = { {0.2, 0.7, 0.1}, {0.9, 0.3, 0.87} }; stats::qf(X,3.0,2.0);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
- Returns
a matrix of quantile values corresponding to the elements of
X
.
Blaze¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, bool To = blaze::columnMajor>
inline BlazeMat<rT, To> qf(const BlazeMat<eT, To> &X, const T1 df1_par, const T2 df2_par)¶ Quantile function of the F-distribution.
Example:
stats::qf(X,3.0,2.0);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
- Returns
a matrix of quantile values corresponding to the elements of
X
.
Eigen¶
-
template<typename eT, typename T1, typename T2, typename rT = common_return_t<eT, T1, T2>, int iTr = Eigen::Dynamic, int iTc = Eigen::Dynamic>
inline EigenMat<rT, iTr, iTc> qf(const EigenMat<eT, iTr, iTc> &X, const T1 df1_par, const T2 df2_par)¶ Quantile function of the F-distribution.
Example:
stats::qf(X,3.0,2.0);
- Parameters
X – a matrix of input values.
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
- Returns
a matrix of quantile values corresponding to the elements of
X
.
Random Sampling¶
Random sampling for the Beta distribution is achieved by simulating two independent \(\chi^2\)-distributed random variables, \(X \sim \chi^2(d_1), Y \sim \chi^2(d_2)\), then returning:
Scalar Output¶
Random number engines
-
template<typename T1, typename T2>
inline common_return_t<T1, T2> rf(const T1 df1_par, const T2 df2_par, rand_engine_t &engine)¶ Random sampling function for the F-distribution.
Example:
stats::rand_engine_t engine(1776); stats::rf(3.0,2.0,engine);
- Parameters
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
engine – a random engine, passed by reference.
- Returns
a pseudo-random draw from the F-distribution.
Seed values
-
template<typename T1, typename T2>
inline common_return_t<T1, T2> rf(const T1 df1_par, const T2 df2_par, const ullint_t seed_val = std::random_device{}())¶ Random sampling function for the F-distribution.
Example:
stats::rf(3.0,2.0,1776);
- Parameters
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.
- Returns
a pseudo-random draw from the F-distribution.
Vector/Matrix Output¶
Random number engines
-
template<typename mT, typename T1, typename T2>
inline mT rf(const ullint_t n, const ullint_t k, const T1 df1_par, const T2 df2_par, rand_engine_t &engine)¶ Random matrix sampling function for the F-distribution.
Example:
stats::rand_engine_t engine(1776); // std::vector stats::rf<std::vector<double>>(5,4,3.0,2.0,engine); // Armadillo matrix stats::rf<arma::mat>(5,4,3.0,2.0,engine); // Blaze dynamic matrix stats::rf<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,3.0,2.0,engine); // Eigen dynamic matrix stats::rf<Eigen::MatrixXd>(5,4,3.0,2.0,engine);
Note
This function requires template instantiation; acceptable output types include:
std::vector
, with element typefloat
,double
, etc., as well as Armadillo, Blaze, and Eigen dense matrices.- Parameters
n – the number of output rows
k – the number of output columns
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
engine – a random engine, passed by reference.
- Returns
a matrix of pseudo-random draws from the F-distribution.
Seed values
-
template<typename mT, typename T1, typename T2>
inline mT rf(const ullint_t n, const ullint_t k, const T1 df1_par, const T2 df2_par, const ullint_t seed_val = std::random_device{}())¶ Random matrix sampling function for the F-distribution.
Example:
// std::vector stats::rf<std::vector<double>>(5,4,3.0,2.0); // Armadillo matrix stats::rf<arma::mat>(5,4,3.0,2.0); // Blaze dynamic matrix stats::rf<blaze::DynamicMatrix<double,blaze::columnMajor>>(5,4,3.0,2.0); // Eigen dynamic matrix stats::rf<Eigen::MatrixXd>(5,4,3.0,2.0);
Note
This function requires template instantiation; acceptable output types include:
std::vector
, with element typefloat
,double
, etc., as well as Armadillo, Blaze, and Eigen dense matrices.- Parameters
n – the number of output rows
k – the number of output columns
df1_par – a degrees of freedom parameter, a real-valued input.
df2_par – a degrees of freedom parameter, a real-valued input.
seed_val – initialize the random engine with a non-negative integral-valued seed.
- Returns
a matrix of pseudo-random draws from the F-distribution.