IMP::isd::MultivariateFNormalSufficient Class Reference

MultivariateFNormalSufficient. More...

#include <IMP/isd/MultivariateFNormalSufficient.h>

Inheritance diagram for IMP::isd::MultivariateFNormalSufficient:

Detailed Description

MultivariateFNormalSufficient.

Probability density function and -log(p) of multivariate normal distribution of N M-variate observations.

\[ p(x_1,\cdots,x_N|\mu,F,\Sigma) = \left((2\pi\sigma^2)^M|\Sigma|\right)^{-N/2} J(F) \exp\left(-\frac{1}{2\sigma^2} \sum_{i=1}^N {}^t(F(\mu) - F(x_i))\Sigma^{-1}(F(\mu)-F(x_i)) \right) \]

which is implemented as

\[ p(x_1,\cdots,x_N|\mu,F,\Sigma) = ((2\pi\sigma^2)^M|\Sigma|)^{-N/2} J(F) \exp\left(-\frac{N}{2\sigma^2} {}^t\epsilon \Sigma^{-1} \epsilon\right) \exp\left(-\frac{1}{2\sigma^2} \text{tr}(W\Sigma^{-1})\right) \]

where

\[\epsilon = (F(\mu)- \overline{F(x)}) \quad \overline{F(x)} = \frac{1}{N} \sum_{i=1}^N F(x_i)\]

and

\[W=\sum_{i=1}^N(F(x_i)-\overline{F(x)}){}^t(F(x_i)-\overline{F(x)}) \]

\(\sigma\) is a multiplicative scaling factor that factors out of the \(\Sigma\) covariance matrix. It is set to 1 by default and its intent is to avoid inverting the \(\Sigma\) matrix unless necessary.

Set J(F) to 1 if you want the multivariate normal distribution. The distribution is normalized with respect to the matrix variable X. The Sufficient statistics are calculated at initialization.

Example: if F is the log function, the multivariate F-normal distribution is the multivariate lognormal distribution with mean \(\mu\) and standard deviation \(\Sigma\).

Note

This is an implementation of the matrix normal distribution for F(X), where rows of F(X) are independent and homoscedastic (they represent repetitions of the same experiment), but columns might be correlated, though the provided matrix.

For now, F must be monotonically increasing, so that J(F) > 0. The program will not check for that. Uses a Cholesky ( \(LDL^T\)) decomposition of \(\Sigma\), which is recomputed when needed.

All observations must be given, so if you have missing data you might want to do some imputation on it first.

References:

• Multivariate Likelihood: Box and Tiao, "Bayesian Inference in Statistical Analysis", Addison-Wesley publishing company, 1973, pp 423.
• Factorization in terms of sufficient statistics: Daniel Fink, "A Compendium of Conjugate Priors", online, May 1997, p.40-41.
• Matrix calculations for derivatives: Petersen and Pedersen, "The Matrix Cookbook", 2008, matrixcookbook.com
• Useful reading on missing data (for the case of varying Nobs): Little and Rubin, "Statistical Analysis with Missing Data", 2nd ed, Wiley, 2002, Chapters 6,7 and 11.

Definition at line 86 of file MultivariateFNormalSufficient.h.

Public Member Functions
	MultivariateFNormalSufficient (const Eigen::MatrixXd &FX, double JF, const Eigen::VectorXd &FM, const Eigen::MatrixXd &Sigma, double factor=1)
	Initialize with all observed data. More...
	MultivariateFNormalSufficient (const Eigen::VectorXd &Fbar, double JF, const Eigen::VectorXd &FM, int Nobs, const Eigen::MatrixXd &W, const Eigen::MatrixXd &Sigma, double factor=1)
	Initialize with sufficient statistics. More...
double	density () const
	Probability density function. More...
double	evaluate () const
	Energy (score) function, aka -log(p) More...
double	evaluate_derivative_factor () const
	derivative wrt scalar factor More...
Eigen::VectorXd	evaluate_derivative_FM () const
	gradient of the energy wrt the mean F(M) More...
Eigen::MatrixXd	evaluate_derivative_Sigma () const
	gradient of the energy wrt the variance-covariance matrix Sigma More...
Eigen::MatrixXd	evaluate_second_derivative_FM_FM () const
	second derivative wrt FM and FM More...
Eigen::MatrixXd	evaluate_second_derivative_FM_Sigma (unsigned l) const
	second derivative wrt FM(l) and Sigma More...
Eigen::MatrixXd	evaluate_second_derivative_Sigma_Sigma (unsigned k, unsigned l) const
	second derivative wrt Sigma and Sigma(k,l) More...
Eigen::VectorXd	get_epsilon () const
double	get_factor () const
Eigen::VectorXd	get_Fbar () const
Eigen::VectorXd	get_FM () const
Eigen::MatrixXd	get_FX () const
double	get_jacobian () const
double	get_log_generalized_variance () const
	return log |^2 | More...
double	get_mean_square_residuals () const
	return transpose(epsilon)*P*epsilon More...
double	get_minus_exponent () const
	return minus exponent More...
double	get_minus_log_jacobian () const
double	get_minus_log_normalization () const
	return minus log normalization More...
Eigen::MatrixXd	get_Sigma () const
double	get_Sigma_condition_number () const
	return Sigma's condition number More...
Eigen::VectorXd	get_Sigma_eigenvalues () const
	return Sigma's eigenvalues from smallest to biggest More...
virtual std::string	get_type_name () const override
virtual ::IMP::VersionInfo	get_version_info () const override
	Get information about the module and version of the object. More...
Eigen::MatrixXd	get_W () const
void	reset_flags ()
	if you want to force a recomputation of all stored variables More...
void	set_factor (double f)
void	set_Fbar (const Eigen::VectorXd &f)
void	set_FM (const Eigen::VectorXd &f)
void	set_FX (const Eigen::MatrixXd &f)
void	set_jacobian (double f)
void	set_minus_log_jacobian (double f)
void	set_Sigma (const Eigen::MatrixXd &f)
void	set_use_cg (bool use, double tol)
	use conjugate gradients (default false) More...
void	set_W (const Eigen::MatrixXd &f)
Eigen::MatrixXd	solve (Eigen::MatrixXd B) const
	Solve for Sigma*X = B, yielding X. More...
Public Member Functions inherited from IMP::Object
virtual void	clear_caches ()
CheckLevel	get_check_level () const
LogLevel	get_log_level () const
void	set_check_level (CheckLevel l)
void	set_log_level (LogLevel l)
	Set the logging level used in this object. More...
void	set_was_used (bool tf) const
void	show (std::ostream &out=std::cout) const
const std::string &	get_name () const
void	set_name (std::string name)

Additional Inherited Members
Protected Member Functions inherited from IMP::Object
	Object (std::string name)
	Construct an object with the given name. More...
virtual void	do_destroy ()

Constructor & Destructor Documentation

IMP::isd::MultivariateFNormalSufficient::MultivariateFNormalSufficient	(	const Eigen::MatrixXd &	FX,
		double	JF,
		const Eigen::VectorXd &	FM,
		const Eigen::MatrixXd &	Sigma,
		double	factor = 1
	)

Initialize with all observed data.

Parameters

[in]	FX	F(X) matrix of observations with M columns and N rows.
[in]	JF	J(F) determinant of Jacobian of F with respect to observation matrix X.
[in]	FM	F(M) mean vector \(F(\mu)\) of size M.
[in]	Sigma	: MxM variance-covariance matrix \(\Sigma\).
[in]	factor	: multiplicative factor (default 1)

IMP::isd::MultivariateFNormalSufficient::MultivariateFNormalSufficient	(	const Eigen::VectorXd &	Fbar,
		double	JF,
		const Eigen::VectorXd &	FM,
		int	Nobs,
		const Eigen::MatrixXd &	W,
		const Eigen::MatrixXd &	Sigma,
		double	factor = 1
	)

Initialize with sufficient statistics.

Parameters

[in]	Fbar	M-dimensional vector of mean observations.
[in]	JF	J(F) determinant of Jacobian of F with respect to observation matrix X.
[in]	FM	F(M) : M-dimensional true mean vector \(\mu\).
[in]	Nobs	number of observations for each variable.
[in]	W	MxM matrix of sample variance-covariances.
[in]	Sigma	MxM variance-covariance matrix Sigma.
[in]	factor	multiplicative factor (default 1)

Member Function Documentation

double IMP::isd::MultivariateFNormalSufficient::density

(

)

const

Probability density function.

double IMP::isd::MultivariateFNormalSufficient::evaluate

(

)

const

Energy (score) function, aka -log(p)

double IMP::isd::MultivariateFNormalSufficient::evaluate_derivative_factor

(

)

const

derivative wrt scalar factor

Eigen::VectorXd IMP::isd::MultivariateFNormalSufficient::evaluate_derivative_FM

(

)

const

gradient of the energy wrt the mean F(M)

Eigen::MatrixXd IMP::isd::MultivariateFNormalSufficient::evaluate_derivative_Sigma

(

)

const

gradient of the energy wrt the variance-covariance matrix Sigma

Eigen::MatrixXd IMP::isd::MultivariateFNormalSufficient::evaluate_second_derivative_FM_FM

(

)

const

second derivative wrt FM and FM

Eigen::MatrixXd IMP::isd::MultivariateFNormalSufficient::evaluate_second_derivative_FM_Sigma

(

unsigned

l

)

const

second derivative wrt FM(l) and Sigma

row and column indices in the matrix returned are for Sigma

Eigen::MatrixXd IMP::isd::MultivariateFNormalSufficient::evaluate_second_derivative_Sigma_Sigma	(	unsigned	k,
		unsigned	l
	)		const

second derivative wrt Sigma and Sigma(k,l)

double IMP::isd::MultivariateFNormalSufficient::get_log_generalized_variance

(

)

const

return log |^2 |

double IMP::isd::MultivariateFNormalSufficient::get_mean_square_residuals

(

)

const

return transpose(epsilon)*P*epsilon

double IMP::isd::MultivariateFNormalSufficient::get_minus_exponent

(

)

const

return minus exponent

\[-\frac{1}{2\sigma^2} \sum_{i=1}^N {}^t(F(\mu) - F(x_i))\Sigma^{-1}(F(\mu)-F(x_i)) \]

double IMP::isd::MultivariateFNormalSufficient::get_minus_log_normalization

(

)

const

return minus log normalization

\[\frac{N}{2}\left(\log(2\pi\sigma^2) + \log |\Sigma|\right) -\log J(F) \]

double IMP::isd::MultivariateFNormalSufficient::get_Sigma_condition_number

(

)

const

return Sigma's condition number

Eigen::VectorXd IMP::isd::MultivariateFNormalSufficient::get_Sigma_eigenvalues

(

)

const

return Sigma's eigenvalues from smallest to biggest

virtual ::IMP::VersionInfo IMP::isd::MultivariateFNormalSufficient::get_version_info ( ) const

overridevirtual

Get information about the module and version of the object.

Reimplemented from IMP::Object.

Definition at line 225 of file MultivariateFNormalSufficient.h.

void IMP::isd::MultivariateFNormalSufficient::reset_flags

(

)

if you want to force a recomputation of all stored variables

void IMP::isd::MultivariateFNormalSufficient::set_use_cg	(	bool	use,
		double	tol
	)

use conjugate gradients (default false)

Eigen::MatrixXd IMP::isd::MultivariateFNormalSufficient::solve

(

Eigen::MatrixXd

B

)

const

Solve for Sigma*X = B, yielding X.

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The documentation for this class was generated from the following file:

• MultivariateFNormalSufficient.h