Detailed Description

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 IMP_Eigen::MatrixXd &FX, double JF, const IMP_Eigen::VectorXd &FM, const IMP_Eigen::MatrixXd &Sigma, double factor=1)

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

double	density () const

double	evaluate () const

double	evaluate_derivative_factor () const

IMP_Eigen::VectorXd	evaluate_derivative_FM () const

IMP_Eigen::MatrixXd	evaluate_derivative_Sigma () const

IMP_Eigen::MatrixXd	evaluate_second_derivative_FM_FM () const

IMP_Eigen::MatrixXd	evaluate_second_derivative_FM_Sigma (unsigned l) const

IMP_Eigen::MatrixXd	evaluate_second_derivative_Sigma_Sigma (unsigned k, unsigned l) const

IMP_Eigen::VectorXd	get_epsilon () const

double	get_factor () const

IMP_Eigen::VectorXd	get_Fbar () const

IMP_Eigen::VectorXd	get_FM () const

IMP_Eigen::MatrixXd	get_FX () const

double	get_jacobian () const

double	get_log_generalized_variance () const

double	get_mean_square_residuals () const

double	get_minus_exponent () const

double	get_minus_log_jacobian () const

double	get_minus_log_normalization () const

IMP_Eigen::MatrixXd	get_Sigma () const

double	get_Sigma_condition_number () const

IMP_Eigen::VectorXd	get_Sigma_eigenvalues () const

virtual std::string	get_type_name () const

virtual ::IMP::base::VersionInfo	get_version_info () const
	Get information about the module and version of the object. More...

IMP_Eigen::MatrixXd	get_W () const

void	reset_flags ()

void	set_factor (double f)

void	set_Fbar (const IMP_Eigen::VectorXd &f)

void	set_FM (const IMP_Eigen::VectorXd &f)

void	set_FX (const IMP_Eigen::MatrixXd &f)

void	set_jacobian (double f)

void	set_minus_log_jacobian (double f)

void	set_Sigma (const IMP_Eigen::MatrixXd &f)

void	set_use_cg (bool use, double tol)

void	set_W (const IMP_Eigen::MatrixXd &f)

IMP_Eigen::MatrixXd	solve (IMP_Eigen::MatrixXd B) const

void	stats () const

Public Member Functions inherited from IMP::base::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::base::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 IMP_Eigen::MatrixXd &	FX,
		double	JF,
		const IMP_Eigen::VectorXd &	FM,
		const IMP_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 IMP_Eigen::VectorXd &	Fbar,
		double	JF,
		const IMP_Eigen::VectorXd &	FM,
		int	Nobs,
		const IMP_Eigen::MatrixXd &	W,
		const IMP_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

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

virtual

Get information about the module and version of the object.

Reimplemented from IMP::base::Object.

Definition at line 228 of file MultivariateFNormalSufficient.h.

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

MultivariateFNormalSufficient.h

Detailed Description

Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

Member Function Documentation