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

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

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 $$ 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.

Inheritance diagram for IMP::isd::MultivariateFNormalSufficient:

List of all members.

Public Member Functions
	MultivariateFNormalSufficient (const MatrixXd &FX, double JF, const VectorXd &FM, const MatrixXd &Sigma, double factor=1)
	MultivariateFNormalSufficient (const VectorXd &Fbar, double JF, const VectorXd &FM, int Nobs, const MatrixXd &W, const MatrixXd &Sigma, double factor=1)
double	density () const
double	evaluate () const
double	evaluate_derivative_factor () const
VectorXd	evaluate_derivative_FM () const
MatrixXd	evaluate_derivative_Sigma () const
MatrixXd	evaluate_second_derivative_FM_FM () const
MatrixXd	evaluate_second_derivative_FM_Sigma (unsigned l) const
MatrixXd	evaluate_second_derivative_Sigma_Sigma (unsigned k, unsigned l) const
double	get_factor () const
VectorXd	get_Fbar () const
VectorXd	get_FM () const
MatrixXd	get_FX () const
double	get_jacobian () const
double	get_mean_square_residuals () const
double	get_minus_exponent () const
double	get_minus_log_jacobian () const
double	get_minus_log_normalization () const
MatrixXd	get_Sigma () const
double	get_Sigma_condition_number () const
VectorXd	get_Sigma_eigenvalues () const
MatrixXd	get_W () const
void	reset_flags ()
void	set_factor (double f)
void	set_Fbar (const VectorXd &f)
void	set_FM (const VectorXd &f)
void	set_FX (const MatrixXd &f)
void	set_jacobian (double f)
void	set_minus_log_jacobian (double f)
void	set_Sigma (const MatrixXd &f)
void	set_use_cg (bool use, double tol)
void	set_W (const MatrixXd &f)
MatrixXd	solve (MatrixXd B) const
void	stats () const
Public Member Functions inherited from IMP::base::Object
std::size_t	__hash__ () const
virtual std::string	get_type_name () const =0
	Return a string identifying the type of the object.
virtual IMP::base::VersionInfo	get_version_info () const =0
	Get information about the module and version of the object.
void	set_check_level (CheckLevel l)
void	set_log_level (LogLevel l)
	Set the logging level used in this object.
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)

Static Public Member Functions
static MultivariateFNormalSufficient *	get_from (IMP::base::Object *o)

Additional Inherited Members
Protected Member Functions inherited from IMP::base::Object
	Object (std::string name)

Constructor & Destructor Documentation

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

Initialize with all observed data

Parameters:

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

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

Initialize with sufficient statistics

Parameters:

in)	Fbar : M-dimensional vector of mean observations.
in)	J(F) determinant of Jacobian of F with respect to observation matrix X.
in)	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)	sigma : multiplicative factor (default 1)

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

MultivariateFNormalSufficient.h

Detailed Description

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation