IMP  2.3.1
The Integrative Modeling Platform
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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 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]FXF(X) matrix of observations with M columns and N rows.
[in]JFJ(F) determinant of Jacobian of F with respect to observation matrix X.
[in]FMF(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]JFJ(F) determinant of Jacobian of F with respect to observation matrix X.
[in]FMF(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: