MultivariateFNormalSufficient.h

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/**
 *  \file IMP/isd/MultivariateFNormalSufficient.h
 *  \brief Normal distribution of Function
 *
 *  Copyright 2007-2020 IMP Inventors. All rights reserved.
 */

#ifndef IMPISD_MULTIVARIATE_FNORMAL_SUFFICIENT_H
#define IMPISD_MULTIVARIATE_FNORMAL_SUFFICIENT_H

#include <IMP/isd/isd_config.h>
#include "internal/timer.h"
#include <IMP/macros.h>
#include <IMP/Model.h>
#include <IMP/constants.h>
#include <IMP/Object.h>
#include <math.h>
#include <Eigen/Dense>
#include <Eigen/Cholesky>
#include <IMP/isd/internal/cg_eigen.h>

IMPISD_BEGIN_NAMESPACE

// IMP_MVN_TIMER_NFUNCS is the number of functions used by the timer
#define IMP_MVN_TIMER_NFUNCS 11

//! MultivariateFNormalSufficient
/** Probability density function and -log(p) of multivariate normal
 * distribution of N M-variate observations.
 *
 * \f[ 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)
 *  \f]
 *  which is implemented as
 * \f[ 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)
 *  \f]
 *  where
 *  \f[\epsilon = (F(\mu)- \overline{F(x)}) \quad
 *  \overline{F(x)} = \frac{1}{N} \sum_{i=1}^N F(x_i)\f]
 *  and
 *  \f[W=\sum_{i=1}^N(F(x_i)-\overline{F(x)}){}^t(F(x_i)-\overline{F(x)}) \f]
 *
 * \f$\sigma\f$ is a multiplicative scaling factor that factors out of the
 * \f$\Sigma\f$ covariance matrix. It is set to 1 by default and its intent is
 * to avoid inverting the \f$\Sigma\f$ 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 \f$\mu\f$ and
 *  standard deviation \f$\Sigma\f$.
 *
 *  \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.
 *
 *  \note For now, F must be monotonically increasing, so that J(F) > 0. The
 *  program will not check for that. Uses a Cholesky (\f$LDL^T\f$)
 *  decomposition of \f$\Sigma\f$, which is recomputed when needed.
 *
 *  \note All observations must be given, so if you have missing data you might
 *  want to do some imputation on it first.
 *
 *  \note 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.
 *
 */

class IMPISDEXPORT MultivariateFNormalSufficient : public Object {

 private:
  Eigen::VectorXd FM_, Fbar_, epsilon_, Peps_;
  double JF_, lJF_, norm_, lnorm_;
  Eigen::MatrixXd P_, W_, Sigma_, FX_, PW_;
  int N_;  // number of repetitions
  int M_;  // number of variables
  // Eigen::LLT<Eigen::MatrixXd, Eigen::Upper> ldlt_;
  Eigen::LDLT<Eigen::MatrixXd, Eigen::Upper> ldlt_;
  // flags are true if the corresponding object is up to date.
  bool flag_FM_, flag_FX_, flag_Fbar_, flag_W_, flag_Sigma_, flag_epsilon_,
      flag_PW_, flag_P_, flag_ldlt_, flag_norms_, flag_Peps_;
  // cg-related variables
  Eigen::MatrixXd precond_;
  bool use_cg_, first_PW_, first_PWP_;
  double cg_tol_;
  double factor_;
  Pointer<internal::ConjugateGradientEigen> cg_;

  internal::CallTimer<IMP_MVN_TIMER_NFUNCS> timer_;

 public:
  //! Initialize with all observed data
  /** \param [in] FX F(X) matrix of observations with M columns and N rows.
      \param [in] JF J(F) determinant of Jacobian of F with respect to
                      observation matrix X.
      \param [in] FM F(M) mean vector \f$F(\mu)\f$ of size M.
      \param [in] Sigma : MxM variance-covariance matrix \f$\Sigma\f$.
      \param [in] factor : multiplicative factor (default 1)
   */
  MultivariateFNormalSufficient(const Eigen::MatrixXd& FX, double JF,
                                const Eigen::VectorXd& FM,
                                const Eigen::MatrixXd& Sigma,
                                double factor = 1);

  //! Initialize with sufficient statistics
  /** \param [in] Fbar M-dimensional vector of mean observations.
      \param [in] JF J(F) determinant of Jacobian of F with respect to
                  observation matrix X.
      \param [in] FM F(M) : M-dimensional true mean vector \f$\mu\f$.
      \param [in] Nobs number of observations for each variable.
      \param [in] W MxM matrix of sample variance-covariances.
      \param [in] Sigma MxM variance-covariance matrix Sigma.
      \param [in] factor multiplicative factor (default 1)
   */
  MultivariateFNormalSufficient(const Eigen::VectorXd& Fbar, double JF,
                                const Eigen::VectorXd& FM, int Nobs,
                                const Eigen::MatrixXd& W,
                                const Eigen::MatrixXd& Sigma,
                                double factor = 1);

  //! Probability density function
  double density() const;

  //! Energy (score) function, aka -log(p)
  double evaluate() const;

  //! gradient of the energy wrt the mean F(M)
  Eigen::VectorXd evaluate_derivative_FM() const;

  //! gradient of the energy wrt the variance-covariance matrix Sigma
  Eigen::MatrixXd evaluate_derivative_Sigma() const;

  //! derivative wrt scalar factor
  double evaluate_derivative_factor() const;

  //! second derivative wrt FM and FM
  Eigen::MatrixXd evaluate_second_derivative_FM_FM() const;

  //! second derivative wrt FM(l) and Sigma
  /** row and column indices in the matrix returned are for Sigma
   */
  Eigen::MatrixXd evaluate_second_derivative_FM_Sigma(unsigned l) const;

  //! second derivative wrt Sigma and Sigma(k,l)
  Eigen::MatrixXd evaluate_second_derivative_Sigma_Sigma(unsigned k,
                                                         unsigned l) const;

  /* change of parameters */
  void set_FX(const Eigen::MatrixXd& f);
  Eigen::MatrixXd get_FX() const;

  void set_Fbar(const Eigen::VectorXd& f);
  Eigen::VectorXd get_Fbar() const;

  Eigen::VectorXd get_epsilon() const;

  void set_jacobian(double f);
  double get_jacobian() const;
  void set_minus_log_jacobian(double f);  //-log(J)
  double get_minus_log_jacobian() const;

  void set_FM(const Eigen::VectorXd& f);
  Eigen::VectorXd get_FM() const;

  void set_W(const Eigen::MatrixXd& f);
  Eigen::MatrixXd get_W() const;

  void set_Sigma(const Eigen::MatrixXd& f);
  Eigen::MatrixXd get_Sigma() const;

  void set_factor(double f);
  double get_factor() const;

  //! if you want to force a recomputation of all stored variables
  void reset_flags();

  //! use conjugate gradients (default false)
  void set_use_cg(bool use, double tol);

  //! return Sigma's eigenvalues from smallest to biggest
  Eigen::VectorXd get_Sigma_eigenvalues() const;

  //! return Sigma's condition number
  double get_Sigma_condition_number() const;

  //! Solve for Sigma*X = B, yielding X
  Eigen::MatrixXd solve(Eigen::MatrixXd B) const;

  //! return transpose(epsilon)*P*epsilon
  double get_mean_square_residuals() const;

  //! return minus exponent
  /**  \f[-\frac{1}{2\sigma^2}
       \sum_{i=1}^N {}^t(F(\mu) - F(x_i))\Sigma^{-1}(F(\mu)-F(x_i)) \f]
   */
  double get_minus_exponent() const;

  //! return minus log normalization
  /** \f[\frac{N}{2}\left(\log(2\pi\sigma^2) + \log |\Sigma|\right)
      -\log J(F) \f]
   */
  double get_minus_log_normalization() const;

  //! return log |\sigma^2 \Sigma|
  double get_log_generalized_variance() const;

  /* remaining stuff */
  IMP_OBJECT_METHODS(MultivariateFNormalSufficient);

 private:
  // conjugate gradient init
  void setup_cg();

  // precision matrix
  Eigen::MatrixXd get_P() const;
  void set_P(const Eigen::MatrixXd& P);

  // precision * W
  Eigen::MatrixXd get_PW() const;
  Eigen::MatrixXd compute_PW_direct() const;
  Eigen::MatrixXd compute_PW_cg() const;
  void set_PW(const Eigen::MatrixXd& PW);

  // precision * epsilon
  Eigen::VectorXd get_Peps() const;
  void set_Peps(const Eigen::VectorXd& Peps);

  // epsilon = Fbar - FM
  void set_epsilon(const Eigen::VectorXd& eps);

  // gets factorization object
  // Eigen::LLT<Eigen::MatrixXd, Eigen::Upper> get_ldlt() const;
  Eigen::LDLT<Eigen::MatrixXd, Eigen::Upper> get_ldlt() const;
  void set_ldlt(
      const Eigen::LDLT<Eigen::MatrixXd, Eigen::Upper>& ldlt);

  // compute determinant and norm
  void set_norms(double norm, double lnorm);
  std::vector<double> get_norms() const;

  /* return trace(W.P) */
  double trace_WP() const;

  /* return P*epsilon*transpose(P*epsilon) */
  Eigen::MatrixXd compute_PTP() const;

  /* return P * W * P, O(M^2) */
  Eigen::MatrixXd compute_PWP() const;

  /*computes the discrepancy vector*/
  void compute_epsilon();
};

IMPISD_END_NAMESPACE

#endif /* IMPISD_MULTIVARIATE_FNORMAL_SUFFICIENT_H */