GaussianProcessInterpolation.h

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/**
 *  \file IMP/isd/GaussianProcessInterpolation.h
 *  \brief Normal distribution of Function
 *
 *  Copyright 2007-2015 IMP Inventors. All rights reserved.
 */

#ifndef IMPISD_GAUSSIAN_PROCESS_INTERPOLATION_H
#define IMPISD_GAUSSIAN_PROCESS_INTERPOLATION_H

#include <IMP/isd/isd_config.h>
#include <IMP/macros.h>
#include <boost/scoped_ptr.hpp>
#include <IMP/isd/univariate_functions.h>
#include <IMP/isd/bivariate_functions.h>
#include <IMP/isd/Scale.h>
#include <IMP/algebra/eigen3/Eigen/Dense>
#include <IMP/algebra/eigen3/Eigen/Cholesky>

IMPISD_BEGIN_NAMESPACE

class GaussianProcessInterpolationRestraint;
class GaussianProcessInterpolationScoreState;

//! GaussianProcessInterpolation
/** This class provides methods to perform bayesian interpolation/smoothing of
 * data using a gaussian process prior on the function to be interpolated.
 * It takes a dataset as input (via its sufficient statistics) along with prior
 * mean and covariance functions. It outputs the value of the posterior mean and
 * covariance functions at points requested by the user.
 */
class IMPISDEXPORT GaussianProcessInterpolation : public base::Object {
 public:
  /** Constructor for the gaussian process
   * \param [in] x : a list of coordinates in N-dimensional space
   *                corresponding to the abscissa of each observation
   * \param [in] sample_mean \f$I\f$ : vector of mean observations at each of
   *                                  the previously defined coordinates
   * \param [in] sample_std \f$s\f$ : vector of sample standard deviations
   * \param [in] n_obs \f$N\f$ : sample size for mean and std
   * \param [in] mean_function \f$m\f$ : a pointer to the prior mean function
   *                                    to use.  Should be compatible with
   *                                    the size of x(i).
   * \param [in] covariance_function \f$w\f$: prior covariance function.
   * \param [in] sigma : ISD Scale (proportionality factor to S)
   * \param [in] sparse_cutoff : when to consider that a matrix entry is zero
   *
   * Computes the necessary matrices and inverses when called.
   */
  GaussianProcessInterpolation(FloatsList x, Floats sample_mean,
                               Floats sample_std, unsigned n_obs,
                               UnivariateFunction *mean_function,
                               BivariateFunction *covariance_function,
                               kernel::Particle *sigma,
                               double sparse_cutoff = 1e-7);

  /** Get posterior mean and covariance functions, at the points requested
   * Posterior mean is defined as
   * \f[\hat{I}(x) = m(x)
   *        + {}^t\mathbf{w}(q)
   *        (\mathbf{W}+\mathbf{S})^{-1}
   *        (\mathbf{I}-\mathbf{m}) \f]
   * Posterior covariance
   * \f[\hat{\sigma}^2(x,x') =
   *    w(x,x') - {}^t\mathbf{w}(x)
   *              (\mathbf{W} + \mathbf{S})^{-1}
   *              \mathbf{w}(x') \f]
   * where \f$\mathbf{m}\f$ is the vector built by evaluating the prior mean
   * function at the observation points; \f$\mathbf{w}(x)\f$ is the vector of
   * covariances between each observation point and the current point;
   * \f$\mathbf{W}\f$ is the prior covariance matrix built by evaluating the
   * covariance function at each of the observations; \f$\mathbf{S}\f$ is the
   * diagonal covariance matrix built from sample_std and n_obs.
   *
   * Both functions will check if the mean or covariance functions have
   *changed
   * since the last call, and will recompute
   * \f$(\mathbf{W} + \mathbf{S})^{-1}\f$ if necessary.
   */
  double get_posterior_mean(Floats x) const;
  double get_posterior_covariance(Floats x1, Floats x2) const;
#ifndef SWIG
  // c++ only
  IMP_Eigen::MatrixXd get_posterior_covariance_matrix(FloatsList x) const;
#endif
  // for Python
  FloatsList get_posterior_covariance_matrix(FloatsList x, bool) const;

#ifndef SWIG
// derivative: d(mean(q))/(dparticle_i)
// IMP_Eigen::VectorXd get_posterior_mean_derivative(Floats x) const;
#endif

#ifndef SWIG
  // derivative: d(cov(q,q))/(dparticle_i)
  IMP_Eigen::VectorXd get_posterior_covariance_derivative(Floats x) const;
#endif
  // for Python
  Floats get_posterior_covariance_derivative(Floats x, bool) const;

#ifndef SWIG
// hessian: d(mean(q))/(dparticle_i dparticle_j)
// IMP_Eigen::MatrixXd get_posterior_mean_hessian(Floats x) const;
#endif

#ifndef SWIG
  // hessian: d(cov(q,q))/(dparticle_i dparticle_j)
  IMP_Eigen::MatrixXd get_posterior_covariance_hessian(Floats x) const;
#endif
  // for Python
  FloatsList get_posterior_covariance_hessian(Floats x, bool) const;

  // needed for restraints using gpi
  kernel::ParticlesTemp get_input_particles() const;
  ContainersTemp get_input_containers() const;

  // call these if you called update() on the mean or covariance function.
  // it will force update any internal variables dependent on these functions.
  void force_mean_update();
  void force_covariance_update();

  // returns the number of particles that control m's values.
  // public for testing purposes
  unsigned get_number_of_m_particles() const;

  // returns true if a particle is optimized
  // public for testing purposes
  bool get_m_particle_is_optimized(unsigned i) const;

  // returns the number of particles that control Omega's values.
  // public for testing purposes
  unsigned get_number_of_Omega_particles() const;

  // returns true if a particle is optimized
  // public for testing purposes
  bool get_Omega_particle_is_optimized(unsigned i) const;

  // returns data
  FloatsList get_data_abscissa() const;
  Floats get_data_mean() const;
  FloatsList get_data_variance() const;

  friend class GaussianProcessInterpolationRestraint;
  friend class GaussianProcessInterpolationScoreState;

  IMP_OBJECT_METHODS(GaussianProcessInterpolation);

 protected:
  // returns updated data vector
  IMP_Eigen::VectorXd get_I() const { return I_; }
  // returns updated prior mean vector
  IMP_Eigen::VectorXd get_m() const;
  // returns dm/dparticle
  IMP_Eigen::VectorXd get_m_derivative(unsigned particle) const;
  // returns d2m/(dparticle_1 dparticle_2)
  IMP_Eigen::VectorXd get_m_second_derivative(unsigned particle1,
                                              unsigned particle2) const;
  // returns updated prior covariance vector
  void add_to_m_particle_derivative(unsigned particle, double value,
                                    DerivativeAccumulator &accum);
  // returns updated prior covariance vector
  IMP_Eigen::VectorXd get_wx_vector(Floats xval) const;
  // returns updated data covariance matrix
  IMP_Eigen::DiagonalMatrix<double, IMP_Eigen::Dynamic> get_S() const {
    return S_;
  }
  // returns updated prior covariance matrix
  IMP_Eigen::MatrixXd get_W() const;
  // returns Omega=(W+S/N)
  IMP_Eigen::MatrixXd get_Omega() const;
  // returns dOmega/dparticle
  IMP_Eigen::MatrixXd get_Omega_derivative(unsigned particle) const;
  // returns d2Omega/(dparticle_1 dparticle_2)
  IMP_Eigen::MatrixXd get_Omega_second_derivative(unsigned particle1,
                                                  unsigned particle2) const;
  // returns updated prior covariance vector
  void add_to_Omega_particle_derivative(unsigned particle, double value,
                                        DerivativeAccumulator &accum);
  // returns LDLT decomp of omega
  IMP_Eigen::LDLT<IMP_Eigen::MatrixXd, IMP_Eigen::Upper> get_ldlt() const;
  // returns updated Omega^{-1}
  IMP_Eigen::MatrixXd get_Omi() const;
  // returns updated Omega^{-1}(I-m)
  IMP_Eigen::VectorXd get_OmiIm() const;

 private:
  // ensures the mean/covariance function has updated parameters. Signals an
  // update by changing the state flags. Returns true if the function has
  // changed. This is used by GaussianProcessInterpolationRestraint.
  void update_flags_mean();
  void update_flags_covariance();

  // compute prior covariance matrix
  void compute_W();
  // compute \f$(\mathbf{W} + \frac{\sigma}{N}\mathbf{S})^{-1}\f$.
  void compute_Omega();
  // compute LDLT decomposition of Omega
  void compute_ldlt();
  // compute \f$(\mathbf{W} + \frac{\sigma}{N}\mathbf{S})^{-1}\f$.
  void compute_Omi();
  // compute (W+sigma*S/N)^{-1} (I-m)
  void compute_OmiIm();

  // compute dw(q)/dparticle_i
  IMP_Eigen::VectorXd get_wx_vector_derivative(Floats q, unsigned i) const;
  // compute dw(q)/(dparticle_i * dparticle_j)
  IMP_Eigen::VectorXd get_wx_vector_second_derivative(Floats q, unsigned i,
                                                      unsigned j) const;

  // compute dcov(q,q)/dw(q)
  IMP_Eigen::VectorXd get_dcov_dwq(Floats q) const;
  // compute dcov(q,q)/dOmega
  IMP_Eigen::MatrixXd get_dcov_dOm(Floats q) const;
  // compute d2cov(q,q)/(dw(q) dw(q)) (independent of q !)
  IMP_Eigen::MatrixXd get_d2cov_dwq_dwq() const;
  // compute d2cov(q,q)/(dw(q)_m dOmega)
  IMP_Eigen::MatrixXd get_d2cov_dwq_dOm(Floats q, unsigned m) const;
  // compute d2cov(q,q)/(dOmega dOmega_mn)
  IMP_Eigen::MatrixXd get_d2cov_dOm_dOm(Floats q, unsigned m, unsigned n) const;

  // compute mean observations
  void compute_I(Floats mean);
  // compute diagonal covariance matrix of observations
  void compute_S(Floats std);
  // compute prior mean vector
  void compute_m();

 private:
  unsigned N_;      // number of dimensions of the abscissa
  unsigned M_;      // number of observations to learn from
  FloatsList x_;    // abscissa
  unsigned n_obs_;  // number of observations
  // pointer to the prior mean function
  IMP::base::PointerMember<UnivariateFunction> mean_function_;
  // pointer to the prior covariance function
  IMP::base::PointerMember<BivariateFunction> covariance_function_;
  IMP_Eigen::VectorXd I_, m_;
  IMP_Eigen::MatrixXd W_, Omega_, Omi_;  // Omi = Omega^{-1}
  IMP_Eigen::DiagonalMatrix<double, IMP_Eigen::Dynamic> S_;
  IMP_Eigen::VectorXd OmiIm_;  // Omi * (I - m)
  bool flag_m_, flag_m_gpir_, flag_Omi_, flag_OmiIm_, flag_W_, flag_Omega_,
      flag_Omega_gpir_, flag_ldlt_;
  IMP::base::Pointer<IMP::kernel::Particle> sigma_;
  double cutoff_;
  double sigma_val_;  // to know if an update is needed
  IMP_Eigen::LDLT<IMP_Eigen::MatrixXd, IMP_Eigen::Upper> ldlt_;
};

IMPISD_END_NAMESPACE

#endif /* IMPISD_GAUSSIAN_PROCESS_INTERPOLATION_H */