root/library/bdm/estim/kalman.h @ 737

/*!
  \file
  \brief Bayesian Filtering for linear Gaussian models (Kalman Filter) and extensions
  \author Vaclav Smidl.

  -----------------------------------
  BDM++ - C++ library for Bayesian Decision Making under Uncertainty

  Using IT++ for numerical operations
  -----------------------------------
*/

#ifndef KF_H
#define KF_H


#include "../math/functions.h"
#include "../stat/exp_family.h"
#include "../math/chmat.h"
#include "../base/user_info.h"
//#include <../applications/pmsm/simulator_zdenek/ekf_example/pmsm_mod.h>

namespace bdm {

/*!
 * \brief Basic elements of linear state-space model

Parameter evolution model:\f[ x_{t+1} = A x_{t} + B u_t + Q^{1/2} e_t \f]
Observation model: \f[ y_t = C x_{t} + C u_t + R^{1/2} w_t. \f]
Where $e_t$ and $w_t$ are mutually independent vectors of Normal(0,1)-distributed disturbances.
 */
template<class sq_T>
class StateSpace {
protected:
        //! Matrix A
        mat A;
        //! Matrix B
        mat B;
        //! Matrix C
        mat C;
        //! Matrix D
        mat D;
        //! Matrix Q in square-root form
        sq_T Q;
        //! Matrix R in square-root form
        sq_T R;
public:
        StateSpace() :  A(), B(), C(), D(), Q(), R() {}
        //!copy constructor
        StateSpace ( const StateSpace<sq_T> &S0 ) :  A ( S0.A ), B ( S0.B ), C ( S0.C ), D ( S0.D ), Q ( S0.Q ), R ( S0.R ) {}
        //! set all matrix parameters
        void set_parameters ( const mat &A0, const  mat &B0, const  mat &C0, const  mat &D0, const  sq_T &Q0, const sq_T &R0 );
        //! validation
        void validate();
        //! not virtual in this case
        void from_setting ( const Setting &set ) {
                UI::get ( A, set, "A", UI::compulsory );
                UI::get ( B, set, "B", UI::compulsory );
                UI::get ( C, set, "C", UI::compulsory );
                UI::get ( D, set, "D", UI::compulsory );
                mat Qtm, Rtm; // full matrices
                if ( !UI::get ( Qtm, set, "Q", UI::optional ) ) {
                        vec dq;
                        UI::get ( dq, set, "dQ", UI::compulsory );
                        Qtm = diag ( dq );
                }
                if ( !UI::get ( Rtm, set, "R", UI::optional ) ) {
                        vec dr;
                        UI::get ( dr, set, "dQ", UI::compulsory );
                        Rtm = diag ( dr );
                }
                R = Rtm; // automatic conversion to square-root form
                Q = Qtm;

                validate();
        }
        //! access function
        const mat& _A() const {
                return A;
        }
        //! access function
        const mat& _B() const {
                return B;
        }
        //! access function
        const mat& _C() const {
                return C;
        }
        //! access function
        const mat& _D() const {
                return D;
        }
        //! access function
        const sq_T& _Q() const {
                return Q;
        }
        //! access function
        const sq_T& _R() const {
                return R;
        }
};

//! Common abstract base for Kalman filters
template<class sq_T>
class Kalman: public BM, public StateSpace<sq_T> {
protected:
        //! id of output
        RV yrv;
        //! Kalman gain
        mat  _K;
        //!posterior
        enorm<sq_T> est;
        //!marginal on data f(y|y)
        enorm<sq_T>  fy;
public:
        Kalman<sq_T>() : BM(), StateSpace<sq_T>(), yrv(), _K(),  est() {}
        //! Copy constructor
        Kalman<sq_T> ( const Kalman<sq_T> &K0 ) : BM ( K0 ), StateSpace<sq_T> ( K0 ), yrv ( K0.yrv ), _K ( K0._K ),  est ( K0.est ), fy ( K0.fy ) {}
        //!set statistics of the posterior
        void set_statistics ( const vec &mu0, const mat &P0 ) {
                est.set_parameters ( mu0, P0 );
        };
        //!set statistics of the posterior
        void set_statistics ( const vec &mu0, const sq_T &P0 ) {
                est.set_parameters ( mu0, P0 );
        };
        //! return correctly typed posterior (covariant return)
        const enorm<sq_T>& posterior() const {
                return est;
        }
        //! load basic elements of Kalman from structure
        void from_setting ( const Setting &set ) {
                StateSpace<sq_T>::from_setting ( set );

                mat P0;
                vec mu0;
                UI::get ( mu0, set, "mu0", UI::optional );
                UI::get ( P0, set,  "P0", UI::optional );
                set_statistics ( mu0, P0 );
                // Initial values
                UI::get ( yrv, set, "yrv", UI::optional );
                UI::get ( rvc, set, "urv", UI::optional );
                set_yrv ( concat ( yrv, rvc ) );

                validate();
        }
        //! validate object
        void validate() {
                StateSpace<sq_T>::validate();
                dimy = this->C.rows();
                dimc = this->B.cols();
                set_dim ( this->A.rows() );

                bdm_assert ( est.dimension(), "Statistics and model parameters mismatch" );
        }
};
/*!
* \brief Basic Kalman filter with full matrices
*/

class KalmanFull : public Kalman<fsqmat> {
public:
        //! For EKFfull;
        KalmanFull() : Kalman<fsqmat>() {};
        //! Here dt = [yt;ut] of appropriate dimensions
        void bayes ( const vec &yt, const vec &cond = empty_vec );
        BM* _copy_() const {
                KalmanFull* K = new KalmanFull;
                K->set_parameters ( A, B, C, D, Q, R );
                K->set_statistics ( est._mu(), est._R() );
                return K;
        }
};
UIREGISTER ( KalmanFull );


/*! \brief Kalman filter in square root form

Trivial example:
\include kalman_simple.cpp

Complete constructor:
*/
class KalmanCh : public Kalman<chmat> {
protected:
        //! @{ \name Internal storage - needs initialize()
        //! pre array (triangular matrix)
        mat preA;
        //! post array (triangular matrix)
        mat postA;
        //!@}
public:
        //! copy constructor
        BM* _copy_() const {
                KalmanCh* K = new KalmanCh;
                K->set_parameters ( A, B, C, D, Q, R );
                K->set_statistics ( est._mu(), est._R() );
                K->validate();
                return K;
        }
        //! set parameters for adapt from Kalman
        void set_parameters ( const mat &A0, const mat &B0, const mat &C0, const mat &D0, const chmat &Q0, const chmat &R0 );
        //! initialize internal parametetrs
        void initialize();

        /*!\brief  Here dt = [yt;ut] of appropriate dimensions

        The following equality hold::\f[
        \left[\begin{array}{cc}
        R^{0.5}\\
        P_{t|t-1}^{0.5}C' & P_{t|t-1}^{0.5}CA'\\
        & Q^{0.5}\end{array}\right]<\mathrm{orth.oper.}>=\left[\begin{array}{cc}
        R_{y}^{0.5} & KA'\\
        & P_{t+1|t}^{0.5}\\
        \\\end{array}\right]\f]

        Thus this object evaluates only predictors! Not filtering densities.
        */
        void bayes ( const vec &yt, const vec &cond = empty_vec );

        void from_setting ( const Setting &set ) {
                Kalman<chmat>::from_setting ( set );
                validate();
        }
        void validate() {
                Kalman<chmat>::validate();
                initialize();
        }
};
UIREGISTER ( KalmanCh );

/*!
\brief Extended Kalman Filter in full matrices

An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean.
*/
class EKFfull : public KalmanFull {
protected:
        //! Internal Model f(x,u)
        shared_ptr<diffbifn> pfxu;

        //! Observation Model h(x,u)
        shared_ptr<diffbifn> phxu;

public:
        //! Default constructor
        EKFfull ();

        //! Set nonlinear functions for mean values and covariance matrices.
        void set_parameters ( const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const mat Q0, const mat R0 );

        //! Here dt = [yt;ut] of appropriate dimensions
        void bayes ( const vec &yt, const vec &cond = empty_vec );
        //! set estimates
        void set_statistics ( const vec &mu0, const mat &P0 ) {
                est.set_parameters ( mu0, P0 );
        };
        //! access function
        const mat _R() {
                return est._R().to_mat();
        }
        void from_setting ( const Setting &set ) {
                BM::from_setting ( set );
                shared_ptr<diffbifn> IM = UI::build<diffbifn> ( set, "IM", UI::compulsory );
                shared_ptr<diffbifn> OM = UI::build<diffbifn> ( set, "OM", UI::compulsory );

                //statistics
                int dim = IM->dimension();
                vec mu0;
                if ( !UI::get ( mu0, set, "mu0" ) )
                        mu0 = zeros ( dim );

                mat P0;
                vec dP0;
                if ( UI::get ( dP0, set, "dP0" ) )
                        P0 = diag ( dP0 );
                else if ( !UI::get ( P0, set, "P0" ) )
                        P0 = eye ( dim );

                set_statistics ( mu0, P0 );

                //parameters
                vec dQ, dR;
                UI::get ( dQ, set, "dQ", UI::compulsory );
                UI::get ( dR, set, "dR", UI::compulsory );
                set_parameters ( IM, OM, diag ( dQ ), diag ( dR ) );

                string options;
                if ( UI::get ( options, set, "options" ) )
                        set_options ( options );
//                      pfxu = UI::build<diffbifn>(set, "IM", UI::compulsory);
//                      phxu = UI::build<diffbifn>(set, "OM", UI::compulsory);
//
//                      mat R0;
//                      UI::get(R0, set, "R",UI::compulsory);
//                      mat Q0;
//                      UI::get(Q0, set, "Q",UI::compulsory);
//
//
//                      mat P0; vec mu0;
//                      UI::get(mu0, set, "mu0", UI::optional);
//                      UI::get(P0, set,  "P0", UI::optional);
//                      set_statistics(mu0,P0);
//                      // Initial values
//                      UI::get (yrv, set, "yrv", UI::optional);
//                      UI::get (urv, set, "urv", UI::optional);
//                      set_drv(concat(yrv,urv));
//
//                      // setup StateSpace
//                      pfxu->dfdu_cond(mu0, zeros(pfxu->_dimu()), A,true);
//                      phxu->dfdu_cond(mu0, zeros(pfxu->_dimu()), C,true);
//
                validate();
        }
        void validate() {
                // check stats and IM and OM
        }
};
UIREGISTER ( EKFfull );


/*!
\brief Extended Kalman Filter in Square root

An approximation of the exact Bayesian filter with Gaussian noices and non-linear evolutions of their mean.
*/

class EKFCh : public KalmanCh {
protected:
        //! Internal Model f(x,u)
        shared_ptr<diffbifn> pfxu;

        //! Observation Model h(x,u)
        shared_ptr<diffbifn> phxu;
public:
        //! copy constructor duplicated - calls different set_parameters
        BM* _copy_() const {
                EKFCh* E = new EKFCh;
                E->set_parameters ( pfxu, phxu, Q, R );
                E->set_statistics ( est._mu(), est._R() );
                return E;
        }
        //! Set nonlinear functions for mean values and covariance matrices.
        void set_parameters ( const shared_ptr<diffbifn> &pfxu, const shared_ptr<diffbifn> &phxu, const chmat Q0, const chmat R0 );

        //! Here dt = [yt;ut] of appropriate dimensions
        void bayes ( const vec &yt, const vec &cond = empty_vec );

        void from_setting ( const Setting &set );

        void validate() {};
        // TODO dodelat void to_setting( Setting &set ) const;

};

UIREGISTER ( EKFCh );
SHAREDPTR ( EKFCh );


//////// INstance

/*! \brief (Switching) Multiple Model
The model runs several models in parallel and evaluates thier weights (fittness).

The statistics of the resulting density are merged using (geometric?) combination.

The next step is performed with the new statistics for all models.
*/
class MultiModel: public BM {
protected:
        //! List of models between which we switch
        Array<EKFCh*> Models;
        //! vector of model weights
        vec w;
        //! cache of model lls
        vec _lls;
        //! type of switching policy [1=maximum,2=...]
        int policy;
        //! internal statistics
        enorm<chmat> est;
public:
        //! set internal parameters
        void set_parameters ( Array<EKFCh*> A, int pol0 = 1 ) {
                Models = A;//TODO: test if evalll is set
                w.set_length ( A.length() );
                _lls.set_length ( A.length() );
                policy = pol0;

                est.set_rv ( RV ( "MM", A ( 0 )->posterior().dimension(), 0 ) );
                est.set_parameters ( A ( 0 )->posterior().mean(), A ( 0 )->posterior()._R() );
        }
        void bayes ( const vec &yt, const vec &cond = empty_vec ) {
                int n = Models.length();
                int i;
                for ( i = 0; i < n; i++ ) {
                        Models ( i )->bayes ( yt );
                        _lls ( i ) = Models ( i )->_ll();
                }
                double mlls = max ( _lls );
                w = exp ( _lls - mlls );
                w /= sum ( w ); //normalization
                //set statistics
                switch ( policy ) {
                case 1: {
                        int mi = max_index ( w );
                        const enorm<chmat> &st = Models ( mi )->posterior() ;
                        est.set_parameters ( st.mean(), st._R() );
                }
                break;
                default:
                        bdm_error ( "unknown policy" );
                }
                // copy result to all models
                for ( i = 0; i < n; i++ ) {
                        Models ( i )->set_statistics ( est.mean(), est._R() );
                }
        }
        //! return correctly typed posterior (covariant return)
        const enorm<chmat>& posterior() const {
                return est;
        }

        void from_setting ( const Setting &set );

        // TODO dodelat void to_setting( Setting &set ) const;

};

UIREGISTER ( MultiModel );
SHAREDPTR ( MultiModel );

//! conversion of outer ARX model (mlnorm) to state space model
/*!
The model is constructed as:
\f[ x_{t+1} = Ax_t + B u_t + R^{1/2} e_t, y_t=Cx_t+Du_t + R^{1/2}w_t, \f]
For example, for:
Using Frobenius form, see [].

For easier use in the future, indeces theta_in_A and theta_in_C are set. TODO - explain
*/
//template<class sq_T>
class StateCanonical: public StateSpace<fsqmat> {
protected:
        //! remember connection from theta ->A
        datalink_part th2A;
        //! remember connection from theta ->C
        datalink_part th2C;
        //! remember connection from theta ->D
        datalink_part th2D;
        //!cached first row of A
        vec A1row;
        //!cached first row of C
        vec C1row;
        //!cached first row of D
        vec D1row;

public:
        //! set up this object to match given mlnorm
        void connect_mlnorm ( const mlnorm<fsqmat> &ml ) {
                //get ids of yrv
                const RV &yrv = ml._rv();
                //need to determine u_t - it is all in _rvc that is not in ml._rv()
                RV rgr0 = ml._rvc().remove_time();
                RV urv = rgr0.subt ( yrv );

                //We can do only 1d now... :(
                bdm_assert ( yrv._dsize() == 1, "Only for SISO so far..." );

                // create names for
                RV xrv; //empty
                RV Crv; //empty
                int td = ml._rvc().mint();
                // assuming strictly proper function!!!
                for ( int t = -1; t >= td; t-- ) {
                        xrv.add ( yrv.copy_t ( t ) );
                        Crv.add ( urv.copy_t ( t ) );
                }

                // get mapp
                th2A.set_connection ( xrv, ml._rvc() );
                th2C.set_connection ( Crv, ml._rvc() );
                th2D.set_connection ( urv, ml._rvc() );

                //set matrix sizes
                this->A = zeros ( xrv._dsize(), xrv._dsize() );
                for ( int j = 1; j < xrv._dsize(); j++ ) {
                        A ( j, j - 1 ) = 1.0;    // off diagonal
                }
                this->B = zeros ( xrv._dsize(), 1 );
                this->B ( 0 ) = 1.0;
                this->C = zeros ( 1, xrv._dsize() );
                this->D = zeros ( 1, urv._dsize() );
                this->Q = zeros ( xrv._dsize(), xrv._dsize() );
                // R is set by update

                //set cache
                this->A1row = zeros ( xrv._dsize() );
                this->C1row = zeros ( xrv._dsize() );
                this->D1row = zeros ( urv._dsize() );

                update_from ( ml );
                validate();
        };
        //! fast function to update parameters from ml - not checked for compatibility!!
        void update_from ( const mlnorm<fsqmat> &ml ) {

                vec theta = ml._A().get_row ( 0 ); // this

                th2A.filldown ( theta, A1row );
                th2C.filldown ( theta, C1row );
                th2D.filldown ( theta, D1row );

                R = ml._R();

                A.set_row ( 0, A1row );
                C.set_row ( 0, C1row + D1row ( 0 ) *A1row );
                D.set_row ( 0, D1row );

        }
};
/*!
State-Space representation of multivariate autoregressive model.
The original model:
\f[ y_t = \theta [\ldots y_{t-k}, \ldots u_{t-l}, \ldots z_{t-m}]' + \Sigma^{-1/2} e_t \f]
where \f$ k,l,m \f$ are maximum delayes of corresponding variables in the regressor.

The transformed state is:
\f[ x_t = [y_{t} \ldots y_{t-k-1}, u_{t} \ldots u_{t-l-1}, z_{t} \ldots z_{t-m-1}]\f]

The state accumulates all delayed values starting from time \f$ t \f$ .


*/
class StateFromARX: public StateSpace<chmat> {
protected:
        //! remember connection from theta ->A
        datalink_part th2A;
        //! remember connection from theta ->B
        datalink_part th2B;
        //!function adds n diagonal elements from given starting point r,c
        void diagonal_part ( mat &A, int r, int c, int n ) {
                for ( int i = 0; i < n; i++ ) {
                        A ( r, c ) = 1.0;
                        r++;
                        c++;
                }
        };
        //! similar to ARX.have_constant
        bool have_constant;
public:
        //! set up this object to match given mlnorm
        //! Note that state-space and common mpdf use different meaning of \f$ _t \f$ in \f$ u_t \f$.
        //!While mlnorm typically assumes that \f$ u_t \rightarrow y_t \f$ in state space it is \f$ u_{t-1} \rightarrow y_t \f$
        //! For consequences in notation of internal variable xt see arx2statespace_notes.lyx.
        void connect_mlnorm ( const mlnorm<chmat> &ml, RV &xrv, RV &urv ) {

                //get ids of yrv
                const RV &yrv = ml._rv();
                //need to determine u_t - it is all in _rvc that is not in ml._rv()
                const RV &rgr = ml._rvc();
                RV rgr0 = rgr.remove_time();
                urv = rgr0.subt ( yrv );

                // create names for state variables
                xrv = yrv;

                int y_multiplicity = -rgr.mint ( yrv );
                int y_block_size = yrv.length() * ( y_multiplicity ); // current yt + all delayed yts
                for ( int m = 0; m < y_multiplicity - 1; m++ ) { // ========= -1 is important see arx2statespace_notes
                        xrv.add ( yrv.copy_t ( -m - 1 ) ); //add delayed yt
                }
                //! temporary RV for connection to ml.rvc, since notation of xrv and ml.rvc does not match
                RV xrv_ml = xrv.copy_t ( -1 );

                // add regressors
                ivec u_block_sizes ( urv.length() ); // no_blocks = yt + unique rgr
                for ( int r = 0; r < urv.length(); r++ ) {
                        RV R = urv.subselect ( vec_1 ( r ) ); //non-delayed element of rgr
                        int r_size = urv.size ( r );
                        int r_multiplicity = -rgr.mint ( R );
                        u_block_sizes ( r ) = r_size * r_multiplicity ;
                        for ( int m = 0; m < r_multiplicity; m++ ) {
                                xrv.add ( R.copy_t ( -m - 1 ) ); //add delayed yt
                                xrv_ml.add ( R.copy_t ( -m - 1 ) ); //add delayed yt
                        }
                }
                // add constant
                if ( any ( ml._mu_const() != 0.0 ) ) {
                        have_constant = true;
                        xrv.add ( RV ( "bdm_reserved_constant_one", 1 ) );
                } else {
                        have_constant = false;
                }


                // get mapp
                th2A.set_connection ( xrv_ml, ml._rvc() );
                th2B.set_connection ( urv, ml._rvc() );

                //set matrix sizes
                this->A = zeros ( xrv._dsize(), xrv._dsize() );
                //create y block
                diagonal_part ( this->A, yrv._dsize(), 0, y_block_size - yrv._dsize() );

                this->B = zeros ( xrv._dsize(), urv._dsize() );
                //add diagonals for rgr
                int active_x = y_block_size;
                for ( int r = 0; r < urv.length(); r++ ) {
                        diagonal_part ( this->A, active_x + urv.size ( r ), active_x, u_block_sizes ( r ) - urv.size ( r ) );
                        this->B.set_submatrix ( active_x, 0, eye ( urv.size ( r ) ) );
                        active_x += u_block_sizes ( r );
                }
                this->C = zeros ( yrv._dsize(), xrv._dsize() );
                this->C.set_submatrix ( 0, 0, eye ( yrv._dsize() ) );
                this->D = zeros ( yrv._dsize(), urv._dsize() );
                this->R.setCh ( zeros ( yrv._dsize(), yrv._dsize() ) );
                this->Q.setCh ( zeros ( xrv._dsize(), xrv._dsize() ) );
                // Q is set by update

                update_from ( ml );
                validate();
        };
        //! fast function to update parameters from ml - not checked for compatibility!!
        void update_from ( const mlnorm<chmat> &ml ) {

                vec Arow = zeros ( A.cols() );
                vec Brow = zeros ( B.cols() );
                //  ROW- WISE EVALUATION =====
                for ( int i = 0; i < ml._rv()._dsize(); i++ ) {

                        vec theta = ml._A().get_row ( i );

                        th2A.filldown ( theta, Arow );
                        if ( have_constant ) {
                                // constant is always at the end no need for datalink
                                Arow ( A.cols() - 1 ) = ml._mu_const() ( i );
                        }
                        this->A.set_row ( i, Arow );

                        th2B.filldown ( theta, Brow );
                        this->B.set_row ( i, Brow );
                }
                this->Q._Ch().set_submatrix ( 0, 0, ml.__R()._Ch() );

        };
        //! access function
        bool _have_constant() const {
                return have_constant;
        }
};

/////////// INSTANTIATION

template<class sq_T>
void StateSpace<sq_T>::set_parameters ( const mat &A0, const  mat &B0, const  mat &C0, const  mat &D0, const  sq_T &Q0, const sq_T &R0 ) {

        A = A0;
        B = B0;
        C = C0;
        D = D0;
        R = R0;
        Q = Q0;
        validate();
}

template<class sq_T>
void StateSpace<sq_T>::validate() {
        bdm_assert ( A.cols() == A.rows(), "KalmanFull: A is not square" );
        bdm_assert ( B.rows() == A.rows(), "KalmanFull: B is not compatible" );
        bdm_assert ( C.cols() == A.rows(), "KalmanFull: C is not compatible" );
        bdm_assert ( ( D.rows() == C.rows() ) && ( D.cols() == B.cols() ), "KalmanFull: D is not compatible" );
        bdm_assert ( ( Q.cols() == A.rows() ) && ( Q.rows() == A.rows() ), "KalmanFull: Q is not compatible" );
        bdm_assert ( ( R.cols() == C.rows() ) && ( R.rows() == C.rows() ), "KalmanFull: R is not compatible" );
}

}
#endif // KF_H