/*!
  \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"

namespace bdm
{

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

Parameter evolution model:\f[ x_t = A x_{t-1} + B u_t + Q^{1/2} e_t \f]
Observation model: \f[ y_t = C x_{t-1} + C u_t + Q^{1/2} w_t. \f]
Where $e_t$ and $w_t$ are independent vectors Normal(0,1)-distributed disturbances.
 */
template<class sq_T>
class StateSpace
{
	protected:
		//! cache of rv.count()
		int dimx;
		//! cache of rvy.count()
		int dimy;
		//! cache of rvu.count()
		int dimu;
		//! 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() : dimx (0), dimy (0), dimu (0), A(), B(), C(), D(), Q(), R() {}
		StateSpace(const StateSpace<sq_T> &S0) : dimx (S0.dimx), dimy (S0.dimy), dimu (S0.dimu), A(S0.A), B(S0.B), C(S0.C), D(S0.D), Q(S0.Q), R(S0.R) {}
		void set_parameters (const mat &A0, const  mat &B0, const  mat &C0, const  mat &D0, const  sq_T &Q0, const sq_T &R0);
		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;
			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
			Q=Qtm; 
			
			validate();
		}		
		//! access function
		int _dimx(){return dimx;}
		//! access function
		int _dimy(){return dimy;}
		//! access function
		int _dimu(){return dimu;}
		//! 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;
		//! id of input
		RV urv;
		//! Kalman gain
		mat  _K;
		//!posterior
		shared_ptr<enorm<sq_T> > est;
		//!marginal on data f(y|y)
		enorm<sq_T>  fy;
	public:
		Kalman<sq_T>() : BM(), StateSpace<sq_T>(), yrv(),urv(), _K(),  est(new enorm<sq_T>){}
		Kalman<sq_T>(const Kalman<sq_T> &K0) : BM(K0), StateSpace<sq_T>(K0), yrv(K0.yrv),urv(K0.urv), _K(K0._K),  est(new enorm<sq_T>(*K0.est)), fy(K0.fy){}
		void set_statistics (const vec &mu0, const mat &P0) {est->set_parameters (mu0, P0); };
		void set_statistics (const vec &mu0, const sq_T &P0) {est->set_parameters (mu0, P0); };
		//! posterior
		const enorm<sq_T>& posterior() const {return *est.get();}
		//! shared posterior
		shared_ptr<epdf> shared_posterior() {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 (urv, set, "urv", UI::optional);
			set_drv(concat(yrv,urv));
			
			validate();
		}
		void validate() {
			StateSpace<sq_T>::validate();
			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 &dt);
		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());
			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 &dt);
		
		void from_setting(const Setting &set){
			Kalman<chmat>::from_setting(set);
			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 &dt);
		//! set estimates
		void set_statistics (const vec &mu0, const mat &P0) {
			est->set_parameters (mu0, P0);
		};
		const mat _R() {
			return est->_R().to_mat();
		}
		void from_setting (const 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 ) );
			
			//connect
			shared_ptr<RV> drv = UI::build<RV> ( set, "drv", UI::compulsory );
			set_drv ( *drv );
			shared_ptr<RV> rv = UI::build<RV> ( set, "rv", UI::compulsory );
			set_rv ( *rv );
			
			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 &dt);

		void from_setting (const Setting &set);

		// 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:
		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 &dt) {
			int n = Models.length();
			int i;
			for (i = 0; i < n; i++) {
				Models (i)->bayes (dt);
				_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());
			}
		}
		//! posterior density
		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));
			}
			
			this->dimx = xrv._dsize();
			this->dimy = yrv._dsize();
			this->dimu = urv._dsize();
			
			// 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(dimx,dimx);
			for (int j=1; j<dimx; j++){A(j,j-1)=1.0;} // off diagonal
				this->B=zeros(dimx,1);
				this->B(0) = 1.0;
				this->C=zeros(1,dimx);
				this->D=zeros(1,urv._dsize());
				this->Q = zeros(dimx,dimx);
			// 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*A1row);
			D.set_row(0,D1row);
			
		}
};

/////////// 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(){
	dimx = A.rows();
	dimu = B.cols();
	dimy = C.rows();
	bdm_assert (A.cols() == dimx, "KalmanFull: A is not square");
	bdm_assert (B.rows() == dimx, "KalmanFull: B is not compatible");
	bdm_assert (C.cols() == dimx, "KalmanFull: C is not square");
	bdm_assert ( (D.rows() == dimy) && (D.cols() == dimu), "KalmanFull: D is not compatible");
	bdm_assert ( (Q.cols() == dimx) && (Q.rows() == dimx), "KalmanFull: Q is not compatible");
	bdm_assert ( (R.cols() == dimy) && (R.rows() == dimy), "KalmanFull: R is not compatible");
}

}
#endif // KF_H


