SS                   package:dse1                   R Documentation

_S_t_a_t_e _S_p_a_c_e _M_o_d_e_l_s

_D_e_s_c_r_i_p_t_i_o_n:

     Construct a

_U_s_a_g_e:

         SS(F.=NULL, G=NULL, H=NULL, K=NULL, Q=NULL, R=NULL, z0=NULL, P0=NULL,
                  description=NULL, names=NULL, input.names=NULL, output.names=NULL)
         is.SS(obj)
         is.innov.SS(obj)
         is.nonInnov.SS(obj)

_A_r_g_u_m_e_n_t_s:

      F.: (nxn) state transition matrix.

       H: (pxn) output matrix.

       Q: (nxn) matrix specifying the system noise distribution.

       R: (pxp) matrix specifying the output (measurement) noise
          distribution.

       G: (nxp) input (control) matrix. G should be NULL if there is no
          input.

       K: (nxp) matrix specifying the Kalman gain.

      z0: vector indicating estimate of the state at time 0. Set to
          zero if not supplied.

      P0: matrix indicating initial tracking error P(t=1|t=0). Set to I
          if not supplied.

description: String. An arbitrary description.

   names: A list with elements input and output, each a vector of 
          strings. Arguments input.names and output.names should not be
          used if argument names is used.

input.names: A vector of character strings indicating input variable
          names. 

output.names: A vector of character strings indicating output variable
          names. 

     obj: an object.

_D_e_t_a_i_l_s:

     State space models have a further sub-class: innov or non-innov,
     indicating an innovations form or a non-innovations form.  

     The state space (SS) model is defined by:

     z(t) =Fz(t-1) + Gu(t) + Qe(t)

     y(t) = Hz(t)  + Rw(t)

     or the innovations model:

     z(t) =Fz(t-1) + Gu(t) + Kw(t-1)

     y(t) = Hz(t)  + w(t)

     Matrices are as specified above in the aguments, and 

_y is the p dimensional output data.

_u is the m dimensional exogenous (input) data.

_z is the n dimensional (estimated) state at time t,   E[z(t)|y(t-1),
     u(t)] denoted E[z(t)|t-1]. An initial value for z can  be
     specified as z0 and an initial one step ahead state tracking 
     error (for non-innovations models) as P0.

_z_0 An initial value for z can be specified as z0.

_P_0 An initial one step ahead state tracking error (for  non-innovations
     models) can be specified as P0.  

_K, _Q, _R For sub-class innov the Kalman gain K is specified but not Q
     and R. For sub-class non-innov Q and R are specified but not the
     Kalman gain K.

_e _a_n_d _w are typically assumed to be white noise in the  non-innovations
     form, in which case the covariance of the system noise is QQ' and
     the covariance of  the measurement noise is RR'. The covariance of
     e and w  can be specified  otherwise in the simulate  method
     'simulate.SS' for this class of model, but the assumption is
     usually maintained when estimating models of this form (although,
     not by all authors).

     Typically, an non-innovations form is harder to identify than an
     innovations form. Non-innovations form would typically be choosen
     when there is considerable theoretical or physical knowledge of
     the system (e.g. the system was built from known components with
     measured physical values).

     By default, elements in parameter matrices are treated as
     constants if they are exactly 1.0 or 0.0, and as parameters
     otherwise. A value of 1.001 would be treated as a parameter, and
     this is the easiest way to initialize an element which is not to
     be treated as a constant of value 1.0. Any matrix elements can be
     fixed to constants of any value using fixConstants.   

_V_a_l_u_e:

     An SS TSmodel

_S_e_e _A_l_s_o:

     'TSmodel' 'ARMA' 'simulate.SS' 'l.SS' 'fixConstants'

_E_x_a_m_p_l_e_s:

         f <- array(c(.5,.3,.2,.4),c(2,2))
         h <- array(c(1,0,0,1),c(2,2))
         k <- array(c(.5,.3,.2,.4),c(2,2))
         ss <- SS(F=f,G=NULL,H=h,K=k)
         is.SS(ss)
         ss

