State Space Models

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Description

Construct a

Usage

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    SS(F.=NULL, G=NULL, H=NULL, K=NULL, Q=NULL, R=NULL, z0=NULL, P0=NULL, rootP0=NULL,
          constants=NULL,
          description=NULL, names=NULL, input.names=NULL, output.names=NULL)
    is.SS(obj)
    is.innov.SS(obj)
    is.nonInnov.SS(obj)

Arguments

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.

rootP0

matrix indicating a square root of the initial tracking error (e.g. chol(P0)).

P0

matrix indicating initial tracking error P(t=1|t=0). Set to I if rootP0 or P0 are not supplied.

constants

NULL or a list of logical matrices with the same names as matices above, indicating which elements should be considered constants.

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.

Details

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 arguments, 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]. Note: In the case where there is no input u this corresponds to what would usually be called the predicted state - not the filtered state. 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. In the object returned by l.ss, state is a time series matrix corresponding to z.

z0

An initial value for z can be specified as z0.

P0

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

rootP0

Alternatively, a square root of P0 can be specified. This can be an upper triangular matrix so that only the required number of parameters are used.

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 and 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 by specifying the list constants. Matrices which are not specified in the list will be treated in the default way. An alternative for fixing constants is the function fixConstants.

Value

An SS TSmodel

References

Anderson, B. D. O. and Moore, J. B. (1979) Optimal Filtering. Prentice-Hall. (note p.39,44.)

See Also

TSmodel ARMA simulate.SS l.SS state smoother fixConstants

Examples

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    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

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