State Space Models
Description
Construct a
Usage
1 2 3 4 5 6 
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=1t=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 subclass: innov or noninnov, indicating an innovations form or a noninnovations form.
The state space (SS) model is defined by:
z(t) =Fz(t1) + Gu(t) + Qe(t)
y(t) = Hz(t) + Rw(t)
or the innovations model:
z(t) =Fz(t1) + Gu(t) + Kw(t1)
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(t1), u(t)] denoted E[z(t)t1]. 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 noninnovations 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 noninnovations 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 subclass
innov
the Kalman gain K is specified but not Q and R. For subclassnoninnov
Q and R are specified but not the Kalman gain K.  e and w
are typically assumed to be white noise in the noninnovations 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 noninnovations form is harder to identify than an innovations form. Noninnovations 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. PrenticeHall. (note p.39,44.)
See Also
TSmodel
ARMA
simulate.SS
l.SS
state
smoother
fixConstants
Examples
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