Construct a
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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. |
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
is the p dimensional output data.
is the m dimensional exogenous (input) data.
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.
An initial value for z can be specified as z0.
An initial one step ahead state tracking error (for non-innovations models) can be specified as P0.
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.
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.
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
.
An SS TSmodel
Anderson, B. D. O. and Moore, J. B. (1979) Optimal Filtering. Prentice-Hall. (note p.39,44.)
TSmodel
ARMA
simulate.SS
l.SS
state
smoother
fixConstants
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