Class: ssm
Backward recursion of state-space models
X = smooth(Mdl,Y)
X = smooth(Mdl,Y,Name,Value)
[X,logL,Output]
= smooth(___)
returns smoothed states (X
= smooth(Mdl
,Y
)X
)
by performing backward recursion of the fully-specified state-space model Mdl
.
That is, smooth
applies the standard Kalman filter using Mdl
and
the observed responses Y
.
uses
additional options specified by one or more X
= smooth(Mdl
,Y
,Name,Value
)Name,Value
pair
arguments.
If Mdl
is not fully specified, then you must
set the unknown parameters to known scalars using the Params
Name,Value
pair
argument.
[
uses any of the input arguments
in the previous syntaxes to additionally return the loglikelihood
value (X
,logL
,Output
]
= smooth(___)logL
) and an output structure array (Output
)
containing:
Smoothed states and their estimated covariance matrix
Smoothed state disturbances and their estimated covariance matrix
Smoothed observation innovations and their estimated covariance matrix
The loglikelihood value
The adjusted Kalman gain
And a vector indicating which data the software used to filter
Mdl
does not store the response
data, predictor data, and the regression coefficients. Supply the
data wherever necessary using the appropriate input or name-value
pair arguments.
To accelerate estimation for low-dimensional, time-invariant
models, set 'Univariate',true
. Using this specification,
the software sequentially updates rather then updating all at once
during the filtering process.
The Kalman filter accommodates missing data by not updating filtered state estimates corresponding to missing observations. In other words, suppose there is a missing observation at period t. Then, the state forecast for period t based on the previous t – 1 observations and filtered state for period t are equivalent.
For explicitly defined state-space models, smooth
applies
all predictors to each response series. However, each response series
has its own set of regression coefficients.
[1] Durbin J., and S. J. Koopman. Time Series Analysis by State Space Methods. 2nd ed. Oxford: Oxford University Press, 2012.