Fit Structural Time Series
Description
Fit a structural model for a time series by maximum likelihood.
Usage
StructTS(x, type = c("level", "trend", "BSM"), init = NULL,
fixed = NULL, optim.control = NULL)
Arguments
x
a univariate numeric time series. Missing values are allowed.
type
the class of structural model. If omitted, a BSM is used
for a time series with frequency(x) > 1, and a local trend
model otherwise. Can be abbreviated.
init
initial values of the variance parameters.
fixed
optional numeric vector of the same length as the total
number of parameters. If supplied, only NA entries in
fixed will be varied. Probably most useful for setting
variances to zero.
optim.control
List of control parameters for
optim . Method "L-BFGS-B" is used.
Details
Structural time series models are (linear Gaussian) state-space models for (univariate) time series based on a decomposition of the series into a number of components. They are specified by a set of error variances, some of which may be zero.
The simplest model is the local level model specified by
type = "level". This has an underlying level \mu_t which
evolves by
\mu_{t+1} = \mu_t + \xi_t, \qquad \xi_t \sim N(0, \sigma^2_\xi)
The observations are
x_t = \mu_t + \epsilon_t, \qquad \epsilon_t \sim N(0, \sigma^2_\epsilon)
There are two parameters, \sigma^2_\xi
and \sigma^2_\epsilon. It is an ARIMA(0,1,1) model,
but with restrictions on the parameter set.
The local linear trend model, type = "trend", has the same
measurement equation, but with a time-varying slope in the dynamics for
\mu_t, given by
\mu_{t+1} = \mu_t + \nu_t + \xi_t, \qquad \xi_t \sim N(0, \sigma^2_\xi)
\nu_{t+1} = \nu_t + \zeta_t, \qquad \zeta_t \sim N(0, \sigma^2_\zeta)
with three variance parameters. It is not uncommon to find
\sigma^2_\zeta = 0 (which reduces to the local
level model) or \sigma^2_\xi = 0, which ensures a
smooth trend. This is a restricted ARIMA(0,2,2) model.
The basic structural model, type = "BSM", is a local
trend model with an additional seasonal component. Thus the measurement
equation is
x_t = \mu_t + \gamma_t + \epsilon_t, \qquad \epsilon_t \sim N(0, \sigma^2_\epsilon)
where \gamma_t is a seasonal component with dynamics
\gamma_{t+1} = -\gamma_t + \cdots + \gamma_{t-s+2} + \omega_t, \qquad
\omega_t \sim N(0, \sigma^2_\omega)
The boundary case \sigma^2_\omega = 0 corresponds
to a deterministic (but arbitrary) seasonal pattern. (This is
sometimes known as the ‘dummy variable’ version of the BSM.)
Value
A list of class "StructTS" with components:
coef
the estimated variances of the components.
loglik
the maximized log-likelihood. Note that as all these
models are non-stationary this includes a diffuse prior for some
observations and hence is not comparable to arima
nor different types of structural models.
loglik0
the maximized log-likelihood with the constant used prior to R 3.0.0, for backwards compatibility.
data
the time series x.
residuals
the standardized residuals.
fitted
a multiple time series with one component for the level,
slope and seasonal components, estimated contemporaneously (that is
at time t and not at the end of the series).
call
the matched call.
series
the name of the series x.
code
the convergence code returned by optim .
model, model0
Lists representing the Kalman filter used in the
fitting. See KalmanLike . model0 is the
initial state of the filter, model its final state.
xtsp
the tsp attributes of x.
Note
Optimization of structural models is a lot harder than many of the
references admit. For example, the AirPassengers data
are considered in Brockwell & Davis (1996): their solution appears to
be a local maximum, but nowhere near as good a fit as that produced by
StructTS. It is quite common to find fits with one or more
variances zero, and this can include \sigma^2_\epsilon.
References
Brockwell, P. J. & Davis, R. A. (1996). Introduction to Time Series and Forecasting. Springer, New York. Sections 8.2 and 8.5.
Durbin, J. and Koopman, S. J. (2001) Time Series Analysis by State Space Methods. Oxford University Press.
Harvey, A. C. (1989) Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
Harvey, A. C. (1993) Time Series Models. 2nd Edition, Harvester Wheatsheaf.
See Also
KalmanLike , tsSmooth ;
stl for different kind of (seasonal) decomposition.
Examples
## see also JohnsonJohnson, Nile and AirPassengers
require(graphics)
trees <- window(treering, start = 0)
(fit <- StructTS(trees, type = "level"))
plot(trees)
lines(fitted(fit), col = "green")
tsdiag(fit)
(fit <- StructTS(log10(UKgas), type = "BSM"))
par(mfrow = c(4, 1)) # to give appropriate aspect ratio for next plot.
plot(log10(UKgas))
plot(cbind(fitted(fit), resids=resid(fit)), main = "UK gas consumption")
## keep some parameters fixed; trace optimizer:
StructTS(log10(UKgas), type = "BSM", fixed = c(0.1,0.001,NA,NA),
optim.control = list(trace = TRUE))