| StructTS {stats} | R Documentation |
Fit a structural model for a time series by maximum likelihood.
StructTS(x, type = c("level", "trend", "BSM"), init = NULL,
fixed = NULL, optim.control = NULL)
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. |
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. |
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 m[t] which
evolves by
m[t+1] = m[t] + xi[t], xi[t] ~ N(0, sigma^2_xi)
The observations are
x[t] = m[t] + eps[t], eps[t] ~ N(0, sigma^2_eps)
There are two parameters, sigma^2_xi and sigma^2_eps. 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
m[t], given by
m[t+1] = m[t] + n[t] + xi[t], xi[t] ~ N(0, sigma^2_xi)
n[t+1] = n[t] + zeta[t], zeta[t] ~ 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] = m[t] + s[t] + eps[t], eps[t] ~ N(0, sigma^2_eps)
where s[t] is a seasonal component with dynamics
s[t+1] = -s[t] - ... - s[t - s + 2] + w[t], w[t] ~ N(0, sigma^2_w)
The boundary case sigma^2_w = 0 corresponds to a deterministic (but arbitrary) seasonal pattern. (This is sometimes known as the ‘dummy variable’ version of the BSM.)
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 with arima
nor different types of structural models. |
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. |
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_eps.
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 JohnsonJohnson, Nile and AirPassengers
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))
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))