Legacy Documentation

So far we have discussed ARMA type of models where the trend and often the seasonal components of a given time series are removed before detailed analysis. A structural time series model, on the other hand, is designed to model trend and seasonal components explicitly. For example, the basic structural model is defined by

where represents the trend and is the cyclic or seasonal component, while is the irregular or noise part. Both and evolve according to their own equations. For example, we can have a linear trend with

(where ₀≠0) and the seasonality, _t, modeled as,

where {_t}, {_t}, {_t}, and {_t} are all white noise variables with zero mean; they are assumed to be independent of each other.

When the seasonal component is absent, we can define a simpler version of the basic structural model called the local linear trend model by

or an even simpler local level model by

In the above models, the "trend" _t (or seasonality _t) is not observable. Often the task is to infer the behavior of the trend or some other quantities from the time series {y_t} we observe. This class of problems is most conveniently treated by casting the models into a state-space form and using the Kalman filter technique.