UCM Procedure
Basic Structural Model
The basic structural model (BSM) is obtained by adding a seasonal component, 
, to the local level model. In order to economize on the space, the state space formulation of a BSM with a relatively short season length, season length = 4 (quarterly seasonality), is considered here. The pattern for longer season lengths such as 12 (monthly) and 52 (weekly) is easy to see.
Let us first consider the dummy form of seasonality. In this case the state and state noise vectors are 
and 
, respectively. The first three elements of the state vector are the irregular, level, and slope components, respectively. The remaining elements, 
, are lagged versions of the seasonal component . 
corresponds to lag zero—that is, the same as , 
to lag 1 and 
to lag 2. The system matrices are
and 
. The distribution of the initial state vector 
is diffuse, with 
and 
.
In the case of the trigonometric type of seasonality, 
and 
. The disturbance sequences, 
, and 
, are independent, zero-mean, Gaussian sequences with variance 
. The system matrices are
and 
. Here 
. The distribution of the initial state vector is diffuse, with and . The parameter vector in both the cases is 
.