UCM Procedure
Modeling a Cycle
A deterministic cycle 
with frequency 
, 
, can be written as
If the argument t is measured on a continuous scale, then is a periodic function with period 
, amplitude 
, and phase 
. Equivalently, the cycle can be written in terms of the amplitude and phase as
Note that when is measured only at the integer values, it is not exactly periodic, unless 
for some integers j and k. The cycles in their pure form are not used very often in practice. However, they are very useful as building blocks for more complex periodic patterns. It is well known that the periodic pattern of any complexity can be written as a sum of pure cycles of different frequencies and amplitudes. In time series situations it is useful to generalize this simple cyclical pattern to a stochastic cycle that has a fixed expected period but time-varying amplitude and phase. The stochastic cycle considered here is motivated by the following recursive formula for computing ,
starting with 
and 
. Note that and 
satisfy the relation
A stochastic generalization of the cycle can be obtained by adding random noise to this recursion and by introducing a damping factor, 
, for additional modeling flexibility. This model can be described as follows,
where 
, and the disturbances 
and 
are independent 
variables. The resulting stochastic cycle has a fixed expected period but time-varying amplitude and phase. The stationarity properties of the random sequence depend on the damping factor . If 
, has a stationary distribution with mean 0 and variance 
. If 
, is nonstationary.
You can incorporate a cycle in a UCM by specifying a CYCLE statement. You can include multiple cycles in the model by using separate CYCLE statements for each included cycle.
As mentioned before, the cycles are very useful as building blocks for constructing more complex periodic patterns. Periodic patterns of almost any complexity can be created by superimposing cycles of different periods and amplitudes. In particular, the seasonal patterns, general periodic patterns with integer periods, can be constructed as sums of cycles. This important topic of modeling the seasonal components is considered next.