The SSM Procedure

Multivariate Cycle

The STATE statement option TYPE=CYCLE specifies a (2*dim)-dimensional $\pmb {\alpha }_{t}$, needed for defining a dim-dimensional cycle. As in the LL case, the first dim elements of $\pmb {\alpha }_{t}$ correspond to the needed dim-dimensional cycle, and the remaining dim elements contain some auxiliary quantities. The cycle model defined in this subsection requires a regular data type—that is, the CT option is not included. Let $\rho $ denote the damping factor, and let $\lambda = 2\pi /$period be the frequency associated with the cycle. The admissible parameter ranges are $0 < \rho \leq 1$ and period $ > 2$, which implies that $0 < \lambda < \pi $. Let $\mb{C} = \rho (\cos (\lambda ) \;  \sin (\lambda ) , \;  -\sin (\lambda ) \;  \cos (\lambda ) ) $, a $2 \times 2$ matrix, and let $\mb{T} = \mb{C} \bigotimes \mb{I}_{dim}$, a $2*dim \times 2*dim$ matrix. With this notation, the transition equation associated with $\pmb {\alpha }_{t}$ is

\[  \pmb {\alpha }_{t+1} = \mb{T} \pmb {\alpha }_{t} + \pmb {\eta }_{t+1}  \]

where $ \pmb {\eta }_{t}$ is a sequence of zero mean, independent, $(2*dim)$-dimensional Gaussian vectors with covariance $\mr{Diag}(\pmb {\Sigma }, \; \pmb {\Sigma })$. If $\rho = 1$, the initial condition is fully diffuse ($\mb{Q}_{1} = 0$ and $\mb{A}_{1} =\mb{I}_{2*dim}$). Otherwise, it is nondiffuse: $\mb{Q}_{1} = \frac{1}{(1 - \rho ^{2})}\mr{Diag}(\pmb {\Sigma }, \; \pmb {\Sigma })$ and $\mb{A}_{1} =0$.

The multivariate cycle is useful for capturing periodic behavior for multivariate time series data. The cycle term for the ith response variable is defined by a component that simply picks the ith element of $\pmb {\alpha }_{t}$. For example, the component cycle_i defined as follows can be used as a cycle term in the MODEL statement of the ith response variable:

     state cycleState(dim) type=cycle  ...;
     component cycle_2 = cycleState[2];