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 $i$ th response variable is defined by a component that simply picks the $i$ th 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 $i$ th response variable:

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