SSM Procedure

Multivariate Cycle

The STATE statement option TYPE=CYCLE specifies a (2*dim)-dimensional alpha alpha Subscript t, needed for defining a dim-dimensional cycle. As in the LL case, the first dim elements of alpha alpha Subscript 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 lamda equals 2 pi slashperiod be the frequency associated with the cycle. The admissible parameter ranges are 0 less-than rho less-than-or-equal-to 1 and period greater-than 2, which implies that 0 less-than lamda less-than pi. Let bold upper C equals rho left-parenthesis cosine left-parenthesis lamda right-parenthesis sine left-parenthesis lamda right-parenthesis comma minus sine left-parenthesis lamda right-parenthesis cosine left-parenthesis lamda right-parenthesis right-parenthesis, a 2 times 2 matrix, and let bold upper T equals bold upper C circled-times bold upper I Subscript d i m, a 2 asterisk d i m times 2 asterisk d i m matrix. With this notation, the transition equation associated with alpha alpha Subscript t is

alpha alpha Subscript t plus 1 Baseline equals bold upper T alpha alpha Subscript t Baseline plus eta eta Subscript t plus 1

where eta eta Subscript t is a sequence of zero mean, independent, left-parenthesis 2 asterisk d i m right-parenthesis-dimensional Gaussian vectors with covariance normal upper D normal i normal a normal g left-parenthesis normal upper Sigma normal upper Sigma comma normal upper Sigma normal upper Sigma right-parenthesis. If rho equals 1, the initial condition is fully diffuse (bold upper Q 1 equals 0 and bold upper A 1 equals bold upper I Subscript 2 asterisk d i m). Otherwise, it is nondiffuse: bold upper Q 1 equals StartFraction 1 Over left-parenthesis 1 minus rho squared right-parenthesis EndFraction normal upper D normal i normal a normal g left-parenthesis normal upper Sigma normal upper Sigma comma normal upper Sigma normal upper Sigma right-parenthesis and bold upper A 1 equals 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 alpha alpha Subscript 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];
Last updated: June 19, 2025