SSM Procedure

Continuous-Time Cycle

The STATE statement option TYPE=CYCLE(CT) specifies a two-dimensional alpha alpha Subscript t, needed for defining a univariate continuous time cycle. In this case the nominal dimension, dim, must be 1. In particular, normal upper Sigma normal upper Sigma becomes one-dimensional, which is denoted by sigma squared. This cycle can be used for any data type. As before, the parameters of the cycle are a damping factor rho, 0 less-than rho less-than-or-equal-to 1, and period greater-than 0. Unlike in the discrete-time cycle described in the section Multivariate Cycle, the period is not required to be larger than 2. Let lamda equals 2 pi slash sans-serif-italic period, and let h Subscript t Baseline equals left-parenthesis tau Subscript t plus 1 Baseline minus tau Subscript t Baseline right-parenthesis denote the difference between successive time points. In this case, the system matrices bold upper T and bold upper Q that govern alpha alpha Subscript t depend on h Subscript t. They are as follows:

StartLayout 1st Row 1st Column bold upper T 2nd Column equals 3rd Column rho Superscript h Baseline left-parenthesis cosine left-parenthesis lamda h right-parenthesis sine left-parenthesis lamda h right-parenthesis comma minus sine left-parenthesis lamda h right-parenthesis cosine left-parenthesis lamda h right-parenthesis right-parenthesis 2nd Row 1st Column bold upper Q 2nd Column equals 3rd Column StartFraction sigma squared left-parenthesis 1 minus rho Superscript 2 h Baseline right-parenthesis Over minus 2 ln left-parenthesis rho right-parenthesis EndFraction asterisk bold upper I 2 if rho less-than 1 3rd Row 1st Column bold upper Q 2nd Column equals 3rd Column sigma squared h bold upper I 2 if rho equals 1 EndLayout

If rho less-than 1, the initial condition is nondiffuse: bold upper Q 1 equals StartFraction sigma squared Over minus 2 ln left-parenthesis rho right-parenthesis EndFraction bold upper I 2. For rho equals 1, the initial condition is fully diffuse.

The first element of alpha alpha Subscript t corresponds to the needed cycle, and the second element is an auxiliary quantity. You can define a cycle term based on this state as follows: 

  state cycleState(1) type=cycle(CT)  ...;
  component cycle = cycleState[1];

The CT option must be included in the use of TYPE=CYCLE.

Last updated: June 19, 2025