Cycle component

Introduction

A Cycle component is a special case of an AR2 component, which contains two complex conjugate roots.

More precisely, it is defined by a period $\lambda$ and a dumping factor $\rho$. If $\gamma = \frac{2\pi}{\lambda}$, we have that

\[\begin{pmatrix} y_{t+1} \\ y^*_{t+1}\end {pmatrix} = \begin{pmatrix} \rho \cos{\gamma} && \rho \sin{\gamma}\\ -\rho \sin{\gamma} && \rho \cos{\gamma}\end {pmatrix}\begin{pmatrix} y_t \\ y_t^*\end {pmatrix}+\begin{pmatrix} \epsilon_t \\ \epsilon_t^*\end {pmatrix}\]

Using those notations, the state-space model can be written as follows :

State vector:

\[\alpha_t= \begin{pmatrix} y_t \\ y_t^* \end{pmatrix}\]

Dynamics

\[T_t = \begin{pmatrix} \rho \cos{\gamma} && \rho \sin{\gamma}\\ -\rho \sin{\gamma} && \rho \cos{\gamma}\end {pmatrix}\] \[S_t = \sigma_c \begin{pmatrix} 1 && 0\\ 0 && 1\end {pmatrix}\] \[V_t = \sigma_c^2\begin{pmatrix} 1 && 0\\ 0 && 1\end {pmatrix}\]

Measurement

\[Z_t = \begin{pmatrix} 1 & 0 \end{pmatrix}\] \[h_t = 0\]

Initialization

\[\alpha_{-1} = \begin{pmatrix}0 \\ 0 \end{pmatrix}\] \[P_{*} = \sigma_c^2\begin{pmatrix} \frac{1}{1-\rho^2} && 0\\ 0 && \frac{1}{1-\rho^2}\end {pmatrix}\]

Implementation

Cycle models are implemented in the class jdplus.ssf.CyclicalComponent

Bibliograhpy

Durbin J. and Koopman S.J [2012], Time Series Analysis by State Space Methods, second edition, Oxford University Press, §3.2.4