Methods-AR

The $p$th-order vector autoregressive process can be expressed as:

$${\mathbf{x}}_t=\sum_{j=1}^p {\mathbf{A}}_j {\mathbf{x}}_{t-j}+{\mathbf{e}}_t,$$ (1)

where ${\mathbf{A}}_j$ are $m \times m$ matrices of model coefficients, ${\mathbf{x}}_t$ is the vector of the multichannel signal, and $m$ is the number of channels. In the stochastic linear interpretation of this model, ${\mathbf{e}}_t$ is the vector of multivariate zero mean uncorrelated white noise.

Equation (1) can be converted to a state space representation by introducing a $mp$-dimensional state vector:

$${\mathbf{y}}_t = ({\mathbf{x}}_{t-1}, {\mathbf{x}}_{t-2}, ..., {\mathbf{x}}_{t-p})^T.$$ (2)

The evolution of this state vector is governed by the state transition equation:

$${\mathbf{y}}_{t+1}={\mathbf{F}}{\mathbf{y}}_t + {\mathbf{z}}_t,$$ (3)

where ${\mathbf{z}}_t =({\mathbf{e}}_t, {\mathbf{0}}, \dots, {\mathbf{0}})^T$ and the state transition matrix ${\mathbf{F}}$ is in a block-canonical form with $m \times m$ identity matrices ${\mathbf{I}}$ along the under-diagonal and the parameter matrices ${\mathbf{A}}_j$ on the top:

$${\mathbf{F}}= \left( \begin{array}{cccc} {\mathbf{A}}_1 &... ...}& \dots & {\mathbf{I}}& {\mathbf{0}} \par\end{array} \right).$$ (4)

Equation (3) is a first order difference equation describing a Markov process in $mp$-dimensional space. This type of conversion to a state space is referred to as embedding in the dynamical-system literature.

The evolution of a dynamic system is governed by the deterministic state transition equation:

$$\frac{{\rm d} {\mathbf{x}}(t)}{{\rm d}t}={\mathbf{G}}({\mathbf{x}}(t)),$$ (5)

where the function ${\mathbf{G}}$ can be highly non-linear. It can be shown that, periodic systems (limit cycles) and quasi-periodic systems (tori) can be thought of as autoregressive processes in the limiting case of driving white noise variance equal to zero. The number of degrees of freedom in this case is equal to two$\times$number of frequencies. Thus, the state vector of system evolves on the attractor according to (3) with random part ${\mathbf{z}}_t$ equal to zero.

In case the attractor is not regular (e.g. chaotic) and function ${\mathbf{G}}$ in (5) is nonlinear, one can locally linearize the system by developing ${\mathbf{G}}$ in a Taylor series and retaining only the first-order terms. In this case the ${\mathbf{z}}_t$ is not equal to zero, and the residuals ${\mathbf{e}}_t$ contain information about non-periodic, non-linear portion of signal. In previous work (Franaszczuk and Bergey, 1999) we used the residual covariance $m \times m$ matrix ${\mathbf{V}}_e$ of the vector ${\mathbf{e}}_t$ as a measure of goodness of fit of a linear model to the data. In this study we perform a time-frequency decomposition of ${\mathbf{e}}_t$ to better analyze low amplitude transients.