States

The assumption that we are using at present is that it suffices to study time series with linear dependencies so that if we take vector values or time blocks or both, we have an output vector valued time series v produced by an input vector valued times series u (which might be uncorrelated gaussian white noise or might not) and with an autoregressive component. Formally,

v(n+1) = A v(n) + B u(n)

for vectors $u(n), v(n), n \in {\fam11\tenbbb Z}$, and matrices A,B. It is usual to introduce another complication by supposing that there is an internal state which is produced by such a process and that this is observed by some means which may introduce more noise. Thus we may have two equations:

Here, z is some internal state which is not observed, it is `hidden', and which gives rise to v which is observed. The output vectors lie in some affine subspace of the output space in general, and finding it and its dimension is a part of the problem of filtering the process given by the above equations.

In even more generality, we may have the case where the matrices A,B,C,D are allowed to change in time, although not, one hopes, too quickly.