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Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, and preferably sparse) matrices arise naturally?

I am especially interested in problems that can be mapped onto a setup in which for each event of a reasonably nice point process on $\mathbb{R}$ (the simplest two such processes would be a Poisson or discrete-time process) there is an associated pair $(j,k) \in \{1,\dots,d\}^2$. In this case time-windowed sums $N_{jk}(t)$ of the various pairs can be formed in an obvious way (although there may be plenty of subtlety or freedom in the windowing itself): these supply such a matrix time series.

Each such pair $(j,k)$ could be regarded as a transition from server $j$ to another (possibly identical) server $k$ in a closed queue with $d$ servers and infinitely many clients. It is not hard to see that in the setting of communication networks, this framework amounts to a very general form of traffic analysis. Such an application should not be considered for an answer: it's already been covered.

A slightly more restrictive but simpler example is where the pairs $(j,k)$ are inherited from a cadlag random walk on the root lattice

$A_{d-1} :=\left \{x \in \mathbb{Z}^d : \sum_{j=1}^d x_j = 0\right \}$.

Examples of this sort would also be of considerable interest to me.

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  • $\begingroup$ I should also mention that I'm only interested in real-valued matrices, and preferably nonnegative ones. $\endgroup$
    – Steve Huntsman
    Commented Mar 21, 2010 at 4:01
  • $\begingroup$ As another potential framework for examples, consider a sequence $s \in \{1,\dots,d\}^\mathbb{Z}$. Each doublet $(s_\ell,s_{\ell+1})$ provides a pair. This ties into constructions involving generalized de Bruijn structures. $\endgroup$
    – Steve Huntsman
    Commented Mar 21, 2010 at 15:45
  • $\begingroup$ This is completely out of my area so this is a comment rather than an answer, but you might try the optics problem of direct ray tracing which involves a large number of small matrices. $\endgroup$
    – Jason Dyer
    Commented May 25, 2010 at 3:07

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A very significant application in the context of communications engineering is the modelling of multiple-input-multiple-output (MIMO) communications channels.

These channels are typically modeled by complex $n \times m$ matrices where $n$ is the number of receive antennas and $m$ is the number transmit antennas. The $(i,j)$ entry in the matrix describes the channel between the $i$th transmit antenna and the $j$th receive antenna. In most applications $n$ and $m$ are reasonably small, less than 16. Also, in most real world applications the channel (and hence the matrix) changes over time. This gives you your time series of matrices. In some situations the matrix will even be sparse because some transmit antennas might not see some receive antennas.

There is a seriously huge amount of literature on the MIMO channel and a large amount of it deals with the static case, i.e. for the sake of simplicity it is assumed that the channel doesn't change with time. However there are also many papers that deal with the time varying case. For example:

Chen and Su, "MIMO Channel Estimation in Correlated Fading Environments"

I unfortunately am not an expert in MIMO, but I do know some people who are and could ask them for more details if you were interested.

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