# Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For instance, the trajectories (realizations) of three processes $A$, $B$ and $C$ could be described as:

$A: 0, 0.4s, 0.8s, 1.3s, 2.1s, 2.5s, 3.1s, 4.5s, 4.6s, 5.8s, \ldots$ $B: 0, 1.8s, 1.9s, 2.7s, 2.8s, 2.8s, 2.9s, 4.7s, 6.5s, \ldots$ $C: 0, 0.5s, 0.9s, 1.4s, 2.2s, 2.6s, 3.2s, 4.6s, 4.7s, 5.9s, \ldots$

We can say that $A$ and $C$ are correlated, or might be generated by the same source.

How can I study the correlation between two processes $A$ and $B$?

Thank you!

• This question seems to general. Could you be more specific, like, give the particular processes you're interested in. Feb 10 '15 at 10:09
• I have added an example. Thank you! Feb 10 '15 at 10:29

So you want to study the dependence structure of multivariate point processes. Such problem arises frequently in neuroscience in the study of neural spike trains. (DISCLAIMER: I have a few papers in this area, and they appear below.)

There are many different ways the two processes can be dependent, and there are many different measures of dependence. In many cases you need multiple realizations to be able to correctly quantify the relation (or you'll need to assume some form of ergodicity.) Here is a partial list:

1. Cross-correlation

As a first step, you can plot the cross-correlogram $g(\tau) = E[X(t)Y(t-\tau)]$ and look at it (assuming ergodicity of sorts). If you don't have enough data, you might have to smooth it in various ways (either with or without binning time).

1. Spike train distance measures

There are distance/dissimilarity metrics that are sensitive to various features.

• van Rossum, M. C. W. (2001). A novel spike distance. Neural Computation, 13:751-763.
• Victor, J. D. and Purpura, K. P. (1997). Metric-space analysis of spike trains: theory, algorithms and application. Network: Computation in Neural Systems, 8(2):127-164.
• Paiva, A. R. C., Park, I., and Príncipe, J. (2008). Reproducing kernel Hilbert spaces for spike train analysis. In IEEE International Conference on Acoustics, Speech, and Signal Processing.
• Houghton, C. and Kreuz, T. (2013). Measures of spike train synchrony: From single neurons to populations. pages 277-297.

1. Point process divergence measures

If you have access to multiple realizations, you should really look at dependence and divergence measures.

• Il Memming Park, Sohan Seth, Murali Rao, José C. Príncipe. Strictly Positive Definite Spike Train Kernels for Point Process Divergences. Neural Computation Volume 24, Issue 8, August 2012
• Naud, R., Gerhard, F., Mensi, S., and Gerstner, W. (2011). Improved similarity measures for small sets of spike trains. Neural Computation, 23(12):3016-3069.

1. Fit a multivariate point process model

For example, you can use a coupled generalized linear model, and analyze their coupling strength.

• Pillow, J. W., Shlens, J., Paninski, L., Sher, A., Litke, A. M., Chichilnisky, E. J., and Simoncelli, E. P. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995-999.