Testing for Independence between a Point Process and an Analog Signal

Abstract:

Point processes are enjoying a resurgence of interest due to demand from a number of significant application areas.

In neural coding, electrodes are used to record spike trains (on a ms time scale) from the brains (neurons) of awake animals such as rats, cats and monkeys. Point process models are then fundamental for understanding information

flow in the brain. In high frequency finance the times of trades (event times) are recorded on a very fine (ms) time scale. And so over the last decade or so point process models have been increasingly used to model the consequent market dynamics. In genomics, because of the advent of high throughput data, in the last 5-7 years point process models have begun to be used to jointly model the positions of transcription regulatory elements along a genome. But in all these applications covariate series are usually available and typically take the form of analog (ie continuous valued) signals.

Thus the issue of joint modelling of analog signals and point processes arises. We pursue a basic question in such a setting for the first time; namely testing for independence. We formulate a joint state space model and analyse its stability using Dynkin's theorem. We then develop a Lagrange multiplier test, discuss its asymptotics and give some simulations and a small neural coding data analysis example. Future work is also discussed.