# Similarity measures for point processes

I have multiple measurements of a point process: vectors of 0's and 1's.

I'm trying to gauge the similarity of the measurements, but have no idea how to proceed. Any suggestions?

Thanks!

• Assuming you have a Poisson point process the simplest possible would be the rate parameter $\lambda$. – usεr11852 says Reinstate Monic Apr 3 '14 at 18:05

I think the most direct way to determine if your measurements are similar is to compute the wait time distribution for each measurement.

The wait time is just the time elapsed between events, where an event is, presumably, signal equals $1$. This will give you a series of wait times. You can then plot these as a distribution (a histogram) to get frequencies. What do you do with these?

If your question is solely, "are these measurements of a point process similar", the wait time distributions are one thing you can examine. If the measurements are the same, their wait time distributions should be the same. Obviously if each measurement is very short then you won't get nice wait time distributions and it will be harder to determine if they are the same. With enough events though, the wait time distributions should converge on each other (if the measurements are equivalent).

The nature of the wait time distribution is itself very informative about the process you're studying though, and some analysis will be required if the wait time distributions are not the same. If the point process is "memory-less", i.e., seeing a $1$ has no influence on when you see the next $1$, then the wait time distribution is expected to be exponentially distributed. This is called a Poisson process and is the simplest possible process you could be observing. As user11852 was implying, a Poisson process has only one parameter $\lambda$ which tracks the average rate of the event. If the wait time distribution is in fact exponential, and the mean wait time is $4$, then $\lambda=1/4$. So if your point process measurements all have exponentially distributed wait times, you can ask how similar their $\lambda$ parameters are.

• While everyone provided useful answers, I chose this one as the best because it made me think about the problem in a new way. Thanks! – monkeybiz7 Apr 15 '14 at 18:51

i would try coherence

another idea is to apply haar wavelet analysis and compare the measurements in frequency space

• Is it possible to compute the coherence for multiple signals? – monkeybiz7 Apr 4 '14 at 17:17
• yes, pairwise. coherence is useful when the signals have a structure in frequency domain. random noise will not show any coherence – Aksakal Apr 4 '14 at 17:23
• So you'd suggest computing n*(n-1)/2 coherences, and take the average? (n is the number of measurements) – monkeybiz7 Apr 4 '14 at 17:28
• i'd first look at the sample coherence plots, and see if there's anything interesting. – Aksakal Apr 4 '14 at 17:39