Measure to capture clustering in time based event I am searching for a good measure to capture clustering in time based events. Say that for a given interval of 5 minutes, cars randomly pass in front of house ... say one car at every 10 seconds (for a total of 30 cars). For another interval of 5 minutes, 3 cars passes at every thirty seconds, and the time between the three cars is 0 (again 30 cars over 5 minutes). The difference between both intervals is that for the second intervals, the arrival of cars "clusters". What can be a good measure to capture clustering? I would like a measure that indicates that there was more clustering in the second interval than the first one. 
 A: I think the measure you are looking for is called "clumpiness," or non-constant propensity, or rather temporary elevations of propensity, characterized by serial dependence and non-stationarity. Take a look at the draft version of New Measures of Clumpiness for Incidence Data by Yao Zhang, Eric Bradlow, and Dylan Small for some alternative ways to measure this sort of thing. The paper starts out defining some desirable properties of clumpiness measures:


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*Minimum - the measure should be at the minimum if the events are equally spaced

*Maximum - the measure should be at the maximum if all the events are gathered together 

*Continuity - shifting event timing by a very small amount should only change the measure by a small amount

*Convergence - as events move closer/further apart, the measure should increase/decrease


Then it evaluates several existing measures and proposes 4 new ones. It also evaluates them using some simulation with various types of DGPs. There's also an empirical illustration using Hulu data.
A: If the cars pass following a Poisson distribution of 6 cars per minute, the time between cars is exponential rate 1/6 of a minute. Compare both examples to the exponential distribution using the Kolmogorov-Smirnov test. The clustered one will be significantly different (provided you have enough data). The regularly spaced one should also be significantly different as it isn't random, though I don't expect you mean the cars pass precisely every ten seconds.
