[R] correlating irregular time series

[Christophe Pouzat]

Hi Paul,

Here is how an amateur statistician deals with this problem when 
analyzing spike trains from simultaneously recorded neurons.

Start by estimating the "hazard function" h(t) of your several point 
processes (if you have a copy of MASS, check out the chapter 13, If you 
have a copy of Jim Lindsey, "The Statistical Analysis of Stochastic 
Processes in Time", check out chap 3 & 4; the hazard function is also 
called the "conditional intensity" or the "stochastic intensity").

In practice if you have a renewal process, meaning that the successive 
intervals between your events times are independent, you can first 
estimate the "Inter Event Interval" pdf, f(t), and its cumulative 
distribution function F(t). h(t) is then given by:

h(t) = f(t) / (1-F(t)),

where the quantity S(t) = 1-F(t) is often called the survivor function.

Fine, now if your processes are well approximated by renewal processes, 
you can look for the distribution of "time to next event" (TTN) and 
"time to former event" (TTF). By that I mean that for each of the black 
events of your figure, you must get the interval separating it from the 
last red event preceding it (the time to former) and the next red event 
following it (the time to next). Under the null hypothesis of no 
correlation these to random variables have the same pdf given by:

TTN(i) = S(i) / <IEI>,

where S(i) in that case is the survivor function of the red (test) 
process and <IEI> is its inter event interval expected value.
Using this approach I typically estimate the TTN and TTF pdfs with 
histograms and compare these histograms to their expected values under 
the null hypothesis. A warning though, I have most of the time much more 
events than you seem to have on your figure.

Let me know if any of this makes sense.

Christophe.

paul sorenson wrote:

>I have some time stamped events that are supposed to be unrelated.
>
>I have plotted them and that assumption does not appear to be valid. 
>http://metrak.com/tmp/sevents.png is a plot showing three sets of events 
>over time.  For the purpose of this exercise, the Y value is irrelevant. 
>  The series are not sampled at the same time and are not equispaced 
>(just events in a log file).
>
>The plot is already pretty convincing but requires a human-in-the-loop 
>to zoom in on "hot" areas and then visually interpret the result.  I 
>want to calculate some index of the events' temporal relationship.
>
>I think the question I am trying to ask is something like: "If event B 
>occurs, how likely is it that an event A occurred at almost the same time?".
>
>Can anyone suggest an established approach that could provide some 
>further insight into this relationship?  I can think of a fairly basic 
>approach where I start out with the ecdf of the time differences but I 
>am guessing I would be reinventing some wheel.
>
>Any tips would be most appreciated.
>
>cheers
>
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>


-- 
A Master Carpenter has many tools and is expert with most of them.If you
only know how to use a hammer, every problem starts to look like a nail.
Stay away from that trap.
Richard B Johnson.
--

Christophe Pouzat
Laboratoire de Physiologie Cerebrale
CNRS UMR 8118
UFR biomedicale de l'Universite Paris V
45, rue des Saints Peres
75006 PARIS
France

tel: +33 (0)1 42 86 38 28
fax: +33 (0)1 42 86 38 30
web: www.biomedicale.univ-paris5.fr/physcerv/C_Pouzat.html