It seems to me you that have a sequence (``series'') of random times,
rather than
a sequence of values of a random variable observed at a irregularly
spaced times.
Hence I would say that point process modelling, rather than time
series modelling,
would be more appropriate.
You could consider yourself to have two related point processes ---
the process of
starting times and the process of stopping times. Or you could
consider the process,
of starting times only, as a marked point process, with the marks
being the interval
lengths (i.e. tj_2 - tj_1).
How you would go about analyzing such data, I don't know. In the
point process context
``regularity'' would amount to having a homogeneous Poisson process
(with the marks,
i.e. the interval lengths, being independent of the [starting]
points). There may be
tests for this sort of null hypothesis, against an unspecified
alternative, out there.
I would start by having a look at the book (2 volumes) by Daley and
Vere-Jones (second ed.).
One way to proceed might be to fit some sort of conditional intensity
function (conditional
on the past, including the past marks) and test this model against
the null model by
a likelihood ratio test. The problem, it seems to me, is to specify
an appropriate
and sufficiently alternative general conditional intensity function.
The fitting could then be
done using the Berman-Turner device (see ``Approximating point
process likelihoods using GLIM'',
M. Berman and T. R. Turner, Applied Statistics vol. 41, 1992, pp. 31
-- 38. See also
the paper by Ogata cited therein.)
HTH.
cheers,
Rolf Turner
On 4/09/2008, at 4:44 PM, Alexy Khrabrov wrote:
Greetings -- I've got some sensor data of the form
t1_1, t1_2
t2_1, t2_2
...
tN_1,tN_2
-- time intervals measuring starts and stops of sensor activity.
I'd like to see whether there's any regularity in it. Seems
natural to consider these data timeseries -- except most of the
timeseries packages and models assume regular ones, with a fixed
frequency.
I wonder what's a good way to apply existing regular timeseries
packages to these data, and perhaps try some others? I like David
Stoffer's book a lot, yet he uses R's own ts methods (with some
extras). I also like the zoo package, which allows for irregular
timeseries, yet I'm not sure how to apply the "usual" models to zoo
objects -- even though zoo strives to be compatible with ts... Is
zoo directly usable for ts-like time domain and spectral analysis
as per Stoffer?
Another way I was pondering is to map the above to a an artificial
index 1:n and consider it multivariate timeseries. Is it something
done in irregular timeseries analysis?
Cheers,
Alexy
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