# HMM with variable time delay

1. Apr 5, 2014

### rynlee

Hello All,

I'm building a HMM for some data where there are two different time delays. Specifically, I collect data in consecutive traces, such that the time delay between each data point within a trace is approximately fixed, and the time delay between each trace is approximately fixed, but the two are not equal.

If the time delay between traces was very large, the influence of the last state of one trace on the first state of the next would be negligible (exponential decay of state duration), and I could treat it as multiple observation sequences. That is not the case however - looking at the correlation of observation values between points, I find that the first and last points correlate, implying that indeed the time delay is short enough for information to be contained between the two.

So I was wondering - does anyone have a sense of how to develop a HMM where the time delay between integer time points changes? One way would be to take the time delay as the least common factor between the intra-trace and inter-trace delays, and somehow have observations 'skipped'. For example, if I make measurements every second, maybe the time delay between 'observations' would be 200 ms, but I would only actually make an observation every 5 time points. Is there a way to have a HMM with such 'unknown' points? Is there another way to approach the problem?

2. Apr 5, 2014

### rynlee

My idea above, in the third paragraph, could be described as a HMM with missing observations.

I've looked a bit at the literature on null observations, but that is really pertaining to when observations are made on the arcs/transitions, not at regular time intervals.

3. Apr 5, 2014

### rynlee

Alright, i think I've figured it out.

While null observations typically pertain to HMMs where you treat observations as made on the transitions as opposed to the regular time intervals (i.e. when you set self-transitions, a_ii in common nomenclature, to zero, and instead assign duration pdfs to each state), you can also treat the case where you make observations of a markov chain, but find that some intermittent observations are missing. See [Yeh, Chan, Symanski, 'Intermittent Missing Observations in Discrete-Time Hidden Markov Models', Communications in statistics 2011]. It's actually quite simple - when no observation is made, the probability of being in any state you can set to unity, which in the forward-backward parameters has the effect of skipping the step.