Finding frequencies in a noisy, "uneven" dataset

Question

I'm working on a problem where frequency analysis applies (decomposition of a signal into frequencies, that is), but it's noisy and the samples are unevenly spaced.

Specifically: given a list of items purchased at a bar/restaurant, try to estimate the number of guests on the check based on distinct "frequencies" of purchase. The logic is that if there are N guests on a check, then it's reasonable to see N frequencies of drinks being purchased, one person buying every 10 minutes, another every 15, etc. (Plenty of other properties of the check should be included, but here I'm focusing specifically on estimating distinct frequencies).

So more formally: given a noisy, unevenly spaced time series, find the smallest number of frequencies which reproduce the signal while minimizing the error (... for some sensible definition of how to minimize both the error and the number of frequencies simultaneously).

This is more a machine learning problem than signal processing. I realize it's also an open question, but can anyone point me in the right direction? Is there a particular method or algorithm that applies here?

Answer 1

Just to point you in one possible direction: you could treat this problem as one of probabilistic mixture modeling.

Imagine that each person's drink ordering is governed by a probability distribution. That distribution may be characterized by the time since their last drink order. As time passes, the probability that they will order another drink increases until eventually they order another drink and the time resets.

One possible model for the time between drink orders for a single person is the exponential distribution. If you consider many people ordering, the resultant times would likely be a mixture of exponentials. The problem then comes down to fitting a mixture of exponentials to the table's drink data. You would likely have to supply some additional prior data as well to get a meaningful model (otherwise it's difficult to tell the difference between 5 people ordering 2 drinks a piece or 1 person ordering 10 drinks).