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I'm currently exploring applications of wavelets to time series and stochastic processes, particularly in application to biological systems which exhibit quasi-periodic behavior. In many cases, we see what looks like a clear rhythm emerge from the system we're interested in studying, but this apparent rhythm is actually not (using the narrow, mathematical sense of the word) periodic, emerging from a complicated network of underlying stochastic processes operating on timescales several orders of magnitude faster, in some cases as little as ten-thousandths of a second. Using something like Fourier analysis, with its very strong assumptions of periodicity and stationarity, yields estimators and analyses that are at best fragile and at worse profoundly and unpredictably biased. Wavelets and similar compactly-supported bases allow us to evaluate more local behavior at the cost of frequency resolution. Ongoing questions my work addresses are: - Can we find stochastic analogues of metrics that are used for deterministic system, such as the phase response curve?
- How can we use these results to classify oscillators, and infer network topologies for systems of coupled oscillators?
- How well do stochastic diffusions model the behavior of biological systems? Do networks of these oscillators have the same properties as networks of SCN neurons?
- What are the limits of inference that can be performed on the unobserved states of a biological oscillator given the limited outputs we can observe?
If any of this sounds interesting, by all means get in touch!
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