Publication detail

Diffusion processes are commonly used in the modeling of time series that exhibit stationarity and Markovianity. However, those two properties do not guarantee that a diffusive process is sufficient for the time series. In this paper, we develop a test for the sufficiency of a diffusion process for an observed time series. To develop the test we capitalize on the Kramers--Moyal (KM) expansion: a Taylor expansion of the integral form of the master equation that describes Markov continuous-time processes. In the idealized case, if the observed data indeed arise from a true diffusion process, then the KM expansion should truncate naturally after the second term. In theory, this means that any higher-order (>= 3) KM coefficients should be zero. However, in practice, the discrete nature of measurement introduces artificial higher-order KM coefficients, even when the underlying process is truly diffusive. Nonetheless, for genuinely diffusive systems, it is expected that the sampling distribution of a statistic associated with higher-order coefficients will be different than non-diffusive ones. This is a viable avenue for testing the appropriateness of a diffusion model given an observed time series. We take advantage of this and propose a meaningful statistic that could inform whether or not a diffusion model is sufficient for a given time series. We then build a test that involves reconstructing the diffusion equation to generate surrogate paths, yielding a bootstrap distribution against which the observed statistic could be compared. We evaluate the sensitivity and selectivity of the proposed test in Monte Carlo studies.

@article{medriano_vandekerckhove:preprint:reconstruct,
    title   = {{A} reconstruct-then-bootstrap test for the sufficiency of diffusion processes},
    author  = {Medriano, Kathleen and Vandekerckhove, Joachim},
    year    = {preprint},
    journal = {PsyArXiv}
}