The Influence of Autocorrelation on Extremal Properties of Gaussian Random Processes, Correlated Randomness and Stochastic Periodicity

Ogunshemi, Alessandro (2022) The Influence of Autocorrelation on Extremal Properties of Gaussian Random Processes, Correlated Randomness and Stochastic Periodicity. PhD thesis, University of Nottingham.

[img] PDF (Thesis - as examined) - Repository staff only - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Available under Licence Creative Commons Attribution.
Download (19MB)


Characterising the behaviour of a random process with respect to returns to previous states is a perennial concern of stochastic process theory, with applications of results spanning the applied sciences—from reliability in physical systems and networks, to the detection of surfaces in image analysis. The Gaussian stochastic process is entirely specified by its autocorrelation, a function that measures memory or time dependence in the process and whose form can lead to changes in higher order properties of the process. For a zero-mean Gaussian process the most common returns are to the zero-level, meaning an exploration of the zero-crossings—as influenced by the autocorrelation function—is a route for studying the interplay between randomness and dependence.

In this thesis, autocorrelation with two forms of periodic modulation is used to study changes in the zero-crossings of stationary Gaussian processes when the frequency of periodicity is increased. Realisations of the process are simulated for a variety of autocorrelations with either exponential or power-law decay, and the time intervals between zero-crossings are shown to be well-approximated by either finite or compound mixture distributions—the latter formulation pertaining to cases of strong dependence in crossings. Sufficient conditions for observing three different kinds of zero-crossing behaviour are determined and tested through a combination of simulation and modelling tools. Critical values of the autocorrelation's frequency and respective modulations result in three distinct power spectrum profiles, particularly so at extremely small/large cases of that dominant frequency. (1) Without the periodic modulation, power is concentrated at small frequencies, with zero as the peak frequency, so that crossing intervals have a large variance and are weakly correlated. (2) With a cosine modulation the origin is a minimum point, and the power spectrum is maximal away from the origin, resulting in wave-like sample functions with strongly correlated crossings. (3) With a cosine-squared modulation the spectrum contains at least three maxima including the origin, and at least two minima, leading to long sequences of regular crossings punctuated by rare but significantly large periods of no axis crossings.

Mixture distributions provide an alternative to using Markov chains and stochastic integrals to model the dependence between crossings. This approach provides the added benefit for tail estimation that expressions are all either explicit or asymptotically integrable. A closer investigation of the behaviour at extreme cases when the autocorrelation's periodicity parameter a≥0 is either zero or very large further reveals patterns in the zero-crossing sequence reminiscent of stochastic periodicity in random dynamical systems. Lacunarity, a measure of spatial heterogeneity of the crossings, reveals that at very small and large timescales there is Poisson-like behaviour, and within that range, non-Poissonian features are observed. These are moderately dissimilar when a=0, but for a≫1 the deviations are significant, persist at large timescales relative to the correlation length, and contain oscillations typical of a deterministic process. Further proof of the proposed tripartite classification and ensuing analyses is demonstrated in a case study involving magnetoencephalography signal data that are approximately Gaussian distributed; level-crossings of the mean values recognise a change between two brain states associated with before and after trial participants perform a voluntary action.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Hopcraft, Keith
Preston, Simon
Keywords: zero-crossing analysis, Gaussian processes, dependent behaviour, stationarity, periodicity, lacunarity, inter-event density, persistence, count distributions, time series, simulation methods, mixture modelling, box counting
Subjects: Q Science > QA Mathematics > QA273 Probabilities
Q Science > QA Mathematics > QA276 Mathematical statistics
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 69153
Depositing User: Ogunshemi, Mr Alessandro
Date Deposited: 13 Sep 2023 08:37
Last Modified: 13 Sep 2023 08:37

Actions (Archive Staff Only)

Edit View Edit View