title: Analysis of the non-Markov parameter in continuous-time signal processing.
abstract: The use of statistical complexity metrics has yielded a number of successful methodologies to differentiate and identify signals from complex systems where the underlying dynamics cannot be calculated. The Mori-Zwanzig framework from statistical mechanics forms the basis for the generalized non-Markov parameter (NMP). The NMP has been used to successfully analyze signals in a diverse set of complex systems. In this paper we show that the Mori-Zwanzig framework masks an elegantly simple closed form of the first NMP, which, for C(1) smooth autocorrelation functions, is solely a function of the second moment (spread) and amplitude envelope of the measured power spectrum. We then show that the higher-order NMPs can be constructed in closed form in a modular fashion from the lower-order NMPs. These results provide an alternative, signal processing-based perspective to analyze the NMP, which does not require an understanding of the Mori-Zwanzig generating equations. We analyze the parametric sensitivity of the zero-frequency value of the first NMP, which has been used as a metric to discriminate between states in complex systems. Specifically, we develop closed-form expressions for three instructive systems: band-limited white noise, the output of white noise input to an idealized all-pole filter,f and a simple harmonic oscillator driven by white noise. Analysis of these systems shows a primary sensitivity to the decay rate of the tail of the power spectrum.
authors.affiliation: University of Queensland, Brisbane, Australia.
authors.forename: J J
authors.initials: JJ
authors.lastname: Varghese
authors.affiliation: University of Queensland, Brisbane, Australia.
authors.forename: P A
authors.initials: PA
authors.lastname: Bellette
authors.affiliation: University of Queensland, Brisbane, Australia.
authors.forename: K J
authors.initials: KJ
authors.lastname: Weegink
authors.affiliation: University of Queensland, Brisbane, Australia.
authors.forename: A P
authors.initials: AP
authors.lastname: Bradley
authors.affiliation: University of Queensland, Brisbane, Australia.
authors.forename: P A
authors.initials: PA
authors.lastname: Meehan
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008390:Markov Chains; D015233:Models, Statistical; D009991:Oscillometry / Q000379:methods*; D012815:Signal Processing, Computer-Assisted