Analysis of the non-Markov parameter in continuous-time signal processing.

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.

Theoretical & experimental analysis of the Non Markov Parameter to detect low frequency synchronisation in time series analysis.

A theoretical investigation into the behaviour of the Non-Markov Parameter is performed from a signal processing perspective in contrast to previous methodologies based on stochastic processes theory. The results indicate that the NMP can be regarded as an informational metric which is indicative of the degree of low frequency synchronisation in a complex system. These results have deep implications for physiological analysis of biological systems where the presence of sychronisation is often a marker of pathological functioning. The NMP measure is then applied to in vivo micro-electrode recordings from the subthalamic nucleus.