Disentangling the stochastic behavior of complex time series

Abstract

Complex systems involving a large number of degrees of freedom, generally exhibit non-stationary dynamics, which can result in either continuous or discontinuous sample paths of the corresponding time series. The latter sample paths may be caused by discontinuous events – or jumps – with some distributed amplitudes, and disentangling effects caused by such jumps from effects caused by normal diffusion processes is a main problem for a detailed understanding of stochastic dynamics of complex systems. Here we introduce a non-parametric method to address this general problem. By means of a stochastic dynamical jump-diffusion modelling, we separate deterministic drift terms from different stochastic behaviors, namely diffusive and jumpy ones, and show that all of the unknown functions and coefficients of this modelling can be derived directly from measured time series. We demonstrate appli- cability of our method to empirical observations by a data-driven inference of the deterministic drift term and of the diffusive and jumpy behavior in brain dynamics from ten epilepsy patients. Particularly these different stochastic behaviors provide extra information that can be regarded valuable for diagnostic purposes.

Characterizing abrupt transitions in stochastic dynamics

“…0 is a scalar Wiener (Brownian) motion, and a(x, t) and ( ) b x t , 2 denote the state-dependent deterministic drift and the diffusion functions. A process x(t) generated with equation (1) is a continuous diffusion process if a(x, t) and b(x, t) are smooth and do not change dramatically over a short time interval t d [16]. The unknown functions a(x, t) and b(x, t) can be found non-parametrically [17]-i.e.…”

“…x t x . For this type of process and using the conditional probability distribution, one can show that x(t) satisfies Lindeberg's continuity condition, given some δ>0 [16]…”

“…In general, for a given time series, the non-vanishing of higher-order (>2) KM coefficients can be related to the existence of jumps in the time series [16]. In this case, the KM coefficients provide an upper limit for the continuity condition (equation (3)):…”

Stochastic analysis of time series for the spatial positions of particles trapped in optical tweezers

“…11 with D ( k ) ( x , t ) = 0 for k  ≥ 3. We note that non-vanishing higher order KM coefficients D ( k ) ( x , t ) with k  ≥ 3 have implication for the presence of jump events in the time series 15 .…”

Normal Behaviour Models for Wind Turbine Vibrations: Comparison of Neural Networks and a Stochastic Approach

“…The full implementation in one-and two-dimensions of this approach is already public available [29]. Improvements on the noise term are beyond this paper, but were already addressed previously [30,31].…”