A distance-based dynamical transition analysis of time series signals and application to biological systems

Abstract

This study demonstrates an application of distance-based numerical measures to the phase space of time series signals, in order to obtain a temporal analysis of complex dynamical systems. This method is capable of detecting alterations appearing in the characters of the deterministic dynamical systems and provides a simple tool for the real-time analysis of time series data obtained from a complex dynamical system even with black box functionality. The study presents a possible application of the method in the dynamical transition analysis of real EEG records from epilepsy patients.

Keywords: Chaos; Dynamical system analysis; Epilepsy seizure prediction from EEG signal.

Fig. 1

a Illustration of trajectory curvatures in two-dimensional state space . b Trajectory 1 is an escaping type trajectory with D_E = 0.36, V_E = 0.0028 and D_EN = 128.67. Trajectory 2 is a slightly enwrapping type trajectory with D_E = 0.31, V_E = 0.0028 and D_EN = 109.84. Trajectory 3 is an enwrapping type trajectory with D_E = 0.23, V_E = 0.0053 and D_EN = 43.51 (length of trajectory and size of time window (w = 500) are equal)

Fig. 2

a An orbit in the state space and distance vectors for states on a trajectory. b A representation of the complex dynamical system and the corresponding phase space from time-delayed samples of the time series signal u(n)

Fig. 3

Logarithmic STAD in a, logarithmic STV in b and NAD in c for the logistic difference equation with the parameters of A(0) = 0.1 and various values of r. Response of methods to stepwise temporal variation in the values of r in d (other parameters are w = 500, = 3, p = 100, window sampling period T_s = 300)

Fig. 4

a Sum of the single-channel EEG records from patient A. b WV of cumulative EEG in a. c STAD in logarithmic scale. d STV in logarithmic scale. e NAD (parameter settings: w = 10000, = 1, p = 1000)

Fig. 5

a Sum of the single-channel EEG records from patient B. b WV of cumulative EEG in a.c STAD in logarithmic scale. d STV in logarithmic scale. e NAD (parameter Settings: w = 10000, = 1, p = 1000)

Fig. 6

Logarithmic STAD, logarithmic STV, NAD, and 10×WV are plotted versus various SNR levels of noise. STAD, STV, and NAD are more robust against noise than WV