Chaotic Modeling in Network Spinal Analysis™: Nonlinear Canonical Correlation with Alternating Conditional Expectation (ACE): A Preliminary Report

Stephan Bohacek3 and Edmund Jonckheere, Ph.D.

Journal of Vertebral Subluxation Research
Volume 2 Number 4, December 1998
p.188-195

Abstract: - This paper presents a preliminary non-linear mathematical analysis of surface electromyographic (sEMG) signals from a subject receiving Network Spinal Analysis™ (NSA). The unfiltered sEMG data was collected over a bandwidth of 10-500Hz and stored on a PC compatible computer. Electrodes were placed at the level of C1/2, T6, L5 and S2 and voltage signals were recorded during the periods in which the patient was experiencing the "somatopsychic" wave, characteristic of NSA care. The intent of the preliminary study was to initiate mathematical characterization of the wave phenomenon relative to its "chaotic" and/or nonlinear nature. In the present study the linear and nonlinear Canonical Correlation Analyses (CCA) have been used. The latter nonlinear CCA is coupled to specific implementation referred to as Alternating Conditional Expectation (ACE). Preliminary findings obtained by comparing canonical correlation coefficients (CCC’s) indicate that the ACE nonlinear functions of the sEMG waveform data lead to a smaller expected prediction error than if linear functions are used. In particular, the preliminary observations of larger nonlinear CCC’s compared to linear CCC’s indicate that there is some nonlinearity in the data representing the "Somatopsychic" waveform. Further analysis of linear and nonlinear predictors indicates that 4th order nonlinear predictors perform 20% better than linear predictors, and 10th order nonlinear predictors perform 30% better than linear predictors. This suggests that the waveform possess a nonlinear "attractor" with a dimension between 4 and 10. Continued refinement of the ACE algorithm to allow for detection of more non-linear distortions is expected to further clarify the extent to which the sEMG signal associated with the somatopsychic wave™ form of NSA is differentiated as nonlinear as opposed to random.