Nonlinear systems identification: autocorrelation vs. autoskewness

Autocorrelation function (C1) or autoregressive model parameters are often estimated for temporal analysis of physiological measurements. However, statistical approximations truncated at linear terms are unlikely to be of sufficient accuracy for patients whose homeostatic control systems cannot be presumed to be stable local to a single equilibrium. Thus a quadratic variant of C1 [autoskewness function (C2)] is introduced to detect nonlinearities in an output signal as a function of time delays. By use of simulations of nonlinear autoregressive models, C2 is shown to identify only those nonlinearities that "break" the symmetry of a system, altering the mean and skewness of its outputs. Case studies of patients with cardiopulmonary dysfunction demonstrate a range of ventilatory patterns seen in the clinical environment; whereas testing of C1 reveals their breath-by-breath minute ventilation to be significantly autocorrelated, the C2 test concludes that the correlation is nonlinear and asymmetrically distributed. Higher-order functionals [e.g., autokurtosis (C3)] are necessary for global analysis of metastable systems that continuously "switch" between multiple equilibrium states and unstable systems exhibiting nonequilibrium dynamics.