Nonlinear system identification: Density of states approach

With computational machinery advancing at an accelerated rate, researchers are able to create increasingly sophisticated models of complex phenomenon. Many phenomenon driven by nonlinear dynamic processes, however, still remain impenetrable to current methods that leverage raw computational power. New, more powerful methods must be developed to model these phenomenon. A novel method, “weighted density of state models”, is presented here to detect and classify in near real-time multiple states of a complex system, which have onsets by nonlinear switching events.; Weighted density of state models are built using the localized dynamical structure of a nonlinear system to draw up a detailed and comprehensive portrait of multiple behavioral states. The continuous variation of density across each system's state and an estimate of the vector field of that state provide the foundation for a holistic model of the system's local flow characteristics. Weighted density of state models are presented as a means of comparing the similarities between the localized dynamics of two separate systems or two states of the same system. For each point in a test trajectory, the densities contributed by neighboring points of the model are weighted by a function of the normalized dot product of the vector fields.; Weighted density of state models are applicable to a wide assortment of nonlinear switching problems. Three applications are given here to demonstrate its utility. Large scale short-time parameter modulations in the Lorenz system are successfully tracked in near real time. Previously indecipherable hyperchaotic key shift encryption codes are cracked with no unmasking errors. Epileptic seizures are anticipated using intracranial EEG data in 85% of the seizures studied with a mean anticipation time of 22.6 ± 4.5 minutes and a false-positive rate of 0.9 ± 0.5 false-positives per hour. These seizure anticipation results show marked improvements over a widely cited state of the art method by the criteria of a higher detection rate, a longer mean anticipation time, and a lower false-positive rate. Despite these advances weighted density of state models have too high a false-positive rate to be considered for clinical use at their present level of development.