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Knowledge discovery and information extraction of large and complex datasets has attracted great attention in wide-ranging areas from statistics and biology to medicine. Tools from machine learning, data mining, and neurocomputing have been extensively explored and utilized to accomplish such compelling data analytics tasks. However, for time-series data presenting active dynamic characteristics, many of the state-of-the-art techniques may not perform well in capturing the inherited temporal structures in these data. In this paper, integrating the Koopman operator and linear dynamical systems theory with support vector machines (SVMs), we develop a novel dynamic data mining framework to construct low-dimensional linear models that approximate the nonlinear flow of high-dimensional time-series data generated by unknown nonlinear dynamical systems. This framework then immediately enables pattern recognition, e.g., classification, of complex time-series data to distinguish their dynamic behaviors by using the trajectories generated by the reduced linear systems. Moreover, we demonstrate the applicability and efficiency of this framework through the problems of time-series classification in bioinformatics and healthcare, including cognitive classification and seizure detection with fMRI and EEG data, respectively. The developed Koopman dynamic learning framework then lays a solid foundation for effective dynamic data mining and promises a mathematically justified method for extracting the dynamics and significant temporal structures of nonlinear dynamical systems. 

Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON is robust and versatile enough to reliably reveal full and partial resonance among fast chemical oscillators, coherent circadian rhythms among hundreds of cells, and functional connectivity mediating social synchronization of circadian rhythmicity among mice over weeks.