Identification and filtering of nonlinear systems using canonical variate analysis

States for a nonlinear time series are constructed directly from a nonlinear canonical variate analysis (CVA) of the past and future of the process. Such states can be computed sequentially by solution of the maximal correlation problem. A state space innovations representation for the Markov process is given in terms of the canonical variable states. Computational algorithms are developed for determination of the canonical variable states, state space model fitting, and construction of nonlinear stochastic filters. The performance of the computational procedures are demonstrated on simulated data of the Lorenz chaotic attractor, a multiple equilibria nonlinear system, including process excitation noise. From observation of only one of the three states of the Lorenz attractor, the full dynamics of the system are determined. The filtered state estimate is accurate, and the identified nonlinear system has the same nonlinear character as the true process including chaos and multiple equilibria.