Bilinear stochastic processes and time series

Graduation date: 1988In engineering, biology, ecology, medicine, economics and social\ud science, some processes are essentially bilinear, and some could be\ud approximated accurately by bilinear processes under certain conditions.\ud In this thesis the bilinear stochastic process and bilinear time series\ud are discussed.\ud Bilinear models essentially are nonlinear; the superposition rule\ud is not valid. A useful property, which characterizes the bilinear\ud feature among the nonlinear ones, is emphasized. The solutions of\ud deterministic bilinear systems and bilinear stochastic processes are\ud given. The direct method uses the Lie algebraic structure. For bilinear\ud stochastic processes, the decomposition to a cascade form is a\ud generalization of the Volterra-series expansion. Because a correction\ud term exists in bilinear stochastic differential equations, the\ud decomposition has two different forms; both of them are convergent. The\ud lth -order stationarity and asymptotic stationarity of bilinear\ud stochastic processes and time series are well defined, and the\ud conditions on parameters for lth -order stationarity are derived.\ud Affine bilinear models in time-series form are shown to be more\ud general than bilinear models, and more readily fit certain data. A\ud special high-order scalar affine bilinear time-series model can be\ud transferred to a first-order, vector, affine, bilinear model, but need\ud higher dimension than the linear ARMA model. For first-order affine\ud bilinear time series two possible methods of parameter estimation are\ud presented. The moment method uses the relationships between the\ud parameters, and the second and third-moments to estimate parameters. The\ud inverse method uses the output data to estimate the input, which is\ud compared with the standard white Gaussian random sequence, and the\ud method chooses the parameters of the model to make certain criterion\ud optimal. For the general non-Gaussian time series an identification\ud procedure using the inverse method is proposed.\ud Some examples of analysis and parameter estimation of bilinear\ud models are provided