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A dynamic process with challenge and recovery is an important branch in the family of stochastic processes. The dependent data of such processes are often observed over time, and hence, are time dependent. The purpose of this dissertation is to develop methods to characterize a dynamic process with challenge and recovery under different dimensionalities and error assumptions. In this dissertation, a univariate dynamic process under Gaussian assumption is discussed first and a bi-logistic model is developed by three different methods: compartment, additive, and Bayesian. Then the discussion is extended to a bivariate hysteresis system with challenge and recovery. Three methods: linear, nonlinear, and two-step simple harmonic, were developed to study hysteresis under the independent bivariate Gaussian assumptions. Finally, to be more general, a multivariate cylinder distribution was developed to analyze a multivariate dynamic process with challenge and recovery under more general error assumptions. In this case, the dimensionality could be any positive integers and the error assumptions are not necessarily independent Gaussian. The cylinder method is applied to the hysteretic system and the results show that the cylinder method can be used in various scenarios to obtain the least biased and most efficient parameter estimates.
Advisor: Anne M. Parkhurst