NONLINEAR STATISTICAL MODELING AND MODEL EVALUATION

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Modeling and model evaluation are crucial issues in scientific data analysis. Models are used for understanding the structure of system or process and for making trustworthy predictions in various fields of natural and social sciences. The data contain the information about the system under consideration and the objective of statistical analysis is to express this information in an understandable form using a statistical model. We also make inferences about the unknown aspects of random phenomena and seek an understanding of causal relationships. Now the wide availability of fast and inexpensive computers enables us to construct various types of nonlinear models for analyzing data with complex structure. Through this research project we have investigated the problem of constructing various types of nonlinear models and obtained the results in the following : (1) Model evaluation and selection criteria ; information-theoretic criterion which enables us to evaluate various types of nonli … More near statistical models. (2) Nonlinear modeling based on kernel, spline, neural network. (3) Defining the embedding dimension and delay time in the chaotic time series with additive dynamic noise. (4) The estimation of the Lyapunov exponent for the nonlinear autoregressive time series models. (5) Techniques for the numerical vertification of solutions of differential equations. (6) A method to estimate the guaranteed error bounds of the finite element solutions for the Stokes problem. (7) Mathematical model of the earth's mantle movement. (8) A finite element analysis for a thermal convection problem with the infinite Parandl number on the earth's mantle convection. (9) The motion and expansion of a viscous vortex ring ; various dynamical aspects, such as formation, traveling speed, waves, instability, interactions. (10) The motion of an axisymmetric vortex ring of small cross-section in a viscous in compressible fluid. (11) Theory for normalizing transformations of asymptotic U-statistics. (12) The use of Weighted bootstrap methods. Less