Nonlinear Data Analysis with Algebraic Curve Fitting

本研究的目的是建立一套非线性数据分析的方法。代数曲线拟合的数据分析方法是研究的基本方法。我们首先研究了代数曲线拟合。我们开发了一种算法来推导点和代数曲线之间的精确距离。提出了一种寻找数据中代数曲线结构的方法。根据研究的进展，本文还对广义主成分分析和主曲线进行了研究。在相关的数据分析方法中，研究了切片逆回归、函数数据分析和主点法。提出了投影寻踪切片逆回归方法，数值算例表明了该方法的优越性。提出了一种非线性函数回归分析方法。

The aim of the research is to construct a series of methods for nonlinear data analysis. Data analysis methods with algebraic curve fitting are fundamental methods in the research. We had studied algebraic curve fitting at first. We developed an algorithm to derive the exact distance between a point and algebraic curve. A method for finding an algebraic curve structures in data is proposed. According to the progress of the research, we also study Generalized Principal Components Analysis and Principal Curves. For related data analysis methods, Sliced Inverse Regression, Functional Data Analysis and Principal Points are studied. Sliced Inverse Regression with Projection Pursuit is proposed and numerical examples show the advantages of the proposed method. A method of nonlinear functional regression analysis is developed.