A Stydy of Noulineer Data Analysis by Using Connectinonist Model

本文研究的是非线性映射模型（如多层神经网络）的性质与模型中参数数量的关系。建立非线性框架下统计数据分析的基本思想方法。为了做到这一点。定义了非线性映射的微分几何特征，并将其应用于多变量数据分析的扩展。判别分析与回归分析的函数逼近。在研究过程中。引入了更为重要和普遍的概念“映射的特征密度”。越过边界的微分几何。可以表示各种几何性质。通过对上述概念的扩展，得到微分几何特征。全文的研究成果总结如下：1。非线性映射中映射特征密度的介绍。考虑在一个映射的输出空间中张成的流形的几何性质的一个可数的量，更多的例子是，一个模型（即映射族）的“映射能力”是。在某种意义上。由直方图表示，该直方图是通过在整个参数空间上累积上述数量改变模型中包含的参数而生成的。这种直方图或密度函数称为映射特征密度（MFD）。从Kullbach散度的角度比较了几种映射模型的MFD。几何相似性也可以估计。函数逼近和mfd当一个全局曲率时，绝对曲率在曲线上的积分值。用作上述数量。MFD可以用来评价一输入一输出神经网络的函数逼近能力。预计通过增加隐藏单元的数量，映射容量会变得更大。理论和实验都证实了这一点。n阶多项式函数的MFD在n阶上也有相同的趋势，通过比较这两组MFD得到了一些显著的结论。判别分析和MFD将上述全局曲率作为MFD的量，用于估计特征空间中两类边界的几何复杂度。这涉及到一个非线性判别分析问题。假设每个类别包含几个“核心”，每个核心都由高斯分布组成。空间-类别映射的最大功能域是核数和特征空间维数的函数。在隐藏单元的数量方面，得到了与三层感知器相同的趋势。通过对两组MFD的比较，得出了一些显著的结论。该结果可用于非线性判别问题中最优模型的估计。少

This research is to analyze the properties of nonlinear mapping model (e.g.Multilayr Neural Network) in conjunction with number of parameters in the model.and to establish funda mental methodology of statistical data analysis in nonlinear framework. In order to do this.differential geometrical feature of nonlinear mapping is defined and utilized for expansion of multivariate data analysis such as.discriminant analysis and function approximation for regression analysis. In the course of research.more significant and general concept is introduced called "Mapping's Feature Density" which.over the boundary of differential geometry.can express various geometrical properties.by expanding the above concept of differential geometrical feature.Summary of the whole content resulted by the research are ;1. Introduction of Mapping Feature DensityIn nonlinear mappings.considering a certain countable quantity representing the geometrical property of a manifold spanned in the output space of a mappin … More g.the "mapping capacity" of a model (i.e.family of mappings) is.in a sense.indicated by a histogram which is generated by accumulating the above quantity varying the parameters contained in a model over the whole parameter space.This histogram or the density function is called Mapping Feature Density (MFD). By comparing MFD of several mapping model in terms of Kullbach's divergence.geometrical similarity can also be estimated.2. Function approximation and MFDWhen a global curvature.integrated value of absolute curvature over the curve.is used as an above quantity.the MFD can evaluate capability of function approximation of one-input one-output neural networks.It is expected that the mapping capacity becomes larger by increasing the number of hidden units.which is confirmed by both of theoretical and experimental means.MFD of the polynomial function with n'th order also has the same tendency in terms of the order n.Some remarkable conclusions are derived by comparing these two sets of MFD.3. Discriminant analysis and MFDThe above global curvature is used as the quantity of MFD for application of estimation of the geometrical complexity of a boundary between two categories in the feature space.this is involved in a problem of nonlinear discriminant analysis.Assuming that each category contains several "cores", each of which consists of a Gaussian distribution.MFD of space-to-category mapping is a function of number of cores and dimension (of feature space). The same tendency is obtained as for three layr Perceptrons in terms of number of hidden units.By using this, some remarkable conclusions are derived by comparing these two sets of MFD.The result can be applied for estimation of the optimal model in problem of nonlinear discrimination. Less