A Stydy of Noulineer Data Analysis by Using Connectinonist Model

本研究旨在分析非线性映射模型(如多层神经网络)的性质，并结合模型中的参数个数，建立非线性框架下统计数据分析的基本思路。为此，定义了非线性映射的微分几何特征，并将其用于多元数据分析的扩展，如判别分析和回归分析的函数逼近。在研究过程中，在微分几何的边界上引入了一个更有意义和更一般的概念--映射特征密度，它可以表示各种几何性质。通过对上述微分几何特征概念的扩展，本文的研究内容概括如下：1.在非线性映射中引入映射特征密度。考虑到某一可数量代表一个映射在…的输出空间中的几何性质更重要的是，一个模型(即映射族)的“映射能力”在某种意义上是由一个直方图表示的，该直方图是通过在整个参数空间上累加改变模型中包含的参数而产生的。该直方图或密度函数被称为映射特征密度(MFD)。通过比较几种映射模型在库尔巴赫散度方面的MFD，也可以估计几何相似性。函数逼近与MFD当用曲线上的全局曲率、绝对曲率的积分值作为上述指标时，MFD可以评价单输入单输出神经网络的函数逼近能力。理论和实验都证实了这一点。理论和实验都证实了这一点。n阶多项式函数的MFD在n阶上也具有相同的趋势。通过对这两组MFD的比较，得出了一些显著的结论。判别分析和MFD上述全局曲率被用作MFD的量，用于估计特征空间中两个类别之间边界的几何复杂性。这涉及到一个非线性判别分析问题。假设每个类别包含多个“核”，每个核由高斯分布组成。空间到类别映射的MFD是核数和(特征空间的)维度的函数。对三层感知器的隐含单元数有相同的趋势，通过比较这两组多层感知器得到了一些值得注意的结论，其结果可用于非线性判别问题的最优模型估计。较少

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