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A Stydy of Noulineer Data Analysis by Using Connectinonist Model

应用Connectinonist模型的Noulineer数据分析研究

基本信息

批准号：
05808028
负责人：
YONEKURA Tatsuhiro
金额：
$ 0.96万
依托单位：
Ibaraki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1993
资助国家：
日本
起止时间：
1993 至 1994
项目状态：
已结题

项目摘要

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

本研究旨在结合模型中的参数数量来分析非线性映射模型（例如多层神经网络）的特性，并建立非线性框架下统计数据分析的基本方法。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，例如用于回归分析的判别分析和函数逼近。在研究过程中，通过扩展上述微分几何特征的概念，引入了一个更有意义、更普遍的概念“映射特征密度”，它超越了微分几何的边界，可以表达各种几何性质。整个研究内容概括如下： 1．映射特征密度的介绍在非线性映射中。考虑表示在映射输出空间中跨越的流形的几何属性的某个可数量…更多g.模型（即映射族）的“映射容量”在某种意义上由直方图表示，该直方图是通过累积上述量来改变模型中包含的参数在整个参数上生成的该直方图或密度函数称为映射特征密度（MFD）。通过比较几种映射模型的MFD，用库尔巴赫散度来估计几何相似度。 2．函数逼近和MFD当以全局曲率、曲线上绝对曲率的积分值作为上述量时，MFD可以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，映射能力会变大。这一点得到了理论和实验手段的证实。n阶多项式函数的MFD也有通过比较这两组MFD，得出了一些显着的结论。3．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的边界的几何复杂性。这涉及到一个非线性判别分析的问题。假设每个类别包含多个“核心”，每个核心都由高斯分布组成。空间到类别映射的 MFD 是核心数量和维度（的特征空间）。三层感知器在隐藏单元数量方面也得到了相同的趋势。利用这一点，通过比较这两组MFD，得出了一些显着的结论。结果可以应用于非线性判别问题中最优模型的估计。较少的