| Project/Area Number |
05808028
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Statistical science
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| Research Institution | Ibaraki University |
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Principal Investigator |
YONEKURA Tatsuhiro Ibaraki Univ.Faculty of Eng.Assoc.Prof., 工学部, 助教授 (70240372)
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| Project Period (FY) |
1993 – 1994
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| Project Status |
Completed (Fiscal Year 1994)
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| Budget Amount *help |
¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 1994: ¥300,000 (Direct Cost: ¥300,000)
Fiscal Year 1993: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | Multilayr Neural Network / Noulinear Mapping / Multivariate data analysis / Differential Geometry / Geometrical Property / Mapping Feature Density (MFD) / Global Curvature / Mapping Capacity / 多変量解析 / 写像特徴密度 / 多次元データ解析 / 全曲率 / 汎化能力 |
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| Research Abstract |
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 Density
In 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 MFD
When 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 MFD
The 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
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