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Section 5: Radial Basis Function (RBF) Networks Course: Introduction to Neural Networks Instructor: Jeen-Shing Wang Department of Electrical Engineering National Cheng Kung University Fall, 2005
Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Origin: Cover’s Theorem ,[object Object],Cover, T. M., 1965. “Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition.” IEEE transactions on Electronic Computers , EC-14, 326-334
Cover’s Theorem ,[object Object],[object Object],[object Object],where  ( x ) = [  1 ( x ),  2 ( x ), …,  M ( x )] T.
Cover’s Theorem (cont’d) ,[object Object],[object Object],[object Object],(a) (b) (c) Linear Separable Spherically Separable Quadrically Separable x x x x x x x x
Interpolation Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Interpolation Problem (cont’d) ,[object Object],[object Object],where
Michelli’s Theorem ,[object Object],[object Object]
Learning: Ill and Well Posed Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Regularization Theory ,[object Object]
Solution to the Regularization Problem ,[object Object],Standard error term Regularizing term where  is the regularization parameter
Solution to the Regularization Problem ,[object Object],[object Object]
Interpolation vs Regularization ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Generalized RBF Network (GRBFN) ,[object Object],[object Object],[object Object],[object Object],Possible solution -> Generalized RBFN approach Adjustable parameters
D-Dimensional Gaussian Distribution ,[object Object],( ind. each other) (General form)
D-Dimensional Gaussian Distribution 2-Dimensional Gaussian Distribution
Radial Basis Function Networks
RBFN: Universal Approximation ,[object Object],[object Object],[object Object],[object Object],[object Object]
Kernel Regression ,[object Object],[object Object],[object Object],[object Object]
Kernel Regression ,[object Object]
Kernel Regression ,[object Object]
Kernel Regression ,[object Object]
Nadaraya-Watson Regression Estimator ,[object Object],[object Object],[object Object]
Normalized RBF Network ,[object Object],[object Object]
Normalized RBF Network ,[object Object],[object Object]
Multivariate Gaussian Distribution ,[object Object],[object Object]
Multivariate Gaussian Distribution ,[object Object],[object Object]
RBFN vs MLP ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Learning Strategies ,[object Object],[object Object],[object Object],[object Object],[object Object]
Computation of Centers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
K -Means Clustering ,[object Object],[object Object],[object Object]
Computation of Widths ,[object Object],[object Object],[object Object],[object Object]
Computation of Widths (cont’d.) Red-dotted line: Estimated distribution Blue-solid line: Actual distribution
Computation of Weights (SVD) ,[object Object],[object Object],[object Object],Keep Constant
Computation of Weights (Gradient Descent) ,[object Object],[object Object],[object Object]
Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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Section5 Rbf

  • 1. Section 5: Radial Basis Function (RBF) Networks Course: Introduction to Neural Networks Instructor: Jeen-Shing Wang Department of Electrical Engineering National Cheng Kung University Fall, 2005
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  • 16. D-Dimensional Gaussian Distribution 2-Dimensional Gaussian Distribution
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  • 33. Computation of Widths (cont’d.) Red-dotted line: Estimated distribution Blue-solid line: Actual distribution
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Hinweis der Redaktion

  1. Good morning. First of all, I would like to thank you for attending my today’s presentation. My name is Jeen-Shing Wang and the topic for my today’s presentation is “ on the structure and learning of self-adaptive neuro-fuzzy inference systems .