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Introduction to Radial Basis Function PowerPoint PPT Presentation


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Introduction to Radial Basis Function. Mark J. L. Orr. Radial Basis Function Networks. Linear model. Radial functions. Gassian RBF: c : center, r : radius. monotonically decreases with distance from center. Multiquadric RBF. monotonically increases with distance from center.

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Introduction to Radial Basis Function

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Introduction to radial basis function l.jpg

Introduction to Radial Basis Function

Mark J. L. Orr


Radial basis function networks l.jpg

Radial Basis Function Networks

  • Linear model


Radial functions l.jpg

Radial functions

  • Gassian RBF:c : center, r : radius

  • monotonically decreases with distance from center

  • Multiquadric RBF

  • monotonically increases with distance from center


Slide4 l.jpg

Gaussian RBF

multiqradric RBF


Least squares l.jpg

Least Squares

  • model

  • training data : {(x1, y1), (x2, y2), …, (xp, yp)}

  • minimize the sum-squared-error


Example l.jpg

Example

  • Sample points (noisy) from the curve y = x : {(1, 1.1), (2, 1.8), (3, 3.1)}

  • linear model : f(x) = w1h1(x) + w2h2(x),where h1(x) = 1, h2(x) = x

  • estimate the coefficient w1, w2


Slide7 l.jpg

  • f(x) = x


Slide8 l.jpg

  • New model : f(x) = w1h1(x) + w2h2(x) + w3h3(x)where h1(x) = 1, h2(x) = x, h3(x) = x2


Slide9 l.jpg

  • absorb all the noise : overfit

  • If the model is too flexible, it will fit the noise

  • If it is too inflexible, it will miss the target


The optimal weight vector l.jpg

The optimal weight vector

  • model

  • sum-squared-error

  • cost function : weight penalty term is added


Example15 l.jpg

Example

  • Sample points (noisy) from the curve y = x : {(1, 1.1), (2, 1.8), (3, 3.1)}

  • linear model : f(x) = w1h1(x) + w2h2(x),where h1(x) = 1, h2(x) = x

  • estimate the coefficient w1, w2


The projection matrix l.jpg

The projection matrix

  • At the optimal weight:the value of cost function C = yTPythe sum-squared-error S = yTP2y


Model selection criteria l.jpg

Model selection criteria

  • estimates of how well the trained model will perform on future input

  • standard tool : cross validation

  • error variance


Cross validation l.jpg

Cross validation

  • leave-one-out (LOO) cross-validation

  • generalized cross-validation


Ridge regression l.jpg

Ridge regression

  • mean-squared-error


Global ridge regression l.jpg

Global ridge regression

  • Use GCV

  • re-estimation formula

  • initialize 

  • re-estimate , until convergence


Local ridge regression l.jpg

Local ridge regression

  • research problem


Example23 l.jpg

Example


Selection the rbf l.jpg

Selection the RBF

  • forward selection

    • starts with an empty subset

    • added one basis function at a time

    • most reduces the sum-squared-error

    • until some chosen criterion stops

  • backward elimination

    • starts with the full subset

    • removed one basis function at a time

    • least increases the sum-squared-error

    • until the chosen criterion stops decreasing


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