introduction to radial basis function
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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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radial functions
Radial functions
  • Gassian RBF:c : center, r : radius
  • monotonically decreases with distance from center
  • Multiquadric RBF
  • monotonically increases with distance from center
slide4

Gaussian RBF

multiqradric RBF

least squares
Least Squares
  • model
  • training data : {(x1, y1), (x2, y2), …, (xp, yp)}
  • minimize the sum-squared-error
example
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
slide9
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
The optimal weight vector
  • model
  • sum-squared-error
  • cost function : weight penalty term is added
example15
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
The projection matrix
  • At the optimal weight:the value of cost function C = yTPythe sum-squared-error S = yTP2y
model selection criteria
Model selection criteria
  • estimates of how well the trained model will perform on future input
  • standard tool : cross validation
  • error variance
cross validation
Cross validation
  • leave-one-out (LOO) cross-validation
  • generalized cross-validation
ridge regression
Ridge regression
  • mean-squared-error
global ridge regression
Global ridge regression
  • Use GCV
  • re-estimation formula
  • initialize 
  • re-estimate , until convergence
local ridge regression
Local ridge regression
  • research problem
selection the rbf
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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