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Forward & Backward selection in hybrid network

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Introduction

A training algorithm for an hybrid neural network for regression.

Hybrid neural network has hidden layer that has RBF or projection units (Perceptrons).

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When is it good?

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Hidden Units

RBF:

MLP:

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Overall algorithm

Divide input space and assign units to each sub-region.

Optimize parameters. Prune un-necessary weights using

Bayesian Information Criteria.

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Forward leg

Divide the input space into sub-regions

Select type of hidden unit for each sub-region

Stop when error goal or maximum number of units is achieved.

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Input space division

Like CART using

Maximum reduction in

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Unit type selection (RBF)

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Unit type selection (projection)

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Units parameters

RBF unit: center at maximum point.

Projection unit: weight normalized of maximum point

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ML estimate for unit type

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Pruning

Target function values corrupted with Gaussian noise

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BIC approximation

Schwartz, Kass and Raftery

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Evidence for the model

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Evidence for unit type1

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Evidence for unit type cont2’

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Evidence fore unit type cont3’

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Evidence Unit Type alg4.

Initialize alfa and beta Loop: compute w,wo Recompute alfa and beta Until difference in the evidence is

low.

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Pumadyn data set DELVE archive Dynamic of a puma robot arm. Target: annular acceleration of one

of the links. Inputs: various joint angles,

velocities and torques. Large Guassian noise. Data set non linear. Input dimension: 8, 32.

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Results pumadyn-32nh

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Results pumadyn-8nh

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Related work

Hassibi et al. with Optimal Brain Surgeon

Mackey with Bayesian inference of weights and regularization parameters.

HME Jordan and Jacob, division on input space.

Kass & Raftery Schwarz with BIC.

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Discussion

Pruning removes 90% of parameters. Pruning reduces variance of estimator. The pruning algorithm is slow. PRBFN better then MLP of RBF alone. Bayesian techniques disadvantage: the

prior distribution parameter. Bayesian techniques are better then LRT. Unit type selection is a crucial element in

PRBFN Curse of dimensionality is well seen on

pumadyn data sets.