Radial Basis Networks:An Implementation

of Adaptive Centers

Nivas Durairaj

ECE539 Final Project

Brief Description of RBF Networks

• Consists of 3 layers (input, hidden, output)• Input layer made up of nodes that connect

network to environment• At input of each neuron (hidden layer), distance

between neuron center & input vector is calculated

• Apply RBF (Gaussian bell function) to form output of the neurons.

• Output layer is linear and supplies response of network to activation function.

Project Overview

Purpose: Develop a Radial Basis Network with a supervised selection of centers

Question: Are there any disadvantages or advantages between a fixed center RBF network and an adaptive RBF network?

A RBF network with multiple outputs

Adaptation Formulas

RBF with supervised selection of centers require the following formulas:

)(

)()()1( 1 nw

nEnwnw

iii

1. Linear Weights (output layer)

2. Positions of centers (hidden layer)

)(

)()()1( 2 nt

nEntnt

iii

3. Spreads of centers (hidden layer)

)(

)()()1(

1311

n

nEnn

iii

W: 1x1

T: 1xm vector

: mxm matrix

M is the feature dimension

1 i

Programming

• Used Matlab to implement RBF Network with Adaptive Centers

• Sample code for calculation of linear weights given below:

%Calculation of linear weights weightdiff=0; for j=1:n g=exp(-0.5((x(j,:)-t(i,:)))*covinv(:,:,i)*((x(j,:)-t(i,:))')); weightdiff = weightdiff + e(j)*g; end w(i)=w(i) - (eta1*weightdiff);

)(

)()()1( 1 nw

nEnwnw

iii

Testing & Comparison

• Tested Adaptive Center RBF against Fixed Center RBF.

• Used data for three functions, namely sinusoidal, piecewise-linear, and polynomial functions.

• Made use of the cost function given below analyze differences between two networks

N

jjeE

1

2

2

1

M

iCijij

jjj

itxGwd

xFde

1

)(

)(*Cost Function

where

Sinusoidal Function Testing

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-1.5

-1

-0.5

0

0.5

1RBF with Adaptive Centers

test samplesapproximated curvetrain samplesradial basis

Sinosoid Function Data

0

0.1

0.2

0.3

0.4

0.5

0.6

2 3 4 5 6 7

No. of Radial Basis Functions

Co

st F

un

ctio

n O

utp

ut

Fixed Center RBF Network

Adaptive Center RBF Network

For fewer radial basis functions, adaptive center RBF network seems to perform a bit better. However, after number of RBFs increase, results in cost function are negligible.

Piecewise Linear Function Testing

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5RBF with Adaptive Centers

test samplesapproximated curvetrain samplesradial basis

Piecewise-Linear Function Data Chart

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

2 3 4 5 6 7 8 9 10

No. Of Radial Basis Functions

Co

st F

un

ctio

n O

utp

ut

Fixed Center RBF Network

Adaptive Center RBF Network

Adaptive center RBF network performed better till the number of radial basis functions reached 6. I found that at higher numbers of radial basis functions (9 and above), both RBF networks were providing similar approximations of piecewise-linear function.

Polynomial Function Testing

The adaptive center RBF network was clearly the winner in the approximation of the polynomial function. Differences in cost function for higher numbers of RBFs were too small for Excel to plot.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1RBF with Adaptive Centers

test samplesapproximated curvetrain samplesradial basis

Polynomial Function Data Chart

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

7.00E-04

8.00E-04

2 3 4 5 6

No. of Radial Basis Functions

Cos

t Fun

ctio

n O

utpu

ts

Fixed Center RBF Network

Adaptive Center RBF Network

Conclusion

• Results show RBF network with adaptive centers performs slightly better than fixed-center RBF.

• Advantage of Adaptive RBF: Performs better with fewer RBFs

• Disadvantage of Adaptive RBF: Takes longer to run.

• Unless situation is known, one cannot say with certainty that one model is better than other.