Artificial Neural Networks 0909.560.01/0909.454.01 Fall 2004

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Artificial Neural Networks 0909.560.01/0909.454.01 Fall 2004. Lecture 6 October 18, 2004. Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring04/ann/. Plan. Radial Basis Function Networks RBF Formulation Network Implementation

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### Artificial Neural Networks0909.560.01/0909.454.01Fall 2004

Lecture 6October 18, 2004

Shreekanth Mandayam

ECE Department

Rowan University

http://engineering.rowan.edu/~shreek/spring04/ann/

Plan
• RBF Formulation
• Network Implementation
• Matlab Implementation
• Design Issues
• Center Selection: K-means Clustering Algorithm
• Input data processing
• Selection of training and test data - cross-validation
• Pre-processing: Feature Extraction
• Lab Project 3
RBF Principle

Transform to

“higher”-dimensional

vector space

Non-linearly

separable classes

Linearly

separable classes

j2(x)

x2

j1(x)

x1

Example: X-OR Problem

Decision

Boundary

RBF Formulation

Problem Statement

• Given a set of N distinct real data vectors (xj; j=1,2,…,N) and a set of N real numbers (dj; j=1,2,…,N), find a function that satisfies the interpolating condition

F(xj) = dj; j=1,2,…,N

j

1

j

1

1

1

j

1

j

1

0.5

0

-5

5

RBF Network

Hidden

Layer

Input

Layer

Output

Layer

x1

y1

Outputs

x2

Inputs

y2

x3

wij

1

j(t)

t

Matlab Implementation

%S. Mandayam/ECE Dept./Rowan University

%Neural Nets/Fall 04

clear;close all;

%generate training data (input and target)

p = [0:0.25:4];

t = sin(p*pi);

%Define and train RBF Network

net = newrb(p,t);

plot(p,t,'*r');hold;

%generate test data

p1 = [0:0.1:4];

%test network

y = sim(net,p1);

plot(p1,y,'ob');

legend('Training','Test');

xlabel('input, p');

ylabel('target, t')

Matlab Demos

» demorb1

» demorb3

» demorb4

x2

x1

Centers

Data points

RBF - Center Selection
K-means Clustering Algorithm
• N data points, xi; i = 1, 2, …, N
• At time-index, n, define K clusters with cluster centers cj(n); j = 1, 2, …, K
• Initialization: At n=0, let cj(n)= xj; j = 1, 2, …, K(i.e. choose the first K data points as cluster centers)
• Compute the Euclidean distance of each data point from the cluster center, d(xj , cj(n)) = dij
• Assign xj to cluster cj(n)if dij = mini,j {dij}; i = 1, 2, …, N, j = 1, 2, …, K
• For each cluster j = 1, 2, …, K, update the cluster center cj(n+1)= mean {xjcj(n)}
• Repeat until ||cj(n+1)- cj(n)||< e

Train

Train

Train

Test

Train

Train

Test

Train

Train

Test

Train

Train

Test

Train

Train

Train

Selection of Training and Test Data: Method of Cross-Validation
• Vary network parameters until total mean squared error is minimum for all trials
• Find network with the least mean squared output error

Trial 1

Trial 2

Trial 3

Trial 4

Feature Extraction

Objective:

• Increase information content
• Decrease vector length
• Parametric invariance
• Invariance by structure
• Invariance by training
• Invariance by transformation
Lab Project 3: Radial Basis Function Neural Networks

http://engineering.rowan.edu/~shreek/fall04/ann/lab3.html