Radial Basis Function

CSC 367 2.0 Mathematical Computing

Assignment 3

Radial Basis Functions

AS2010377

M.K.H.Gunasekara

Special Part 1 Department of Computer Science

UNIVERSITY OF SRI JAYEWARDENEPURA

M.K.H.Gunasekara - AS2010377 CSC 367 2.0 Mathematical Computing

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Table of Contents

-

Introduction ............................................................................................................................................ 2

Methodology ........................................................................................................................................... 3

Implementation ...................................................................................................................................... 5

Results ..................................................................................................................................................... 6

Discussion.............................................................................................................................................. 10

Appendices ............................................................................................................................................ 11


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Introduction

Neural Networks offer a powerful framework for representing nonlinear mappings from

several inputs to one or more outputs.

An important application of neural networks is regression. Instead of mapping the inputs

into a discrete class label, the neural network maps the input variables into continuous

values. A major class of neural networks is the radial basis function (RBF) neural network.

We will look at the architecture of RBF neural networks, followed by its applications in both

regression and classification.

In this report Radial Basis function is discussed for clustering as unsupervised learning

algorithm. Radial basis function is simulated to cluster three flowers in a given data set

which is available in http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data.

http://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data


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Methodology Radial Basis Function

Figure 01 : One hidden layer with Radial Basis Activation Functions

Radial basis function (RBF) networks typically have three layers

1. Input Layer

2. A hidden layer with a non-linear RBF activation function

3. Output Layer

Where N is the number of neurons in the hidden layer, is the center vector for neuron i, and is

the weight of neuron i in the linear output neuron. Functions that depend only on the distance from

a center vector are radially symmetric about that vector, hence the name radial basis function. In the

basic form all inputs are connected to each hidden neuron. The norm is typically taken to be the

Euclidean distance and the radial basis function is commonly taken to be Gaussian Function

( ) ( ‖ ‖

) ------ (1)

There are some other Radial Basis functions

Logistic Basis Function

( )

( )

Multi-quadratics

( )

√


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Input nodes connected by weights to a set of RBF neurons fire proportionately to the distance

between the input and the neuron in the weight space

The activation of these nodes is used as inputs to the second layer. The second layer (output layer) is

treated as a simple Perceptron network

Training the RBF Network This can be done positioning the RBF nodes and using the activation of RBF nodes to train the linear

outputs.

Positioning RBF nodes can be done in two ways; First method is randomly picking some of the data

points to act as basis functions. And the second method is trying to position the nodes so that they

are representative of typical inputs, like using k-means clustering algorithm.

In Activation function there is standard deviation parameter.

One option is, giving all nodes the same size, and testing lots of different sizes using a validation set

to select one that works. Alternatively we can select the size of RBF nodes so that the whole space is

coved by the receptive fields. So the width of the Gaussian should be set according to the maximum

distance between the locations of the hidden nodes (d), and the number of hidden nodes (M)

√ ------ (2)

We can use this normalized Gaussian function also.

( ) ( ‖ ‖

)

∑ ( ‖ ‖

)

------ (3)

Outputs of the RBF Network: ( ‖ ‖

)

Training the Perceptron Network We can train Pereceptron Network by using supervised learning method. Therefore we train the

MLP Network according to targets.


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Implementation

Implementation was done using MATLAB 7.10 (2010). Implementation was done according to

following methods

1. Locate RBF nodes into centers

2. Calculate for the Gaussian function

3. Calculate outputs of the RBF layer – Unsupervised Training

4. Make Perceptron Network for second layer –( I used MLP network without a hidden layer)

5. Train MLP Network according to targets and inputs (inputs are the output of RBF network) –

Supervised Training

6. Simulate the network

I have implement RBF Network with different strategies to compare the results

Using Randomly selected centers

Using K-Means Cluster centers

Using Non-normalized Gaussian function

Using Normalized Gaussian function

Using SVM for second layer


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Results

sepal length sepal width petal length petal width Expected Target Actual Output

5.1 3.5 1.4 0.2 Iris-setosa Iris-setosa

4.9 3 1.4 0.2 Iris-setosa Iris-setosa



5 3.6 1.4 0.2 Iris-setosa Iris-setosa


















4.6 3.6 1 0.2 Iris-setosa Iris-setosa



5 3 1.6 0.2 Iris-setosa Iris-setosa














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7 3.2 4.7 1.4 Iris-versicolor FALSE

6.4 3.2 4.5 1.5 Iris-versicolor Iris-versicolor

6.9 3.1 4.9 1.5 Iris-versicolor FALSE

5.5 2.3 4 1.3 Iris-versicolor Iris-versicolor




4.9 2.4 3.3 1 Iris-versicolor Iris-versicolor



5 2 3.5 1 Iris-versicolor Iris-versicolor

5.9 3 4.2 1.5 Iris-versicolor Iris-versicolor

6 2.2 4 1 Iris-versicolor Iris-versicolor















6.7 3 5 1.7 Iris-versicolor FALSE

6 2.9 4.5 1.5 Iris-versicolor Iris-versicolor




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6 2.7 5.1 1.6 Iris-versicolor FALSE


6 3.4 4.5 1.6 Iris-versicolor Iris-versicolor








5 2.3 3.3 1 Iris-versicolor Iris-versicolor







6.3 3.3 6 2.5 Iris-virginica Iris-virginica

5.8 2.7 5.1 1.9 Iris-virginica Iris-virginica

7.1 3 5.9 2.1 Iris-virginica Iris-virginica




4.9 2.5 4.5 1.7 Iris-virginica FALSE




6.5 3.2 5.1 2 Iris-virginica Iris-virginica



5.7 2.5 5 2 Iris-virginica Iris-virginica






6 2.2 5 1.5 Iris-virginica Iris-virginica






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6 3 4.8 1.8 Iris-virginica FALSE









6.5 3 5.2 2 Iris-virginica Iris-virginica



I found best results using RBF Network with Non-Normalized Gaussian activation function with 9

mismatches. And I found best results using MLP Network with 4 mismatches.

MLP Network as Second Layer

Random Center K Means Center

Non-Normalized Gaussian function

9 9

Normalized Gaussian function 11 11

Support Vector Machine as Second Layer

Random Center K Means Center

Non-Normalized Gaussian function

14 10

Normalized Gaussian function 14 17


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Discussion

1. There are some drawbacks of unsupervised center selection in radial basis functions

2. We can use an SVM for the second layer instead of a perceptron but it is not efficient for more

than 2 classes classification


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Appendices MATLAB Sourcecode for RBF Network with MLP Network

clc clear all % M.K.H. Gunasekara % AS2010377 % Machine Learning % Radial Basis Function [arr tx] = xlsread('data.xls'); Centers=zeros(3,4);

% I found centers as mean of the same cluster values

for i=1:50 Centers(1,1)=arr(i,1)+Centers(1,1); Centers(1,2)=arr(i,2)+Centers(1,2); Centers(1,3)=arr(i,3)+Centers(1,3); Centers(1,4)=arr(i,4)+Centers(1,4); end



for j= 1:3 Centers(j,1)=Centers(j,1)/50; Centers(j,2)=Centers(j,2)/50; Centers(j,3)=Centers(j,3)/50; Centers(j,4)=Centers(j,4)/50;

end

Centers

% OR we can use k means algorithms calculate cluster centers k=3; %number of clusters

[IDX,C]=kmeans(arr,k); C %RBF centres

%Uncomment following line to use k means %Centers=C;


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% distance between hidden nodes

%distance between hidden node 1 & 2 dist1= sqrt((Centers(1,1)-Centers(2,1))^2 + (Centers(1,2)-Centers(2,2))^2 +

(Centers(1,3)-Centers(2,3))^2 + (Centers(1,4)-Centers(2,4))^2);





% finding maximum distance maxdist=0; if ( dist1>dist2) & (dist1>dist3) maxdist=dist1; end

if ( dist2>dist1) & (dist2>dist3) maxdist=dist2; end


% calculating width sigma= maxdist/sqrt(2*3);

maxdist;

% Gaussian

%calculating outputs of RBF networks RBFoutput=zeros(150,3); d1=zeros(1,4); Centers; d=zeros(1,3); %Unnormalized method % calculate output for gaussian function

%Uncomment following lines (98-106) to use Non-Normalized Activation %functions % for i=1:150 for j=1:3 d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2;

RBFoutput(i,j)= exp(-(d(1,j)/(2*(sigma^2)))); end

end


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% %

%Normalized method %Summation

%Uncomment following lines (114-130) to use Gaussian Normalized Activation

functions % RBFNormSum=zeros(150,1); % for i=1:150 % for j=1:3 % d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2; % RBFNormSum(i,1)= exp(-(d(1,j)/(2*(sigma^2))))+ RBFNormSum(i,1); % end % % d=[0 0 0]; % end % % % calculate output for gaussian function % for i=1:150 % for j=1:3 % d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2; % % RBFoutput(i,j)= exp(-(d(1,j)/(2*(sigma^2))))/RBFNormSum(i,1); % end % % d=[0 0 0]; % end

RBFoutput

RBFo=RBFoutput.' % making MLP network % T=zeros(1,150);

T=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3]

S=[3 1]

;

R=[0 1;0 1;0 1]

% used feedforward neural network as MLP [3 1] MLPnet=newff(RBFo,S);

MLPnet.trainParam.epochs = 500; MLPnet.trainParam.lr = 0.1; MLPnet.trainParam.mc = 0.9; MLPnet.trainParam.show = 40;


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MLPnet.trainParam.perf = 'mse'; MLPnet.trainParam.goal = 0.001; MLPnet.trainParam.min_grad = 0.00001; MLPnet.trainParam.max_fail=4;

MLPnet = train(MLPnet,RBFo,T);

%simulating neural network y=sim(MLPnet,RBFo); output=round(y.'); Target=T.'; compare= [T.' output] count=0; for i=1:150 if(output(i)~=Target(i)) count=count+1; end end Unmatched=count

MATLAB Source code for RBF Network with SVM

clc clear all % M.K.H. Gunasekara % AS2010377 % Machine Learning % Radial Basis Function with Support Vector Machine [arr tx] = xlsread('data.xls'); Centers=zeros(3,4);

% I found centers as mean of the same cluster values



for i=101:150 Centers(3,1)=arr(i,1)+Centers(3,1); Centers(3,2)=arr(i,2)+Centers(3,2); Centers(3,3)=arr(i,3)+Centers(3,3);


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Centers(3,4)=arr(i,4)+Centers(3,4); end

for j= 1:3 Centers(j,1)=Centers(j,1)/50; Centers(j,2)=Centers(j,2)/50; Centers(j,3)=Centers(j,3)/50; Centers(j,4)=Centers(j,4)/50;

end

Centers

% OR we can use k means algorithms calculate cluster centers k=3; %number of clusters

[IDX,C]=kmeans(arr,k); C %RBF centres

%Uncomment following line to use k means Centers=C;

% distance between hidden nodes







% finding maximum distance maxdist=0; if ( dist1>dist2) & (dist1>dist3) maxdist=dist1; end



% calculating width sigma= maxdist/sqrt(2*3);


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maxdist;

% Gaussian

%calculating outputs of RBF networks RBFoutput=zeros(150,3); d1=zeros(1,4); Centers;

%Unnormalized method % calculate output for gaussian function

%Uncomment following lines (98-106) to use Non-Normalized Activation %functions d=zeros(1,3); for i=1:150 for j=1:3 d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2;

RBFoutput(i,j)= exp(-(d(1,j)/(2*(sigma^2)))); end % d=[0 0 0]; end %

%Normalized method %Summation

%Uncomment following lines (114-130) to use Gaussian Normalized Activation

functions % RBFNormSum=zeros(150,1); % for i=1:150 % for j=1:3 % d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2; % RBFNormSum(i,1)= exp(-(d(1,j)/(2*(sigma^2))))+ RBFNormSum(i,1); % end % % d=[0 0 0]; % end % % % calculate output for gaussian function % for i=1:150 % for j=1:3 % d(1,j)= (arr(i,1)- Centers(j,1))^2 + (arr(i,2)- Centers(j,2))^2 +

(arr(i,3)- Centers(j,3))^2 + (arr(i,4)- Centers(j,4))^2; % % RBFoutput(i,j)= exp(-(d(1,j)/(2*(sigma^2))))/RBFNormSum(i,1); % end % % d=[0 0 0]; % end

RBFoutput

RBFo=RBFoutput.' % making SVM network


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group=cell(3,1) group{1,1}=zeros(150,1);

for n=1:150; tclass(n,1)=tx(n,5); end

group{1,1}=ismember(tclass,'Iris-setosa') group{2,1}=ismember(tclass,'Iris-versicolor') group{3,1}=ismember(tclass,'Iris-virginica')

[train, test] = crossvalind('holdOut',group{1,1}); cp = classperf(group{1,1});

for i=1:3 %svmStruct(i) =

svmtrain(RBFoutput(train,:),group{i,1}(train),'showplot',true); svmStruct(i) = svmtrain(RBFoutput,group{i,1},'showplot',true); end

for j=1:size(RBFoutput) for k=1:3 if(svmclassify(svmStruct(k),RBFoutput(j,:))) break; end end result(j) = k; end

T=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3]

compare=[T.' result.']

Target=T.' output=result.' count=0; for i=1:150 if(output(i)~=Target(i)) count=count+1; end end Unmatched=count

MATLAB Source Code MLP Network

clc clear all % M.K.H. Gunasekara % AS2010377 % Machine Learning % MLP Network [arr tx] = xlsread('data.xls');


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inputs=arr.'; T=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

3]

%Multilayer network with hidden layer with 3 nodes MLPnet=newff(inputs,[4 3 1]);

MLPnet.trainParam.epochs = 500; MLPnet.trainParam.lr = 0.1; MLPnet.trainParam.mc = 0.9; MLPnet.trainParam.show = 40; MLPnet.trainParam.perf = 'mse'; MLPnet.trainParam.goal = 0.001; MLPnet.trainParam.min_grad = 0.00001; MLPnet.trainParam.max_fail=4;

MLPnet = train(MLPnet,inputs,T);

%simulating neural network y=sim(MLPnet,inputs); output=round(y.'); Target=T.'; compare= [T.' output] count=0; for i=1:150 if(output(i)~=Target(i)) count=count+1; end end Unmatched=count

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