A RBFNN is an artificial neural network that uses radial basis functions as activation functions. Figure shows the structure of the RBFNN. The RBFNN is three layered feedforward neural network. The first layer is linear and only distributes the input signal, while the next layer is nonlinear and uses Gaussian functions. The third layer linearly combines the Gaussian outputs. Only the tap weights between the hidden layer and the output layer are modified during training.
RBFNN have 5 parameters for optimization:
1 The weights between the hidden layer and the output layer.
2 The activation function.
3 The center of activation functions.
4 The distribution of center of activation functions.
5 The number of hidden neurons.
The weights between the hidden layer and the output layer are calculated by using MoorePenrose generalized pseudoinverse. This algorithm overcomes many issues in traditional gradient algorithms such as stopping criterion, learning rate, number of epochs and local minima. Due to its shorter training time and generalization ability, it is suitable for realtime applications.
The radial basis function selected is usually a Gaussian kernel for pattern recognition application.
Generally the center and distribution of activation functions should have characteristic similar to data. Here, the center and width of Gaussians are selected using Kmeans clustering algorithm.
Based on universal approximation theory center and distribution of activation functions are not deterministic if the numbers of hidden neurons being sufficient enough, one can say that the single hidden layer feedforward network with sufficient number of hidden neurons can approximate any function to any arbitrary level of accuracy.

This code is used in the following papers:
1. A. Asvadi, M. Karami, Y. Baleghi, “Efficient Object Tracking Using Optimized Kmeans Segmentation and Radial Basis Function Neural Networks,” International Journal of Information and Communication Technology Research (IJICT), vol. 4, no. 1, pp. 2939, December 2011.
2. A. Asvadi, M. Karami, Y. Baleghi, H. Seyyedi, “Improved Object Tracking Using Radial Basis Function Neural Networks,” in: Proceedings of 7th Iranian Machine Vision and Image Processing (MVIP2011), Tehran, Iran, November 2011.

"mvnrnd" from "Statistics and Machine Learning Toolbox" is used for generating sample data points.
Alireza (2020). Radial Basis Function Neural Networks (with parameter selection using Kmeans) (https://www.mathworks.com/matlabcentral/fileexchange/52580radialbasisfunctionneuralnetworkswithparameterselectionusingkmeans), MATLAB Central File Exchange. Retrieved .
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Inspired by: Kmeans clustering
Inspired: KMRBF Tracker, RBF Neural Networks with random selection of parameters
Create scripts with code, output, and formatted text in a single executable document.
Very helpful and concise manner of implementing an RBF neural network.
Better to be aware of the comment of "Alan Yu" on "9 Dec 2017" that is listed below.
How can I use this code "Radial Basis Function neural networks with parameter selection using Kmeans" for facial recognition ? I want to train the network first with a set of trainnind data then simulate it with a set of test data ? please help
If this code does not work for that, can anyone tell me what neural network with radial basis function can work for my problem? thank you
Dear Mr Alireza ,
how can i use your code "Radial Basis Function Neural Networks with parameter selection using Kmeans" for regression task ?
Regards
There is a problem in rbfn_test.m. If the values of output Y are close, then it would produce the same classification(wrong result).
My code to fix the problem is as follows.
function Y = rbfn_test(F, W, K, MU, SIGMA)
%RBFN_TEST Summary of this function goes here
% Detailed explanation goes here
%% Test RBF
N = size(F,1); % n = Number of Pixels
H = zeros(N,K);
for k = 1:K
h = RBFKernel(F,MU,SIGMA,N,k);
H(:,k) = h;
end
Y = H*W; % Output
[M,I] = max(Y,[],2);
IV = ind2vec(I');
Y = IV';
end
Besides, you use the diagonal matrices to represent the kernel widths SIGMA(rbfn_train.m). Is there any theory behind this formula?
Good work!
Can you give more explanation how to find kernel function of RBF? That is, give more description on RBFKernel.m.
I'm learning Matlab and RBF now. My question is: how can I use this code with data from a csv, consisting of three columns?
Thanks!
@pradeep s v it is due to random initialization of the kmeans clustering algorithm
accuracy is varying for every time running without any change
sir this code give me an eror in this line [data1,Y1,Xt1]=generate_two_moons(rad,width,dist,(num_samp./2),seed,0)
random_data=[data1,Y1]';random_seq = randperm(num_samp);
it said Error in Untitled (line 15)
[data1,Y1,Xt1]=generate_two_moons(rad,width,dist,(num_samp./2),seed,0)
please solve this issue
num_tr = 3000;
num_te = 500;
num_hd = 20;
num_ou = 1;
num_in = 2;
num_hd = 30;
num_samp = num_tr+num_te;
width = 4;
dist = 4;
seed=3000;
rand('seed',seed);
[data1,Y1,Xt1]=half double_moons(rad,width,dist,(num_samp./2),)
random_data=[data1,Y1]';random_seq = randperm(num_samp);random_data = random_data(:,random_seq);
X = random_data(1:2,1:num_tr)';
[indx,centers] = kmeans(X,num_hd);
pos_vector = indx';
c = centers';
%Calculating the variance for each center
for i = 1 : num_hd;
cluster_number=(i);
variance = 0;
counter2=0;
for j = 1 : num_tr;
if cluster_number==pos_vector(j)
sssss = sqrt(sum((random_data(1:2,j)'c(:,i)').^2))
variance = sssss + variance
counter2=counter2+1;
end
end
sigma(i) = variance/counter2;
end
%Parameters For RLS
w = ones(num_ou,num_hd);
lambda=0.9
P= (lambda)^(1)*eye(num_hd);
spread = sigma;
IN = random_data(1:2,1:num_tr);
T = random_data(3,1:num_tr);
P= (lambda)^(1)*eye(num_hd);
w = ones(num_ou,num_hd);
num_train_min=10;
for epoch =1:50;
for m = 1:(num_trnum_tr./10);
for i = 1:num_hd;
g(i,:) = exp((xc(:,i))'*(xc(:,i))/(2*spread(i)^2));
end
alpha = d  w*g;
w_diffrence = ((P*g)*alpha' )';
if (norm(w_diffrence) < 0.01) && (m >= num_train_min)
break;
end
w = w + w_diffrence ;
P=P((P*g/(g'*P*g))*g'*P) ;
end
end
%kmean/rls
num_tr = 3600;
num_te = 400;
num_hd = 20;
num_ou = 1;
num_in = 2;
num_hd = 30;
num_samp = num_tr+num_te;
rad = 10;
width = 6;
dist = 6;
seed=30000000;
rand('seed',seed);
[data1,Y1,Xt1]=generate_two_moons(rad,width,dist,(num_samp./2),seed,0)
random_data=[data1,Y1]';random_seq = randperm(num_samp);random_data = random_data(:,random_seq);
X = random_data(1:2,1:num_tr)';
[indx,centers] = kmeans(X,num_hd);
pos_vector = indx';
c = centers';
%Calculating the variance for each center
for i = 1 : num_hd;
cluster_number=(i);
variance = 0;
counter2=0;
for j = 1 : num_tr;
if cluster_number==pos_vector(j)
sssss = sqrt(sum((random_data(1:2,j)'c(:,i)').^2))
variance = sssss + variance
counter2=counter2+1;
end
end
sigma(i) = variance/counter2;
end
%Parameters For RLS
w = ones(num_ou,num_hd);
lambda=0.9
P= (lambda)^(1)*eye(num_hd);
spread = sigma;
IN = random_data(1:2,1:num_tr);
T = random_data(3,1:num_tr);
P= (lambda)^(1)*eye(num_hd);
w = ones(num_ou,num_hd);
num_train_min=10;
for epoch =1:50;
for m = 1:(num_trnum_tr./10);
for i = 1:num_hd;
g(i,:) = exp((xc(:,i))'*(xc(:,i))/(2*spread(i)^2));
end
alpha = d  w*g;
w_diffrence = ((P*g)*alpha' )';
if (norm(w_diffrence) < 0.01) && (m >= num_train_min)
break;
end
w = w + w_diffrence ;
P=P((P*g/(g'*P*g))*g'*P) ;
end
end
Thanks Charles for mentioning that. Yes, mvnrnd is used for generating sample data points for classification.
You may want to mention that your set of functions requires mvnrnd which is part of the Statistics and Machine Learning Toolbox.