In the name of God
Institute for advanced studies
in basic sciences
Radial Basis Function Networks
Yousef Akhlaghi
This seminar is an introduction to radial basis function networks as
linear neural networks.
Definition :
Radial basis function (RBF) networks are a special class of single
hidden-layer feed forward neural networks for application to problems
of supervised learning.
Reference: Radial Basis Function Networks, Mark J. Orre,
Linear models have been studied in statistics for about 200 years and the theory
is applicable to RBF networks which are just one particular type of linear model.
However the fashion for neural networks which started in the mid-80’s has given
rise to new names for concepts already familiar to statisticians.
Equivalent terms in statistics and neural networks:
Supervised Learning:
Problem to be solved: Estimate a function from some example input-output
pairs with little or no knowledge of the form of the function.
DIFFERENT NAMES OF THIS PROBLEM:
Nonparametric regression
Function approximation
System identification
Inductive learning
IN NEURAL NETWORK :
Supervised learning
The function is learned from the examples or training set
contains elements which consist of paired values of the
independent input variable and the dependent output variable.
Two main regression problems in statistics:
- parametric regression
- nonparametric regression
Parametric Regression:
In parametric regression the form of the functional relationship between the
dependent and independent variables is known but may contain parameters
whose values are unknown and capable of being estimated from the training
set.
For example fitting a straight line:
parameters
to a bunch of points
Important point: The free parameters as well as the dependent and
independent variables have meaningful interpretations like initial concentration
or rate.
Nonparametric regression:
There is no or very little a priori knowledge about the form of the true function
which is being estimated.
The function is still modeled using an equation containing free parameters but
Typically this involves using many free parameters which have no physical
meaning in relation to the problem. In parametric regression there is typically a
small number of parameters and often they have physical interpretations.
In neural networks including radial basis function networks:
- Models are nonparametric and their weights and other parameters have no
particular meaning in relation to the problems to which they are applied.
-The primary goal is to estimate the underlying function or at least to estimate its
output at certain desired values of the input
Linear Models:
A linear model for a function f (x ) takes the form:
The model ‘f ’ is expressed as a linear combination of a set of ‘m’ fixed
functions often called basis functions by analogy with the concept of a vector
being composed of a linear combination of basis vectors.
Nonlinear Models:
if the basis functions can change during the learning process then the model is
nonlinear.
An example
Almost the simplest polynomial is the straight line:
which is a linear model whose two basis functions are:
and whose parameters (weights) are:
The two main advantages of the linear character of RBF
networks:
1- Keeping the mathematics simple it is just linear algebra (the
linearly weighted structure of RBF networks)
2- There is no optimization by general purpose gradient descent
algorithms (without involving nonlinear optimization).
Radial Functions:
Their characteristic feature is that their response decreases (or increases)
monotonically with distance from a central point. The centre, the distance scale,
and the precise shape of the radial function, are parameters of the model, all
fixed if it is linear.
A typical radial function is the Gaussian:
Its parameters are
c : centre
r : width (spread)
Local modelling with radial basis function networks
B. Walczak , D.L. Massart
Chemometrics and Intelligent Laboratory Systems 50 (2000) 179–198
From exact fitting to RBFNs
RBF methods were originally developed for exact interpolation of a set of
data points in a multidimensional space. The aim of exact interpolation is to
project every input vector
, onto the corresponding target
, to find a
function
such that:
According to the radial basis function approach, exact mapping can be
performed using a set of m basis functions (one for each data point) with the
form
, where
is some nonlinear function, and
denotes
distance between
and
, usually Euclidean distance. Then the output of
the mapping can be presented as linear combinations of these basis functions:
where
denotes weights,
basis function, respectively.
and
denote input object and the center of
In matrix notation the above equation is:
For a large class of functions, the matrix Φ is non-singular, and eq. 3 can be
solved:
The basis functions can have different forms. The most popular among them is
the Gaussian function:
controlling the smoothness properties of the interpolating function.
|| xi – xj ||
Euclidean Distance
of center j
from object i
= [(xi1 - x1j)2 + (xi2–x2j)2 + …+ (xin-xnj)2]0.5
object
center
Distance (x)
f
w
x2
f ( x)  exp( 2 )
2
×
x2
f ( x)  exp( 2 )
2
×
x2
f ( x)  exp( 2 )
2
×
# Hidd. nodes = # objects
Exact Fitting
S
output
In practice, we do not want exact modeling of the training data, as the
constructed model would have a very poor predictive ability, due to fact that all
details noise, outliers are modeled.
To have a smooth interpolating function in which the number of basis functions
is determined by the fundamental complexity of the data structure, some
modifications to the exact interpolation method are required.
1) The number K of basis functions need not equal the number m of data
points, and is typically much less than m.
2) Bias parameters are included in the linear sum.
3) The determination of suitable centers becomes part of the training process.
4) Instead of having a common width parameters, each basis function is
given its own width σj whose value is also determined during training.
By introducing at least the first two modifications to the exact interpolation
method we obtain RBFN.
Architecture of RBFN
RBFN can be presented as a three-layer feedforward structure.
• The input layer serves only as input
distributor to the hidden layer.
• Each node in the hidden layer is a radial
function, its dimensionality being the
same as the dimensionality of the input
data.
• The output is calculated by a linear
combination . i.e. a weighted sum of the
radial basis functions plus the bias,
according to:
In matrix notation:
Nod1
object1
Nod2
Nodk
bias
Training algorithms
RBF network parameters
– The centers of the RBF
activation functions
– The spreads of the
Gaussian RBF activation
functions
– The weights from the hidden
to the output layer
subset selection:
Different subsets of basis functions can be drawn from the same fixed set of
candidates. This is called subset selection in statistics.
• forward selection
– starts with an empty subset
– added one basis function at a time (the one
that most reduces the sum-squared-error)
– until some chosen criterion stops
• backward elimination
– starts with the full subset
– removed one basis function at a time ( the
one that least increases the sum-squarederror)
– until the chosen criterion stops decreasing
Orthonormalization of basis functions:
The Gram–Schmidt orthogonalization procedure is used to replace the k basis
functions
by the set of K orthogonal vectors
describing the same space.
Initially, the basis functions are centered on data objects.
:
The set of vectors ui , which are a linear combination of the functions
, can
be used directly to model y. To model y, the consecutive orthonormal vectors
ui (i =1,2, . . . m) are introduced until the network performance reaches the
desired level of approximation error.
Nonlinear training algorithm:
 Apply the gradient descent method for finding centers,
spread and weights, by minimizing the cost function (in most
cases squared error). Back-propagation adapts iteratively the
network parameters considering the derivatives of the cost
function with respect to those parameters.
Nonlinear neural network
Drawback: Back-propagation algorithm may require several
iterations and can get stuck into a local minima of the cost
function
Radial Basis Function Neural Net in MATLAB nnet-Toolbox
function [net, tr] = newrb (p, t, goal, spread, mn, df)
NEWRB adds neurons to the hidden layer of a radial basis network until it
meets the specified mean squared error goal.
P
T
GOAL
SPREAD
MN
DF
- RxQ matrix of Q input vectors.
- SxQ matrix of Q target class vectors.
- Mean squared error goal, default = 0.0.
- Spread of radial basis functions, default = 1.0.
- Maximum number of neurons, default is Q.
- Number of neurons to add between displays, default = 25.
The following steps are repeated until the network's mean squared error falls
below GOAL or the maximum number of neurons are reached:
1) The network is simulated with random weight
2) The input vector with the greatest error is found
3) A neuron (basis function) is added with weights equal to that vector.
4) The output layer weights are redesigned to minimize error.
THANKS