Journal of Intelligent Learning Systems and Applications, 2011, 3, 17-25
doi:10.4236/jilsa.2011.31003 Published Online February 2011 (http://www.SciRP.org/journal/jilsa)
17
Function Approximation Using Robust Radial
Basis Function Networks
Oleg Rudenko, Oleksandr Bezsonov
The Department of Computer Engineering and Control, Kharkov National University of Radio Electronics, Kharkiv, Ukraine.
Received April 9th, 2010; revised July 20th, 2010; accepted August 4th, 2010
ABSTRACT
Resistant training in radial basis function (RBF) networks is the topic of this paper. In this paper, one modification of
Gauss-Newton training algorithm based on the theory of robust regression for dealing with outliers in the framework of
function approximation, system identification and control is proposed. This modification combines the numerical ro-
bustness of a particular class of non-quadratic estimators known as M-estimators in Statistics and dead-zone. The al-
gorithms is tested on some examples, and the results show that the proposed algorithm not only eliminates the influence
of the outliers but has better convergence rate then the standard Gauss-Newton algorithm.
Keywords: Neural Network, Robust Training, Basis Function, Dead Zone
1. Introduction
Function approximation involves estimating (approxi-
mating) the underlying relationship from a given finite
input-output data set



yxf x

 (1)
where 1M
xR

 is an input vector;

f
 is the arbi-
trary nonlinear fun ction , unknown in the general case;

is the unobserved disturbance with unknown characteris-
tics; has been the fundamental problem for a variety of
applications in system identification, pattern classifica-
tion, data mining and signal reconstruction [1-4].
Feedforward neural networks such as multilayer per-
ceptrons (MLP) have been widely used as an approach to
function approximation since they provide a generic
black-box functional representation and have been shown
to be capable of approximating any continuous function
defined on a compact set in
N
R with arbitrary accuracy
[5-7].
It has been proved that a radial basis function network
(RBF) can approximate arbitrarily well any multivariate
continuous function on a compact domain if a sufficient
number of ra dial basis functi on units are gi ven [8].
In contrast to MLPs, RBF networks use a localized re-
presentation of information. The RBF network requires
less computation time for the learning and more compact
topology than MLP. The network can be co nfigured with
one radial basis function centre at each training data
point. Thus, the complexity of the network is of the same
order as the dimensionality of the training data and the
network has a poor generalization capability. The RBF
decomposition of


f
x is
 
0
ˆ,,
NT
ii
i
f
xwxrwxr 


 (2)
where 1N
wR

 is a vector of linear weights, 1
N
R 


is a vector of RBFs and r is a distance.
An important advantage of RBFN from viewpoint of
practitioners is, therefore, clear and understandable in-
terpretation of the functionality of basis functions.
The traditional RBF basis function is defined by Euc-
lidian distance
E
ij
rxt

 and Gaussian activation
function by




22
exp 0.5
ji E
xr 

 , where i
x
is the
input sample number i, j
t is the center of j-th radial
basis function (radii),

is the standard deviation. If we
use the Mahalanobis distance



1
T
M
ij ij
rxtRxt

 
where 1k
ij
Rr





 is weight matrix, M is the dimension
of input vector i
x
, N is the number of neurons, for the
RBF activation function we have
 
1
exp T
jjjj
x
xtR xt



 



 (3)
where
j
R is the covariance matrix. Geometrically
j
t
represents the center and
j
R the shape of the j-th basis

Function Approximation Using Robust Radial Basis Function Networks 18
function. A hidden unit function can be represented as a
hyper-ellipsoid in the N-dimensional space.
All the network parameters (weights, centers and radii)
may be determined using various learning algorithms
have been used in order to find the most appropriate pa-
rameters for the RBF decomposition.
A network iteratively adjusts parameters of each node
by minimizing some cost function which can be defined
as an ensemble average errors.
  
1
1
,,
k
i
Fek ei
k



 

 (4)
where
 
,ei

 is a scalar loss function;
 
ˆ
, eiyifi   represents the residual error be-
tween the desired

y
i, and the actual network outputs,

ˆ
f
i; i—indicates the index of the series;

comprises
all the unknown parameters of the network,

0111 11
11,1,
11,1,
,,,, , , ,,,
,,,,,,
MMM
T
NNN NN
MMM
kcct trr
ctt rr

 




.
The problem of neural network training (estimating

)
which approximates the function (1) “well”, has essen-
tially been tackled, based on the following two different
assumptions [9]:
(A1) The noise has some probabilistic and/or statistical
properties.
(A2) Regardless of the disturbance nature, a noise
bound is available, i.e. 22
kk

 .
Assumption (A1) leads to different stochastic training
methods that are based on minimization of some loss
function. Different choices of loss functions arise from
various assumptions about the distribution of the noise in
measurement. The most common loss function is the
quadratic function corresponding to a Gaussian noise
model with zero mean, and a standard deviation that does
not depend on the inputs. The Gaussian loss function is
used popularly as it has nice analytical properties. How-
ever, one of the potential difficulties of the standard qua-
dratic loss function is that it receives the large contribu-
tions from outliers that have particularly large errors. The
problems in the neural network training are that when the
training data sets contain outliers, traditional supervised
learning algorithms usually cannot come up acceptable
performance. Since traditional training algorithms also
adopt the least-square cost function (4), those algorithms
are very sensitive to outliers.
Techniques that attempt to solve these problems are
referred to robust statistics [10,11]. In recent years, vari-
ous robust learning algorithms based on M-estimation
have been proposed to overcome the outlier’s problems
[12-17].
The basic idea of M-estimators is to replace the quad-
ratic function in the cost function (4) by the loss function
so that effect of those outliers may be degraded.
Traditional approaches of solving such a problem are
to introduce a robust cost function (4), and then, a steep-
est descent approach is applied. The idea of such an ap-
proach is to identify outliers and then to reduce the effect
of outliers directly.
Alternative approaches have been formulated in a de-
terministic framework based on Assumption (A2). In this
context th e training prob lem is then to find a θ belonging
to the class of models (2) for which the absolute value of
the difference between the function (1) and model is
smaller than k

for all times k.
Three different types of solutions to this problem have
mainly been explored in literature. The first method is to
formulate the estimation problem in a geometrical setting.
Different proposals result from this approach but Fogel
and Huang [18] proposed a minimal volume recursive
algorithm (FHMV) which minimizes the size of an ellip-
soid and was attractive for on-line estimation.
The second alternative is to derive estimation algo-
rithm for stability consideration together with the geo-
metrical (ellipsoidal-outer-bounding algorithm by Lozano-
Leal and Ortega) [19]. The third approach is to obtain
estimation (training) algorithm from modifying the ex-
ponentially weighted recursive least squares algorithm
(EW-RLS) [9].
All these algorithms have a dead zone. The dead zone
scheme guarantees convergence of the neural network
training algorithm in the present of noise 22
kk

 .
It should be noted that this dead zone may serve as
value that limits the accuracy of the obtained solutions,
i.e. determines its acceptable inaccuracy.
The proposed method combines the numerical robust-
ness of a particular class of non-quadratic M-estimators
and dead-zone.
2. Robust Gauss-Newton Training
Algorithm
The estimation ˆ

is the solution of the following set of
equations




1
,
1,0
k
i
jj
Fe ei
ei
k








 (5)
where






 


,
,,,
,
ei
eiei ei
ei






 is the
influence function and




,ei

 is the weight function.
For quadratic function
 
,ei

 in the maximum
likehood estimation case (5) has a closed form solution,
the sample mean. The sample mean is substantially af-

Function Approximation Using Robust Radial Basis Function Networks 19
fected by the presence of outliers.
For many non-quadratic loss functions, the Equation
(5) does not have closed form solution, but can be solved
by the some iterative or recursive m ethods.
The minimization of the criterion (4) can therefore be
performed using Gauss-Newton algorithm.
  

   
ˆ
1
ˆˆ
1ˆˆ
11
T
Pkfkek
kk ek fkPkfk

 

 
 
(6)
 
 


 

1
ˆˆ
11
ˆˆ
11
T
T
Pk Pk
Pkfk fkPkek
ekf kPkf k



 

 
(7)
where
     
  
01111
1,1 1,2
11 1
1,,1,1 ,
ˆˆˆˆˆ
ˆ,,,,,,
ˆ ˆ ˆˆˆ ˆ
,, ,, , , ,,
T
NN NN
MMM MM
fk fk fk fk fk
fk cctrr
fkfkfk fk fkfk
rr ctrr

 





 



 

0
ˆ1
fk
c




ˆ,,
iii
i
fk
x
tR
c



ˆA
i
ii
fk A
ce
tt





ˆA
i
ij ij
mm
fk A
ce
rr




with
 
1
T
j
j
A
xtR xt

 .
The initial value of the matrix

0P is chosen as in
the recursive MLS (RMLS), i.e.

0PI

, where
1

 , and the initial dimension of the identity matrix
I
is given as SS, where

2
11SMM is
number of adjustable parameters of a network contain-
ing 1 neuron. Because after the introduction into the
network a new n-th neuron the dimension of the


Pk
increases, the values of elements in matrix


Pk are
reset and initialized again, then S becomes equal to

2
11SnMM , where n—the current number
of neurons in the network.
The influence function


e  measures the influence
of a datum on the value of the parameter estimate. For
example, for the least-squares with

2
0.5ee  , the
influence function is

ee  , that is, the influence of a
datum on the estimate increases linearly with the size of
its error and without bound, which confirms the non-
robustness of the least-squares estimate.
Huber proposed a robust estimator so-called an M-es-
timator, M for maximum likelihood. M-estimator is the
solution of (5) where different non-quadratic loss func-
tion
 
,ei


are used.
Following Huber [10], a distribution of the noise con-
taminated by outliers expressed by a mixture of two
probability density functions



 
0
1pxp xqx   (8)
where


0
px
is the density of basic distribution of a
measurement noise;


qx is the density of a distribu-
tion of outliers;


0,1   is the probability of occurring
a large error.
Even if the basic


0
px and contaminating


qx
distributions are Gaussian with zero mean and variances
2
1

and 2
2

, 22
12

  hence, than optimal for the
Gaussian distribution estimations (6)-(7), obtained by
choosing


2
0.5ee  , will be unstable.
The density distribution *
p for

—contaminated pro-
bability distributions (8), which gives the minimum Fisher
information, contains a central region




0
1pp

 
and tails with exponentially decreasing density


0
p 

x
ce

. Usage of these distributions allowed to ob-
tain nonlinear robust Maximum likelihood estimates, that
are workable for almost all the noise distributions. This
algorithm combines the conventional least mean
square (LMS) if


2
1
3ek

 and least absolute devia-
tion (LAD) if


2
1
3ek

 stochastic gradient algo-
rithms and called the mixed-norm LMS algorithm
[10,20,21].
On the other hand, the choice of loss function, differ-
ent from the quadratic, ensures the robustness of esti-
mates, i.e. their workability for almost all distributions of
noises. Currently, there are many such functions


e

,
however, keeping in mind that
 




ek ek


 
 is
used in the learning algorithm (6)-(7) it is advisable to
choose such functions 


ek

, which have nonzero
second derivatives. As these functions can be taken, for
example [22,23],
 
2
12
ek
Fekc






(9)
 
2lncosh ek
Fek cc

  


(10)
 

2
322
ek
Fek cek 
  (11)
graphs of which are shown in Table 1.
It should be noted that in case of using functionals as
(9) and (10) a problem of the selection (evaluation) of
parameter с (in Table 1 shapes of the functionals with c =
5 are shown) arises.
The standard deviations 2
1

and 2
2

in (8) are
usually unknown and must be estimated and they can be

Current Distortion Evaluation in Traction 4Q Constant Switching Frequency Converters 20
Table 1. Graphs of functions (9)-(11), their first and second derivatives and weight functions.

F
ek

 
ek 



ek 

 
ek 

1
F
ek


2
F
ek


3
F
ek

taken into account in the learning algorithm. If 2
1

and
2
2

do not change over time, this evaluation can be car-
ried out by stochastic approximation:


  

 

222
11
1
2
11
2
1
1
ˆˆ
11
ˆˆ
for3 1
ˆ1 otherwise
kekk
lk
k ekk
k



 






(12)

  

 

222
22
2
2
21
2
2
1
ˆˆ 11
ˆˆ
for3 1
ˆ1 otherwise
kekk
lk
k ekk
k



 






where




12
lkk l k
  

1
2
2
ˆ
0 31
11 otherwise
for ekk
lk lk






The total variance of noise, calculated as
 



2
11
2
2
2
ˆˆ
for31
ˆ otherwise
k ekk
kk






 (13)
can be used for normalizing the selected functional



*2
2
,
,, ek ek






(14)
It should be noted that as estimation of the parameter с
in the functionals (9) and (10) 3

can be used.
3. Modification of Robust Gauss-Newton
Algorithm with Dead Zone
Dead zone, which determines the degree of permissible
errors, can be set as follows:







**
*
1*
for
,0 for
ek ek
ek
ek


 





(15)
and










**
**
2
**
for
,0 for
for .
ek ek
ek ek
ek ek


 










(16)
The forms of functions (12) and (13) are shown in the
Table 2 (columns 2 and 3, respectively).
In this case, the robust Gauss-Newton algorithm takes
the form

Current Distortion Evaluation in Traction 4Q Constant Switching Frequency Converters 21
Table 2. Graphs of (9)-(11) functions derivatives with dead zones.

F
ek

 
*
1,ek





*
2,ek



1
F
ek


2
F
ek


3
F
ek

 
 



 
*
*
ˆˆ
1
ˆ
1,
ˆˆ
1,1
T
kk
Pkf kek
e kfkPkfk







(17)
 
 


 

*
*
1
ˆˆ
11
,
ˆˆ
1,1
T
T
Pk Pk
Pkf kfkPkek
e kfkPkfk





 


(18)
where









**
**
**
for
,0 for
for
ek ek
ek ek
ek ek


 











*
1 for
0 otherwise
ek







Table 1 (column 3) shows that for the functional (11)
there are areas where 0

. This can lead to instabil-
ity of estimates ˆ

. In this case, in the algorithm (17),
(18) instead of


*,ek


 the weighting function


*,ek


should be used, which, as seen from the
Table 1 (column 4) is always greater than zero. In this
case, algorithm (17, 18) takes the form
 
 



 
*
*
ˆˆ
1
ˆ
1,
ˆˆ
1, 1
T
kk
Pkf kek
ekfkPkfk






(19)




 


 

*
*
1
ˆˆ
11
,
ˆˆ
1, 1
T
T
Pk Pk
Pkfk fkPkek
e kfkPkfk





 

(20)
4. Experimental Results
Consider using an RBF network to approximate the func-
tion [24]
 
12 12
22
12
16 8
0.725sin0.2 0.2
34 4
xx
ykxx k
xx







(21)
where

12
,T
x
xx is an input signal that was generated
using uniformly distributed random data in range [–1, 1].
The additive noise


k  is a Gaussian mixture that
smixes two types of noises, a large portion of normal

Function Approximation Using Robust Radial Basis Function Networks 22
noise with smaller variance and a smaller portion of noise
with higher variance, i.e.
  


12
1kqkqk

  ,
where 00.2

 is a small number to denote the con-
tamination ratio and (


1
qk,


2
qk —normally distrib-
uted noises with variances 2
1

and 2
2

respectively).
50000 training data points were used for investigation of
the given function. A surface described by function (21)
without noise is shown on the Figure 1(a), on the Figure
1(b) the same surface with noise


k

(10.6


and
212

) is shown. On the Figure 2 the cross-sections of
the function (21) are given (dashed line denotes the re-
constructed function). The results of approximation of
the function (21) with different values of

, 2
1

and
2
2

are given in the Table 3. Here are the values of the
RMS error, calculated after train ing the networ k for 2500
reference values using the formula
 

2500 2
*
1
1ˆ
2500 i
yi yi 



where *
y
—the reference value of the output signal in
the absence of interference measurements; ˆ y—real
networks output.
Graphs of the adjustment 1

and 2

estimations at
(a) (b)
Figure 1. A surface described by function (21), (a) without noise


 k; (b) with noise


 k.
Figure 2. The cross-sections of the function (21).

Function Approximation Using Robust Radial Basis Function Networks 23
(a) (b)
Figure 3. Results of the estimation 10.6   and 212 

with 0.2


.
Table 3. The results of function (21) approximation.
Table 4. Estimations of 1

, 2

and N.
Given parameters Estimations

1
ref

2
ref

Real number of outliers 1
est

2
est

Estimated number of outliers N
0.0 0 0 0 - - -
0.1 0.6
3 5061 0.6369 4.0902 4758
6 5008 0.6166 6.7468 4984
12 4991 0.6073 12.5611 4969
0.2 0.6
3 10013 0.7351 4.3658 9957
6 10020 0.6151 6.8815 9897
12 10111 0.6220 12.8381 10005
each step of training the network are shown in Figure 3.
Estimations of 1

, 2

and number of outliers are given
in Table 4.
As seen from the simulation results, the algorithm (12)
gives reasonably accurate estimates of 2
1

and 2
2

(assuming 22
12

  ) that are used in the normalization
of the loss function, which ensures high accuracy of ap-
Given parameters



2
2
ek
Fekc

 






ln coshek Fekcc

  
 




2
32
1
ek
Fek ek 


(with a weight function)

1
ref

2
ref

Number of
outliers

without
dead zone

With dead
zone (15)

with dead
zone (16)

without
dead zone

with dead
zone (15)

with dead
zone (16)

without
dead zone

with dead
zone (15)

with dead
zone (16)
0.0 0 0 0 0.6286 - - - - - - - -
0.1 0.6
3 5061 1.5252 2.7339 2.6047 1.5556 2.4468 2.3937 2.0836 2.8747 2.8137
6 5008 1.6415 2.4909 2.4697 1.6553 2.2052 2.2047 1.8936 2.7882 2.7199
12 4991 1.9389 1.9634 1.9491 1.7256 1.7386 1.7379 1.6365 2.3665 2.3088
0.2 0.6
3 10013 1.6497 2.1061 2.0698 2.3438 3.0111 2.9940 2.9080 2.9365 2.9198
6 10020 2.0402 2.1209 2.0813 2.2875 2.4361 2.4113 2.2054 2.7103 2.5998
12 10111 1.9863 2.2117 2.1887 2.3682 2.7750 2.7217 2.5152 2.7012 2.6260

Function Approximation Using Robust Radial Basis Function Networks 24
proximation of very noisy nonlinear functions. Also it
should be noted that the usage of dead zones has reduced
training time by about 20%.
5. Conclusions
This paper proposes a resistant radial function network
on-line training algorithm based on the theory of robust
regression for dealing with outliers in the framework of
function approximation.
The proposed algorith m minimizes an M-estimate cost
functions instead of the conventional mean square error
and represents one modification of recursive Gauss-
Newton algorithm with dead-zone. These dead zone may
serve as value that limits the accuracy of the obtained
solutions.
Utilization of dead zones can decrease training time of
the network.
If the distribution of the noise contaminated by outliers
expressed by a mixture of two Gaussian distributions with
unknown standard deviations 2
1

and 2
2

, 22
12

 
these can be estimated and taken into account in the
training algorithm.
It is an efficient algorithm for practical using in inves-
tigation of real nonlinear systems. It is expedient to
develop this approach further and to investigate other
robust cost functions and training algorithms such as
Levenberg-Marquardt algorithm.
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