<s>
In	O
the	O
field	O
of	O
mathematical	O
modeling	O
,	O
a	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
network	I-Algorithm
is	O
an	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
that	O
uses	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
as	O
activation	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
The	O
output	O
of	O
the	O
network	O
is	O
a	O
linear	O
combination	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
of	O
the	O
inputs	O
and	O
neuron	O
parameters	O
.	O
</s>
<s>
Radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
networks	I-Algorithm
have	O
many	O
uses	O
,	O
including	O
function	O
approximation	O
,	O
time	O
series	O
prediction	O
,	O
classification	B-General_Concept
,	O
and	O
system	O
control	O
.	O
</s>
<s>
Radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
(	O
RBF	O
)	O
networks	O
typically	O
have	O
three	O
layers	O
:	O
an	O
input	O
layer	O
,	O
a	O
hidden	O
layer	O
with	O
a	O
non-linear	O
RBF	O
activation	B-Algorithm
function	I-Algorithm
and	O
a	O
linear	O
output	O
layer	O
.	O
</s>
<s>
Functions	O
that	O
depend	O
only	O
on	O
the	O
distance	O
from	O
a	O
center	O
vector	O
are	O
radially	O
symmetric	O
about	O
that	O
vector	O
,	O
hence	O
the	O
name	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Given	O
certain	O
mild	O
conditions	O
on	O
the	O
shape	O
of	O
the	O
activation	B-Algorithm
function	I-Algorithm
,	O
RBF	B-Algorithm
networks	I-Algorithm
are	O
universal	B-Algorithm
approximators	I-Algorithm
on	O
a	O
compact	O
subset	O
of	O
.	O
</s>
<s>
This	O
means	O
that	O
an	O
RBF	B-Algorithm
network	I-Algorithm
with	O
enough	O
hidden	O
neurons	O
can	O
approximate	O
any	O
continuous	O
function	O
on	O
a	O
closed	O
,	O
bounded	O
set	O
with	O
arbitrary	O
precision	O
.	O
</s>
<s>
In	O
addition	O
to	O
the	O
above	O
unnormalized	O
architecture	O
,	O
RBF	B-Algorithm
networks	I-Algorithm
can	O
be	O
normalized	O
.	O
</s>
<s>
is	O
known	O
as	O
a	O
normalized	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
RBF	B-Algorithm
networks	I-Algorithm
are	O
typically	O
trained	O
from	O
pairs	O
of	O
input	O
and	O
target	O
values	O
,	O
by	O
a	O
two-step	O
algorithm	O
.	O
</s>
<s>
This	O
step	O
can	O
be	O
performed	O
in	O
several	O
ways	O
;	O
centers	O
can	O
be	O
randomly	O
sampled	O
from	O
some	O
set	O
of	O
examples	O
,	O
or	O
they	O
can	O
be	O
determined	O
using	O
k-means	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
this	O
step	O
is	O
unsupervised	B-General_Concept
.	O
</s>
<s>
A	O
third	O
optional	O
backpropagation	B-Algorithm
step	O
can	O
be	O
performed	O
to	O
fine-tune	O
all	O
of	O
the	O
RBF	B-Algorithm
net	I-Algorithm
's	O
parameters	O
.	O
</s>
<s>
RBF	B-Algorithm
networks	I-Algorithm
can	O
be	O
used	O
to	O
interpolate	O
a	O
function	O
when	O
the	O
values	O
of	O
that	O
function	O
are	O
known	O
on	O
finite	O
number	O
of	O
points	O
:	O
.	O
</s>
<s>
If	O
the	O
purpose	O
is	O
not	O
to	O
perform	O
strict	O
interpolation	O
but	O
instead	O
more	O
general	O
function	O
approximation	O
or	O
classification	B-General_Concept
the	O
optimization	O
is	O
somewhat	O
more	O
complex	O
because	O
there	O
is	O
no	O
obvious	O
choice	O
for	O
the	O
centers	O
.	O
</s>
<s>
Basis	O
function	O
centers	O
can	O
be	O
randomly	O
sampled	O
among	O
the	O
input	O
instances	O
or	O
obtained	O
by	O
Orthogonal	O
Least	O
Square	O
Learning	O
Algorithm	O
or	O
found	O
by	O
clustering	B-Algorithm
the	O
samples	O
and	O
choosing	O
the	O
cluster	O
means	O
as	O
the	O
centers	O
.	O
</s>
<s>
After	O
the	O
centers	O
have	O
been	O
fixed	O
,	O
the	O
weights	O
that	O
minimize	O
the	O
error	O
at	O
the	O
output	O
can	O
be	O
computed	O
with	O
a	O
linear	O
pseudoinverse	B-Algorithm
solution	O
:	O
</s>
<s>
where	O
the	O
entries	O
of	O
G	O
are	O
the	O
values	O
of	O
the	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
evaluated	O
at	O
the	O
points	O
:	O
.	O
</s>
<s>
The	O
existence	O
of	O
this	O
linear	O
solution	O
means	O
that	O
unlike	O
multi-layer	O
perceptron	O
(	O
MLP	O
)	O
networks	O
,	O
RBF	B-Algorithm
networks	I-Algorithm
have	O
an	O
explicit	O
minimizer	O
(	O
when	O
the	O
centers	O
are	O
fixed	O
)	O
.	O
</s>
<s>
Another	O
possible	O
training	O
algorithm	O
is	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
In	O
gradient	B-Algorithm
descent	I-Algorithm
training	O
,	O
the	O
weights	O
are	O
adjusted	O
at	O
each	O
time	O
step	O
by	O
moving	O
them	O
in	O
a	O
direction	O
opposite	O
from	O
the	O
gradient	O
of	O
the	O
objective	O
function	O
(	O
thus	O
allowing	O
the	O
minimum	O
of	O
the	O
objective	O
function	O
to	O
be	O
found	O
)	O
,	O
</s>
<s>
The	O
basic	O
properties	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
can	O
be	O
illustrated	O
with	O
a	O
simple	O
mathematical	O
map	O
,	O
the	O
logistic	B-Algorithm
map	I-Algorithm
,	O
which	O
maps	O
the	O
unit	O
interval	O
onto	O
itself	O
.	O
</s>
<s>
The	O
logistic	B-Algorithm
map	I-Algorithm
can	O
be	O
used	O
to	O
explore	O
function	O
approximation	O
,	O
time	O
series	O
prediction	O
,	O
and	O
control	O
theory	O
.	O
</s>
<s>
The	O
value	O
of	O
x	O
at	O
time	O
t+1	O
is	O
a	O
parabolic	O
function	O
of	O
x	O
at	O
time	O
t	O
.	O
This	O
equation	O
represents	O
the	O
underlying	O
geometry	O
of	O
the	O
chaotic	O
time	O
series	O
generated	O
by	O
the	O
logistic	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
The	O
examples	O
here	O
illustrate	O
the	O
inverse	O
problem	O
;	O
identification	O
of	O
the	O
underlying	O
dynamics	O
,	O
or	O
fundamental	O
equation	O
,	O
of	O
the	O
logistic	B-Algorithm
map	I-Algorithm
from	O
exemplars	O
of	O
the	O
time	O
series	O
.	O
</s>
<s>
where	O
the	O
learning	B-General_Concept
rate	I-General_Concept
is	O
taken	O
to	O
be	O
0.3	O
.	O
</s>
<s>
The	O
rms	B-Algorithm
error	I-Algorithm
is	O
0.15	O
.	O
</s>
<s>
where	O
the	O
learning	B-General_Concept
rate	I-General_Concept
is	O
again	O
taken	O
to	O
be	O
0.3	O
.	O
</s>
<s>
The	O
rms	B-Algorithm
error	I-Algorithm
on	O
a	O
test	O
set	O
of	O
100	O
exemplars	O
is	O
0.084	O
,	O
smaller	O
than	O
the	O
unnormalized	O
error	O
.	O
</s>
