<s>
A	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
(	O
RBF	O
)	O
is	O
a	O
real-valued	O
function	O
whose	O
value	O
depends	O
only	O
on	O
the	O
distance	O
between	O
the	O
input	O
and	O
some	O
fixed	O
point	O
,	O
either	O
the	O
origin	O
,	O
so	O
that	O
,	O
or	O
some	O
other	O
fixed	O
point	O
,	O
called	O
a	O
center	O
,	O
so	O
that	O
.	O
</s>
<s>
They	O
are	O
often	O
used	O
as	O
a	O
collection	O
which	O
forms	O
a	O
basis	O
for	O
some	O
function	B-Algorithm
space	I-Algorithm
of	O
interest	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
Sums	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
are	O
typically	O
used	O
to	O
approximate	O
given	O
functions	O
.	O
</s>
<s>
This	O
approximation	O
process	O
can	O
also	O
be	O
interpreted	O
as	O
a	O
simple	O
kind	O
of	O
neural	B-Architecture
network	I-Architecture
;	O
this	O
was	O
the	O
context	O
in	O
which	O
they	O
were	O
originally	O
applied	O
to	O
machine	O
learning	O
,	O
in	O
work	O
by	O
David	O
Broomhead	O
and	O
David	O
Lowe	O
in	O
1988	O
,	O
which	O
stemmed	O
from	O
Michael	O
J	O
.	O
D	O
.	O
Powell	O
's	O
seminal	O
research	O
from	O
1977	O
.	O
</s>
<s>
RBFs	O
are	O
also	O
used	O
as	O
a	O
kernel	B-Algorithm
in	O
support	B-Algorithm
vector	I-Algorithm
classification	I-Algorithm
.	O
</s>
<s>
The	O
technique	O
has	O
proven	O
effective	O
and	O
flexible	O
enough	O
that	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
are	O
now	O
applied	O
in	O
a	O
variety	O
of	O
engineering	O
applications	O
.	O
</s>
<s>
When	O
paired	O
with	O
a	O
metric	O
on	O
a	O
vector	O
space	O
a	O
function	O
is	O
said	O
to	O
be	O
a	O
radial	O
kernel	B-Algorithm
centered	O
at	O
.	O
</s>
<s>
Commonly	O
used	O
types	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
include	O
(	O
writing	O
and	O
using	O
to	O
indicate	O
a	O
shape	O
parameter	O
that	O
can	O
be	O
used	O
to	O
scale	O
the	O
input	O
of	O
the	O
radial	O
kernel	B-Algorithm
)	O
:	O
</s>
<s>
where	O
the	O
approximating	O
function	O
is	O
represented	O
as	O
a	O
sum	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
,	O
each	O
associated	O
with	O
a	O
different	O
center	O
,	O
and	O
weighted	O
by	O
an	O
appropriate	O
coefficient	O
The	O
weights	O
can	O
be	O
estimated	O
using	O
the	O
matrix	O
methods	O
of	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
because	O
the	O
approximating	O
function	O
is	O
linear	O
in	O
the	O
weights	O
.	O
</s>
<s>
can	O
also	O
be	O
interpreted	O
as	O
a	O
rather	O
simple	O
single-layer	O
type	O
of	O
artificial	B-Architecture
neural	I-Architecture
network	I-Architecture
called	O
a	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
network	I-Algorithm
,	O
with	O
the	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
taking	O
on	O
the	O
role	O
of	O
the	O
activation	O
functions	O
of	O
the	O
network	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
any	O
continuous	O
function	O
on	O
a	O
compact	O
interval	O
can	O
in	O
principle	O
be	O
interpolated	O
with	O
arbitrary	O
accuracy	O
by	O
a	O
sum	O
of	O
this	O
form	O
,	O
if	O
a	O
sufficiently	O
large	O
number	O
of	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
is	O
used	O
.	O
</s>
<s>
The	O
weights	O
could	O
thus	O
be	O
learned	O
using	O
any	O
of	O
the	O
standard	O
iterative	O
methods	O
for	O
neural	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
Using	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
in	O
this	O
manner	O
yields	O
a	O
reasonable	O
interpolation	O
approach	O
provided	O
that	O
the	O
fitting	O
set	O
has	O
been	O
chosen	O
such	O
that	O
it	O
covers	O
the	O
entire	O
range	O
systematically	O
(	O
equidistant	O
data	O
points	O
are	O
ideal	O
)	O
.	O
</s>
<s>
However	O
,	O
without	O
a	O
polynomial	O
term	O
that	O
is	O
orthogonal	O
to	O
the	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
,	O
estimates	O
outside	O
the	O
fitting	O
set	O
tend	O
to	O
perform	O
poorly	O
.	O
</s>
<s>
Radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
are	O
used	O
to	O
approximate	O
functions	O
and	O
so	O
can	O
be	O
used	O
to	O
discretize	O
and	O
numerically	O
solve	O
Partial	O
Differential	O
Equations	O
(	O
PDEs	O
)	O
.	O
</s>
<s>
Different	O
numerical	O
methods	O
based	O
on	O
Radial	B-Algorithm
Basis	I-Algorithm
Functions	I-Algorithm
were	O
developed	O
thereafter	O
.	O
</s>
