<s>
The	O
Volterra	B-Algorithm
series	I-Algorithm
is	O
a	O
model	O
for	O
non-linear	O
behavior	O
similar	O
to	O
the	O
Taylor	O
series	O
.	O
</s>
<s>
In	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
,	O
the	O
output	O
of	O
the	O
nonlinear	O
system	O
depends	O
on	O
the	O
input	O
to	O
the	O
system	O
at	O
all	O
other	O
times	O
.	O
</s>
<s>
Thus	O
,	O
it	O
is	O
sometimes	O
considered	O
a	O
non-parametric	B-General_Concept
model	I-General_Concept
.	O
</s>
<s>
In	O
mathematics	O
,	O
a	O
Volterra	B-Algorithm
series	I-Algorithm
denotes	O
a	O
functional	O
expansion	O
of	O
a	O
dynamic	O
,	O
nonlinear	O
,	O
time-invariant	B-Algorithm
functional	O
.	O
</s>
<s>
The	O
Volterra	B-Algorithm
series	I-Algorithm
are	O
frequently	O
used	O
in	O
system	O
identification	O
.	O
</s>
<s>
The	O
Volterra	B-Algorithm
series	I-Algorithm
,	O
which	O
is	O
used	O
to	O
prove	O
the	O
Volterra	O
theorem	O
,	O
is	O
an	O
infinite	O
sum	O
of	O
multidimensional	O
convolutional	O
integrals	O
.	O
</s>
<s>
The	O
Volterra	B-Algorithm
series	I-Algorithm
is	O
a	O
modernized	O
version	O
of	O
the	O
theory	O
of	O
analytic	O
functionals	O
from	O
the	O
Italian	O
mathematician	O
Vito	O
Volterra	O
,	O
in	O
his	O
work	O
dating	O
from	O
1887	O
.	O
</s>
<s>
The	O
use	O
of	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
for	O
system	O
analysis	O
originated	O
from	O
a	O
restricted	O
1942	O
wartime	O
report	O
of	O
Wiener	O
's	O
,	O
who	O
was	O
then	O
a	O
professor	O
of	O
mathematics	O
at	O
MIT	O
.	O
</s>
<s>
As	O
a	O
general	O
method	O
of	O
analysis	O
of	O
nonlinear	O
systems	O
,	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
came	O
into	O
use	O
after	O
about	O
1957	O
as	O
the	O
result	O
of	O
a	O
series	O
of	O
reports	O
,	O
at	O
first	O
privately	O
circulated	O
,	O
from	O
MIT	O
and	O
elsewhere	O
.	O
</s>
<s>
The	O
name	O
itself	O
,	O
"	O
Volterra	B-Algorithm
series	I-Algorithm
,	O
"	O
came	O
into	O
use	O
a	O
few	O
years	O
later	O
.	O
</s>
<s>
The	O
theory	O
of	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
can	O
be	O
viewed	O
from	O
two	O
different	O
perspectives	O
:	O
</s>
<s>
The	O
latter	O
functional	O
mapping	O
perspective	O
is	O
more	O
frequently	O
used	O
due	O
to	O
the	O
assumed	O
time-invariance	B-Algorithm
of	O
the	O
system	O
.	O
</s>
<s>
The	O
function	O
is	O
called	O
the	O
n-th-order	O
Volterra	B-Algorithm
kernel	I-Algorithm
.	O
</s>
<s>
For	O
the	O
representation	O
to	O
be	O
unique	O
,	O
the	O
kernels	B-Algorithm
must	O
be	O
symmetrical	O
in	O
the	O
n	O
variables	O
.	O
</s>
<s>
If	O
it	O
is	O
not	O
symmetrical	O
,	O
it	O
can	O
be	O
replaced	O
by	O
a	O
symmetrized	O
kernel	B-Algorithm
,	O
which	O
is	O
the	O
average	O
over	O
the	O
n	O
!	O
</s>
<s>
The	O
causality	O
condition	O
:	O
Since	O
in	O
any	O
physically	O
realizable	O
system	O
the	O
output	O
can	O
only	O
depend	O
on	O
previous	O
values	O
of	O
the	O
input	O
,	O
the	O
kernels	B-Algorithm
will	O
be	O
zero	O
if	O
any	O
of	O
the	O
variables	O
are	O
negative	O
.	O
</s>
<s>
Fréchet	O
's	O
approximation	O
theorem	O
:	O
The	O
use	O
of	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
to	O
represent	O
a	O
time-invariant	B-Algorithm
functional	O
relation	O
is	O
often	O
justified	O
by	O
appealing	O
to	O
a	O
theorem	O
due	O
to	O
Fréchet	O
.	O
</s>
<s>
This	O
theorem	O
states	O
that	O
a	O
time-invariant	B-Algorithm
functional	O
relation	O
(	O
satisfying	O
certain	O
very	O
general	O
conditions	O
)	O
can	O
be	O
approximated	O
uniformly	O
and	O
to	O
an	O
arbitrary	O
degree	O
of	O
precision	O
by	O
a	O
sufficiently	O
high	O
finite-order	O
Volterra	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
are	O
called	O
discrete-time	O
Volterra	B-Algorithm
kernels	I-Algorithm
.	O
</s>
<s>
If	O
a	O
,	O
b	O
and	O
P	O
are	O
finite	O
,	O
the	O
series	O
operator	O
is	O
called	O
doubly	O
finite	O
Volterra	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
We	O
can	O
always	O
consider	O
,	O
without	O
loss	O
of	O
the	O
generality	O
,	O
the	O
kernel	B-Algorithm
as	O
symmetrical	O
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
the	O
commutativity	O
of	O
the	O
multiplication	O
it	O
is	O
always	O
possible	O
to	O
symmetrize	O
it	O
by	O
forming	O
a	O
new	O
kernel	B-Algorithm
taken	O
as	O
the	O
average	O
of	O
the	O
kernels	B-Algorithm
for	O
all	O
permutations	O
of	O
the	O
variables	O
.	O
</s>
<s>
Estimating	O
the	O
Volterra	O
coefficients	O
individually	O
is	O
complicated	O
,	O
since	O
the	O
basis	O
functionals	O
of	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
are	O
correlated	O
.	O
</s>
<s>
the	O
Wiener	B-Algorithm
series	I-Algorithm
,	O
and	O
then	O
recomputing	O
the	O
coefficients	O
of	O
the	O
original	O
Volterra	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
The	O
Volterra	B-Algorithm
series	I-Algorithm
main	O
appeal	O
over	O
the	O
orthogonalized	O
series	O
lies	O
in	O
its	O
intuitive	O
,	O
canonical	O
structure	O
,	O
i.e.	O
</s>
<s>
To	O
allow	O
identification	O
orthogonalization	O
,	O
Volterra	B-Algorithm
series	I-Algorithm
must	O
be	O
rearranged	O
in	O
terms	O
of	O
orthogonal	O
non-homogeneous	O
G	O
operators	O
(	O
Wiener	B-Algorithm
series	I-Algorithm
)	O
:	O
</s>
<s>
The	O
main	O
drawback	O
of	O
this	O
technique	O
is	O
that	O
the	O
estimation	O
errors	O
,	O
made	O
on	O
all	O
elements	O
of	O
lower-order	O
kernels	B-Algorithm
,	O
will	O
affect	O
each	O
diagonal	O
element	O
of	O
order	O
p	O
by	O
means	O
of	O
the	O
summation	O
,	O
conceived	O
as	O
the	O
solution	O
for	O
the	O
estimation	O
of	O
the	O
diagonal	O
elements	O
themselves	O
.	O
</s>
<s>
Once	O
the	O
Wiener	B-Algorithm
kernels	I-Algorithm
were	O
identified	O
,	O
Volterra	B-Algorithm
kernels	I-Algorithm
can	O
be	O
obtained	O
by	O
using	O
Wiener-to-Volterra	O
formulas	O
,	O
in	O
the	O
following	O
reported	O
for	O
a	O
fifth-order	O
Volterra	B-Algorithm
series	I-Algorithm
:	O
</s>
<s>
In	O
the	O
traditional	O
orthogonal	O
algorithm	O
,	O
using	O
inputs	O
with	O
high	O
has	O
the	O
advantage	O
of	O
stimulating	O
high-order	O
nonlinearity	O
,	O
so	O
as	O
to	O
achieve	O
more	O
accurate	O
high-order	O
kernel	B-Algorithm
identification	O
.	O
</s>
<s>
As	O
a	O
drawback	O
,	O
the	O
use	O
of	O
high	O
values	O
causes	O
high	O
identification	O
error	O
in	O
lower-order	O
kernels	B-Algorithm
,	O
mainly	O
due	O
to	O
nonideality	O
of	O
the	O
input	O
and	O
truncation	O
errors	O
.	O
</s>
<s>
On	O
the	O
contrary	O
,	O
the	O
use	O
of	O
lower	O
in	O
the	O
identification	O
process	O
can	O
lead	O
to	O
a	O
better	O
estimation	O
of	O
lower-order	O
kernel	B-Algorithm
,	O
but	O
can	O
be	O
insufficient	O
to	O
stimulate	O
high-order	O
nonlinearity	O
.	O
</s>
<s>
This	O
phenomenon	O
,	O
which	O
can	O
be	O
called	O
locality	O
of	O
truncated	O
Volterra	B-Algorithm
series	I-Algorithm
,	O
can	O
be	O
revealed	O
by	O
calculating	O
the	O
output	O
error	O
of	O
a	O
series	O
as	O
a	O
function	O
of	O
different	O
variances	O
of	O
input	O
.	O
</s>
<s>
To	O
overcome	O
this	O
limitation	O
,	O
a	O
low	O
value	O
should	O
be	O
used	O
for	O
the	O
lower-order	O
kernel	B-Algorithm
and	O
gradually	O
increased	O
for	O
higher-order	O
kernels	B-Algorithm
.	O
</s>
<s>
This	O
is	O
not	O
a	O
theoretical	O
problem	O
in	O
Wiener	B-Algorithm
kernel	I-Algorithm
identification	O
,	O
since	O
the	O
Wiener	O
functional	O
are	O
orthogonal	O
to	O
each	O
other	O
,	O
but	O
an	O
appropriate	O
normalization	O
is	O
needed	O
in	O
Wiener-to-Volterra	O
conversion	O
formulas	O
for	O
taking	O
into	O
account	O
the	O
use	O
of	O
different	O
variances	O
.	O
</s>
<s>
The	O
traditional	O
Wiener	B-Algorithm
kernel	I-Algorithm
identification	O
should	O
be	O
changed	O
as	O
follows	O
:	O
</s>
<s>
In	O
the	O
above	O
formulas	O
the	O
impulse	O
functions	O
are	O
introduced	O
for	O
the	O
identification	O
of	O
diagonal	O
kernel	B-Algorithm
points	O
.	O
</s>
<s>
If	O
the	O
Wiener	B-Algorithm
kernels	I-Algorithm
are	O
extracted	O
with	O
the	O
new	O
formulas	O
,	O
the	O
following	O
Wiener-to-Volterra	O
formulas	O
(	O
explicited	O
up	O
the	O
fifth	O
order	O
)	O
are	O
needed	O
:	O
</s>
<s>
As	O
can	O
be	O
seen	O
,	O
the	O
drawback	O
with	O
respect	O
to	O
the	O
previous	O
formula	O
is	O
that	O
for	O
the	O
identification	O
of	O
the	O
n-th-order	O
kernel	B-Algorithm
,	O
all	O
lower	O
kernels	B-Algorithm
must	O
be	O
identified	O
again	O
with	O
the	O
higher	O
variance	O
.	O
</s>
<s>
However	O
,	O
an	O
outstanding	O
improvement	O
in	O
the	O
output	O
MSE	O
will	O
be	O
obtained	O
if	O
the	O
Wiener	O
and	O
Volterra	B-Algorithm
kernels	I-Algorithm
are	O
obtained	O
with	O
the	O
new	O
formulas	O
.	O
</s>
<s>
This	O
method	O
was	O
developed	O
by	O
Wray	O
and	O
Green	O
(	O
1994	O
)	O
and	O
utilizes	O
the	O
fact	O
that	O
a	O
simple	O
2-fully	O
connected	O
layer	O
neural	B-Architecture
network	I-Architecture
(	O
i.e.	O
,	O
a	O
multilayer	B-Algorithm
perceptron	I-Algorithm
)	O
is	O
computationally	O
equivalent	O
to	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
and	O
therefore	O
contains	O
the	O
kernels	B-Algorithm
hidden	O
in	O
its	O
architecture	O
.	O
</s>
<s>
After	O
such	O
a	O
network	O
has	O
been	O
trained	O
to	O
successfully	O
predict	O
the	O
output	O
based	O
on	O
the	O
current	O
state	O
and	O
memory	O
of	O
the	O
system	O
,	O
the	O
kernels	B-Algorithm
can	O
then	O
be	O
computed	O
from	O
the	O
weights	O
and	O
biases	O
of	O
that	O
network	O
.	O
</s>
<s>
It	O
is	O
important	O
to	O
note	O
that	O
this	O
method	O
allows	O
kernel	B-Algorithm
extraction	O
up	O
until	O
the	O
number	O
of	O
input	O
delays	O
in	O
the	O
architecture	O
of	O
the	O
network	O
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
is	O
a	O
standard	O
tool	O
from	O
linear	O
analysis	O
.	O
</s>
<s>
Hence	O
,	O
one	O
of	O
its	O
main	O
advantages	O
is	O
the	O
widespread	O
existence	O
of	O
standard	O
tools	O
for	O
solving	O
linear	B-General_Concept
regressions	I-General_Concept
efficiently	O
.	O
</s>
<s>
It	O
has	O
some	O
educational	O
value	O
,	O
since	O
it	O
highlights	O
the	O
basic	O
property	O
of	O
Volterra	B-Algorithm
series	I-Algorithm
:	O
linear	O
combination	O
of	O
non-linear	O
basis-functionals	O
.	O
</s>
<s>
This	O
method	O
was	O
invented	O
by	O
Franz	O
and	O
Schölkopf	O
and	O
is	O
based	O
on	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
.	O
</s>
<s>
Consequently	O
,	O
this	O
approach	O
is	O
also	O
based	O
on	O
minimizing	O
the	O
empirical	O
error	O
(	O
often	O
called	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
)	O
.	O
</s>
<s>
Franz	O
and	O
Schölkopf	O
proposed	O
that	O
the	O
kernel	B-Algorithm
method	O
could	O
essentially	O
replace	O
the	O
Volterra	B-Algorithm
series	I-Algorithm
representation	O
,	O
although	O
noting	O
that	O
the	O
latter	O
is	O
more	O
intuitive	O
.	O
</s>
