<s>
In	O
statistics	O
,	O
a	O
quadratic	B-General_Concept
classifier	I-General_Concept
is	O
a	O
statistical	B-General_Concept
classifier	I-General_Concept
that	O
uses	O
a	O
quadratic	O
decision	B-General_Concept
surface	I-General_Concept
to	O
separate	O
measurements	O
of	O
two	O
or	O
more	O
classes	O
of	O
objects	O
or	O
events	O
.	O
</s>
<s>
It	O
is	O
a	O
more	O
general	O
version	O
of	O
the	O
linear	B-General_Concept
classifier	I-General_Concept
.	O
</s>
<s>
Statistical	B-General_Concept
classification	I-General_Concept
considers	O
a	O
set	O
of	O
vectors	O
of	O
observations	O
x	O
of	O
an	O
object	O
or	O
event	O
,	O
each	O
of	O
which	O
has	O
a	O
known	O
type	O
y	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
we	O
can	O
state	O
that	O
a	O
quadratic	O
model	O
is	O
a	O
generalization	O
of	O
the	O
linear	O
model	O
,	O
and	O
its	O
use	O
is	O
justified	O
by	O
the	O
desire	O
to	O
extend	O
the	O
classifier	B-General_Concept
's	O
ability	O
to	O
represent	O
more	O
complex	O
separating	O
surfaces	O
.	O
</s>
<s>
Quadratic	O
discriminant	B-General_Concept
analysis	I-General_Concept
(	O
QDA	O
)	O
is	O
closely	O
related	O
to	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
(	O
LDA	O
)	O
,	O
where	O
it	O
is	O
assumed	O
that	O
the	O
measurements	O
from	O
each	O
class	O
are	O
normally	O
distributed	O
.	O
</s>
<s>
When	O
the	O
normality	O
assumption	O
is	O
true	O
,	O
the	O
best	O
possible	O
test	O
for	O
the	O
hypothesis	O
that	O
a	O
given	O
measurement	O
is	O
from	O
a	O
given	O
class	O
is	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
While	O
QDA	O
is	O
the	O
most	O
commonly-used	O
method	O
for	O
obtaining	O
a	O
classifier	B-General_Concept
,	O
other	O
methods	O
are	O
also	O
possible	O
.	O
</s>
<s>
Finding	O
a	O
quadratic	B-General_Concept
classifier	I-General_Concept
for	O
the	O
original	O
measurements	O
would	O
then	O
become	O
the	O
same	O
as	O
finding	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
based	O
on	O
the	O
expanded	O
measurement	O
vector	O
.	O
</s>
<s>
This	O
observation	O
has	O
been	O
used	O
in	O
extending	O
neural	O
network	O
models	O
;	O
the	O
"	O
circular	O
"	O
case	O
,	O
which	O
corresponds	O
to	O
introducing	O
only	O
the	O
sum	O
of	O
pure	O
quadratic	O
terms	O
with	O
no	O
mixed	O
products	O
(	O
)	O
,	O
has	O
been	O
proven	O
to	O
be	O
the	O
optimal	O
compromise	O
between	O
extending	O
the	O
classifier	B-General_Concept
's	O
representation	O
power	O
and	O
controlling	O
the	O
risk	O
of	O
overfitting	O
(	O
Vapnik-Chervonenkis	O
dimension	O
)	O
.	O
</s>
<s>
For	O
linear	B-General_Concept
classifiers	I-General_Concept
based	O
only	O
on	O
dot	O
products	O
,	O
these	O
expanded	O
measurements	O
do	O
not	O
have	O
to	O
be	O
actually	O
computed	O
,	O
since	O
the	O
dot	O
product	O
in	O
the	O
higher-dimensional	O
space	O
is	O
simply	O
related	O
to	O
that	O
in	O
the	O
original	O
space	O
.	O
</s>
<s>
This	O
is	O
an	O
example	O
of	O
the	O
so-called	O
kernel	O
trick	O
,	O
which	O
can	O
be	O
applied	O
to	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
as	O
well	O
as	O
the	O
support	B-Algorithm
vector	I-Algorithm
machine	I-Algorithm
.	O
</s>
