<s>
The	O
resulting	O
combination	O
may	O
be	O
used	O
as	O
a	O
linear	B-General_Concept
classifier	I-General_Concept
,	O
or	O
,	O
more	O
commonly	O
,	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
before	O
later	O
classification	B-General_Concept
.	O
</s>
<s>
LDA	O
is	O
closely	O
related	O
to	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
and	O
regression	O
analysis	O
,	O
which	O
also	O
attempt	O
to	O
express	O
one	O
dependent	O
variable	O
as	O
a	O
linear	O
combination	O
of	O
other	O
features	O
or	O
measurements	O
.	O
</s>
<s>
However	O
,	O
ANOVA	B-General_Concept
uses	O
categorical	O
independent	O
variables	O
and	O
a	O
continuous	O
dependent	O
variable	O
,	O
whereas	O
discriminant	O
analysis	O
has	O
continuous	O
independent	O
variables	O
and	O
a	O
categorical	O
dependent	O
variable	O
(	O
i.e.	O
</s>
<s>
Logistic	O
regression	O
and	O
probit	B-Architecture
regression	I-Architecture
are	O
more	O
similar	O
to	O
LDA	O
than	O
ANOVA	B-General_Concept
is	O
,	O
as	O
they	O
also	O
explain	O
a	O
categorical	O
variable	O
by	O
the	O
values	O
of	O
continuous	O
independent	O
variables	O
.	O
</s>
<s>
LDA	O
is	O
also	O
closely	O
related	O
to	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
and	O
factor	O
analysis	O
in	O
that	O
they	O
both	O
look	O
for	O
linear	O
combinations	O
of	O
variables	O
which	O
best	O
explain	O
the	O
data	O
.	O
</s>
<s>
Discriminant	O
analysis	O
is	O
used	O
when	O
groups	O
are	O
known	O
a	O
priori	O
(	O
unlike	O
in	O
cluster	B-Algorithm
analysis	I-Algorithm
)	O
.	O
</s>
<s>
In	O
simple	O
terms	O
,	O
discriminant	O
function	O
analysis	O
is	O
classification	B-General_Concept
-	O
the	O
act	O
of	O
distributing	O
things	O
into	O
groups	O
,	O
classes	O
or	O
categories	O
of	O
the	O
same	O
type	O
.	O
</s>
<s>
It	O
is	O
different	O
from	O
an	O
ANOVA	B-General_Concept
or	O
MANOVA	B-General_Concept
,	O
which	O
is	O
used	O
to	O
predict	O
one	O
(	O
ANOVA	B-General_Concept
)	O
or	O
multiple	O
(	O
MANOVA	B-General_Concept
)	O
continuous	O
dependent	O
variables	O
by	O
one	O
or	O
more	O
independent	O
categorical	O
variables	O
.	O
</s>
<s>
The	O
classification	B-General_Concept
problem	O
is	O
then	O
to	O
find	O
a	O
good	O
predictor	O
for	O
the	O
class	O
of	O
any	O
sample	O
of	O
the	O
same	O
distribution	O
(	O
not	O
necessarily	O
from	O
the	O
training	O
set	O
)	O
given	O
only	O
an	O
observation	O
.	O
</s>
<s>
Under	O
this	O
assumption	O
,	O
the	O
Bayes-optimal	B-General_Concept
solution	I-General_Concept
is	O
to	O
predict	O
points	O
as	O
being	O
from	O
the	O
second	O
class	O
if	O
the	O
log	O
of	O
the	O
likelihood	O
ratios	O
is	O
bigger	O
than	O
some	O
threshold	O
T	O
,	O
so	O
that	O
:	O
</s>
<s>
Without	O
any	O
further	O
assumptions	O
,	O
the	O
resulting	O
classifier	B-General_Concept
is	O
referred	O
to	O
as	O
quadratic	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
(	O
QDA	O
)	O
.	O
</s>
<s>
LDA	O
instead	O
makes	O
the	O
additional	O
simplifying	O
homoscedasticity	B-General_Concept
assumption	O
(	O
i.e.	O
</s>
<s>
The	O
assumptions	O
of	O
discriminant	O
analysis	O
are	O
the	O
same	O
as	O
those	O
for	O
MANOVA	B-General_Concept
.	O
</s>
<s>
Homogeneity	O
of	O
variance/covariance	O
(	O
homoscedasticity	B-General_Concept
)	O
:	O
Variances	O
among	O
group	O
variables	O
are	O
the	O
same	O
across	O
levels	O
of	O
predictors	O
.	O
</s>
<s>
Can	O
be	O
tested	O
with	O
Box	B-General_Concept
's	I-General_Concept
M	I-General_Concept
statistic	O
.	O
</s>
<s>
It	O
has	O
been	O
suggested	O
,	O
however	O
,	O
that	O
linear	O
discriminant	O
analysis	O
be	O
used	O
when	O
covariances	O
are	O
equal	O
,	O
and	O
that	O
quadratic	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
may	O
be	O
used	O
when	O
covariances	O
are	O
not	O
equal	O
.	O
</s>
<s>
Discriminant	O
analysis	O
then	O
,	O
finds	O
“	O
good	O
”	O
regions	O
of	O
to	O
minimize	O
classification	B-General_Concept
error	O
,	O
therefore	O
leading	O
to	O
a	O
high	O
percent	O
correct	O
classified	O
in	O
the	O
classification	B-General_Concept
table	O
.	O
</s>
<s>
The	O
farther	O
apart	O
the	O
means	O
are	O
,	O
the	O
less	O
error	O
there	O
will	O
be	O
in	O
classification	B-General_Concept
.	O
</s>
<s>
Bayes	O
Discriminant	O
Rule	O
:	O
Assigns	O
to	O
the	O
group	O
that	O
maximizes	O
,	O
where	O
πi	O
represents	O
the	O
prior	O
probability	O
of	O
that	O
classification	B-General_Concept
,	O
and	O
represents	O
the	O
population	O
density	O
.	O
</s>
<s>
The	O
eigenvalue	O
can	O
be	O
viewed	O
as	O
a	O
ratio	O
of	O
SSbetween	O
and	O
SSwithin	O
as	O
in	O
ANOVA	B-General_Concept
when	O
the	O
dependent	O
variable	O
is	O
the	O
discriminant	O
function	O
,	O
and	O
the	O
groups	O
are	O
the	O
levels	O
of	O
the	O
IV	O
.	O
</s>
<s>
Some	O
suggest	O
the	O
use	O
of	O
eigenvalues	O
as	O
effect	B-Application
size	I-Application
measures	O
,	O
however	O
,	O
this	O
is	O
generally	O
not	O
supported	O
.	O
</s>
<s>
Instead	O
,	O
the	O
canonical	O
correlation	O
is	O
the	O
preferred	O
measure	O
of	O
effect	B-Application
size	I-Application
.	O
</s>
<s>
Another	O
popular	O
measure	O
of	O
effect	B-Application
size	I-Application
is	O
the	O
percent	O
of	O
variance	O
for	O
each	O
function	O
.	O
</s>
<s>
Percent	O
correctly	O
classified	O
can	O
also	O
be	O
analyzed	O
as	O
an	O
effect	B-Application
size	I-Application
.	O
</s>
<s>
See	O
“	O
Multiclass	B-General_Concept
LDA	I-General_Concept
”	O
for	O
details	O
below	O
.	O
</s>
<s>
Otsu	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
related	O
to	O
Fisher	O
's	O
linear	O
discriminant	O
,	O
and	O
was	O
created	O
to	O
binarize	O
the	O
histogram	O
of	O
pixels	O
in	O
a	O
grayscale	O
image	O
by	O
optimally	O
picking	O
the	O
black/white	O
threshold	O
that	O
minimizes	O
intra-class	O
variance	O
and	O
maximizes	O
inter-class	O
variance	O
within/between	O
grayscales	O
assigned	O
to	O
black	O
and	O
white	O
pixel	O
classes	O
.	O
</s>
<s>
This	O
generalization	O
is	O
due	O
to	O
C	O
.	O
R	B-Language
.	O
Rao	O
.	O
</s>
<s>
If	O
classification	B-General_Concept
is	O
required	O
,	O
instead	O
of	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
there	O
are	O
a	O
number	O
of	O
alternative	O
techniques	O
available	O
.	O
</s>
<s>
This	O
will	O
result	O
in	O
C	O
classifiers	B-General_Concept
,	O
whose	O
results	O
are	O
combined	O
.	O
</s>
<s>
method	O
is	O
pairwise	O
classification	B-General_Concept
,	O
where	O
a	O
new	O
classifier	B-General_Concept
is	O
created	O
for	O
each	O
pair	O
of	O
classes	O
(	O
giving	O
C( C−1	O
)	O
/2	O
classifiers	B-General_Concept
in	O
total	O
)	O
,	O
with	O
the	O
individual	O
classifiers	B-General_Concept
combined	O
to	O
produce	O
a	O
final	O
classification	B-General_Concept
.	O
</s>
<s>
Either	O
the	O
maximum	O
likelihood	O
estimate	O
or	O
the	O
maximum	B-General_Concept
a	I-General_Concept
posteriori	I-General_Concept
estimate	I-General_Concept
may	O
be	O
used	O
in	O
place	O
of	O
the	O
exact	O
value	O
in	O
the	O
above	O
equations	O
.	O
</s>
<s>
One	O
is	O
to	O
use	O
a	O
pseudo	B-Algorithm
inverse	I-Algorithm
instead	O
of	O
the	O
usual	O
matrix	O
inverse	O
in	O
the	O
above	O
formulae	O
.	O
</s>
<s>
Linear	B-General_Concept
classification	I-General_Concept
in	O
this	O
non-linear	O
space	O
is	O
then	O
equivalent	O
to	O
non-linear	O
classification	O
in	O
the	O
original	O
space	O
.	O
</s>
<s>
The	O
most	O
commonly	O
used	O
example	O
of	O
this	O
is	O
the	O
kernel	B-General_Concept
Fisher	I-General_Concept
discriminant	I-General_Concept
.	O
</s>
<s>
LDA	O
can	O
be	O
generalized	O
to	O
multiple	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
,	O
where	O
c	O
becomes	O
a	O
categorical	O
variable	O
with	O
N	O
possible	O
states	O
,	O
instead	O
of	O
only	O
two	O
.	O
</s>
<s>
Analogously	O
,	O
if	O
the	O
class-conditional	O
densities	O
are	O
normal	O
with	O
shared	O
covariances	O
,	O
the	O
sufficient	O
statistic	O
for	O
are	O
the	O
values	O
of	O
N	O
projections	O
,	O
which	O
are	O
the	O
subspace	O
spanned	O
by	O
the	O
N	O
means	O
,	O
affine	B-Algorithm
projected	I-Algorithm
by	O
the	O
inverse	O
covariance	O
matrix	O
.	O
</s>
<s>
See	O
“	O
Multiclass	B-General_Concept
LDA	I-General_Concept
”	O
above	O
for	O
details	O
.	O
</s>
<s>
Linear	O
discriminant	O
analysis	O
is	O
primarily	O
used	O
here	O
to	O
reduce	O
the	O
number	O
of	O
features	O
to	O
a	O
more	O
manageable	O
number	O
before	O
classification	B-General_Concept
.	O
</s>
<s>
The	O
linear	O
combinations	O
obtained	O
using	O
Fisher	O
's	O
linear	O
discriminant	O
are	O
called	O
Fisher	O
faces	O
,	O
while	O
those	O
obtained	O
using	O
the	O
related	O
principal	B-Application
component	I-Application
analysis	I-Application
are	O
called	O
eigenfaces	B-General_Concept
.	O
</s>
<s>
The	O
data	O
for	O
multiple	O
products	O
is	O
codified	O
and	O
input	O
into	O
a	O
statistical	O
program	O
such	O
as	O
R	B-Language
,	O
SPSS	B-Algorithm
or	O
SAS	B-Language
.	O
</s>
<s>
Use	O
Wilks	O
's	O
Lambda	O
to	O
test	O
for	O
significance	O
in	O
SPSS	B-Algorithm
or	O
F	O
stat	O
in	O
SAS	B-Language
.	O
</s>
<s>
The	O
validation	O
sample	O
is	O
used	O
to	O
construct	O
a	O
classification	B-General_Concept
matrix	O
which	O
contains	O
the	O
number	O
of	O
correctly	O
classified	O
and	O
incorrectly	O
classified	O
cases	O
.	O
</s>
<s>
Geometric	O
anomalies	O
in	O
higher	O
dimensions	O
lead	O
to	O
the	O
well-known	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Data	O
separability	O
by	O
classical	O
linear	O
discriminants	O
simplifies	O
the	O
problem	O
of	O
error	O
correction	O
for	O
artificial	B-Application
intelligence	I-Application
systems	O
in	O
high	O
dimension	O
.	O
</s>
