<s>
Principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
is	O
a	O
popular	O
technique	O
for	O
analyzing	O
large	O
datasets	O
containing	O
a	O
high	O
number	O
of	O
dimensions/features	O
per	O
observation	O
,	O
increasing	O
the	O
interpretability	O
of	O
data	O
while	O
preserving	O
the	O
maximum	O
amount	O
of	O
information	O
,	O
and	O
enabling	O
the	O
visualization	B-Application
of	O
multidimensional	O
data	O
.	O
</s>
<s>
Many	O
studies	O
use	O
the	O
first	O
two	O
principal	B-Application
components	I-Application
in	O
order	O
to	O
plot	O
the	O
data	O
in	O
two	O
dimensions	O
and	O
to	O
visually	O
identify	O
clusters	B-Algorithm
of	O
closely	O
related	O
data	O
points	O
.	O
</s>
<s>
Principal	B-Application
component	I-Application
analysis	I-Application
has	O
applications	O
in	O
many	O
fields	O
such	O
as	O
population	O
genetics	O
,	O
microbiome	O
studies	O
,	O
and	O
atmospheric	O
science	O
.	O
</s>
<s>
The	O
principal	B-Application
components	I-Application
of	O
a	O
collection	O
of	O
points	O
in	O
a	O
real	O
coordinate	O
space	O
are	O
a	O
sequence	O
of	O
unit	O
vectors	O
,	O
where	O
the	O
-th	O
vector	O
is	O
the	O
direction	O
of	O
a	O
line	O
that	O
best	O
fits	O
the	O
data	O
while	O
being	O
orthogonal	O
to	O
the	O
first	O
vectors	O
.	O
</s>
<s>
Principal	B-Application
component	I-Application
analysis	I-Application
is	O
the	O
process	O
of	O
computing	O
the	O
principal	B-Application
components	I-Application
and	O
using	O
them	O
to	O
perform	O
a	O
change	O
of	O
basis	O
on	O
the	O
data	O
,	O
sometimes	O
using	O
only	O
the	O
first	O
few	O
principal	B-Application
components	I-Application
and	O
ignoring	O
the	O
rest	O
.	O
</s>
<s>
In	O
data	O
analysis	O
,	O
the	O
first	O
principal	B-Application
component	I-Application
of	O
a	O
set	O
of	O
variables	O
,	O
presumed	O
to	O
be	O
jointly	O
normally	O
distributed	O
,	O
is	O
the	O
derived	O
variable	O
formed	O
as	O
a	O
linear	O
combination	O
of	O
the	O
original	O
variables	O
that	O
explains	O
the	O
most	O
variance	O
.	O
</s>
<s>
The	O
second	O
principal	B-Application
component	I-Application
explains	O
the	O
most	O
variance	O
in	O
what	O
is	O
left	O
once	O
the	O
effect	O
of	O
the	O
first	O
component	O
is	O
removed	O
,	O
and	O
we	O
may	O
proceed	O
through	O
iterations	O
until	O
all	O
the	O
variance	O
is	O
explained	O
.	O
</s>
<s>
PCA	O
is	O
used	O
in	O
exploratory	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
and	O
for	O
making	O
predictive	B-General_Concept
models	I-General_Concept
.	O
</s>
<s>
It	O
is	O
commonly	O
used	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
by	O
projecting	O
each	O
data	O
point	O
onto	O
only	O
the	O
first	O
few	O
principal	B-Application
components	I-Application
to	O
obtain	O
lower-dimensional	O
data	O
while	O
preserving	O
as	O
much	O
of	O
the	O
data	O
's	O
variation	O
as	O
possible	O
.	O
</s>
<s>
The	O
first	O
principal	B-Application
component	I-Application
can	O
equivalently	O
be	O
defined	O
as	O
a	O
direction	O
that	O
maximizes	O
the	O
variance	O
of	O
the	O
projected	O
data	O
.	O
</s>
<s>
The	O
-th	O
principal	B-Application
component	I-Application
can	O
be	O
taken	O
as	O
a	O
direction	O
orthogonal	O
to	O
the	O
first	O
principal	B-Application
components	I-Application
that	O
maximizes	O
the	O
variance	O
of	O
the	O
projected	O
data	O
.	O
</s>
<s>
For	O
either	O
objective	O
,	O
it	O
can	O
be	O
shown	O
that	O
the	O
principal	B-Application
components	I-Application
are	O
eigenvectors	O
of	O
the	O
data	O
's	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
Thus	O
,	O
the	O
principal	B-Application
components	I-Application
are	O
often	O
computed	O
by	O
eigendecomposition	O
of	O
the	O
data	O
covariance	O
matrix	B-Architecture
or	O
singular	O
value	O
decomposition	O
of	O
the	O
data	O
matrix	B-Architecture
.	O
</s>
<s>
Factor	O
analysis	O
typically	O
incorporates	O
more	O
domain	O
specific	O
assumptions	O
about	O
the	O
underlying	O
structure	O
and	O
solves	O
eigenvectors	O
of	O
a	O
slightly	O
different	O
matrix	B-Architecture
.	O
</s>
<s>
Depending	O
on	O
the	O
field	O
of	O
application	O
,	O
it	O
is	O
also	O
named	O
the	O
discrete	O
Karhunen	O
–	O
Loève	O
transform	O
(	O
KLT	O
)	O
in	O
signal	O
processing	O
,	O
the	O
Hotelling	B-Application
transform	I-Application
in	O
multivariate	O
quality	O
control	O
,	O
proper	B-Algorithm
orthogonal	I-Algorithm
decomposition	I-Algorithm
(	O
POD	O
)	O
in	O
mechanical	O
engineering	O
,	O
singular	O
value	O
decomposition	O
(	O
SVD	O
)	O
of	O
X	O
(	O
invented	O
in	O
the	O
last	O
quarter	O
of	O
the	O
20th	O
century	O
)	O
,	O
eigenvalue	O
decomposition	O
(	O
EVD	O
)	O
of	O
XTX	O
in	O
linear	O
algebra	O
,	O
factor	O
analysis	O
(	O
for	O
a	O
discussion	O
of	O
the	O
differences	O
between	O
PCA	O
and	O
factor	O
analysis	O
see	O
Ch.7	O
of	O
Jolliffe	O
's	O
Principal	B-Application
Component	I-Application
Analysis	I-Application
)	O
,	O
Eckart	O
–	O
Young	O
theorem	O
(	O
Harman	O
,	O
1960	O
)	O
,	O
or	O
empirical	O
orthogonal	O
functions	O
(	O
EOF	O
)	O
in	O
meteorological	O
science	O
(	O
Lorenz	O
,	O
1956	O
)	O
,	O
empirical	O
eigenfunction	O
decomposition	O
(	O
Sirovich	O
,	O
1987	O
)	O
,	O
quasiharmonic	O
modes	O
(	O
Brooks	O
et	O
al.	O
,	O
1988	O
)	O
,	O
spectral	O
decomposition	O
in	O
noise	O
and	O
vibration	O
,	O
and	O
empirical	O
modal	O
analysis	O
in	O
structural	O
dynamics	O
.	O
</s>
<s>
PCA	O
can	O
be	O
thought	O
of	O
as	O
fitting	O
a	O
p-dimensional	O
ellipsoid	O
to	O
the	O
data	O
,	O
where	O
each	O
axis	O
of	O
the	O
ellipsoid	O
represents	O
a	O
principal	B-Application
component	I-Application
.	O
</s>
<s>
Then	O
,	O
we	O
compute	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
data	O
and	O
calculate	O
the	O
eigenvalues	O
and	O
corresponding	O
eigenvectors	O
of	O
this	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
choice	O
of	O
basis	O
will	O
transform	O
the	O
covariance	O
matrix	B-Architecture
into	O
a	O
diagonalized	B-Algorithm
form	O
,	O
in	O
which	O
the	O
diagonal	O
elements	O
represent	O
the	O
variance	O
of	O
each	O
axis	O
.	O
</s>
<s>
Biplots	B-Application
and	O
scree	B-Application
plots	I-Application
(	O
degree	O
of	O
explained	O
variance	O
)	O
are	O
used	O
to	O
explain	O
findings	O
of	O
the	O
PCA	O
.	O
</s>
<s>
PCA	O
is	O
defined	O
as	O
an	O
orthogonal	O
linear	B-Architecture
transformation	I-Architecture
that	O
transforms	O
the	O
data	O
to	O
a	O
new	O
coordinate	O
system	O
such	O
that	O
the	O
greatest	O
variance	O
by	O
some	O
scalar	O
projection	O
of	O
the	O
data	O
comes	O
to	O
lie	O
on	O
the	O
first	O
coordinate	O
(	O
called	O
the	O
first	O
principal	B-Application
component	I-Application
)	O
,	O
the	O
second	O
greatest	O
variance	O
on	O
the	O
second	O
coordinate	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Consider	O
an	O
data	O
matrix	B-Architecture
,	O
X	O
,	O
with	O
column-wise	O
zero	O
empirical	O
mean	O
(	O
the	O
sample	O
mean	O
of	O
each	O
column	O
has	O
been	O
shifted	O
to	O
zero	O
)	O
,	O
where	O
each	O
of	O
the	O
n	O
rows	O
represents	O
a	O
different	O
repetition	O
of	O
the	O
experiment	O
,	O
and	O
each	O
of	O
the	O
p	O
columns	O
gives	O
a	O
particular	O
kind	O
of	O
feature	O
(	O
say	O
,	O
the	O
results	O
from	O
a	O
particular	O
sensor	O
)	O
.	O
</s>
<s>
A	O
standard	O
result	O
for	O
a	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
such	O
as	O
XTX	O
is	O
that	O
the	O
quotient	O
's	O
maximum	O
possible	O
value	O
is	O
the	O
largest	O
eigenvalue	O
of	O
the	O
matrix	B-Architecture
,	O
which	O
occurs	O
when	O
w	O
is	O
the	O
corresponding	O
eigenvector	O
.	O
</s>
<s>
With	O
w(1 )	O
found	O
,	O
the	O
first	O
principal	B-Application
component	I-Application
of	O
a	O
data	O
vector	O
x(i )	O
can	O
then	O
be	O
given	O
as	O
a	O
score	O
t1(i )	O
=	O
x(i )	O
⋅	O
w(1 )	O
in	O
the	O
transformed	O
co-ordinates	O
,	O
or	O
as	O
the	O
corresponding	O
vector	O
in	O
the	O
original	O
variables	O
,	O
{	O
x(i )	O
⋅	O
w(1 )	O
}	O
w(1 )	O
.	O
</s>
<s>
The	O
k-th	O
component	O
can	O
be	O
found	O
by	O
subtracting	O
the	O
first	O
k−1	O
principal	B-Application
components	I-Application
from	O
X	O
:	O
</s>
<s>
The	O
k-th	O
principal	B-Application
component	I-Application
of	O
a	O
data	O
vector	O
x(i )	O
can	O
therefore	O
be	O
given	O
as	O
a	O
score	O
tk(i )	O
=	O
x(i )	O
⋅	O
w(k )	O
in	O
the	O
transformed	O
coordinates	O
,	O
or	O
as	O
the	O
corresponding	O
vector	O
in	O
the	O
space	O
of	O
the	O
original	O
variables	O
,	O
{	O
x(i )	O
⋅	O
w(k )	O
}	O
w(k )	O
,	O
where	O
w(k )	O
is	O
the	O
kth	O
eigenvector	O
of	O
XTX	O
.	O
</s>
<s>
where	O
W	O
is	O
a	O
p-by-p	O
matrix	B-Architecture
of	O
weights	O
whose	O
columns	O
are	O
the	O
eigenvectors	O
of	O
XTX	O
.	O
</s>
<s>
The	O
transpose	O
of	O
W	O
is	O
sometimes	O
called	O
the	O
whitening	B-Algorithm
or	I-Algorithm
sphering	I-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
XTX	O
itself	O
can	O
be	O
recognized	O
as	O
proportional	O
to	O
the	O
empirical	O
sample	O
covariance	O
matrix	B-Architecture
of	O
the	O
dataset	O
XT	O
.	O
</s>
<s>
The	O
sample	O
covariance	O
Q	O
between	O
two	O
of	O
the	O
different	O
principal	B-Application
components	I-Application
over	O
the	O
dataset	O
is	O
given	O
by	O
:	O
</s>
<s>
However	O
eigenvectors	O
w(j )	O
and	O
w(k )	O
corresponding	O
to	O
eigenvalues	O
of	O
a	O
symmetric	O
matrix	B-Architecture
are	O
orthogonal	O
(	O
if	O
the	O
eigenvalues	O
are	O
different	O
)	O
,	O
or	O
can	O
be	O
orthogonalised	O
(	O
if	O
the	O
vectors	O
happen	O
to	O
share	O
an	O
equal	O
repeated	O
value	O
)	O
.	O
</s>
<s>
The	O
product	O
in	O
the	O
final	O
line	O
is	O
therefore	O
zero	O
;	O
there	O
is	O
no	O
sample	O
covariance	O
between	O
different	O
principal	B-Application
components	I-Application
over	O
the	O
dataset	O
.	O
</s>
<s>
Another	O
way	O
to	O
characterise	O
the	O
principal	B-Application
components	I-Application
transformation	O
is	O
therefore	O
as	O
the	O
transformation	O
to	O
coordinates	O
which	O
diagonalise	O
the	O
empirical	O
sample	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
where	O
Λ	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
eigenvalues	O
λ(k )	O
of	O
XTX	O
.	O
</s>
<s>
However	O
,	O
not	O
all	O
the	O
principal	B-Application
components	I-Application
need	O
to	O
be	O
kept	O
.	O
</s>
<s>
where	O
the	O
matrix	B-Architecture
TL	O
now	O
has	O
n	O
rows	O
but	O
only	O
L	O
columns	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
PCA	O
learns	O
a	O
linear	B-Architecture
transformation	I-Architecture
where	O
the	O
columns	O
of	O
matrix	B-Architecture
form	O
an	O
orthogonal	O
basis	O
for	O
the	O
L	O
features	O
(	O
the	O
components	O
of	O
representation	O
t	O
)	O
that	O
are	O
decorrelated	O
.	O
</s>
<s>
By	O
construction	O
,	O
of	O
all	O
the	O
transformed	O
data	O
matrices	O
with	O
only	O
L	O
columns	O
,	O
this	O
score	O
matrix	B-Architecture
maximises	O
the	O
variance	O
in	O
the	O
original	O
data	O
that	O
has	O
been	O
preserved	O
,	O
while	O
minimising	O
the	O
total	O
squared	O
reconstruction	O
error	O
or	O
.	O
</s>
<s>
Such	O
dimensionality	B-Algorithm
reduction	I-Algorithm
can	O
be	O
a	O
very	O
useful	O
step	O
for	O
visualising	O
and	O
processing	O
high-dimensional	O
datasets	O
,	O
while	O
still	O
retaining	O
as	O
much	O
of	O
the	O
variance	O
in	O
the	O
dataset	O
as	O
possible	O
.	O
</s>
<s>
For	O
example	O
,	O
selecting	O
L	O
=	O
2	O
and	O
keeping	O
only	O
the	O
first	O
two	O
principal	B-Application
components	I-Application
finds	O
the	O
two-dimensional	O
plane	O
through	O
the	O
high-dimensional	O
dataset	O
in	O
which	O
the	O
data	O
is	O
most	O
spread	O
out	O
,	O
so	O
if	O
the	O
data	O
contains	O
clusters	B-Algorithm
these	O
too	O
may	O
be	O
most	O
spread	O
out	O
,	O
and	O
therefore	O
most	O
visible	O
to	O
be	O
plotted	O
out	O
in	O
a	O
two-dimensional	O
diagram	O
;	O
whereas	O
if	O
two	O
directions	O
through	O
the	O
data	O
(	O
or	O
two	O
of	O
the	O
original	O
variables	O
)	O
are	O
chosen	O
at	O
random	O
,	O
the	O
clusters	B-Algorithm
may	O
be	O
much	O
less	O
spread	O
apart	O
from	O
each	O
other	O
,	O
and	O
may	O
in	O
fact	O
be	O
much	O
more	O
likely	O
to	O
substantially	O
overlay	O
each	O
other	O
,	O
making	O
them	O
indistinguishable	O
.	O
</s>
<s>
Similarly	O
,	O
in	O
regression	O
analysis	O
,	O
the	O
larger	O
the	O
number	O
of	O
explanatory	O
variables	O
allowed	O
,	O
the	O
greater	O
is	O
the	O
chance	O
of	O
overfitting	B-Error_Name
the	O
model	O
,	O
producing	O
conclusions	O
that	O
fail	O
to	O
generalise	O
to	O
other	O
datasets	O
.	O
</s>
<s>
One	O
approach	O
,	O
especially	O
when	O
there	O
are	O
strong	O
correlations	O
between	O
different	O
possible	O
explanatory	O
variables	O
,	O
is	O
to	O
reduce	O
them	O
to	O
a	O
few	O
principal	B-Application
components	I-Application
and	O
then	O
run	O
the	O
regression	O
against	O
them	O
,	O
a	O
method	O
called	O
principal	B-Algorithm
component	I-Algorithm
regression	I-Algorithm
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
may	O
also	O
be	O
appropriate	O
when	O
the	O
variables	O
in	O
a	O
dataset	O
are	O
noisy	O
.	O
</s>
<s>
If	O
each	O
column	O
of	O
the	O
dataset	O
contains	O
independent	O
identically	O
distributed	O
Gaussian	O
noise	O
,	O
then	O
the	O
columns	O
of	O
T	O
will	O
also	O
contain	O
similarly	O
identically	O
distributed	O
Gaussian	O
noise	O
(	O
such	O
a	O
distribution	O
is	O
invariant	O
under	O
the	O
effects	O
of	O
the	O
matrix	B-Architecture
W	O
,	O
which	O
can	O
be	O
thought	O
of	O
as	O
a	O
high-dimensional	O
rotation	O
of	O
the	O
co-ordinate	O
axes	O
)	O
.	O
</s>
<s>
However	O
,	O
with	O
more	O
of	O
the	O
total	O
variance	O
concentrated	O
in	O
the	O
first	O
few	O
principal	B-Application
components	I-Application
compared	O
to	O
the	O
same	O
noise	O
variance	O
,	O
the	O
proportionate	O
effect	O
of	O
the	O
noise	O
is	O
less	O
—	O
the	O
first	O
few	O
components	O
achieve	O
a	O
higher	O
signal-to-noise	O
ratio	O
.	O
</s>
<s>
PCA	O
thus	O
can	O
have	O
the	O
effect	O
of	O
concentrating	O
much	O
of	O
the	O
signal	O
into	O
the	O
first	O
few	O
principal	B-Application
components	I-Application
,	O
which	O
can	O
usefully	O
be	O
captured	O
by	O
dimensionality	B-Algorithm
reduction	I-Algorithm
;	O
while	O
the	O
later	O
principal	B-Application
components	I-Application
may	O
be	O
dominated	O
by	O
noise	O
,	O
and	O
so	O
disposed	O
of	O
without	O
great	O
loss	O
.	O
</s>
<s>
If	O
the	O
dataset	O
is	O
not	O
too	O
large	O
,	O
the	O
significance	O
of	O
the	O
principal	B-Application
components	I-Application
can	O
be	O
tested	O
using	O
parametric	O
bootstrap	O
,	O
as	O
an	O
aid	O
in	O
determining	O
how	O
many	O
principal	B-Application
components	I-Application
to	O
retain	O
.	O
</s>
<s>
The	O
principal	B-Application
components	I-Application
transformation	O
can	O
also	O
be	O
associated	O
with	O
another	O
matrix	B-Architecture
factorization	O
,	O
the	O
singular	O
value	O
decomposition	O
(	O
SVD	O
)	O
of	O
X	O
,	O
</s>
<s>
Here	O
Σ	O
is	O
an	O
n-by-p	O
rectangular	B-Algorithm
diagonal	I-Algorithm
matrix	I-Algorithm
of	O
positive	O
numbers	O
σ(k )	O
,	O
called	O
the	O
singular	O
values	O
of	O
X	O
;	O
U	O
is	O
an	O
n-by-n	O
matrix	B-Architecture
,	O
the	O
columns	O
of	O
which	O
are	O
orthogonal	O
unit	O
vectors	O
of	O
length	O
n	O
called	O
the	O
left	O
singular	O
vectors	O
of	O
X	O
;	O
and	O
W	O
is	O
a	O
p-by-p	O
matrix	B-Architecture
whose	O
columns	O
are	O
orthogonal	O
unit	O
vectors	O
of	O
length	O
p	O
and	O
called	O
the	O
right	O
singular	O
vectors	O
of	O
X	O
.	O
</s>
<s>
where	O
is	O
the	O
square	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
the	O
singular	O
values	O
of	O
X	O
and	O
the	O
excess	O
zeros	O
chopped	O
off	O
that	O
satisfies	O
.	O
</s>
<s>
Efficient	O
algorithms	O
exist	O
to	O
calculate	O
the	O
SVD	O
of	O
X	O
without	O
having	O
to	O
form	O
the	O
matrix	B-Architecture
XTX	O
,	O
so	O
computing	O
the	O
SVD	O
is	O
now	O
the	O
standard	O
way	O
to	O
calculate	O
a	O
principal	B-Application
components	I-Application
analysis	I-Application
from	O
a	O
data	O
matrix	B-Architecture
,	O
unless	O
only	O
a	O
handful	O
of	O
components	O
are	O
required	O
.	O
</s>
<s>
As	O
with	O
the	O
eigen-decomposition	O
,	O
a	O
truncated	O
score	O
matrix	B-Architecture
TL	O
can	O
be	O
obtained	O
by	O
considering	O
only	O
the	O
first	O
L	O
largest	O
singular	O
values	O
and	O
their	O
singular	O
vectors	O
:	O
</s>
<s>
The	O
truncation	O
of	O
a	O
matrix	B-Architecture
M	O
or	O
T	O
using	O
a	O
truncated	O
singular	O
value	O
decomposition	O
in	O
this	O
way	O
produces	O
a	O
truncated	O
matrix	B-Architecture
that	O
is	O
the	O
nearest	O
possible	O
matrix	B-Architecture
of	O
rank	O
L	O
to	O
the	O
original	O
matrix	B-Architecture
,	O
in	O
the	O
sense	O
of	O
the	O
difference	O
between	O
the	O
two	O
having	O
the	O
smallest	O
possible	O
Frobenius	O
norm	O
,	O
a	O
result	O
known	O
as	O
the	O
Eckart	O
–	O
Young	O
theorem	O
 [ 1936 ] 	O
.	O
</s>
<s>
The	O
singular	O
values	O
(	O
in	O
Σ	O
)	O
are	O
the	O
square	O
roots	O
of	O
the	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
XTX	O
.	O
</s>
<s>
PCA	O
essentially	O
rotates	O
the	O
set	O
of	O
points	O
around	O
their	O
mean	O
in	O
order	O
to	O
align	O
with	O
the	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
PCA	O
is	O
often	O
used	O
in	O
this	O
manner	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
This	O
advantage	O
,	O
however	O
,	O
comes	O
at	O
the	O
price	O
of	O
greater	O
computational	O
requirements	O
if	O
compared	O
,	O
for	O
example	O
,	O
and	O
when	O
applicable	O
,	O
to	O
the	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
,	O
and	O
in	O
particular	O
to	O
the	O
DCT-II	O
which	O
is	O
simply	O
known	O
as	O
the	O
"	O
DCT	B-General_Concept
"	O
.	O
</s>
<s>
Nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
techniques	O
tend	O
to	O
be	O
more	O
computationally	O
demanding	O
than	O
PCA	O
.	O
</s>
<s>
If	O
we	O
have	O
just	O
two	O
variables	O
and	O
they	O
have	O
the	O
same	O
sample	O
variance	O
and	O
are	O
completely	O
correlated	O
,	O
then	O
the	O
PCA	O
will	O
entail	O
a	O
rotation	O
by	O
45°	O
and	O
the	O
"	O
weights	O
"	O
(	O
they	O
are	O
the	O
cosines	O
of	O
rotation	O
)	O
for	O
the	O
two	O
variables	O
with	O
respect	O
to	O
the	O
principal	B-Application
component	I-Application
will	O
be	O
equal	O
.	O
</s>
<s>
But	O
if	O
we	O
multiply	O
all	O
values	O
of	O
the	O
first	O
variable	O
by	O
100	O
,	O
then	O
the	O
first	O
principal	B-Application
component	I-Application
will	O
be	O
almost	O
the	O
same	O
as	O
that	O
variable	O
,	O
with	O
a	O
small	O
contribution	O
from	O
the	O
other	O
variable	O
,	O
whereas	O
the	O
second	O
component	O
will	O
be	O
almost	O
aligned	O
with	O
the	O
second	O
original	O
variable	O
.	O
</s>
<s>
One	O
way	O
of	O
making	O
the	O
PCA	O
less	O
arbitrary	O
is	O
to	O
use	O
variables	O
scaled	O
so	O
as	O
to	O
have	O
unit	O
variance	O
,	O
by	O
standardizing	O
the	O
data	O
and	O
hence	O
use	O
the	O
autocorrelation	O
matrix	B-Architecture
instead	O
of	O
the	O
autocovariance	O
matrix	B-Architecture
as	O
a	O
basis	O
for	O
PCA	O
.	O
</s>
<s>
"	O
mean	O
centering	O
"	O
)	O
is	O
necessary	O
for	O
performing	O
classical	O
PCA	O
to	O
ensure	O
that	O
the	O
first	O
principal	B-Application
component	I-Application
describes	O
the	O
direction	O
of	O
maximum	O
variance	O
.	O
</s>
<s>
If	O
mean	O
subtraction	O
is	O
not	O
performed	O
,	O
the	O
first	O
principal	B-Application
component	I-Application
might	O
instead	O
correspond	O
more	O
or	O
less	O
to	O
the	O
mean	O
of	O
the	O
data	O
.	O
</s>
<s>
A	O
mean	O
of	O
zero	O
is	O
needed	O
for	O
finding	O
a	O
basis	O
that	O
minimizes	O
the	O
mean	B-General_Concept
square	I-General_Concept
error	I-General_Concept
of	O
the	O
approximation	O
of	O
the	O
data	O
.	O
</s>
<s>
Mean-centering	O
is	O
unnecessary	O
if	O
performing	O
a	O
principal	B-Application
components	I-Application
analysis	I-Application
on	O
a	O
correlation	O
matrix	B-Architecture
,	O
as	O
the	O
data	O
are	O
already	O
centered	O
after	O
calculating	O
correlations	O
.	O
</s>
<s>
Since	O
covariances	O
are	O
correlations	O
of	O
normalized	O
variables	O
(	O
Z	O
-	O
or	O
standard-scores	O
)	O
a	O
PCA	O
based	O
on	O
the	O
correlation	O
matrix	B-Architecture
of	O
X	O
is	O
equal	O
to	O
a	O
PCA	O
based	O
on	O
the	O
covariance	O
matrix	B-Architecture
of	O
Z	O
,	O
the	O
standardized	O
version	O
of	O
X	O
.	O
</s>
<s>
However	O
,	O
it	O
has	O
been	O
used	O
to	O
quantify	O
the	O
distance	O
between	O
two	O
or	O
more	O
classes	O
by	O
calculating	O
center	O
of	O
mass	O
for	O
each	O
class	O
in	O
principal	B-Application
component	I-Application
space	O
and	O
reporting	O
Euclidean	O
distance	O
between	O
center	O
of	O
mass	O
of	O
two	O
or	O
more	O
classes	O
.	O
</s>
<s>
The	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
is	O
an	O
alternative	O
which	O
is	O
optimized	O
for	O
class	O
separability	O
.	O
</s>
<s>
Symbol	O
Meaning	O
Dimensions	O
Indices	O
data	O
matrix	B-Architecture
,	O
consisting	O
of	O
the	O
set	O
of	O
all	O
data	O
vectors	O
,	O
one	O
vector	O
per	O
row	O
the	O
number	O
of	O
row	O
vectors	O
in	O
the	O
data	O
set	O
scalar	O
the	O
number	O
of	O
elements	O
in	O
each	O
row	O
vector	O
(	O
dimension	O
)	O
scalar	O
the	O
number	O
of	O
dimensions	O
in	O
the	O
dimensionally	O
reduced	O
subspace	O
,	O
scalar	O
vector	O
of	O
empirical	O
means	O
,	O
one	O
mean	O
for	O
each	O
column	O
j	O
of	O
the	O
data	O
matrix	B-Architecture
vector	O
of	O
empirical	O
standard	B-General_Concept
deviations	I-General_Concept
,	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
each	O
column	O
j	O
of	O
the	O
data	O
matrix	B-Architecture
vector	O
of	O
all	O
1	O
's	O
deviations	B-General_Concept
from	O
the	O
mean	O
of	O
each	O
column	O
j	O
of	O
the	O
data	O
matrix	B-Architecture
z-scores	O
,	O
computed	O
using	O
the	O
mean	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
for	O
each	O
row	O
m	O
of	O
the	O
data	O
matrix	B-Architecture
covariance	O
matrix	B-Architecture
correlation	O
matrix	B-Architecture
matrix	B-Architecture
consisting	O
of	O
the	O
set	O
of	O
all	O
eigenvectors	O
of	O
C	O
,	O
one	O
eigenvector	O
per	O
column	O
diagonal	B-Algorithm
matrix	I-Algorithm
consisting	O
of	O
the	O
set	O
of	O
all	O
eigenvalues	O
of	O
C	O
along	O
its	O
principal	B-Algorithm
diagonal	I-Algorithm
,	O
and	O
0	O
for	O
all	O
other	O
elements	O
(	O
note	O
used	O
above	O
)	O
matrix	B-Architecture
of	O
basis	O
vectors	O
,	O
one	O
vector	O
per	O
column	O
,	O
where	O
each	O
basis	O
vector	O
is	O
one	O
of	O
the	O
eigenvectors	O
of	O
C	O
,	O
and	O
where	O
the	O
vectors	O
in	O
W	O
are	O
a	O
sub-set	O
of	O
those	O
in	O
V	O
matrix	B-Architecture
consisting	O
of	O
n	O
row	O
vectors	O
,	O
where	O
each	O
vector	O
is	O
the	O
projection	O
of	O
the	O
corresponding	O
data	O
vector	O
from	O
matrix	B-Architecture
X	O
onto	O
the	O
basis	O
vectors	O
contained	O
in	O
the	O
columns	O
of	O
matrix	B-Architecture
W	O
.	O
</s>
<s>
where	O
is	O
a	O
q-element	O
vector	O
and	O
is	O
a	O
(	O
q	O
×	O
p	O
)	O
matrix	B-Architecture
,	O
and	O
let	O
be	O
the	O
variance-covariance	O
matrix	B-Architecture
for	O
.	O
</s>
<s>
Then	O
,	O
perhaps	O
the	O
main	O
statistical	O
implication	O
of	O
the	O
result	O
is	O
that	O
not	O
only	O
can	O
we	O
decompose	O
the	O
combined	O
variances	O
of	O
all	O
the	O
elements	O
of	O
into	O
decreasing	O
contributions	O
due	O
to	O
each	O
PC	O
,	O
but	O
we	O
can	O
also	O
decompose	O
the	O
whole	O
covariance	O
matrix	B-Architecture
into	O
contributions	O
from	O
each	O
PC	O
.	O
</s>
<s>
This	O
can	O
be	O
cured	O
by	O
scaling	O
each	O
feature	O
by	O
its	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
so	O
that	O
one	O
ends	O
up	O
with	O
dimensionless	O
features	O
with	O
unital	O
variance	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
coordinate	O
transformations	O
can	O
restore	O
the	O
linearity	O
assumption	O
and	O
PCA	O
can	O
then	O
be	O
applied	O
(	O
see	O
kernel	B-Algorithm
PCA	I-Algorithm
)	O
.	O
</s>
<s>
Another	O
limitation	O
is	O
the	O
mean-removal	O
process	O
before	O
constructing	O
the	O
covariance	O
matrix	B-Architecture
for	O
PCA	O
.	O
</s>
<s>
As	O
an	O
alternative	O
method	O
,	O
non-negative	O
matrix	B-Architecture
factorization	O
focusing	O
only	O
on	O
the	O
non-negative	O
elements	O
in	O
the	O
matrices	O
,	O
which	O
is	O
well-suited	O
for	O
astrophysical	O
observations	O
.	O
</s>
<s>
See	O
more	O
at	O
Relation	O
between	O
PCA	O
and	O
Non-negative	O
Matrix	B-Architecture
Factorization	O
.	O
</s>
<s>
PCA	O
transforms	O
original	O
data	O
into	O
data	O
that	O
is	O
relevant	O
to	O
the	O
principal	B-Application
components	I-Application
of	O
that	O
data	O
,	O
which	O
means	O
that	O
the	O
new	O
data	O
variables	O
cannot	O
be	O
interpreted	O
in	O
the	O
same	O
ways	O
that	O
the	O
originals	O
were	O
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
results	O
in	O
a	O
loss	O
of	O
information	O
,	O
in	O
general	O
.	O
</s>
<s>
PCA-based	O
dimensionality	B-Algorithm
reduction	I-Algorithm
tends	O
to	O
minimize	O
that	O
information	O
loss	O
,	O
under	O
certain	O
signal	O
and	O
noise	O
models	O
.	O
</s>
<s>
that	O
is	O
,	O
that	O
the	O
data	O
vector	O
is	O
the	O
sum	O
of	O
the	O
desired	O
information-bearing	O
signal	O
and	O
a	O
noise	O
signal	O
one	O
can	O
show	O
that	O
PCA	O
can	O
be	O
optimal	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
from	O
an	O
information-theoretic	O
point-of-view	O
.	O
</s>
<s>
In	O
particular	O
,	O
Linsker	O
showed	O
that	O
if	O
is	O
Gaussian	O
and	O
is	O
Gaussian	O
noise	O
with	O
a	O
covariance	O
matrix	B-Architecture
proportional	O
to	O
the	O
identity	O
matrix	B-Architecture
,	O
the	O
PCA	O
maximizes	O
the	O
mutual	O
information	O
between	O
the	O
desired	O
information	O
and	O
the	O
dimensionality-reduced	O
output	O
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
transform	O
a	O
given	O
data	O
set	O
X	O
of	O
dimension	O
p	O
to	O
an	O
alternative	O
data	O
set	O
Y	O
of	O
smaller	O
dimension	O
L	O
.	O
Equivalently	O
,	O
we	O
are	O
seeking	O
to	O
find	O
the	O
matrix	B-Architecture
Y	O
,	O
where	O
Y	O
is	O
the	O
Karhunen	O
–	O
Loève	O
transform	O
(	O
KLT	O
)	O
of	O
matrix	B-Architecture
X	O
:	O
</s>
<s>
Place	O
the	O
row	O
vectors	O
into	O
a	O
single	O
matrix	B-Architecture
X	O
of	O
dimensions	O
n	O
×	O
p	O
.	O
</s>
<s>
Mean	O
subtraction	O
is	O
an	O
integral	O
part	O
of	O
the	O
solution	O
towards	O
finding	O
a	O
principal	B-Application
component	I-Application
basis	O
that	O
minimizes	O
the	O
mean	B-General_Concept
square	I-General_Concept
error	I-General_Concept
of	O
approximating	O
the	O
data	O
.	O
</s>
<s>
Subtract	O
the	O
empirical	O
mean	O
vector	O
from	O
each	O
row	O
of	O
the	O
data	O
matrix	B-Architecture
X	O
.	O
</s>
<s>
Store	O
mean-subtracted	O
data	O
in	O
the	O
n	O
×	O
p	O
matrix	B-Architecture
B	O
.	O
</s>
<s>
This	O
step	O
affects	O
the	O
calculated	O
principal	B-Application
components	I-Application
,	O
but	O
makes	O
them	O
independent	O
of	O
the	O
units	O
used	O
to	O
measure	O
the	O
different	O
variables	O
.	O
</s>
<s>
Find	O
the	O
p	O
×	O
p	O
empirical	O
covariance	O
matrix	B-Architecture
C	O
from	O
matrix	B-Architecture
B	O
:	O
where	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
operator	O
.	O
</s>
<s>
If	O
B	O
consists	O
entirely	O
of	O
real	O
numbers	O
,	O
which	O
is	O
the	O
case	O
in	O
many	O
applications	O
,	O
the	O
"	O
conjugate	B-Algorithm
transpose	I-Algorithm
"	O
is	O
the	O
same	O
as	O
the	O
regular	O
transpose	O
.	O
</s>
<s>
The	O
reasoning	O
behind	O
using	O
instead	O
of	O
n	O
to	O
calculate	O
the	O
covariance	O
is	O
Bessel	B-General_Concept
's	I-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
Compute	O
the	O
matrix	B-Architecture
V	O
of	O
eigenvectors	O
which	O
diagonalizes	B-Algorithm
the	O
covariance	O
matrix	B-Architecture
C	O
:	O
where	O
D	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
eigenvalues	O
of	O
C	O
.	O
This	O
step	O
will	O
typically	O
involve	O
the	O
use	O
of	O
a	O
computer-based	O
algorithm	O
for	O
computing	O
eigenvectors	O
and	O
eigenvalues	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
readily	O
available	O
as	O
sub-components	O
of	O
most	O
matrix	B-Architecture
algebra	O
systems	O
,	O
such	O
as	O
SAS	B-Language
,	O
R	B-Language
,	O
MATLAB	B-Language
,	O
Mathematica	B-Language
,	O
SciPy	B-Application
,	O
IDL	B-Language
(	O
Interactive	B-Language
Data	I-Language
Language	I-Language
)	O
,	O
or	O
GNU	B-Language
Octave	I-Language
as	O
well	O
as	O
OpenCV	B-Language
.	O
</s>
<s>
Matrix	B-Architecture
V	O
,	O
also	O
of	O
dimension	O
p	O
×	O
p	O
,	O
contains	O
p	O
column	O
vectors	O
,	O
each	O
of	O
length	O
p	O
,	O
which	O
represent	O
the	O
p	O
eigenvectors	O
of	O
the	O
covariance	O
matrix	B-Architecture
C	O
.	O
</s>
<s>
Matrix	B-Architecture
V	O
denotes	O
the	O
matrix	B-Architecture
of	O
right	O
eigenvectors	O
(	O
as	O
opposed	O
to	O
left	O
eigenvectors	O
)	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
matrix	B-Architecture
of	O
right	O
eigenvectors	O
need	O
not	O
be	O
the	O
(	O
conjugate	O
)	O
transpose	O
of	O
the	O
matrix	B-Architecture
of	O
left	O
eigenvectors	O
.	O
</s>
<s>
Sort	O
the	O
columns	O
of	O
the	O
eigenvector	O
matrix	B-Architecture
V	O
and	O
eigenvalue	O
matrix	B-Architecture
D	O
in	O
order	O
of	O
decreasing	O
eigenvalue	O
.	O
</s>
<s>
Make	O
sure	O
to	O
maintain	O
the	O
correct	O
pairings	O
between	O
the	O
columns	O
in	O
each	O
matrix	B-Architecture
.	O
</s>
<s>
That	O
is	O
,	O
the	O
first	O
column	O
of	O
is	O
the	O
projection	O
of	O
the	O
data	O
points	O
onto	O
the	O
first	O
principal	B-Application
component	I-Application
,	O
the	O
second	O
column	O
is	O
the	O
projection	O
onto	O
the	O
second	O
principal	B-Application
component	I-Application
,	O
etc	O
.	O
</s>
<s>
We	O
want	O
to	O
find	O
a	O
orthonormal	O
transformation	O
matrix	B-Architecture
P	O
so	O
that	O
PX	O
has	O
a	O
diagonal	O
covariance	O
matrix	B-Architecture
(	O
that	O
is	O
,	O
PX	O
is	O
a	O
random	O
vector	O
with	O
all	O
its	O
distinct	O
components	O
pairwise	O
uncorrelated	O
)	O
.	O
</s>
<s>
Hence	O
holds	O
if	O
and	O
only	O
if	O
were	O
diagonalisable	B-Algorithm
by	O
.	O
</s>
<s>
This	O
is	O
very	O
constructive	O
,	O
as	O
cov(X )	O
is	O
guaranteed	O
to	O
be	O
a	O
non-negative	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
and	O
thus	O
is	O
guaranteed	O
to	O
be	O
diagonalisable	B-Algorithm
by	O
some	O
unitary	O
matrix	B-Architecture
.	O
</s>
<s>
In	O
practical	O
implementations	O
,	O
especially	O
with	O
high	O
dimensional	O
data	O
(	O
large	O
)	O
,	O
the	O
naive	O
covariance	O
method	O
is	O
rarely	O
used	O
because	O
it	O
is	O
not	O
efficient	O
due	O
to	O
high	O
computational	O
and	O
memory	O
costs	O
of	O
explicitly	O
determining	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
covariance-free	O
approach	O
avoids	O
the	O
operations	O
of	O
explicitly	O
calculating	O
and	O
storing	O
the	O
covariance	O
matrix	B-Architecture
,	O
instead	O
utilizing	O
one	O
of	O
matrix-free	O
methods	O
,	O
for	O
example	O
,	O
based	O
on	O
the	O
function	O
evaluating	O
the	O
product	O
at	O
the	O
cost	O
of	O
operations	O
.	O
</s>
<s>
One	O
way	O
to	O
compute	O
the	O
first	O
principal	B-Application
component	I-Application
efficiently	O
is	O
shown	O
in	O
the	O
following	O
pseudo-code	O
,	O
for	O
a	O
data	O
matrix	B-Architecture
with	O
zero	O
mean	O
,	O
without	O
ever	O
computing	O
its	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
power	B-Language
iteration	I-Language
algorithm	O
simply	O
calculates	O
the	O
vector	O
,	O
normalizes	O
,	O
and	O
places	O
the	O
result	O
back	O
in	O
.	O
</s>
<s>
The	O
eigenvalue	O
is	O
approximated	O
by	O
,	O
which	O
is	O
the	O
Rayleigh	O
quotient	O
on	O
the	O
unit	O
vector	O
for	O
the	O
covariance	O
matrix	B-Architecture
.	O
</s>
<s>
If	O
the	O
largest	O
singular	O
value	O
is	O
well	O
separated	O
from	O
the	O
next	O
largest	O
one	O
,	O
the	O
vector	O
gets	O
close	O
to	O
the	O
first	O
principal	B-Application
component	I-Application
of	O
within	O
the	O
number	O
of	O
iterations	O
,	O
which	O
is	O
small	O
relative	O
to	O
,	O
at	O
the	O
total	O
cost	O
.	O
</s>
<s>
The	O
power	B-Language
iteration	I-Language
convergence	O
can	O
be	O
accelerated	O
without	O
noticeably	O
sacrificing	O
the	O
small	O
cost	O
per	O
iteration	O
using	O
more	O
advanced	O
matrix-free	O
methods	O
,	O
such	O
as	O
the	O
Lanczos	O
algorithm	O
or	O
the	O
Locally	O
Optimal	O
Block	O
Preconditioned	O
Conjugate	O
Gradient	O
(	O
LOBPCG	B-Application
)	O
method	O
.	O
</s>
<s>
Subsequent	O
principal	B-Application
components	I-Application
can	O
be	O
computed	O
one-by-one	O
via	O
deflation	O
or	O
simultaneously	O
as	O
a	O
block	O
.	O
</s>
<s>
In	O
the	O
former	O
approach	O
,	O
imprecisions	O
in	O
already	O
computed	O
approximate	O
principal	B-Application
components	I-Application
additively	O
affect	O
the	O
accuracy	O
of	O
the	O
subsequently	O
computed	O
principal	B-Application
components	I-Application
,	O
thus	O
increasing	O
the	O
error	O
with	O
every	O
new	O
computation	O
.	O
</s>
<s>
The	O
latter	O
approach	O
in	O
the	O
block	O
power	B-Language
method	I-Language
replaces	O
single-vectors	O
and	O
with	O
block-vectors	O
,	O
matrices	O
and	O
.	O
</s>
<s>
Every	O
column	O
of	O
approximates	O
one	O
of	O
the	O
leading	O
principal	B-Application
components	I-Application
,	O
while	O
all	O
columns	O
are	O
iterated	O
simultaneously	O
.	O
</s>
<s>
Implemented	O
,	O
for	O
example	O
,	O
in	O
LOBPCG	B-Application
,	O
efficient	O
blocking	O
eliminates	O
the	O
accumulation	O
of	O
the	O
errors	O
,	O
allows	O
using	O
high-level	O
BLAS	B-Application
matrix-matrix	O
product	O
functions	O
,	O
and	O
typically	O
leads	O
to	O
faster	O
convergence	O
,	O
compared	O
to	O
the	O
single-vector	O
one-by-one	O
technique	O
.	O
</s>
<s>
Non-linear	O
iterative	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
NIPALS	O
)	O
is	O
a	O
variant	O
the	O
classical	O
power	B-Language
iteration	I-Language
with	O
matrix	B-Architecture
deflation	O
by	O
subtraction	O
implemented	O
for	O
computing	O
the	O
first	O
few	O
components	O
in	O
a	O
principal	B-Application
component	I-Application
or	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
analysis	O
.	O
</s>
<s>
The	O
non-linear	O
iterative	O
partial	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
NIPALS	O
)	O
algorithm	O
updates	O
iterative	O
approximations	O
to	O
the	O
leading	O
scores	O
and	O
loadings	O
t1	O
and	O
r1T	O
by	O
the	O
power	B-Language
iteration	I-Language
multiplying	O
on	O
every	O
iteration	O
by	O
X	O
on	O
the	O
left	O
and	O
on	O
the	O
right	O
,	O
that	O
is	O
,	O
calculation	O
of	O
the	O
covariance	O
matrix	B-Architecture
is	O
avoided	O
,	O
just	O
as	O
in	O
the	O
matrix-free	O
implementation	O
of	O
the	O
power	B-Language
iterations	I-Language
to	O
,	O
based	O
on	O
the	O
function	O
evaluating	O
the	O
product	O
.	O
</s>
<s>
The	O
matrix	B-Architecture
deflation	O
by	O
subtraction	O
is	O
performed	O
by	O
subtracting	O
the	O
outer	O
product	O
,	O
t1r1T	O
from	O
X	O
leaving	O
the	O
deflated	O
residual	O
matrix	B-Architecture
used	O
to	O
calculate	O
the	O
subsequent	O
leading	O
PCs	O
.	O
</s>
<s>
For	O
large	O
data	O
matrices	O
,	O
or	O
matrices	O
that	O
have	O
a	O
high	O
degree	O
of	O
column	O
collinearity	O
,	O
NIPALS	O
suffers	O
from	O
loss	O
of	O
orthogonality	O
of	O
PCs	O
due	O
to	O
machine	O
precision	O
round-off	B-Algorithm
errors	I-Algorithm
accumulated	O
in	O
each	O
iteration	O
and	O
matrix	B-Architecture
deflation	O
by	O
subtraction	O
.	O
</s>
<s>
A	O
Gram	B-Algorithm
–	I-Algorithm
Schmidt	I-Algorithm
re-orthogonalization	O
algorithm	O
is	O
applied	O
to	O
both	O
the	O
scores	O
and	O
the	O
loadings	O
at	O
each	O
iteration	O
step	O
to	O
eliminate	O
this	O
loss	O
of	O
orthogonality	O
.	O
</s>
<s>
NIPALS	O
reliance	O
on	O
single-vector	O
multiplications	O
cannot	O
take	O
advantage	O
of	O
high-level	O
BLAS	B-Application
and	O
results	O
in	O
slow	O
convergence	O
for	O
clustered	O
leading	O
singular	O
values	O
—	O
both	O
these	O
deficiencies	O
are	O
resolved	O
in	O
more	O
sophisticated	O
matrix-free	O
block	O
solvers	O
,	O
such	O
as	O
the	O
Locally	O
Optimal	O
Block	O
Preconditioned	O
Conjugate	O
Gradient	O
(	O
LOBPCG	B-Application
)	O
method	O
.	O
</s>
<s>
When	O
analyzing	O
the	O
results	O
,	O
it	O
is	O
natural	O
to	O
connect	O
the	O
principal	B-Application
components	I-Application
to	O
the	O
qualitative	O
variable	O
species	O
.	O
</s>
<s>
For	O
each	O
center	O
of	O
gravity	O
and	O
each	O
axis	O
,	O
p-value	O
to	O
judge	O
the	O
significance	O
of	O
the	O
difference	O
between	O
the	O
center	O
of	O
gravity	O
and	O
origin	B-Application
.	O
</s>
<s>
This	O
is	O
the	O
case	O
of	O
that	O
historically	O
,	O
following	O
the	O
work	O
of	O
Ludovic	O
Lebart	O
,	O
was	O
the	O
first	O
to	O
propose	O
this	O
option	O
,	O
and	O
the	O
R	B-Language
package	O
.	O
</s>
<s>
These	O
were	O
known	O
as	O
'	O
social	O
rank	O
 '	O
(	O
an	O
index	O
of	O
occupational	O
status	O
)	O
,	O
'	O
familism	O
 '	O
or	O
family	O
size	O
,	O
and	O
'	O
ethnicity	O
 '	O
;	O
Cluster	B-Algorithm
analysis	I-Algorithm
could	O
then	O
be	O
applied	O
to	O
divide	O
the	O
city	O
into	O
clusters	B-Algorithm
or	O
precincts	O
according	O
to	O
values	O
of	O
the	O
three	O
key	O
factor	O
variables	O
.	O
</s>
<s>
In	O
2000	O
,	O
Flood	O
revived	O
the	O
factorial	O
ecology	O
approach	O
to	O
show	O
that	O
principal	B-Application
components	I-Application
analysis	I-Application
actually	O
gave	O
meaningful	O
answers	O
directly	O
,	O
without	O
resorting	O
to	O
factor	O
rotation	O
.	O
</s>
<s>
The	O
principal	B-Application
components	I-Application
were	O
actually	O
dual	O
variables	O
or	O
shadow	O
prices	O
of	O
'	O
forces	O
 '	O
pushing	O
people	O
together	O
or	O
apart	O
in	O
cities	O
.	O
</s>
<s>
About	O
the	O
same	O
time	O
,	O
the	O
Australian	O
Bureau	O
of	O
Statistics	O
defined	O
distinct	O
indexes	O
of	O
advantage	O
and	O
disadvantage	O
taking	O
the	O
first	O
principal	B-Application
component	I-Application
of	O
sets	O
of	O
key	O
variables	O
that	O
were	O
thought	O
to	O
be	O
important	O
.	O
</s>
<s>
The	O
first	O
principal	B-Application
component	I-Application
was	O
subject	O
to	O
iterative	O
regression	O
,	O
adding	O
the	O
original	O
variables	O
singly	O
until	O
about	O
90%	O
of	O
its	O
variation	O
was	O
accounted	O
for	O
.	O
</s>
<s>
In	O
1978	O
Cavalli-Sforza	O
and	O
others	O
pioneered	O
the	O
use	O
of	O
principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
to	O
summarise	O
data	O
on	O
variation	O
in	O
human	O
gene	O
frequencies	O
across	O
regions	O
.	O
</s>
<s>
Genetics	O
varies	O
largely	O
according	O
to	O
proximity	O
,	O
so	O
the	O
first	O
two	O
principal	B-Application
components	I-Application
actually	O
show	O
spatial	O
distribution	O
and	O
may	O
be	O
used	O
to	O
map	O
the	O
relative	O
geographical	O
location	O
of	O
different	O
population	O
groups	O
,	O
thereby	O
showing	O
individuals	O
who	O
have	O
wandered	O
from	O
their	O
original	O
locations	O
.	O
</s>
<s>
In	O
any	O
consumer	O
questionnaire	O
,	O
there	O
are	O
series	O
of	O
questions	O
designed	O
to	O
elicit	O
consumer	O
attitudes	O
,	O
and	O
principal	B-Application
components	I-Application
seek	O
out	O
latent	O
variables	O
underlying	O
these	O
attitudes	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Oxford	O
Internet	O
Survey	O
in	O
2013	O
asked	O
2000	O
people	O
about	O
their	O
attitudes	O
and	O
beliefs	O
,	O
and	O
from	O
these	O
analysts	O
extracted	O
four	O
principal	B-Application
component	I-Application
dimensions	O
,	O
which	O
they	O
identified	O
as	O
'	O
escape	O
 '	O
,	O
'	O
social	O
networking	O
 '	O
,	O
'	O
efficiency	O
 '	O
,	O
and	O
'	O
problem	O
creating	O
 '	O
.	O
</s>
<s>
The	O
first	O
principal	B-Application
component	I-Application
represented	O
a	O
general	O
attitude	O
toward	O
property	O
and	O
home	O
ownership	O
.	O
</s>
<s>
In	O
quantitative	O
finance	O
,	O
principal	B-Application
component	I-Application
analysis	I-Application
can	O
be	O
directly	O
applied	O
to	O
the	O
risk	O
management	O
of	O
interest	O
rate	O
derivative	O
portfolios	O
.	O
</s>
<s>
Trading	O
multiple	O
swap	O
instruments	O
which	O
are	O
usually	O
a	O
function	O
of	O
30	O
–	O
500	O
other	O
market	O
quotable	O
swap	O
instruments	O
is	O
sought	O
to	O
be	O
reduced	O
to	O
usually	O
3	O
or	O
4	O
principal	B-Application
components	I-Application
,	O
representing	O
the	O
path	O
of	O
interest	O
rates	O
on	O
a	O
macro	O
basis	O
.	O
</s>
<s>
A	O
second	O
is	O
to	O
enhance	O
portfolio	O
return	O
,	O
using	O
the	O
principal	B-Application
components	I-Application
to	O
select	O
stocks	O
with	O
upside	O
potential	O
.	O
</s>
<s>
A	O
variant	O
of	O
principal	B-Application
components	I-Application
analysis	I-Application
is	O
used	O
in	O
neuroscience	O
to	O
identify	O
the	O
specific	O
properties	O
of	O
a	O
stimulus	O
that	O
increases	O
a	O
neuron	O
's	O
probability	O
of	O
generating	O
an	O
action	B-Algorithm
potential	I-Algorithm
.	O
</s>
<s>
This	O
technique	O
is	O
known	O
as	O
spike-triggered	O
covariance	O
analysis	O
.	O
</s>
<s>
In	O
a	O
typical	O
application	O
an	O
experimenter	O
presents	O
a	O
white	O
noise	O
process	O
as	O
a	O
stimulus	O
(	O
usually	O
either	O
as	O
a	O
sensory	O
input	O
to	O
a	O
test	O
subject	O
,	O
or	O
as	O
a	O
current	O
injected	O
directly	O
into	O
the	O
neuron	O
)	O
and	O
records	O
a	O
train	O
of	O
action	B-Algorithm
potentials	I-Algorithm
,	O
or	O
spikes	B-Algorithm
,	O
produced	O
by	O
the	O
neuron	O
as	O
a	O
result	O
.	O
</s>
<s>
Presumably	O
,	O
certain	O
features	O
of	O
the	O
stimulus	O
make	O
the	O
neuron	O
more	O
likely	O
to	O
spike	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
extract	O
these	O
features	O
,	O
the	O
experimenter	O
calculates	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
spike-triggered	O
ensemble	O
,	O
the	O
set	O
of	O
all	O
stimuli	O
(	O
defined	O
and	O
discretized	O
over	O
a	O
finite	O
time	O
window	O
,	O
typically	O
on	O
the	O
order	O
of	O
100	O
ms	O
)	O
that	O
immediately	O
preceded	O
a	O
spike	B-Algorithm
.	O
</s>
<s>
The	O
eigenvectors	O
of	O
the	O
difference	O
between	O
the	O
spike-triggered	O
covariance	O
matrix	B-Architecture
and	O
the	O
covariance	O
matrix	B-Architecture
of	O
the	O
prior	O
stimulus	O
ensemble	O
(	O
the	O
set	O
of	O
all	O
stimuli	O
,	O
defined	O
over	O
the	O
same	O
length	O
time	O
window	O
)	O
then	O
indicate	O
the	O
directions	O
in	O
the	O
space	O
of	O
stimuli	O
along	O
which	O
the	O
variance	O
of	O
the	O
spike-triggered	O
ensemble	O
differed	O
the	O
most	O
from	O
that	O
of	O
the	O
prior	O
stimulus	O
ensemble	O
.	O
</s>
<s>
Specifically	O
,	O
the	O
eigenvectors	O
with	O
the	O
largest	O
positive	O
eigenvalues	O
correspond	O
to	O
the	O
directions	O
along	O
which	O
the	O
variance	O
of	O
the	O
spike-triggered	O
ensemble	O
showed	O
the	O
largest	O
positive	O
change	O
compared	O
to	O
the	O
variance	O
of	O
the	O
prior	O
.	O
</s>
<s>
Since	O
these	O
were	O
the	O
directions	O
in	O
which	O
varying	O
the	O
stimulus	O
led	O
to	O
a	O
spike	B-Algorithm
,	O
they	O
are	O
often	O
good	O
approximations	O
of	O
the	O
sought	O
after	O
relevant	O
stimulus	O
features	O
.	O
</s>
<s>
In	O
neuroscience	O
,	O
PCA	O
is	O
also	O
used	O
to	O
discern	O
the	O
identity	O
of	O
a	O
neuron	O
from	O
the	O
shape	O
of	O
its	O
action	B-Algorithm
potential	I-Algorithm
.	O
</s>
<s>
Spike	B-Algorithm
sorting	I-Algorithm
is	O
an	O
important	O
procedure	O
because	O
extracellular	O
recording	O
techniques	O
often	O
pick	O
up	O
signals	O
from	O
more	O
than	O
one	O
neuron	O
.	O
</s>
<s>
In	O
spike	B-Algorithm
sorting	I-Algorithm
,	O
one	O
first	O
uses	O
PCA	O
to	O
reduce	O
the	O
dimensionality	O
of	O
the	O
space	O
of	O
action	B-Algorithm
potential	I-Algorithm
waveforms	O
,	O
and	O
then	O
performs	O
clustering	B-Algorithm
analysis	I-Algorithm
to	O
associate	O
specific	O
action	B-Algorithm
potentials	I-Algorithm
with	O
individual	O
neurons	O
.	O
</s>
<s>
PCA	O
as	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
technique	O
is	O
particularly	O
suited	O
to	O
detect	O
coordinated	O
activities	O
of	O
large	O
neuronal	O
ensembles	O
.	O
</s>
<s>
It	O
is	O
traditionally	O
applied	O
to	O
contingency	B-Application
tables	I-Application
.	O
</s>
<s>
CA	O
decomposes	O
the	O
chi-squared	B-General_Concept
statistic	I-General_Concept
associated	O
to	O
this	O
table	O
into	O
orthogonal	O
factors	O
.	O
</s>
<s>
Because	O
CA	O
is	O
a	O
descriptive	O
technique	O
,	O
it	O
can	O
be	O
applied	O
to	O
tables	O
for	O
which	O
the	O
chi-squared	B-General_Concept
statistic	I-General_Concept
is	O
appropriate	O
or	O
not	O
.	O
</s>
<s>
Several	O
variants	O
of	O
CA	O
are	O
available	O
including	O
detrended	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
and	O
canonical	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
One	O
special	O
extension	O
is	O
multiple	B-Algorithm
correspondence	I-Algorithm
analysis	I-Algorithm
,	O
which	O
may	O
be	O
seen	O
as	O
the	O
counterpart	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
for	O
categorical	O
data	O
.	O
</s>
<s>
Principal	B-Application
component	I-Application
analysis	I-Application
creates	O
variables	O
that	O
are	O
linear	O
combinations	O
of	O
the	O
original	O
variables	O
.	O
</s>
<s>
Factor	O
analysis	O
is	O
similar	O
to	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
in	O
that	O
factor	O
analysis	O
also	O
involves	O
linear	O
combinations	O
of	O
variables	O
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
correlation	O
matrix	B-Architecture
,	O
this	O
corresponds	O
with	O
focusing	O
on	O
explaining	O
the	O
off-diagonal	O
terms	O
(	O
that	O
is	O
,	O
shared	O
co-variance	O
)	O
,	O
while	O
PCA	O
focuses	O
on	O
explaining	O
the	O
terms	O
that	O
sit	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
It	O
has	O
been	O
asserted	O
that	O
the	O
relaxed	O
solution	O
of	O
-means	O
clustering	O
,	O
specified	O
by	O
the	O
cluster	O
indicators	O
,	O
is	O
given	O
by	O
the	O
principal	B-Application
components	I-Application
,	O
and	O
the	O
PCA	O
subspace	O
spanned	O
by	O
the	O
principal	O
directions	O
is	O
identical	O
to	O
the	O
cluster	O
centroid	O
subspace	O
.	O
</s>
<s>
Non-negative	O
matrix	B-Architecture
factorization	O
(	O
NMF	O
)	O
is	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
method	O
where	O
only	O
non-negative	O
elements	O
in	O
the	O
matrices	O
are	O
used	O
,	O
which	O
is	O
therefore	O
a	O
promising	O
method	O
in	O
astronomy	O
,	O
in	O
the	O
sense	O
that	O
astrophysical	O
signals	O
are	O
non-negative	O
.	O
</s>
<s>
The	O
residual	O
fractional	O
eigenvalue	O
plots	O
,	O
that	O
is	O
,	O
as	O
a	O
function	O
of	O
component	O
number	O
given	O
a	O
total	O
of	O
components	O
,	O
for	O
PCA	O
has	O
a	O
flat	O
plateau	O
,	O
where	O
no	O
data	O
is	O
captured	O
to	O
remove	O
the	O
quasi-static	O
noise	O
,	O
then	O
the	O
curves	O
dropped	O
quickly	O
as	O
an	O
indication	O
of	O
over-fitting	B-Error_Name
and	O
captures	O
random	O
noise	O
.	O
</s>
<s>
The	O
FRV	O
curves	O
for	O
NMF	O
is	O
decreasing	O
continuously	O
when	O
the	O
NMF	O
components	O
are	O
constructed	O
sequentially	O
,	O
indicating	O
the	O
continuous	O
capturing	O
of	O
quasi-static	O
noise	O
;	O
then	O
converge	O
to	O
higher	O
levels	O
than	O
PCA	O
,	O
indicating	O
the	O
less	O
over-fitting	B-Error_Name
property	O
of	O
NMF	O
.	O
</s>
<s>
It	O
is	O
often	O
difficult	O
to	O
interpret	O
the	O
principal	B-Application
components	I-Application
when	O
the	O
data	O
include	O
many	O
variables	O
of	O
various	O
origins	O
,	O
or	O
when	O
some	O
variables	O
are	O
qualitative	O
.	O
</s>
<s>
The	O
iconography	B-General_Concept
of	I-General_Concept
correlations	I-General_Concept
,	O
on	O
the	O
contrary	O
,	O
which	O
is	O
not	O
a	O
projection	O
on	O
a	O
system	O
of	O
axes	O
,	O
does	O
not	O
have	O
these	O
drawbacks	O
.	O
</s>
<s>
The	O
principle	O
of	O
the	O
diagram	O
is	O
to	O
underline	O
the	O
"	O
remarkable	O
"	O
correlations	O
of	O
the	O
correlation	O
matrix	B-Architecture
,	O
by	O
a	O
solid	O
line	O
(	O
positive	O
correlation	O
)	O
or	O
dotted	O
line	O
(	O
negative	O
correlation	O
)	O
.	O
</s>
<s>
A	O
particular	O
disadvantage	O
of	O
PCA	O
is	O
that	O
the	O
principal	B-Application
components	I-Application
are	O
usually	O
linear	O
combinations	O
of	O
all	O
input	O
variables	O
.	O
</s>
<s>
Sparse	B-Algorithm
PCA	I-Algorithm
overcomes	O
this	O
disadvantage	O
by	O
finding	O
linear	O
combinations	O
that	O
contain	O
just	O
a	O
few	O
input	O
variables	O
.	O
</s>
<s>
It	O
extends	O
the	O
classic	O
method	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
for	O
the	O
reduction	O
of	O
dimensionality	O
of	O
data	O
by	O
adding	O
sparsity	O
constraint	O
on	O
the	O
input	O
variables	O
.	O
</s>
<s>
The	O
methodological	O
and	O
theoretical	O
developments	O
of	O
Sparse	B-Algorithm
PCA	I-Algorithm
as	O
well	O
as	O
its	O
applications	O
in	O
scientific	O
studies	O
were	O
recently	O
reviewed	O
in	O
a	O
survey	O
paper	O
.	O
</s>
<s>
Most	O
of	O
the	O
modern	O
methods	O
for	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
find	O
their	O
theoretical	O
and	O
algorithmic	O
roots	O
in	O
PCA	O
or	O
K-means	B-Algorithm
.	O
</s>
<s>
See	O
also	O
the	O
elastic	B-Algorithm
map	I-Algorithm
algorithm	O
and	O
principal	B-Algorithm
geodesic	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Another	O
popular	O
generalization	O
is	O
kernel	B-Algorithm
PCA	I-Algorithm
,	O
which	O
corresponds	O
to	O
PCA	O
performed	O
in	O
a	O
reproducing	O
kernel	O
Hilbert	O
space	O
associated	O
with	O
a	O
positive	O
definite	O
kernel	O
.	O
</s>
<s>
In	O
multilinear	O
subspace	O
learning	O
,	O
PCA	O
is	O
generalized	O
to	O
multilinear	B-Algorithm
PCA	I-Algorithm
(	O
MPCA	O
)	O
that	O
extracts	O
features	O
directly	O
from	O
tensor	O
representations	O
.	O
</s>
<s>
N-way	O
principal	B-Application
component	I-Application
analysis	I-Application
may	O
be	O
performed	O
with	O
models	O
such	O
as	O
Tucker	B-Algorithm
decomposition	I-Algorithm
,	O
PARAFAC	O
,	O
multiple	O
factor	O
analysis	O
,	O
co-inertia	O
analysis	O
,	O
STATIS	O
,	O
and	O
DISTATIS	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
data	B-Application
mining	I-Application
algorithms	O
like	O
correlation	B-Algorithm
clustering	I-Algorithm
,	O
the	O
assignment	O
of	O
points	O
to	O
clusters	B-Algorithm
and	O
outliers	O
is	O
not	O
known	O
beforehand	O
.	O
</s>
<s>
Outlier-resistant	O
variants	O
of	O
PCA	O
have	O
also	O
been	O
proposed	O
,	O
based	O
on	O
L1-norm	O
formulations	O
(	O
L1-PCA	B-General_Concept
)	O
.	O
</s>
<s>
Robust	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
RPCA	O
)	O
via	O
decomposition	O
in	O
low-rank	O
and	O
sparse	O
matrices	O
is	O
a	O
modification	O
of	O
PCA	O
that	O
works	O
well	O
with	O
respect	O
to	O
grossly	O
corrupted	O
observations	O
.	O
</s>
<s>
Independent	B-Algorithm
component	I-Algorithm
analysis	I-Algorithm
(	O
ICA	O
)	O
is	O
directed	O
to	O
similar	O
problems	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
but	O
finds	O
additively	O
separable	O
components	O
rather	O
than	O
successive	O
approximations	O
.	O
</s>
<s>
Given	O
a	O
matrix	B-Architecture
,	O
it	O
tries	O
to	O
decompose	O
it	O
into	O
two	O
matrices	O
such	O
that	O
.	O
</s>
<s>
The	O
justification	O
for	O
this	O
criterion	O
is	O
that	O
if	O
a	O
node	O
is	O
removed	O
from	O
the	O
regulatory	O
layer	O
along	O
with	O
all	O
the	O
output	O
nodes	O
connected	O
to	O
it	O
,	O
the	O
result	O
must	O
still	O
be	O
characterized	O
by	O
a	O
connectivity	O
matrix	B-Architecture
with	O
full	O
column	O
rank	O
.	O
</s>
<s>
Discriminant	B-General_Concept
analysis	I-General_Concept
of	O
principal	B-Application
components	I-Application
(	O
DAPC	O
)	O
is	O
a	O
multivariate	O
method	O
used	O
to	O
identify	O
and	O
describe	O
clusters	B-Algorithm
of	O
genetically	O
related	O
individuals	O
.	O
</s>
<s>
Linear	O
discriminants	O
are	O
linear	O
combinations	O
of	O
alleles	O
which	O
best	O
separate	O
the	O
clusters	B-Algorithm
.	O
</s>
<s>
In	O
DAPC	O
,	O
data	O
is	O
first	O
transformed	O
using	O
a	O
principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
and	O
subsequently	O
clusters	B-Algorithm
are	O
identified	O
using	O
discriminant	B-General_Concept
analysis	I-General_Concept
(	O
DA	O
)	O
.	O
</s>
<s>
A	O
DAPC	O
can	O
be	O
realized	O
on	O
R	B-Language
using	O
the	O
package	O
Adegenet	O
.	O
</s>
<s>
Like	O
PCA	O
,	O
it	O
allows	O
for	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
improved	O
visualization	B-Application
and	O
improved	O
interpretability	O
of	O
large	O
data-sets	O
.	O
</s>
<s>
Also	O
like	O
PCA	O
,	O
it	O
is	O
based	O
on	O
a	O
covariance	O
matrix	B-Architecture
derived	O
from	O
the	O
input	O
dataset	O
.	O
</s>
<s>
Analytica	B-Language
–	O
The	O
built-in	O
EigenDecomp	O
function	O
computes	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
ELKI	B-Language
–	O
includes	O
PCA	O
for	O
projection	O
,	O
including	O
robust	O
variants	O
of	O
PCA	O
,	O
as	O
well	O
as	O
PCA-based	O
clustering	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
Gretl	B-Application
–	O
principal	B-Application
component	I-Application
analysis	I-Application
can	O
be	O
performed	O
either	O
via	O
the	O
pca	O
command	O
or	O
via	O
the	O
princomp( )	O
function	O
.	O
</s>
<s>
KNIME	B-Language
–	O
A	O
java	O
based	O
nodal	O
arranging	O
software	O
for	O
Analysis	O
,	O
in	O
this	O
the	O
nodes	O
called	O
PCA	O
,	O
PCA	O
compute	O
,	O
PCA	O
Apply	O
,	O
PCA	O
inverse	O
make	O
it	O
easily	O
.	O
</s>
<s>
Maple	B-Language
(	O
software	O
)	O
–	O
The	O
PCA	O
command	O
is	O
used	O
to	O
perform	O
a	O
principal	B-Application
component	I-Application
analysis	I-Application
on	O
a	O
set	O
of	O
data	O
.	O
</s>
<s>
Mathematica	B-Language
–	O
Implements	O
principal	B-Application
component	I-Application
analysis	I-Application
with	O
the	O
PrincipalComponents	O
command	O
using	O
both	O
covariance	O
and	O
correlation	O
methods	O
.	O
</s>
<s>
MathPHP	O
–	O
PHP	B-Application
mathematics	O
library	O
with	O
support	O
for	O
PCA	O
.	O
</s>
<s>
MATLAB	B-Language
-	O
The	O
SVD	O
function	O
is	O
part	O
of	O
the	O
basic	O
system	O
.	O
</s>
<s>
In	O
the	O
Statistics	O
Toolbox	O
,	O
the	O
functions	O
princomp	O
and	O
pca	O
(	O
R2012b	O
)	O
give	O
the	O
principal	B-Application
components	I-Application
,	O
while	O
the	O
function	O
pcares	O
gives	O
the	O
residuals	O
and	O
reconstructed	O
matrix	B-Architecture
for	O
a	O
low-rank	O
PCA	O
approximation	O
.	O
</s>
<s>
Matplotlib	B-Language
–	O
Python	B-Language
library	O
have	O
a	O
PCA	O
package	O
in	O
the	O
.mlab	O
module	O
.	O
</s>
<s>
mlpack	B-Language
–	O
Provides	O
an	O
implementation	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
in	O
C++	B-Language
.	O
</s>
<s>
-	O
A	O
high	O
performance	O
math	O
library	O
for	O
Delphi	B-Language
and	O
FreePascal	B-Operating_System
can	O
perform	O
PCA	O
;	O
including	O
robust	O
variants	O
.	O
</s>
<s>
NAG	O
Library	O
–	O
Principal	B-Application
components	I-Application
analysis	I-Application
is	O
implemented	O
via	O
the	O
g03aa	O
routine	O
(	O
available	O
in	O
both	O
the	O
Fortran	O
versions	O
of	O
the	O
Library	O
)	O
.	O
</s>
<s>
NMath	B-Language
–	O
Proprietary	O
numerical	O
library	O
containing	O
PCA	O
for	O
the	O
.NET	B-Application
Framework	I-Application
.	O
</s>
<s>
GNU	B-Language
Octave	I-Language
–	O
Free	B-Application
software	I-Application
computational	O
environment	O
mostly	O
compatible	O
with	O
MATLAB	B-Language
,	O
the	O
function	O
princomp	O
gives	O
the	O
principal	B-Application
component	I-Application
.	O
</s>
<s>
PCA	O
displays	O
a	O
scree	B-Application
plot	I-Application
(	O
degree	O
of	O
explained	O
variance	O
)	O
where	O
user	O
can	O
interactively	O
select	O
the	O
number	O
of	O
principal	B-Application
components	I-Application
.	O
</s>
<s>
Origin	B-Application
–	O
Contains	O
PCA	O
in	O
its	O
Pro	O
version	O
.	O
</s>
<s>
Qlucore	B-Application
–	O
Commercial	O
software	O
for	O
analyzing	O
multivariate	O
data	O
with	O
instant	O
response	O
using	O
PCA	O
.	O
</s>
<s>
R	B-Language
–	O
Free	B-Application
statistical	O
package	O
,	O
the	O
functions	O
princomp	O
and	O
prcomp	O
can	O
be	O
used	O
for	O
principal	B-Application
component	I-Application
analysis	I-Application
;	O
prcomp	O
uses	O
singular	O
value	O
decomposition	O
which	O
generally	O
gives	O
better	O
numerical	O
accuracy	O
.	O
</s>
<s>
Some	O
packages	O
that	O
implement	O
PCA	O
in	O
R	B-Language
,	O
include	O
,	O
but	O
are	O
not	O
limited	O
to	O
:	O
ade4	O
,	O
vegan	O
,	O
ExPosition	O
,	O
dimRed	O
,	O
and	O
FactoMineR	O
.	O
</s>
<s>
scikit-learn	B-Application
–	O
Python	B-Language
library	O
for	O
machine	O
learning	O
which	O
contains	O
PCA	O
,	O
Probabilistic	O
PCA	O
,	O
Kernel	B-Algorithm
PCA	I-Algorithm
,	O
Sparse	B-Algorithm
PCA	I-Algorithm
and	O
other	O
techniques	O
in	O
the	O
decomposition	O
module	O
.	O
</s>
<s>
Scilab	B-Application
–	O
Free	B-Application
and	O
open-source	O
,	O
cross-platform	O
numerical	O
computational	O
package	O
,	O
the	O
function	O
princomp	O
computes	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
the	O
function	O
pca	O
computes	O
principal	B-Application
component	I-Application
analysis	I-Application
with	O
standardized	O
variables	O
.	O
</s>
<s>
SPSS	B-Algorithm
–	O
Proprietary	O
software	O
most	O
commonly	O
used	O
by	O
social	O
scientists	O
for	O
PCA	O
,	O
factor	O
analysis	O
and	O
associated	O
cluster	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Weka	B-Language
–	O
Java	O
library	O
for	O
machine	O
learning	O
which	O
contains	O
modules	O
for	O
computing	O
principal	B-Application
components	I-Application
.	O
</s>
