<s>
Functional	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
(	O
FPCA	O
)	O
is	O
a	O
statistical	O
method	O
for	O
investigating	O
the	O
dominant	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
of	O
functional	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
Using	O
this	O
method	O
,	O
a	O
random	O
function	O
is	O
represented	O
in	O
the	O
eigenbasis	O
,	O
which	O
is	O
an	O
orthonormal	B-Algorithm
basis	O
of	O
the	O
Hilbert	O
space	O
L2	O
that	O
consists	O
of	O
the	O
eigenfunctions	O
of	O
the	O
autocovariance	O
operator	O
.	O
</s>
<s>
FPCA	O
represents	O
functional	B-General_Concept
data	I-General_Concept
in	O
the	O
most	O
parsimonious	O
way	O
,	O
in	O
the	O
sense	O
that	O
when	O
using	O
a	O
fixed	O
number	O
of	O
basis	O
functions	O
,	O
the	O
eigenfunction	O
basis	O
explains	O
more	O
variation	O
than	O
any	O
other	O
basis	O
expansion	O
.	O
</s>
<s>
If	O
the	O
observations	O
are	O
sparse	O
,	O
one	O
needs	O
to	O
smooth	O
the	O
data	O
pooled	O
from	O
all	O
observations	O
to	O
obtain	O
the	O
mean	O
estimate	O
,	O
using	O
smoothing	O
methods	O
like	O
local	O
linear	O
smoothing	O
or	O
spline	B-Algorithm
smoothing	I-Algorithm
.	O
</s>
<s>
FPCA	O
can	O
be	O
applied	O
for	O
displaying	O
the	O
modes	B-Algorithm
of	I-Algorithm
functional	I-Algorithm
variation	I-Algorithm
,	O
in	O
scatterplots	O
of	O
FPCs	O
against	O
each	O
other	O
or	O
of	O
responses	O
against	O
FPCs	O
,	O
for	O
modeling	O
sparse	O
longitudinal	O
data	O
,	O
or	O
for	O
functional	O
regression	O
and	O
classification	O
,	O
e.g.	O
,	O
functional	O
linear	O
regression	O
.	O
</s>
<s>
Functional	B-General_Concept
Principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
has	O
varied	O
applications	O
in	O
time	O
series	O
analysis	O
.	O
</s>
<s>
Due	O
to	O
dimensionality	B-Algorithm
reduction	I-Algorithm
as	O
well	O
as	O
its	O
accuracy	O
to	O
represent	O
data	O
,	O
there	O
is	O
a	O
wide	O
scope	O
for	O
further	O
developments	O
of	O
functional	O
principal	B-Application
component	I-Application
techniques	O
in	O
the	O
financial	O
field	O
.	O
</s>
<s>
The	O
following	O
table	O
shows	O
a	O
comparison	O
of	O
various	O
elements	O
of	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
and	O
FPCA	O
.	O
</s>
<s>
The	O
two	O
methods	O
are	O
both	O
used	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
First	O
,	O
the	O
order	O
of	O
multivariate	O
data	O
in	O
PCA	O
can	O
be	O
permuted	B-Algorithm
,	O
which	O
has	O
no	O
effect	O
on	O
the	O
analysis	O
,	O
but	O
the	O
order	O
of	O
functional	B-General_Concept
data	I-General_Concept
carries	O
time	O
or	O
space	O
information	O
and	O
cannot	O
be	O
reordered	O
.	O
</s>
<s>
Third	O
,	O
regular	O
PCA	O
does	O
not	O
work	O
for	O
high-dimensional	O
data	O
without	O
regularization	O
,	O
while	O
FPCA	O
has	O
a	O
built-in	O
regularization	O
due	O
to	O
the	O
smoothness	O
of	O
the	O
functional	B-General_Concept
data	I-General_Concept
and	O
the	O
truncation	O
to	O
a	O
finite	O
number	O
of	O
included	O
components	O
.	O
</s>
