<s>
Functional	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
(	O
FDA	O
)	O
is	O
a	O
branch	O
of	O
statistics	O
that	O
analyses	O
data	O
providing	O
information	O
about	O
curves	O
,	O
surfaces	O
or	O
anything	O
else	O
varying	O
over	O
a	O
continuum	O
.	O
</s>
<s>
Functional	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
has	O
roots	O
going	O
back	O
to	O
work	O
by	O
Grenander	O
and	O
Karhunen	O
in	O
the	O
1940s	O
and	O
1950s	O
.	O
</s>
<s>
They	O
considered	O
the	O
decomposition	O
of	O
square-integrable	B-Algorithm
continuous	O
time	O
stochastic	O
process	O
into	O
eigencomponents	O
,	O
now	O
known	O
as	O
the	O
Karhunen-Loève	O
decomposition	O
.	O
</s>
<s>
The	O
term	O
"	O
Functional	B-General_Concept
Data	I-General_Concept
Analysis	I-General_Concept
"	O
was	O
coined	O
by	O
James	O
O	O
.	O
Ramsay	O
.	O
</s>
<s>
In	O
the	O
Hilbert	O
space	O
viewpoint	O
,	O
one	O
considers	O
an	O
-valued	O
random	O
element	O
,	O
where	O
is	O
a	O
separable	O
Hilbert	O
space	O
such	O
as	O
the	O
space	O
of	O
square-integrable	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
This	O
formulation	O
is	O
the	O
Pettis	B-Algorithm
integral	I-Algorithm
but	O
the	O
mean	O
can	O
also	O
be	O
defined	O
as	O
Bochner	O
integral	O
.	O
</s>
<s>
Truncating	O
this	O
infinite	O
series	O
to	O
a	O
finite	O
order	O
underpins	O
functional	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Functional	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
(	O
FPCA	O
)	O
is	O
the	O
most	O
prevalent	O
tool	O
in	O
FDA	O
,	O
partly	O
because	O
FPCA	O
facilitates	O
dimension	B-Algorithm
reduction	I-Algorithm
of	O
the	O
inherently	O
infinite-dimensional	O
functional	O
data	O
to	O
finite-dimensional	O
random	O
vector	O
of	O
scores	O
.	O
</s>
<s>
More	O
specifically	O
,	O
dimension	B-Algorithm
reduction	I-Algorithm
is	O
achieved	O
by	O
expanding	O
the	O
underlying	O
observed	O
random	O
trajectories	O
in	O
a	O
functional	O
basis	O
consisting	O
of	O
the	O
eigenfunctions	O
of	O
the	O
covariance	O
operator	O
on	O
.	O
</s>
<s>
The	O
Karhunen	O
–	O
Loève	O
expansion	O
facilitates	O
dimension	B-Algorithm
reduction	I-Algorithm
in	O
the	O
sense	O
that	O
the	O
partial	O
sum	O
converges	O
uniformly	O
,	O
i.e.	O
,	O
as	O
and	O
thus	O
the	O
partial	O
sum	O
with	O
a	O
large	O
enough	O
yields	O
a	O
good	O
approximation	O
to	O
the	O
infinite	O
sum	O
.	O
</s>
<s>
Thereby	O
,	O
the	O
information	O
in	O
is	O
reduced	O
from	O
infinite	O
dimensional	O
to	O
a	O
-dimensional	O
vector	O
with	O
the	O
approximated	O
process：	O
Other	O
popular	O
bases	O
include	O
spline	B-Algorithm
,	O
Fourier	O
series	O
and	O
wavelet	O
bases	O
.	O
</s>
<s>
Important	O
applications	O
of	O
FPCA	O
include	O
the	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
and	O
functional	O
principal	O
component	O
regression	O
.	O
</s>
<s>
Functional	O
linear	O
models	O
can	O
be	O
viewed	O
as	O
an	O
extension	O
of	O
the	O
traditional	O
multivariate	B-General_Concept
linear	I-General_Concept
models	I-General_Concept
that	O
associates	O
vector	O
responses	O
with	O
vector	O
covariates	O
.	O
</s>
<s>
The	O
traditional	O
linear	O
model	O
with	O
scalar	O
response	O
and	O
vector	O
covariate	O
can	O
be	O
expressed	O
aswhere	O
denotes	O
the	O
inner	O
product	O
in	O
Euclidean	O
space	O
,	O
and	O
denote	O
the	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
and	O
is	O
a	O
zero	O
mean	O
finite	O
variance	O
random	O
error	O
(	O
noise	O
)	O
.	O
</s>
<s>
Replacing	O
the	O
vector	O
covariate	O
and	O
the	O
coefficient	O
vector	O
in	O
model	O
(	O
)	O
by	O
a	O
centered	O
functional	O
covariate	O
and	O
coefficient	O
function	O
for	O
and	O
replacing	O
the	O
inner	O
product	O
in	O
Euclidean	O
space	O
by	O
that	O
in	O
Hilbert	O
space	O
,	O
one	O
arrives	O
at	O
the	O
functional	O
linear	O
modelThe	O
simple	O
functional	O
linear	O
model	O
(	O
)	O
can	O
be	O
extended	O
to	O
multiple	O
functional	O
covariates	O
,	O
,	O
also	O
including	O
additional	O
vector	O
covariates	O
,	O
where	O
,	O
bywhere	O
is	O
regression	B-General_Concept
coefficient	I-General_Concept
for	O
,	O
the	O
domain	O
of	O
is	O
,	O
is	O
the	O
centered	O
functional	O
covariate	O
given	O
by	O
,	O
and	O
is	O
regression	B-General_Concept
coefficient	I-General_Concept
function	O
for	O
,	O
for	O
.	O
</s>
<s>
Direct	O
nonlinear	O
extensions	O
of	O
the	O
classical	O
functional	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
(	O
FLMs	O
)	O
still	O
involve	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
,	O
but	O
combine	O
it	O
with	O
a	O
nonlinear	O
link	O
function	O
,	O
analogous	O
to	O
the	O
idea	O
of	O
generalized	O
linear	O
model	O
from	O
the	O
conventional	O
linear	O
model	O
.	O
</s>
<s>
Developments	O
towards	O
fully	O
nonparametric	O
regression	O
models	O
for	O
functional	O
data	O
encounter	O
problems	O
such	O
as	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Functional	O
polynomial	O
regression	O
models	O
may	O
be	O
viewed	O
as	O
a	O
natural	O
extension	O
of	O
the	O
Functional	O
Linear	O
Models	O
(	O
FLMs	O
)	O
with	O
scalar	O
responses	O
,	O
analogous	O
to	O
extending	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
to	O
polynomial	O
regression	O
model	O
.	O
</s>
<s>
However	O
,	O
for	O
,	O
this	O
model	O
is	O
problematic	O
due	O
to	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
A	O
functional	O
linear	O
model	O
with	O
scalar	O
responses	O
(	O
see	O
(	O
)	O
)	O
can	O
thus	O
be	O
written	O
as	O
follows	O
,	O
One	O
form	O
of	O
FAMs	O
is	O
obtained	O
by	O
replacing	O
the	O
linear	O
function	O
of	O
in	O
the	O
above	O
expression	O
(	O
i.e.	O
,	O
)	O
by	O
a	O
general	O
smooth	O
function	O
,	O
analogous	O
to	O
the	O
extension	O
of	O
multiple	O
linear	B-General_Concept
regression	I-General_Concept
models	I-General_Concept
to	O
additive	O
models	O
and	O
is	O
expressed	O
as	O
,	O
where	O
satisfies	O
for	O
.	O
</s>
<s>
Link	O
function	O
connecting	O
the	O
conditional	O
mean	O
and	O
the	O
linear	B-General_Concept
predictor	I-General_Concept
through	O
.	O
</s>
<s>
For	O
vector-valued	O
multivariate	O
data	O
,	O
k-means	B-Algorithm
partitioning	I-Algorithm
methods	I-Algorithm
and	O
hierarchical	B-Algorithm
clustering	I-Algorithm
are	O
two	O
main	O
approaches	O
.	O
</s>
<s>
For	O
clustering	O
of	O
functional	O
data	O
,	O
k-means	B-Algorithm
clustering	I-Algorithm
methods	O
are	O
more	O
popular	O
than	O
hierarchical	B-Algorithm
clustering	I-Algorithm
methods	O
.	O
</s>
<s>
For	O
k-means	B-Algorithm
clustering	I-Algorithm
on	O
functional	O
data	O
,	O
mean	O
functions	O
are	O
usually	O
regarded	O
as	O
the	O
cluster	O
centers	O
.	O
</s>
<s>
Besides	O
k-means	B-Algorithm
type	O
clustering	O
,	O
functional	O
clustering	O
based	O
on	O
mixture	O
models	O
is	O
also	O
widely	O
used	O
in	O
clustering	O
vector-valued	O
multivariate	O
data	O
and	O
has	O
been	O
extended	O
to	O
functional	O
data	O
clustering	O
.	O
</s>
<s>
Furthermore	O
,	O
Bayesian	O
hierarchical	B-Algorithm
clustering	I-Algorithm
also	O
plays	O
an	O
important	O
role	O
in	O
the	O
development	O
of	O
model-based	O
functional	O
clustering	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
generalized	O
functional	O
linear	B-General_Concept
regression	I-General_Concept
model	I-General_Concept
based	O
on	O
the	O
FPCA	B-General_Concept
approach	I-General_Concept
is	O
used	O
.	O
</s>
<s>
The	O
simplest	O
case	O
of	O
a	O
family	O
of	O
warping	O
functions	O
to	O
specify	O
phase	O
variation	O
is	O
linear	B-Architecture
transformation	I-Architecture
,	O
that	O
is	O
,	O
which	O
warps	O
the	O
time	O
of	O
an	O
underlying	O
template	O
function	O
by	O
subjected-specific	O
shift	O
and	O
scale	O
.	O
</s>
<s>
The	O
set	O
of	O
linear	B-Architecture
transformation	I-Architecture
is	O
contained	O
in	O
the	O
set	O
of	O
diffeomorphisms	O
.	O
</s>
<s>
Earlier	O
approaches	O
include	O
dynamic	B-Algorithm
time	I-Algorithm
warping	I-Algorithm
(	O
DTW	O
)	O
used	O
for	O
applications	O
such	O
as	O
speech	B-Application
recognition	I-Application
.	O
</s>
<s>
There	O
are	O
Python	B-Language
packages	O
to	O
work	O
with	O
functional	O
data	O
,	O
and	O
its	O
representation	O
,	O
perform	O
exploratory	O
analysis	O
,	O
or	O
preprocessing	O
,	O
and	O
among	O
other	O
tasks	O
such	O
as	O
inference	O
,	O
classification	O
,	O
regression	O
or	O
clustering	O
of	O
functional	O
data	O
.	O
</s>
