<s>
In	O
statistics	O
,	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
are	O
a	O
continuously	O
indexed	O
set	O
of	O
vectors	O
or	O
functions	O
that	O
are	O
centered	O
at	O
a	O
mean	O
and	O
are	O
used	O
to	O
depict	O
the	O
variation	O
in	O
a	O
population	O
or	O
sample	O
.	O
</s>
<s>
Typically	O
,	O
variation	O
patterns	O
in	O
the	O
data	O
can	O
be	O
decomposed	O
in	O
descending	O
order	O
of	O
eigenvalues	O
with	O
the	O
directions	O
represented	O
by	O
the	O
corresponding	O
eigenvectors	O
or	O
eigenfunctions	B-Algorithm
.	O
</s>
<s>
Modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
provide	O
a	O
visualization	O
of	O
this	O
decomposition	O
and	O
an	O
efficient	O
description	O
of	O
variation	O
around	O
the	O
mean	O
.	O
</s>
<s>
Both	O
in	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	B-Application
)	O
and	O
in	O
functional	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
(	O
FPCA	B-General_Concept
)	O
,	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
play	O
an	O
important	O
role	O
in	O
visualizing	O
and	O
describing	O
the	O
variation	O
in	O
the	O
data	O
contributed	O
by	O
each	O
eigencomponent	O
.	O
</s>
<s>
In	O
real-world	O
applications	O
,	O
the	O
eigencomponents	O
and	O
associated	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
aid	O
to	O
interpret	O
complex	O
data	O
,	O
especially	O
in	O
exploratory	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
(	O
EDA	O
)	O
.	O
</s>
<s>
Modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
are	O
a	O
natural	O
extension	O
of	O
PCA	B-Application
and	O
FPCA	B-General_Concept
.	O
</s>
<s>
where	O
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
whose	O
columns	O
are	O
the	O
eigenvectors	O
of	O
,	O
and	O
is	O
a	O
diagonal	O
matrix	O
whose	O
entries	O
are	O
the	O
eigenvalues	O
of	O
.	O
</s>
<s>
Consider	O
realizations	O
of	O
a	O
square-integrable	B-Algorithm
random	O
function	O
with	O
the	O
mean	O
function	O
and	O
the	O
covariance	O
function	O
.	O
</s>
<s>
Functional	B-General_Concept
principal	I-General_Concept
component	I-General_Concept
analysis	I-General_Concept
provides	O
methods	O
for	O
the	O
estimation	O
of	O
and	O
in	O
detail	O
,	O
often	O
involving	O
point	O
wise	O
estimate	O
and	O
interpolation	B-Algorithm
.	O
</s>
<s>
Modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
are	O
useful	O
to	O
visualize	O
and	O
describe	O
the	O
variation	O
patterns	O
in	O
the	O
data	O
sorted	O
by	O
the	O
eigenvalues	O
.	O
</s>
<s>
In	O
real-world	O
applications	O
,	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
associated	O
with	O
eigencomponents	O
allow	O
to	O
interpret	O
complex	O
data	O
,	O
such	O
as	O
the	O
evolution	O
of	O
function	O
traits	O
and	O
other	O
infinite-dimensional	O
data	O
.	O
</s>
<s>
To	O
illustrate	O
how	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
work	O
in	O
practice	O
,	O
two	O
examples	O
are	O
shown	O
in	O
the	O
graphs	O
to	O
the	O
right	O
,	O
which	O
display	O
the	O
first	O
two	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
.	O
</s>
<s>
The	O
dashed	O
,	O
dot-dashed	O
,	O
and	O
dotted	O
curves	O
correspond	O
to	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
with	O
and	O
,	O
respectively	O
.	O
</s>
<s>
The	O
first	O
graph	O
displays	O
the	O
first	O
two	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
of	O
female	O
mortality	O
data	O
from	O
41	O
countries	O
in	O
2003	O
.	O
</s>
<s>
Compared	O
to	O
female	O
mortality	O
data	O
,	O
modes	B-Algorithm
of	I-Algorithm
variation	I-Algorithm
of	O
male	O
mortality	O
data	O
shows	O
higher	O
mortality	O
after	O
around	O
age	O
20	O
,	O
possibly	O
related	O
to	O
the	O
fact	O
that	O
life	O
expectancy	O
for	O
women	O
is	O
higher	O
than	O
that	O
for	O
men	O
.	O
</s>
