<s>
In	O
statistical	B-Application
graphics	I-Application
,	O
the	O
functional	B-Application
boxplot	I-Application
is	O
an	O
informative	O
exploratory	O
tool	O
that	O
has	O
been	O
proposed	O
for	O
visualizing	O
functional	O
data	O
.	O
</s>
<s>
Analogous	O
to	O
the	O
classical	O
boxplot	B-Application
,	O
the	O
descriptive	B-General_Concept
statistics	I-General_Concept
of	O
a	O
functional	B-Application
boxplot	I-Application
are	O
:	O
the	O
envelope	O
of	O
the	O
50%	O
central	O
region	O
,	O
the	O
median	O
curve	O
and	O
the	O
maximum	O
non-outlying	O
envelope	O
.	O
</s>
<s>
To	O
construct	O
a	O
functional	B-Application
boxplot	I-Application
,	O
data	O
ordering	O
is	O
the	O
first	O
step	O
.	O
</s>
<s>
In	O
functional	B-General_Concept
data	I-General_Concept
analysis	I-General_Concept
,	O
each	O
observation	O
is	O
a	O
real	O
function	O
,	O
therefore	O
,	O
different	O
from	O
the	O
classical	O
boxplot	B-Application
where	O
data	O
are	O
simply	O
ordered	O
from	O
the	O
smallest	O
sample	O
value	O
to	O
the	O
largest	O
,	O
in	O
a	O
functional	B-Application
boxplot	I-Application
,	O
functional	O
data	O
,	O
e.g.	O
</s>
<s>
Having	O
the	O
ranks	O
of	O
functional	O
data	O
,	O
the	O
functional	B-Application
boxplot	I-Application
is	O
a	O
natural	O
extension	O
of	O
the	O
classical	O
boxplot	B-Application
.	O
</s>
<s>
In	O
the	O
classical	O
boxplot	B-Application
,	O
the	O
box	O
itself	O
represents	O
the	O
middle	B-General_Concept
50%	I-General_Concept
of	O
the	O
data	O
.	O
</s>
<s>
Since	O
the	O
data	O
ordering	O
in	O
the	O
functional	B-Application
boxplot	I-Application
is	O
from	O
the	O
center	O
outwards	O
,	O
the	O
50%	O
central	O
region	O
is	O
defined	O
by	O
the	O
band	O
delimited	O
by	O
the	O
50%	O
of	O
deepest	O
,	O
or	O
the	O
most	O
central	O
observations	O
.	O
</s>
<s>
border	O
of	O
the	O
50%	O
central	O
region	O
is	O
defined	O
as	O
the	O
envelope	O
representing	O
the	O
box	O
in	O
a	O
classical	O
boxplot	B-Application
.	O
</s>
<s>
region	O
is	O
the	O
analog	O
to	O
the	O
"	O
interquartile	B-General_Concept
range	I-General_Concept
"	O
(	O
IQR	B-General_Concept
)	O
and	O
gives	O
a	O
useful	O
indication	O
of	O
the	O
spread	O
of	O
the	O
central	O
50%	O
of	O
the	O
curves	O
.	O
</s>
<s>
Outliers	O
can	O
be	O
detected	O
in	O
a	O
functional	B-Application
boxplot	I-Application
by	O
the	O
1.5	O
times	O
the	O
50%	O
central	O
region	O
empirical	O
rule	O
,	O
analogous	O
to	O
the	O
1.5	O
IQR	B-General_Concept
empirical	O
rule	O
for	O
classical	O
boxplots	B-Application
.	O
</s>
<s>
When	O
each	O
observation	O
is	O
simply	O
a	O
point	O
,	O
the	O
functional	B-Application
boxplot	I-Application
degenerates	O
to	O
a	O
classical	O
boxplot	B-Application
,	O
and	O
it	O
is	O
different	O
from	O
the	O
pointwise	O
boxplots	B-Application
.	O
</s>
<s>
By	O
introducing	O
the	O
concept	O
of	O
central	O
regions	O
,	O
the	O
functional	B-Application
boxplot	I-Application
can	O
be	O
generalized	O
to	O
an	O
enhanced	O
functional	B-Application
boxplot	I-Application
where	O
the	O
25%	O
and	O
75%	O
central	O
regions	O
are	O
provided	O
as	O
well	O
.	O
</s>
<s>
Spatio-temporal	B-General_Concept
data	I-General_Concept
can	O
be	O
viewed	O
as	O
a	O
temporal	O
curve	O
at	O
each	O
spatial	O
location	O
,	O
or	O
a	O
spatial	O
surface	O
at	O
each	O
time	O
.	O
</s>
<s>
In	O
the	O
latter	O
case	O
,	O
a	O
volume-based	O
surface	O
band	O
depth	O
can	O
be	O
used	O
to	O
order	O
sample	O
surfaces	O
and	O
leads	O
to	O
a	O
three-dimensional	O
surface	O
boxplot	B-Application
with	O
similar	O
characteristics	O
as	O
the	O
functional	B-Application
boxplots	I-Application
.	O
</s>
<s>
The	O
surface	O
boxplot	B-Application
is	O
a	O
natural	O
extension	O
of	O
the	O
functional	B-Application
boxplot	I-Application
to	O
R3	O
.	O
</s>
<s>
The	O
command	O
fbplot	O
for	O
functional	B-Application
boxplots	I-Application
is	O
in	O
fda	O
R	B-Language
package	O
,	O
and	O
MATLAB	B-Language
code	O
is	O
also	O
available	O
.	O
</s>
<s>
The	O
Python	O
library	O
statsmodels	B-Application
makes	O
functional	B-Application
boxplots	I-Application
available	O
via	O
the	O
fboxplot	O
function	O
.	O
</s>
<s>
One	O
could	O
also	O
use	O
the	O
boxplot	B-Application
function	O
in	O
package	O
.	O
</s>
