<s>
In	O
statistics	O
,	O
sufficient	B-Algorithm
dimension	I-Algorithm
reduction	I-Algorithm
(	O
SDR	O
)	O
is	O
a	O
paradigm	O
for	O
analyzing	O
data	O
that	O
combines	O
the	O
ideas	O
of	O
dimension	B-Algorithm
reduction	I-Algorithm
with	O
the	O
concept	O
of	O
sufficiency	O
.	O
</s>
<s>
Dimension	B-Algorithm
reduction	I-Algorithm
has	O
long	O
been	O
a	O
primary	O
goal	O
of	O
regression	O
analysis	O
.	O
</s>
<s>
A	O
dimension	B-Algorithm
reduction	I-Algorithm
is	O
a	O
function	O
that	O
maps	O
to	O
a	O
subset	O
of	O
,	O
k	O
<	O
p	O
,	O
thereby	O
reducing	O
the	O
dimension	O
of	O
.	O
</s>
<s>
A	O
dimension	B-Algorithm
reduction	I-Algorithm
is	O
said	O
to	O
be	O
sufficient	O
if	O
the	O
distribution	O
of	O
is	O
the	O
same	O
as	O
that	O
of	O
.	O
</s>
<s>
For	O
instance	O
,	O
one	O
may	O
consider	O
a	O
scatter	B-Application
plot	I-Application
of	O
versus	O
one	O
or	O
more	O
of	O
the	O
predictors	O
.	O
</s>
<s>
A	O
scatter	B-Application
plot	I-Application
that	O
contains	O
all	O
available	O
regression	O
information	O
is	O
called	O
a	O
sufficient	O
summary	O
plot	O
.	O
</s>
<s>
Even	O
three-dimensional	O
scatter	B-Application
plots	I-Application
must	O
be	O
viewed	O
via	O
a	O
computer	O
program	O
,	O
and	O
the	O
third	O
dimension	O
can	O
only	O
be	O
visualized	O
by	O
rotating	O
the	O
coordinate	O
axes	O
.	O
</s>
<s>
However	O
,	O
if	O
there	O
exists	O
a	O
sufficient	B-Algorithm
dimension	I-Algorithm
reduction	I-Algorithm
with	O
small	O
enough	O
dimension	O
,	O
a	O
sufficient	O
summary	O
plot	O
of	O
versus	O
may	O
be	O
constructed	O
and	O
visually	O
interpreted	O
with	O
relative	O
ease	O
.	O
</s>
<s>
Hence	O
sufficient	B-Algorithm
dimension	I-Algorithm
reduction	I-Algorithm
allows	O
for	O
graphical	O
intuition	O
about	O
the	O
distribution	O
of	O
,	O
which	O
might	O
not	O
have	O
otherwise	O
been	O
available	O
for	O
high-dimensional	O
data	O
.	O
</s>
<s>
Most	O
graphical	O
methodology	O
focuses	O
primarily	O
on	O
dimension	B-Algorithm
reduction	I-Algorithm
involving	O
linear	O
combinations	O
of	O
.	O
</s>
<s>
Suppose	O
is	O
a	O
sufficient	B-Algorithm
dimension	I-Algorithm
reduction	I-Algorithm
,	O
where	O
is	O
a	O
matrix	B-Architecture
with	O
rank	O
.	O
</s>
<s>
Then	O
the	O
subspace	O
is	O
defined	O
to	O
be	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
(	O
DRS	O
)	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
smallest	O
dimension	B-Algorithm
reduction	I-Algorithm
that	O
is	O
still	O
sufficient	O
maps	O
to	O
a	O
subset	O
of	O
.	O
</s>
<s>
If	O
a	O
subspace	O
is	O
a	O
DRS	O
for	O
,	O
and	O
if	O
for	O
all	O
other	O
DRSs	O
,	O
then	O
it	O
is	O
a	O
central	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
,	O
or	O
simply	O
a	O
central	O
subspace	O
,	O
and	O
it	O
is	O
denoted	O
by	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
central	O
subspace	O
for	O
exists	O
if	O
and	O
only	O
if	O
the	O
intersection	O
of	O
all	O
dimension	B-Algorithm
reduction	I-Algorithm
subspaces	O
is	O
also	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
,	O
and	O
that	O
intersection	O
is	O
the	O
central	O
subspace	O
.	O
</s>
<s>
However	O
,	O
if	O
does	O
exist	O
,	O
then	O
it	O
is	O
also	O
the	O
unique	O
minimum	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
.	O
</s>
<s>
Let	O
and	O
be	O
dimension	B-Algorithm
reduction	I-Algorithm
subspaces	O
for	O
.	O
</s>
<s>
If	O
has	O
density	O
for	O
all	O
and	O
everywhere	O
else	O
,	O
where	O
is	O
convex	O
,	O
then	O
the	O
intersection	O
is	O
also	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
.	O
</s>
<s>
There	O
are	O
many	O
existing	O
methods	O
for	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
both	O
graphical	O
and	O
numeric	O
.	O
</s>
<s>
For	O
example	O
,	O
sliced	B-Algorithm
inverse	I-Algorithm
regression	I-Algorithm
(	O
SIR	O
)	O
and	O
sliced	O
average	O
variance	O
estimation	O
(	O
SAVE	O
)	O
were	O
introduced	O
in	O
the	O
1990s	O
and	O
continue	O
to	O
be	O
widely	O
used	O
.	O
</s>
<s>
More	O
recent	O
methods	O
for	O
dimension	B-Algorithm
reduction	I-Algorithm
include	O
likelihood-based	O
sufficient	B-Algorithm
dimension	I-Algorithm
reduction	I-Algorithm
,	O
estimating	O
the	O
central	O
subspace	O
based	O
on	O
the	O
inverse	O
third	O
moment	O
(	O
or	O
kth	O
moment	O
)	O
,	O
estimating	O
the	O
central	O
solution	O
space	O
,	O
graphical	O
regression	O
,	O
envelope	O
model	O
,	O
and	O
the	O
principal	O
support	O
vector	O
machine	O
.	O
</s>
<s>
Principal	B-Application
components	I-Application
analysis	I-Application
(	O
PCA	O
)	O
and	O
similar	O
methods	O
for	O
dimension	B-Algorithm
reduction	I-Algorithm
are	O
not	O
based	O
on	O
the	O
sufficiency	O
principle	O
.	O
</s>
<s>
Hence	O
,	O
the	O
span	O
of	O
is	O
a	O
dimension	B-Algorithm
reduction	I-Algorithm
subspace	O
.	O
</s>
<s>
The	O
OLS	B-General_Concept
estimate	O
of	O
is	O
consistent	O
,	O
and	O
so	O
the	O
span	O
of	O
is	O
a	O
consistent	O
estimator	O
of	O
.	O
</s>
