<s>
In	O
statistics	O
,	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
,	O
sometimes	O
also	O
called	O
the	O
influence	O
matrix	O
or	O
hat	B-Algorithm
matrix	I-Algorithm
,	O
maps	O
the	O
vector	O
of	O
response	O
values	O
(	O
dependent	O
variable	O
values	O
)	O
to	O
the	O
vector	O
of	O
fitted	O
values	O
(	O
or	O
predicted	O
values	O
)	O
.	O
</s>
<s>
The	O
diagonal	O
elements	O
of	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
are	O
the	O
leverages	O
,	O
which	O
describe	O
the	O
influence	O
each	O
response	O
value	O
has	O
on	O
the	O
fitted	O
value	O
for	O
that	O
same	O
observation	O
.	O
</s>
<s>
As	O
is	O
usually	O
pronounced	O
"	O
y-hat	O
"	O
,	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
is	O
also	O
named	O
hat	B-Algorithm
matrix	I-Algorithm
as	O
it	O
"	O
puts	O
a	O
hat	O
on	O
"	O
.	O
</s>
<s>
The	O
formula	O
for	O
the	O
vector	O
of	O
residuals	O
can	O
also	O
be	O
expressed	O
compactly	O
using	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
the	O
case	O
of	O
linear	B-Algorithm
models	I-Algorithm
with	O
independent	O
and	O
identically	O
distributed	O
errors	O
in	O
which	O
,	O
this	O
reduces	O
to	O
:	O
</s>
<s>
Suppose	O
that	O
we	O
wish	O
to	O
estimate	O
a	O
linear	B-Algorithm
model	I-Algorithm
using	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
where	O
is	O
a	O
matrix	O
of	O
explanatory	O
variables	O
(	O
the	O
design	B-Algorithm
matrix	I-Algorithm
)	O
,	O
β	O
is	O
a	O
vector	O
of	O
unknown	O
parameters	O
to	O
be	O
estimated	O
,	O
and	O
ε	O
is	O
the	O
error	O
vector	O
.	O
</s>
<s>
A	O
few	O
examples	O
are	O
linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
smoothing	B-Algorithm
splines	I-Algorithm
,	O
regression	O
splines	O
,	O
local	O
regression	O
,	O
kernel	B-Language
regression	I-Language
,	O
and	O
linear	O
filtering	O
.	O
</s>
<s>
and	O
again	O
it	O
may	O
be	O
seen	O
that	O
,	O
though	O
now	O
it	O
is	O
no	O
longer	O
symmetric	B-Algorithm
.	O
</s>
<s>
The	O
projection	B-Algorithm
matrix	I-Algorithm
has	O
a	O
number	O
of	O
useful	O
algebraic	O
properties	O
.	O
</s>
<s>
In	O
the	O
language	O
of	O
linear	B-Language
algebra	I-Language
,	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
is	O
the	O
orthogonal	O
projection	B-Algorithm
onto	O
the	O
column	O
space	O
of	O
the	O
design	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Some	O
facts	O
of	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
in	O
this	O
setting	O
are	O
summarized	O
as	O
follows	O
:	O
</s>
<s>
is	O
symmetric	B-Algorithm
,	O
and	O
so	O
is	O
.	O
</s>
<s>
The	O
projection	B-Algorithm
matrix	I-Algorithm
corresponding	O
to	O
a	O
linear	B-Algorithm
model	I-Algorithm
is	O
symmetric	B-Algorithm
and	O
idempotent	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
not	O
always	O
the	O
case	O
;	O
in	O
locally	O
weighted	O
scatterplot	O
smoothing	O
(	O
LOESS	O
)	O
,	O
for	O
example	O
,	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
is	O
in	O
general	O
neither	O
symmetric	B-Algorithm
nor	O
idempotent	O
.	O
</s>
<s>
For	O
linear	B-Algorithm
models	I-Algorithm
,	O
the	O
trace	O
of	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
is	O
equal	O
to	O
the	O
rank	O
of	O
,	O
which	O
is	O
the	O
number	O
of	O
independent	O
parameters	O
of	O
the	O
linear	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
For	O
other	O
models	O
such	O
as	O
LOESS	O
that	O
are	O
still	O
linear	O
in	O
the	O
observations	O
,	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
can	O
be	O
used	O
to	O
define	O
the	O
effective	O
degrees	O
of	O
freedom	O
of	O
the	O
model	O
.	O
</s>
<s>
Practical	O
applications	O
of	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
in	O
regression	O
analysis	O
include	O
leverage	O
and	O
Cook	O
's	O
distance	O
,	O
which	O
are	O
concerned	O
with	O
identifying	O
influential	O
observations	O
,	O
i.e.	O
</s>
<s>
Suppose	O
the	O
design	B-Algorithm
matrix	I-Algorithm
can	O
be	O
decomposed	O
by	O
columns	O
as	O
.	O
</s>
<s>
Define	O
the	O
hat	O
or	O
projection	B-Algorithm
operator	I-Algorithm
as	O
.	O
</s>
<s>
Then	O
the	O
projection	B-Algorithm
matrix	I-Algorithm
can	O
be	O
decomposed	O
as	O
follows	O
:	O
</s>
<s>
Another	O
use	O
is	O
in	O
the	O
fixed	B-General_Concept
effects	I-General_Concept
model	I-General_Concept
,	O
where	O
is	O
a	O
large	O
sparse	B-Algorithm
matrix	I-Algorithm
of	O
the	O
dummy	O
variables	O
for	O
the	O
fixed	B-General_Concept
effect	I-General_Concept
terms	O
.	O
</s>
<s>
One	O
can	O
use	O
this	O
partition	O
to	O
compute	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
of	O
without	O
explicitly	O
forming	O
the	O
matrix	O
,	O
which	O
might	O
be	O
too	O
large	O
to	O
fit	O
into	O
computer	O
memory	O
.	O
</s>
