<s>
In	O
statistics	O
and	O
in	O
machine	O
learning	O
,	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
is	O
a	O
linear	O
function	O
(	O
linear	O
combination	O
)	O
of	O
a	O
set	O
of	O
coefficients	O
and	O
explanatory	O
variables	O
(	O
independent	O
variables	O
)	O
,	O
whose	O
value	O
is	O
used	O
to	O
predict	O
the	O
outcome	O
of	O
a	O
dependent	O
variable	O
.	O
</s>
<s>
This	O
sort	O
of	O
function	O
usually	O
comes	O
in	O
linear	B-General_Concept
regression	I-General_Concept
,	O
where	O
the	O
coefficients	O
are	O
called	O
regression	B-General_Concept
coefficients	I-General_Concept
.	O
</s>
<s>
However	O
,	O
they	O
also	O
occur	O
in	O
various	O
types	O
of	O
linear	B-General_Concept
classifiers	I-General_Concept
(	O
e.g.	O
</s>
<s>
logistic	O
regression	O
,	O
perceptrons	B-Algorithm
,	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
,	O
and	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
)	O
,	O
as	O
well	O
as	O
in	O
various	O
other	O
models	O
,	O
such	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
and	O
factor	O
analysis	O
.	O
</s>
<s>
where	O
,	O
for	O
k	O
=	O
1	O
,	O
...	O
,	O
p	O
,	O
is	O
the	O
value	O
of	O
the	O
k-th	O
explanatory	O
variable	O
for	O
data	O
point	O
i	O
,	O
and	O
are	O
the	O
coefficients	O
(	O
regression	B-General_Concept
coefficients	I-General_Concept
,	O
weights	O
,	O
etc	O
.	O
)	O
</s>
<s>
For	O
each	O
data	O
point	O
i	O
,	O
an	O
additional	O
explanatory	O
pseudo-variable	O
xi0	O
is	O
added	O
,	O
with	O
a	O
fixed	O
value	O
of	O
1	O
,	O
corresponding	O
to	O
the	O
intercept	B-Algorithm
coefficient	O
β0	O
.	O
</s>
<s>
This	O
makes	O
it	O
possible	O
to	O
write	O
the	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
as	O
follows	O
:	O
</s>
<s>
where	O
is	O
a	O
disturbance	B-General_Concept
term	I-General_Concept
or	O
error	B-General_Concept
variable	I-General_Concept
—	O
an	O
unobserved	O
random	O
variable	O
that	O
adds	O
noise	O
to	O
the	O
linear	O
relationship	O
between	O
the	O
dependent	O
variable	O
and	O
predictor	O
function	O
.	O
</s>
<s>
The	O
matrix	O
X	O
is	O
known	O
as	O
the	O
design	B-Algorithm
matrix	I-Algorithm
and	O
encodes	O
all	O
known	O
information	O
about	O
the	O
independent	O
variables	O
.	O
</s>
<s>
The	O
variables	O
are	O
random	O
variables	O
,	O
which	O
in	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
are	O
distributed	O
according	O
to	O
a	O
standard	O
normal	O
distribution	O
;	O
they	O
express	O
the	O
influence	O
of	O
any	O
unknown	O
factors	O
on	O
the	O
outcome	O
.	O
</s>
<s>
This	O
makes	O
it	O
possible	O
to	O
find	O
optimal	O
coefficients	O
through	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
using	O
simple	O
matrix	O
operations	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
optimal	O
coefficients	O
as	O
estimated	O
by	O
least	B-Algorithm
squares	I-Algorithm
can	O
be	O
written	O
as	O
follows	O
:	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
data	B-General_Concept
analyst	I-General_Concept
is	O
free	O
to	O
transform	O
the	O
explanatory	O
variables	O
in	O
arbitrary	O
ways	O
,	O
including	O
creating	O
multiple	O
copies	O
of	O
a	O
given	O
explanatory	O
variable	O
,	O
each	O
transformed	O
using	O
a	O
different	O
function	O
.	O
</s>
<s>
An	O
example	O
is	O
polynomial	O
regression	O
,	O
which	O
uses	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
to	O
fit	O
an	O
arbitrary	O
degree	O
polynomial	O
relationship	O
(	O
up	O
to	O
a	O
given	O
order	O
)	O
between	O
two	O
sets	O
of	O
data	O
points	O
(	O
i.e.	O
</s>
<s>
and	O
then	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
is	O
run	O
.	O
</s>
<s>
This	O
example	O
shows	O
that	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
can	O
actually	O
be	O
much	O
more	O
powerful	O
than	O
it	O
first	O
appears	O
:	O
It	O
only	O
really	O
needs	O
to	O
be	O
linear	O
in	O
the	O
coefficients	O
.	O
</s>
<s>
An	O
example	O
of	O
this	O
is	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
(	O
RBF	O
's	O
)	O
,	O
which	O
compute	O
some	O
transformed	O
version	O
of	O
the	O
distance	O
to	O
some	O
fixed	O
point	O
:	O
</s>
<s>
That	O
is	O
,	O
the	O
application	O
of	O
the	O
radial	B-Algorithm
basis	I-Algorithm
functions	I-Algorithm
will	O
pick	O
out	O
the	O
nearest	O
point	O
,	O
and	O
its	O
regression	B-General_Concept
coefficient	I-General_Concept
will	O
dominate	O
.	O
</s>
<s>
The	O
result	O
will	O
be	O
a	O
form	O
of	O
nearest	B-General_Concept
neighbor	I-General_Concept
interpolation	O
,	O
where	O
predictions	O
are	O
made	O
by	O
simply	O
using	O
the	O
prediction	O
of	O
the	O
nearest	O
observed	O
data	O
point	O
,	O
possibly	O
interpolating	O
between	O
multiple	O
nearby	O
data	O
points	O
when	O
they	O
are	O
all	O
similar	O
distances	O
away	O
.	O
</s>
<s>
This	O
type	O
of	O
nearest	B-General_Concept
neighbor	I-General_Concept
method	I-General_Concept
for	O
prediction	O
is	O
often	O
considered	O
diametrically	O
opposed	O
to	O
the	O
type	O
of	O
prediction	O
used	O
in	O
standard	O
linear	B-General_Concept
regression	I-General_Concept
:	O
But	O
in	O
fact	O
,	O
the	O
transformations	O
that	O
can	O
be	O
applied	O
to	O
the	O
explanatory	O
variables	O
in	O
a	O
linear	B-General_Concept
predictor	I-General_Concept
function	I-General_Concept
are	O
so	O
powerful	O
that	O
even	O
the	O
nearest	B-General_Concept
neighbor	I-General_Concept
method	I-General_Concept
can	O
be	O
implemented	O
as	O
a	O
type	O
of	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
Linear	B-General_Concept
regression	I-General_Concept
and	O
similar	O
techniques	O
could	O
be	O
applied	O
and	O
will	O
often	O
still	O
find	O
the	O
optimal	O
coefficients	O
,	O
but	O
their	O
error	O
estimates	O
and	O
such	O
will	O
be	O
wrong	O
.	O
</s>
<s>
This	O
allows	O
for	O
separate	O
regression	B-General_Concept
coefficients	I-General_Concept
to	O
be	O
matched	O
for	O
each	O
possible	O
value	O
of	O
the	O
discrete	O
variable	O
.	O
</s>
<s>
This	O
causes	O
problems	O
for	O
a	O
number	O
of	O
methods	O
,	O
such	O
as	O
the	O
simple	O
closed-form	O
solution	O
used	O
in	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
