<s>
The	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
is	O
a	O
standard	O
approach	O
in	O
regression	O
analysis	O
to	O
approximate	O
the	O
solution	O
of	O
overdetermined	O
systems	O
(	O
sets	O
of	O
equations	O
in	O
which	O
there	O
are	O
more	O
equations	O
than	O
unknowns	O
)	O
by	O
minimizing	O
the	O
sum	O
of	O
the	O
squares	O
of	O
the	O
residuals	O
(	O
a	O
residual	O
being	O
the	O
difference	O
between	O
an	O
observed	O
value	O
and	O
the	O
fitted	O
value	O
provided	O
by	O
a	O
model	O
)	O
made	O
in	O
the	O
results	O
of	O
each	O
individual	O
equation	O
.	O
</s>
<s>
The	O
most	O
important	O
application	O
is	O
in	O
data	B-Algorithm
fitting	I-Algorithm
.	O
</s>
<s>
When	O
the	O
problem	O
has	O
substantial	O
uncertainties	O
in	O
the	O
independent	O
variable	O
(	O
the	O
x	O
variable	O
)	O
,	O
then	O
simple	O
regression	O
and	O
least-squares	B-Algorithm
methods	I-Algorithm
have	O
problems	O
;	O
in	O
such	O
cases	O
,	O
the	O
methodology	O
required	O
for	O
fitting	O
errors-in-variables	B-Algorithm
models	I-Algorithm
may	O
be	O
considered	O
instead	O
of	O
that	O
for	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Least	B-Algorithm
squares	I-Algorithm
problems	I-Algorithm
fall	O
into	O
two	O
categories	O
:	O
linear	O
or	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
and	O
nonlinear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
,	O
depending	O
on	O
whether	O
or	O
not	O
the	O
residuals	O
are	O
linear	O
in	O
all	O
unknowns	O
.	O
</s>
<s>
The	O
linear	O
least-squares	B-Algorithm
problem	I-Algorithm
occurs	O
in	O
statistical	O
regression	O
analysis	O
;	O
it	O
has	O
a	O
closed-form	O
solution	O
.	O
</s>
<s>
Polynomial	O
least	B-Algorithm
squares	I-Algorithm
describes	O
the	O
variance	O
in	O
a	O
prediction	O
of	O
the	O
dependent	O
variable	O
as	O
a	O
function	O
of	O
the	O
independent	O
variable	O
and	O
the	O
deviations	O
from	O
the	O
fitted	O
curve	O
.	O
</s>
<s>
for	O
normal	O
,	O
exponential	O
,	O
Poisson	O
and	O
binomial	O
distributions	O
)	O
,	O
standardized	O
least-squares	B-Algorithm
estimates	O
and	O
maximum-likelihood	O
estimates	O
are	O
identical	O
.	O
</s>
<s>
The	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
can	O
also	O
be	O
derived	O
as	O
a	O
method	O
of	O
moments	O
estimator	O
.	O
</s>
<s>
The	O
following	O
discussion	O
is	O
mostly	O
presented	O
in	O
terms	O
of	O
linear	O
functions	O
but	O
the	O
use	O
of	O
least	B-Algorithm
squares	I-Algorithm
is	O
valid	O
and	O
practical	O
for	O
more	O
general	O
families	O
of	O
functions	O
.	O
</s>
<s>
Also	O
,	O
by	O
iteratively	O
applying	O
local	O
quadratic	O
approximation	O
to	O
the	O
likelihood	O
(	O
through	O
the	O
Fisher	O
information	O
)	O
,	O
the	O
least-squares	B-Algorithm
method	I-Algorithm
may	O
be	O
used	O
to	O
fit	O
a	O
generalized	O
linear	O
model	O
.	O
</s>
<s>
The	O
least-squares	B-Algorithm
method	I-Algorithm
was	O
officially	O
discovered	O
and	O
published	O
by	O
Adrien-Marie	O
Legendre	O
(	O
1805	O
)	O
,	O
though	O
it	O
is	O
usually	O
also	O
co-credited	O
to	O
Carl	O
Friedrich	O
Gauss	O
(	O
1795	O
)	O
who	O
contributed	O
significant	O
theoretical	O
advances	O
to	O
the	O
method	O
and	O
may	O
have	O
previously	O
used	O
it	O
in	O
his	O
work	O
.	O
</s>
<s>
The	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
grew	O
out	O
of	O
the	O
fields	O
of	O
astronomy	O
and	O
geodesy	O
,	O
as	O
scientists	O
and	O
mathematicians	O
sought	O
to	O
provide	O
solutions	O
to	O
the	O
challenges	O
of	O
navigating	O
the	O
Earth	O
's	O
oceans	O
during	O
the	O
Age	O
of	O
Discovery	O
.	O
</s>
<s>
This	O
approach	O
was	O
notably	O
used	O
by	O
Tobias	O
Mayer	O
while	O
studying	O
the	O
librations	O
of	O
the	O
moon	O
in	O
1750	O
,	O
and	O
by	O
Pierre-Simon	O
Laplace	O
in	O
his	O
work	O
in	O
explaining	O
the	O
differences	O
in	O
motion	O
of	O
Jupiter	O
and	O
Saturn	B-Device
in	O
1788	O
.	O
</s>
<s>
The	O
method	O
came	O
to	O
be	O
known	O
as	O
the	O
method	O
of	O
least	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
The	O
first	O
clear	O
and	O
concise	O
exposition	O
of	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
was	O
published	O
by	O
Legendre	O
in	O
1805	O
.	O
</s>
<s>
Within	O
ten	O
years	O
after	O
Legendre	O
's	O
publication	O
,	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
had	O
been	O
adopted	O
as	O
a	O
standard	O
tool	O
in	O
astronomy	O
and	O
geodesy	O
in	O
France	O
,	O
Italy	O
,	O
and	O
Prussia	O
,	O
which	O
constitutes	O
an	O
extraordinarily	O
rapid	O
acceptance	O
of	O
a	O
scientific	O
technique	O
.	O
</s>
<s>
In	O
that	O
work	O
he	O
claimed	O
to	O
have	O
been	O
in	O
possession	O
of	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
since	O
1795	O
.	O
</s>
<s>
However	O
,	O
to	O
Gauss	O
's	O
credit	O
,	O
he	O
went	O
beyond	O
Legendre	O
and	O
succeeded	O
in	O
connecting	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
with	O
the	O
principles	O
of	O
probability	O
and	O
to	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
The	O
only	O
predictions	O
that	O
successfully	O
allowed	O
Hungarian	O
astronomer	O
Franz	O
Xaver	O
von	O
Zach	O
to	O
relocate	O
Ceres	O
were	O
those	O
performed	O
by	O
the	O
24-year-old	O
Gauss	O
using	O
least-squares	B-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
In	O
1810	O
,	O
after	O
reading	O
Gauss	O
's	O
work	O
,	O
Laplace	O
,	O
after	O
proving	O
the	O
central	O
limit	O
theorem	O
,	O
used	O
it	O
to	O
give	O
a	O
large	O
sample	O
justification	O
for	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
and	O
the	O
normal	O
distribution	O
.	O
</s>
<s>
In	O
1822	O
,	O
Gauss	O
was	O
able	O
to	O
state	O
that	O
the	O
least-squares	B-Algorithm
approach	O
to	O
regression	O
analysis	O
is	O
optimal	O
in	O
the	O
sense	O
that	O
in	O
a	O
linear	O
model	O
where	O
the	O
errors	O
have	O
a	O
mean	O
of	O
zero	O
,	O
are	O
uncorrelated	O
,	O
and	O
have	O
equal	O
variances	O
,	O
the	O
best	O
linear	O
unbiased	O
estimator	O
of	O
the	O
coefficients	O
is	O
the	O
least-squares	B-Algorithm
estimator	O
.	O
</s>
<s>
The	O
idea	O
of	O
least-squares	B-Algorithm
analysis	I-Algorithm
was	O
also	O
independently	O
formulated	O
by	O
the	O
American	O
Robert	O
Adrain	O
in	O
1808	O
.	O
</s>
<s>
In	O
the	O
next	O
two	O
centuries	O
workers	O
in	O
the	O
theory	O
of	O
errors	O
and	O
in	O
statistics	O
found	O
many	O
different	O
ways	O
of	O
implementing	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
The	O
objective	O
consists	O
of	O
adjusting	O
the	O
parameters	O
of	O
a	O
model	O
function	O
to	O
best	B-Algorithm
fit	I-Algorithm
a	O
data	O
set	O
.	O
</s>
<s>
The	O
least-squares	B-Algorithm
method	I-Algorithm
finds	O
the	O
optimal	O
parameter	O
values	O
by	O
minimizing	O
the	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
,	O
:	O
</s>
<s>
In	O
the	O
simplest	O
case	O
and	O
the	O
result	O
of	O
the	O
least-squares	B-Algorithm
method	I-Algorithm
is	O
the	O
arithmetic	O
mean	O
of	O
the	O
input	O
data	O
.	O
</s>
<s>
See	O
linear	O
least	B-Algorithm
squares	I-Algorithm
for	O
a	O
fully	O
worked	O
out	O
example	O
of	O
this	O
model	O
.	O
</s>
<s>
This	O
regression	O
formulation	O
considers	O
only	O
observational	O
errors	O
in	O
the	O
dependent	O
variable	O
(	O
but	O
the	O
alternative	O
total	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
regression	O
can	O
account	O
for	O
errors	O
in	O
both	O
variables	O
)	O
.	O
</s>
<s>
It	O
is	O
therefore	O
logically	O
consistent	O
to	O
use	O
the	O
least-squares	B-Algorithm
prediction	O
rule	O
for	O
such	O
data	O
.	O
</s>
<s>
In	O
standard	O
regression	O
analysis	O
that	O
leads	O
to	O
fitting	O
by	O
least	B-Algorithm
squares	I-Algorithm
there	O
is	O
an	O
implicit	O
assumption	O
that	O
errors	O
in	O
the	O
independent	O
variable	O
are	O
zero	O
or	O
strictly	O
controlled	O
so	O
as	O
to	O
be	O
negligible	O
.	O
</s>
<s>
When	O
errors	O
in	O
the	O
independent	O
variable	O
are	O
non-negligible	O
,	O
models	B-Algorithm
of	I-Algorithm
measurement	I-Algorithm
error	I-Algorithm
can	O
be	O
used	O
;	O
such	O
methods	O
can	O
lead	O
to	O
parameter	O
estimates	O
,	O
hypothesis	O
testing	O
and	O
confidence	O
intervals	O
that	O
take	O
into	O
account	O
the	O
presence	O
of	O
observation	O
errors	O
in	O
the	O
independent	O
variables	O
.	O
</s>
<s>
An	O
alternative	O
approach	O
is	O
to	O
fit	O
a	O
model	O
by	O
total	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
;	O
this	O
can	O
be	O
viewed	O
as	O
taking	O
a	O
pragmatic	O
approach	O
to	O
balancing	O
the	O
effects	O
of	O
the	O
different	O
sources	O
of	O
error	O
in	O
formulating	O
an	O
objective	O
function	O
for	O
use	O
in	O
model-fitting	O
.	O
</s>
<s>
The	O
gradient	O
equations	O
apply	O
to	O
all	O
least	B-Algorithm
squares	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Letting	O
and	O
putting	O
the	O
independent	O
and	O
dependent	O
variables	O
in	O
matrices	O
and	O
,	O
respectively	O
,	O
we	O
can	O
compute	O
the	O
least	B-Algorithm
squares	I-Algorithm
in	O
the	O
following	O
way	O
.	O
</s>
<s>
There	O
is	O
,	O
in	O
some	O
cases	O
,	O
a	O
closed-form	O
solution	O
to	O
a	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
–	O
but	O
in	O
general	O
there	O
is	O
not	O
.	O
</s>
<s>
These	O
are	O
the	O
defining	O
equations	O
of	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
model	O
function	O
,	O
f	O
,	O
in	O
LLSQ	O
(	O
linear	O
least	B-Algorithm
squares	I-Algorithm
)	O
is	O
a	O
linear	O
combination	O
of	O
parameters	O
of	O
the	O
form	O
The	O
model	O
may	O
represent	O
a	O
straight	O
line	O
,	O
a	O
parabola	O
or	O
any	O
other	O
linear	O
combination	O
of	O
functions	O
.	O
</s>
<s>
In	O
NLLSQ	O
(	O
nonlinear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
)	O
the	O
parameters	O
appear	O
as	O
functions	O
,	O
such	O
as	O
and	O
so	O
forth	O
.	O
</s>
<s>
If	O
analytical	O
expressions	O
are	O
impossible	O
to	O
obtain	O
either	O
the	O
partial	O
derivatives	O
must	O
be	O
calculated	O
by	O
numerical	O
approximation	O
or	O
an	O
estimate	O
must	O
be	O
made	O
of	O
the	O
Jacobian	O
,	O
often	O
via	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
LLSQ	O
solutions	O
can	O
be	O
computed	O
using	O
direct	O
methods	O
,	O
although	O
problems	O
with	O
large	O
numbers	O
of	O
parameters	O
are	O
typically	O
solved	O
with	O
iterative	O
methods	O
,	O
such	O
as	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
These	O
differences	O
must	O
be	O
considered	O
whenever	O
the	O
solution	O
to	O
a	O
nonlinear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
is	O
being	O
sought	O
.	O
</s>
<s>
There	O
are	O
many	O
methods	O
we	O
might	O
use	O
to	O
estimate	O
the	O
unknown	O
parameter	O
k	O
.	O
Since	O
the	O
n	O
equations	O
in	O
the	O
m	O
variables	O
in	O
our	O
data	O
comprise	O
an	O
overdetermined	O
system	O
with	O
one	O
unknown	O
and	O
n	O
equations	O
,	O
we	O
estimate	O
k	O
using	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
After	O
having	O
derived	O
the	O
force	O
constant	O
by	O
least	B-Algorithm
squares	I-Algorithm
fitting	I-Algorithm
,	O
we	O
predict	O
the	O
extension	O
from	O
Hooke	O
's	O
law	O
.	O
</s>
<s>
In	O
a	O
least	B-Algorithm
squares	I-Algorithm
calculation	O
with	O
unit	O
weights	O
,	O
or	O
in	O
linear	O
regression	O
,	O
the	O
variance	O
on	O
the	O
jth	O
parameter	O
,	O
</s>
<s>
where	O
the	O
true	O
error	O
variance	O
σ2	O
is	O
replaced	O
by	O
an	O
estimate	O
,	O
the	O
reduced	B-General_Concept
chi-squared	I-General_Concept
statistic	I-General_Concept
,	O
based	O
on	O
the	O
minimized	O
value	O
of	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
(	O
objective	O
function	O
)	O
,	O
S	O
.	O
The	O
denominator	O
,	O
n−m	O
,	O
is	O
the	O
statistical	O
degrees	O
of	O
freedom	O
;	O
see	O
effective	O
degrees	O
of	O
freedom	O
for	O
generalizations	O
.	O
</s>
<s>
In	O
a	O
linear	O
model	O
in	O
which	O
the	O
errors	O
have	O
expectation	O
zero	O
conditional	O
on	O
the	O
independent	O
variables	O
,	O
are	O
uncorrelated	O
and	O
have	O
equal	O
variances	O
,	O
the	O
best	O
linear	O
unbiased	O
estimator	O
of	O
any	O
linear	O
combination	O
of	O
the	O
observations	O
,	O
is	O
its	O
least-squares	B-Algorithm
estimator	O
.	O
</s>
<s>
"	O
Best	O
"	O
means	O
that	O
the	O
least	B-Algorithm
squares	I-Algorithm
estimators	O
of	O
the	O
parameters	O
have	O
minimum	O
variance	O
.	O
</s>
<s>
If	O
the	O
errors	O
belong	O
to	O
a	O
normal	O
distribution	O
,	O
the	O
least-squares	B-Algorithm
estimators	O
are	O
also	O
the	O
maximum	O
likelihood	O
estimators	O
in	O
a	O
linear	O
model	O
.	O
</s>
<s>
A	O
special	O
case	O
of	O
generalized	O
least	B-Algorithm
squares	I-Algorithm
called	O
weighted	O
least	B-Algorithm
squares	I-Algorithm
occurs	O
when	O
all	O
the	O
off-diagonal	O
entries	O
of	O
Ω	O
(	O
the	O
correlation	O
matrix	O
of	O
the	O
residuals	O
)	O
are	O
null	O
;	O
the	O
variances	O
of	O
the	O
observations	O
(	O
along	O
the	O
covariance	O
matrix	O
diagonal	O
)	O
may	O
still	O
be	O
unequal	O
(	O
heteroscedasticity	B-General_Concept
)	O
.	O
</s>
<s>
In	O
simpler	O
terms	O
,	O
heteroscedasticity	B-General_Concept
is	O
when	O
the	O
variance	O
of	O
depends	O
on	O
the	O
value	O
of	O
which	O
causes	O
the	O
residual	O
plot	O
to	O
create	O
a	O
"	O
fanning	O
out	O
"	O
effect	O
towards	O
larger	O
values	O
as	O
seen	O
in	O
the	O
residual	O
plot	O
to	O
the	O
right	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
homoscedasticity	B-General_Concept
is	O
assuming	O
that	O
the	O
variance	O
of	O
and	O
variance	O
of	O
are	O
equal	O
.	O
</s>
<s>
The	O
first	O
principal	B-Application
component	I-Application
about	O
the	O
mean	O
of	O
a	O
set	O
of	O
points	O
can	O
be	O
represented	O
by	O
that	O
line	O
which	O
most	O
closely	O
approaches	O
the	O
data	O
points	O
(	O
as	O
measured	O
by	O
squared	O
distance	O
of	O
closest	O
approach	O
,	O
i.e.	O
</s>
<s>
In	O
contrast	O
,	O
linear	O
least	B-Algorithm
squares	I-Algorithm
tries	O
to	O
minimize	O
the	O
distance	O
in	O
the	O
direction	O
only	O
.	O
</s>
<s>
Thus	O
,	O
although	O
the	O
two	O
use	O
a	O
similar	O
error	O
metric	O
,	O
linear	O
least	B-Algorithm
squares	I-Algorithm
is	O
a	O
method	O
that	O
treats	O
one	O
dimension	O
of	O
the	O
data	O
preferentially	O
,	O
while	O
PCA	O
treats	O
all	O
dimensions	O
equally	O
.	O
</s>
<s>
Notable	O
statistician	O
Sara	O
van	O
de	O
Geer	O
used	O
empirical	B-General_Concept
process	I-General_Concept
theory	I-General_Concept
and	O
the	O
Vapnik	O
–	O
Chervonenkis	O
dimension	O
to	O
prove	O
a	O
least-squares	B-Algorithm
estimator	O
can	O
be	O
interpreted	O
as	O
a	O
measure	O
on	O
the	O
space	O
of	O
square-integrable	B-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
In	O
some	O
contexts	O
a	O
regularized	O
version	O
of	O
the	O
least	B-Algorithm
squares	I-Algorithm
solution	O
may	O
be	O
preferable	O
.	O
</s>
<s>
Tikhonov	O
regularization	O
(	O
or	O
ridge	O
regression	O
)	O
adds	O
a	O
constraint	O
that	O
,	O
the	O
squared	O
-norm	O
of	O
the	O
parameter	O
vector	O
,	O
is	O
not	O
greater	O
than	O
a	O
given	O
value	O
to	O
the	O
least	B-Algorithm
squares	I-Algorithm
formulation	O
,	O
leading	O
to	O
a	O
constrained	O
minimization	O
problem	O
.	O
</s>
<s>
This	O
is	O
equivalent	O
to	O
the	O
unconstrained	O
minimization	O
problem	O
where	O
the	O
objective	O
function	O
is	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
plus	O
a	O
penalty	O
term	O
and	O
is	O
a	O
tuning	O
parameter	O
(	O
this	O
is	O
the	O
Lagrangian	O
form	O
of	O
the	O
constrained	O
minimization	O
problem	O
)	O
.	O
</s>
<s>
An	O
alternative	O
regularized	O
version	O
of	O
least	B-Algorithm
squares	I-Algorithm
is	O
Lasso	B-Algorithm
(	O
least	B-Algorithm
absolute	I-Algorithm
shrinkage	I-Algorithm
and	I-Algorithm
selection	I-Algorithm
operator	I-Algorithm
)	O
,	O
which	O
uses	O
the	O
constraint	O
that	O
,	O
the	O
L1-norm	O
of	O
the	O
parameter	O
vector	O
,	O
is	O
no	O
greater	O
than	O
a	O
given	O
value	O
.	O
</s>
<s>
(	O
One	O
can	O
show	O
like	O
above	O
using	O
Lagrange	O
multipliers	O
that	O
this	O
is	O
equivalent	O
to	O
an	O
unconstrained	O
minimization	O
of	O
the	O
least-squares	B-Algorithm
penalty	O
with	O
added	O
.	O
)	O
</s>
<s>
The	O
optimization	O
problem	O
may	O
be	O
solved	O
using	O
quadratic	B-Algorithm
programming	I-Algorithm
or	O
more	O
general	O
convex	O
optimization	O
methods	O
,	O
as	O
well	O
as	O
by	O
specific	O
algorithms	O
such	O
as	O
the	O
least	O
angle	O
regression	O
algorithm	O
.	O
</s>
<s>
One	O
of	O
the	O
prime	O
differences	O
between	O
Lasso	B-Algorithm
and	O
ridge	O
regression	O
is	O
that	O
in	O
ridge	O
regression	O
,	O
as	O
the	O
penalty	O
is	O
increased	O
,	O
all	O
parameters	O
are	O
reduced	O
while	O
still	O
remaining	O
non-zero	O
,	O
while	O
in	O
Lasso	B-Algorithm
,	O
increasing	O
the	O
penalty	O
will	O
cause	O
more	O
and	O
more	O
of	O
the	O
parameters	O
to	O
be	O
driven	O
to	O
zero	O
.	O
</s>
<s>
This	O
is	O
an	O
advantage	O
of	O
Lasso	B-Algorithm
over	O
ridge	O
regression	O
,	O
as	O
driving	O
parameters	O
to	O
zero	O
deselects	O
the	O
features	O
from	O
the	O
regression	O
.	O
</s>
<s>
Thus	O
,	O
Lasso	B-Algorithm
automatically	O
selects	O
more	O
relevant	O
features	O
and	O
discards	O
the	O
others	O
,	O
whereas	O
Ridge	O
regression	O
never	O
fully	O
discards	O
any	O
features	O
.	O
</s>
<s>
Some	O
feature	B-General_Concept
selection	I-General_Concept
techniques	O
are	O
developed	O
based	O
on	O
the	O
LASSO	B-Algorithm
including	O
Bolasso	O
which	O
bootstraps	O
samples	O
,	O
and	O
FeaLect	O
which	O
analyzes	O
the	O
regression	O
coefficients	O
corresponding	O
to	O
different	O
values	O
of	O
to	O
score	O
all	O
the	O
features	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
Lasso	B-Algorithm
and	O
its	O
variants	O
are	O
fundamental	O
to	O
the	O
field	O
of	O
compressed	O
sensing	O
.	O
</s>
