<s>
Least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
(	O
LAD	O
)	O
,	O
also	O
known	O
as	O
least	B-General_Concept
absolute	I-General_Concept
errors	I-General_Concept
(	O
LAE	O
)	O
,	O
least	B-General_Concept
absolute	I-General_Concept
residuals	I-General_Concept
(	O
LAR	O
)	O
,	O
or	O
least	B-General_Concept
absolute	I-General_Concept
values	I-General_Concept
(	O
LAV	O
)	O
,	O
is	O
a	O
statistical	O
optimality	B-General_Concept
criterion	I-General_Concept
and	O
a	O
statistical	O
optimization	O
technique	O
based	O
minimizing	O
the	O
sum	B-General_Concept
of	I-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
(	O
sum	O
of	O
absolute	O
residuals	O
or	O
sum	O
of	O
absolute	O
errors	O
)	O
or	O
the	O
L1	O
norm	O
of	O
such	O
values	O
.	O
</s>
<s>
It	O
is	O
analogous	O
to	O
the	O
least	B-Algorithm
squares	I-Algorithm
technique	O
,	O
except	O
that	O
it	O
is	O
based	O
on	O
absolute	O
values	O
instead	O
of	O
squared	O
values	O
.	O
</s>
<s>
It	O
attempts	O
to	O
find	O
a	O
function	O
which	O
closely	O
approximates	O
a	O
set	B-General_Concept
of	I-General_Concept
data	I-General_Concept
by	O
minimizing	O
residuals	O
between	O
points	O
generated	O
by	O
the	O
function	O
and	O
corresponding	O
data	O
points	O
.	O
</s>
<s>
Though	O
the	O
idea	O
of	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
regression	O
is	O
just	O
as	O
straightforward	O
as	O
that	O
of	O
least	B-Algorithm
squares	I-Algorithm
regression	O
,	O
the	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
is	O
not	O
as	O
simple	O
to	O
compute	O
efficiently	O
.	O
</s>
<s>
Unlike	O
least	B-Algorithm
squares	I-Algorithm
regression	O
,	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
regression	O
does	O
not	O
have	O
an	O
analytical	O
solving	O
method	O
.	O
</s>
<s>
The	O
following	O
is	O
an	O
enumeration	O
of	O
some	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
solving	O
methods	O
.	O
</s>
<s>
Because	O
the	O
problem	O
is	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
any	O
of	O
the	O
many	O
linear	B-Algorithm
programming	I-Algorithm
techniques	O
(	O
including	O
the	O
simplex	B-Algorithm
method	I-Algorithm
as	O
well	O
as	O
others	O
)	O
can	O
be	O
applied	O
.	O
</s>
<s>
Simplex-based	B-Algorithm
methods	I-Algorithm
are	O
the	O
“	O
preferred	O
”	O
way	O
to	O
solve	O
the	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
problem	O
.	O
</s>
<s>
A	O
Simplex	B-Algorithm
method	I-Algorithm
is	O
a	O
method	O
for	O
solving	O
a	O
problem	O
in	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
most	O
popular	O
algorithm	O
is	O
the	O
Barrodale-Roberts	O
modified	O
Simplex	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Checking	O
all	O
combinations	O
of	O
lines	O
traversing	O
any	O
two	O
(	O
x	O
,	O
y	O
)	O
data	O
points	O
is	O
another	O
method	O
of	O
finding	O
the	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
.	O
</s>
<s>
Since	O
it	O
is	O
known	O
that	O
at	O
least	O
one	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
traverses	O
at	O
least	O
two	O
data	O
points	O
,	O
this	O
method	O
will	O
find	O
a	O
line	O
by	O
comparing	O
the	O
SAE	O
(	O
Smallest	O
Absolute	O
Error	O
over	O
data	O
points	O
)	O
of	O
each	O
line	O
,	O
and	O
choosing	O
the	O
line	O
with	O
the	O
smallest	O
SAE	O
.	O
</s>
<s>
The	O
problem	O
can	O
be	O
solved	O
using	O
any	O
linear	B-Algorithm
programming	I-Algorithm
technique	O
on	O
the	O
following	O
problem	O
specification	O
.	O
</s>
<s>
Since	O
this	O
version	O
of	O
the	O
problem	O
statement	O
does	O
not	O
contain	O
the	O
absolute	O
value	O
operator	O
,	O
it	O
is	O
in	O
a	O
format	O
that	O
can	O
be	O
solved	O
with	O
any	O
linear	B-Algorithm
programming	I-Algorithm
package	O
.	O
</s>
<s>
There	O
exist	O
other	O
unique	O
properties	O
of	O
the	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
set	O
of	O
(	O
x	O
,	O
y	O
)	O
data	O
,	O
the	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
will	O
always	O
pass	O
through	O
at	O
least	O
two	O
of	O
the	O
data	O
points	O
,	O
unless	O
there	O
are	O
multiple	O
solutions	O
.	O
</s>
<s>
If	O
multiple	O
solutions	O
exist	O
,	O
then	O
the	O
region	O
of	O
valid	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
solutions	O
will	O
be	O
bounded	O
by	O
at	O
least	O
two	O
lines	O
,	O
each	O
of	O
which	O
passes	O
through	O
at	O
least	O
two	O
data	O
points	O
.	O
</s>
<s>
The	O
"	O
latching	O
"	O
also	O
helps	O
to	O
understand	O
the	O
"	O
robustness	O
"	O
property	O
:	O
if	O
there	O
exists	O
an	O
outlier	O
,	O
and	O
a	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
line	O
must	O
latch	O
onto	O
two	O
data	O
points	O
,	O
the	O
outlier	O
will	O
most	O
likely	O
not	O
be	O
one	O
of	O
those	O
two	O
points	O
because	O
that	O
will	O
not	O
minimize	O
the	O
sum	B-General_Concept
of	I-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
in	O
most	O
cases	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
table	O
contrasting	O
some	O
properties	O
of	O
the	O
method	O
of	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
with	O
those	O
of	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
for	O
non-singular	O
problems	O
)	O
.	O
</s>
<s>
The	O
method	O
of	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
finds	O
applications	O
in	O
many	O
areas	O
,	O
due	O
to	O
its	O
robustness	O
compared	O
to	O
the	O
least	B-Algorithm
squares	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
is	O
robust	O
in	O
that	O
it	O
is	O
resistant	O
to	O
outliers	O
in	O
the	O
data	O
.	O
</s>
<s>
LAD	O
gives	O
equal	O
emphasis	O
to	O
all	O
observations	O
,	O
in	O
contrast	O
to	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
(	O
OLS	O
)	O
which	O
,	O
by	O
squaring	O
the	O
residuals	O
,	O
gives	O
more	O
weight	O
to	O
large	O
residuals	O
,	O
that	O
is	O
,	O
outliers	O
in	O
which	O
predicted	O
values	O
are	O
far	O
from	O
actual	O
observations	O
.	O
</s>
<s>
If	O
it	O
is	O
important	O
to	O
give	O
greater	O
weight	O
to	O
outliers	O
,	O
the	O
method	B-Algorithm
of	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
is	O
a	O
better	O
choice	O
.	O
</s>
<s>
The	O
case	O
of	O
gives	O
the	O
standard	O
regression	O
by	O
least	B-General_Concept
absolute	I-General_Concept
deviations	I-General_Concept
and	O
is	O
also	O
known	O
as	O
median	O
regression	O
.	O
</s>
<s>
The	O
least	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
problem	O
may	O
be	O
extended	O
to	O
include	O
multiple	O
explanators	O
,	O
constraints	O
and	O
regularization	O
,	O
e.g.	O
,	O
a	O
linear	O
model	O
with	O
linear	O
constraints	O
:	O
</s>
<s>
Regularization	O
with	O
LASSO	B-Algorithm
(	O
least	B-Algorithm
absolute	I-Algorithm
shrinkage	I-Algorithm
and	I-Algorithm
selection	I-Algorithm
operator	I-Algorithm
)	O
may	O
also	O
be	O
combined	O
with	O
LAD	O
.	O
</s>
