<s>
Weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
WLS	O
)	O
,	O
also	O
known	O
as	O
weighted	O
linear	B-General_Concept
regression	I-General_Concept
,	O
is	O
a	O
generalization	O
of	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
and	O
linear	B-General_Concept
regression	I-General_Concept
in	O
which	O
knowledge	O
of	O
the	O
variance	O
of	O
observations	O
is	O
incorporated	O
into	O
the	O
regression	O
.	O
</s>
<s>
A	O
special	O
case	O
of	O
generalized	O
least	O
squares	O
called	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
can	O
be	O
used	O
when	O
all	O
the	O
off-diagonal	O
entries	O
of	O
Ω	O
,	O
the	O
covariance	O
matrix	O
of	O
the	O
errors	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	B-Algorithm
diagonal	I-Algorithm
)	O
may	O
still	O
be	O
unequal	O
(	O
heteroscedasticity	B-General_Concept
)	O
.	O
</s>
<s>
The	O
normal	O
equations	O
can	O
then	O
be	O
written	O
in	O
the	O
same	O
form	O
as	O
ordinary	B-General_Concept
least	I-General_Concept
squares	I-General_Concept
:	O
</s>
<s>
This	O
is	O
a	O
type	O
of	O
whitening	B-Algorithm
transformation	I-Algorithm
;	O
the	O
last	O
expression	O
involves	O
an	O
entrywise	O
division	O
.	O
</s>
<s>
For	O
non-linear	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
systems	O
a	O
similar	O
argument	O
shows	O
that	O
the	O
normal	O
equations	O
should	O
be	O
modified	O
as	O
follows	O
.	O
</s>
<s>
For	O
this	O
feasible	O
generalized	O
least	O
squares	O
(	O
FGLS	O
)	O
techniques	O
may	O
be	O
used	O
;	O
in	O
this	O
case	O
it	O
is	O
specialized	O
for	O
a	O
diagonal	O
covariance	O
matrix	O
,	O
thus	O
yielding	O
a	O
feasible	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
solution	O
.	O
</s>
<s>
where	O
wi	O
>	O
0	O
is	O
the	O
weight	O
of	O
the	O
ith	O
observation	O
,	O
and	O
W	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
such	O
weights	O
.	O
</s>
<s>
This	O
method	O
is	O
used	O
in	O
iteratively	B-Algorithm
reweighted	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
an	O
expression	O
for	O
the	O
estimated	O
variance-covariance	O
matrix	O
of	O
the	O
parameter	O
estimates	O
can	O
be	O
obtained	O
by	O
error	B-General_Concept
propagation	I-General_Concept
from	O
the	O
errors	O
in	O
the	O
observations	O
.	O
</s>
<s>
When	O
unit	O
weights	O
are	O
used	O
(	O
W	O
=	O
I	O
,	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
)	O
,	O
it	O
is	O
implied	O
that	O
the	O
experimental	O
errors	O
are	O
uncorrelated	O
and	O
all	O
equal	O
:	O
M	O
=	O
σ2I	O
,	O
where	O
σ2	O
is	O
the	O
a	O
priori	O
variance	O
of	O
an	O
observation	O
.	O
</s>
<s>
In	O
any	O
case	O
,	O
σ2	O
is	O
approximated	O
by	O
the	O
reduced	B-General_Concept
chi-squared	I-General_Concept
:	O
</s>
<s>
The	O
standard	B-General_Concept
deviation	I-General_Concept
is	O
the	O
square	O
root	O
of	O
variance	O
,	O
,	O
and	O
the	O
correlation	O
coefficient	O
is	O
given	O
by	O
.	O
</s>
<s>
It	O
is	O
often	O
assumed	O
,	O
for	O
want	O
of	O
any	O
concrete	O
evidence	O
but	O
often	O
appealing	O
to	O
the	O
central	O
limit	O
theorem	O
—	O
see	O
Normal	O
distribution	O
#Occurrence	O
and	O
applications	O
—	O
that	O
the	O
error	O
on	O
each	O
observation	O
belongs	O
to	O
a	O
normal	O
distribution	O
with	O
a	O
mean	O
of	O
zero	O
and	O
standard	B-General_Concept
deviation	I-General_Concept
.	O
</s>
<s>
When	O
the	O
number	O
of	O
observations	O
is	O
relatively	O
small	O
,	O
Chebychev	O
's	O
inequality	O
can	O
be	O
used	O
for	O
an	O
upper	O
bound	O
on	O
probabilities	O
,	O
regardless	O
of	O
any	O
assumptions	O
about	O
the	O
distribution	O
of	O
experimental	O
errors	O
:	O
the	O
maximum	O
probabilities	O
that	O
a	O
parameter	O
will	O
be	O
more	O
than	O
1	O
,	O
2	O
,	O
or	O
3	O
standard	B-General_Concept
deviations	I-General_Concept
away	O
from	O
its	O
expectation	O
value	O
are	O
100%	O
,	O
25%	O
and	O
11%	O
respectively	O
.	O
</s>
<s>
where	O
H	O
is	O
the	O
idempotent	O
matrix	O
known	O
as	O
the	O
hat	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
and	O
I	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
