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ya ax ax
...
ax
0
1
1
2
2
nn
between the observed variables
x
i
= 1, 2, …,
n
, and the resulting variable to be
estimated
y
. Also here, using the above equation and the collected data, the
problem is to determine the values of the coefficients
a
0
,
a
1
,
a
2
, ...,
a
n
that will
guarantee the best fitting of the regression line to the experimental data. This is
verified through correlation analysis.
The compact form of multiple regression is
,
yAx
H
where
T
yyy y
xxx x
HHHH
[, , ]
[, , ]
[
12
n
T
12
n
T
,
,...,
]
12
n
are the corresponding vectors and
ª
º
aa
...
a
11
12
1
n
«
»
«
aa
...
... ... ... ...
...
a
»
A
«
21
22
2
n
»
«
»
«
»
aa
a
¬
¼
nn
1
2
n
the corresponding parameter matrix.
To apply the
least-squares estimator
to find the best estimation value of
ˆ
x
we
first build the error value
H(
x
) =
(
yAx
ˆ
)
and try to find the
x
value that minimizes the product
ˆ
)
T
ˆ
(
yAx
(
yAx
)
.
Using the least-squares estimation procedure with respect to
ˆ
x
the equation
T
ˆ
T
2
AAx
2
Ay
0
is obtained, from which the estimated value of
ˆ
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