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system (7.16). Therefore, given the linear independence of the column vectors of
W
, the matrix
W
T
W
is invertible, and the asserted value for
z
0
can be computed.
7.7.1
The
k
-Variable Multiple Regression Model
In this section we will recall the classical
k-variable multiple regression
. The reader
can find more details and statistical motivations in Aczel and Sounderpandian's
topic [216], from which we adopt the notation.
In statistics, the regression analysis provides techniques for finding the relation-
ship between a
dependent variable Y
and one or more
independent variables X
1
,
X
2
,
...,
X
k
. The following equation is the general form of a linear regression:
Y
=
β
0
+
β
1
X
1
+
β
2
X
2
+
...
+
β
k
X
k
+
ε
.
(7.17)
When
k
1 the regression equation above is called a
multiple regression model
.
The correctness of the regression model is subjected to the following assumptions:
>
1. For each observation, the observation error
ε
is normally distributed with mean
zero and standard deviation
σ
and is independent from the errors associated with
all other observations;
2. The variables
X
i
are independent from the error
ε
.
Ta b l e 7 . 6
The three deviations associated with the data points (see also Fig. 7.6). We indicate
with
Y
the value of the dependent variable, with
Y
its predicted value by means of the multiple
regression model, and with
Y
the average of the values of
Y
. Finally, we indicate with
n
the
total number of data points used during the regression
Y
Y
−
Y
Y
−
Y
Y
−
=
+
Total
Unexplained
Explained
deviation
deviation (error)
deviation (regression)
n
j
=
1
(
y
[
j
]
−
y
)
n
j
=
1
(
y
[
j
]
−
y
[
j
])
n
j
=
1
(
y
[
j
]
−
y
)
2
2
2
=
+
∑
∑
∑
SST
SSE
SSR
Sum of Squared
Sum of Squared
Sum of Squared
Total deviations
Errors
Regression deviations
If the assumptions given above are satisfied, then we can compute the coefficients
c
i
,
i
=
0
,...,
k
in terms of least squares estimations of the regression parameters
β
i
,
as the values providing the best approximation
Y
to the value
Y
:
Y
=
c
0
+
c
1
X
1
+
c
2
X
2
+
...
+
c
k
X
k
(7.18)
