Digital Signal Processing Reference
In-Depth Information
model is
+
β
11
x
1
+
β
22
x
2
y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
12
x
1
x
2
+
ε
we can create an equivalent representation
y
=
β
0
+
β
1
x
1
+
β
2
x
2
+
β
3
x
3
+
β
4
x
4
+
β
5
x
5
+
ε
by defining
≡
β
5
,x
1
≡
x
3
,x
2
β
11
≡
β
3
,β
22
≡
β
4
,β
12
≡
x
4
,
and
x
1
x
2
≡
x
5
By expressing the model this way, we can recast it in matrix form:
y
=
X
β
+
ε
(14-4)
where
y
1
y
2
.
y
n
=
is the
n
×
y
1 vector of observed responses
1
x
11
x
12
···
x
1
k
1
x
21
x
22
···
x
2
k
X
=
is the
n
×
k
matrix of inputs
.
.
.
.
1
x
n
1
x
n
2
···
x
nk
β
0
β
1
.
β
k
β
=
is the
k
×
1 vector of model coefficients
ε
1
ε
2
.
ε
n
ε
=
is the
n
×
1 vector of random errors
and
n
is the number of observations for fitting the model and
k
is the number of
model terms.
Each column of the input matrix corresponds to a model term. The first column
represents the intercept for the model, from which the
β
0
coefficient is estimated.
Each row of the input matrix corresponds to an experimental observation. The
response vector contains a row for each observation, as does the residual vector.
The model coefficient vector contains a row for each term in the model.
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