Information Technology Reference
In-Depth Information
n
X
.˛
y
i
/
2
:
F.˛/
D
(5.49)
i D1
We compute the value ˛ where F is minimized by solving the equation
F
0
.˛/
D
0:
(5.50)
Since
n
X
F
0
.˛/
D
2
.˛
y
i
/;
(5.51)
i D1
the solution of (
5.50
)is
n
X
1
n
˛
D
y
i
;
(5.52)
i D1
which we recognize as the arithmetic average.
Approximation by a Linear Function
Next, we seek a linear function
p
1
.t /
D
˛
C
ˇt
(5.53)
approximating the data (
5.47
) in the sense of least squares. We define
n
X
y
i
/
2
:
F.˛/
D
.˛
C
ˇt
i
(5.54)
i D1
and determine the constants ˛ and ˇ by requiring that
@F
@˛
@F
@ˇ
D
D
0:
(5.55)
Since
n
X
@F
@˛
D
2
.˛
C
ˇt
i
y
i
/
(5.56)
i D1
and
n
X
@F
@ˇ
D
2
.˛
C
ˇt
i
y
i
/t
i
;
(5.57)
i D1
