Page 357 - Fundamentals of Probability and Statistics for Engineers

340 Fundamentals of Probability and Statistics for Engineers
^
The least-square estimate of q, q, is found by minimizing Q. Applying the
variational principle discussed in Section 9.3.1.1, we have
dQ ˆ dq C y Cq† y Cq† Cdq
T
T
T
T
T
ˆ 2dq C y Cq†:
^
Setting dQ ˆ 0, the solution for q is obtained from normal equation
^
T
C y Cq†ˆ 0; 11:14†
or
T ^
T
C Cq ˆ C y;
which gives
^ T 1 T 11:15†
q ˆ C C† C y:
T
In the above, the inverse of matrix C C exists if there are at least two distinct
values of x i represented in the sample.
We can easily check that Equation (11.15) is identical to Equations (11.7)
and (11.8) by noting that
1 x 1
2 3
2 3
1 1 1 6 x 2 7 n
6 1 7 n nx
T
C C ˆ 6 . 7 ˆ 4 P 2 ;
5
x 1 x 2 x n 6 . . . . 7 nx x i
iˆ1
4 . 5
1 x n
2 3
y 1
ny
2 3
1 1 1 6 y 2 7
6 7
T
n
C y ˆ 6 7 ˆ 4 P 5 ;
6
x 1 x 2 x n 4 . . 7 x i y i
. 5
iˆ1
y n
and
1
n nx ny
2 3 2 3
^ T 1 T 4 P 5 4 n 5
n
q ˆ C C† C y ˆ
nx x 2 i P x i y i
iˆ1 iˆ1
2 1 3
n n
2
y P x i y i nxy P x nx 2 x
i
6 7
iˆ1 iˆ1
6 7
ˆ 6
1 7
n n
4 5
P P 2 2
x i y i nxy x nx
i
iˆ1 iˆ1
2 ^ 3
y x
n n 5:
6 1 7
ˆ 4
P P 2
x i x† y i y† x i x†
iˆ1 iˆ1
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