324 Data Analysis and Statistics for Geography, Environmental Science, and Engineering
Solving for the error in Equation 9.55
(9.59)
Now, we note that the sum of all square errors can be written as the major product matrix e
ee
T
=
=
=
∑
ee e
e
e
e
e
n
n
i
i
n
12
1
2
2
1
...
...
(9.60)
As we know we need to nd b that minimizes this total square error:
min[ ] min[( )( )]
bb
ee yxbyxb
=−−
(9.61)
This is achieved by the b obtained by solving the unknown b from the system of equations y = xb,
where y and x are known. To do this premultiply both sides by x
T
xy=x xb
(9.62)
Note that x
T
x is a 2 × 2 matrix
xx
T
=
1111
1
1
1
1
123
1
2
3
...
...
... ...
xxxx
x
x
x
x
n
n
=
∑
i
2
(9.63)
which is the same as matrix S
x
in Equation 9.43. The entr ...