Geoscience Reference
In-Depth Information
77
77
68
71
83
98
51
84
97
95
82
81
90
90
70
78
87
93
61
89
74
45
96
81
70
50
80
77
75
60
76
70
75
68
74
76
93
75
96
85
76
82
58
80
71
72
58
68
61
46
69
65
Sums
2206
1987
2003
2179
Means (
n
= 28)
78.7857
70.9643
71.5387
77.8214
With these data we would like to fit an equation of the form
YabX
=+ +
bX
+
bX
33
11
22
According to the principle of least squares, the best estimates of the
X
coefficients can be obtained
by solving the set of least squares normal equations.
(
)
+
(
)
+
(
)
equation:
∑∑ ∑
∑
2
b
x
b
x xb
xx b
=
x
y
1
1
1
1
22
13
3
1
(
)
+
(
)
+
(
)
∑
∑
∑
∑
2
b
equation:
x xb
xb
xx
b
=
x y
2
1
21
2
2
2
3
3
2
(
)
+
(
)
(
)
∑
∑
∑∑
2
b
equation:
x xb
xx b
+
x
b
=
x y
3
1
31
23
2
3
3
3
where
−
(
)(
)
∑∑
∑∑
XX
n
i
j
xx
=
XY
ij
ij
Having solved for the
X
coefficients (
b
1
,
b
2
, and
b
3
), we obtain the constant term by solving
aY bX
=− +
bX
−
bX
33
11
22
Derivation of the least squares normal equations requires a knowledge of differential calculus.
However, for the general linear mode with a constant term
YabX
=+ +
bX
22
…
+
b
kk
11
the normal equations can be written quite mechanically once their pattern has been recognized.
Every term in the first row contains an
x
1
, every term in the second row an
x
2
, and so forth down
to the
k
th row, every term of which will have an
x
k
. Similarly, every term in the first column has an
x
1
and a
b
1
, every term in the second column has an
x
2
and a
b
2
, and so on through the
k
th column,
every term of which has an
x
k
and a
b
k
. On the right side of the equations, each term has a
y
times the
x
that is appropriate for a particular row. So, for the general linear model given above, the normal
equations are
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