In matrix form we can write
V
«
A
X
-
L
6,1
6,3
3,1
6,1
where
V =
6,1
v3
\
v6
x -
3,1
H.
» L =
6,1
6.16 12.57
6Л1
1.09 11.58
5-07
and the design matrix, A5is
A «
6,3
0 |
1 |
0 |
0 |
0 |
1 |
0 |
-1 |
1 |
1 |
0 |
0 |
-1 |
0 |
1 |
-1 |
1 |
0 |
1
Since we have no information about the correlation between
h., we will treat them as uncorrelated. Hence, the variance-
l
covariance matrix of the observed quantities will be:
Li
Zj- = diag (k9 2, 2, h-9 2, k) 6,6
understanding that* the constant factor к is assumed one. The corresponding weight matrix is given as:
P = diag (0.25, 0.5, 0.5, 0.25, 0.5, 0.25) 6, 6
The normal equations are
N X * U 3,3 3,1 3,1
yielding the solution
-1
X ~ N U 3,1 " 3,3 3,1
where
|
|
|
N |
|
т А Р |
|
А |
|
т |
|
|
|
|
|
|
|
|
3,3 |
|
3,6 6,( |
|
6, |
3 |
|
|
|
|
|
|
Thus: |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Г 0 |
0 |
0 |
1 |
-1 |
-1 |
|
0. |
25 |
0 |
0 |
0 |
0 |
0 |
N = |
l |
0 |
-1 |
0 |
0 |
1 |
|
|
0 |
0.5 |
0 |
0 |
0 |
0 |
|
0 |
1 |
1 |
0 |
1 |
0 |
|
|
0 |
0 |
0.5 |
0 |
0 |
0 |
|
|
|
|
|
|
|
|
|
0 |
0 |
0 |
0.25 |
0 |
0 |
|
|
|
|
|
|
|
|
|
0 |
0 |
0 |
0 |
0.5 |
0 |
|
|
|
|
|
|
|
|
|
0 |
0 |
0 |
0 |
0 |
0., |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
-1 |
1 |
1 |
0 |
0 |
-1 |
0 |
1 |
-1 |
1 |
0 |
and N =
ООО 0.25 0 -0.5 О 0.5 0.5
0.25
О О
-0.5
О
0.5
-0.25 0.25
О
0 |
1 |
0 |
0 |
0 |
1 |
0 |
-1 |
1 |
1 |
0 |
0 |
-1 |
0 |
1 |
-1 |
1 |
0 |
Finally:
N 3,3
1.00 -0.25 -0.5
-0.25 .1.00 -0.50
-0.5 -0.5
1.50
Note that N is a symmetricjpositive-definite matrix.
Hence:
-1
H 3,3
1.6 0.8 0.8 1.6 0.8 0.8
0.8 0.8 1.2
Computing U = A PL , we get
U = 3,1
ООО 0.25 0.25 0 -0.5 О О 0.5 0.5 О
-0.5
О
0.5
-0.25 0.25
О
6.16 12.57
6Л1
1.09 11.58
5.07
and
U = ЗД
-6.7850 -0.3975 15.2800
Performing the multiplication N U, |
we |
get |
X as: |
||||||||
|
1.6 |
0.8 |
0.8 |
|
-6.7850 |
|
|
1.05 |
|||
X = |
0.8 |
1.6 |
0.8 |
|
-0.3975 |
= |
|
6.16 |
|||
3,1 |
0.8 |
0.8 |
1.2, |
|
15,2800 |
|
|
12.59 . |
|||
Therefore, we have obtained: the following estimates ^ = 1.05
m.
H = 6.16 с
m ,
EL
12.59
m.
By substituting the values of X we get the residual vector V for the observed Ik from the equation
V = A X - L .
Namely:
V = 6,1
1 0.00 m 0.02 m 0.02 m -0.04 m -0.04 m 0.04 m
The adjusted observations h are computed from:
h
=
and we get:
h. l |
= h. + l |
vi ' |
i = 1 |
, 2 / . |
6.16 |
|
r 0.00 |
|
6.16 |
12.57 |
|
0.02 |
|
12.59 |
6.41 |
|
0.02 |
|
6.43 |
1.09 |
+ |
-0.04 |
|
1.05 |
11.58 |
|
-0.04 |
|
11.54 |
5.07 |
|
°-04 |
|
5.11 |