Global Positioning System Reference
In-Depth Information
x
1
x
2
N
µ
1
µ
2
,
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Σ
11
O
∼
(4.225)
O
Σ
22
th
en
x
1
and
x
2
are stochastically independent. If one set of normally distributed ran-
do
m variables is uncorrelated with the remaining variables, the two sets are indepen-
de
nt. The proof of the above theorem follows from the fact that the density function
ca
n be written as a product of
f
1
(
x
1
)
and
f
2
(
x
2
)
because of the special form of the
de
nsity function (4.217).
4.9.2 Distribution of
v
T
Pv
Th
e derivation of the distribution is based on the assumption that the observations
ha
ve a multivariate normal distribution. The dimension of the distribution equals the
nu
mber of observations. In the subsequent derivations the observation equation model
is
used. However, these statistical derivations could just as well have been carried out
wi
th the mixed model.
The observation equations are
[13
Lin
—
1.7
——
Lon
*PgE
v
=
Ax
+
0
−
b
(4.226)
=
Ax
+
A first assumption is that the residuals are randomly distributed, i.e., the probability
for a positive or negative residual of the equal magnitude is the same. From this
assumption it follows that
[13
E(
v
)
=
o
(4.227)
Because
x
and
0
are constant vectors, it further follows that the mean and variance-
covariance matrix, respectively, are
E
b
=
0
+
Ax
(4.228)
E
vv
T
=
E
b
−
b
)
b
−
b
)
T
=
Σ
b
= σ
0
P
−
1
E(
E(
(4.229)
Th
e second basic assumption refers to the type of distribution of the observations. It
is
assumed that the distribution is multivariate normal. Using the mean (4.228) and
th
e covariance matrix (4.229), the
n
-dimensional multivariate normal distribution of
b
is written as
N
n
0
+
Σ
b
b
∼
Ax
,
(4.230)
Al
ternative expressions are
N
n
−
Σ
b
∼
Ax
,
(4.231)
