Global Positioning System Reference
In-Depth Information
r
r
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
z
i
1
/
√
λ
i
2
x
)
T
N
(
x
z
T
z
i
(
x
−
−
x
)
=
Λ
z
=
λ
i
=
(4.288)
i
=
1
i
=
1
Co
mbining Equations (4.285) and (4.288), we can write the
r
-dimensional ellipsoid,
or
the
r
-dimensional confidence region, in the principal axes form:
P
z
1
1
z
r
σ
0
rF
r,n
−
r,
α
/
λ
1
2
+···+
≤
=
1
− α
(4.289)
σ
0
rF
r,n
−
r,
α
/
r
2
λ
Th
e confidence region is centered at
x
. Whenever the zero hypothesis
H
0
of (4.282)
is
accepted, the point
x
T
falls within the confidence region. The probability that the
ell
ipsoid contains the true parameters
x
T
,is1
. For these reasons, one naturally
wo
uld like the ellipsoid to be small. Equation (4.289) shows that the semimajor axes
ar
e proportional to the inverse of the eigenvalues of the normal matrix. It is exactly this
re
lationship that makes us choose the eigenvalues of
N
as large as possible, provided
th
at we have a choice through appropriate network design variation. As an eigenvalue
ap
proaches zero, the respective axis of the confidence ellipsoid approaches infinity;
th
is is an undesirable situation, both from a statistical point of view and because of
th
e numerical difficulties encountered during the inversion of the normal matrix.
− α
[14
Lin
—
1.4
——
No
PgE
4.
9.5 Ellipses as Confidence Regions
Confidence ellipses are statements of precision. They are frequently used in con-
nection with two-dimensional networks in order to make the directional precision
of station location visible. Ellipses of confidence follow from Section 4.9.4 simply
by limiting the hypothesis (4.282) to two parameters, i.e., the Cartesian coordinates
of a station. Of course, in a three-dimensional network one can compute three-
dimensional ellipsoids or several ellipses, e.g., one for the horizontal and others for
the vertical. Confidence ellipses or ellipsoids are not limited to the specific applica-
tion of networks. However, in networks the confidence regions can be referenced with
respect to the coordinate system of the network and thus can provide an integrated
view of the geometry of the confidence regions and the network.
Consider the following hypothesis:
[14
H
0
:
x
i
−
x
i,T
=
o
(4.290)
where the notation
x
2
]
T
x
i
=
[
x
1
(4.291)
is used. The symbols
x
1
and
x
2
denote the Cartesian coordinates of a two-dimensional
network station
P
i
. The test of this hypothesis follows the outline given in the previous
section. The
A
2
matrix is of size 2
u
because there are two separate equations in the
hypothesis and
u
components in
x
. The elements of
A
2
are zero except those elements
×
