Global Positioning System Reference
In-Depth Information
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The zero hypothesis states that the parameters equal a certain (true) value
x
T
. From
(4.278) it follows that
A
2
=
I
and
2
=−
x
T
. Using these specifications we can use
T
=
N
in (4.279), and the statistic (4.280) becomes
x
∗
−
x
T
T
N
x
∗
−
x
T
∼
F
r, n
1
−
r,
α
(4.284)
2
0
r
σ
w
here the a posteriori variance of unit weight (first group only) has been substituted
fo
r
v
T
Pv
∗
. Once the adjustment of the first group (4.277) is completed, the values
fo
r the adjusted parameters and the a posteriori variance of unit weight are entered
in
(4.284), and the fraction is computed and compared with the
F
value (taking the
pr
oper degrees of freedom and the desired significance level into account). Rejection
or
acceptance of the zero hypothesis follows rule (4.281).
Note that one of the degrees of freedom in (4.284) is
r
[14
R(
N
)<u
, instead of
u
,
which equals the number of parameters, even though Equation (4.282) expresses
u
conditions. Because of the possible rank defect of the normal matrix
N
, the distri-
bu
tion of
=
Lin
—
0.7
——
Nor
*PgE
v
T
Pv
in (4.279) is a chi-square distribution with
r
degrees of freedom.
Co
nsider the derivation leading to (4.273). The
u
components of
z
3
are transformed
to
r
stochastically independent unit variate normal distributions that are then squared
an
d summed to yield the distribution of
∆
v
T
Pv
. The interpretation is that (4.282)
re
presents one hypothesis on all parameters
x
, and not
u
hypotheses on the
u
compo-
ne
nts on
x
.
Expression (4.284) can be used to define the
r
-dimensional confidence region.
Re
place the particular
x
T
by the unknown parameter
x
, and drop the asterisk; then
P
(
x
∆
[14
F
r,n
1
−
r,
α
x
)
T
−
N
(
x
−
x
)
≤
F
r,n
1
−
r,
α
=
F
r,n
1
−
r
dF
=
1
− α
(4.285)
2
0
r
σ
0
The probability region described by the expression on the left side of Equation
(4.285) is an
R(
N
)
-dimensional ellipsoid. The probability region is an ellipsoid,
because the normal matrix
N
is positive definite or, at least, semipositive definite.
If one identifies the center of the ellipsoid with
)
probability
that the unknown point
x
lies within the ellipsoid. The orientation and the size of
this ellipsoid are a function of the eigenvectors and eigenvalues of the normal matrix,
the rank of the normal matrix, and the degree of freedom. Consider the orthonormal
transformation
x
, then there is
(
1
− α
F
T
(
x
z
=
−
x
)
(4.286)
with
F
as specified in (A.52) and containing the normalized eigenvectors of
N
, then
F
T
NF
=
Λ
(4.287)
with
Λ
containing the
r
eigenvalues of
N
, and
