The GNSS integer ambiguities: estimation and validation

5.6 Optimal Integer Aperture estimation
The method of penalized IA estimation as described in section 5.5 is some way regulates
the probabilities of success and failure. As mentioned in the beginning of this chapter,
this is why the new class of integer estimators was defined. A disadvantage of PIA
estimation is that the user has to choose the values for the penalties that will be assigned
to the three cases success, failure and undecided. A user would like to choose the
penalties in such a way that he can be sure that the fail rate is below a certain value.
All other IA estimators described in previous sections are not optimal in the sense of
maximization of the success rate. Therefore, still another approach could be based on
the optimization problem defined as:
max Ps subject to: Pf = β (5.49)
Ω0⊂S0
where β is a chosen value for the fail rate, (Teunissen 2003g). In words, this means that
the aperture space is defined such that the success rate is maximized subject to a fixed
fail rate. The solution of the optimization problem is given by:
Ω0 = {x ∈ S0 |
fâ(x + z) ≤ µfâ(x + a)} (5.50)
z∈Z n
The proof was given in Teunissen (2004d), and is as follows.
Lemma 5.6.1 Let f(x) and g(x) be integrable functions over R n . Then the region
ˆΩ = {x ∈ R n |
f(x + z) ≥ λ
g(x + z), λ ∈ R} (5.51)
z∈Z n
z∈Z n
solves the constrained maximization problem:
max
Ω⊂Rn
Ω
subject to:
Ω⊂R n
f(x)dx (5.52)
g(x)dx = c
Ω = Ω + z, ∀z ∈ Z n
if λ ∈ R is chosen such that the integral constraint of equation (5.53) is satisfied.
(5.53)
Note the similarity with the Neyman-Pearson lemma, cf. (Rao 1965). The proof of
lemma 5.6.1 is as follows.
Optimal Integer Aperture estimation 111