3.2
Deriving the Shape Function
An exhaustive and detailed treatment of PIMs can be found in [Fai15, 7], but for the sake of thoroughness we will provide in this section a brief description. In principle, what we require is a transformation
y k = Φk
syk, lky , (3.4)
with inverse function
" sky ly k # = Φ−1 k (yk) (3.5) ="φ−1k (yk) ϕ∗ k(yk) # ,
which represents a reparametrization of the measurement y
k into a new rep- resentationhsyk, lky
i
, where syk describes where in the shape the true source is, and lky denotes how distant y
k is to the shape (Figure 3.1). The objective is that, when this term is later rewritten in the probabilistic model, the effect of skyfades or becomes minimal, thus eliminating the effect of any error in the source approximation. In turn, this yields the following advantages. On the one hand, we only need to take into account the shape function value lky = ϕ∗k(y
k) which is usually easy to obtain. On the other hand, we can also avoid having to find a path parametrization φ
k(sk), a task which for many shapes is not straightforward.
From (3.3) and (3.5) we observe a conceptual link between lkyand νk, i.e., the measured lkycan be interpreted as a realization of the random variable νk. From this, it follows that p(lky| xk)= p(νk| xk), and thus, once Φk(·) is obtained, we can obtain information about the distribution of νk simply by analyzing how
3 Partial Information Models
lkybehaves. By plugging (2.1) into (3.5), we can describe the generative model of y k in function of s y k and l y k, i.e., h sky, lky i> = Φ−1 k (yk) (3.6) = Φ−1 k (zk+ vk).
By propagating vk through this term, we obtain the pdf p(sky, lky| xk), which can be rewritten as p(syk, lky| xk)= p(sy k | xk) · p(l y k | s y k, xk).
In order to minimize the importance of the unknown true source, we need a reparametrization which causes the resulting skyand lky to be as independent from each other as possible. If that were the case, we could write
p(lky| xk)= p(ly k | s
y
k, xk). (3.7)
This would mean that the probability of lkybehaves the same independently of the assumed source, and thus, we could simply say for any arbitrary p
k ∈ S x k that νk = ϕ∗k(p k + vk),
and we would be finished. Unfortunately, as [7] showed, this independence only holds for simple shapes such as lines or planes. It is generally impossible to apply these ideas to arbitrary shapes, as no general reparametrization exists that fulfills (3.7) for all syk and lkyand any covariance matrix Cv
k, except in very limited cases.
In order to solve this issue, [7] proposed for the reparametrization the following compromise. First, we need to find a source which closely approximates the true source of y
k, for example, by finding the source π ∗
k(yk) which most likely generated it according to the generative model, as explained in Section 3.3. Second, we use as shape function the signed Mahalanobis distance from (2.16), i.e., ϕ∗k = ϕsmk , using as weight Σ= Cvk. Finally, by applying the approximation
3.2 Deriving the Shape Function
z k ≈π
∗
k(yk) and plugging this into (3.6), we obtain as bias correction term the random variable
νk ≈ϕ∗ k(π
∗
k(yk)+ vk). This leads to the bias-corrected measurement equation
0= ϕ∗k(y
k) −νk (3.8)
:= h(xk, yk, νk),
Note that νk acts as a new scalar noise term, replacing the old term vk. By denoting the distribution of the correction term as fkν(νk) := p(νk| xk), we can obtain a likelihood function by interpreting (3.8) probabilistically, i.e.,
p(y k| xk) ≈ f ν k(ϕ ∗ k(yk)). (3.9)
and treating the state xk, which contains the parameters of ϕ∗k(·), as the free variable. Note that this parametrization does not guarantee a complete inde- pendence between sykand lkyall around the shape, as this is usually impossible. Instead, we can only say that both variables are independent in an infinite- simally small neighborhood around the true source. Thus, the quality of the bias reduction depends highly on the ability of π∗
k(yk) to approximate zk. Still, as [7] showed, the proposed approach still provides very accurate results.
Figure 3.1: Change of coordinates. A measurement y
k= [y (0) k , y (1) k ] is reparametrized as [s y k, l y k].
The coordinate syk describes where on the shape the most likely source is, while lky shows how ‘distant’ the measurement is. The subindex k is omitted for legibility.
3 Partial Information Models