A track from the joint posterior distribution may reveal information that is not immediately evident from the marginal posterior distributions.
Formally the joint posterior distribution for all positions (often referred to merely as the joint posterior distribution) is dened as
P(ξ|YN), (3.7)
where ξ= [X0=x0, . . . ,Xj =xj]T denotes a track given by the positions at all time steps.
Sampling from the joint posterior distribution is done recursively by apply-ing Bayes' rule. The samplapply-ing scheme runs backwards in time, initialised by sampling the terminal position atτN from the distribution
P(XN =xN|YN),
and thereby obtaining xsN, the sampled terminal position.
The position precedingτj+1 is sampled from
P(Xj=xj|YN,XN =xsN, . . . ,Xj+1=xsj+1).
This can be rewritten by applying the Markov property and Bayes' rule to obtain P(Xj =xj|YN,XN =xsN, . . . ,Xj+1=xsj+1)
=P(Xj =xj|Yj,Xj+1=xsj+1)
=P(Xj+1=xsj+1|Xj =xj) P(Xj=xj|Yj)
P(Xj+1=xsj+1|Yj). (3.8) The formula (3.8) uses the reconstruction,P(Xj =xj|Yj), and updates it with the information of the previous (in an iterative not temporal sense) sample point P(Xj+1 =xsj+1|Xj =xj). The term, P(Xj+1=xsj+1|Yj), is considered a normalisation constant in the implementation that makes the distribution sum to one. The Markov assumption is essential to this sampling method that would otherwise require a more complex simultaneous sampling from the joint distribution.
3.5 Finding the Most Probable Track
Another perhaps more interesting representation of the joint posterior distribu-tion is the Most Probable Track (MPT). Previous studies employing the Kalman
Filter (Sibert et al., 2003) or particle lter technique (Nielsen, 2004) have sug-gested using a track that connects the conditional mean of all time steps. In the linear framework of the Kalman lter, the choice is rational. However, the potential multi modal distributions of a particle ltering can produce erroneous tracks in a nonlinear environment, possibly locating the most probable position on dry land. An environment is termed nonlinear if it contains islands or shores that cause the Gaussianity assumption to be violated.
The joint posterior distribution is given by (3.7). The mode in this distri-bution is de facto the most probable of all possible tracks in the outcome space and entitled the Most Probable Track. The non-trivial task of nd-ing this track is solved by application of the Viterbi algorithm (Viterbi, 1967;
Viterbi, 2006). The algorithm was developed for information theory and deep space communication and have found wide applications most prominently in speech recognition. It was later shown to be a computationally ecient tech-nique for determining the most probable sequence in a hidden Markov model (Forney, 1973).
As it is a novel approach to track estimation in a geolocation context, the technique is here presented in some detail.
A track ending at a given positionxe at timeτj, is written ξ(xej) = [X0=x0, . . . ,Xj=xej].
Furthermore the branch metric is dened
B(xj−1,xj;yj) =P(Xj =xj|Xj−1=xj−1)
as a product of the likelihood for the observationyj, given the new positionxj
and the transition probability for jumping fromxj−1 toxj. A likelihood measure for a track is dened as
L[ξ(xej)] =B(xj−1,xej;yj)
j−1Y
k=1
B(xk−1,xk;yk), which is proportional to the probability ofξ(exj).
The state metric at a position,xe at timeτj, is given by S(exj) = max
e xj
L[ξ(exj)],
meaning the likelihood of the most probable track leading toxej.
3.5 Finding the Most Probable Track 27 As a consequence of the Markov property the maximisation can be done re-cursively
S(exj) = max
e xj−1
{S(exj−1)B(exj−1,xej;yj)}.
The algorithm sequentially nds the current state metric by maximising the product of the previous state metric and the attached branch metric. For allxej, S(xej)contains the likelihood of the most probable track leading toxej. Logging the most probable track, in each recursion, for eachxej is a simple way to obtain the Most Probable Track, ξb. The track leading to xbN = arg maxxNS(xN)is ξb.
Chapter 4
Geolocation of simulated sh
Before applying the lter methods described in Chapter3to a real dataset, the performance of the method is investigated in a simulation study. Emphasis is put on clarication of bias and uncertainty on the parameter estimation ofD.
The track representation of a geolocation result is evaluated by determining a mean track, a mode track and the Most Probable Track according to Section3.5.
The simulation model is not an attempt to make an entirely realistic model of reality, it is merely a mean to assess and illustrate the properties of the ltering technique.
4.1 Construction of the model
The simulated geolocation will rely on depth measurements in an articial do-main. Environmental variables such as light, temperature and tidal information add a complexity to the model that is unwanted in this simulation and are therefore not included.
4.1.1 Bathymetry
The domain is constructed in close resemblance to the bathymetry of a real life situation e.g. with islands and varying depth gradient in order to achieve a non-trivial simulation.
The articial lake that is used for most simulations is constructed in Mat-lab based on the surface created by the command peaks. This is modied by means of simple arithmetic operations to become a lake as shown in Figure4.1.
The lake is discretised with101×101 grid points.
Figure 4.1: Bathymetry for simulation.
The domain contains three small islands that serves as a test for the handling of nonlinearities. The lake has very shallow areas close to the border of the domain and deeper areas near the middle. These gradients in depth and their eect on the uncertainty of the geolocation will be revealed from this bathymetry as well.
4.1.2 Simulation of random walk in the domain
The movement model for the sh is a two dimensional homogeneous random walk with transition probabilities according to (2.10) for each coordinate di-rection. The value of r is assumed to be constant in time. With this scheme each coordinate can maximally increase with hover a time step ofk. For this simulation, the values of the increment parameters for time and space are for simplicity dened as
k = 1, h = 1.
4.2 Likelihood estimation of D 31