Undergrad Multiple hypothesis testing for radar tracking in clutter

[RichardJ]

Hello All,

The goal is to formulate a multiple hypothesis test for a radar tracking problem when false alarms are occurring and to apply a particle filter on this update step, however I first need to come to/understand the multiple hypothesis formulation in this problem.

1. Say we are interested in the next state (i.e. position and velocity) based on the current data. $$p(s_{k+1}|Z_k)$$
2. We can write this down something like:$$ p(s_k|z_k,H_0) p(H_0|z_k) $$$$ p (s_k|z_k, H_i) p(H_i|z_k)$$

for i = 1,..., M
In this case $H_0$ is the hypothesis that there were no measurements from the target and $H_i$ are the M-1 hypothesis were there was a measurement from the target.

3. So apparently we can write this down as $$p(z_k^1,...,z_k^M|s_k) = \sum_{i=0}^M p(z_k^1,...,z_k^M|s_k,H_i) p(H_i)$$​

The final goal is to derive some equations from step 3. to obtain an expression for the likelihoods based on a Poission clutter model, probability of detection, false alarm rate and validated measurements. The only term left should then be something like: $p(z_k^i|s_k^j )$ The likelihood function (where j stands for the particle, which I already have an expression for) to be able to run the recursion with a particle filter.
Furthermore $$ p(H_0) = (1-P_D) P_{FA} (m=M) $$
where $P_{FA}$ is a poission distribution for 'FA' (or M in this case) false alarms. and $$ p(z_k^1,...,z_k^M|s_k,H_0)=(1/V)^M $$ because under H_0 we have no targets and one assumes false alarms are distributed uniformly.

4. $p(H_i) = P_D P_{FA}(M-1)$
5. For the other terms i = 1,...,M something like $$ p(z_k^1,...,z_k^M|s_k,H_i) -> p(z_k^i|s_k)(1/V)^{M-1} $$​

should hold. but not sure how to write down all the terms.

So my questions are (for now):

1. How to obtain step 2, the multiple hypothesis formulation
2. How is step 3 actually related to 1 and 2.
3. Why does relation 4 hold?
4. How to write down/solve 5 for all hypothesis.

 

[RPinPA]

Could you define some of your variables please? What are the $s$'s? What are the $z_k$? What are the $M-1$ hypotheses? You said they are "hypotheses where there was a measurement from the target" but why does "there is a measurement" have more than one variable? It's true or it isn't.