- #1
RichardJ
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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.
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.
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.
So my questions are (for now):
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.
- Say we are interested in the next state (i.e. position and velocity) based on the current data. $$p(s_{k+1}|Z_k)$$
- We can write this down something like:$$ p(s_k|z_k,H_0) p(H_0|z_k) $$$$ p (s_k|z_k, H_i) p(H_i|z_k)$$
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.5. For the other terms i = 1,...,M something like $$ p(z_k^1,...,z_k^M|s_k,H_i) -> p(z_k^i|s_k)(1/V)^{M-1} $$
So my questions are (for now):
- How to obtain step 2, the multiple hypothesis formulation
- How is step 3 actually related to 1 and 2.
- Why does relation 4 hold?
- How to write down/solve 5 for all hypothesis.
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