How do I find the threshold in a Poisson-distributed process that gives a desired False Alarm rate?
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Given that I am trying to find signals in a Poisson-distributed background process, how do I find the threshold that will reduce my [Poisson-]noise false alarm rate to a desired level? Say I know the mean for the Poisson process (lamda) and I want a false alarm rate of no more 10^(-N), how do I find the appropriate threshold for the non-closed solution Poisson distribution? Should I just create a lamda-dependent look-up table? (e.g; TH = f(lamda, N) where FAR = 10 ^(-N)?) If so, could someone recommend a reliable source for those values? (I see some online but many do not cover a broad enough set of possible values.)
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So, just so I understand your terminology and symbols, a "false alarm" is said to occur any time one in a sequence of the N Poisson samples in the process goes above your threshold, TH ? And all the samples in the process are i.i.d. with known parameter lambda?
If so, then you want
Prob(X(i)<=TH, i=1...N) >= 1-10^(-N)
Using the i.i.d. assumption this becomes,
CDF(TH) >= log(1-10^-N)^(1/N)
where CDF(TH) is the cumulative distribution function Prob(X(1)<=TH; lambda).
So, yes if you have a table of the CDF values, you could find the above threshold. Or, you could use FMINSEARCH to minimize
f(TH) = abs(CDF(TH) - log(1-10^-N)^(1/N) )
With FMINSEARCH, you do not need a closed form expression for f(TH).
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