How do I find the threshold in a Poisson-distributed process that gives a desired False Alarm rate?

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.)

Answer 1

[Matt J]

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).

Answer 2

[David]

Matt -

Thank you very much for your help. This will help me in the design of a new medical device that, if successful, will save many lives in the future. FYI: the problem arises in flow cytometry when laser excitement causes background fluorescence events which are Poisson-distributed.

Thanks again, David

[Matt J]

Well, I hope it works then ;)