# Re: [R] Overlapping distributions (populations) - assigning an individual to a population?

From: Ruben Roa Ureta <rroa_at_udec.cl>
Date: Tue, 08 Apr 2008 20:38:39 -0400 (CLT)

> Rolf,
>
>
> On Wed, 2008-04-09 at 10:57 +1200, Rolf Turner wrote:
>> On 9/04/2008, at 10:30 AM, Phil Rhoades wrote:
>>
>> > People,
>> >
>> > Say a particular measure of an attribute for individuals in different
>> > populations gives a set of overlapping normal distributions (one
>> > distribution per population). If I then measure this attribute in
>> > a new
>> > individual - how do I assess the likelihood of this new individual
>> > belonging to each of the different populations?
>>
>> You have a mixture of distributions. Let the density be
>>
>> k
>> f(x) = SUM lambda_i * f_i(x)
>> i=1
>>
>> where the f_i(x) are the densities for the individual components in
>> the mixture,
>> and the lambda_i are the mixing probabilities.
>>
>> The probability that an individual with observation x is from
>> component i is
>>
>> lambda_i * f_i(x)
>> -----------------
>> f(x)
>
>
> Thanks for the quick response but I think I need to put some numbers on
> this so I can see what you mean. Say I have two pops with individual
> values:
>
> 1 2 3 4 5
>
> 3 4 5 6 7
>
> and a new individual with value 5 - what is the likelihood of assignment
> to each of the populations?

Phil, for an application and more detailed explanation you can check the article:
A Test for Long-Term Cyclical Clustering of Stock Market Regimes John Powell, Rubén Roa, Jing Shi, Viliphonh Xayavong Australian Journal of Management, vol. 32(2), 2007, available for free download from the journal website: http://www.agsm.edu.au/~eajm/current.html I provide there a quotation to a book by Hamilton on time series, where this technique is further explained.
By the way, the computation suggested is a conditional probability. Rubén

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