From: Ruben Roa Ureta <rroa_at_udec.cl>

Date: Tue, 08 Apr 2008 20:38:39 -0400 (CLT)

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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Wed 09 Apr 2008 - 00:41:48 GMT

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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