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Subject:

A challenging problem

From:

"Khayat, Etienne" <[log in to unmask]>

Reply-To:

Khayat, Etienne

Date:

Thu, 20 Oct 2005 14:39:36 +0100

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (78 lines)

Hi everyone, 

I have been struggling with this problem since a while now with no luck in
solving it. I find it very challenging, but also, I don't consider myself as
an expert in probability and statistics. I really hope and appreciate very
much if someone can help me with a hint or a solution for it. It is problem
summarised below.

Many thanks

Etienne

****************************************************************************
**********
We need to classify the risk of some computer processes Pi (where i in
[1..n], where n is an integer of finite value, say 20), in risk levels.

As the risk level of every Pi is not known in advance, we choose to classify
them those Pi in a maximum of n risk levels, in case every one of those
processes turned out was in a different risk level. I informally call a risk
level as a risk band, or simply a band. Band goes from 1 to n, also.

The data about the risks of those processes is available in the following
forms:

(Pi<Pj, 0.6) meaning that the risk level of Pi is lower than that of Pj with
a degree of confidence of 60%. Informally, the risk analyst is 60% sure of
this statement (a kind of fuzziness). 

Also, for the same i and j, we have:

(Pi=Pj, 0.3) and (Pi>Pj, 0.1). The sum of the assertion of the 3 different
statements about every relation between two permissions is always 1
(0.6+0.3+0.1).

So we are provided with such statements about processes in the system in the
way shown. Not every relation between Pi and Pj (for all i and j) has risk
data available, but we have a good subset of relations between many of the
system's processes.

Since there is no total assertion of the relations between the risk level of
processes, there isn't also a total assertion about which risk level every
process lies in. There is a kind of distribution of the membership of every
process within the risk levels (bands). For instance, Pi can be found to be
in band j with a membership value of 0.25, and in band k with a membership
value of 0.46.

A fact is, since every Pi can be in any of those n bands mean that for j=1
to n: 

Sum(membership (Pi in band j) =1

The goal is to calculate the degree of memberships of every Pi (i=1 to n) in
every Band k (k=1 to n). To be more concrete, I consider n=20.

For this I have adopted a probabilistic style approach, because the degree
of membership of a process in a band is similar to the idea of the
likelihood of the process to be in this band. 

I have tried many solutions, but I couldn't get anywhere. Also the number of
scenarios to consider is very large and impossible to calculate on a PC. So,
I have tried to do some simulations, with no luck, because the conditions
for the simulation technique I used (Monte Carlo) were not met.

 
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