Dear Bugs' users
I wonder if somebody can help me to solve a modeling problem.
I am studying the degradation of water meters. My idea is that the meter,
during its lifetime, passes trough 4 differents states:
1) Perfect
2) Fair
3) Bad
4) Very Bad
The jump from a state to another is ruled by a Markov model in which the
transition probabilities depend on the age of the meter (t):
Transition State i --> State j : logit(Pij(t)) = Aij + Bij*t
Data consist in samples of meters of the same age, the states of which are
observed.
For the age t I have N(t) observation:
N1(t) meters are perfect, N2(t) are fair, N3(t) are bad and N4(t) are very
bad, whith SUM(Ni(t))=N(t)
So the vector [N1(t),N2(t),N3(t),N4(t)] is a result of a multinomial
sampling of parameters [P1(t),P2(t),P3(t),P4(t)] and N(t).
The probabilities Pi(t) are functions of the Pij(t).
My biggest problem is generating, given t and i, 4 transition probabilities
Pij(t) the sum of which is equal to one. I cannot use a Dirichlet prior
because pij(t) are not the parameters of the multinomial, and that is the
only case in which Dirichlet distribution is allowed. Furthermore I do not
know how to develop a sort of rejecting rule who looks like: "IF Sum(Pij) >
1 THEN Go Back and recalculate" in WinBugs environment.
What I do is generating 4 independent pseudo-probabilities Qij and after
calculating Pij as a function of Qijs. For instance:
logit(Q11(t)) = C11 + D11*t
(...)
logit(Q14(t) = C14 + D14*t
and after trivially:
P11(t) = Q11(t)/(Q11(t)+Q12(t)+Q13(t)+Q14(t))
(...)
P14(t) = Q14(t)/(Q11(t)+Q12(t)+Q13(t)+Q14(t))
Does somebody has a more "elegant" solution to solve the problem? Or maybe
suggesting a paper in which a similar problem is discussed?
Many thanks
Alberto.
------------------------------------------------------------
Alberto Pasanisi, Doctorant.
Ecole Nationale du Génie Rural, des Eaux et des Forets.
Laboratoire GRESE - Gestion du Risque en Sciences de l'Eau.
19 Avenue du Maine. 75732 PARIS CEDEX 15
Tél. 01 45 49 89 21 - Fax 01 45 49 88 27
http://www.engref.fr
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