Modelos para Dados de Contagem com Estrutura Temporal
Modelos para Dados de Contagem com Estrutura Temporal
Modelos para Dados de Contagem com Estrutura Temporal
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A.1.3<br />
Mo<strong>de</strong>lo Poisson-Lognormal Dinâmico<br />
• µ = (µ 1 , . . . , µ T ):<br />
p(µ | D T , Ψ −µ)<br />
• µ t , <strong>para</strong> t = 1, . . . , T − 1:<br />
∝<br />
T∏<br />
exp[− exp(µ t )δ t ] exp(µ t y t )<br />
t=1<br />
×<br />
T∏<br />
t=1<br />
(A.18a)<br />
{<br />
exp − 1<br />
}<br />
2W (µ t − µ t−1 ) 2 ; (A.18b)<br />
p(µ t | D T , Ψ −µt ) ∝ exp[− exp(µ t )δ t ] exp(µ t y t ) (A.19a)<br />
{<br />
× exp − 1<br />
}<br />
2W (µ t − µ t−1 ) 2 (A.19b)<br />
{<br />
× exp − 1<br />
}<br />
2W (µ t+1 − µ t ) 2 ; (A.19c)<br />
• µ T :<br />
p(µ T | D T , Ψ −µT ) ∝ exp[− exp(µ T )δ T ] exp(µ T y T ) (A.20a)<br />
{<br />
× exp − 1<br />
}<br />
2W (µ T − µ T −1 ) 2 ; (A.20b)<br />
• µ 0 :<br />
p(µ 0 | D T , Ψ −µ0 ) ∝ exp<br />
{<br />
− C 0 + W<br />
2C 0 W<br />
[<br />
µ 0 − C ] } 2<br />
0µ 1 + W m 0<br />
, (A.21)<br />
C 0 + W<br />
o que implica que<br />
• W :<br />
µ 0 | D T , Ψ −µ0 ∼ N<br />
( )<br />
C0 µ 1 + W m 0<br />
C 0 + W , C 0 W<br />
; (A.22)<br />
C 0 + W<br />
p(W | D T , Ψ −W ) ∝ W −(T/2+α W +1)<br />
{ [ T∑<br />
× exp − 1 W<br />
t=1<br />
(µ t − µ t−1 ) 2<br />
2<br />
+ β W<br />
]}<br />
(A.23a)<br />
, (A.23b)<br />
o que implica que<br />
W | D T , Ψ −W ∼ IG<br />
(<br />
T<br />
2 + α W ,<br />
T∑ (µ t − µ t−1 ) 2<br />
t=1<br />
2<br />
+ β W<br />
)<br />
; (A.24)<br />
132