Biomedical Engineering Reference
In-Depth Information
P oissonf 0 (t)g; for r = 1; : : : ; k1, let f r fi; jjI r (t)gdenote the prob-
abilityfB r (t) = i; D r (t) = jgfrom the multinomial distribution
fB r (t); D r (t)gML[I r (t); b r (t); d r (t)] and let h r fjji r ; j r ; I r (t)gbe the
probability of (M r (t) = j) from the binomial distribution M r (t)
BinomialfI r (t)i r j r ;
r (t)
1b r (t)d r (t) g: Then, from results in Section (4.1),
the probability density function of X =f X (1); : : : ; X (t M )gis
t M
Y
P (X) =
fP [ X (t)j X (t1)]g
t=1
and for t = 0; 1; : : : ; t M 1,
I 1 (t)
I 1 (t)i 1
X
X
Pf X (t + 1)j X (t)g=
f 1 fi 1 ; j 1 jI 1 (t)gg 0 fa 1 (t); 0 (t)g
i 1 =0
j 1 =0
I 2 (t)
I 2 (t)i 2
X
X
f 2 fi 2 ; j 2 jI 2 (t)gh 1 fa 2 (t)ji 2 ; j 2 ; I 2 (t)g
i 2 =0
j 2 =0
I k1 (t)
I k1 (t)i k1
X
X
f k1 fi k1 ; j k1 jI k1 (t)g
i k1 =0
j k1 =0
h k2 fa k1 (t)ji k1 ; j k1 ; I k1 (t)g;
(4)
where a r (t) = Max(0; I r+1 (t)I r (t)i r + j r ).
5. A State Space Model for the Extended Multi-Event
Model of Carcinogenesis
State space model is a stochastic models which consists of two sub-
models: The stochastic system model which is the stochastic model of the
system and the observation model which is a statistical model based on
available observed data from the system. Hence it takes into account the
basic mechanisms of the system and the random variation of the system
through its stochastic system model and incorporate all these into the ob-
served data from the system; furthermore, it validates and upgrades the
stochastic model through its observation model and the observed data of
the system. Thus the state space model adds one more dimension to the
stochastic model and to the statistical model by combining both of these
models into one model. As illustrated in 56 , Chapters 8-9, the state space
model has many advantages over both the stochastic model and the statis-
tical model when used alone since it combines information and advantages
from both of these models.
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