Biomedical Engineering Reference
In-Depth Information
5.4.1
Transition probabilities
Assuming that we know the current state of the system, it is useful to predict the state
that the system might be in at some time t later. Let
X
(
t
)
denote the state occupied
,
=
>
by the mechanism at time
t
then, for
i
j
1to
m
and
t
0, we define transition
(
)
probabilities
p
ij
t
by
p
ij
(
t
)=
P
(
X
(
t
)=
j
|
X
(
0
)=
i
)
Then by the total probability theorem, if we consider the situation at times
t
and
t
+
∆
t
,
)=
∑
k
p
ij
(
t
+
∆
t
p
ik
(
t
)
Prob
(
X
(
t
+
∆
t
)=
j
|
X
(
0
)=
i
and
X
(
t
)=
k
)
But the Markov property implies that the condition in the second factor of this
expression can be replaced by the condition
X
(
)=
t
k
only (the most recent thing
known) so that we have, approximately,
)=
∑
k
=
j
p
ik
(
t
)
q
kj
∆
p
ij
(
t
+
∆
t
t
+
p
ij
(
t
)
{
1
+
q
jj
t
}
Then the derivative
p
ij
(
=
∑
k
t
)=
lim
0
{
p
ij
(
t
+
∆
t
)
−
p
ij
(
t
)
}/
∆
t
p
ik
(
t
)
q
kj
t
→
or, if the square matrix
P
(
t
)
has elements
p
ij
(
t
)
we have the matrix differential equa-
tion
P
(
t
)=
P
(
t
)
Q
(5.6)
which has formal solution
P
(
t
)=
exp
(
Q
t
)
,
t
>
0
(5.7)
The initial value is
P
(
0
)=
I
, an identity matrix, because
p
ii
(
0
)=
1 as the system
cannot move anywhere in zero time.
So what is the exponential of a matrix? We can define it by a matrix version of the
usual series expansion
∞
∑
n
=
1
t
n
n
!
Q
n
exp
(
Q
t
)=
I
+
A 'cookbook' approach to programming the calculations of this matrix function is
provided in [32] but, theoretically, the nicest result arises from the so-called spectral
expansion of the matrix
Q
m
i
=
−
∑
i
A
i
, where
i
the are the eigenvalues of the
=
1
matrix
Q
. The spectral matrices
A
i
may be calculated from the eigenvectors of
Q
.
The details need not concern us here but the important thing to note is the property
that
−
A
i
=
A
i
A
j
=
0
,
i
=
j
;
A
i
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