Probabilistic Performance Analysis of Fault Diagnosis Schemes

from J k by taking column-sums. If Q † k is the pseudoinverse of Q k [46], then
(J k Q † q k )(i , j ) = ∑
J k (i ,l)Q † k (l, j )
l=0
= J k (i , j )Q † k (j, j )
{ P(Di ,k ∩ H j,k ) P(H j,k ) −1 , if P(H j,k ) ≠ 0,
=
0, otherwise
= P(D i ,k | H j,k )
= C k (i , j ),
for all i , j ∈ D, so C k = J k Q † k . Hence, the pair (C k,Q k ) provides an alternate means of
quantifying performance that is numerically equivalent to the performance matrix J k .
Remark 3.13. At a high level, evaluating (C k ,Q k ) requires the same amount of effort as
evaluating J k , in the sense that both formulations have the same number of independent
quantities to compute. Indeed, the j th column-sum of C k is
q∑
q∑
C k (i , j ) =
i=0
i=0
( q )
⋃
P(D i ,k | H j,k ) = P D i ,k | H j,k = P(Ω | H j,k ) = 1,
i=0
so C k has (q + 1) 2 − (q + 1) independent entries. Also, the sum of all the elements of Q k is
q∑
q∑
Q k (i ,i ) =
i=0
i=0
( q )
⋃
P(H i ,k ) = P H i ,k = P(Ω) = 1,
i=0
so Q k has q independent entries. Therefore, in total, there are (q + 1) 2 − 1 quantities that
must be computed to obtain C k and Q k , which is the same as the number of independent
entries of J k . However, it is often the case that computing a single entry of J k is more
straightforward.
3.6.2 Bayesian Risk
As in the fault detection case, we can define a loss matrix L ∈ R (q+1)×(q+1) with nonnegative
entries, such that L i j reflects the subject loss of deciding d k = j when hypothesis H i ,k is
true. The corresponding Bayesian risk is given by
R k (Q,V ) =
q∑
i=0 j =0
q∑
q∑
L i j P(D j,k ∩ H i ,k ) =
i=0 j =0
q∑
q∑
L i j J k (j,i ) =
i=0 j =0
Of course, a different loss matrix L k can be used at each time step.
q∑
L i j C k (j,i )Q k (i ,i ).
42