Probabilistic Performance Analysis of Fault Diagnosis Schemes

In general, as Q 0,k decreases, the slope of the boundary line that delineates the set of
acceptable performance points also decreases. Hence, from a Bayesian risk perspective,
when Q 0,k is large and faults are unlikely to occur, it is more important to avoid false alarms.
On the other hand, when Q 0,k is small and faults are likely to occur, it is more important
to detect faults. Figure 3.8 shows a typical plot of the evolution of the Bayesian risk bound
through time.
3.6 Extension to Fault Isolation and Identification
In this section, we extend our performance analysis to fault isolation and identification problems.
As in the fault detection case, there is a set of joint probabilities that fully characterizes
the performance, and a set of conditional probabilities that characterize the performance
relative to the marginal probabilities of the hypotheses being considered. We show that
these sets of performance metrics are equivalent. We also show how the concept of Bayesian
risk is defined in the multi-hypothesis case. Finally, we provide some brief comments on
how the roc curve can be extended, as well.
3.6.1 Quantifying Accuracy
Consider the general fault isolation problem, where the parameter space is partitioned as
Θ = Θ 0 ⊔ Θ 1 ⊔ ··· ⊔ Θ q ,
for some q > 1. As in the simpler fault detection case, Θ 0 = {0} represents the nominal
parameter value, while the set Θ i , for i > 0, represents the i th class of faulty behavior. If Θ
is finite, fault identification can be achieved by taking each Θ i to be a singleton set. The
corresponding set of decisions is
D := {0,1,..., q}.
At each time k, define the events
D i ,k := {d k = i } and H j,k := {θ k ∈ Θ j },
for all i , j ∈ D. The performance metrics (3.1)–(3.4) are extended to the multi-hypothesis
case by the performance matrix J k ∈ R (q+1)×(q+1) , which is defined as
J k (i , j ) := P(D i ,k ∩ H j,k ), i , j ∈ D.
Hence, J k can be viewed as a confusion matrix for the multi-hypothesis case. Because
{D 0,k ,D 1,k ,...,D q,k } and {H 0,k , H 1,k ,..., H q,k } form partitions of the sample space Ω, the
performance matrix satisfies identities analogous to those in equations (3.6)–(3.10). As in
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