Probabilistic Performance Analysis of Fault Diagnosis Schemes

1
P d,k
P f,k
β
Q 0,k
Probability
α
0
0 N
Time, k
Figure 3.6. Visualization of a constraint on the performance metrics {P f,k } and {P d,k } over time. Here,
the constraint is P d,k > β and P f,k < α, for k = 0,1,..., N . The marginal probability that the system is
in the nominal mode, denoted {Q 0,k }, is shown for reference.
steady-state performance metrics if k m is large enough.
3.5.2 Bound on Bayesian Risk
As discussed in Section 3.3.3, the Bayesian risk provides a general linear framework for
aggregating the performance of a fault detection scheme into a single performance metric.
For the sake of simplicity, assume that the loss matrix L ∈ R 2 is constant for all time. Given a
sequence { ¯R k }, such that ¯R k > 0 for all k, the bound on the Bayesian risk at time k is
R k (Q,V ) = L 00 Q 0,k + L 01 Q 1,k + (L 01 − L 00 )P f,k Q 0,k + (L 11 − L 10 )P d,k Q 1,k < ¯R k .
At each k, the set of performance points (P f,k ,P d,k ) satisfying this bound is the intersection
of some half-space in R 2 with the roc space [0,1] 2 (see Figure 3.8). The boundary of this
half-space is determined the loss matrix L and the probability Q 0,k . Clearly, if the ideal
performance point (0,1) does not lie in this half-space at time k, then the bound R k < ¯R k is
too stringent.
Note that as Q 0,k → 1, the bound on risk approaches
L 00 + (L 01 − L 00 )P f,k < ¯R ⇐⇒ P f,k < ¯R − L 00
L 01 − L 00
.
Similarly, as Q 0,k → 0, the bound approaches
L 01 + (L 11 − L 10 )P d,k < ¯R ⇐⇒ P d,k > L 01 − ¯R
L 10 − L 11
.
38