Probabilistic Performance Analysis of Fault Diagnosis Schemes

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ε increasing
ε = 0
Pd,k
Probability of Detection,
0.5
0
ε → ∞
0 0.5 1
Probability of False Alarm,
P f,k
Figure 3.4. Set of performance points achieved by the family of tests given in equation (3.18). Varying
the threshold ε yields a curve of performance points passing through (0,0) and (1,1). Randomization
can be used to achieve any performance in the convex hull of this curve (shaded region).
availability is to require that
P tn,k > a,
for k = 0,1,..., N . This type of bound is shown in Figure 3.5(a), where the constraint fails to
hold for k > k f . In terms of the performance metrics, the availability may be written as
P tn,k = (1 − P f,k )Q 0,k ,
for all k. Thus, the lower bound on availability can be translated to a time-varying upper
bound on P f,k , as follows:
P f,k < 1 −
a ,
Q 0,k
for k = 0,1,..., N . This type of bound is shown in Figure 3.5(b). Note that no fault detection
scheme can satisfy the bound on availability once Q 0,k ≤ a.
Given β > α > 0, another natural performance criterion is to assert that the performance
metrics P f,k and P d,k satisfy the constraints
P f,k < α and P d,k > β,
for all k. A visualization of this type of bound is shown in Figure 3.6. In Figure 3.7, this
constraint can be visualized in roc space by plotting the roc curves at a number of time
steps {k 0 ,k 1 ,...,k m }. Unlike P tn,k which eventually converges to 0, the metrics P f,k and
P d,k often converge to steady-state values, so the visualization in Figure 3.7 can depict the
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