Probabilistic Performance Analysis of Fault Diagnosis Schemes

1
ideal
roc
Pd,k
Probability of Detection,
0.5
(α,β)
W k
( 12
, 1 2
)
(1 − α,1 − β)
0
0 0.5 1
Probability of False Alarm,
P f,k
Figure 3.3. Visual summary of Facts 3.6–3.8. At each time k, the set W k is convex, it contains the
extreme points (0,0) and (1,1), and it is symmetric about the point ( 1
2 , 1 2
)
.
Fact 3.8. The set W k is symmetric about the point ( 1
2 , 1 2)
, in the sense that if (α,β) ∈ Wk , then
(1 − α,1 − β) ∈ W k , as well.
Proof. Let (α,β) ∈ W k and take V ∈ V to be a test whose performance, at time k, is given by
(α,β). Define ¯V to be the test that always decides the opposite of what V decides. Then, the
probability of a false alarm for ¯V is 1 − α, and the probability of detection for V is 1 − β.
To summarize, at each time k, the set of achievable performance points W k is a convex
set that is symmetric about the point ( 1
2 , 1 2)
and contains the extreme points (0,0) and (1,1)
(see Figure 3.3). Although Facts 3.6–3.8 are well known and can be found in the literature
(e.g., [61]), the brief proofs provided here provide some insight into the structure of the
sets {W k } k≥0 .
3.4.2 Receiver Operating Characteristic
The ideal performance point (P f,k ,P d,k ) = (0,1) is achieved by a test that always chooses
the correct hypothesis. However, such perfect tests rarely exist, because the test statistic
(u 0:k , y 0:k ) contains only partial information about the parameter θ k . Indeed, the test statistic
is related to the parameter through the dynamics of the system G θ , which is unlikely to
yield a one-to-one relation. Moreover, the exogenous noise process {v k } corrupts the limited
information available about θ k . Therefore, the set W k of achievable performance points is
separated from the ideal (0,1) by a curve passing through (0,0) and (1,1).
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