Probabilistic Performance Analysis of Fault Diagnosis Schemes

1
(α 3 ,β 3 )
Pd,k
Probability of Detection,
0.5
(α 2 ,β 2 )
(α 1 ,β 1 )
(α,β)
(α 4 ,β 4 )
0
0 0.5 1
Probability of False Alarm,
P f,k
Figure 3.2. Illustration of Example 3.5 showing the range of performance points (shaded region)
achievable by randomly selecting the decision made by one of four deterministic tests.
probability of a false alarm for this test is
P f,k (V p ) = P f,k (V 1 )γ + P f,k (V 2 )(1 − γ) = α 1 γ + α 2 (1 − γ) = α.
Similarly, the probability of detection is
P d,k (V p ) = P d,k (V 1 )γ + P d,k (V 2 )(1 − γ) = β 1 γ + β 2 (1 − γ) = β.
Hence, V p has the desired performance at time k, and W k is convex.
Fact 3.7. The set W k contains the points (0,0) and (1,1).
Proof. Let V no ∈ V be the test makes the decision d k = 0, for all k. Similarly, let V yes ∈ V be
the test that makes the decision d k = 1, for all k. The performance of the test V no is clearly
(0,0), while the performance of V yes is (1,1).
Since W k is convex and always contains the points (0,0) and (1,1), W k also contains the
point (γ,γ), for any γ ∈ (0,1). One test that achieves performance (γ,γ), is the randomized
test that uses V no with probability 1 − γ and V yes with probability γ. Since such tests make
random decisions, independent of the value of the test statistic (u 0:k , y 0:k ), they are often
called uninformative tests [73]. Hence, we are mostly concerned with tests whose performance
point is above the diagonal (i.e., P d,k > P f,k ). However, the following fact shows that
a test whose performance point falls below the diagonal can also be useful.
32