Probabilistic Performance Analysis of Fault Diagnosis Schemes

deterministic tests.
Example 3.5. One common way to produce a randomized test is to randomly select a test from
some finite collection of deterministic tests {V 1 ,V 2 ,...,V m } ⊂ V and use the decision produced by
that test. More precisely, let p be a point in the simplex
{
S m := p ∈ R m : p i ≥ 0,
m∑
i=1
}
p i = 1 ,
and define λ to be a random variable that takes values in the set {1,2,...,m}, such that
Let the randomized test V p be defined by
P(λ = i ) = p i .
V p (u 0:k , y 0:k ) := V λ (u 0:k , y 0:k ), (3.14)
for all k and all (u 0:k , y 0:k ). Then, probability of a false alarm for V p at time k is
P f,k (V p ) = P(D 1,k | H 0,k )
m∑
= P(D 1,k | H 0,k ,λ = i ) P(λ = i )
=
i=1
m∑
P f,k (V i ) p i .
By a similar calculation, the probability of detection for V p at time k is
i=1
P d,k (V p ) =
m∑
P d,k (V i ) p i .
i=1
The case m = 4 is shown in Figure 3.2, where the shaded region represents the performance
points achieved by the family of randomized tests {V p } p∈S4 obtained using this method.
Fact 3.6. The set of achievable performance points W k is convex.
Proof. Let (α 1 ,β 1 ) and (α 2 ,β 2 ) be any two points in W k , and let V 1 and V 2 , respectively, be
tests in V that achieve these performance points at time k. Let γ ∈ [0,1]. To show that W k is
convex, we must exhibit a test with performance
(α,β) := γ(α 1 ,β 1 ) + (1 − γ)(α 2 ,β 2 ),
at time k. Since the point p := (γ,1 − γ) is in the simplex S 2 , we can use the procedure
outlined in Example 3.5 to construct a randomized test V p that utilizes V 1 and V 2 . The
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