Probabilistic Performance Analysis of Fault Diagnosis Schemes

4.4 Decision Functions
The final step in evaluating the performance metrics is to compute the probabilities
∫
P(d k = j | θ 0:k = ϑ 0:k ) = p r |θ (r k | θ 0:k = ϑ 0:k ) dr k , (4.21)
E j,k
where
E j,k := {r k : δ(k,r k ) = j }.
Assuming that the dynamics are conditionally linear-Gaussian, as in Section 4.3, the conditional
distribution p r |θ (r k | θ 0:k = ϑ 0:k ) is the Gaussian N ( ˆr k ,Σ k ). Although these assumptions
generally make computation easier, the set E j,k must be simple enough to enable
computation of the integral (4.21). In this section, we provide some practical examples of
decision functions for which computation is tractable.
4.4.1 Threshold Decision Functions
First, consider the case where {r k } is scalar-valued. One common decision function, used
frequently in fault detection [9, 32], is a time-varying threshold function of the form
⎧
⎨0, if |r k | < ε k ,
δ(k,r k ) :=
⎩1, otherwise,
where ε k > 0, for all k. Hence, E 0,k = [−ε k ,ε k ], and the integral (4.21) can be written in terms
of the density of N ( ˆr k ,Σ k ) as
∫ εk
1
P(D 0,k | θ 0:k = ϑ 0:k ) = exp
(− (r k − ˆr k ) 2 )
dr k . (4.22)
−ε k 2πΣk 2Σ k
Since r k is scalar, the error function, defined in Section 2.2.6, can be used to write the
conditional cumulative distribution function of r k ∼ N ( ˆr k ,Σ k ) as
P(r k < c | θ 0:k = ϑ 0:k ) = 1 2
[ ( c − ˆrk
1 + erf
)],
2Σk
for all c ∈ R. Similarly, the integral (4.22) can be written as
P(D 0,k | θ 0:k = ϑ 0:k ) = 1 2
[ ( ) (
εk − ˆr k −εk − ˆr k
erf − erf
)].
2Σk 2Σk
Since the error function can be approximated by a rational function with a maximum relative
error less than 6 × 10 −19 [17], this expression can be evaluated accurately in O(1) time.
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