Probabilistic Performance Analysis of Fault Diagnosis Schemes

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1 Pd,k Probability of Detection, β 0.5 (α,β) k increasing 0 0 α 0.5 1 Probability of False Alarm, P f,k Figure 3.7. Visualization of a constraint on the performance metrics P f,k and P d,k in roc space. Unlike Figure 3.6, which shows the performance for a single test V = (F,δ), this visualization shows the performance over an entire family of tests. However, it is less clear in this visualization which curve corresponds to a given point in time. 1 Pd,k Probability of Detection, 0.5 k increasing (Q 0,k decreasing) 0 0 0.5 1 Probability of False Alarm, P f,k Figure 3.8. Visualization of a constraint on Bayesian risk in roc space. Each blue line represents the Bayesian risk bound at a different time step. Note that as Q 0,k decreases with time, the slope of the bound decreases and the probability of detection P d,k plays a more significant role in satisfying the constraint. A roc curve corresponding to a single time step is plotted for reference. 39
In general, as Q 0,k decreases, the slope of the boundary line that delineates the set of acceptable performance points also decreases. Hence, from a Bayesian risk perspective, when Q 0,k is large and faults are unlikely to occur, it is more important to avoid false alarms. On the other hand, when Q 0,k is small and faults are likely to occur, it is more important to detect faults. Figure 3.8 shows a typical plot of the evolution of the Bayesian risk bound through time. 3.6 Extension to Fault Isolation and Identification In this section, we extend our performance analysis to fault isolation and identification problems. As in the fault detection case, there is a set of joint probabilities that fully characterizes the performance, and a set of conditional probabilities that characterize the performance relative to the marginal probabilities of the hypotheses being considered. We show that these sets of performance metrics are equivalent. We also show how the concept of Bayesian risk is defined in the multi-hypothesis case. Finally, we provide some brief comments on how the roc curve can be extended, as well. 3.6.1 Quantifying Accuracy Consider the general fault isolation problem, where the parameter space is partitioned as Θ = Θ 0 ⊔ Θ 1 ⊔ ··· ⊔ Θ q , for some q > 1. As in the simpler fault detection case, Θ 0 = {0} represents the nominal parameter value, while the set Θ i , for i > 0, represents the i th class of faulty behavior. If Θ is finite, fault identification can be achieved by taking each Θ i to be a singleton set. The corresponding set of decisions is D := {0,1,..., q}. At each time k, define the events D i ,k := {d k = i } and H j,k := {θ k ∈ Θ j }, for all i , j ∈ D. The performance metrics (3.1)–(3.4) are extended to the multi-hypothesis case by the performance matrix J k ∈ R (q+1)×(q+1) , which is defined as J k (i , j ) := P(D i ,k ∩ H j,k ), i , j ∈ D. Hence, J k can be viewed as a confusion matrix for the multi-hypothesis case. Because {D 0,k ,D 1,k ,...,D q,k } and {H 0,k , H 1,k ,..., H q,k } form partitions of the sample space Ω, the performance matrix satisfies identities analogous to those in equations (3.6)–(3.10). As in 40
Page 1 and 2: Probabilistic Performance Analysis
Page 3 and 4: Abstract Probabilistic Performance
Page 5 and 6: Soli Deo gloria. i
Page 7 and 8: 3 Probabilistic Performance Analysi
Page 9 and 10: List of Figures 2.1 “Bathtub” s
Page 11 and 12: List of Tables 4.1 Time-complexity
Page 13 and 14: Acknowledgements When I started wri
Page 15 and 16: tic metrics that rigorously quantif
Page 17 and 18: tion. • Complexity of Markov Chai
Page 19 and 20: Given an event B ∈ F with P(B) >
Page 21 and 22: Note that E ( f (x) | y ) is a rand
Page 23 and 24: for all c ∈ R, where erf(c) := 2
Page 25 and 26: Hazard Rate λ 0 0 break-in 0 t 1 t
Page 27 and 28: malfunction — an intermittent irr
Page 29 and 30: where the matrix Q is chosen to app
Page 31 and 32: v w u G θ y F r δ V d Figure 2.3.
Page 33 and 34: These relationships can be used to
Page 35 and 36: ε Residual 0 0 T f T d Time Figure
Page 37 and 38: Chapter 3 Probabilistic Performance
Page 39 and 40: the worst-case performance under a
Page 41 and 42: For example, P tp,k = P(D 1,k ∩ H
Page 43 and 44: where the subscript k has been omit
Page 45 and 46: 1 (α 3 ,β 3 ) Pd,k Probability of
Page 47 and 48: Definition 3.9. The upper boundary
Page 49 and 50: 1 ε increasing ε = 0 Pd,k Probabi
Page 51: 1 P d,k P f,k β Q 0,k Probability
Page 55 and 56: from J k by taking column-sums. If
Page 57 and 58: Chapter 4 Computational Framework 4
Page 59 and 60: Fact 4.1. Given a Markov chain θ
Page 61 and 62: v 1 v 2 v 3 v 4 Figure 4.1. Simple
Page 63 and 64: for some p ∈ (0,1). Then, the cor
Page 65 and 66: Proof. Let ϑ 0:l be a possible pat
Page 67 and 68: the matrix A as in Theorem 4.12. Th
Page 69 and 70: every 1-bit of b i is a 1-bit of b
Page 71 and 72: Assumed Structure of the Residual G
Page 73 and 74: Therefore, conditional on the event
Page 75 and 76: In the non-scalar case (i.e., r k
Page 77 and 78: probability matrix is given by ( Λ
Page 79 and 80: s 2 . .. s 0 s 1 s q Figure 4.4. St
Page 81 and 82: metrics at time k are defined as Ĵ
Page 83 and 84: Proposition 4.32. Let N be the fina
Page 85 and 86: Algorithm 4.2. Procedure for comput
Page 87 and 88: Second, we show that the running ti
Page 89 and 90: Table 4.1. Time-complexity of compu
Page 91 and 92: For p ∈ [1,∞], the l p -norm ba
Page 93 and 94: linear time-varying uncertainties
Page 95 and 96: for the appropriate choice of k f a
Page 97 and 98: Hence, it is straightforward to sho
Page 99 and 100: v w f (θ) u G θ y . F r Figure 5.
Page 101 and 102: Maximizing the Probability of False
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β ∆ α v f (θ) u G θ y . F r F
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β ∆ α β ∆ 1 . .. ∆q α (a)
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if and only if [ T (βi ) T T (M i
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Fix a parameter sequence θ = ϑ an
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As in Section 5.2.2, if the matrix
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Chapter 6 Applications 6.1 Introduc
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p t p s v t + f t (θ) φ γ ˆV ĥ
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300 18 V (m/s) 200 h (km) 12 Airspe
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1 P tn,k 0.8 P fp,k P fn,k (a) Prob
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0.4 Probability of False Alarm, P
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The following matrices correspond t
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6.4.2 Applying the Framework The ma
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P ⋆ f Probability of False Alarm,
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Chapter 7 Conclusions & Future Work
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measurement noise. Hence, the appli
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[12] R. H. Chen, D. L. Mingori, and
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[40] R. L. Graham, D. E. Knuth, and
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[68] L. A. Mironovski, Functional d
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[95] H. B. Wang, J. L. Wang, and J.
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Probabilistic Performance Analysis of Fault Diagnosis Schemes

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