- Page 1: Probabilistic Performance Analysis
- 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 and 52: 1 P d,k P f,k β Q 0,k Probability
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In general, as Q 0,k decreases, the
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from J k by taking column-sums. If
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Chapter 4 Computational Framework 4
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Fact 4.1. Given a Markov chain θ
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v 1 v 2 v 3 v 4 Figure 4.1. Simple
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for some p ∈ (0,1). Then, the cor
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Proof. Let ϑ 0:l be a possible pat
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the matrix A as in Theorem 4.12. Th
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every 1-bit of b i is a 1-bit of b
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Assumed Structure of the Residual G
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Therefore, conditional on the event
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In the non-scalar case (i.e., r k
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probability matrix is given by ( Λ
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s 2 . .. s 0 s 1 s q Figure 4.4. St
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metrics at time k are defined as Ĵ
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Proposition 4.32. Let N be the fina
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Algorithm 4.2. Procedure for comput
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Second, we show that the running ti
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Table 4.1. Time-complexity of compu
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For p ∈ [1,∞], the l p -norm ba
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linear time-varying uncertainties
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for the appropriate choice of k f a
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Hence, it is straightforward to sho
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v w f (θ) u G θ y . F r Figure 5.
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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.