Fault Detection and Diagnosis Using Information Measures

Transcription

1 Fault Detection and Diagnosis Using Information Measures Rudolf Kulhavý Honewell Technolog Center & Institute of Information Theor and Automation Prague, Cech Republic

2 Outline Probabilit-based inference revisited Fundamentals of information geometr Finite-memor inference Minimum Relative Entrop MRE approximation Implementation Marov Chain Monte Carlo MCMC methods Brute-Force Alternative Monte Carlo Again: Weighted Bootstrap

3 Lielihood-based Inference General regression Model Lielihood function Posterior densit,, m m m m m m m m s c u u q l + + = = = m m u + + = =,,,,, Κ n R T s, 0 l p p = c

4 Information-based Inference Empirical densit r Conditional inaccurac Lielihood l K, = δ, = c exp K r Posterior densit = r : s r, s : = log d d s p l p = c 0 + m log s = m+

5 r r s r r s r s r K d d log, d d log, d d log, : + = = Conditional Inaccurac conditional relative entrop conditional Shannon entrop

6 Example: Random-Coefficient AR Model µ + v + e Assumptions µ is v e = is is constant 0, σ 0, σ 2 v 2 e distributed distributed 2 2 = µ, σ v, σ e Theoretical densit s = exp µ 2 2 2π σ 2σ histor dependent 2 2 e variance σ = σ σ 2 v!

7 Empirical vs Theoretical Densit 4 scatter plot histogram = =

8 Testing of Various Hpotheses 4 = = 0.03 σ v 4 2 = 0.03 σ e

9 Minimum Inaccurac MI min n λ R s K r : s S r, s,,λˆ K unnormalied inaccurac s = r, log d s = log l + const. r : d S = exponential envelope, = c s exp h s,, λ λ MI coincides with Maximum Lielihood!

10 Minimum Relative Entrop MRE s min r R D r s r, s,,λˆ R = h-compatible set R r, h, d d = = h, = h D unnormalied relative entrop r, r s = r, log d s d MRE generalies Maximum Entrop!

11 Unnormalied Relative Entrop D r s = r, log r s, d d = r r log r s d d r log r d conditional relative entrop marginal Shannon entrop

12 Information Geometr r, h-projection s S R s, h, d d s,,λˆ, ˆ λ =, h, d d h r = K r : s = K r : s ˆ D s ˆ, λ +, λ s Pthagorean theorem

13 Outline Probabilit-based inference revisited Fundamentals of information geometr Finite-memor inference Minimum Relative Entrop MRE approximation Implementation Marov Chain Monte Carlo MCMC methods Brute-Force Alternative Monte Carlo Again: Weighted Bootstrap

14 MRE Approximation r, R choose h, so that r const. K : s, ˆ λ for expected values of s S s,,λˆ 2 approximate K r : s 3 approximate posterior densit via minimum relative entrop pˆ = c p0 exp D R s D R s = min D r s r R

15 MRE Algorithm Convex optimiation problem eas part D R s = min [ ψ, λ λ h n λ R ] Logarithm of normaliing divisor difficult part exp h, s ψ, λ = log λ d d

16 Choice of Statistic Differencing h, = log s Differentiation h i i Weighted integration h i i+ log s, = ω grad log s i = i i, w log s d i w i d = 0

17 Two Simple Hpotheses s 0 s 0 s s log, 0 s s h = implies r r, exp 0 h s c s λ λ = log 0 l l h = exponential envelope

18 Two Composite Hpotheses r H 0 s λˆ H exponential famil enveloping H 0, H

19 Construction of h-statistic: Differencing = + e = + v h Cauch noise = arctan + e = sin + e h h

20 Construction of h-statistic: Differentiation h h = 0. = 0.4 h = sin + e = 0.2

21 Example: Sensor Validation Monitoring of signal differences e = Model = mixture of 3 normal distributions 0, v + 0,0.0* v + 0,00 * v f g f g normal operation froen sensor gross errors Unnown parameters probabilities, Statistic chosen s hi e = log s i f g e e = [0,0] = [,0] 2 = [0,] 3 = [/ 3,/ 3]

22 Signal Difference 5 e

23 Relative Entrop log D R s 2

24 Posterior Densit ˆ p 2

25 Outline Probabilit-based inference revisited Fundamentals of information geometr Finite-memor inference Minimum Relative Entrop MRE approximation Implementation Marov Chain Monte Carlo MCMC methods Brute-Force Alternative Monte Carlo Again: Weighted Bootstrap

26 MRE Algorithm Dual optimiation tas ψ, λ D R s = max [ λ h log s n λ R umerical integration necessar exp λ h, d d ] sample,, Κ,, from, ernel estimate M M sˆ, λ, from D s sˆ = ε 0 it follows ψ, λ, λ M i i s exp h, log λ λ M i i i= sˆ, λ s, λ i, i

27 MRE Implementation Metropolis sampler, Κ, M MRE Optimiation Metropolis sampler Tilted model densit Model densit x R s D, Κ, x s, λ x s x i x =,

28 Sample-based Computations expectation E covariance Cov probabilit of the event Rao-Blacwellied estimate M sˆ = s i M i P A = A p d marginal densit of a given = a, b predictive densit direct sampling sample from s i i=

29 Metropolis Sampler I. Sample x from π x. Accept + x i = x w.p. α. π x px α = min p x p x / π x / π x i i,

30 Metropolis Sampler II. Random wal x = x i + n. Accept + x i = x w.p. α. px α = min p x p x i,

31 Example: Metropolis Sampling scatter plot histogram

32 Outline Probabilit-based inference revisited Fundamentals of information geometr Finite-memor inference Minimum Relative Entrop MRE approximation Implementation Marov Chain Monte Carlo MCMC methods Brute-Force Alternative Monte Carlo Again: Weighted Bootstrap

33 Weighted Bootstrap Filtering model time update data update calculate normalied weights resample M-times from the discrete distribution over with probabilit mass w i associated with element i,, v x g w x f x = = M i w x f x i i i,,,, Κ = = },, : { M i x i Κ = = = M j j i i x p x p w

34 Stochastic Simulation measured data RESAMPLIG current filtered state Model of Process Dnamics new predicted state Model of Sensors predicted sensor response

35 Example: onisothermal CSTR c Af, T f F CSTR model dc dt A = c A T c A + c Af, Q c V c A,T c A,T F dt dt = T β T c T = 0 exp E / RT A + T Reaction rate Arrhenius relation f χ, Ref: Seborg, Edgar, Mellichamp 989, Exercise 5.2

36 Variable Feed 0.86 Variations in feed concentration [lb mole/ft 3 ] 0.84 c Af Variations in feed temperature [ o F] 50 T f

37 Cooling Effect 5 Periodic cooling 4 χ Temperature [ o F] 60 T

38 State Estimation 0.03 Concentration [lb mole/ft 3 ] c A Temperature [ o F] T

39 Measurement Prediction 0.03 Concentration measurements vs predictions c A Temperature measurements vs predictions 60 T

40 State Estimation with Sensor Validation 0.03 Concentration [lb mole/ft 3 ] c A Temperature [ o F] 60 T

41 Conclusions Theor: Information geometr ields additional insight. Information geometr is tolerant to approximations and cheating. Algorithm: Iterative sampling and importance resampling Monte Carlo schemes offer powerful tools to manage the curse of dimensionalit. Benefit: Fine description of uncertaint results in lower missed & false alarm rates, and shorter dela in detection.

42 Further Reading T.M. Cover and J.A. Thomas 99. Elements of Information Theor. Wile, ew Yor. R. E. Blahut 987. Principles and Practice of Information Theor. Addison-Wesle, Reading, MA. L. Tierne 994. Marov chains for exploring posterior distributions. Ann. Statist., 22, A.F.M. Smith and A.E. Gelfand 992. Baesian statistics without tears: a sampling-resampling perspective. Amer. Statist., 46, R. Kulhavý 996. Recursive onlinear Estimation: A Geometric Approach. Springer-Verlag, London.