An Adaptive Generalized Likelihood Ratio Control Chart for Detecting an Unknown Mean Pattern

Transcription

1 An adapive GLR conrol char... 1/32 An Adapive Generalized Likelihood Raio Conrol Char for Deecing an Unknown Mean Paern GIOVANNA CAPIZZI and GUIDO MASAROTTO Deparmen of Saisical Sciences Universiy of Padua Ialy 2ND INTERNATIONAL SYMPOSIUM ON STATISTICAL PROCESS CONTROL Rio de Janeiro, Brazil July 13-14, 2011

2 Ouline An adapive GLR conrol char... 2/32 1 Problem/Reference model 2 Lieraure review Known faul signaure (CUSCORE, GLR, Opimal Linear Filer) Unknown faul signaure (reference free CUSCORE, Weighed CUSUM,... ) 3 A novel conrol char 4 Comparisons CUSUM Weighed CUSUM 5 Non parameric version 6 Conclusions

3 Mos saisical process conrol mehods... An adapive GLR conrol char... 3/32... are focused on he deecion of a consan and persisen shif. MEAN PATTERN ASSUMING A CONSTANT SHIFT AT = 51

4 An adapive GLR conrol char... 4/32 However, i is more realisic o consider also he possibiliy of a ime varying mean afer a faul. SOME STYLIZED MEAN PATTERNS Gradual degradaion Inermien faul Parial recover (feedback) Vibraion in a mechanical sysem

5 Daa preprocessing... An adapive GLR conrol char... 5/32... can inroduce (or modify) a dynamic paern. auocorrelaed daa self-saring char conrol EXAMPLES compuaion of residuals from a ime series model rasformaion of he original observaions o eliminae he unknown parameers

6 Daa preprocessing (example) An adapive GLR conrol char... 6/32 MEANS OF OBSERVATIONS AND RESIDUALS y = 1.13y y 2 + u + 0.9u 1 (Model 1, Apley and Shi, IIE Trans., 1999) Original observaions Residuals

7 Daa preprocessing (example) An adapive GLR conrol char... 6/32 MEANS OF OBSERVATIONS AND RESIDUALS y = 1.13y y 2 + u + 0.9u 1 (Model 1, Apley and Shi, IIE Trans., 1999) Original observaions Residuals

8 Reference model An adapive GLR conrol char... 7/32 INDEPENDENT DATA y 1,y 2,... Original observaions, residuals from a suiable ime series model,... DISTRIBUTION y { N(µ,σ 2 ) if < τ (in conrol) N(µ + σγ,σ 2 ) if τ (ou of conrol) PROBLEM AT TIME H 0 = {he process is in conrol} { < τ} H 1 = {he process is ou of conrol} { τ}

9 Lieraure review: summary An adapive GLR conrol char... 8/32 Mehods differ for he assumed level of knowledge on he ou of conrol mean paern γ ( = 1,2,...). SCENARIO Compleely known paern Parially known paern (only he shape) Unknown one-sided paern (all γ i eiher greaer or lesser han zero) Unknown oscillaory paern SCHEMES CUSCORE Generalized Likelihood Raio (GLR) and Opimal Linear Filer Adapive CUSCORE (differen versions)???

10 An adapive GLR conrol char... 9/32 CUSCORE Box and Ramirez (1992),QREI,... Sequenial likelihood es for H 0 : γ i = 0 agains H 1 : γ i = g i (i = 1,...,) for a known sequence g i (and, hence, for a known τ). The conrol saisic is ( C = max 0,C 1 + log f ) 1(y ) f 0 (y ) where f 0 ( ) and f 1 ( ) are he densiies compued under H 0 and H 1, respecively. A resaring procedure can be used o avoid he assumpion of known τ.

11 Generalized Likelihood Raio Apley and Shi (1999), IIE Trans.,... I is assumed ha γ = δs τ ( τ) where δ is unknown while he faul signaure s i is a known sequence. A ime, he conrol saisic is max sup M+1<τ δ where M is a suiable ineger. log f 1(y,,y M+1 ;τ,δ). f 0 (y,,y M+1 ) An adapive GLR conrol char... 10/32

12 Opimal Linear Filer Chin and Apley (2006), Tech.; Apley and Chin (2007), JQT I based on he GLR assumpions (known faul signaure, unknown direcion and size). The conrol saisic is M w i y i i=0 where he weighs w i are opimal for deecing he specified faul signaure. An adapive GLR conrol char... 11/32

13 Adapive CUSCOREs An adapive GLR conrol char... 12/32 SCHEME reference free CUSCORE (Han and Tsung, 2006, JASA) MEAN PATTERN ESTIMATE ˆγ = y µ σ, ˆγ = ˆγ + Weighed CUSUM (Shu e al., 2008, JQT) ˆγ + = EWMA µ σ, ˆγ = ˆγ + Adapive CUSUM (Jiang e al., 2008, IIE Trans.) ˆγ + ˆγ ( = max d, AEWMA ) µ σ ( = min d, AEWMA ) µ σ

14 An adapive GLR conrol char... 13/32 Quesions 1 How can an adapive GLR conrol char be defined? 2 How does i compare wih he CUSUM (EWMA,... ) for deecing persisen mean shifs? 3 How does i compare wih he adapive CUSCORE conrol chars for deecing arbirary one-sided paerns? 4 Can i be designed also for deecing arbirary oscillaory ou-of-conrol mean paerns? 5 Can i be modified o cope wih non Gaussian daa?

15 An adapive GLR conrol char An adapive GLR conrol char... 14/32 Conrol saisic One-sided faul signaure Generic faul signaure aglr = max{glr (ŝ 1 ),GLR (ŝ 2 )} ( λ ŝ 1, = max k q 2 λ, EWMA µ σ ) (y,...,y M+1 ) T (ŝ 2,,...,ŝ 2, M+1 ) T = (DWT)+(THRESHOLDING)+(IDWT)

16 Comparisons: generaliies SCHEMES 1 adapive GLR 2 sandard CUSUM 3 weighed CUSUM PERFORMANCE IN CONTROL: ARL = 500; OUT OF CONTROL: E(RL 200 RL τ = 201) (I can be viewed as an approximaion of he seady sae ou of conrol ARL). An adapive GLR conrol char... 15/32

17 An adapive GLR conrol char... 16/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. CUSUM(k = 0.5) CONSTANT SHIFT Faul signaure δ Seady-sae Average Run Lengh

18 An adapive GLR conrol char... 17/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 3.5,k w = 3.5) vs. CUSUM(k = 1) CONSTANT SHIFT Faul signaure δ Seady-sae Average Run Lengh

19 An adapive GLR conrol char... 17/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 3.5,k w = 3.5) vs. CUSUM(k = 1) CONSTANT SHIFT δ aglr CUSUM Faul signaure Seady-sae Average Run Lengh

20 An adapive GLR conrol char... 18/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. CUSUM(k = 0.5) A ONE-SIDED SCENARIO Faul signaure δ Seady-sae Average Run Lengh

21 An adapive GLR conrol char... 18/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. CUSUM(k = 0.5) A ONE-SIDED SCENARIO δ aglr CUSUM Faul signaure Seady-sae Average Run Lengh

22 An adapive GLR conrol char... 19/32 Comparisons: CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. CUSUM(k = 0.5) AN OSCILATORY FAULT SIGNATURE Faul signaure δ Seady-sae Average Run Lengh

23 An adapive GLR conrol char... 20/32 Comparisons: Weighed CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. four WCUSUMs CONSTANT SHIFT Faul signaure δ Seady-sae Average Run Lengh

24 An adapive GLR conrol char... 21/32 Comparisons: Weighed CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. four WCUSUMs A ONE-SIDED SCENARIO Faul signaure δ Seady-sae Average Run Lengh

25 An adapive GLR conrol char... 22/32 Comparisons: Weighed CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. four WCUSUMs AN OSCILLATORY FAULT SIGNATURE Faul signaure δ Seady-sae Average Run Lengh

26 An adapive GLR conrol char... 23/32 Comparisons: Weighed CUSUM aglr(m = 128,λ = 0.1,k q = 1.5,k w = 3.5) vs. four WCUSUMs ANOTHER OSCILLATORY FAULT SIGNATURE Faul signaure δ Seady-sae Average Run Lengh

27 A non parameric version: framework An adapive GLR conrol char... 24/32 REFERENCE MODEL y 1,y 2,... are indipenden such ha { F(a) if < τ (in conrol) P(y a) = F(a γ ) if τ (ou of conrol) τ and γ ( τ) are unknown. INFORMATION ON F( ) F( ) is unknown bu a reference sample of n preliminary in conrol observaions, say y (IC) 1,...,y n (IC), is available.

28 A non parameric version: key idea An adapive GLR conrol char... 25/32 CONTROL CHART Deec he change poin applying an adapive GLR o ( ) y = φ 1 nf n (IC) (y ) + 1 n + 2 where F n (IC) ( ) is he empirical c.d.f. of he reference sample and φ( ) is he c.d.f. of a sandard normal r.v. REMARK nf n (IC) (y ) + 1 is he rank of he curren observaion y in he combined sample (y,y (IC) 1,...,y n (IC) ).

29 Four disribuions An adapive GLR conrol char... 26/ In each case, he mean and he variance are equal o 0 and 1, respecively.

30 Effec of he reference sample size In conrol run lengh disribuions n = 100 n = 250 n = 500 n = 1000 n = a b c d a b c d a b c d a b c d all In every case, he in conrol ARL is equal o 500 An adapive GLR conrol char... 27/32

31 Effec of he reference sample size Ou of conrol performance 1 For some mean paern, he reference sample size only slighly affecs he performance. E(RL 200 RL τ = 201) - γ = 1 for 201 Reference Sample Size Disribuion Normal Skew Normal (λ = 5) Skew Normal (λ = 20) of Suden (df = 3) An adapive GLR conrol char... 28/32

32 Effec of he reference sample size Ou of conrol performance 2 Bu in oher cases, performance increases as n increases. E(RL 200 RL τ = 201) - γ = 2cos(2π/8) for 201 Reference Sample Size Disribuion Normal Skew Normal (λ = 5) Skew Normal (λ = 20) of Suden (df = 3) An adapive GLR conrol char... 29/32

33 A non parameric version: recommendaions An adapive GLR conrol char... 30/32 1 Monioring can be sared wih as few as 50/100 in conrol observaions. 2 The reference se should be updaed as more observaions are gahered.

34 An adapive GLR conrol char... 31/32 Conclusions 1 Adapive GLR conrol chars perform well in a variey of ou of conrol scenarios one-sided paerns boh consan and ime-varying; oscillaory paerns; differen shif sizes. 2 A disribuion free version is available. 3 I is relaively simple o design since a single scheme provides a good performance in a variey of ou of conrol scenarios.

35 An adapive GLR conrol char... 32/32 THANKS... SINCE 1222 UNIVERSA UNIVERSIS PATAVINA LIBERTAS (Paduan Freedom is Complee and for Everyone) Observaory (1374) Palazzo Bo (1405)