An Adaptive Generalized Likelihood Ratio Control Chart for Detecting - PowerPoint PPT Presentation

An Adaptive Generalized Likelihood Ratio Control Chart for Detecting an Unknown Mean Pattern G IOVANNA C APIZZI and G UIDO M ASAROTTO Department of Statistical Sciences University of Padua Italy 2 ND I NTERNATIONAL S YMPOSIUM ON S TATISTICAL P ROCESS C ONTROL Rio de Janeiro, Brazil July 13-14, 2011 An adaptive GLR control chart. . . 1/32

Outline Problem/Reference model 1 Literature review 2 Known fault signature (CUSCORE, GLR, Optimal Linear Filter) Unknown fault signature (reference free CUSCORE, Weighted CUSUM,. . . ) A novel control chart 3 Comparisons 4 CUSUM Weighted CUSUM Non parametric version 5 Conclusions 6 An adaptive GLR control chart. . . 2/32

Most statistical process control methods. . . . . . are focused on the detection of a constant and persistent shift. M EAN P ATTERN A SSUMING A C ONSTANT S HIFT AT t = 51 1.0 0.5 0.0 −0.5 −1.0 20 40 60 80 100 t An adaptive GLR control chart. . . 3/32

However, it is more realistic. . . . . . to consider also the possibility of a time varying mean after a fault. S OME S TYLIZED M EAN P ATTERNS 1.0 40 0.5 20 0.0 0 −20 −0.5 −40 −1.0 20 40 60 80 100 20 40 60 80 100 t t Gradual degradation Intermittent fault 1.0 1.0 0.5 0.5 0.0 0.0 −0.5 −0.5 −1.0 −1.0 20 40 60 80 100 20 40 60 80 100 t t Partial recover (feedback) Vibration in a mechanical system An adaptive GLR control chart. . . 4/32

Data preprocessing. . . . . . can introduce (or modify) a dynamic pattern. E XAMPLES computation of residuals autocorrelated data from a time series model trasformation of the origi- self-starting control nal observations to elimi- chart nate the unknown parame- ters An adaptive GLR control chart. . . 5/32

Data preprocessing (example) M EANS OF O BSERVATIONS AND R ESIDUALS y t = 1 . 13 y t − 1 − 0 . 64 y t − 2 + u t + 0 . 9 u t − 1 (Model 1, Apley and Shi, IIE Trans., 1999) 1.0 1.0 0.5 0.5 0.0 0.0 −0.5 −0.5 −1.0 −1.0 20 40 60 80 100 20 40 60 80 100 t t Original observations Residuals An adaptive GLR control chart. . . 6/32

Data preprocessing (example) M EANS OF O BSERVATIONS AND R ESIDUALS y t = 1 . 13 y t − 1 − 0 . 64 y t − 2 + u t + 0 . 9 u t − 1 (Model 1, Apley and Shi, IIE Trans., 1999) 1.0 1.0 0.5 0.5 0.0 0.0 −0.5 −0.5 −1.0 −1.0 20 40 60 80 100 20 40 60 80 100 t t Original observations Residuals An adaptive GLR control chart. . . 6/32

Reference model I NDEPENDENT DATA y 1 , y 2 ,... Original observations, residuals from a suitable time series model,. . . D ISTRIBUTION � N ( µ , σ 2 ) if t < τ (in control) y t ∼ N ( µ + σγ t , σ 2 ) if t ≥ τ (out of control) P ROBLEM AT T IME t H 0 = { the process is in control } ⇐ ⇒ { t < τ } H 1 = { the process is out of control } ⇐ ⇒ { t ≥ τ } An adaptive GLR control chart. . . 7/32

Literature review: summary Methods differ for the assumed level of knowledge on the out of control mean pattern γ t ( t = 1 , 2 ,... ). S CENARIO S CHEMES Completely known pattern CUSCORE Partially known pattern Generalized Likelihood Ra- (only the shape) tio (GLR) and Optimal Lin- ear Filter Unknown one-sided pattern Adaptive CUSCORE (dif- (all γ i either greater or ferent versions) lesser than zero) Unknown oscillatory pat- ??? tern An adaptive GLR control chart. . . 8/32

CUSCORE Box and Ramirez (1992),QREI, . . . Sequential likelihood test for H 0 : γ i = 0 against H 1 : γ i = g i ( i = 1 ,..., t ) for a known sequence g i (and, hence, for a known τ ). The control statistic is � � 0 , C t − 1 + log f 1 ( y t ) C t = max f 0 ( y t ) where f 0 ( · ) and f 1 ( · ) are the densities computed under H 0 and H 1 , respectively. A “restarting” procedure can be used to avoid the assumption of known τ . An adaptive GLR control chart. . . 9/32

Generalized Likelihood Ratio Apley and Shi (1999), IIE Trans., . . . It is assumed that γ t = δ s t − τ ( t ≥ τ ) where δ is unknown while the “fault signature” s i is a known sequence. At time t , the control statistic is log f 1 ( y t , ··· , y t − M + 1 ; τ , δ ) t − M + 1 < τ ≤ t sup max . f 0 ( y t , ··· , y t − M + 1 ) δ where M is a suitable integer. An adaptive GLR control chart. . . 10/32

Optimal Linear Filter Chin and Apley (2006), Tech.; Apley and Chin (2007), JQT It based on the GLR assumptions (known fault signature, unknown direction and size). The control statistic is t − M ∑ w i y t − i i = 0 where the weights w i are optimal for detecting the specified fault signature. An adaptive GLR control chart. . . 11/32

Adaptive CUSCOREs S CHEME M EAN PATTERN ESTIMATE reference free CUSCORE � y t − µ � γ − γ + (Han and Tsung, 2006, � � ˆ � , ˆ t = − ˆ γ t = � � t σ � JASA) Weighted CUSUM (Shu et � � EWMA t − µ γ + γ − γ + � � ˆ � , ˆ t = − ˆ t = al. , 2008, JQT) � � t σ � Adaptive CUSUM (Jiang et � d , AEWMA t − µ � γ + ˆ t = max al. , 2008, IIE Trans.) σ � − d , AEWMA t − µ � γ − ˆ t = min σ An adaptive GLR control chart. . . 12/32

Questions How can an adaptive GLR control chart be defined? 1 How does it compare with the CUSUM (EWMA,. . . ) 2 for detecting persistent mean shifts ? How does it compare with the adaptive CUSCORE 3 control charts for detecting arbitrary one-sided patterns ? Can it be designed also for detecting arbitrary 4 oscillatory out-of-control mean patterns? Can it be modified to cope with non Gaussian data ? 5 An adaptive GLR control chart. . . 13/32

An adaptive GLR control chart Control aGLR t = max { GLR t (ˆ s 1 ) , GLR t (ˆ s 2 ) } statistic One-sided � � � � EWMA t − µ � λ ˆ � � s 1 , t = max k q 2 − λ , fault � � σ � � signature Generic T → (ˆ s 2 , t ,..., ˆ ( y t ,..., y t − M + 1 ) − s 2 , t − M + 1 ) fault signature T = (DWT)+(THRESHOLDING)+(IDWT) An adaptive GLR control chart. . . 14/32

Comparisons: generalities S CHEMES adaptive GLR 1 standard CUSUM 2 weighted CUSUM 3 P ERFORMANCE I N CONTROL : ARL = 500; O UT OF CONTROL : E ( RL − 200 | RL ≥ τ = 201 ) (It can be viewed as an approximation of the steady state out of control ARL ). An adaptive GLR control chart. . . 15/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 0 . 5 ) C ONSTANT S HIFT 1.0 100 200 400 0.5 50 0.0 20 10 −0.5 5 2 −1.0 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 16/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 3 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 1 ) C ONSTANT S HIFT 1.0 100 200 400 0.5 50 0.0 20 10 −0.5 5 2 −1.0 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 17/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 3 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 1 ) C ONSTANT S HIFT 1.0 δ aGLR CUSUM 0.25 110.01 246.92 0.5 0.50 33.63 80.93 0.0 0.75 17.18 30.74 1.00 10.58 14.48 −0.5 −1.0 20 40 60 80 100 t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 17/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 0 . 5 ) A O NE -S IDED S CENARIO 100 200 400 1.0 0.5 50 0.0 20 10 −0.5 5 −1.0 2 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 18/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 0 . 5 ) A O NE -S IDED S CENARIO δ aGLR CUSUM 1.0 0.50 160.6 218.5 0.5 0.75 56.8 94.1 0.0 1.00 15.2 25.0 −0.5 −1.0 20 40 60 80 100 t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 18/32

Comparisons: CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. CUSUM ( k = 0 . 5 ) A N O SCILATORY F AULT S IGNATURE 100 200 400 1.0 0.5 50 0.0 20 10 −0.5 5 −1.0 2 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 19/32

Comparisons: Weighted CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. four WCUSUMs C ONSTANT S HIFT 1.0 100 200 400 0.5 50 0.0 20 10 −0.5 5 2 −1.0 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 20/32

Comparisons: Weighted CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. four WCUSUMs A O NE -S IDED S CENARIO 100 200 400 1.0 0.5 50 0.0 20 10 −0.5 5 −1.0 2 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 21/32

Comparisons: Weighted CUSUM aGLR ( M = 128 , λ = 0 . 1 , k q = 1 . 5 , k w = 3 . 5 ) vs. four WCUSUMs A N O SCILLATORY F AULT S IGNATURE 1.0 100 200 400 0.5 50 0.0 20 10 −0.5 5 2 −1.0 1 20 40 60 80 100 0 1 2 3 4 5 6 δ t Fault signature Steady-state Average Run Length An adaptive GLR control chart. . . 22/32