A Least Angle Regression Control Chart for Multidimensional Data 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 A Least Angle Regression control chart. . . 1/ 22
Outline Multivariate Statistical Monitoring 1 Variable-selection Methods in SPC 2 Proposed Procedure 3 Reference Model Least Angle Regression Algorithm LAR-EWMA control chart Simulation Results 4 Concluding Remarks and Future Research 5 A Least Angle Regression control chart. . . 2/ 22
Synopsis Framework: Phase II monitoring of a normal multivariate vector of product variables. Fault type: Persistent change in the process mean and/or increase in the total dispersion. Problem: Simultaneous monitoring of several variables: how many and which variables are really changed? Proposal: Combination of a variable-selection method (Least Angle Regression) with a multivariate control chart (multivariate EWMA). Results: The LAR-based EWMA is a very competitive statistical tool for handling several change- point scenarios in the high-dimensional framework. A Least Angle Regression control chart. . . 3/ 22
Multivariate Statistical Monitoring Possible Frameworks U NSTRUCTURED : multivariate vector of quality characteristics S TRUCTURED : analytical models describing process quality linear and non linear profiles 1 multistage processes 2 A Least Angle Regression control chart. . . 4/ 22
Multivariate Statistical Monitoring Possible Approaches Monitoring the stability of the whole set of product variables (low sensitivity in a high-dimensional context) Monitoring a reduced set of out-of-control product variables. But the shifted components are obviously unknown! A Least Angle Regression control chart. . . 5/ 22
Variable-selection Methods in SPC State of the art Promising approach: combine a suitable variable selection method with the multivariate statistical monitoring Forward search algorithm with a Shewhart-type control chart (Wang and Jiang, 2009) for handling mean changes in the unstructured scenario. LASSO algorithm with a multivariate EWMA, for detecting mean changes in the unstructured case (Zou and 1 Qiu, 2009) changes in multivariate linear profiles (Zou et al. 2 2010) A Least Angle Regression control chart. . . 6/ 22
Variable-selection Methods in SPC Our proposal L EAST A NGLE R EGRESSION WITH A M ULTIVARIATE EWMA general model formulation unifying the unstructured 1 and structured framework use of a variable selection method yet unexplored in 2 the SPC framework competitive procedure for detecting changes in 3 process mean and total dispersion for a wide variety of change point-scenarios relatively simple tool for fault identification 4 A Least Angle Regression control chart. . . 7/ 22
Reference Model Characterization of mean and variance changes C HANGE - POINT MODEL � N n ( µ µ µ , Σ Σ Σ) if t < τ ( in-control ) y t ∼ N n ( µ µ µ + δ δ δ , Ω Ω if t ≥ τ Ω) ( out-of-control ) M EAN SHIFT δ δ δ = F β β β F n × p : matrix of known constants; β β β p × 1 : vector of unknown parameters . D ISPERSION INCREASE E OC [ ξ 2 t ] > n ⇐ Ω Ω Ω − Σ Σ Σ positive definite matrix µ ) ′ . Σ − 1 ( y t − µ with ξ 2 t = ( y t − µ µ µ )Σ Σ µ A Least Angle Regression control chart. . . 8/ 22
Reference Model Frameworks and characterization of δ δ δ U NSTRUCTURED [ δ i ] = [ β i ] ⇐ ⇒ F = I n P ROFILE � p � ∑ [ δ i ] = [ g ( x i )] = β j f j ( x i ) j = 1 e.g. δ i = β 1 + β 2 x i + β 3 x 2 i for i = 1 ,..., n . M ULTISTAGE P ROCESS Standard state space representations of a process with n stages can be written in the desired form. � y t , i = µ i + c i x t , i + v t , i ( i = 1 ,..., n ) x t , i = d i x t , i − 1 + β i I { t ≥ τ } + w t , i A Least Angle Regression control chart. . . 9/ 22
Reference Model β Monitoring stability ← → Testing hypotheses on β β Multivariate EWMA control statistic z t = ( 1 − λ ) z t − 1 + λ ( y t − µ µ µ ) with z 0 = 0 n , 0 < λ ≤ 1. Linear model β z t = F β β + a t , with a t ∼ N n ( 0 n , λ / ( 1 − λ )Σ Σ Σ) M ONITORING STABILITY OF PROCESS MEAN ⇒ { β β β = 0 } H 0 = { the process is in control } ⇐ H 1 = { the process is out of control } ⇐ ⇒ { How many β j are non zero? One, two....all? } A Least Angle Regression control chart. . . 10/ 22
Least Angle Regression algorithm Main steps Start with all the p coefficients equal to zero. 1 Build up estimates of the unknown mean in 2 successive steps: each step adding one variable. Reach the full least square solution in p steps (using 3 all the variables). S TEP k { j 1 ,..., j k } ⇐ ⇒ { β j 1 ,..., β j k } � �� � � �� � variables selected by LAR just k parameters are assumed � = 0 A Least Angle Regression control chart. . . 11/ 22
LAR-EWMA control chart Hypotheses systems and control statistics M EAN C HANGE H YPOTHESIS C ONTROL S TATISTIC . β j 1 � = 0 ,..., β j k � = 0 , Likelihood ratio test: β j k + 1 = ··· = β j p = 0 S t , k ( k = 1 ,..., p ) V ARIATION I NCREASE H YPOTHESIS C ONTROL S TATISTIC S t , p + 1 = � � 1 , ( 1 − λ ) S t − 1 , p + 1 + λ ξ 2 β = 0 p and E [ ξ 2 t β β t ] > n x max n with S 0 , p + 1 = 1 A Least Angle Regression control chart. . . 12/ 22
LAR-EWMA control chart Control statistic, stopping rule, fault identification S t , k − a k Control W t = max b k statistic k = 1 ,..., p + 1 t ⋆ = min { t : W t > h } Alarm time � � Fault k : S t ⋆ , k − a k k ⋆ = min > h Identification b k k ⋆ ≤ p : { plausible mean shift } k ⋆ = p + 1 : { plausible dispersion increase } A Least Angle Regression control chart. . . 13/ 22
LAR-EWMA control chart Control chart design S MOOTHING C ONSTANT λ (0.1, 0.3) normal distribution (0.03, 0.05) skewed and heavy-tailed distributions C ONTROL LIMIT h h is determined to obtain a desired value of the in-control ARL via simulation, using the Polyak-Ruppert stochastic approximation algorithm. A Least Angle Regression control chart. . . 14/ 22
Simulation Results Competitive control charts M EAN CHANGE AND NO VARIANCE CHANGE U NSTRUCTURED SCENARIO M ULTISTAGE P ROCESSES REWMA (Hawkins, MEWMA (Lowry et al.,1992) 1991,1993) MEWMA (Lowry et al.,1992) DEWMA (Zou and Tsung, 2008) LEWMA (Zou and Qiu, 2009) M EAN CHANGE AND VARIANCE INCREASE P ARAMETRIC P ROFILE N ONPARAMETRIC P ROFILE KMW (Kim, et al. 2003)) NEWMA (Zou et al., 2008) PEWMA (Zou et al.,2007) A Least Angle Regression control chart. . . 15/ 22
Simulation Results Investigation of a wide variety of out-of-control scenarios (unstructured and structured) large range of size shifts 1 different combinations of shifted components or 2 locations: one, more than one. mean shifts and/or variance increases 3 performance measure: Average Run Length summary performance measure: the Relative Mean Index (Han and Tsung, 2006). Very small RMI values = ⇒ best or close to the best chart. A Least Angle Regression control chart. . . 16/ 22
Simulation Results The Relative Mean Index (Han and Tsung, 2006) The RMI is a summary performance measure defined as N N ARL δ δ l − MARL δ RMI = 1 RMI l = 1 δ δ δ l ∑ ∑ N N MARL δ δ δ l l = 1 l = 1 N total number of shifts 1 δ l out-of-control ARL for detecting δ δ δ l ARL δ 2 δ MARL δ δ l smallest out-of-control ARL, among the 3 δ δ compared charts, for detecting δ δ l RMI l relative efficiency of a chart in detecting δ δ δ l 4 compared to the best chart. A Least Angle Regression control chart. . . 17/ 22
Linear Profile IC: y t , i ∼ N ( 0 , 1 ) . OC: y t , i ∼ N ( β 1 + β 2 x i , ω 2 ) , i = 1 ,..., 4, x i = − 1 , − 1 / 3 , 1 / 3 , 1 Shifts PEWMA KMW LAR-EWMA β 1 = 0 . 1 293.08 296.40 272.39 β 1 = 0 . 3 46.16 43.04 39.55 β 1 = 1 4.65 4.29 4.22 β 2 = 0 . 2 180.73 182.79 163.19 β 2 = 0 . 4 46.84 43.66 40.14 β 2 = 1 . 2 5.41 4.99 4.90 β 1 = 0 . 1, β 2 = 0 . 1 230.79 243.60 215.11 β 1 =0.4, β 2 = 0 . 2 20.69 21.14 18.80 β 1 = 0 . 4 , β 2 = 0 . 6 10.39 12.34 10.05 β 1 = 0 . 6, β 2 = 0 . 6 7.11 8.16 6.88 β 1 = 0 . 1, ω = 1 . 2 45.88 50.95 38.78 β 1 = 0 . 3, ω = 1 . 2 21.55 23.77 20.14 β 1 = 1, ω = 1 . 2 4.44 4.29 4.08 β 2 = 0 . 2, ω = 1 . 2 38.68 43.46 33.71 β 2 = 0 . 4, ω = 1 . 2 21.80 23.93 20.13 β 2 = 1 . 2, ω = 1 . 2 5.12 4.97 4.72 β 1 = 0 . 1, β 2 = 0 . 1, ω = 1 . 2 42.47 47.36 36.26 β 1 = 0 . 4, β 2 = 0 . 2, ω = 1 . 2 13.74 15.14 13.02 β 1 = 0 . 4, β 2 = 0 . 6, ω = 1 . 2 8.60 9.96 8.36 β 1 = 0 . 6, β 2 = 0 . 6, ω = 1 . 2 6.41 7.29 6.19 ω = 1 . 2 53.39 58.17 43.24 ω = 1 . 5 11.06 12.35 7.90 ω = 2 4.37 4.71 3.03 RMI 0.13 0.19 0.00 A Least Angle Regression control chart. . . 18/ 22
Simulation Results: RMI ARL 0 = 500, λ = 0 . 2, τ = 1 “Unstructured” LAR-EWMA MEWMA REWMA LEWMA n = 15 0.02 0.24 0.32 0.10 Linear Profile LAR-EWMA PEWMA KMW n = 4 0.00 0.13 0.19 n = 10 0.01 0.07 0.11 Cubic Profile LAR-EWMA PEWMA KMW n = 8 0.00 0.24 0.36 n = 15 0.00 0.23 0.31 Non-parametric profile LAR-EWMA NEWMA n = 20 0.05 0.28 n = 40 0.06 0.30 Multistage process LAR-EWMA MEWMA DEWMA n = 20 ; c i = d i = 1 0.02 0.24 0.10 n = 20 ; c i = 1 . 2 ; d i = 0 . 8 0.06 0.22 0.18 A Least Angle Regression control chart. . . 19/ 22
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