Directional Control Schemes for Multivariate Categorical Processes

Transcription

1 Directional Control Schemes for Multivariate Categorical Processes Nankai University Homepage: math.nankai.edu.cn/ chlzou (Joint work with Mr. Jian Li and Prof. Fugee Tsung)

2 Outline 1 Multivariate Categorical Modeling 2 Directional Control Schemes 3 Performance Assessment

3 Multivariate Categorical Modeling Multivariate Categorical Processes The aluminium electrolytic capacitor (AEC). Capacity (CAP) Dissipation Factor (DF) Leakage Current (LC) (CAP DF LC), 1 for conforming and 2 for nonconforming, 2 3 = 8 level combinations or cross-classifications: (111), (112), (121), (122), (211), (212), (221), (222). (121) means an AEC with conforming CAP and LC, and nonconforming DF 3-way contingency table of size and with 8 cells; Each cell corresponds to one level combination and stores the count under this combination.

4 Multivariate Categorical Modeling Existing works Multivariate binomial chart χ 2 -chart (Patel s 1973) : G MB,k = 1 N ( nmb,k Np (0) ) T ( MB Σ 1 MB nmb,k Np (0) MB). only applicable in multivariate binomial processes The multivariate multinomial chart, a multi-chart comprising p individual charts (Marcucci 1985) G MM,(i)k = 1 N ( nmm,(i)k Np (0) ) T ( MM,(i) Σ 1 MM,(i) nmm,(i)k Np (0) MM,(i)). cumbersome

5 Multivariate Categorical Modeling Some review on multivariate continuous monitoring Hotelling T 2 -based statistic (Lowry et al. 1992): n X Σ 1 X Regression-adjusted statistic (Hawkins 1991): max i=1,...,p v i, v j = n(d jσ 1 X)/(d jσ 1 d j ) 1 2, for j = 1,..., p. MSPC using LASSO or other variable selection techniques

6 Multivariate Categorical Modeling Contingency Table p categorical variables or factors {C 1, C 2,..., C p }. Each classification factor C i takes h i of possible levels. A p-way h 1 h 2... h p cross-classified contingency table with h = p i=1 h i cells. The AEC example: p = 3 and h 1 = h 2 = h 3 = 2. Log-Linear Models For an h 1 h 2 h 3 table and a fixed sample size N, the cell counts follow the multinomial distribution MN(N; p ijk ) with and their expectations m i,j,k = Np i,j,k, and i,j,k p i,j,k = 1. The log-linear model can characterize the relationship between cell probabilities and factor levels, ln p i,j,k = u (0) +u (1) i +u (2) j +u (3) k +u (1,2) i,j +u (1,3) i,k +u (2,3) j,k +u (1,2,3) i,j,k.

7 Multivariate Categorical Modeling Identifiability requires constraints such as u (1) i = u (1,2) i,j = u (1,3) i,k = i i i i u (1,2,3) i,j,k = 0 for the first factor along its index i. Similar equations describe the second and third factors along with their indexes j and k, respectively. Re-parameterizations For example, a 2 3 contingency table u (0) = β 0, u (1) 1 = β 1, u (1) 2 = β 1, u (2) 1 = β 2, u (2) 2 = β 3, u (2) 3 = β 2 β 3, u (1,2) 1,1 = β 4, u (1,2) 1,2 = β 5, u (1,2) 1,3 = β 4 β 5, u (1,2) 2,1 = β 4, u (1,2) 2,2 = β 5, u (1,2) 2,3 = β 4 + β 5.

8 Multivariate Categorical Modeling Log-Linear Model at the effect level Imposed by some constraints, the log-linear model for a general p-way contingency table can be written as 2 p 1 ln p = 1β 0 + X i β i. The subvector β i stands for the ith main or interaction effect. i=1 Log-Linear Model at the coefficient level h 1 ln p = 1β 0 + x i β i i=1 The number β i stands for the ith coefficient, either part or totality of an effect.

9 Directional Control Schemes One-to-one correspondence between coefficient subvectors and factor effects Shifts in the marginal dist. of one factor represent deviations of the coefficient subvector corresponding to its main effect; Shifts in the dependence among multiple factors represent deviations of the coefficient subvector reflecting their interaction effect.

10 Outline 1 Multivariate Categorical Modeling 2 Directional Control Schemes 3 Performance Assessment

11 Directional Control Schemes Problem Formulation F (X; β): ln p = 1β 0 + Xβ and p T 1 = 1 X = [X 1,..., X 2 p 1] = [x 1, x 2,..., x h 1 ] The jth on-line multivariate sampling observation vector, n (j) of size h 1, collected over time from the change-point model { n (j) i.i.d. F (X; β (0) ), for j = 1,..., τ, F (X; β (1) ), for j = τ + 1,...

12 Directional Control Schemes Shift directions Shifts with all possibilities H 0 : β = β (0) H 1 : β β (0). Only the coefficient β i (1 i h 1) adds by an unknown constant δ i H 0 : β = β (0) H 1 : β = β (0) + d i δ i. Only one coefficient may deviate with unknown location H 0 : β = β (0) H 1 : β = β (0) + d 1 δ 1 or... or β = β (0) + d h 1 δ h 1.

13 Directional Control Schemes Shift directions (cont d) Most shifts are inclined to occur in lower-order effects. Most real applications care about only means and variances, namely moments of the first two orders. The monitoring task is to only answer yes or no about whether process is IC, rather than telling the locations of shifted effects. Focus on merely effects of the first two orders (i.e., the main effects and the two-factor interaction effects). H 0 : β = β (0) ) H 1 : (β = β (0) + d i δ i. i I 2

14 Directional Control Schemes LLD chart Given the IC probability vector p (0), Phase II sample size N, observation vectors n j (j = 1, 2,...), we construct an EWMA control chart based on likelihood ratio test with its charting statistic ( 1 ( R k = max z (k) Np (0)) T ( xi x T i I 2 N i Σ (0) ) 1x ( T x i i n Np (0) )). z (k) = (1 λ)z (k 1) + λn (k) EWMA observation vector Σ (0) = diag ( p (0)) p (0) (p (0) ) T x i ith column of X Log-Linear Directional (LLD) control chart.

15 Directional Control Schemes Diagnostic Procedures The real one-coefficient-shift direction index ζ can be identified as ( 1 ( ˆζ = arg max zη Np (0)) T ( ) xj x T 1x ( T j I 3 N j Σxj j zη Np (0))). Σ = diag (ˆp ) ˆpˆp T with ˆp = z η /N. z η is the EWMA observation vector at the signal time point η. We confine the candidate subset of diagnostic shift directions to effects in the first three orders, which is still safe in case a shift appears in a three-factor interaction effect.

16 Outline 1 Multivariate Categorical Modeling 2 Directional Control Schemes 3 Performance Assessment

17 Performance Assessment LLD chart VS MBE chart Multivariate binomial EWMA (MBE) chart, the EWMA version of Patel s (1973) χ 2 -chart. Multivariate Binomial Processes 5 characteristics each with 2 levels, a 5-way contingency table with 2 5 = 32 cells. Table: OC ARLs of LLD and MBE, λ = 0.1, N = 1, 000, ARL 0 =370 δ LLD MBE LLD MBE LLD MBE β (3) β (5) β (1,4) β (2,3) β (2,5) β (3,4)

18 Performance Assessment LLD chart VS MME chart The multivariate multinomial EWMA (MME) chart, a multi-chart comprising p individual charts. Each is the EWMA version of the generalized p-chart in Marcucci (1985). Multivariate Multinomial Processes. 4 characteristics of 2, 2, 3, 3 levels, a 4-way table. Table: OC ARLs of LLD and MME, λ = 0.1, N = 1, 000, ARL 0 =370 δ LLD MME LLD MME LLD MME β (32 ) β (41 ) β (1,2) β (1,32 ) β (2,31 ) β (31,4 1 )

19 Performance Assessment Diagnostic Performance The same setting in the 5-way table as the comparison with MBE. Matching probability P( ˆζ = ζ) Table: Observed matching probability for diagnosing shift direction δ β (2) β (4) β (1,3) β (4,5) β (1,4,5) β (2,3,5) NOTE: λ = 0.1. N = 1, 000.

20 Conclusion The superiority of the LLD chart lies in one-coefficient-shifts or shifts in high-order interaction effects. The LLD chart can work well within the unified framework of multivariate binomial and multivariate multinomial processes. The diagnostic schemes show good performance in estimating fault directions.

21 Thank you for your attention!