Comparison of Phase II Control Charts Based on Variable Selection Methods

Abstract

In recent years, control charts based on variable selection (VS) algorithms have been suggested for monitoring multivariate data. These charts share the common idea that process faults usually affect a small fraction of the monitored quality characteristics. Thus, VS methods can be used to identify the subset of the variables for which the shift may have occurred. However, the suggested VS-based control charts differ in many aspects such as the particular VS algorithm and the type of control statistic. In this paper, we compare VS-based control charts in various out-of-control scenarios characterizing modern manufacturing environments such as high-dimensional data, profile, and multistage process monitoring. The main aim of this paper is to provide practical guidelines for choosing a suitable VS-based monitoring scheme.

Appendix

In the following, we provide details on the OC scenarios listed in the x axes of Fig. 1. When only variables or stages with an even (odd) index are subject to a shift of size δ, the OC scenario is indicated with either Even[δ] or Odd[δ].

• 1. Unstructured model: $1 = (\beta _{1} = 0.5)$, $2 = (\beta _{1} = 1)$, $3 = (\beta _{3} = 0.5)$, $4 = (\beta _{3} = 1)$, $5 = (\beta _{1} = 0.5,\;\beta _{2} = 0.25)$, $6 = (\beta _{1} = 0.5,\;\beta _{2} = 0.5)$, $7 = (\beta _{1} = 0.5,\;\beta _{2} = 0.75)$, $8 = (\beta _{1} = 0.5,\;\beta _{3} = 0.25)$ $9 = (\beta _{1} = 0.5,\;\beta _{3} = 0.5)$, $10 = (\beta _{1} = 0.5,\;\beta _{3} = 0.75)$, $11 = (\beta _{3} = 0.5,\;\beta _{8} = 0.25)$, $12 = (\beta _{3} = 0.5,\;\beta _{8} = 0.5)$, $13 = (\beta _{3} = 0.5,\;\beta _{8} = 0.75)$, $14 =\beta _{1} = 0.5,\;\beta _{2} = 0.25,\;\beta _{3} = 0.25)$, $15 = (\beta _{1} = 0.25,\;\beta _{2} = 0.25,\;\beta _{3} = 0.5)$, $16 = (\beta _{2} = 0.5,\;\beta _{3} = 0.25,\;\beta _{8} = 0.25)$, $17 = (\beta _{2} = 0.25,\;\beta _{3} = 0.25,\;\beta _{8} = 0.5)$, $18 = (\beta _{7} = 0.5,\;\beta _{8} = 0.25,\;\beta _{9} = 0.5)$, $19 = (\beta _{7} = 0.25,\;\beta _{8} = 0.75,\;\beta _{9} = 0.5)$, $20 = (\beta _{6} = 0.5,\;\beta _{8} = 0.25,\;\beta _{10} = 0.5)$, $21 = (\beta _{6} = 0.25,\;\beta _{8} = 0.75,\;\beta _{10} = 0.5)$, 22 = (Even[0. 25]), 23 = (Even[0. 5]), 24 = (Odd[0. 25]), 25 = (Even[0. 5]), 26 = (Even[0. 5], Odd[0. 25]), 27 = (Odd[0. 5], Even[0. 25]), $28 = (\omega = 1.2)$, $29 = (\omega = 1.5)$, $30 = (\omega = 2)$.

• 2. Linear profiles: $1 = (\beta _{1} = 0.1)$, $2 = (\beta _{1} = 0.3)$, $3 = (\beta _{2} = 0.2)$, $4 = (\beta _{2} = 0.5)$, $5 = (\beta _{3} = 0.2)$, $6 = (\beta _{3} = 0.5)$, $7 = (\beta _{1} = 0.1,\;\beta _{2} = 0.1)$, $8 = (\beta _{1} = 0.4,\;\beta _{2} = 0.2)$ $9 = (\beta _{1} = 0.4,\;\beta _{2} = 0.6)$, $10 = (\beta _{1} = 0.6,\;\beta _{2} = 0.6)$, $11 = (\beta _{1} = 0.1,\;\omega = 1.2)$, $12 = (\beta _{1} = 0.3,\;\omega = 1.2)$, $13 = (\beta _{1} = 1,\;\omega = 1.2)$, $14 = (\beta _{2} = 0.1,\;\omega = 1.2)$, $15 = (\beta _{2} = 0.4,\;\omega = 1.2)$, $16 = (\beta _{1} = 1.2,\;\omega = 1.2)$, $17 = (\beta _{1} = 0.1,\;\beta _{2} = 0.1,\;\omega = 1.2)$, $18 = (\beta _{1} = 0.4,\;\beta _{2} = 0.2,\;\omega = 1.2)$, $19 = (\beta _{1} = 0.4,\;\beta _{2} = 0.6,\;\omega = 1.2)$, $20 = (\beta _{1} = 0.6,\;\beta _{2} = 0.6,\;\omega = 1.2)$, $21 = (\omega = 1.2)$, $22 = (\omega = 1.5)$, $23 = (\omega = 2)$.

• 3. Cubic profiles: $1 = (\beta _{1} = 0.1)$, $2 = (\beta _{1} = 0.3)$, $3 = (\beta _{1} = 1)$, $4 = (\beta _{2} = 0.2)$, $5 = (\beta _{2} = 0.4)$, $6 = (\beta _{2} = 1.2)$, $7 = (\beta _{4} = 0.2)$, $8 = (\beta _{4} = 0.5)$, $9 = (\beta _{1} = 0.1,\;\beta _{2} = 0.1)$, $10 = (\beta _{1} = 0.1,\;\beta _{3} = 0.2)$ $11 = (\beta _{1} = 0.1,\;\beta _{4} = 0.2)$, $12 = (\beta _{2} = 0.2,\;\beta _{3} = 0.2)$, $13 = (\beta _{2} = 0.4,\;\beta _{4} = 0.2)$, $14 = (\beta _{3} = 0.5,\;\beta _{4} = 0.5)$, $15 = (\beta _{1} = 0.1,\;\beta _{2} = 0.1,\;\beta _{3} = 0.1)$, $16 = (\beta _{1} = 0.1,\;\beta _{3} = 0.2,\;\beta _{4} = 0.1)$, $17 = (\beta _{2} = 0.1,\;\beta _{3} = 0.3,\;\beta _{4} = 0.2)$, $18 = (\beta _{1} = 0.1,\;\beta _{2} = 0.1,\;\beta _{3} = 0.1,\;\beta _{4} = 0.1)$, $19 = (\beta _{1} = 0.1,\;\beta _{2} = 0.2,\;\beta _{3} = 0.1,\;\beta _{4} = 0.2)$, $20 = (\beta _{1} = 0.2,\;\beta _{2} = 0.1,\;\beta _{3} = 0.2,\;\beta _{4} = 0.1)$, $21 = (\omega = 1.2)$, $22 = (\omega = 1.5)$, $23 = (\omega = 2)$.

• 4. Non parametric profiles. The following OC scenarios are referred to possible shifts in the regression coefficients of models I, II and III. $1 = (\mathrm{I},\beta _{1} = 1.00,\;\beta _{2} = 1.30)$, $2 = (\mathrm{I},\beta _{1} = 1.00,\;\beta _{2} = 1.50)$, $3 = (\mathrm{I},\beta _{1} = 1.10,\;\beta _{2} = 1.00)$ $4 = (\mathrm{I},\beta _{1} = 1.30,\;\beta _{2} = 1.00)$, $5 = (\mathrm{I},\beta _{1} = 1.20,\;\beta _{2} = 1.00,\;\omega = 1.10)$, $6 = (\mathrm{I},\beta _{1} = 1.00,\;\beta _{2} = 1.20,\;\omega = 1.30)$, $7 = (\mathrm{II},\beta _{1} = 0.10,\;\beta _{2} = 3.00)$, $8 = (\mathrm{II},\beta _{1} = 0.30,\;\beta _{2} = 3.00)$, $9 = (\mathrm{II},\beta _{1} = 0.10,\;\beta _{2} = 2.00)$ $10 = (\mathrm{II},\beta _{1} = 0.30,\;\beta _{2} = 2.00)$, $11 = (\mathrm{II},\beta _{1} = 0.20,\;\beta _{2} = 4.00,\;\omega = 1.10)$, $12 = (\mathrm{II},\beta _{1} = 0.20,\;\beta _{2} = 4.00,\;\omega = 1.30)$ $13 = (\mathrm{III},\beta _{1} = 2.00,\;\beta _{2} = 0.90)$, $14 = (\mathrm{III},\beta _{1} = 4.00,\;\beta _{2} = 0.90)$, $15 = (\mathrm{III},\beta _{1} = 2.00,\;\beta _{2} = 0.75)$, $16 = (\mathrm{III},\beta _{1} = 4.00,\;\beta _{2} = 0.75)$, $17 = (\mathrm{III},\beta _{1} = 2.00,\;\beta _{2} = 0.90,\;\omega = 1.20)$, $18 = (\mathrm{III},\beta _{1} = 4.00,\;\beta _{2} = 0.75,\;\omega = 1.20)$.

• 5. Multistage process. In the following, β _j indicates the shift of size β, occurring at the j-th stage, j = 1, …, 20. $1 = (\beta _{1} = 0.75)$, $2 = (\beta _{5} = 0.75)$, $3 = (\beta _{10} = 0.75)$, $4 = (\beta _{1} = 1.5)$, $5 = (\beta _{5} = 1.5)$, $6 = (\beta _{10} = 1.5)$, $7 = (\beta _{2} = 0.6,\;\beta _{8} = 0.6)$, $8 = (\beta _{4} = 0.6,\;\beta _{5} = 0.6)$, $9 = (\beta _{2} = 1.2,\;\beta _{8} = 1.2)$, $10 = (\beta _{4} = 1.2,\;\beta _{5} = 1.2)$, $11 = (\beta _{2} = 1.8,\;\beta _{8} = 1.8)$, $12 = (\beta _{4} = 1.8,\;\beta _{5} = 1.8)$, $13 = (\beta _{1} =\beta _{5} =\beta _{10} = 0.2,\;\beta _{3} =\beta _{7} = 0.4)$, $14 = (\beta _{3} =\beta _{5} =\beta _{7} = 0.2,\;\beta _{4} =\beta _{6} = 0.4)$, $15 = (\beta _{1} =\beta _{5} =\beta _{10} = 0.6,\;\beta _{3} =\beta _{7} = 0.4)$, $16 = (\beta _{3} =\beta _{5} =\beta _{7} = 0.6,\;\beta _{4} =\beta _{6} = 0.4)$, 17 = (Even[0. 1], Odd[0. 2]), 18 = (Even[0. 5], Odd[0. 25]).