Literature Review

Statistical process control (SPC) is often utilized to monitor an industrial process.

How to construct a control chart is an important issue in SPC. The control chart is used to determine whether a manufacturing or business process is in a state of statis-tical control.

The control chart can be used in both phases I and II. In phase I, some reference data are collected and analyzed to assess whether they are in control. Then the in-control process parameters and in-control limits are estimated from the in-in-control data identified from those reference data. In phase II, the process is monitored over time to see whether it is in control by using control limits. The average run length (ARL) is usually used to appraise the process performance.

Shewhart (1931) proposed the ¯X control chart which has been used to monitor the process mean and has good performance for a large sample size or for detecting a large shift in the process mean.

Page (1954) proposed the cumulative sum (CUSUM) control chart whose perfor-mance is better than that of the Shewhart control chart in detecting a small sustained shift in the process mean.

The EWMA control chart was proposed by Roberts (1959) for detecting a small sus-tained shift in the process mean. Its performance for detecting a small sussus-tained shift in the process mean is better than that of the Shewhart control chart. As the in-control

eters and derived its run-length (RL) distribution. Castagliola et al. (2006) reviewed the EWMA control chart for monitoring the process position and variability. Jensen et al. (2006) reviewed the effect of parameter estimation and proposed some recommen-dations for future research.

Sometimes, we are interested in the relationship between a response variable and one or more explanatory variables in the process. Kim et al. (2003) proposed a method based on three EWMA control charts, where these three charts were used for different process parameters in simple linear profiles assuming the in-control process parameters are known. Zou et al. (2006) proposed an LR-based control chart for a change-point model to monitor simple linear profiles assuming the in-control process parameters are unknown. Zou et al. (2007) proposed an MEWMA control chart for monitoring general linear profiles assuming the in-control process parameters are known. Zou et al. (2009) compared five control schemes for monitoring the process mean subject to drifts. Zou et al. (2010) proposed a single chart that integrated the EWMA procedure with the LR test statistics for monitoring both the process mean and variance. Huang (2012) proposed an EWMA control chart based on LR test statistics for monitoring general linear profiles.

Kim et al. (2003) proposed three EWMA control charts for monitoring simple linear profiles as follows: Suppose that data {(xi, yij) : i = 1, 2, . . . , n} are available at time j = 1, 2, . . . , τ , where xis are not all the same and an out-of-control signal occurs at

where εijs are independent standard normal random variables. Model (1.1) is equivalent to

y_ij = β₀⁰ + β₁⁰x⁰_i+ σ ε_ij, i = 1, 2, . . . , n, (1.2)

where β⁰₀ = β₀ + β₁x, β¯ ₁⁰ = β₁, and x⁰_i = x_i − ¯x with ¯x = Σⁿ_i=1x_i/n. At time j, the least-squares estimator of β₀⁰, β₁⁰ , and σ² are

b_0j = ¯y_j,

b_1j = Σⁿ_i=1(xi− ¯x)y_ij Σⁿ_i=1(xi− ¯x)² , and

MSE_j = 1 n − 2

Xn i=1

(y_ij − b_0j − b_1jx⁰_i)²,

where ¯yj = Σⁿ_i=1yi/n. Since b0j, b1j, and MSEj are independent random variables, they proposed three EWMA control charts

EWMAI(j) = λb0j + (1 − λ)EWMAI(j − 1), EWMA_S(j) = λb_1j + (1 − λ)EWMA_S(j − 1),

and

EWMA_E(j) = max{λ ln(MSE_j) + (1 − λ)EWMA_E(j − 1), ln(σ²)},

where EWMA_I(0) = β₀⁰, EWMA_S(0) = β₁⁰, EWMA_E(0) = ln(σ²), and λ is a smoothing parameter in (0, 1]. Those three EWMA control charts are proposed to monitor β₀⁰, β₁⁰,

profiles as follows: Suppose that data (X, yj) are available at time j = 1, 2, . . . , τ , where an out-of-control signal occurs at time τ . The process is called in control at time j if

y_j = Xβ + σ ε_j, j = 1, 2, . . . , τ, (1.3)

where y_j is an n × 1 response vector and X is an n × p known model (or design) matrix of rank p (< nj), β = (β0, β1, . . . , βp−1)^T is a known in-control p × 1 process regression parameter vector, σ is a known in-control positive process scale parameter, and ε_js are independent standardized error vectors with ε_j ∼ N_n(0_n×1, I_n). Set

Z_j(β) = βˆ_j− β σ

and

Zj(σ) = Φ⁻¹ µ

F_χ²_n−p

µ(n − p)ˆσ²_j σ²

¶¶

,

where ˆβ_j = (X^TX)⁻¹X^Ty_j, ˆσ_j² = (y_j− X ˆβ_j)^T(y_j− X ˆβ_j)/(n − p), Φ⁻¹(·) is the inverse function of the standard normal cumulative distribution function (c.d.f.), and F_χ²_n−p(·) is the chi-squared c.d.f. with n − p degrees of freedom. Set Z_j ≡ (Z^T_j(β), Z_j(σ))^T, a (p + 1) × 1 random vector. When the process is in control at time j, Z_j is multivariate normally distributed with mean vector 0_(p+1)×1 and covariance matrix

 

Then the MEWMA sequence is defined as

W_j = λZ_j+ (1 − λ)W_j−1, j = 1, 2, . . . ,

where W₀ ≡ 0_(p+1)×1 and λ is a smoothing parameter in (0, 1]. An out-of-control signal occurs at time j if

WjΣ⁻¹Wj > L λ 2 − λ,

where L (> 0) is chosen to achieve a specified in-control ARL.

In Section 2, general linear profiles are described and then an EWMA control chart based on score test statistics is proposed for monitoring general linear profiles. In Section 3, a simulation study is presented to illustrate the proposed methodology. In Section 4, conclusions are given. In Section 5, some potential future works is suggested.

2 An EWMA control chart for monitoring general linear profiles

In this section, general linear profiles are described and then an EWMA control chart based on score test statistics is proposed for monitoring linear profiles.

2.1 Model

Suppose that data {(y_ij, x_ij): i = 1, 2, . . . , n_j} are available at time j = 1, 2, . . . , τ , where y_ij is the ith response variable at time j, x_ij is its corresponding explanatory variable(s), and an out-of-control signal occurs at time τ . Assume that

yij = βj0u0(xij) + βj1u1(xij) + · · · + βj,p−1up−1(xij) + σjεij, i = 1, 2, . . . , nj, (2.1)

where β_j0, β_j1, . . . , β_j,p−1 are unknown real-valued process regression parameters at time j ; u₀(·), u₁(·), . . . , u_p−1(·) are known real-valued functions; σ_j is an unknown positive process scale parameter at time j ; and ε_ijs are i.i.d. N(0, 1) standardized errors.

Example 1: Model (2.1) has the form

yij = βj0+ βj1xij + · · · + βj,p−1x^p−1_ij + σjεij, i = 1, 2, . . . , nj,

for simple linear profiles if p = 2 or for polynomial profiles if p ≥ 3.

Example 2: Model (2.1) has the form

y_ij = β_j0+ Xk u=1

β_jux_iju+ Xk

u=1

β_juux²_iju+ X

1≤u<u⁰≤k

β_juu⁰x_ijux_iju⁰+ σ_jε_ij, i = 1, 2, . . . , n_j,

with x_ij ≡ (x_ij1, . . . , x_ijk)^T for quadratic polynomial profiles if k ≥ 2.

For simplicity of notation, model (2.1) is rewritten as

y_j = X_jβ_j + σ_jε_j, (2.2)

where y_j (≡ (y_j1, y_j2, ..., y_jnj)^T) is an n_j× 1 response vector at time j, X_j is an known n_j × p model (or design) matrix of full rank p (< n_j) at time j, β_j

(≡ (β_j0, β_j1, . . . , β_j,p−1)^T) is a p × 1 parameter vector of unknown real-valued process regression parameters at time j, σj is an unknown positive process scale parameter at time j, and εjs are independent standardized error vectors with εj ∼ Nnj(0nj×1, Inj).

Set θ_j ≡ (β^T_j, σ_j)^T (∈ R^p× (0, ∞) ≡ Θ), the process parameter vector at time j. Set θ ≡ (β^T, σ)^T (∈ Θ), the in-control process parameter vector.