SPC refers to statistical methods which are extensively used to monitor and improve the quality of industrial processes and others operations. Most of the studies on SPC are focused on the charting skill which is used to monitor processes in order to distinguish
special significant reasons of variation from general allocation reasons of variation in processes. In the Phase I stage, a set of process data is gathered to construct trial control limits that determine whether or not the process has been in control over the period of time, and then to model the in-control process so that reliable control limits of the control chart can be established for the later Phase II stage. In the Phase II stage, process data are compared with a pre-established standard from the previous Phase I stage to determine whether the process is in control or not.
Shewhart (1931) proposed a Shewhart ¯X control chart which is briefly introduced as follows: At time j (≥ 1), let Xj1, Xj2, . . . , Xjnbe i.i.d. N(µj, σ2) observations, where µj
denotes the unknown process mean at time j and σ the known positive process standard deviation. Set ¯Xj ≡ Pn
i=1Xji/n for j ≥ 1. Then an out-of-control signal occurs at time j (≥ 1) if√
n| ¯Xj− µ|/σ > L, where µ denotes the known in-control process mean and L is chosen to achieve a specified in-control average run length (ARL).
Page (1954) proposed a cumulative sum (CUSUM) control chart which is briefly introduced as follows: Let X1, X2, . . . be independent observations such that Xj ∼ N(µj, σ2), where µj denotes the unknown process mean at time j and σ the known positive process standard deviation. For j ≥ 1, two statistics Cj+and Cj− are iteratively defined as
Cj+ ≡ max{0, Cj−1+ + Xj − (µ + K)}
and
Cj− ≡ max{0, Cj−1− − Xj + (µ − K)},
of the specified shift in process mean that should be quickly detected by the scheme.
Then an out-of-control signal occurs at time j (≥ 1) if max{Cj+, Cj−} ≥ Hσ, where H (> 0) is chosen to achieve a specified in-control ARL.
Roberts (1959) proposed an EWMA, originally called geometric moving average, con-trol chart which is briefly introduced as follows: At time j (≥ 1), let Xj1, Xj2, . . . , Xjn
be i.i.d. N(µj, σ2) observations, where µj denotes the unknown process mean at time j and σ the known positive process standard deviation. The EWMA sequence is
Wj ≡ (1 − λ)Wj−1+ λ( ¯Xj − µ) = λ
j−1
X
k=0
(1 − λ)k( ¯Xj−k − µ), j = 1, 2, . . . ,
where W0 ≡ 0, µ denotes the known in-control process mean, and λ is a smoothing parameter chosen in (0, 1]. Observe that the standard deviation of Wj is
σj = s
λ[1 − (1 − λ)2j] n(2 − λ) σ →
s λ
n(2 − λ)σ
as j → ∞. Then out-of-control signal occurs at time j (≥ 1) if |Wj|/σj > L, where L is chosen to achieve a specified in-control ARL.
An EWMA control chart is typically designed in the Phase II stage for a manu-facturing process. Stoumbos et al. (2003) investigated the performance of Shewhart, EWMA, and CUSUM control charts for detecting sustained shifts and drifts in the pro-cess mean and variance. In practice, the in-control propro-cess distribution is rarely known exactly, and thus control charts are usually constructed using an approximate in-control process distribution estimated from some historical in-control process data. Jones et al. (2001) utilized a numerical procedure to study the run-length (RL) distribution of
an EWMA control chart using estimated in-control process parameters. Jensen et al.
(2006) provided a review of the literature that considered the effect of in-control process parameter estimation on control charts and concluded that the influence of in-control process parameter estimation on control charts should not be ignored.
In most SPC applications, it is assumed that the quality of a process can be suit-ably represented by the joint distribution of quality characteristics. However, in many situations, the quality of a process may be better characterized and summarized by the relationship between the response variable and one or more explanatory variables. Sev-eral methods for monitoring linear profiles have been proposed in literature, e.g., Kim et al. (2003) proposed a control chart for monitoring simple linear profiles in the known in-control process parameter case, and Zou et al. (2006) proposed a control chart based on the LR test statistics for monitoring simple linear profiles to detect a sustained shift in the unknown in-control process parameter case.
Through the modern technology that allows simultaneously monitoring all key qual-ity characteristics during a manufacturing process, the monitored qualqual-ity characteris-tics are generally dependent each other. The purpose of multivariate techniques is to study whether quality characteristics are simultaneously in control or not. Lowry et al.
(1992) proposed a multivariate exponentially weighted moving average (MEWMA) con-trol chart giving guidelines for designing this easy-to-implement multivariate procedure.
The performance of their control chart is similar to that of a multivariate cumulative sum (MCUSUM) control chart (Crosier, 1988) in detecting a small shift in the process mean.
Zou et al. (2007) [13] proposed an MEWMA control chart for monitoring general lin-ear profiles in the known in-control process parameter case, which is briefly introduced as follows: The process is called in control at time j (≥ 1) if
yj = Xβ + σεj,
where yj is the n × 1 response vector at time j, X is the known n × p model matrix of full rank p (< n), β (≡ (β0, β1, ..., βp−1)T) is the known p × 1 vector of real-valued in-control process regression parameters, σ is the known positive in-control process scale parameter, and εjs are i.i.d. Nn(0n×1, In) standardized error vectors. Set Zj ≡ (ZjT(β), Zj(σ))T, where Zj(β) ≡ ( ˆβj − β)/σ and Zj(σ) ≡ Φ−1(Fχ2
n−p((n − p)ˆσj2/σ2)) with ˆβj ≡ (XTX)−1XTyj, ˆσj2 ≡ (yj − X ˆβj)T(yj − X ˆβj)/(n − p), Φ−1(·) denoting the inverse of the standard normal cumulative distribution function (c.d.f.), and Fχ2
n−p(·) the c.d.f. of the chi-squared distribution with n − p degrees of freedom. The MEWMA sequence is
Wj ≡ λZj+ (1 − λ)Wj−1 = λ
j−1
X
k=0
(1 − λ)kZj−k, j = 1, 2, . . . ,
where W0 ≡ 0(p+1)×1 and λ is a smoothing parameter chosen in (0, 1]. Observe that the in-control covariance matrix of Wj is
λ[1 − (1 − λ2j)]
2 − λ Σ → λ
2 − λΣ
as j → ∞, where
Σ =
(XTX)−1 0p×1 0Tp×1 1
.
Then an out-of-control signal occurs at time j (≥ 1) if (2−λ)WjTΣ−1Wj/λ > L, where L is chosen to achieve a specified in-control ARL.
Zou et al. (2007) [16] proposed a self-starting control chart based on recursive resid-uals for monitoring simple linear profiles to detect a sustained shift in the process intercept, slope, or standard deviation. Although current SPC methods mostly focus on detecting a sustained shift in the process mean, time-varying shifts in the process mean frequently occur in industrial applications. Thus, Zou et al. (2009) investigated several control charts for detecting drifts in the process mean. Recently, Zou et al.
(2010) proposed an EWMA control chart based on the LR test statistics for monitoring the process mean and variance.