An Enhanced Auxiliary Information-Based EWMA-t Chart for Monitoring the Process Mean

: The exponentially weighted moving average t chart using auxiliary information (AIB-EWMA-t chart) is an e ﬀ ective approach for monitoring small process mean shifts when the process standard deviation is unstable or poorly estimated. To further enhance the sensitivity of the AIB-EWMA-t chart, in this study, we propose an AIB generally weighted moving average (GWMA) t chart (AIB-GWMA-t chart) to monitor the process mean. The existing EWMA-t, GWMA-t, and AIB-EWMA-t charts are special cases of the AIB-GWMA-t chart. Numerical simulation studies indicate that the AIB-GWMA-t chart performs uniformly and substantially better than the EWMA-t and GWMA-t charts in terms of average run length. Moreover, the AIB-GWMA-t chart with large design and adjustment parameters also outperforms the AIB-EWMA-t chart when the correlation coe ﬃ cients are within a certain range. An illustrative example is provided to highlight the e ﬃ ciency of the proposed AIB-GWMA-t chart in detecting small process mean shifts.


Introduction
Memory-type control charts, which utilize both current and past information to calculate their plotting statistics, are well known to be sensitive tools in detecting small process mean shifts in statistical process control. One memory-type control chart is the exponentially weighted moving average (EWMA) chart introduced by Roberts [1] to effectively improve the detection ability of the Shewhart control chart when monitoring small process mean shifts. Since then, numerous studies aiming to improve the sensitivity and detection ability of EWMA charts have been proposed, such as Crowder [2], Ng and Case [3], Lucas and Saccucci [4], Steiner [5], and Capizzi and Masarotto [6]. In pioneering work, adopting design and adjustment parameters, Sheu and Lin [7] developed a generally weighted moving average (GWMA) chart and found that it outperforms the classical Shewhart and EWMA charts in detecting small process mean shifts.
Subsequently, the popularity of GWMA-type charts has risen rapidly owing to their various quality characteristics. Sheu and Chiu [8] developed a GWMA chart for monitoring Poisson observations. Sheu and Lu [9] proposed a GWMA chart for which observational data show significant autocorrelation. Lu [10] developed a non-parametric GWMA sign chart to improve detection ability in small process mean shifts. Lu [11] proposed a mixed GWMA-Cumulative Sum (GWMA-CUSUM) chart and its reverse CUSUM-GWMA chart to enhance detection sensitivity over existing counterparts. Other studies of GWMA-type charts include Sheu and Yang [12], Sheu and Hsieh [13], Huang et al. [14], Chakraborty et al. [15], and Aslam et al. [16].

EWMA-t Control Charts
Let X be the quality characteristic of interest and assume that X is a normally distributed random variable with mean µ X + δσ X and variance σ 2 X , that is, X ∼ N(µ X + δσ X , σ 2 X ). The process is in the control state when δ = 0; otherwise, the process has shifted. Let X ij , i = 1, 2, . . ., j = 1, 2, . . . , n, be a random sample of size n drawn from the process at time i. Let X i and S 2 X,i be the sample mean and sample variance of the ith subgroup, respectively, where X i = n j=1 X ij /n and S 2 X,i = n j=1 (X ij − X i ) 2 /(n − 1).
X i follows a normal random variable with mean µ X + δσ X and variance σ 2 X /n, whereas (n − 1)S 2 X,i /σ 2 X is a chi-squared random variable with n − 1 degrees of freedom. X i and S 2 X,i are mutually independent random variables for an in-control process. Suppose that the process is in control (δ = 0); then, the statistic T i is defined by: is a Student's t-distribution with n − 1 degrees of freedom and S X,i is the sample standard deviation. Therefore, the EWMA-t statistic Z i at time i can be defined as: where T i is a t distributed random variable at time i and λ is the smoothing constant satisfying 0 < λ ≤ 1. The initial value of Z i is usually set to zero (i.e., Z 0 = 0). The upper control limit (UCL), central line (CL), and the lower control limit (LCL) for the well-known EWMA-t control chart based on Z i are: where L e is the control limit constant chosen to match the desired in-control ARL. When statistics Z i exceeds the UCL or LCL, the EWMA-t control chart initiates an out-of-control signal; otherwise, the process remains in-control. For more details on the EWMA-t chart, see Zhang et al. [22] and Haq et al. [25].

AIB-EWMA-t Control Charts
Assume a corresponding auxiliary variable Y accompanies the quality characteristic X of interest. Let (X, Y) follow a bivariate normally distributed process with means (µ X , µ Y ) and variances (σ 2 X , σ 2 Y ); ρ is the correlation between X and Y, that is, . . , n, is a random sample of size n taken from the process at time i, for i = 1, 2, . . .. Let X i = n j=1 X ij /n and Y i = n j=1 Y ij /n be the sample mean based on (X i1 , X i2 , . . . , X in ) and (Y i1 , Y i2 , . . . , Y in ), respectively. Then, following Abbas et al. [20], the regression estimator of the process mean µ X is given by: The mean and variance of X * i are: Assume that the underlying process remains in an in-control state; then, the statistic T * i is defined by: As this is also a Student's t-distribution with n − 1 degrees of freedom, the AIB-EWMA-t statistic Z * i at time i can be defined as: where λ is the smoothing constant satisfying 0 < λ ≤ 1. The initial value of Z * i is usually set to zero (i.e., Z * i = 0). The asymptotic control limits for the well-known AIB-EWMA-t control chart based on Z * i are: where L ae is the control limit constant which is similar to that of the EWMA-t control chart. The AIB-EWMA-t control chart initiates an out-of-control signal whenever statistic Z * i > UCL or Z * i < LCL; otherwise, the process remains in-control. For more details on the AIB-EWMA-t chart, see Haq et al. [25].

Proposed Control Charts
Sheu and Lin [7] first extended the EWMA chart to the GWMA chart by adding design parameter q and adjustment parameter α. Owing to this novel feature, the GWMA chart is capable of monitoring small process changes. The GWMA statistic H i at time i can be represented as: where X i is the quality characteristic mentioned as above and H 0 is usually set to zero.

Generally Weighted Moving Average (GWMA-t) Control Charts
This study further extends the EWMA-t chart by adding design and adjustment parameters to enhance detection capability. The statistic T i in Equation (1) is used instead of the statistic X i in Equation (10); hence, the novel statistic for the GWMA-t chart is G i , defined as follows: where the initial value of G i is set to zero (i.e., G 0 = 0). For the in-control case, we express the mean and variance of G i as: where  is an asymptotic value. We use the asymptotic control limits instead of the time-varying control limits in this study to simplify the control chart. Assuming that L g denotes the width of the control limit, the GWMA-t chart is: Appl. Sci. 2020, 10, 2252 When the statistic G i remains inside the control limits (LCL, UCL), the process stays in control. However, if G i > UCL or G i < LCL, the process mean shifts. We can determine the value of L g using a numerical simulation to achieve the desired in-control ARL. Since the design parameter q is a constant, we can use parameter α to adjust the kurtosis of the weighting function slightly. In particular, the EWMA-t chart is a special case of the GWMA-t chart when α = 1 and q = 1 − λ.

AIB-GWMA-t Control Charts
This study also extends the AIB-EWMA-t chart by adding design and adjustment parameters to enhance detection capability. Now, we use the statistic T * i in Equation (7) to replace the statistic X i in Equation (10); hence, the statistic for the AIB-GWMA-t chart is G * i , which we define as follows: where the initial value of G * i is equal to zero (i.e., G * i = 0). For the in-control case, the mean and variance of G * i are the same as those of G i in Equations (12) and (13). The chart based on G * i is called the AIB-GWMA-t chart. The constant control limits of the AIB-GWMA-t chart are:  is an asymptotic value. L ag is the control limit constant, as in L ae in Equation (9) or L g in Equation (14), which is used to determine the desired in-control ARL. When the statistic G * i moves beyond the control limit, it indicates a process mean shift. Figure 1 depicts the relationship among the AIB-GWMA-t chart, AIB-EWMA-t chart, GWMA-t chart, and EWMA-t chart.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 16 is an asymptotic value. ag L is the control limit constant, as in ae L in Equation (9) or g L in Equation (14), which is used to determine the desired incontrol ARL . When the statistic * i G moves beyond the control limit, it indicates a process mean shift.  In Figure 1, the AIB-EWMA-t and EWMA-t charts are special cases of the AIB-GWMA-t and GWMA-t charts, respectively when adjustment parameter   1 and design parameter Moreover, the AIB-GWMA-t and AIB-EWMA-t charts reduce to the GWMA-t and EWMA-t charts, respectively when correlation coefficient   0 .

In-Control ARL Profiles
The ARL (Average Run Length) is a popular indicator for evaluating the performance of control charts. When the process operates in an in-control state, the in-control ARL , termed 0 ARL , of a AIB-GWMA-t chart is a function of ( , , ) ag q L  and this is expected to be sufficiently large to avoid false alarms. To compute the ARL values, the Monte Carlo simulation was recommended In Figure 1, the AIB-EWMA-t and EWMA-t charts are special cases of the AIB-GWMA-t and GWMA-t charts, respectively when adjustment parameter α = 1 and design parameter q = 1 − λ. Moreover, the AIB-GWMA-t and AIB-EWMA-t charts reduce to the GWMA-t and EWMA-t charts, respectively when correlation coefficient ρ = 0.

In-Control ARL Profiles
The ARL (Average Run Length) is a popular indicator for evaluating the performance of control charts. When the process operates in an in-control state, the in-control ARL, termed ARL 0 , of a AIB-GWMA-t chart is a function of (q, α, L ag ) and this is expected to be sufficiently large to avoid false alarms. To compute the ARL values, the Monte Carlo simulation was recommended by Sheu and Lin [7] instead of the Markov chain or integral equation approaches since the GWMA statistic is complicated. An algorithm in R has been developed in Appendix A to calculate the ARL values, which are 50,000 run lengths on average. Without loss of generality, the random samples (X ij , Y ij ), j = 1, 2, . . . , n, for i = 1, 2, . . ., are drawn from a bivariate normal distribution, that is, , and the known correlation coefficient ρ, we find that the underlying process is in-control. The control limit coefficient L ag under various combinations of n, q, and α can be adjusted to match the desired ARL 0 . Table 1 presents the L ag values for the AIB-GWMA-t charts. The various parameter combinations of q ∈ {0.1, 0.2, . . . , 0.9} and α ∈ {0.1, 0.2, . . . , 1.0}, correlation coefficient value of ρ = 0, and sample size n ∈ {5, 10} are considered to achieve an in-control ARL at approximately 500. Note that sample sizes n = 5 and 10 are selected for brief discussion to compare with that of Haq et al. [25] and explore the effect of different sample sizes on L ag . When ρ = 0, the results in Table 1 correspond to the proposed GWMA-t charts.  Table 1 shows that when q and α are fixed, a large sample size corresponds to a small control limit coefficient and vice versa. The L ag values decrease as design parameter q or adjustment parameter α increases. In particular, the difference among the L ag values is large for larger q or α values, whereas the difference among the L ag values become small for smaller q or α values. For example, when n = 5, the values of L ag for q = 0.1 and 0.9 at α = 0.9 are 5.060 and 3.145, respectively; however, those for q = 0.1 and 0.9 at α = 0.1 are 5.079 and 5.060, respectively. Moreover, when α = 1.0, the L ag value of the Appl. Sci. 2020, 10, 2252 7 of 15 AIB-GWMA-t chart at q = 0.9 is 3.047, which is close to 3.046 at λ = 0.1 for the existing AIB-EWMA-t chart, as shown by Haq et al. [25].

Performance Comparison
The out-of-control ARL, denoted by ARL 1 , is expected to be sufficiently small to detect shifts early when the process is out of control. It is customary to set an ARL 0 value and then compute the ARL 1 values. The smaller the ARL 1 value, the better is the statistical performance. To compare the performance of the AIB-GWMA-t charts, this study keeps ARL 0 close to 500 and the ARL 1 s are evaluated for specific process mean shifts. To ensure the discussion remains brief, design parameter q = {0.5, 0.7, 0.9, 0.95}, adjustment parameter α = {0.5, 0.7, 0.9, 1.0}, correlation coefficient ρ = {0.00, 0.25, 0.50, 0.75, 0.95}, and process mean shifts δ = {0.1, 0.2, 0.4, 0.6, 1.0, 2.0} are used to compute the ARL values of the AIB-GWMA-t charts with n = 5 and n = 10 (see Tables 2 and 3, respectively). In particular, α = 1.0 and q = 1 − λ reduce to the AIB-EWMA-t chart proposed by Haq et al. [25]. Furthermore, when ρ = 0, the AIB-GWMA-t and AIB-EWMA-t charts reduce to the GWMA-t and EWMA-t charts, respectively.
The boldface values in Tables 2 and 3 indicate smaller ARL 1 values than those of the AIB-EWMA-t and EWMA-t charts under various process mean shifts with the specific design parameter q and correlation coefficient ρ. The main findings from Tables 2 and 3 are as follows: (1) For fixed q and α, the values of L ag are close and unaffected by the choices of ρ, a finding consistent with that of Haq and Abidin [21] and Haq et al. [25]. (2) A larger sample size n results in a smaller ARL 1 value at fixed parameter combinations of (q, α, ρ).
(3) For fixed qα, and ρ, the ARL 1 values decrease as the value of δ increases. Moreover, for fixed α, δ, and ρ, the ARL 1 values tend to decrease as the value of q increases. Similarly, the ARL 1 value is a decreasing function of ρ when q, α, and δ are fixed. (4) The AIB-GWMA-t and AIB-EWMA-t charts uniformly perform better than the GWMA-t and EWMA-t charts, respectively. This result reveals that the use of auxiliary information enhances the performance of the GWMA-t and EWMA-t charts, especially for large values of ρ. (5) To detect small process mean shifts, the AIB-GWMA-t chart with large q and α performs better than the AIB-EWMA-t chart with 0.25 ≤ ρ ≤ 0.75. However, the AIB-GWMA-t chart performs comparably to the AIB-GWMA-t chart at ρ = 0.95.
Facing an auxiliary variable with different correlations, suitable parameter combinations of (q, α) facilitate the use of our proposed charts to monitor small process mean shifts. Figure 2 depicts the ARL 1 curves of the AIB-GWMA-t charts at different correlation coefficients with specific parameter combinations. It shows that when we apply the proposed GWMA-t or AIB-GWMA-t charts to monitor small process mean shifts (δ = 0.1), large values of q and α under ρ ≤ 0.75 are recommended for practical applications. Moreover, the performance of the AIB-GWMA-t chart is comparable to the AIB-EWMA-t chart at q = 0.95 regardless of the α value for ρ = 0.95. Table 2. Values for the AIB-GWMA-t charts when n = 5.  ARL curves of the AIB-GWMA-t charts at different correlation coefficients with specific parameter combinations. It shows that when we apply the proposed GWMA-t or AIB-GWMA-t charts to monitor small process mean shifts ( 0.1)   , large values of q and  under 0.75   are recommended for practical applications. Moreover, the performance of the AIB-GWMA-t chart is comparable to the AIB-EWMA-t chart at 0.95 q  regardless of the  value for 0.95   .  The near optimal parameters of the proposed AIB-GWMA-t charts at various correlation coefficients are next investigated. Considering ARL 0 ≈ 500 and correlation coefficient ρ = {0.00, 0.25, 0.50, 0.75, 0.95}, Table 4 proposes the near optimal parameters of the proposed charts under a small shift δ = 0.1, median shift δ = 0.6, and large shift δ = 2.0 to reach the minimum ARL 1 . Table 4. The near optimal parameters under various process mean shifts at ARL 0 ≈ 500. As shown in Table 4, when ρ = 0 and δ = 0.1, in the GWMA-t chart with the near optimal parameters q * = 0.95, α * = 0.9, and L * ag = 2.750, for n = 5, the minimum ARL 1 is 130.698. Similarly, for the near optimal parameters q * = 0.97, α * = 0.9, and L * ag = 2.448, for n = 10, the minimum ARL 1 is 58.390. A significant result is that the GWMA-t chart outperforms the EWMA-t chart at detecting small process mean shifts. When the auxiliary variable is related to the quality characteristic between ρ = 0.25 and ρ = 0.75, the AIB-GWMA-t chart with q = 0.95 and α = 0.9 has the minimum ARL 1 at δ = 0.1. However, the AIB-EWMA-t chart with large q performs better for small process mean shifts at ρ = 0.95.

Illustrative Example
A simulated dataset is used to demonstrate the implementation of the existing EWMA-t and AIB-EWMA-t charts as well as the proposed GWMA-t and AIB-GWMA-t charts when detecting process mean shifts. For this purpose, 50 bivariate samples, each of size n = 5, are generated from a bivariate normal distribution with µ X = µ Y = 0, σ X = σ Y = 1, and ρ = 0.75. The first 20 samples are referred to as in-control. Moreover, assuming the last 30 samples suffered from some assignable causes, the process mean shifted from µ X to µ X + δσ X , where δ = 0.2, which is referred to as out-of-control. Table 5 lists these 50 simulated samples when ρ = 0.75.
To investigate the detection ability of these existing and proposed charts, their in-control ARL values are both set to 500. From Table 2, the parameter combinations (q, α, L e(g) ) for the EWMA-t and GWMA-t charts are (0.9, 1.0, 3.047) and (0.9, 0.9, 3.146), respectively. When the auxiliary variable is available and correlated with the study variable at ρ = 0.75, the parameter combinations (q, α, L ae(ag) ) for the AIB-EWMA-t and AIB-GWMA-t charts are (0.9, 1.0, 3.042) and (0.9, 0.9, 3.142), respectively. Moreover, Z i , G i , Z * i , and G * i represent the EWMA-t, GWMA-t, AIB-EWMA-t, and AIB-GWMA-t statistics, respectively. Table 6 lists the related upper control limits of these charts (see also Figure 3).   Figure 3 shows that the process remains in control during the first 20 samples. However, when an assignable cause produces small shifts in the process mean ( 0.2)   , the EWMA-t chart triggers an out-of-control signal in the 50 th sample, whereas the GWMA-t chart triggers an out-of-control signal after the 49 th sample. When an auxiliary variable exists, the first outof-control signal is detected with the AIB-EWMA-t chart in sample 48, while 29 samples are required for the   Figure 3 shows that the process remains in control during the first 20 samples. However, when an assignable cause produces small shifts in the process mean (δ = 0.2), the EWMA-t chart triggers an out-of-control signal in the 50th sample, whereas the GWMA-t chart triggers an out-of-control signal after the 49th sample. When an auxiliary variable exists, the first out-of-control signal is detected with the AIB-EWMA-t chart in sample 48, while 29 samples are required for the AIB-GWMA-t chart. The simulation results suggest that the AIB-GWMA-t chart is more sensitive for detecting small process mean shifts than the AIB-EWMA-t chart and its counterpart, the GWMA-t chart. Table 6. Simulation dataset of the EWMA-t and GWMA-t charts when ρ = 0.0 and AIB-EWMA-t and AIB-GWMA-t charts when ρ = 0.75 at process mean shift δ = 0.2.