Nonparametric Double EWMA Control Chart for Process Monitoring

In monitoring process parameters, we assume normality of the quality characteristic of interest, which is an ideal assumption. In many practical situations, we may not know the distributional behavior of the data, and hence, the need arises use nonparametric techniques. In this study, a nonparametric double EWMA control chart, namely the NPDEWMA chart, is proposed to ensure efficient monitoring of the location parameter. The performance of the proposed chart is evaluated in terms of different run length properties, such as average, standard deviation and percentiles. The proposed scheme is compared with its recent existing counterparts, namely the nonparametric EWMA and the nonparametric CUSUM schemes. The performance measures used are the average run length (ARL), standard deviation of the run length (SDRL) and extra quadratic loss (EQL). We observed that the proposed chart outperforms the said existing schemes to detect shifts in the process mean level. We also provide an illustrative example for practical considerations.


Introduction
Process monitoring plays an important role in improving quality of the final output.Control charts are important statistical process monitoring tools that help in differentiating un-natural variations from natural.The design structures of control charts may be memory-less (cf.Shewhart, Shewhart 1931) or memory (cf.CUSUM, Page 1954) and (cf.EWMA, Roberts 1959).The memory-less control charts are meant for larger shifts while the others are meant for the shifts of smaller magnitudes.The application of control charting techniques is not only limited the to manufacturing industry but it covers a wide range of disciplines, such as nuclear engineering (cf.Hwang, Lin, Liang, Yau, Yenn & Hsu 2008), health care (cf.Woodall 2006), education (cf.Wang & Liang 2008) and analytical laboratories (cf.Masson 2007, Abbasi 2010), among many others.
There are different types of classifications of control charts, such as parametric versus non-parametric, variable versus attributes, univariate versus multivariate and bayesian versus classical.The parametric charting structures rely on the assumption that the parent distribution of the quality characteristic of interest is known (more specifically normally distributed in many cases).This assumption is not very realistic as, in practice, one may not know the distributional form of the quality characteristic of interest.This limitation restricts the use of parametric control charts for the monitoring of process parameters.In these scenarios, the parametric charts may lead to misleading results, and hence, we prefer to use the non-parametric control charts.The relevant literature is written by Chakraborti & Graham (2007), Das (2009), Qiu, Zou & Wang (2010), Human, Chakraborti & Smit (2010), Khilare & Shirke (2010), and Pawar & Shirke (2010), Abbasi & Miller (2013) and the references therein.The robust charting structures also provide a good alternative for nonnormal situations (cf.Stoumbos & Reynolds 2000, Stoumbos & Sullivan 2002, and the references therein) There are different types of non-parametric Shewhart, EWMA and CUSUM control charts.For an efficient detection of small shifts, nonparametric EWMA (NPEWMA) and nonparametric CUSUM (NPCUSUM) control charts have also been proposed in the literature.In reference to several of these; Li, Tang & Ng (2010) proposed NPEWMA and NPCUSUM charts based on the Mann-Whitney statistic; Zou & Tsung (2011) proposed a multivariate EWMA control chart using the weighted version of the sign test; Graham, Chakraborti & Human 2011a, 2011b proposed nonparametric EWMA sign and signed-rank control charts to monitor the location parameter; Yang & Cheng (2011) proposed two NPEWMA control charts, namely the nonparametric EWMA sign (NPS E ) chart and the nonparametric Arcsine EWMA sign (NPAS E ) chart; Yang & Cheng (2011) proposed a nonparametric CUSUM (NPS C ) chart, using the sign statistics, for quick detection of shifts from the process target.Abbasi (2012) may also be seen in this direction.
The double EWMA (DEWMA) concept was investigated by Shamma & Shamma (1992), Zhang & Chen (2005), and Khoo, Teh & Wu (2010) for a normally distributed quality characteristic of interest.Zhang, Govindaraju, Lai & Bebbington (2003) also investigated the DEWMA chart for Poisson processes.This study is aimed at proposing a new nonparametric DEWMA chart, namely the NPDEWMA chart to efficiently monitor process location.The organization of the rest of the article is given as follows: Section 2 describes the design structure of the proposed NPDEWMA control chart.Section 3 evaluates the performance of the proposed chart using different run length characteristics.Section 4 provides a comparison of the NPDEWMA chart with recently proposed NPS E , NPAS E and NPS C charts.Section 5 presents an example to illustrate the application of the proposed, and finally, the article ends with the concluding remarks in Section 6.

The Proposed Nonparametric Double EWMA (NPDEWMA) Chart
This section provides the design structure of the proposed NPDEWMA control chart to monitor the location parameter.Assume that X 1 ,X 2 ,. . .,X n represent a random sample of size n from a process with the location parameter µ.Let us define a transformed random variable Y as Y i = X i -µ.Let p be a probability measure defined as: p=pr(Y i >0).We assume that the process location is incontrol if p takes the value p 0 and otherwise out-of-control for any other value.We intend to control the location parameter by monitoring the stability of p with reference to p 0 .
In order to do this, we introduce here an indicator variable I,as I = 1 if Y i > 0 and zero otherwise.Based on this indicator variable, Yang, Lin & Cheng (2011) defined M = n i=1 I i , which follows a binomial distribution with parameters n and p 0 .The arcsine transformation for M and its approximate distribution is given as Based on the above mentioned variable, the exponentially weighted moving average (EWMA) statistic is defined as: where λ is a smoothing constant (0 < λ < 1), and W 0 = sin −1 √ p 0 .
We define here another EWMA statistic for W t as: where λ is defined above and D 0 = sin −1 √ p 0 .
We refer to this new statistic as the double EWMA (DEWMA) statistic.The DEWMA statistic defined above may be expressed in an alternative from, as (cf.Roberts 1959 andMontgomery 2009) have described By substitutions and further simplifications we have For the DEWMA statistic D t , the expressions for the mean and the variance are respectively given as (cf.Zhang et al. 2003 andZhang &Chen 2005) Based on these quantities, the L-sigma limits for the proposed NPDEWMA chart are defined as where L is the control limits coefficient that helps in fixing the average run length (ARL) for an in-control situation denoted by ARL 0 .The values of L depend on the choices of n, λ and ARL 0 for the NPDEWMA control chart.The values of L are worked out for different combinations of the aforementioned quantities.Figure 1 presents plots of ARL 0 versus L for some representative values of n at λ = 0.05 for ARL 0 values ranging from 50 to 500.These plots help to choose the appropriate control chart multiplier to fix the ARL 0 for the NPDEWMA chart.The ARL 0 plots versus L may be obtained on the similar lines for other choices of n, λ and ARL 0 .
By using the above mentioned structure of the control limits for the proposed NPDEWMA chart, an out-of-control signal (if any) is detected when D t plots beyond the (LCL, UCL) limits.The NPDEWMA chart makes use of current as well as the past information to efficiently monitor the location shits (particularly of a smaller magnitude) that accumulate over time.

Performance Evaluations of the Proposed Chart
This section provides the performance evaluations of the proposed NPDEWMA chart to monitor the location parameter.The performance measures used in this study are the average run length (ARL) and extra quadratic loss (EQL).The ARL measure is defined as the average number of samples required before an out-ofcontrol signal is issued in the process.It is classified by two forms, namely ARL 0 (when process is in-control) and ARL 1 (when process is out-of-control).Also, it is evaluated shift by shift (we will use δ to refer to the amount of shift in standard deviation units).A good chart should have larger values for ARL 0 and smaller values for ARL 1 .The other measure, namely EQL is evaluated over the whole range of δ values (from the smallest (min) to the largest (max)).It is defined as (see cf. Ahmad, Lin, Abbasi & Riaz 2012, 2013a, 2013b) and the references therein for further details) where ARL(δ) is the average run length of a particular chart at δ.It is generally desirable to have smaller values of EQL for an efficient charting structure.
In order to evaluate the performance of the proposed NPDEWMA chart for different numbers of shifts, p = 0.5 is taken as the in-control value while p = 0.5 is the out-of-control value.Moreover, by computing the EQL, we define δ as a deviation of p from 0.5 (the in-control level) to some other value of p in standard deviation units.In this study, a Monte Carlo simulation with 10 4 iterations is used for the run length distribution of the proposed NPDEWMA chart with an acceptable error rate (cf.Lucas & Saccucci 1990, Maravelakis, Panaretos & Psarakis 2005, Abbasi 2010and Abbasi & Miller 2013, Kim 2005and Schaffer & Kim 2007).The run length properties investigated in this study include ARL, standard deviation of the run length (SDRL) and different quantile points (Q i = i th percentile point) of the run length distribution.It should to be noted that Q 0.50 refers to median run length (MRL), which is also a useful run length measure.
The summary of the run length properties (in-control and out-of-control) for the proposed NPDEWMA chart is reported in the following form: ARL and SDRL (along with the controlling coefficient L) (cf.Table 1), different percentiles (cf.Table 2) and EQL (cf.Table 5) for some representative choices of n and at ARL 0 = 370.For the other choices, the results can be obtained on similar lines.The relative standard errors of the results reported in the above mentioned tables are observed to be around 1.5%, and they are checked by repeating the simulations.This is quite acceptable in control chart studies (cf.Kim 2005 andSchaffer &Kim 2007).

Comparative Analysis of the Results
In this section we provide a comparative discussion of the results obtained in Section 3 for the proposed NPDEWMA chart.Moreover, the efficiency of the proposed chart is also compared with the recently proposed NPS E , NPAS E and NPS C charts in the literature.The summarized information of these charts, along with the proposed NPDEWMA chart, is provided in Table 3.For the details of these charting structures, see Yang et al. (2011), Yang & Cheng (2011) and Abbasi (2012).We evaluate performance of these competing charts for some selective values of n.The control charting parameters of different control charts (like K, H, k, L) are set in such a way that the ARL 0 value of all the charts is fixed at 370 for valid comparisons among different charts.The resulting ARL values for the competing charts are provided in Table 4.The EQL values are also evaluated for these competing charts on the same lines as the proposed NPDEWMA chart.The results for all charts are provided in Table 5.
Revista Colombiana de Estadística 39 (2016) 167-184 Revista Colombiana de Estadística (2016) 167-184  For a comparative discussion of the different choices for the proposed NPDEW-MA chart and the other competing charts, we have created some useful graphs for different values of n and λ at ARL 0 = 370.The similar graphs may be observed for the other choices of the said quantities.These graphs are provided in Figures 2-5 where p is plotted on horizontal axis and the ARL on the vertical axis.They are in logarithmic scale for better visual inspection.In terms of different measures, the performance evaluations of the proposed and the other competing control charts, advocate the following: i) the proposed NPDEWMA chart is easily implemented in practice for process monitoring as it simply involves the computation of the D t statistic given in ( 4), and it is plotted it against the limits given in (7) (without relying on normality); ii) the proposed NPDEWMA chart has the ability to efficiently detect smaller (for smaller choices of λ) as well as larger shifts (for larger choices of λ) (cf.Table 1 and Figures 2-3); iii) the proposed structure is effective for both directions, i.e. increasing and decreasing shifts in the process parameter) (cf.iv) the run length distribution of NPDEWMA is positively skewed; v) the NPDEWMA chart shows decreasing behavior of ARL, SDRL and the percentile points with a decrease in the value of λ and an increase in the value of n and δ (cf.Tables 1-2 and Figures 2-3); vi) the proposed NPDEWMA chart performs better than the competing counterparts NPS E , NPAS E and NPS C charts in terms of run length efficiency (cf.Tables 1 and 4 and Figures 4-5); vii) the proposed NPDEWMA chart outperforms the other competing charts in terms of its overall performance measure EQL (cf.Table 5).
viii) The order and superiority (in terms of ARL and EQL measures) of the different charts being researched in the study is: NPDEWMA, NPAS E /NPS E and NPS C .As the run-length distribution is skewed to the right, many researchers recommend examining a different number of percentiles including the 5 th , 25 th , median, 75 th and 95 th percentiles in order to better characterize the run-length distribution.Therefore, we have provided some additional measures for the proposed chart (cf.Table 2).In addition, we have provided some useful properties of the proposed scheme for smaller values of λ, including λ = 0.01, 0.025 and 0.10 (for n = 5) to highlight the effectiveness of the proposal for smaller shifts.These results can be seen in Table 6; they support the performance of the proposal for smaller and smaller shift values.
Moreover, motivated by Gan (1994), we have also set the design parameters by fixing the MDRL value at a specific level, MDRL 0 for example and have evaluated the performance of the proposed scheme.The resulting performance measures (with the desired nominal MDRL is equal to 350 i.e.MDRL 0 = 350, in the form of ARL, MDRL and SDRL) are provided in Table 7.This means that there is at least a 50% chance that the first out-of-control signal will be observed by the 350 th sample despite the process actually being in-control.The results of Table 7 offer attractive properties for the proposed chart by fixing the in-control run-lengths to be greater than or equal to 350.These results are given for λ =0.01, 0.025 and 0.10 for n = 5.Similar results may be obtained for other choices of n,λ and MDRL 0 values.

Illustrative Example
In this section, we provide an application of the proposed NPDEWMA chart for a real data set to illustrate how it is implemented.The data set is based on an important quality characteristic in a production process, namely the fill volume of soft-drink beverage bottles (cf.Montgomery 2009, Yang et al. 2011and Yang & Cheng 2011).A coded scale is used to measure the volume by placing a gauge over the crown and comparing the height of the liquid in the neck of the bottle.The correct fill height is indicated by a reading of zero on this scale.The data consists of fifteen samples of size n = 10 that are measured on the fill volume.Figure 6 represents the data in graph form.For the aforementioned data set, the charting statistic (D t ) values of the proposed NPDEWMA chart are computed using λ= 0.25.The control limits are set such that ARL 0 = 370 for which the control limits multiplier for the proposed NPDEWMA chart L = 0.284 is used.The charting statistics are plotted against these control limits.The same is done for the dataset under discussion, in the three other competing control charts NPS E , NPAS E and NPS C of this study.The resulting control chart displays are shown in Figure 7 for all the four charts.We observed that the proposed NPDEWMA chart gives six out-of-control signals for the location parameter.The other charts show the out-of-control signals as: NPS E three, NPAS E one and NPS C none.This detection ability in the example data set is in agreement with the dominance order that the previous section establishes.

Concluding Remarks
This study has proposed a nonparametric double EWMA control chart for efficiently monitor to the location parameter that has an unknown distributional behavior.The design structure of the proposed NPDEWMA chart depends on the sample size n and the smoothing constant λ.The controlling coefficients L are worked out in irder to fix ARL 0 at the prespecified level.Also, the control limits are developed for the proposed chart to improve the monitoring of process location when the parametric distribution of the quality characteristic of interest is not known.The performance of the proposed chart is evaluated in terms of different run length properties, including ARL, SDRL and some useful percentile points.The EQL measure is also computed as an overall performance measure of the proposed chart.
Comparisons of the proposal were also carried out with some existing counterparts, namely the NPS E , NPAS E and NPS C charts that were proposed in the literature.The comparisons revealed the superiority of the proposed NPDEWMA chart over the competing counterparts in terms of improved run length properties.For example, when p shifts to 0.55 at n = 5, λ = 0.05 and ARL 0 = 370, the ARL 1 performance of the proposed NPDEWMA chart is 67.10.This is followed by NPS E with ARL 1 = 86.33,then NPAS E with ARL 1 = 90.98,and finally NPS C with ARL 1 = 94.90.For this particular change in p a gain of almost 22%, 26% and 29% for the proposed NPDEWMA chart in comparison to NPS E , NPAS E and NPS C charts, respectively.The similar gains may be seen at other values of p for different combinations of n and λ.
The quality control practitioners may benefit from the proposal for better detection of out-of-control signals during in the monitoring of process location parameter.The scope of the study may be extended to double CUSUM charts.Moreover, multivariate extensions of these charts to monitor the vector of location parameters for more than one quality characteristics of interest is another potential topic for future research.

Figure 1 :
Figure 1: ARL0 plots versus L for the proposed NPDEWMA chart.

Figure 2 :Figure 3 :
Figure 2: ARL comparison of the NPDEWMA chart for different values of n at ARL0 = 370.

Figure 6 :
Figure 6: Graphical display for the example's Fill Volume data.

Figure 7 :
Figure 7: Control chart plots of the NPDEWMA, NPSE, NPASE and NPSC charts for the example dataset.

Table 1 :
Run length characteristics of the proposed NPDEWMA control chart when ARL0 = 370.

Table 2 :
Percentile points of the run length distribution of the proposed NPDEWMA control chart when ARL0 = 370.

Table 3 :
Design structure of different control charts.

Table 4 :
ARL of N P SE, N P ASE and N P SC charts when ARL0 = 370.

Table 6 :
ARL, MDRL and SDRL of the DNPEWMA chart at low levels of Lambda when ARL0 = 370 and n = 5.

Table 7 :
ARL, MDRL and SDRL of the DNPEWMA chart at low levels of when MDRL0 = 350 and n = 5.