The Statistical Process Control charts (SPCC) was laid by Dr. Walter Shewhart in 1920. Many statisticians defined the SPCC, typical of those, Montgomery (2005) defined SPCC is the powerful tool to optimize the amount of information needed for use in making decisions in different aspects of the life for instance management, industrial, education,â€¦,etc. Likewise, Nedumaran and Pignatiello (2000) defined SPC control charts in other way such as," SPCC are used to monitor the state of a process by distinguishing between random causes and assignable causes of variability".
For this reason, SPC uses statistical tools to observe the accomplishment of the production process. Rather, Nedumaran and Pignatiello (2000) said that the Shewhart control chart is one of the highly developed tools from the univariate control charts. Thus, the univariate control charts are used to monitor the process of manufactures products with a single quality characteristic. In practice the overall quality of a product is usually determined by more than one quality characteristic. On such, the quality of a certain type of tablets may be determined by weight, degree of hardness, thickness, width and length (Liu, 1995). In conclusion, several characteristics of a manufactured component need to be monitored simultaneously; therefore, the multivariate Shewhart-type 2 or the Hotelling's control charts are the most suitable charts to use.
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On the contrary, the goal of SPC charts are to detect the occurrence of assignable causes of method shifts, therefore, the investigation of the method and corrective action may be undertaken before many nonconforming units are manufactured, and the control chart as monitoring technique widely used for this purpose. Furthermore, Pignatiello and Simpson (2000) said that the most important aspect to indicate that the control chart is working well when it detects or gives quickly react when there are any change in the process.
Multivariate SPC gives us a set of enhanced techniques for monitoring and controlling of the working performance. On the other hand, these techniques reduce the information contained within the batch and continuous processes the variables into two or three composite metrics by the application of the statistical modeling. This implies that, these composite metrics can then be monitored easily in order to highlight potential problems and target process performance. As result, these things will give a framework for continuous enhancement of the process operation (Bersimis, Pnaretos and Psarakis, 2006).
Moreover, Multivariate statistical process control charts have been increased as the data collection methods, and have become more sophisticated. Typically, Oktay and Aricigil (2001) indicated to three popular types of multivariate control charts such as Hotelling's control charts, MEWMA (Multivariate exponentially weighted moving average) and MCUSUM (Multivariate Cumulative Sum).
Alternatively, control charts are used as the initially testing whether the process is in control and that happened through two distinct phases of control chart Phase I and II. Phase I is used in case of the first observations were being drawn. In other words, the control charts in this phase are used to the practitioners in bringing the process into a state of statistically in control, which makes the practitioners to study the process very well. Whereas, in phase II, the control charts are used to determine whether the future observations make the process remain in control or out of control. On other words, the control charts are used here to help the practitioners in monitoring the process for any change of an in control state. This means that the practitioners are monitoring the process with respect to the parameters of the process and where known or estimated. In conclusion, the Hotelling's control charts take into a consideration the correlation between the variables or the characteristics in building the control charts. For this reason, these charts enable us to detect the outliers, shift in the mean vector and other deviations from control distributions. As a result, an important aspect of the Hotelling's control charts is determining the variance covariance matrix that is used in the calculation of Hotelling's statistic of the chart (Woodall & Ncube, 1985; Williams, Woodall, Birch & Sullivan, 2006).
The Hotelling's statistic is one of the most common methods in the multivariate statistical control charts (Alt 1985; Montgomery, 2005). Furthermore, the Hotelling's statistic is the multivariate generalization of the Student's t- statistic. In fact, the Hotelling's statistic is the extended case after taking the square for the two sides of the equation of the Student's t- statistic, that is to say Hotelling statistic is equal to
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[1.1] where the sample standard covariance matrix refers to as a scatter matrix, and gives us information on the relationships between the observations vectors of variables, such as, the diagonal elements of represent the values of the variances of each variable, and the other elements in matrix represent the covariance between pairs of variables. On other words, the Hotelling's statistic value of an observation vector is measuring how far the observation from the process center relative (Mason & Young, 2002, pp20-21). Also, the importance of this control charts are coming from the numerous discussions about it for the most popular statisticians in the world (Young and Bai, 2004).
The multivariate expansion to the univariate Shewhart chart is the Hotelling's control chart. And there are two types of the Hotelling's control charts one for the individual observations and the other one for the subgroups data (Cheng, Away and Hasan, 2006). Furthermore, Hotelling's control charts are very versatile multivariate control chart, it is used to detect the multiple outliers, mean shift and any deviation from the in control distribution (Williams, Woodall, Birich and Sullivan, 2006). The multiple quality characteristics are jointly distributed as variate normal whereas if the parametersand are known then the Hotellingcontrol charts are suitable for the proceeding control charts. As a result, the statistical shift in the mean occurs if the value of
is greater than the particular upper control limits (UCL) for where is the number of the subgroups, n is the sample size of each subgroup, is the vector of known means, is the known variance covariance matrix and is the vector of samples means for subgroup. In the same way, ifand are unknown, then the Hotelling's control chart is suitable for proceeding control chart. The Hotelling's statistic equation is
For where is the mean vector overall sample means of subgroups, and is the mean of the sample variance covariance matrix overall subgroups. In conclusion,And2 statistics represent the distance of any observation from the process mean under in control condition. Also, the 2 statistic follows the chi-square distribution with degrees of freedom. Whereas statistic follows
When subgroups are independent and the joint probability distribution of variables is following the multivariate normal distribution.
Alternatively, if each one of the samples has size n=1 then the subgroups multivariate observations change into the individual multivariate observations and its statistic will be as follows:
Whereand are the sample mean vector and the sample variance covariance matrix respectively.
According to Mason and Young (2002) the suitable upper control limits UCL for the individual observations of random variables in case of unknown parameters is that to have the distinguishing between if the random variables of vector X are independent or dependent. Such as, the independent variables mean the observations are not including in calculating of the estimated of the parameters, but the dependent of variables mean the observations of the random variables are including in calculating the estimated of the unknown parameters. As a result of that, there are three cases for the distribution of Hotelling's statistic and these are as follows:
If the parameters Âµ and of the underlying multivariate normal distribution are known then the Hotelling's statistics and its distribution for the individual observation vector X as follows:
If the parameters Âµ and are unknown, then the alternatives estimated sample mean vector and variance covariance matrix are then used, and the distribution of the statistics in this case as follows:
If the vector are independent, then the Hotelling's statistic and its distribution as the following
Where is an F distribution with and () degrees of freedoms.
Another possibility, if the vector is dependent then the formula of the Hotelling's statistic and its distribution as follows:
Where represents the beta distribution with parameters and .
Ultimately, the process for any products will pass during two phases I and II. Such as phase I for calculating of the estimators of the sample mean vectors and the sample variance covariance matrix or the robust location and scale estimators. Whereas, phase II is used for the new observation which is generated from in control parameters when the empirical type I error rates (false alarms) are calculated. Likewise, a new observation is generated from out of control distribution when the probability of detection of outliers will be calculated. Thus, the calculation of Hotelling's control charts for the new observations for the two measurements the empirical type I error rates (false alarms) or for the probability of detection outliers are based on the traditional and robust estimators that are calculated in phase I. Consequently, by using the values of Hotelling's statistics for the new additional observation the percentages of type I error or probability detection of outliers are calculated by counting the number of that values of Hotelling's statistics that are greater than the UCL or in other words the Hotelling's statistics that are rejected and divided its number by the whole numbers of generated new observations.
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Despite of offering, an easy way for monitoring the quality of a number of characteristics simultaneously by using Hotelling'scontrol chart. Although there are some drawbacks for using these charts, such as these charts depend on the estimated parametric statistics, the mean vector and variance covariance matrix where these two estimators are sensitive for the outliers. However, these two sensitive estimators may be affected if the underlying assumptions are not satisfied. Likewise, the other important drawback is that when the traditional Hotelling'scontrol chart are used then the data must be come from the normal distribution and without outliers. However by the huge of collecting data and the sophisticated of the manufacturing process nowadays, this condition is easy to abuse. Therefore, to rectify this situation, another possible way can be used such as the using of robust statistics. This study introduced the development of modified Hotelling's control charts using robust estimators as alternatives to the parametric estimates in the traditional Hotelling's control charts.
Consequently, the Hotelling's control charts are efficient and suitable when the data are coming from normal distribution. However, when there are extreme values among the measurements of the characteristics, the traditional Hotelling's become inefficient in such cases. So, this study proposed nine modified control charts for the traditional Hotellingcontrol charts by using some types of robust locations such as the trimmed mean, winsorized modified one step M-estimator () and the Hodges-Lehmann, whereas the types of the scale estimators are the Median absolute deviation and . Chernobai and Rachev (2005) also mentioned many possible methods in cleaning the outliers from the data, the most important methods are, the truncation, winsorizing, trimming and screening. Likewise this study has been chosen the robust location estimators according to trimming and without trimming. Typical of this, according to the trimming, there are two types of the trimming, symmetry and asymmetry, for instance Î±-trimmed mean is used as a symmetry trimming, whereas the winsorized modified one step M-estimator is used as asymmetry trimming. While as, the Hodges-Lehmann is used as an example about the robust location estimators that does not depend on the trimming of outliers.
The location estimators which will be used in this study are the winsorized using the criterion of the modified one step M-estimator () to proceed the trimming, -Trimmed is proceeded by using modified Mahalonobis distance to trim the outliers and ultimately the Hodges-Lehmann estimator is used as a method for the robust location estimator without trimming. All these previous robust location estimators are replaced instead of the sample mean vector. While as, the scale estimators Median absolute deviation ,, and the trimmed variance covariance matrix are will be replaced instead of the sample variance covariance matrix as the following way: ,, are used in two forms first one is as a modified on the Modified one step M-estimator and the Mahalonobis distance in case of using the two scale estimators the winsorized MOM and variance covariance matrices respectively, where as the second form of using the robust scale estimators ,, as alternatives of the classical variance covariance matrix in the traditional Hotelling's control charts when the modified robust charts using the location estimator Hodges-Lehmann as alternative of arithmetic mean vector in the traditional Hotelling's control charts.
To investigate from the performances of these new nine modified robust Hotelling control charts, the outliers are put in the data, where the original data are generated from the standard normal distribution, then the empirical type I error rates (false alarms) and the probability of detection of outliers are calculated to assess the robustness and the performances of these alternatives robust Hotellingcharts.
For all that, Besterfield (2004) confirmed and defined the quality as the excellence of achievements in goods, services or other things which exceed the expectations. Likewise, Oakland (2003) defined the quality "as meeting the requirements of the customer". Consequently, the competition is in presentation of kind of the productions. The scientists take advantage of the growth of technology and online computer to collect more data. Equally, they study these kinds of products from several quality characteristics simultaneously. As result of that, there are a huge of data were collected. And so, these products need to study from several quality characteristics. And we shall bear in mind, that the multivariate robust Hotelling T2 control charts are the most suitable to study the data. As a result, the using of the alternatives robust charts will present high quality of the products.
The robustness theories can be viewed as stability theories of statistical inference. Likewise, it can be defined in other words such as, it means what is desired of an inference procedure which in some logic does well possible if the assumption is true, but does not perform well within a range of alternatives to the assumptions. Frequently in discussions of robustness, the assumed distribution is normal; therefore, the type of robustness of interest is robustness to non-normality.
Also Crosier (1988) indicated that the robustness against multivariate outliers is an even more essential aspect in a multivariate quality control plan. Furthermore, he described multivariate outliers as being observations that have separated the values which cause unreliable Hotelling'sstatistic. As a result the outliers or data errors occur more likely in the multivariate case.
For all that, Hampel, Ronchetti, Rousseeuw, and Stahel, (1986) have concluded the reasons of using the robust Hotelling's control charts as follow:
Robust methods produce better results than the traditional methods.
Robust methods are not influenced as much by slight deviations in the underlying distribution nor to small number of outliers.
3) The control limits based on robust methods that trim outliers allow the user to distinguish control problems earlier since they have smaller limits.
4) Lesser time has taken to distinguish the assignable causes.
In order to tackle the effect of outliers and trying to explore better methods to deal with this problem, this study proposed nine alternatives modified robust Hotelling's charts, where these methods are known to be insensitive to the violations of assumptions. Ultimately, according to the Bradley's (1978) inequality liberal criterion the robustness these new alternatives robust charts can be measured, such as if the empirical rate of Type I error (false alarms) within this inequality the charts are regarded robust.
1.1 Problem Statement
Various multivariate SPC charts based on multivariate methods have been proposed. The advantage of multivariate control charts is that they can take into account the nature of the control scheme and the correlation structure among the characteristics simultaneously. However, most of the proposed multivariate control charts are based on Hotelling'sstatistic, which assumes that quality characteristics are jointly distributed as a -variate normal (Nandini, 2009). Moreover, Zou (2005) indicated to the importance of the sample mean and sample variance, where he likened them as the cornerstone of a multivariate analysis. Most importantly, they are the best in the case that the data is normal, and be bad in case of data unnatural and extreme. Furthermore, he confirmed about the important idea that is these estimators are very sensitive for the outliers' data.
According to Tiku and Singh (1982) when the data are departure from the normality, the traditional Hotelling's statistic's resistance for the non normality is very weak and ineffective. Likewise, Johnson (1987) confirmed on the same subject about the robustness of the traditional Hotelling's statistic does not exist against the nonormal data. Moreover, Everitt (1979) concluded that the non normality of the distribution severely affected on the traditional Hotelling's statistic. Also, Brooks (1985) noted that the possibility for the data errors increase with sophisticated manufacturing systems because of the big huge mass of collected data. Besides of, the assumptions of normality are difficult to satisfy in the multivariate case rather than in the univariate case, for all that the robustness is very necessary in case of the multivariate quality control method (Crosier, 1988). Likewise, Peña and Due (2001) said the existences of the data errors make the values of the joint traditional estimators, location and scale estimators meaningless values. In other words, the existence of the outliers in the univariate data is easy to detect, because the outliers data in simply way are the largest and the lowest values that are furthest from the other homogenous data. Furthermore, the values of the sample statistics like the mean and the variance indicate to the existence of the outliers if there are any inflation of their values, however, in case of multivariate data the distortion of the sample variance covariance matrix can hide the effectiveness of the outliers, especially when there are multiple outliers in the different direction (Filler, 2005). According to Das (2009) the performances of the Hotelling'scontrol charts are deteriorated when in reality the multivariate normality is "impossible" to achieve. Therefore, the data errors can frequently occur and are even more likely in the multivariate case.
Likewise, Alfaro and Ortega (2009) found in the process and especially, in phase I, to construct the traditional Hotelling's control charts, the historical data sets need to analyze for many times to eliminate the outliers. As result, if the occurrence of the multiple outliers or the masking effect in the data, the traditional Hotelling's control charts cannot detect like these types of multiple outliers or the masking effect. For all that, the multivariate control charts based on some types of robust statistics will be very useful and necessary when implement the statistical process control charts.
1.2 Research Objectives
The goal of this study is to obtain better alternatives for traditional Hotelling's charts, whereas, these alternatives are effective in detecting the occurrence of assignable causes of process shifts simultaneously by controlling on the probability of Type I error, (false alarms) and detect the occurrence of outliers under non-normal distributions. Consequently, to achieve this goal, the following objectives need to be accomplished:
To modify the traditional Hotelling's by integrated some of highly efficient robust location and scale estimators to these methods.
To assess the performance of the modified Hotelling's control charts by using two methods, first one the probability of Type I error (percentage of the false alarms) and the second method is the probability of detection outliers.
To compare the modified Hotelling's control charts with the traditional Hotelling control chart in terms of Type I error and the probability of detection outliers.
To make comparisons amongst the modified Hotelling's control charts to recommend which one the best of these alternatives robust charts.
1.3 Significance of the Study
This study will contribute towards knowledge development in statistical process control charts (SPCC), especially in the manufacturing sectors. Statisticians are aware that the use of Hotelling's statistics depends on the assumption of normality. However, in the real world, data are not always normally distributed. Through the previous studies of all the robust univariate alternative control charts achieved better than the traditional in the presence of the contaminated data or multiple outliers. From above reasons the using of robust estimators may give us better conclusions. The significance of this study is that with these proposed methods, the researchers (in various fields of quality control) will not be concentrated or given any attention to the assumption of normality. They can instead of that work with the original data without having to worry about the shape of the distributions or the existence of the outlier's data. Moreover, no need to analyze the historical data in phase I to eliminate the outliers and then use the traditional Hotelling's control charts.
1.4 Organization of the Thesis
I will delay this section to the last of the thesis.