Identifying Variables Contributing to Outliers in Phase I

When a process is monitored with a T ² control chart in a Phase II setting, the MYT decomposition is a valuable diagnostic tool for interpreting signals in terms of the process variables. The decomposition splits a signaling T ² statistic into independent components that can be associated with either individual variables or groups of variables. Since these components are T ² statistics with known distributions, they can be used to determine which of the process variable(s) contribute to the signal. However, this procedure cannot be applied directly to Phase I since the distributions of the individual components are unknown. In this article, we develop the MYT decomposition procedure for a Phase I operation, when monitoring a random sample of individual observations and identifying outliers. We use a relationship between the T ² statistic in Phase I with the corresponding T ² statistic resulting when an observation is omitted from this sample to derive the distributions of these components and demonstrate the Phase I application of the MYT decomposition.     

Hotelling’s T2 control chart with double warning lines

Recent studies have shown that the T ² control chart with variable sampling intervals (VSI) and/or variable sample sizes (VSS) detects process shifts faster than the traditional T ² chart. This article extends these studies for processes that are monitored with VSI and VSS using double warning lines (T ² —DWL). It is assumed that the length of time the process remains in control has exponential distribution. The properties of T ² —DWL chart are obtained using Markov chains. The results show that the T ² —DWL chart is quicker than VSI and/or VSS charts in detecting almost all shifts in the process mean.     

Identification of out of control quality characteristics in a multivariate manufacturing environment

There are many instances in which the quality of a product or constancy of a process is determined by the joint levels of several attributes or properties. During the conduct of such a process or the production of such a product, one wishes to detect as quickly as possible any departure from a satisfactory state, while at the same time identifying which attributes are responsible for the deviation. In most cases of practical interest, however, there exist correlations among the several properties of interest; this makes it advisable to monitor certain aggregate characteristics of the process, rather than observing its various components separately. When the mean vector of the quality attributes is the major concern, this aggregate monitoring function is most commonly implemented via a T ² chart. The dependencies among attributes, however, complicate the determination of which are responsible when a deviation occurs. This paper presents an approach to help identify aberrant variables when Shewhart type multivariate control charts based on Hotelling's T ² are in use.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Average run length comparison of multivariate control charts

In this paper we use Monte Carlo Simulation methodology to compare the effectiveness of five multivariate quality control methods, namely Hotelling T ², Multivariate Shewhart Char, Discriminant Analysis, Decomposition Method, and Multivariate Ridge Residual Chart-developed by Authors-, for controlling the mean vector in a multivariate process. P-dimensional multivariate normal data generated using different covariance structures. Various amount of shift in the mean vector is induced and the resulting Average Run Length (ARL) is computed. The effectiveness of each method with regard to ARL is discussed.     

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT

We present an alternative sampling scheme for the Hotelling's T² control chart with variable parameters (VP T²) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

Integration of classification algorithms and control chart techniques for monitoring multivariate processes

We propose new multivariate control charts that can effectively deal with massive amounts of complex data through their integration with classification algorithms. We call the proposed control chart the ‘Probability of Class (PoC) chart’ because the values of PoC, obtained from classification algorithms, are used as monitoring statistics. The control limits of PoC charts are established and adjusted by the bootstrap method. Experimental results with simulated and real data showed that PoC charts outperform Hotelling's T ² control charts. Further, a simulation study revealed that a small proportion of out-of-control observations are sufficient for PoC charts to achieve the desired performance.     

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

ON THE ECONOMIC DESIGN OF MULTIVARIATE CONTROL CHARTS

ABSTRACT

It is an increasingly common practice to monitor several related quality characteristics of a product or process using a multivariate control chart procedure. Several types of multivariate control charts, including Hotelling's χ ² and T ² control charts, have been developed in attempts to improve monitoring by using the correlation structure that exists between quality characteristics. The purpose of this paper is to summarize the assumptions made regarding the out-of-control process shift in the economic design of multivariate control charts and to address their consequences. We study the average run length (ARL) properties of the χ ² control chart using a numerical example and show that this chart can perform ineffectively under the assumed out-of-control conditions when designed using the economic approach. Following Healy,[1] Healy, J.D. 1987. A Note on the Multivariate CUSUM Procedures. Technometrics, 29: 409–412. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] we offer an alternative procedure that has improved ARL properties and overall performance. These results can be important to researchers and practitioners who are interested in using the economic design of multivariate control procedures.     

Multivariate control chart based on the highest possibility region

The T ² control chart is widely adopted in multivariate statistical process control. However, when dealing with asymmetrical or multimodal distributions using the traditional T ² control chart, some points with relatively high occurrence possibility might be excluded, while some points with relatively low occurrence possibility might be accepted. Motived by the thought of the highest posterior density credible region, we develop a control chart based on the highest possibility region to solve this problem. It is shown that the proposed multivariate control chart will not only meet the false alarm requirement, but also ensure that all the in-control points are with relatively high occurrence possibility. The advantages and effectiveness of the proposed control chart are demonstrated by some numerical examples in the end.     

On constructing T 2 control charts for retrospective examination

In this paper we consider the issue of constructing retrospective T ² control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt's control limit and to the standard x ² control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt's control limit performs better that the Bonferroni-adjusted x ² control limit. Furthermore, it appears that the Bonferroni-adjusted Alt's control limit is more than adequate for controlling the overall false alarm probability at a specified value.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

A Multivariate Synthetic Control Chart for Monitoring Process Mean Vector

In this article, a multivariate synthetic control chart is developed for monitoring the mean vector of a normally distributed process. The proposed chart is a combination of the Hotelling's T ² chart and Conforming Run Length chart. The operation, design, and performance of the chart are described. Average run length comparisons between some other existing control charts and the synthetic T ² chart are presented. They indicate that the synthetic T ² chart outperforms Hotelling's T ² chart and T ² chart with supplementary runs rules.     

Adaptive Hotelling's T 2 Control Charts with Run Rules

This study investigates the statistical properties of the adaptive Hotelling's T ² charts with run rules in which the sample size and sampling interval are allowed to vary according on the current and past sampling points. The adaptive charts include variable sample size (VSS), variable sampling interval (VSI), and variable sample size and sampling interval (VSSI) charts. The adaptive Hotelling's T ² charts with run rules are compared with the fixed sampling rate Hotelling's T ² chart with run rules. The numerical results show that the VSS, VSI, and VSSI features improve the performance of the Hotelling's T ² chart with run rules.     

A Coefficient of Determination for Generalized Linear Models

The coefficient of determination, a.k.a. R², is well-defined in linear regression models, and measures the proportion of variation in the dependent variable explained by the predictors included in the model. To extend it for generalized linear models, we use the variance function to define the total variation of the dependent variable, as well as the remaining variation of the dependent variable after modeling the predictive effects of the independent variables. Unlike other definitions that demand complete specification of the likelihood function, our definition of R² only needs to know the mean and variance functions, so applicable to more general quasi-models. It is consistent with the classical measure of uncertainty using variance, and reduces to the classical definition of the coefficient of determination when linear regression models are considered.