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.     

One-Class Classification-Based Control Charts for Monitoring Autocorrelated Multivariate Processes

In recent years, statistical process control (SPC) of multivariate and autocorrelated processes has received a great deal of attention. Modern manufacturing/service systems with more advanced technology and higher production rates can generate complex processes in which consecutive observations are dependent and each variable is correlated. These processes obviously violate the assumption of the independence of each observation that underlies traditional SPC and thus deteriorate the performance of its traditional tools. The popular way to address this issue is to monitor the residuals—the difference between the actual value and the fitted value—with the traditional SPC approach. However, this residuals-based approach requires two steps: (1) finding the residuals; and (2) monitoring the process. Also, an accurate prediction model is necessary to obtain the uncorrelated residuals. Furthermore, these residuals are not the original values of the observations and consequently may have lost some useful information about the targeted process. The main purpose of this article is to examine the feasibility of using one-class classification-based control charts to handle multivariate and autocorrelated processes. The article uses simulated data to present an analysis and comparison of one-class classification-based control charts and the traditional Hotelling's T ² chart.     

CUSUM Control Charts for Multivariate Poisson Distribution

A cumulative sum control chart for multivariate Poisson distribution (MP-CUSUM) is proposed. The MP-CUSUM chart is constructed based on log-likelihood ratios with in-control parameters, Θ₀, and shifts to be detected quickly, Θ₁. The average run length (ARL) values are obtained using a Markov Chain-based method. Numerical experiments show that the MP-CUSUM chart is effective in detecting parameter shifts in terms of ARL. The MP-CUSUM chart with smaller Θ₁ is more sensitive than that with greater Θ₁ to smaller shifts, but more insensitive to greater shifts. A comparison shows that the proposed MP-CUSUM chart outperforms an existing MP chart.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.     

Multivariate CUSUM and EWMA Control Charts for Skewed Populations Using Weighted Standard Deviations

This article proposes a heuristic method of constructing multivariate cumulative sum and exponentially weighted moving average control charts for skewed populations based on the weighted standard deviation method which adjusts the variance–covariance matrix of quality characteristics and approximates the probability density function using several multivariate normal distributions. These control charts, however, reduce to the conventional control charts when the underlying distribution is symmetric. In-control and out-of-control average run lengths of the proposed control charts are compared with those of the conventional control charts for multivariate lognormal and Weibull distributions. Simulation results show that considerable improvements over the standard method can be achieved when the underlying distribution is skewed.     

Tolerance Limits Based on the Multivariate MaxMin Chart

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises y_n such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences x_n which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.     

The Design and Performance of the Multivariate Synthetic-T 2 Control Chart

One of the objectives of research in statistical process control is to obtain control charts that show few false alarms but, at the same time, are able to detect quickly the shifts in the distribution of the quality variables employed to monitor a productive process. In this article, the synthetic-T ² control chart is developed, which consists of the simultaneous use of a CRL chart and a Hotelling's T ² control chart. The ARL is calculated employing Markov chains for steady and zero-state scenarios. A procedure of optimization has been developed to obtain the optimum parameters of the synthetic-T ², for zero and steady cases, given the values of in-control ARL and magnitude of shift which needs to be detected rapidly. A comparison between (standard T ², MEWMA, T ² with variable sample size, and T ² with double sampling) charts reveals that the synthetic-T ² chart always performs better than the standard T ² chart. The comparison with the remaining charts demonstrate in which cases the performance of this new chart makes it interesting to employ in real applications.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.     

Diagnosis of Multivariate Control Chart Signal Based on Dummy Variable Regression Technique

Abstract

It is common to monitor several correlated quality characteristics using the Hotelling's T ² statistic. However, T ² confounds the location shift with scale shift and consequently it is often difficult to determine the factors responsible for out of control signal in terms of the process mean vector and/or process covariance matrix. In this paper, we propose a diagnostic procedure called ‘D-technique’ to detect the nature of shift. For this purpose, two sets of regression equations, each consisting of regression of a variable on the remaining variables, are used to characterize the ‘structure’ of the ‘in control’ process and that of ‘current’ process. To determine the sources responsible for an out of control state, it is shown that it is enough to compare these two structures using the dummy variable multiple regression equation. The proposed method is operationally simpler and computationally advantageous over existing diagnostic tools. The technique is illustrated with various examples.     

空气质量指数监控与预测的控制图方法

世界上多数国家都采用空气质量指数这一指标衡量空气质量状况,对空气质量的有效监测和预警是解决空气污染的重要参考依据.本研究使用ARMA模型拟合空气污染指数(API)时序数据,通过模型残差建立控制图,根据控制图的变化监控并预警.研究采用2010年上海世博会API作为可控状态建立控制限,以2011年1～8月上海API数据建立ARMA(1,1)模型,通过2011年9月上海API模型预测与残差控制图证实模型和控制图的有效性.     

Application of the Generalized Likelihood Ratio Test for Detecting Changes in the Mean of Multivariate GARCH Processes

We derive several multivariate control charts to monitor the mean vector of multi-variate GARCH processes under the presence of changes, by means of maximizing the generalized likelihood ratio. This presentation is rounded up by a comparative performance study based on extensive Monte Carlo simulations. An empirical illustration shows how the obtained results can be applied to real data.     

Multivariate Quality Control Chart for Autocorrelated Processes

Traditional multivariate statistical process control (SPC) techniques are based on the assumption that the successive observation vectors are independent. In recent years, due to automation of measurement and data collection systems, a process can be sampled at higher rates, which ultimately leads to autocorrelation. Consequently, when the autocorrelation is present in the data, it can have a serious impact on the performance of classical control charts. This paper considers the problem of monitoring the mean vector of a process in which observations can be modelled as a first-order vector autoregressive VAR (1) process. We propose a control chart called Z-chart which is based on the single step finite intersection test (Timm, 1996). An important feature of the proposed method is that it not only detects an out of control status but also helps in identifying variable(s) responsible for the out of control situation. The proposed method is illustrated with the help of suitable illustrations.     

Progressive Variance Control Charts for Monitoring Process Dispersion

In a process, the deviation from location or scale parameters affects the quality of the process and waste resources. So it is essential to monitor such processes for possible changes due to any assignable causes. Control charts are the most famous tool used to meet this intention. It is useless to monitor process location until the assurance that process dispersion is in-control. This study proposes some new two-sided memory control charts named as progressive variance (PV) control charts which are based on sample variance to monitor changes in process dispersion assuming normality of quality characteristic to be monitored. Simulation studies are made, and an example is discussed to evaluate the performance of the proposed charts. The comparison of the proposed chart is made with exponentially weighted moving average- and cumulative sum-type charts for process dispersion. The study shows that performance of the proposed charts are uniformly better than its competitors for detecting positive shifts while for detecting negative shift in the variance their performance is better for small shifts and reasonably good for moderated shifts.     

Economic-Statistical Design of EWMA Control Charts Based on Taguchi's Loss Function

This paper proposes an economic-statistical design of the EWMA chart with time-varying control limits in which the Taguchi's quadratic loss function is incorporated into the economic-statistical design based on Lorenzen and Vance's economical model. A nonlinear programming with statistical performance constraints is developed and solved to minimize the expected total quality cost per unit time. This model, which is divided into three parts, depends on whether production continues during the period when the assignable cause is being searched for and/or repaired. Through a computational procedure, the optimal decision variables, including the sample size, the sampling interval, the control limit width, and the smoothing constant, can be solved for by each model. It is showed that the optimal economic-statistical design solution can be found from the set of optimal solutions obtained from the statistical design, and both the optimal sample size and sampling interval always decrease as the magnitude of shift increases.     

Multivariate Exponentially Weighted Moving Average Charts for a Mean Based on Its Prediction Distribution

This article develops a control chart for a mean vector when it is monitored by a quadratic form in the exponentially weighted observation vector. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model.     

A Distribution-free Multivariate Change-point Model for Statistical Process Control

This article develops a new distribution-free multivariate procedure for statistical process control based on minimal spanning tree (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on MST and change-point model. Simulation results show that our proposed procedure is quite robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially moderate to large shifts, which is one of the main drawbacks of most distribution-free procedures in the literature. The proposed procedure is particularly useful in start-up situations. Comparison results and a real data example show that our proposed procedure has great potential for application.     

Using Control Charts to Corroborate Bribery in Jai Alai

Statistical process control charts were used in the State of Florida District Court to help establish the guilt of an individual who was alleged to have affected the outcome of jai alai contests by bribing some of the contestants to lose. By placing wagers on the nonbribed contestants the briber gains an increased chance of winning, which is to the detriment of the other bettors. This paper gives an example of how statistical process control techniques can be employed to detect the unusually high bets that generally accompany bribery of the contestants. If the management of the jai alai gaming facility had been using control charts on a regular basis, the game fixing might have been detected much sooner.     

Attribute Control Chart for Multivariate Poisson Distribution

In this study, a control chart is constructed to monitor multivariate Poisson count data, called the MP chart. The control limits of the MP chart are developed by an exact probability method based on the sum of defects or non conformities for each quality characteristic. Numerical examples are used to illustrate the MP chart. The MP chart is evaluated by the average run length (ARL) in simulation. The result indicates that the MP chart is more appropriate than the Shewhart-type control chart when the correlation between variables exists.