Principal component analysis-based control charts for multivariate nonnormal distributions

Robust adaptive multivariate Hotelling's T2 control chart based on kernel density estimation for intrusion detection system

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.

Traditional multivariate control charts such as Hotelling’s χ2 and T2 control charts are designed to monitor vectors of variable quality characteristics. However, in certain situations, data are expressed in linguistic terms and, under these circumstances, variable or attribute multivariate control charts are not suitable choices for monitoring purposes. Fuzzy multivariate control charts such as fuzzy Hotelling’s T2 could be considered as efficient tools to overcome the problems of linguistic observations. The purpose of this paper is to develop a fuzzy multivariate exponentially weighted moving average (F-MEWMA) control chart. In this paper, multivariate statistical quality control and fuzzy set theory are combined to develop the proposed method. Fuzzy sets and fuzzy logic are powerful mathematical tools for modeling uncertain systems in industry, nature, and humanity. Through a numerical example, the performance of the proposed control chart was compared to the fuzzy Hotelling’s T2 control chart. Results indicate uniformly superior performance of the F-MEWMA control chart over Hotelling’s T2 control chart.

In recent decades, multivariate statistical methods for monitoring and controlling of industrial processes have received increasing attention. Multivariate process control charts such as Hotelling's T2 and multivariate exponentially weighted moving average have gained acceptance as manufacturing process control tools especially for the semiconductor industry. We propose in this paper a novel multivariate nonparametric control chart based on the permutation and nonparametric combination (NPC) methodology. The proposed NPC chart is always an exact inferential procedure for every finite sample size and overcomes some limitations of traditional approaches such as the ability to include a large number of variables. Moreover, NPC chart offers several advantages because it is a robust solution with respect to the true underlying random error distribution, and it is not affected by the problem of the loss of degrees of freedom when keeping fixed the number of observations. Unlike traditional methods, when the number of informative variables increases its power monotonically increases as well (finite sample consistency property of NPC test). As confirmed by an extensive simulation study and by an application to some real case studies in the field of microelectronics, the NPC charts are certainly good alternatives with respect to the traditional multivariate control charts such as Hotelling's T2 especially in cases of asymmetric or heavy‐tailed error distributions. Finally, NPC chart represents a suitable solution for monitoring a multivariate process when the number of observed process variables may be larger than the number of observations. Copyright © 2013 John Wiley & Sons, Ltd.

Gower distance-based multivariate control charts for a mixture of continuous and categorical variables

A new multivariate statistical process variability control scheme, the multivariate double sampling (MDS) |S| control chart scheme for controlling shifts in a variance–covariance matrix, is proposed. An MDS |S| chart is a double-stage extension of a single-stage control chart based on the sample generalized variance. The statistical efficiency in terms of average run length of the MDS |S| charts is compared with that of traditional and adaptive sample size control charts based on the sample generalized variance for a bivariate case. The results of the comparison show that the MDS |S| charts provide a significant improvement in efficiency for detecting shifts of any range over these schemes. The performance of the MDS |S| charts in terms of detecting power is also compared with that of the modified likelihood ratio test, the sum of standardized variances for principal components charts and a decomposition procedure. The results of the comparison show that for certain out-of-control situations, the performance of the MDS |S| chart is among those of the best schemes.

Analyzing out-of-control signals of T2 control chart for compositional data using artificial neural networks

Control charts are widely used to identify changes in a production process. Nonparametric or distribution-free charts can be useful when there is a lack of underlying process distribution. A nonparametric exponentially weighted moving average (EWMA) control chart based on sign test using ranked set sampling (RSS) is proposed to monitor the possible small shifts in the process mean. The performance of the proposed chart is evaluated in terms of average run length, median run length, and standard deviation of the run length distribution. It has been observed that the proposed version of the EWMA sign chart, using RSS shows better detection ability than that version of the EWMA sign chart and the parametric EWMA control chart using simple random sampling scheme. An application with real data-set is also provided to explain the proposal for practical considerations.

Data mining model-based control charts for multivariate and autocorrelated processes

Statistical Process Control (SPC) charts are frequently used in the semiconductor manufacturing environment to monitor process quality and detect special-cause variations, hence, to take corrective actions when necessary. The important aspects of control charts to consider on production floors are identifying the primary objective of implementing control charts, the type of data to monitor and the most appropriate control limits to establish. When the quality data is a type of attribute data like the proportion of defectives from a production lot, a p-chart approach is most suitable. In p-chart applications, although the assumption of normally distributed process data is not mandatory, the widespread practice is to assume the normal distribution of process data when establishing the control limits. Yet again, the reality of industrial settings is that process data are most likely influenced by outliers, resulting in highly skewed distributions. This paper addresses these issues by proposing robust SPC charting techniques to detect special-cause variations in the semiconductor manufacturing processes. Here, we present a case study of a semiconductor company in Malaysia, Dominant Opto Technologies Sdn. Bhd. to propose three robust statistical approaches for monitoring the proportion of defectives in production lots. We apply M-estimates, median, and interquartile range to calculate the upper control limits (UCL) and found that robust estimators are more effective in detecting early process deterioration and capturing the out-of-control (OOC) conditions better than traditional control charts. By proposing robust methods, this study enlightens the practical aspects of process quality improvement for real-life manufacturing setups. Because a high OOC rate may impact manufacturing productivity, we recommend the decision-makers choose the types of control charts based on the implications of each robust approach toward quality and productivity. The significance of this study includes providing insights into setting up the appropriate attribute control charts for detecting defective proportions for professionals and SPC researchers working in these areas.

In recent years, many supplementary designs have been developed incorporating the assumption that data follow the particular non-normal distribution. The V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -control chart is one such design proposed to monitor the parameter σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of the single parameter Rayleigh distributed data. Commonly, in authentic situations, practitioners need to estimate the scale parameter σ in the observed processes instead of σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . However, the positive square root of the V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -statistic used in the existing design of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -control chart is not an unbiased estimator of σ and thus could not be practiced to monitor the scale parameter σ of the Rayleigh distributed process. A new structure of the V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</sub> -control chart namely V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SQR</sub> for monitoring the scale parameter of the Rayleigh distributed data has been originally developed in this study. The statistical basis of this newly V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SQR</sub> design in terms of average run length (ARL), characteristic function and power curve have been derived. The analytical results are utilized further to determine the parameters of V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SQR</sub> -chart and in comparing the performance of the proposed control chart with existing competitors. Comparative results illustrate the effectiveness of the proposed design in view of statistical power. Finally, the computational procedure of this newly V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SQR</sub> -chart has been demonstrated using simulated data and real data on the breaking strength of carbon fibers.

Multivariate Control Chart is an effective tool in Statistical Process Control to identify either an out-of-control process or in-control process. Hotelling T2 control chart is a quite popular and widely used technique in this field. However, its performance is deteriorated when the underlying distribution of the quality characteristics is not following the multivariate normal distribution. Multivariate control chart usually recommends a procedure in the phase-in Hotelling T2 control chart although there is difficulty in interpreting the signals from multivariate control charts more work is needed on data reduction methods and graphics techniques. Basically, the multivariate control chart refers to the theory of prediction interval. Therefore, it’s called predictive multivariate control charts. The purpose of this research is to construct what so-called predictive multivariate control chart both classical in this part in the phase-in Hotelling T2 and Copula-based ones. We argue that appropriate joint distribution function may be well estimated by employing Copula. A numerical analysis is carried out to illustrate that an Application Copula-based Multivariate control chart outperforms than bivariate control chart and others.

Different control charts are proposed by many researchers for the aim of detecting process shifts in the mean and assuring quality of the product. Exponentially weighted moving average (EWMA) control charts is one of the famous charts which performs well in detecting small shifts in the process mean. In this paper, a new statistic based on different smoothing parameters is proposed to improve the performance of the exponentially weighted moving average (EWMA) control chart in detecting large shifts as well as small and medium shifts. The performance of the developed scheme based on the proposed smoothing parameter is evaluated in terms of average run length criterion in comparison with the traditional EWMA control chart. The results show the better performance of the proposed scheme rather than the traditional EWMA control charts in most cases.

Control charts that are based on assumption(s) of a specific form for the underlying process distribution are referred to as parametric control charts. There are many applications where there is insufficient information to justify such assumption(s) and, consequently, control charting techniques with a minimal set of distributional assumption requirements are in high demand. To this end, nonparametric or distribution-free control charts have been proposed in recent years. The charts have stable in-control properties, are robust against outliers and can be surprisingly efficient in comparison with their parametric counterparts. Chakraborti and some of his colleagues provided review papers on nonparametric control charts in 2001, 2007 and 2011, respectively. These papers have been received with considerable interest and attention by the community. However, the literature on nonparametric statistical process/quality control/monitoring has grown exponentially and because of this rapid growth, an update is deemed necessary. In this article, we bring these reviews forward to 2017, discussing some of the latest developments in the area. Moreover, unlike the past reviews, which did not include the multivariate charts, here we review both univariate and multivariate nonparametric control charts. We end with some concluding remarks.

A multivariate T 2 chart is developed based on the Generalized Likelihood Ratio Test (GLRT) without a priori information about potential mean deviations. Identification of response variables from a T 2 chart is challenging and has received considerable attention recently. By highlighting the intrinsic relationship between various multivariate control charts and statistical hypothesis testing, this paper presents a theoretical framework for various individual multivariate control charts including the T 2 chart, regression-adjusted chart and M chart. The performance of these control charts is compared under different correlation structures among variables and different mean deviations. A hybrid control chart is also proposed based on the GLRT and union-intersection test, which can serve as a complementary diagnosis tool for the T 2 chart.