Nonparametric Multivariate Control Chart Research Articles

An efficient adaptive r-MD multivariate single control chart for nonlinear multiple quality characteristics

An efficient multivariate single control chart to simultaneously monitor ‘location or mean’ and ‘scale or variabilities’, based on individual observations of multiple quality characteristics (MQC), is a continuous research endeavour. Compared to rational subgroups, time-ordered individual sample observations of MQC are common in manufacturing. However, few researchers proposed parametric and nonparametric multivariate control charts to monitor individual observations in this context. The proposed parametric approach has limited applicability due to the restrictive distribution assumption of MQC. In addition, most of the suggested parametric or nonparametric approaches recommend multiple charts to monitor process ‘location’ and ‘scale’ parameters simultaneously. Multiple charts can increase the control complexities and overall false-alarm-rate . MQC can also have significant nonlinear correlation structures and follow unknown or nonnormal distributions in real-life scenarios. Single parametric or nonparametric multivariate charts for individual observations can exhibit poor performance in such situations. In this study, an adaptive Mahalanobis depth and k-nearest neighbour (k-NN) rule-based r-AMD chart is proposed and verified to address the lacuna mentioned above. Monte-Carlo simulation studies are used to evaluate the performance of the r-AMD chart. Four real-life manufacturing scenarios are also considered to validate the suitability and superiority of r-AMD over the existing ‘r-MD’ and other competing charts.

A Shewhart-type nonparametric multivariate depth-based control chart for monitoring location

Parametric multivariate control charts like Hotelling’s rely on certain distributional assumptions that if these assumptions are not met, it leads us to an excessive number of false alarms and reduces the effectiveness of the monitoring strategy. In contrast, nonparametric multivariate control charts, specifically data-depth-based charts and related inferences, do not require any distributional assumptions and no constraints on data dimensions. In this paper, based on a family of multivariate affine invariant distribution-free data-depth-based tests for the multivariate one-sample location problem, a Shewhart-type nonparametric multivariate phase-II control chart is proposed. The performance of the proposed control chart is evaluated with a Monte Carlo simulation study using the average run length measure and is compared to that of competitors. It is observed that the proposed control chart maintained a desirable performance and leads to quicker detection of shifts. It is also a close competitor to Hotelling’s chart when the underlying distribution of the process is bivariate normal. Moreover, it performs better than its nonparametric competitor under the bivariate non-normal elliptical direction class of distributions, specifically heavy-tailed, light-tailed, and t-distributions, in a variety of settings. The control chart procedure is illustrated as an application to real-life data.

Nonparametric multivariate covariance chart for monitoring individual observations

Parametric and nonparametric multivariate control charts that are proven very useful in monitoring the covariance matrix of multivariate normally or “nearly” normally distributed continuous datasets have been proposed in statistical process control (SPC) literature. However, in many recent practical applications of SPC, the underlying systems or processes are characterised by discrete or a mixture of discrete and continuous multivariate random variables. In such cases, the available multivariate control charts for monitoring the covariance matrix of continuous processes are inadequate. We propose a multivariate nonparametric Shewhart-type chart for monitoring shifts in the covariance matrix of multivariate discrete or mixture of discrete and continuous random variables. The proposed chart first projects the multivariate dataset into Euclidean space. It then uses the Alt’s likelihood ratio obtained from the least absolute shrinkage and selection operator estimator that guarantees a well-conditioned estimate of the covariance matrix as the monitoring statistic. The proposed scheme does not require any parametric model assumptions and can be based on any distance measure of choice. It has the advantage of handling multivariate datasets of any type, including discrete, continuous or a mixture of discrete and continuous random variables. It uses the relationships among the process variables to build new variables that capture the dominant structure among the original variables. A bootstrap procedure is employed to obtain the control limit of the proposed chart for a suitable distance-based model through time. Simulation results show the advantage of the proposed chart in monitoring shifts in the covariance matrix. An illustrative example involving monitoring covariance structures of the lapping process in wafer semiconductor manufacturing and diagnosis single-proton emission computed tomography are provided to show the applications of the proposed chart.

A multivariate non-parametric control chart based on run test

Multivariate control charts have a wide range of applications in the field of quality management. This paper proposes a new multivariate non-parametric control chart based on run test. First, the shortest Hamiltonian path of the observations is determined by means of Kruskal algorithm. Then an EWMA control chart with sliding window based on the number of runs of the shortest Hamiltonian path is proposed. The selection of the design parameters and the determination of control limit are discussed in detail. The effect of different dimensions, shift magnitude, reference data size and the underlying distribution are investigated via Monte Carlo simulation. Compared with three recent non-parametric multivariate control charts, the newly proposed chart has superior performance when mean shift magnitude is large and the underlying distribution is non-normal. Also, the chart proposed is more suitable for monitoring in high-dimensional situations. A real case study is also presented to show the implementation procedures of the chart.

On the non-parametric multivariate control charts in fuzzy environment

Multivariate control chats are generally used in situations where the simultaneous monitoring or control of two or more related quality characteristics is necessary. In most processes in the real world, distribution of the process characteristics are unknown or at least non-normal, so the non-parametric or distribution-free charts are desirable. Most non-parametric statistical process-control techniques depend on ranks. In this survey, we apply the fuzzy set theory to deal with the circumstances thatthe values of each characteristic are presented in linguistic form, so we propose non-parametric multivariate control charts based on sign and Wilcoxon signed-rank tests.The performance of the proposed charts is investigated in a simulation study. Numerical examples are used to demonstrate the effectiveness and performance of the proposed charts.

Spatial rank-based high-dimensional monitoring through random projection

High-dimensional process monitoring becomes increasingly important in many applications where the joint distribution of process variables is usually unknown and not normal, and requires nonparametric methods for analysis and monitoring. However, when the process dimension is much larger than the reference sample size, most traditional nonparametric multivariate control charts fail due to the curse of dimensionality. Furthermore, when the process goes out of control, only a few (sparse) dimensions will be influenced, which increases the difficulty for both detection and diagnosis. To address these problems, this article proposes a new nonparametric monitoring scheme for high-dimensional processes. This scheme first projects the high-dimensional process into several subprocesses using ensemble random projections for dimension reduction. Then, for each subprocess a local nonparametric control chart is constructed based on the spatial rank test. Finally, all the local charts are fused together for decision making. Furthermore, after an out-of-control alarm is triggered, a diagnostic framework is proposed based on the square-root LASSO algorithm. Numerical studies together with real-data examples demonstrate the efficacy and applicability of the proposed methodology.

Generalized run tests for statistical process control

ABSTRACTIn a sequence of elements, a run is defined as a maximal subsequence of like elements. The number of runs or the length of the longest run has been widely used to test the randomness of an ordered sequence. Based on two different sampling methods and two types of test statistics used, run tests can be classified into one of four cases. Numerous researchers have derived the probability distributions in many different ways, treating each case separately. In the paper, we propose a unified approach which is based on recurrence arguments of two mutually exclusive sub-sequences. We also consider the sequence of nominal data that has more than two classes. Thus, the traditional run tests for a binary sequence are special cases of our generalized run tests. We finally show that the generalized run tests can be applied to many quality management areas, such as testing changes in process variation, developing non-parametric multivariate control charts, and comparing the shapes and locations of more than two process distributions.

Nonparametric multivariate control chart based on density-sensitive novelty weight for non-normal processes

ABSTRACTThe control chart is one of the most important statistical process control tools and the traditional control charts mostly require a specific probability distribution to set up their control limits. However, in modern manufacturing, the processes often have a lot of complexity and variability. Most existing control charts cannot efficiently handle situations with nonlinear or multi-modal patterns of observations. One recent trend about control charts is based on the novelty score algorithms that can effectively describe and reflect the characteristics of the monitored data. In this paper, we propose a density-sensitive novelty weight (DNW) control chart using k-nearest neighbors (kNN) algorithm that can efficiently monitor a process when the distribution of observations is unknown. More importantly, our chart can fully utilize the in-control local density information which can be regarded as the amount of in-control process knowledge. We demonstrated the usefulness of the proposed chart in experiments with simulated data in terms of average run length, and showed that the proposed chart generated proper portions of false alarms in the dense and sparse regions in Phase-I monitoring procedure in a real-life data, the E.coli dataset from the University of California, Irvine (UCI) repository.

On data depth and the application of nonparametric multivariate statistical process control charts

The purpose of this article is to summarize recent research results for constructing nonparametric multivariate control charts with main focus on data depth‐based control charts. Data depth provides dimension reduction to high‐dimensional problems in a completely nonparametric way. Several depth measures including Tukey depth are shown to be particularly effective for purposes of statistical process control in case that the data deviate normality assumption. For detecting small or moderate shifts in the process target mean, the multivariate version of the exponentially weighted moving average chart is generally robust to non‐normal data, so that nonparametric alternatives may be less often required. Copyright © 2016 John Wiley & Sons, Ltd.

Nonparametric multivariate statistical process control charts: a hypothesis testing-based approach

Nonparametric multivariate control charts are highly sought-after due to their flexibility to adapt to different distribution assumptions. However, most existing nonparametric multivariate control charts involve some tuning parameter, which needs to be pre-specified to implement those control charts. To choose the appropriate tuning parameter to achieve optimal performance, it usually requires the information about the out-of-control distribution. However, in practice, it is rarely known in advance what the out-of-control distribution is. In this paper, we propose a new nonparametric multivariate phase-II control chart using a hypothesis testing-based approach when a body of reference data (phase-I data) is available. The proposed control chart does not depend on any tuning parameter, and can be considered as a natural generalisation of the generalised likelihood ratio chart to the nonparametric setting. Our simulation study and real data analysis show that the proposed control chart performs well across a broad range of settings, and compares favourably with existing nonparametric multivariate control charts.

Bearing fault detection through multiscale wavelet scalogram-based SPC

Vibration-based fault detection and condition monitoring of rotating machinery, using statistical process control (SPC) combined with statistical pattern recognition methodology, has been widely investigated by many researchers. In particular, the discrete wavelet transform (DWT) is considered as a powerful tool for feature extraction in detecting fault on rotating machinery. Although DWT significantly reduces the dimensionality of the data, the number of retained wavelet features can still be significantly large. Then, the use of standard multivariate SPC techniques is not advised, because the sample covariance matrix is likely to be singular, so that the common multivariate statistics cannot be calculated. Even though many feature-based SPC methods have been introduced to tackle this deficiency, most methods require a parametric distributional assumption that restricts their feasibility to specific problems of process control, and thus limit their application. This study proposes a nonparametric multivariate control chart method, based on multiscale wavelet scalogram (MWS) features, that overcomes the limitation posed by the parametric assumption in existing SPC methods. The presented approach takes advantage of multi-resolution analysis using DWT, and obtains MWS features with significantly low dimensionality. We calculate Hotelling\' s T2-type monitoring statistic using MWS, which has enough damage-discrimination ability. A bootstrap approach is used to determine the upper control limit of the monitoring statistic, without any distributional assumption. Numerical simulations demonstrate the performance of the proposed control charting method, under various damage-level scenarios for a bearing system.

Multivariate Control Charts Based on Hybrid Novelty Scores

We propose a new nonparametric multivariate control chart that integrates a novelty score. The proposed control chart uses as its monitoring statistic a hybrid novelty score, calculated based on the distance to local observations as well as on the distance to the convex hull constructed by its neighbors. The control limits of the proposed control chart were established based on a bootstrap method. A rigorous simulation study was conducted to examine the properties of the proposed control chart under various scenarios and compare it with existing multivariate control charts in terms of average run length (ARL) performance. The simulation results showed that the proposed control chart outperformed both the parametric and nonparametric Hotelling's T 2 control charts, especially in nonnormal situations. Moreover, experimental results with real semiconductor data demonstrated the applicability and effectiveness of the proposed control chart. To increase the capability to detect small mean shift, we propose an exponentially weighted hybrid novelty score control chart. Simulation results indicated that exponentially weighted hybrid score charts outperformed the hybrid novelty score based control charts.

A Directional Multivariate Control Chart for Monitoring Univariate Autocorrelated Processes

A new nonparametric multivariate control chart, based on a spatial-sign test and integrating the directional information from processes with the exponentially weighted moving average (EWMA) scheme, is developed for monitoring the mean of a univariate autocorrelated process. Simulation studies show that it has robustness in in-control (IC) performance, and it is more sensitive to the small and moderate mean shifts for non-normality underlying process than other existing multivariate chart methods.

Nonparametric multivariate control charts based on a linkage ranking algorithm

AbstractControl charts have been widely recognized as important and critical tools in system monitoring for detection of abnormal behavior and quality improvement. In particular, multivariate control charts have been effectively used when a process involves a number of correlated quality characteristics. Most existing multivariate control charts were developed using the assumption of normally distributed quality characteristics. However, process data from modern industries often do not follow the normal distribution. Despite the great need for nonparametric control charts that can control the error rate regardless of the underlying distribution, few efforts have been made in this direction. In this paper, we propose a new nonparametric control chart (called the kLINK chart) based on a k‐linkage ranking algorithm that calculates the ranking of a new observation relative to the in‐control training data. A simulation study was performed to demonstrate the effectiveness of our kLINK chart and its superiority over the traditional Hotelling's T2 chart and the ranking depth control chart in nonnormal situations. In addition, to enable increased sensitivity to small shifts, we present an exponentially weighted moving average version of a kLINK chart. Copyright © 2010 John Wiley & Sons, Ltd.

A New Multivariate Non-Parametric Control Chart Based on Sign Test

Multivariate statistical process control deserves particular attention in the recent scenario. Though, Hotelling T2 control chart is quite popular and widely used technique in this field but its performance is deteriorated when the underlying distribution of the quality characteristics is not following multivariate normal distribution. Hence the need of developing a non-parametric multivariate control chart arises which does not require any underlying distribution. In this work a multivariate non-parametric control chart based on bivariate sign test is proposed. Its performance in both in control and out of control state was evaluated by simulating data from multivariate normal and multivariate t distribution and compared with those of existing multivariate parametric control chart.

Robust kernel distance multivariate control chart using support vector principles

It is important to monitor manufacturing processes in order to improve product quality and reduce production cost. Statistical Process Control (SPC) is the most commonly used method for process monitoring, in particular making distinctions between variations attributed to normal process variability to those caused by ‘special causes’. Most SPC and multivariate SPC (MSPC) methods are parametric in that they make assumptions about the distributional properties and autocorrelation structure of in-control process parameters, and, if satisfied, are effective in managing false alarms/-positives and false-negatives. However, when processes do not satisfy these assumptions, the effectiveness of SPC methods is compromised. Several non-parametric control charts based on sequential ranks of data depth measures have been proposed in the literature, but their development and implementation have been rather slow in industrial process control. Several non-parametric control charts based on machine learning principles have also been proposed in the literature to overcome some of these limitations. However, unlike conventional SPC methods, these non-parametric methods require event data from each out-of-control process state for effective model building. The paper presents a new non-parametric multivariate control chart based on kernel distance that overcomes these limitations by employing the notion of one-class classification based on support vector principles. The chart is non-parametric in that it makes no assumptions regarding the data probability density and only requires ‘normal’ or in-control data for effective representation of an in-control process. It does, however, make an explicit provision to incorporate any available data from out-of-control process states. Experimental evaluation on a variety of benchmarking datasets suggests that the proposed chart is effective for process monitoring.

DDMA-charts: Nonparametric multivariate moving average control charts based on data depth

This paper studies the DDMA–chart, a data depth based moving–average control chart for monitoring multivariate data. This chart is nonparametric and it can detect simultaneously location and scale changes in the process. It improves upon the existing r– and Q–chart in the efficiency of detecting location changes. Both theoretical justifications and simulation studies are provided. Comparisons with some existing multivariate control charts via simulation results are also provided. Some applications of the DDMA–chart to the analysis of airline performance data (collected by the FAA) are demonstrated. The results indicate that the DDMA–chart is an effective nonparametric multivariate control chart.