Process Mean Vector Research Articles

Monitoring the mean with least-squares support vector data description

Abstract: Multivariate control charts are essential tools in multivariate statistical process control (MSPC). “Shewhart-type” charts are control charts using rational subgroupings which are effective in the detection of large shifts. Recently, the one-class classification problem has attracted a lot of interest. Three methods are typically used to solve this type of classification problem. These methods include the k−center method, the nearest neighbor method, one-class support vector machine (OCSVM), and the support vector data description (SVDD). In industrial applications, like statistical process control (SPC), practitioners successfully used SVDD to detect anomalies or outliers in the process. In this paper, we reformulate the standard support vector data description and derive a least squares version of the method. This least-squares support vector data description (LS-SVDD) is used to design a control chart for monitoring the mean vector of processes. We compare the performance of the LS-SVDD chart with the SVDD and T2 chart using out-of-control Average Run Length (ARL) as the performance metric. The experimental results indicate that the proposed control chart has very good performance.

Multivariate cumulative sum control chart and measure of process capability based on bivariate ranked set schemes

In this article we have proposed multivariate cumulative sum control chart based on bivariate ranked set schemes for quick identification of small variation in the process mean vector. Also, we have offered multivariate measure of process capability based on bivariate ranked set schemes for testing the customer requirements. In the construction of control chart, we have designed plotting statistic, and derived control limit. Regarding the multivariate measure of process capability, we have defined an estimator and then computed the critical values for inference purposes. In order to compare the performance of existing and proposed control charts, we have obtained various performance measures. Results reveal that performance of proposed control chart based on bivariate ranked set schemes depends on the choice of the factors such as sampling scheme, sample size, magnitude of association between concomitant variable and study variables, and size of the shift. Furthermore, comparative analysis shows that the performance of the proposed control chart based on bivariate ranked set schemes outperforms the existing methods. Finally, real life example is included in which we have applied proposed and existing control charts for monitoring calcium–magnesium and residual sodium contents in irrigation water. In addition, the implementation of the proposed multivariate measure of process capability ensures the level of calcium–magnesium and residual sodium contents in irrigation water to satisfy the requirements of customers or engineering tolerance.

Robust T2 control chart using median‐based estimators

AbstractOne of the most widely used multivariate control charts is the Hotelling T2. In order to construct a Hotelling T2 control chart, the mean vector (μ) and the variance–covariance matrix (Σ) must be first estimated. The classical estimators of μ and Σ are usually used to design Hotelling T2 control chart. The classical estimators are sensitive to the presence of outliers. One way to deal with outliers is to use robust estimators. In this study, a robust T2 control chart is proposed. The mean vector is obtained from the sample median. The median absolute deviation and the comedian are used as the estimates of the elements of the variance–covariance matrix. The proposed robust estimators of the mean vector and the variance–covariance matrix are compared with the sample mean vector and the sample variance–covariance matrix, and the M estimator of these parameters, through efficiency and robustness measures. The performances of the proposed robust T2 control chart and the classical and the M estimators are also compared by means of average run length. Simulation results reveal that the proposed robust T2 control chart has much better performance than the traditional Hotelling T2 and similar performance to the M estimator in detecting shifts in process mean vector. Use of other robust estimators to estimate the process parameters is an area for further research.

On the Efficient Monitoring of Multivariate Processes with Unknown Parameters

Control charts are commonly used tools that deal with monitoring of process parameters in an efficient manner. Multivariate control charts are more practical and are of greater importance for timely detection of assignable causes in multiple quality characteristics. This study deals with multivariate memory control charts to address smaller shifts in process mean vector. By adopting a new homogeneous weighting scheme, we have designed an efficient structure for multivariate process monitoring. We have also investigated the effect of an estimated variance covariance matrix on the proposed chart by considering different numbers and sizes of subgroups. We have evaluated the performance of the newly proposed multivariate chart under different numbers of quality characteristics and varying sample sizes. The performance measures used in this study include average run length, standard deviation run length, extra quadratic loss, and relative average run length. The performance analysis revealed that the proposed control chart outperforms the usual scheme under both known and estimated parameters. An application of the study proposal is also presented using a data set related to Olympic archery, for the monitoring of the location of arrows over the concentric rings on the archery board.

An improved process monitoring by mixed multivariate memory control charts: An application in wind turbine field

Memory control chart such as multivariate CUSUM (MCUSUM) and multivariate EWMA (MEWMA) control charts are considered superior for the detection of small-to-moderate variation in the process mean vector. In this article, we have proposed two advanced forms of memory multivariate charts to identify the small amount of shifts in the process mean vector. The proposed control charts methodologies are based on the mixed features of the MCUSUM, MEWMA, classical EWMA chart, and chart based on principal component analysis. Monte Carlo simulation technique is used to simulate numerical results. To evaluate the performance of proposed control charts, we have used average run length for a single shift, extra quadratic loss function, relative average run length, and performance comparison index measures for certain range of shifts for overall performance. Results elaborate that proposed charts have outstanding performance for detection of small shifts in mean vector as compared to the various existing such as MCUSUM, MEWMA, etc. charts. For practical purpose, implementation of the proposed control charts with a real-life data in the field of wind turbine has included to make clear the advantages of proposed control chart(s) over other control charts for early detection of shifts.

Change detection using least squares one-class classification control chart

ABSTRACT One-class classification can be thought as a special type of two-class classification problem, where data only from one class, the target class, are available for training the classifier (referred to as one-class classifier). The problem of classifying positive (or target) cases in the absence of appropriately characterized negative cases (or outliers) has gained increasing attention in recent years. Several methods are available to solve the one-class classification problem. Three methods are commonly used: density estimation, boundary methods, and reconstruction methods. This paper focuses on boundary methods which include k–center method, nearest neighbor method, one-class support vector machine (OCSVM), and support vector data description (SVDD). In statistical process control (SPC), practitioners successfully used SVDD to detect anomalies or outliers in the process. In this paper, we reformulate the standard OCSVM by a least squares version of the method. This least squares one-class support vector machine (LS-OCSVM) is used to design a control chart for monitoring the mean vector of processes. We compare the performance of the LS-OCSVM chart with the SVDD and chart. The experimental results indicate that the proposed control chart has very good performances.

Misleading signals in joint schemes for the mean vector and covariance matrix

AbstractIn multivariate statistical process control, it is recommendable to run two individual charts: one for the process mean vector and another one for the covariance matrix. The resulting joint scheme provides a way to satisfy Shewhart's dictum that proper process control implies monitoring both process location and spread.The multivariate quality characteristic is deemed to be out of control whenever a signal is triggered by either individual chart of the joint scheme. Consequently, a shift in the mean vector can be misinterpreted as a shift in the covariance matrix and vice versa. Compelling results are provided to give the quality control practitioner an idea of how joint schemes for the mean vector and covariance matrix are prone to trigger misleading signals that will likely lead to a incorrect diagnostic of which parameter has changed.

One-sided and two one-sided MEWMA charts for monitoring process mean

ABSTRACTThis study focuses on the development of new one-sided and two one-sided MEWMA charts for monitoring the mean of a multivariate normal process. The one-sided MEWMA chart detects either increases or decreases while the two one-sided MEWMA chart detects both increases and decreases, in the process mean vector. The idea is to first truncate either positive or negative normalized observations to zero, and then apply the one-sided MEWMA charts to transformed data. This helps increase the sensitivities of one-sided and two one-sided MEWMA charts. The Monte Carlo simulation method is used to compute the run length characteristics of the proposed multivariate charts. A detailed run length comparison reveals that the proposed one-sided and two one-sided MEWMA charts are uniformly more sensitive than the existing counterparts when detecting moderate-to-large shifts in the process mean. In addition, with reasonable assumptions, the proposed two one-sided MEWMA chart outperforms the conventional MEWMA chart. Real datasets are considered to explain the implementation of the proposed multivariate charts.

Memory‐type multivariate charts with fixed and variable sampling intervals for process mean when covariance matrix is unknown

AbstractMemory‐type multivariate charts have been widely recognized as a potentially powerful process monitoring tool because of their excellent speed in detecting small‐to‐moderate shifts in the mean vector of a multivariate normally distributed process, namely, the multivariate EWMA (MEWMA), double MEWMA, Crosier multivariate CUSUM (MCUSUM), and Pignatiello and Runger MCUSUM charts. These multivariate charts are based on the assumption that the covariance matrix is known in advance; but, it may not be known in practice. It is thus not possible to use these multivariate charts unless a large Phase I dataset is available from an in‐control process. In this paper, we propose multivariate charts with fixed and variable sampling intervals for the process mean vector when the covariance matrix is estimated from sample. Using the Monte Carlo simulation method, the run length characteristics of the multivariate charts are computed. It is shown that the in‐control and out‐of‐control run length performances of the proposed multivariate charts are robust to the changes in the process covariance matrix, while the existing multivariate charts are not. A real dataset is taken to explain the implementation of the proposed multivariate charts.

A Multivariate Homogeneously Weighted Moving Average Control Chart

This paper presents a multivariate homogeneously weighted moving average (MHWMA) control chart for monitoring a process mean vector. The MHWMA control chart statistic gives a specific weight to the current observation, and the remaining weight is evenly distributed among the previous observations. We present the design procedure and compare the average run length (ARL) performance of the proposed chart with multivariate Chi-square, multivariate EWMA, and multivariate cumulative sum control charts. The ARL comparison indicates superior performance of the MHWMA chart over its competitors, particularly for the detection of small shifts in the process mean vector. Examples are also provided to show the application of the proposed chart.

Multivariate coefficient of variation control charts in phase I of SPC

Multivariate control charts are mostly available for monitoring the process mean vector or the covariance matrix. Recently, work has been done on monitoring the multivariate coefficient of variation (CV) in phase II of the statistical process control (SPC). However, no study has investigated the performance of the multivariate CV charts in phase I. The phase I procedures are more important and involve the estimation of the charts’ limits from a historical or reference dataset that represents the in-control state of the process. In real life, contaminations are mostly present in the historical samples; hence, the phase I procedures are mostly adopted to get rid of these contaminated samples. In this study, we investigate the performance of a variety of multivariate CV charts in phase I considering both diffuse symmetric and localized CV disturbance scenarios, using probability to signal as a performance measure. A real-life application, concerning carbon fiber tubing, is also provided to show the implementation of the proposed charts in phase I. The findings of this study will be useful for practitioners in their selection of an efficient phase I control chart for monitoring multivariate CV.

Multi-Objective Genetic Algorithm for Economic Statistical Design of the T2 Control Chart with Variable Sample Size: The Updated Markov Chain Approach

Abstract T 2 control charts are used to primarily monitor the mean vector of quality characteristics of a process. Recent studies have shown that using variable sample size (VSS) schemes results in charts with more statistical power for detecting small to moderate shifts in the process mean vector. In this study, we have presented a multiple-objective economic statistical design of VSS T2 control chart with the adjusted average time to signal (AATS) as the statistical objective and the expected cost per hour as the economic objective. Then, a multi-objective genetic algorithm for economic statistical design is proposed for identifying the Pareto optimal solutions of control chart design. Through an illustrative example, the advantages of the proposed approach are shown by providing a list of viable optimal solutions and graphical representations, which indicate the advantage of flexibility and adaptability of our approach.

Monitoring bivariate and trivariate mean vectors with a Shewhart chart

AbstractIn this article, we propose the use of the mean chart to control multivariate processes. The basic idea is to control the mean vector of bivariate (X, Y) and trivariate (X, Y, Z) processes by alternating the charting statistic of the Shewhart chart. If the mean of X observations was the charting statistic to obtain the current sample point, then the mean of Y observations will be the charting statistic to obtain the next sample point (for the trivariate case, the mean of Z observations will be the charting statistic to obtain the sample point subsequent to the next one). As a Shewhart chart, the signal is given anytime a sample point is plotted beyond the control limits, independent of the charting statistic in use. A fair comparison between the proposed chart and the Hotelling chart is based on an equal number of measurements per sample. The Shewhart chart with alternated charting statistic (ACS) always outperforms the Hotelling chart, except for specific types of disturbances in quality characteristics highly correlated (ρ = 0.7). The ACS chart is substantially easier to operate and faster than the Hotelling chart in signaling changes in the mean vector of bivariate and trivariate processes. Even with fewer measurements per sample, the trivariate ACS chart outperforms the Hotelling chart.

Optimal statistical design of variable sample size multivariate exponentially weighted moving average control chart based on median run-length

Conventionally, a standard control chart implements fixed sample size in process monitoring. In this study, we propose an optimal statistical design for the variable sample size (VSS) multivariate exponentially weighted moving average (MEWMA) chart based on the median run-length (MRL). The proposal is based on the fact that the percentiles of the run-length distribution, especially the MRL, are more reflective and reliable for performance evaluation with respect to a skewed run-length distribution. The MRL for the VSS MEWMA chart computed using the Markov chain approach is verified with Monte Carlo simulation. For benchmarking purposes, the performance of the VSS MEWMA chart is compared against the standard MEWMA chart and the synthetic T2 chart, in terms of the MRL. The numerical results show that the VSS MEWMA chart performs better than the standard MEWMA chart and the synthetic T2 chart, in detecting shifts in the process mean vector. Finally, an application is provided as an illustration for the implementation of the VSS MEWMA chart based on the MRL.

On the effect of measurement errors in simultaneous monitoring of mean vector and covariance matrix of multivariate processes

In this paper, we investigate the misleading effect of measurement errors on simultaneous monitoring of the multivariate process mean and variability. For this purpose, we incorporate the measurement errors into a hybrid method based on the generalized likelihood ratio (GLR) and exponentially weighted moving average (EWMA) control charts. After that, we propose four remedial methods to decrease the effects of measurement errors on the performance of the monitoring procedure. The performance of the monitoring procedure as well as the proposed remedial methods is investigated through extensive simulation studies and a real data example.

Some Group Runs Based Multivariate Control Charts for Monitoring the Process Mean Vector

In this article, we propose two control charts namely, the “Multivariate Group Runs’ (MV-GR-M)” and the “Multivariate Modified Group Runs’ (MV-MGR-M)” control charts, based on the multivariate normal processes, for monitoring the process mean vector. Methods to obtain the design parameters and operations of these control charts are discussed. Performances of the proposed charts are compared with some existing control charts. It is verified that, the proposed charts give a significant reduction in the out-of-control “Average Time to Signal” (ATS) in the zero state, as well in the steady state compared to the Hotelling’s T2 and the synthetic T2 control charts.

The T2 chart with mixed samples to control bivariate autocorrelated processes

In this paper, we propose the use of the T2 chart with the mixed sampling strategy (MS) to monitor the mean vector of bivariate processes with observations that fit to a first-order vector autoregressive model. With the MS, rational subgroups of size n are taken from the process and the selected units are regrouped to form the mixed samples. The units of the mixed samples are units selected from the last two rational subgroups. The aim of the proposed sampling strategy is to reduce the negative effect of the autocorrelation on the performance of the T2 chart. When the two variables are autocorrelated, the MS always enhances the T2 chart performance, however, the mixed samples are not recommended for bivariate processes with only one autocorrelated variable which is rarely affected by the assignable cause.

On the misleading signals in simultaneous schemes for the mean vector and covariance matrix of multivariate i.i.d. output

The performance of a product often depends on several quality characteristics. Simultaneous schemes for the process mean vector $$(\varvec{\mu })$$ and the covariance matrix ( $$\varvec{\Sigma }$$ ) are essential to determine if unusual variation in the location and dispersion of a multivariate normal vector of quality characteristics has occurred. Misleading signals (MS) are likely to happen while using such simultaneous schemes and correspond to valid signals that lead to a misinterpretation of a shift in $$\varvec{\mu }$$ (resp. $$\varvec{\Sigma }$$ ) as a shift in $$\varvec{\Sigma }$$ (resp. $$\varvec{\mu }$$ ). This paper focuses on numerical illustrations that show that MS are fairly frequent, and on the use of stochastic ordering to qualitatively assess the impact of changes in $$\varvec{\mu }$$ and $$\varvec{\Sigma }$$ in the probabilities of misleading signals in simultaneous schemes for these parameters while dealing with multivariate normal i.i.d. output.

Generalized Inferences about the Mean Vector of Several Multivariate Gaussian Processes

We consider in this paper the problem of comparing the means of several multivariate Gaussian processes. It is assumed that the means depend linearly on an unknown vector parameterθand that nuisance parameters appear in the covariance matrices. More precisely, we deal with the problem of testing hypotheses, as well as obtaining confidence regions forθ. Both methods will be based on the concepts of generalizedpvalue and generalized confidence region adapted to our context.

Non-dominated Sorting Genetic Algorithm (NSGA-II) Approach to the Economic Statistical Design of Variable Sampling Interval T2 Control Charts

T control charts are used to primarily monitor the mean vector of quality characteristics of a process. Recent studies have shown that using variable sampling interval (VSI) schemes results in charts with more statistical power for detecting small to moderate shifts in the process mean vector. In this study, we have presented a multiple-objective economic statistical design of VSI T control chart when the in-control process mean vector and process covariance matrix are unknown. Then we exert to find the Pareto-optimal designs in which the two objectives are minimized simultaneously by using the Non-dominated sorting genetic algorithm. Through an illustrative example, the advantages of the proposed approach is shown by providing a list of viable optimal solutions and graphical representations, thereby bolding the advantage of flexibility and adaptability.