Multivariate Exponentially Weighted Moving Average Research Articles

Monitoring covariance in multivariate time series: Comparing machine learning and statistical approaches

AbstractIn complex systems with multiple variables monitored at high‐frequency, variables are not only temporally autocorrelated, but they may also be nonlinearly related or exhibit nonstationarity as the inputs or operation changes. One approach to handling such variables is to detrend them prior to monitoring and then apply control charts that assume independence and stationarity to the residuals. Monitoring controlled systems is even more challenging because the control strategy seeks to maintain variables at prespecified mean levels, and to compensate, correlations among variables may change, making monitoring the covariance essential. In this paper, a vector autoregressive model (VAR) is compared with a multivariate random forest (MRF) and a neural network (NN) for detrending multivariate time series prior to monitoring the covariance of the residuals using a multivariate exponentially weighted moving average (MEWMA) control chart. Machine learning models have an advantage when the data's structure is unknown or may change. We design a novel simulation study with nonlinear, nonstationary, and autocorrelated data to compare the different detrending models and subsequent covariance monitoring. The machine learning models have superior performance for nonlinear and strongly autocorrelated data and similar performance for linear data. An illustration with data from a reverse osmosis process is given.

Dynamic probability control limits for the adaptive multivariate EWMA chart

AbstractEffective monitoring of the control charts requires the establishment of appropriate control limits. Various methods have been proposed to determine the fixed control limits (FCLs) based on a specified in‐control average run‐length value, such as Markov chains, integral equations, and Monte Carlo simulations.With the FCLs, the conditional false alarm rate (CFAR) of a control chart varies over time in an unexpected and undesirable way, where the CFAR refers to the probability of a false alarm at a particular time given no previous false alarm. To address this issue, dynamic probability control limits (DPCLs) can be employed to keep the CFAR constant over time. In this study, we determine the DPCLs using fixed and time‐varying sample sizes for the adaptive multivariate EWMA chart when the underlying process is assumed to follow a multivariate normal distribution. The DPCLs are designed to automatically adjust to changes in the probability distribution of the sample size, while maintaining a consistent CFAR. This results in a closely matched run‐length performance for an in‐control process compared to that of the geometric distribution.

Generalized multivariate EWMA control chart supplemented with bivariate ranked set techniques

ABSTRACT Existing univariate exponential weighted moving average (EWMA) control charts with ranked set schemes are used for monitoring related characteristics independently. Also, existing multivariate EWMA control charts are generally applicable for simple random sampling (SRS) based processes. In this work, a generalized multivariate EWMA control chart is proposed with bivariate ranked set techniques (BVRST) for simultaneously monitoring. Furthermore, an extensive comparative analysis between proposed generalized multivariate EWMA and classical control chart is conducted. In this regard, we have used various performance measures, which cover average run length, median of run length, relative average run length, performance comparison index, and extra quadratic loss. The performance measures are obtained using the Monte Carlo simulation (MCS) approach. Results revealed that the proposed generalized multivariate EWMA control chart has an outstanding detection ability relative to the existing multivariate control charts. Additionally, existing multivariate control charts are found special cases of the proposed control chart. A real-life example from the agriculture field is included in which chemical properties (residual sodium and calcium-magnesium contents) of irrigation water are monitored through the control charts. The results revealed that the proposed control chart is proficient in diagnosing small variations in the chemical characteristics relative to the existing control charts.

Control charts for high-dimensional time series with estimated in-control parameters

In this article, we study the effect of misspecification caused by fitting the target process in the Phase I analysis of the monitoring procedure on the behavior of several types of multivariate exponentially weighted moving average (MEWMA) control charts in the high-dimensional setting. In particular, the classical MEWMA control charts, whose control statistics are based on the exact and asymptotic Mahalanobis distance, are considered together with the novel approaches where the Euclidean distance and the diagonalized Euclidean distance are employed in the construction of control statistics. The high-dimensional distributions of the control statistics are deduced at each time. These results are later used to assess the performance of the considered control charts under misspecification. Both theoretical and empirical findings lead to the conclusion that the control charts based on the Euclidean distance and the diagonalized Euclidean distance are robust to misspecification for moderate dimensions of the data-generating model, whereas they tend to overestimate the in-control average run lengths (ARLs) in the case of larger dimensions. On the other hand, the control schemes based on the Mahalanobis distance are considerably affected by the estimation of the parameters of the target process, and their application results in drastically smaller values of the ARLs, especially when the dimension of the data-generating model is large.

Evolutionary support vector regression for monitoring Poisson profiles

Many researchers have shown interest in profile monitoring; however, most of the applications in this field of research are developed under the assumption of normal response variable. Little attention has been given to profile monitoring with non-normal response variables, known as general linear models which consists of two main categories (i.e., logistic and Poisson profiles). This paper aims to monitor Poisson profile monitoring problem in Phase II and develops a new robust control chart using support vector regression by incorporating some novel input features and evolutionary training algorithm. The new method is quicker in detecting out-of-control signals as compared to conventional statistical methods. Moreover, the performance of the proposed scheme is further investigated for Poisson profiles with both fixed and random explanatory variables as well as non-parametric profiles. The proposed monitoring scheme is revealed to be superior to its counterparts, including the likelihood ratio test (LRT), multivariate exponentially weighted moving average (MEWMA), LRT-EWMA and other machine learning-based schemes. The simulation results show superiority of the proposed method in profiles with fixed explanatory variables and non-parametric models in nearly all situations while it is not able to be the best in all the simulations when there are with random explanatory variables. A diagnostic method with machine learning approach is also used to identify the parameters of change in the profile. It is shown that the proposed profile diagnosis approach is able to reach acceptable results in comparison with other competitors. A real-life example in monitoring Poisson profiles is also provided to illustrate the implementation of the proposed charting scheme.

Economic design of residuals MEWMA control chart with variable sampling intervals and sample size

In order to improve the monitoring efficiency of the multivariable autocorrelation process and reduce the cost of process control, an economic design method of residual Multivariate Exponentially Weighted Moving Average (MEWMA) control charts with variable sample size and sampling intervals (VSSIs) was proposed. First, residual MEWMA control charts with VSSI were designed for monitoring multivariable autocorrelation processes. Second, the economic model of the VSSI residual MEWMA control chart was established, and a data example was given. The optimal design parameters of the control chart were obtained by using a genetic algorithm to minimize the expected cost per unit time. Third, using orthogonal experimental design and regression analysis, a sensitivity analysis of the economic model was carried out to get the relationship between the model parameters and the design parameters of the control chart. Finally, through the optimality analysis, it was verified that the proposed economic model was superior to the models established by the variable sampling intervals strategy and the variable sample size strategy and had less expected cost per unit time.

New statistical and machine learning based control charts with variable parameters for monitoring generalized linear model profiles

In this research, we develop three statistical based control charts: the Hotelling’s T2, MEWMA (multivariate exponentially weighted moving average), and LRT (likelihood ratio test) as well as three machine learning (ML) based control charts: the ANN (artificial neural network), SVR (support vector regression), and RFR (random forest regression), for monitoring generalized linear model (GLM) profiles. We train these ML models with two different training methods to get a linear (regression) output and then apply our classification technique to see if the process is in-control or out-of-control, at each sampling time. In addition to developing the FP (fixed parameters) schemes, for the first time in GLM profiles, we design an adaptive VP (variable parameters) scheme for each control chart as well to increase the charts’ sensitivity in detecting shifts. We develop some algorithms with which the values of the control chart parameters in both FP and VP schemes can be obtained. Then, we develop two algorithms to measure the charts’ performance in both FP and VP schemes, by using the run-length and time-to-signal based performance measures. This is also the first control chart-related research that develops an algorithm to compute the performance measures that applies to any VP adaptive control scheme.After designing the control charts as well as performance measures, we perform extensive simulation studies and evaluate and compare all our control charts under different shift sizes and scenarios, and in three different simulation environments. Finally, we present a numerical example regarding a drug dose-response study to show how the proposed control charts can be implemented in real practice.

A phase II multivariate EWMA chart for monitoring multi-dimensional ratios of process means with individual observations

In recent years, there has been a resurgence in the development of control charts for monitoring the mean of the ratio of two correlated variables. However, most of the existing research has focused on the univariate mean of the ratio of two correlated variables under the assumption that the process follows a bivariate normal distribution. Furthermore, most of the existing research utilize biased estimators of the mean of the ratio of two correlated variables to develop control charts. More importantly, in certain applications, critical quality characteristics to be closely monitored are actually the ratios of the means of correlated variables, in that the process is considered to be stable as long as the ratios of the means of correlated variables remain constant at given levels, regardless of how each variable changes. We are thus motivated in this study to develop an exponentially weighted moving average (EWMA) based Phase II control chart for monitoring multi-dimensional ratios of the means of correlated variables.In this study, we provide a general framework for estimating parameters and control limits which is applicable without having to assume that the process follows a multivariate normal distribution. The performance of the proposed chart is evaluated under different multivariate distributions. Finally, a real data application of the proposed chart is presented to illustrate the practicality of the proposed chart.

Performance evaluation of multivariate statistical techniques using edge-enabled optimisation for change detection in activity monitoring

The monitoring of human activities using simple body worn sensors is an important and emerging area of research in machine learning. The sensors capture a large amount of data in a short period of Time in a relatively un-obtrusive manner. The sensor data might have different transitions to be used for deification of different user activities. Therefore, change point detection can be used to classify the transition from one underlying distribution to another. The automatic and accurate change point detection is not only used for different events, however, can also be used for generating real world datasets and responding to changes in patient vital signs in critical situation. Moreover, the huge amount of data can use the current state-of-the-art cloud and edge computing platforms to process the change detection locally and more efficiently. In this paper, we used multivariate exponentially weighted moving Average (MEWMA) for online change point detection. Additionally, genetic algorithm (GA) and particle swarm optimization (PSO) is used to automatically identify an optimal parameter set by maximizing the F-measure. The optimisation approach is implemented over an edge cloud platform so that the data can be processed locally and more accurately. Furthermore, we evaluate our approach against multivariate cumulative sum (MCUSUM) from state-of the-art in terms of different metric measures such as accuracy, precision, sensitivity, G-means and F-measure. Results have been evaluated based on real data set collected using accelerometer for a set of 9 distinct activities performed by 10 users for total period of 35 minutes with achieving high accuracy from 99.3% to 99.9% and F-measure up to 62.94%.

A Framework for Multivariate Statistical Quality Monitoring of Additive Manufacturing: Fused Filament Fabrication Process

Advances in additive manufacturing (AM) processes have increased the number of relevant applications in various industries. To keep up with this development, the process stability of AM processes should be monitored, which is conducted through the assessment of the outputs or product characteristics. However, the use of univariate control charts to monitor an AM process might lead to misleading results, as most additively manufactured products have more than one correlated quality characteristic (QC). This paper proposes a framework for monitoring the multivariate quality characteristics of AM processes, and the proposed framework was applied to monitor a fused filament fabrication (FFF) process. In particular, specimens were designed and produced using the FFF process, and their QCs were identified. Then, critical quality characteristic data were collected using a precise measurement system. Furthermore, we propose a transformation algorithm to ensure the normality of the collected data. After examining the correlations between the investigated quality characteristics, a multivariate exponential weighted moving average (MEWMA) control chart was used to monitor the stability of the process. Furthermore, the MEWMA parameters were optimized using a novel heuristic technique. The results indicate that the majority of the collected data are not normally distributed. Consequently, the efficacy of the proposed transformation technique is demonstrated. In addition, our findings illustrate the correlations between the QCs. It is worth noting that the MEWMA optimization results confirm that the considered AM process (i.e., FFF) is relatively stable.

Penerapan Bagan Kendali MEWMA-MEWMV pada Pengendalian Kualitas Lulusan Prodi Statistika FMIPA Universitas Syiah Kuala

Statistical technique used to analyze quality problems and improve process performance is Statistical Process Control (SPC). In this study, multivariate control chart is used to control the quality of the graduates of Statistics Study Program, FMIPA USK, namely using Multivariate Exponentially Weighted Moving Average (MEWMA) control chart to control process average, Multivariate Exponentially Weighted Moving Variance (MEWMV) to control process variability, and analysis Process Capability to assess the entire process. Data used is secondary data, namely GPA data, duration of thesis preparation, and length of study for graduates of Statistics Study Program, FMIPA USK in 2016-2021 as many as 122 people. Results of the study, using MEWMA control chart, it was found that the average process for quality of graduates was statistically controlled in phase II. Selection of the most optimal weighting is λ=0.9. Meanwhile, for application of the MEWMV control chart, it was found that the process variability in the quality of graduates was also statistically controlled. The selection of the most optimal weights is λ =0.9 and ω=0.3. The results of the calculation of process capability, multivariately the GPA variable, length of the final project preparation, and length of the study show that all three are not capable.

A stage-wise non-Gaussian statistical process control strategy

The article delves into the development of a Non-Gaussian Process Monitoring Strategy for a Copper Cathode Manufacturing Unit (CCMU). The monitoring strategy being devised highlighted the issue of multi-stage process monitoring via the usage of Multi-block Independent Component Analysis (MBICA) techniques. MBICA is the multi-block variant of ICA technique which is prevalently used for process laden with non-Gaussian or non-normal data. Development of the monitoring strategy involved detection of fault(s) and their subsequent diagnosis. Detection of fault(s) was carried out by employment of I2 control chart whose control limit was established via Bootstrap procedure. The diagnosis of the detected fault was carried out by employment of fault diagnostic statistic. An amalgamation of MBICA and Multivariate Exponentially Weighted Moving Average (MEWMA) are also known as MBICA-MEWMA approach was also proposed for detection of incipient fault(s). The monitoring strategy thus developed was showcased for a CCMU which specialised in the manufacture of copper cathode which has got varied practical applications. The monitoring strategy thus devised was able to detect and diagnose the faults with appreciable accuracy.

Copula-based multivariate EWMA control charts for monitoring the mean vector of bivariate processes using a mixture model

Control charts are the main tools in the statistical process monitoring area to investigate how the quality of production remains stable or changes over the time. This article develops Hotelling’s T 2 and multivariate exponentially weighted moving average (MEWMA) control charts for monitoring the mean vector of processes when observations come from a mixture copula model including, Gumbel, Clayton, and Frank copulas. The proposed mixture model enables the quality inspectors to cover several dependence levels of observations, from weak and moderate to strong in positive values by Kendall’s tau. To assess the performance of the proposed charts, extensive Monte-Carlo simulations were conducted based on the average run length (ARL) metric for both in-control and out-of-control states by considering a bivariate process with normal marginals. For more illustration, the step-by-step procedure of the proposed monitoring technique implementation has been investigated in a computer manufacturing process with both Phases I and II analysis.

Multivariate auto‐correlated process control by a residual‐based mixed CUSUM‐EWMA model

AbstractMultivariate auto‐correlated process control issues in industrial systems are a concern for statistical process monitoring (SPM). Traditional control charts produce large false alarms and/or miss timely detections of quality deterioration because they are unable to recognize the signals from multivariate auto‐correlated response variables. To track multivariate auto‐correlated processes, this paper presents a new residual‐based mixed multivariate control chart using cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) approaches. Using in‐control data, the multi‐output least square support vector regression model's optimal hyper‐parameters are determined, and a bootstrap method is used to estimate the upper control limit of the proposed control chart. The suggested control chart has strong detection performance for a small magnitude mean vector shift based on the average run length (ARL) performance for a particular range of shifts. Experimental result elaborates that the proposed control chart is more sensitive to detecting the mean vector shift compared with the existing commonly used models, such as multivariate CUSUM and multivariate EWMA control charts. The proposed control chart model and corresponding computational algorithm are successfully applied to SPM in an electronic conductor production line with multivariate auto‐correlated attributes.

PENERAPAN DIAGRAM KONTROL MEWMA DALAM PENGENDALIAN KUALITAS PRODUKSI KERIPIK SINGKONG PADA UMKM DI KOTA SEMARANG

Quality is the main thing that needs to be considered by every company. Ceriping Bintang Putra Bu Slamet is an UMKM (Usaha Mikro, Kecil dan Menengah) that produces cassava chips. During production, there are three quality characteristics, namely large crumbs defects, small crumbs, and chips sticking together. It is important to control these defects to produce quality products according to customer needs. This research was conducted from July to August 2021. The purpose of this study was to control the production quality of cassava chips using the Multivariate Exponentially Weighted Moving Average (MEWMA) control chart and multivariate process capability analysis. The MEWMA control chart is used to detect the shift in the process average which is more sensitive using weights (λ), while the process capability analysis is used to determine the process performance. The implementation of the MEWMA control chart is carried out in two stages, namely phase I control to obtain the optimal weighting and control limits so that it can be used in phase II control to monitor the average process for the next period. Based on the results of the analysis, the optimal weighting is λ =0,4 with BKA=201,7434, GT=113,538, and BKB=0 in phase I control. Then, the results of phase II control show a shift in the average process in a better direction. In addition, the results of the process capability analysis show an improvement in the performance of the production process from July 2021 to August 2021 with MCpm values of 0,535 and 1,147

Neutrosophic multivariate EWMA control chart

The MEWMA chart is one of the traditional multivariate charts which are widely employed in inspecting the quality of manufacturing and services. This chart is created through monitoring the small shifts of mean vectors of variable quality characteristics. Often in practice, the measurement of a quality characteristic produces uncertain, incomplete values, so that ambiguous numbers are obtained. In this condition, a neutrosophic-based control chart can overcome the problem resulting from the ambiguous data. The paper’s objective is to construct a new multivariate monitoring scheme based on a neutrosophic chart, namely the neutrosophic Multivariate EWMA (NMEWMA). Furthermore, the performance of the new multivariate monitoring scheme is evaluated in detecting process shifts employing the Average Run Length (ARL) and Standard Deviation Run Length (SDRL). This control chart is an innovation in the quality monitoring of uncertain data. The research result obtained indicates that the NMEWMA chart performs better than the MEWMA in finding the small mean shifts as well as in the real case application.

A Monitoring Kualitas Kaca Panasab dengan Diagram Kontrol MEWMA

Panasab glass is one type of glass that can absorb some of the heart's heat so that the temperature in the room is more comfortable. One of the national glass companies that produce hot glass monitors the silvering coating process. The silvering method is considered less effective because it is expensive. Therefore, in this paper, the process of controlling the quality of hot glass is carried out on the quality characteristics of the left and right CD edge distortion, which is the length of the roll marks on the right and left sides of the net glass that affect the presence or absence of distortion in the visual check results. The two quality characteristics correlate with each other. Therefore, quality control is carried out using a Multivariate Exponentially Weighted Moving Average (MEWMA) control chart to monitor process variability and averages. MEWMA control chart statistics are affected by the weighting value . The quality characteristic data of panasab glass is divided into phase 1 and phase 2. Phase 1 is used to obtain an in-control parameter estimate (IC) which is then used for monitoring phase 2. Quality monitoring in phase 1, the process has been statistically controlled after iterations have been carried out. By using =0.4. Meanwhile, phase 2, based on the MEWMA control chart, shows that the producton procrss of panasab glass production has not been controlled statistically. Based on process capability analysis, the results obtained that the level of precision and accuracy of the quality of panasab glass products is still low. So it can be concluded that the production process of panasav glass is not yet capable.

A bivariate exponentially weighted moving average control chart based on exceedance statistics

Nonparametric control charts are more practical tools for statistical process control (SPC), as they are robust in situations in which the underlying distribution is unknown. Comprehensibility and simplicity of exceedance statistics provide great convenience to analysts in multivariate SPC applications. By using the exceedance statistics, analysts save time and avoid complex calculations. Therefore, in this study, a bivariate nonparametric exponentially weighted moving average (BEWMA-EX) control chart is proposed based on the exceedance statistics to detect the shifts in the location parameter. The performance of the proposed BEWMA-EX chart is compared with the multivariate sign EWMA (MSEWMA) control chart under some well-known bivariate distributions, such as bivariate normal, t, and gamma distributions. The BEWMA-EX chart outperforms the MSEWMA control chart in terms of run-length properties. To highlight the importance of the stated study, the BEWMA-EX chart is implemented on industrial engineering datasets related to coal power plant and aluminum electrolytic capacitor manufacturing processes. The findings are promising and support the simulated results.

Monitoring of Linear Profiles Using Linear Mixed Model in the Presence of Measurement Errors

In the application of control charts, most of the research in profile monitoring is based on accurate measurements. Measurement errors, however, often exist in many manufacturing and service environments. In this paper, we apply linear mixed models in the presence of measurement errors in fixed effects. We discuss three modified multivariate charts, namely Hotelling’s T2, multivariate exponential weighted moving average (MEWMA) control chart, and multivariate cumulative sum (MCUSUM) control chart. Performance comparisons are made in terms of the average run length (ARL) and average extra quadratic loss (AEQL). Finally, a real data example on healthcare expenditures is used to illustrate the implementation of the proposed monitoring schemes.

Sequential monitoring of high‐dimensional time series

AbstractIn the paper we derive new types of multivariate exponentially weighted moving average (EWMA) control charts which are based on the Euclidean distance and on the distance defined by using the inverse of the diagonal matrix consisting of the variances. The design of the proposed control schemes does not involve the computation of the inverse covariance matrix and, thus, it can be used in the high‐dimensional setting. The distributional properties of the control statistics are obtained and are used in the determination of the new control procedures. Within an extensive simulation study, the new approaches are compared with the multivariate EWMA control charts which are based on the Mahalanobis distance.