Multivariate Cumulative Sum Control Charts Research Articles

An equivalence between multivariate cumulative sum control charts

ABSTRACT We show that an auxiliary information based (AIB) cumulative sum (CUSUM) chart is equivalent to a special case of the multivariate CUSUM control chart proposed by J. D. Healy. Due to this equivalence, the limitations and restrictions of the AIB-CUSUM chart also apply to Healy’s multivariate CUSUM chart.

Bayesian multivariate control charts for multivariate profiles monitoring

ABSTRACT In many topical applications, the product’s quality can be well described in terms of statistical regression relationships between one or more response and a set of explanatory variables. In the literature, various types of regression models have been proposed for profile monitoring applications, and each of those regression models can be implemented and applied in its standard frequentist’s and its Bayesian variant. We formulate two popular Phase II multivariate cumulative sum control charts for monitoring multivariate linear profiles in terms of Bayesian regression models, and we show empirically that the resulting new Bayesian control charts perform better than the corresponding non-Bayesian control charts. For the comparative evaluation of the control charts we employ the average run length criterion. Moreover, we propose a new Bayesian approach, which we refer to as the informative prior generation method. The key idea of this method is to make use of historical datasets to generate informative prior distributions. The advantage of this method is that we do not ignore the historical data from Phase I. Instead we re-use it to construct informative prior distributions for Phase II monitoring. The applicability and the superiority of the proposed Bayesian control charts are illustrated through extensive simulation studies.

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.

Estimating risk probabilities for sickness from behavioural patterns to identify health challenges in dairy cows with multivariate cumulative sum control charts

Dairy cattle housing is characterised by increasing herd sizes and the need for assisting technical tools to monitor the cows’ health. This study investigated the combination of logistic regression models with multivariate cumulative sum (MCUSUM) control charts in healthmonitoring of dairy cattle. Sensor information of 618 cows with 791 lactations (138 438 cow days), nine behavioural variables were included as parts of the behavioural patterns: physical activity (“neck activity”, “leg activity”, “walking duration”), resting (“lying duration”, “standing duration”, “transitions from lying to standing”) and feeding (“feeding duration”, “rumination duration”, “inactivity duration”) behaviour. For each of these behavioural patterns, a logistic regression model with the health status (sick vs not sick) as a dependent variable was designed after a variable selection (herd level) based on the herd dataset with 618 cows (618 lactations; 115 547 cow days), which included the variables of each behaviour pattern and the stage of lactation nested in the number of lactations as explanatory variables. The explanatory variables were added stepwise to the model, with the final model being selected with respect to the lowest values of Akaike’s and Bayes’ information criteria. Each model was then applied to a dataset with 173 cows (22 891 cow days) at cow level, resulting in individual daily risk probabilities for getting sick. Thus, risk probabilities of each behavioural pattern were estimated and included in the MCUSUM control charts to identify cows at risk of disease. The performance of the MCUSUM control charts was cross-validated to identify the best fitting reference value k and the threshold value h. Alerts given within 5 days prior to diagnosis were counted as detected sicknesses. The performance resulted in a block sensitivity of 70.9–81.4%, specificity of 87.9–94.2% and a false-positive rate of 5.8–12.1%. The performance was confirmed while testing the entire algorithm resulting in a mean area under the receiver operating characteristics curve of 0.89. Calculating precision and the F1-score resulted in a precision of 49.0–60.9% (training: 48.8–63.5%) and an F1-score of 61.1–65.7% in testing (training: 61.0–67.0%). The precision-recall curve (PRC) was derived from precision and recall with an area under the PRC of 0.70 in training and testing. In summary, the present study was able to develop an algorithm showing good classification potential for the online monitoring of sickness behaviour.

An Adaptive EWMA Control Chart Based on Principal Component Method to Monitor Process Mean Vector

The special causes of variations, which is also known as a shift, can occur in a single or more than one related process characteristics. Statistical process control tools such as control charts are useful to monitor shifts in the process parameters (location and/or dispersion). In real-life situation, the shift is emerging in different sizes, and it is hard to identify it with classical control charts. Moreover, more than one process of characteristics required special attention because they must monitor jointly due to the association among them. This study offers two adaptive control charts to monitor the different sizes of a shift in the process mean vector. The novelty behind this study is to use dimensionally reduction techniques such as principal component analysis (PCA) and an adaptive method such as Huber and Bi-square functions. In brief, the multivariate cumulative sum control chart based on PCA is designed, and its plotting statistic is utilized as an input in the classical exponentially weighted moving average (EWMA) control chart. The run length (RL) properties of the proposed and other control charts are obtained by designing algorithms in MATLAB through a Monte Carlo simulation. For a single shift, the performance of the control charts is assessed through an average of RL, standard deviation of RL, and standard error of RL. Likewise, overall performance measures such as extra quadratic loss, relative ARL, and the performance comparison index are also used. The comparison reveals the superiority over other control charts. Furthermore, to emphasize the application process and benefits of the proposed control charts, a real-life example of the wind turbine process is included.

Multivariate cumulative sum control chart for compositional data with known and estimated process parameters

AbstractThis article uses the classic multivariate cumulative sum () chart scheme proposed by Crossier (1988) to present a new modified chart for compositional data (). For this purpose, the data are first transformed using isometric log‐ratio () coordinates representation to eliminate the constant sum constraint of . The ‐ control chart has been defined along with the performance measures of the proposed chart using the average run length (). Besides, the Markov chain method has been used to study the performance of the proposed chart. Assuming that the transformed data are normally distributed, the proposed ‐ charts have been compared with existing competitors such as ‐ and ‐ charts. The comparison shows that the proposed chart has better performance than the ‐ control charts, while the performance of the proposed chart is comparable with the ‐ chart. The effect of the estimated mean vector and variance‐co‐variance matrix on run‐length characteristics of the proposed ‐ control chart has also been studied in this paper. For the performance of ‐ with estimated parameters Monte Carlo simulation has been adopted. The effect of the number of variables , sample size , and subgroup size has also been studied on the data's upper control limit () and . In the end, two illustrative examples of the particle size distribution of plants and production of muesli are provided to represent the practical implementation of the ‐ chart.

The impact of a random vector with variables from normal and non‐normal distributions on multivariate control charts

AbstractMultivariate control charts are widely used to simultaneously monitor several quality characteristics for detecting the mean changes in manufacturing industry processes. A multivariate Shewhart, multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) are types of multivariate control charts which have been discussed extensively. For those control charts, a multivariate normal distribution is an important assumption that is used to describe a behavior of a set of quality characteristics. However, sometimes the normal assumption is violated. This research explores the sensitivity of an average run length (ARL), the standard deviation of run length (SDRL) and the shape of distribution of run length for multivariate Shewhart, MEWMA, and MCUSUM control charts when the distribution of some quality characteristics (or variables) in a random vector are not normal. The results show how robust or sensitive multivariate control charts are to violations of the multivariate normal assumption.

Evaluating and monitoring distribution network efficiency with multivariate process control methods

In this paper, we present a methodology, by utilising a nonparametric multivariate cumulative sum control chart and data envelopment analysis in tandem, to evaluate and monitor distribution network performance by simultaneously considering multiple performance factors and their interrelationships. While these methods were previously applied individually, we demonstrate that their combined use can provide more powerful results in performance assessment, monitoring and improvement. Our approach contributes to the domain of supply chain analytics and risk management as it can provide new insights into network performance based on real time and past data. We present the application of our approach on data obtained from The Dow Chemical Company.

Combining multivariate cumulative sum control charts with principal component analysis and partial least squares model to detect sickness behaviour in dairy cattle

The present study investigated the suitability of latent variables, generated by principal component analysis (PCA) and partial least squares regression (PLS), for the early detection of behavioural changes due to developing sickness. Therefore, behavioural information was collected from 480 milking cows between September 2018 and May 2019 on a German dairy farm. All animals were equipped with two sensor systems delivering information about the behavioural patterns resting, activity, feeding and rumination. In addition, performance parameters were provided by the milking parlour. The sensor information was combined in seven different ways to create scenarios (C1-C7) that are potentially available on-farm. Diagnoses, treatments and claw trimmings were provided by the farm’s veterinarian and claw trimmer. 298 cows with 44,865 days of observations were selected from all the milking cows in consideration of different data restrictions such as missing values; hence a data set was created that included 154 healthy and 144 sick cows with 300 sickness events. For the analyses, the data set was subdivided into ten training data sets (90% of the cows) which were used to set the necessary number of principal components (PCs) and PLS-factors, respectively, and ten testing (10% of the cows) data sets. After selecting PCs and PLS-factors from each scenario, the training data sets were used to train the reference value (k) and threshold value (h) of the multivariate cumulative sum control charts (MCUSUM). The best performing combination of k and h was then used for testing accuracy of the approaches. Hence, 2 (C1) to 6 (C7) PCs were chosen that jointly explained ≥ 70% of the data’s variance. Within the PLS approach, 3 (C1) to 10 (C7) PLS-factors were selected that explained the variation of the health status. The PCA-MCUSUMs showed consistent sickness detection as the block sensitivities showed a range from 69.9% to 77.2% (training: 71.0% to 75.8%) and specificities varied from 85.3% to 89.3% (training: 85.2% to 89.4%). The PLS-MCUSUMs showed some irregularities. Here, scenarios C5 and C7 detected > 83% and > 94% sickness events in training and testing, thus causing a decrease of specificities and therefore increased false positive rates of ≥ 20%. In summary, both approaches could be applicable in practice, although the results of the PCA are more consistent and could be more reliable in comparison to the PLS approach.

Advanced multivariate cumulative sum control charts based on principal component method with application

AbstractExisting multivariate cumulative sum (MCUSUM) control charts involve entire associated variables of a process to monitor variations in the mean vector. In this study, we have offered MCUSUM control charts with principal component method (PCM). The proposed MCUSUM control charts with PCM capture the whole process variations using fewer latent variables (principal components) while preserving as much data variability as possible. To show the significance of proposed MCUSUM control charts with PCM, various performance measures are considered including average run length, extra quadratic loss, relative average run length, and performance comparison index. Furthermore, performance measures are calculated through advanced Monte Carlo simulation method to explore the behavior of proposed MCUSUM control charts and to conduct comparative analysis with existing models. Results revealed that proposed MCUSUM control charts with PCM are efficient to detect variations timely by involving smaller number of principal components instead of considering entire associated variables. Also, proposed MCUSUM control charts have the ability to accommodate the features of existing control charts, which are illustrated as the special cases. Besides, to highlight the implementation mechanism and advantages of proposed MCUSUM control charts with PCM, a real‐life example from wind turbine process is included.

Variable selection for monitoring sickness behavior in lactating dairy cattle with the application of control charts

The present observational study investigated the application of multivariate cumulative sum (MCUSUM) control charts by including variables selected by principal component analysis and partial least squares (PLS) regression to identify sickness behavior in dairy cattle. Therefore, sensor information (24 variables) was collected from 480 milking cows on a German dairy farm between September 2018 and December 2019. These variables were gathered in potentially different scenarios on farm. In total, data from 749 animals were available for evaluation. Variables were chosen based on the information of 499 cows (62 healthy; 437 sick) with 93,598 observations. The available diagnoses were collected together to form 1,025 sickness events. Hence, the different numbers of selected variables were included into the MCUSUM control charts. The performance of the MCUSUM control charts was evaluated by a 10-fold cross validation; hence, 90% of the original data set (749 cows) represented the training data, and the remaining 10% was used to test the training results. On average, the 10 training data sets included 124,871 observations with 1,392 sickness events, and the 10 testing data sets included, on average, 13,704 observations with 153 sickness events. The MCUSUM generated from the variables selected by principal component analysis showed comparable results in training and testing in all scenarios; therefore, 70.0 to 97.4% of the sickness events were detected. The false-positive rates ranged from 8.5 to 29.6%, and thus they created at least 2.6 false-positive alerts per day in testing. The variables selected by the PLS regression approach showed comparable sickness detection rates (70.0-99.9%) as well as false-positive rates (8.2-62.8%) in most scenarios. The best performing scenario produced 2.5 false-positive alerts in testing. Summarizing, both approaches showed potential for practical implementation; however, the PLS variable selection approach showed fewer false positives. Therefore, the PLS regression approach could generate a more reliable sickness detection algorithm, if combined with MCUSUM control charts, and considered for practical implementation.

The effects of constructed bivariate copulas on multivariate control charts effectiveness

AbstractThe average control chart monitors the shifts in the process. The familiar multivariate control charts are used to detect the mean vector of the process such as multivariate cumulative sum (MCUSUM) and Hotelling's T2 control charts. In this paper, the effects of constructing bivariate copulas on multivariate control charts, that is, MCUSUM and Hotelling's T2 control charts are intensively investigated when observations are drawn from the exponential distribution. Moreover, the dependence levels of observations are classified to be weak, moderate, and strong in both positive and negative values by Kendall's tau. The numerical results were obtained by Monte Carlo simulation to explore the average run length (ARL). The simulation results show that the performance of Hotelling's T2 control chart is superior to the MCUSUM control chart for all shifts in the mean vector of process. Furthermore, from applying the presented control chart to two sets of real data, data set of the strength of 1.5 cm glass fibers measured at the National Physical Laboratory, England and data set of the strength of glass of the aircraft window, it was found that for a small shift (), the MCUSUM control chart is better than Hotelling's T2 control chart.

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.

A multivariate CUSUM control chart for monitoring Gumbel's bivariate exponential data

AbstractExponentially distributed data are commonly encountered in high‐quality processes. Control charts dedicated to the univariate exponential distribution have been extensively studied by many researchers. In this paper, we investigate a multivariate cumulative sum (MCUSUM) control chart for monitoring Gumbel's bivariate exponential (GBE) data. Some tables are provided to determine the optimal design parameters of the proposed MCUSUM GBE chart. Furthermore, both zero‐state and steady‐state properties of the proposed MCUSUM GBE chart for the raw and the transformed GBE data are compared with the multivariate exponentially weighted moving average (MEWMA) chart and the paired individual cumulative sum (CUSUM) chart. The results show that the proposed MCUSUM GBE chart outperforms the other two types of control charts for most shift domains. In addition, an extension to Gumbel's multivariate exponential (GME) distribution is also investigated. Finally, an illustrative example is provided in order to explain how the proposed MCUSUM GBE chart can be implemented in practice.

Cement Compressive Strength Control Using CUSUM and MCUSUM Control Chart

Compressive strength is one of the test factors used to determine whether cement production is in a controlled state or not. Portland type composite cement or PCC is the cement that is widely used in infrastructure development. The Cement of 3-days compressive strength, 7-days compressive strength, and 28-days compressive strength are the variables that will be controlled in this study. The normal distribution test and correlation test show that the data on each variable is normally distributed, and each variable has a strong correlation. Univariate cement control using the cumulative sum control chart (CUSUM) and multivariate control using a multivariate cumulative sum (MCUSUM) control chart is performed to obtain the best control results. Correlated variables show that control using a multivariate control chart results in fewer outs of control observations compared to a univariate control chart. This explains that the MCUSUM control chart is more sensitive than the CUSUM control chart in controlling observations of data out of control.

Dynamic Field Monitoring Based on Multitask Learning in Sensor Networks.

Field monitoring serves as an important supervision tool in a variety of engineering domains. An efficient monitoring would trigger an alarm timely once it detects an out-of-control event by learning the state change from the collected sensor data. However, in practice, multiple sensor data may not be gathered appropriately into a database for some unexpected reasons, such as sensor aging, wireless communication failures, and data reading errors, which leads to a large number of missing data as well as inaccurate or delayed detection, and poses a great challenge for field monitoring in sensor networks. This fact motivates us to develop a multitask-learning based field monitoring method in order to achieve an efficient detection when considerable missing data exist during data acquisition. Specifically, we adopt a log likelihood ratio (LR)-based multivariate cumulative sum (MCUSUM) control chart given spatial correlation among neighboring regions within the monitored field. To deal with the missing data problem, we integrate a multitask learning model into the LR-based MCUSUM control chart in the sensor network. Both simulation and real case studies are conducted to validate our proposed approach and the results show that our approach can achieve an accurate and timely detection for an out-of-control state when a large number of missing data exist in the sensor database. Our model provides an effective field monitoring strategy for engineering applications to accurately and timely detect the products with abnormal quality during production and reduce product losses.

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.

A wavelet‐based nonparametric CUSUM control chart for autocorrelated processes with applications to network surveillance

AbstractStatistical process control (SPC) has natural applications in data network surveillance. However, network data are commonly autocorrelated, which presents challenges to the basic SPC methods. Most existing SPC methods for correlated data assume parametric models to account for the correlation structure within the data. Those model assumptions can be difficult to justify in practice. In this paper, we propose a nonparametric cumulative sum (CUSUM) control chart for autocorrelated processes. In our proposed approach, we incorporate a wavelet decomposition and a nonparametric multivariate CUSUM control chart to obtain a robust procedure for autocorrelated processes without distribution assumptions. Extensive simulations show that the procedure appropriately controls the in‐control average run length and also has good sensitivity for detecting location shifts.

An effective multivariate control chart for detecting small mean shifts using support vector data description

Conventional multivariate cumulative sum control charts are more sensitive to small shifts than [Formula: see text] control charts, but they cannot get the knowledge of manufacturing process through the learning of in-control data due to the characteristics of their own structures. To address this issue, a modified multivariate cumulative sum control chart based on support vector data description for multivariate statistical process control is proposed in this article, which is named [Formula: see text] control chart. The proposed control chart will have both advantages of the multivariate cumulative sum control charts and the support vector data description algorithm, namely, high sensitivities to small shifts and learning abilities. The recommended values of some key parameters are also given for a better application. Based on these, a bivariate simulation experiment is conducted to evaluate the performance of the [Formula: see text] control chart. A real industrial case illustrates the application of the proposed control chart. The results also show that the [Formula: see text] control chart is more sensitive to small shifts than other traditional control charts (e.g. [Formula: see text] and multivariate cumulative sum) and a D control chart based on support vector data description.

Weighted adaptive multivariate CUSUM control charts

AbstractAn adaptive multivariate cumulative sum (AMCUSUM) control chart has received considerable attention because of its ability to dynamically adjust the reference parameter whereby achieving a better performance over a range of mean shifts than the conventional multivariate cumulative sum (CUSUM) charts. In this paper, we introduce a progressive mean–based estimator of the process mean shift and then use it to devise new weighted AMCUSUM control charts for efficiently monitoring the process mean. These control charts are easy to design and implement in a computerized environment compared with their existing counterparts. Monte Carlo simulations are used to estimate the run‐length characteristics of the proposed control charts. The run‐length comparison results show that the weighted AMCUSUM charts perform substantially and uniformly better than the classical multivariate CUSUM and AMCUSUM charts in detecting a range of mean shifts. An example is used to illustrate the working of existing and proposed multivariate CUSUM control charts.