Multivariate Cumulative Sum Charts Research Articles

Evaluating multivariate charts: A first‐to‐signal criterion approach

AbstractThis study advances the application of the first‐to‐signal (FTS) concept to evaluate the effectiveness of multivariate charts when detecting shifts in the mean vector of a multivariate normally distributed process. Using Monte Carlo simulations, we estimate steady‐state average run‐length profiles and the probabilities of multivariate charts being the FTS an out‐of‐control state. Notably, our findings indicate that while the multivariate cumulative sum (MCUSUM) chart serves as a viable alternative to the multivariate exponentially weighted moving average (MEWMA) chart when detecting small shifts in the process mean vector, the latter surpasses the former in detecting moderate‐to‐large shifts. In addition, the Hotelling's chart outperforms the MEWMA and MCUSUM charts when detecting large shifts.

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.

A new multivariate CUSUM chart for monitoring of covariance matrix with individual observations under estimated parameter

AbstractMultivariate charts for process dispersion detect changes in the variance‐covariance matrix of a process. Most of the existing multivariate charts for monitoring the dispersion of individual observations were designed based on exponentially weighted moving average (EWMA) charting schemes. However, an alternative to the EWMA scheme is the cumulative sum (CUSUM) control chart, which has proven to be better in some cases. In the last decades, few studies have been conducted on methods based on multivariate CUSUM (MCUSUM) schemes to monitor the covariance matrix of individual observations. Consequently, we propose a new MCUSUM dispersion chart. Besides, most of the existing methods have been developed by assuming that the process parameters are known and that the process distribution is normal; these assumptions are not always true in practice. Hence, we compare the performance of the proposed chart and its counterparts based on the estimation effects under normal and non‐normal distributions. The results show that the proposed chart outperforms the other charts in terms of minor shifts in the process. Similarly, the proposed chart is the most robust to the normality assumption among the compared charts. The average value of the conditional average run length was used as the performance measure. Finally, the proposed method was also implemented with a simulated dataset to support the stated proposal findings.

A novel local temperature change detection approach in a 3D thermal field

ABSTRACT Local thermal field monitoring is of great importance in industrial engineering practice. In view of the complex thermodynamics of a 3D thermal field, the local temperature change beneath the global thermal field is relatively challenging to detect using existing approaches. To fill this gap, we propose a thermal field monitoring method that decomposes the dynamic thermal field into global trend and local variability parts. Specifically, a universal kriging model that characterises the spatial correlation at each time epoch is developed to model the local variability using sensing data. To efficiently monitor the thermal field changes, a multivariate cumulative sum chart that uses the estimated parameters is designed. An actual case study of the thermal field of a granary is used to validate the performance of the proposed method.

A new multivariate CUSUM chart using principal components with a revision of Crosier's chart

ABSTRACTIn this article, we introduce a new multivariate cumulative sum chart, where the target mean shift is assumed to be a weighted sum of principal directions of the population covariance matrix. This chart provides an attractive performance in terms of average run length (ARL) for large-dimensional data and it also compares favorably to existing multivariate charts including Crosier's benchmark chart with updated values of the upper control limit and the associated ARL function. In addition, Monte Carlo simulations are conducted to assess the accuracy of the well-known Siegmund's approximation of the average ARL function when observations are normal distributed. As a byproduct of the article, we provide updated values of upper control limits and the associated ARL function for Crosier's multivariate CUSUM chart.

Vibrational spectroscopy and multivariate control charts: A new strategy for monitoring the stability of captopril in the pharmaceutical industry

The quality control of pharmaceutical production has focused on controlling impurity formation and product degradation. High temperatures, humidity, and the presence of hygroscopic excipients affect the stability of captopril. These conditions accelerate the degradation of captopril and favor the formation of captopril disulfide. In this study, multivariate control charts were combined with vibrational spectroscopy to monitor the stability of captopril. Fifty spectra of different batches of normal captopril were recorded in the mid-infrared (MIR) and near-infrared (NIR) spectral regions, and they were used in the training phase of multivariate charts. The validation phase consisted of 20 lots of normal captopril and 10 lots past their expiration dates. A stability study was performed by subjecting one batch of captopril to a temperature of 40±2°C and relative humidity of 75±5% for a period of six months in a climatic chamber. The captopril samples were collected weekly from the climatic chamber during this period, for a total of 24 spectra. Principal component analysis was performed on the MIR and NIR datasets, and the scores of the principal components were used in the construction of Hotelling's T2 chart, whereas the squared prediction error (SPE) chart was built with the residuals of the model. In the validation phase, the SPE chart adequately identified expired lots in these two infrared regions. A strategy based on the successive projections algorithm in combination with a multivariate cumulative sum chart was developed to improve the sensitivity of Hotelling's T2 charts in the MIR region in the monitoring step of the stability study. Our results indicate that vibrational spectroscopy combined with control charts is a useful tool for monitoring changes in the stability pattern of captopril.

Implementation of multivariate cumulative sum control charts in mastitis and lameness monitoring

This study analyzed the methodology and applicability of multivariate cumulative sum (MCUSUM) charts for early mastitis and lameness detection. Data used were recorded on the Karkendamm dairy research farm, Germany, between August 2008 and December 2010. Data of 328 and 315 cows in their first 200d in milk were analyzed for mastitis and lameness detection, respectively. Mastitis as well as lameness was specified according to veterinary treatments. Both diseases were defined as disease blocks. Different disease definitions for mastitis and lameness (2 for mastitis and 3 for lameness) varied solely in the sequence length of the blocks. Only the days before the treatment were included in the disease blocks. Milk electrical conductivity, milk yield, and feeding patterns (feed intake, number of trough visits, and feeding time) were used for the recognition of mastitis. Pedometer activity and feeding patterns were used for lameness detection. To exclude biological trends and obtain independent observations, the values of each input variable were either preprocessed by wavelet filters or a multivariate vector autoregressive model. The residuals generated between the observed and filtered or observed and forecast values, respectively, were then transferred to a classic or self-starting MCUSUM chart. The combination of the 2 preprocessing methods with each of the 2 MCUSUM sum charts resulted in 4 combined monitoring systems. For mastitis as well as lameness detection requiring a block sensitivity of at least 70%, all 4 of the combined monitoring systems used revealed similar results within each of the disease definitions. Specificities of 73 to 80% and error rates of 99.6% were achieved for mastitis. The results for lameness showed that the definitions used obtained specificities of up to 81% and error rates of 99.1%. The results indicate that the monitoring systems with these study characteristics have appealing features for mastitis and lameness detection. However, they are not yet directly applicable for practical implementations.

The performance of multivariate CUSUM control charts with estimated parameters

In this paper, we study the effect of estimating the vector of means and the variance–covariance matrix on the performance of two of the most widely used multivariate cumulative sum (CUSUM) control charts, the MCUSUM chart proposed by Crosier [Multivariate generalizations of cumulative sum quality-control schemes, Technometrics 30 (1988), pp. 291–303] and the MC1 chart proposed by Pignatiello and Runger [Comparisons of multivariate CUSUM charts, J. Qual. Technol. 22 (1990), pp. 173–186]. Using simulation, we investigate and compare the in-control and out-of-control performances of the competing charts in terms of the average run length measure. The in-control and out-of-control performances of the competing charts deteriorate significantly if the estimated parameters are used with control limits intended for known parameters, especially when only a few Phase I samples are used to estimate the parameters. We recommend the use of the MC1 chart over that of the MCUSUM chart if the parameters are estimated from a small number of Phase I samples.

Cumulative Sum Control Charts for Simultaneously Monitoring Means and Variances of Multiple Quality Variables

Multivariate cumulative sum (CUSUM) control charts for simultaneously monitoring both means and variances under multivariate normal process are investigated. Performances of multivariate CUSUM schemes are evaluated for matched fixed sampling interval (FSI) and variable sampling interval (VSI) features in terms of average time to signal (ATS), average number of samples to signal (ANSS). Multivariate Shewhart charts are also considered to compare the properties of multivariate CUSUM charts. Numerical results show that presented CUSUM charts are more efficient than the corresponding Shewhart chart for small or moderate shifts and VSI feature with two sampling intervals is more efficient than FSI feature. When small changes in the production process have occurred, CUSUM chart with small reference values will be recommended in terms of the time to signal.

A Comparison on the MRL Performances of Optimal MEWMA and Optimal MCUSUM Control Charts

The MEWMA (called the multivariate exponentially weighted moving average) chart and the MCUSUM (called the multivariate cumulative sum) chart are used in process monitoring when a quick detection of small or moderate shifts in the mean vector is desired. The primary objective of this study is to compare the performances of the optimal MEWMA and optimal MCUSUM charts based on their median run length (MRL) profiles. The number of quality characteristics considered is p = 2. Two cases are studied, i.e., Case 1 (a shift in only one variable) and Case 2 (a shift in two variables). A Monte Carlo simulation is conducted using Statistical Analysis Software (SAS) to study and compare the MRL performances for various magnitudes of mean shifts when the process is normally distributed. Overall, the results show that the MRL performances of the MEWMA and MCUSUM charts are comparable.

Implementation of multivariate control charts in a clinical setting

In most clinical monitoring cases there is a need to track more than one quality characteristic. If separate univariate charts are used, the overall probability of a false alarm may be inflated since correlation between variables is ignored. In such cases, multivariate control charts should be considered. This paper considers the implementation and performance of the T(2), multivariate exponentially weighted moving average (MEWMA) and multivariate cumulative sum (MCUSUM) charts in light of the challenges faced in clinical settings. We discuss how to handle incomplete records and non-normality of data, and we provide recommendations on chart selection. Our discussion is supported by a case study involving the monitoring of radiation delivered to patients undergoing diagnostic coronary angiogram procedures at St Andrew's War Memorial Hospital, Australia. We also perform a simulation study to investigate chart performance for various correlation structures, patterns of mean shifts, amounts of missing data and methods of imputation. The MEWMA and MCUSUM charts detect small to moderate shifts quickly, even when the quality characteristics are uncorrelated. The T(2) chart performs less well overall, although it is useful for rapid detection of large shifts. When records are incomplete, we recommend using multiple imputation.

Support vector regression based residual MCUSUM control chart for autocorrelated process

Traditional control charts assume that processes are serially independent, and autocorrelation among variables makes them unreliable. To handle this problem alternative charts estimate the time series structure of the process and use residuals for control. While in previous studies, estimation is performed using classical statistical methods or artificial neural networks, this study proposes to apply support vector regression (SVR) method for construction of a residuals Multivariate Cumulative Sum (MCUSUM) control chart, for monitoring changes in the process mean vector. Using simulated data, analysis and comparison of the proposed control chart with other charts show that SVR-based control chart is more effective in detecting small shifts in the mean vector. This fact makes the proposed chart a very promising method since the MCUSUM chart is, in practice, designed to detect small shifts in the process parameters.

Multivariate multiple sampling charts

A new multivariate statistical process control scheme, the Multivariate Multiple Sampling (MMS) control chart scheme, is proposed in this paper. A MMS chart is a multivariate extension of a double sampling X-bar control chart with at least two sampling stages. In the paper, a statistical design optimization procedure to design the MMS chart is presented and the performance of the MMS chart is investigated. The statistical efficiency in terms of average run length of the MMS chart is compared with that of the Hotelling chart both with and without variable sampling schemes, a multivariate CUMulative SUM (CUSUM) chart, and a multivariate Exponentially Weighted Moving Average (EWMA) chart. The ability of the MMS chart to handle the worst-case scenario is also investigated and compared with that of the multivariate EWMA and CUSUM charts. The results of the investigation show that even with only two sampling stages, the MMS chart provides an improvement in efficiency in detecting small shifts over the Hotelling chart without variable sampling schemes. When the number of sampling stages is equal to two, the MMS chart is better in detecting large shifts and the multivariate EWMA and CUSUM charts are better in detecting relatively small shifts. As the number of sampling stages is increased beyond two, the improvement in sensitivity of the MMS chart in detecting the small shifts increases. When the number of sampling stages ≥3, the MMS chart begins to give a better performance than a Hotelling chart with a variable sampling scheme for small shifts and is also better than a multivariate EWMA chart for both small and large shifts. As the number of sampling stages ≥4, the MMS chart begins to give a better performance than a multivariate CUSUM chart for both small and large shifts. The results of the investigation also show that the MMS chart outperforms the multivariate EWMA and CUSUM charts in the worst-case scenario situation.

A Multivariate Exponentially Weighted Moving Average Control Chart

A multivariate extension of the exponentially weighted moving average (EWMA) control chart is presented, and guidelines given for designing this easy-to-implement multivariate procedure. A comparison shows that the average run length (ARL) performance of this chart is similar to that of multivariate cumulative sum (CUSUM) control charts in detecting a shift in the mean vector of a multivariate normal distribution. As with the Hotelling's χ2 and multivariate CUSUM charts, the ARL performance of the multivariate EWMA chart depends on the underlying mean vector and covariance matrix only through the value of the noncentrality parameter. Worst-case scenarios show that Hotelling's χ2 charts should always be used in conjunction with multivariate CUSUM and EWMA charts to avoid potential inertia problems. Examples are given to illustrate the use of the proposed procedure.

Cusum quality control-multivariate approach

A new multivariate approach to quality control is presented. On the basis of sequential probability ratio tests, a multivariate cumulative sum chart is derived. Using an approximation to the noncentral x2 distribution, a linear decision rule is obtained.