Simultaneous detection of shifts in the marginal and dependence terms for multivariate quality control

A control chart for detecting shifts in the variance of a process is developed for the case where the nominal value of the variance is unknown. As our approach does not require that the in-control variance be known a priori, it avoids the need for a lengthy Phase I data-gathering step before charting can begin. The method is a variance-change-point model, based on the likelihood ratio test for a change in variance with the conventional Bartlett correction, adapted for repeated sequential use. The chart may be used alone in settings where one wishes to monitor one-degree-of-freedom chi-squared variates for departure from control; or it may be used together with a parallel change-point methodology for the mean to monitor process data for shifts in mean and/or variance. In both the solo use and as the scale portion of a combined scheme for monitoring changes in mean and/or variance, the approach has good performance across the range of possible shifts.

Process monitoring using PCA-based GLR methods: A comparative study

An integrated MEWMA-ANN scheme towards balanced monitoring and accurate diagnosis of bivariate process mean shifts

Monitoring multivariate process variability via eigenvalues

Quality control with multivariate data the multivariate normal distribution in quality control quality control with externally assigned targets quality control with internal targets - multivariate process capability studies quality control with targets from a reference sample analyzing data with multivariate control charts detection of out-of-control characteristics the statistical tolerance regions approach multivariate quality control with units in batches applications of principal components additional graphical techniques for multivariate quality control implementing multivariate quality control appendix 1 - MINITAB (TM) macros for multivariate quality control appendix 2 - the data from case studies appendix 3 - review of matrix algebra for statistics with MINIATB (TM) computations.

Multivariate quality characteristics are often monitored using a single statistic or a few statistics. However, it is difficult to determine the causes of an out-of-control signal based on a few summary statistics. Therefore, if a control chart for the mean detects a change in the mean, the quality engineer needs to determine which means shifted and the directions of the shifts to facilitate identification of root causes. We propose a Bayesian approach that gives a direct answer to this question. For each mean, an indicator variable that indicates whether the mean shifted upward, shifted downward, or remained unchanged is introduced. Prior distributions for the means and indicators capture prior knowledge about mean shifts and allow for asymmetry in upward and downward shifts. The mode of the posterior distribution of the vector of indicators or the mode of the marginal posterior distribution of each indicator gives the most likely scenario for each mean. Evaluation of the posterior probabilities of all possible values of the indicators is avoided by employing Gibbs sampling. This renders the computational cost more affordable for high-dimensional problems. This article has supplementary materials online.

On line detection of mean and variance shift using neural networks and support vector machine in multivariate processes

Differential expression analysis is crucial in genomics, yet existing methods primarily focus on detecting mean shifts. Variance shifts in gene expression are well-documented in studies of cellular signaling pathways, and more recently they have characterized aging, thus motivating the need for flexible detection approaches that include tests of expression variance changes. In this work, we present QRscore (Quantile Rank Score), a general method for detecting distributional shifts in gene expression by extending the Mann-Whitney test into a flexible family of rank-based tests. Here, we focus on implementing QRscore to detect shifts in mean and variance in gene expression, using weights designed from negative binomial (NB) and zero-inflated negative binomial (ZINB) models to combine the strengths of parametric and non-parametric approaches. We show through simulations that QRscore not only achieves high statistical power while controlling the false discovery rate (FDR), but also outperforms existing methods in detecting variance shifts and mean shifts. Applying QRscore to bulk RNA-seq data from the Genotype-Tissue Expression (GTEx) project, we identified numerous differentially dispersed genes and differentially expressed genes across 33 tissues. Notably, many genes have significant variance shifts but non-significant mean shifts. QRscore augments the genome bioinformatics toolkit by offering a powerful and flexible approach for differential expression analysis. QRscore is available in R, at https://github.com/songlab-cal/QRscore.

When performing quality control in a situation in which measures are made of several possibly related variables, it is desirable to use methods that capitalize on the relationship between the variables to provide controls more sensitive than those that may be made on the variables individually. The most common methods of multivariate quality control that assess the vector of variables as a whole are those based on the Hotelling T 2 between the variables and the specification vector. Although T 2 is the optimal single-test statistic for a general multivariate shift in the mean vector, it is not optimal for more structured mean shifts-for example, shifts in only some of the variables. Measures based on quadratic forms (like T 2) also confound mean shifts with variance shifts and require quite extensive analysis following a signal to determine the nature of the shift. This article proposes Shewhart and cumulative sum (CUSUM) controls based on the vector Z of scaled residuals from the regression of each varia...

A new methodology is proposed in this paper to both monitor an overall mean shift and classify the states of a multivariate quality control system. Based on the Bayesian rule (Montgomery, Introduction to statistical quality control, 5th edn. Wiley, New York, USA, 2005), the belief that each quality characteristic is in an out-of-control state is first updated in an iterative approach and the proof of its convergence is given. Next, the decision-making process of the detection and classification the process mean shift is modeled. Numerical examples by simulation are provided in order to understand the proposed methodology and to evaluate its performance. Moreover, the in-control and out-of-control average run length (Montgomery, Introduction to statistical quality control, 5th edn. Wiley, New York, USA, 2005) of the proposed method are compared with the ones from the well-known Multivariate Cumulative Sum (MCUSUM), Multivariate Exponentially Weighted Moving Average (MEWMA) and Hotelling T2 methods in different scenarios of mean shifts. The results of the simulation study show that the proposed methodology performs better than other methods for all shifts of the process mean. Additionally, the estimated probabilities of making correct classifications by the proposed approach are encouraging.

We propose a composite multivariate quality control (CMQC) system to control simultaneously measured variables. This system is designed to detect unacceptable trends and systematic error in one or more variables, unacceptable random error in one or more variables, and unacceptable changes in the correlation structure in any pair of variables. It is also designed to be tolerant of missing data, to be capable of rejecting as few as one or as many as all variables in a run, and to provide the analyst with control statistics and graphics that logically relate to sources of analytical error. Quality control rules for univariate, multivariate, and correlation conditions are incorporated in the system, as are plots displaying CMQC statistic values and control limits for univariate, multivariate, and correlation parameters. We also discuss advantages of the CMQC over the T2 and principal component multivariate quality control methods. We demonstrate the CMQC procedure using data from a laboratory process in which 40 variables were measured during 40 characterization runs and 23 runs analyzing unknowns.

In a production process, when quality of the product depends on more than one characteristic, 'Multivariate Quality Control' (MQC) techniques are efficiently used. Many MQC techniques have been developed to control the multivariate variable processes, but no much work has been reported to control the multivariate attribute processes. In this article, to detect a change in the vector of fraction non-conforming, we develop 'Likelihood Ratio based Multi-Attribute Control Chart' (LR-MACC) using the exact joint distribution and the LR-test under multinomial setup. It is verified that, in some situations, LR-MACC is superior to the MNP chart proposed by Lu et al.7 When MACC gives a signal, the attributes responsible are not readily identifiable. Therefore, a procedure to detect the responsible attributes is also developed.

We compared the interannual variability of annual daily maximum and minimum extreme water levels in Lake Ontario and the St Lawrence River (Sorel station) from 1918 to 2010, using several statistical tests. The interannual variability of annual daily maximum extreme water levels in Lake Ontario is characterized by a positive long‐term trend showing two shifts in mean (1929–1930 and 1942–1943) and a single shift in variance (in 1958–1959). In contrast, for the St Lawrence River, this interannual variability is characterized by a negative long‐term trend with a single shift in mean, which occurred in 1955–1956. As for annual daily minimum extreme water levels, their interannual variability shows no significant long‐term change in trend. However, for Lake Ontario, the interannual variability of these water levels shows two shifts in mean, which are synchronous with those for maximum water levels, and a single shift in variance, which occurred in 1965–1966. These changes in trend and stationarity (mean and variance) are thought to be due to factors both climatic (the Great Drought of the 1930s) and human (digging of the Seaway and construction of several dams and locks during the 1950s). Despite this change in means and variance, the four series are clearly described by the generalized extreme value distribution. Finally, annual daily maximum and minimum extreme water levels in the St Lawrence and Lake Ontario are negatively correlated with Atlantic multidecadal oscillation over the period from 1918 to 2010. Copyright © 2013 John Wiley & Sons, Ltd.

A review of the literature on control charts for multivariate quality control (MQC) is given, with a concentration on developments occurring since the mid-1980s. Multivariate cumulative sum (CUSUM) control procedures and a multivariate exponentially weighted moving average (EWMA) control chart are reviewed and recommendations are made regarding their use. Several recent articles that give methods for interpreting an out-of-control signal on a multivariate control chart are analyzed and discussed. Other topics such as the use of principal components and regression adjustment of variables in MQC, as well as frequently used approximations in MQC, are discussed.

In this study, we compared the frequency and timing of drought and wetness indices of annual mean water levels in the North American Great Lakes as they relate to teleconnection indices over the period from 1918 to 2012. In terms of timing, drought occurred in the Great Lakes watershed during the 1920, 1930 and 2000 decades, and was very intense in the East during the 1930’s and in the West during the 2000 decade. The main cause of extreme drought episodes in the 1920’s and 1930’s was a decrease in precipitation, while the 2000 decade drought is thought to be caused by increased water temperature (enhanced evaporation) due to a significant decrease in winter ice cover. The 1970 and 1980 decades were very wet over the whole watershed as a result of increased precipitation in the region. The succession of these dry and wet episodes did not have the same impacts on the stationarity of annual mean water levels in the five Great Lakes. Lake Superior shows an abrupt shift in mean in 1999, but a smoothed shift in variance since 1994, whereas Lake Erie shows four abrupt shifts in mean. Lake Ontario also shows the two first abrupt shift in mean and one abrupt change in variance. Extreme drought indices are negatively correlated with the North Atlantic Oscillation (NAO) for the two shallowest lakes (Ontario and Erie). In contrast, extreme wetness indices are positively correlated with PDO (positive correlation) and SOI (negative correlation) for Lake Superior only.