Diagnosis Aids in Multivariate Multiple Linear Regression Profiles Monitoring

Diagnosis aids in addition to detecting the out-of-control state is an important issue in multivariate multiple linear regression profiles monitoring; because a large number of parameters and profiles in this structure are involved. In this paper, we specifically concentrate on identification of profile(s) and parameter(s) which have changed during the process in multivariate multiple linear regression profiles structure in Phase II. We demonstrate the effectiveness of our proposed approaches through Monte Carlo simulations and a real case study in terms of accuracy percent.     

Phase I Monitoring of Polynomial Profiles

In some statistical process control applications, quality of a process or product is characterized by a relationship between two or more variables which is referred to as profile. In many practical situations, a profile can be modeled as a polynomial regression. In this article, three methods are developed for monitoring polynomial profiles in Phase I. Their performance is evaluated using power criterion. Furthermore, a method based on likelihood ratio test is developed to identify the location of shifts. Numerical simulation is used to evaluate the performance of the developed method.     

Change Point Estimation of Multivariate Linear Profiles Under Linear Drift

In this paper, maximum likelihood estimators (MLE) for both step and linear drift changes in the regression parameters of multivariate linear profiles are developed. Performance of the proposed estimators is compared under linear drift changes in the regression parameters when a combined MEWMA and Chi-square control charts method signals an out-of-control condition. The effect of smoothing parameter of MEWMA control charts, missing data, and multiple drift changes on the performance of the both estimators is also evaluated. The application of the proposed estimators is also investigated thorough a numerical example resulted from a real case.     

Diagnosis of Multivariate Control Chart Signal Based on Dummy Variable Regression Technique

Abstract

It is common to monitor several correlated quality characteristics using the Hotelling's T ² statistic. However, T ² confounds the location shift with scale shift and consequently it is often difficult to determine the factors responsible for out of control signal in terms of the process mean vector and/or process covariance matrix. In this paper, we propose a diagnostic procedure called ‘D-technique’ to detect the nature of shift. For this purpose, two sets of regression equations, each consisting of regression of a variable on the remaining variables, are used to characterize the ‘structure’ of the ‘in control’ process and that of ‘current’ process. To determine the sources responsible for an out of control state, it is shown that it is enough to compare these two structures using the dummy variable multiple regression equation. The proposed method is operationally simpler and computationally advantageous over existing diagnostic tools. The technique is illustrated with various examples.     

Identifying Variables Contributing to Outliers in Phase I

When a process is monitored with a T ² control chart in a Phase II setting, the MYT decomposition is a valuable diagnostic tool for interpreting signals in terms of the process variables. The decomposition splits a signaling T ² statistic into independent components that can be associated with either individual variables or groups of variables. Since these components are T ² statistics with known distributions, they can be used to determine which of the process variable(s) contribute to the signal. However, this procedure cannot be applied directly to Phase I since the distributions of the individual components are unknown. In this article, we develop the MYT decomposition procedure for a Phase I operation, when monitoring a random sample of individual observations and identifying outliers. We use a relationship between the T ² statistic in Phase I with the corresponding T ² statistic resulting when an observation is omitted from this sample to derive the distributions of these components and demonstrate the Phase I application of the MYT decomposition.     

Maximum Value of Hotelling's T 2 Statistics Based on the Successive Differences Covariance Matrix Estimator

In statistical process control applications, the multivariate T ² control chart based on Hotelling's T ² statistic is useful for detecting the presence of special causes of variation. In particular, use of the T ² statistic based on the successive differences covariance matrix estimator has been shown to be very effective in detecting the presence of a sustained step or ramp shift in the mean vector. However, the exact distribution of this statistic is unknown. In this article, we derive the maximum value of the T ² statistic based on the successive differences covariance matrix estimator. This distributional property is crucial for calculating an approximate upper control limit of a T ² control chart based on successive differences, as described in Williams et al. (2006 Williams , J. D. , Woodall , W. H. , Birch , J. B. , Sullivan , J. H. ( 2006 ). On the distribution of T ² statistics based on successive differences . J. Qual. Technol. 38 : 217 – 229 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

Using a Truncated C p Statistic for Variable Selection in Multiple Linear Regression

In multiple linear regression analysis each lower-dimensional subspace L of a known linear subspace M of ? ⁿ corresponds to a non empty subset of the columns of the regressor matrix. For a fixed subspace L, the C _p statistic is an unbiased estimator of the mean square error if the projection of the response vector onto L is used to estimate the expected response. In this article, we consider two truncated versions of the C _p statistic that can also be used to estimate this mean square error. The C _p statistic and its truncated versions are compared in two example data sets, illustrating that use of the truncated versions may result in models different from those selected by standard C _p .     

基于加权似然比检验的监控线性曲线的控制图

文章提出了一种基于加权似然比检验的阶段二监控线性曲线的控制图,称为WLRT图,并通过平均运行长度来衡量控制图的性能表现。模拟结果表明,WLRT图对于线性曲线的截距、斜率、标准差的变化及截距和斜率同时变化都具有很好的检测能力。通过与其他几种控制图的性能比较,得出WLRT图能较快地发现过程变化,而且设计简单、操作方便。     

A change point method for Phase II monitoring of generalized linear profiles

In this article, we adopt the change point approach to monitor the generalized linear profiles in phase II Statistical process control (SPC). Generalized linear profiles include a large class of profiles defined in one framework. In contrast to the conventional change point approach, we adopt the Rao score test rather than the likelihood ratio test. Simulated results show that our approach has a good performance over any possible single step change in process parameters for two special cases of generalized linear profiles, namely Poisson and binomial profiles. Some diagnostic aids are also given and a real example is introduced to shed light on the merits of our approach in real applications.     

Statistical Monitoring of Autocorrelated Simple Linear Profiles Based on Principal Components Analysis

In this article, a transformation method using the principal component analysis approach is first applied to remove the existing autocorrelation within each profile in Phase I monitoring of autocorrelated simple linear profiles. This easy-to-use approach is independent of the autocorrelation coefficient. Moreover, since it is a model-free method, it can be used for Phase I monitoring procedures. Then, five control schemes are proposed to monitor the parameters of the profile with uncorrelated error terms. The performances of the proposed control charts are evaluated and are compared through simulation experiments based on different values of autocorrelation coefficient as well as different shift scenarios in the parameters of the profile in terms of probability of receiving an out-of-control signal.     

Selection of Variables in Multivariate Regression Models for Large Dimensions

The Akaike information criterion, AIC, and Mallows’ C _p statistic have been proposed for selecting a smaller number of regressors in the multivariate regression models with fully unknown covariance matrix. All of these criteria are, however, based on the implicit assumption that the sample size is substantially larger than the dimension of the covariance matrix. To obtain a stable estimator of the covariance matrix, it is required that the dimension of the covariance matrix is much smaller than the sample size. When the dimension is close to the sample size, it is necessary to use ridge-type estimators for the covariance matrix. In this article, we use a ridge-type estimators for the covariance matrix and obtain the modified AIC and modified C _p statistic under the asymptotic theory that both the sample size and the dimension go to infinity. It is numerically shown that these modified procedures perform very well in the sense of selecting the true model in large dimensional cases.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

Average run length comparison of multivariate control charts

In this paper we use Monte Carlo Simulation methodology to compare the effectiveness of five multivariate quality control methods, namely Hotelling T ², Multivariate Shewhart Char, Discriminant Analysis, Decomposition Method, and Multivariate Ridge Residual Chart-developed by Authors-, for controlling the mean vector in a multivariate process. P-dimensional multivariate normal data generated using different covariance structures. Various amount of shift in the mean vector is induced and the resulting Average Run Length (ARL) is computed. The effectiveness of each method with regard to ARL is discussed.     

Monotonic change point estimation of generalized linear model-based regression profiles

In this article, a maximum likelihood estimator is derived in the generalized linear model-based regression profiles under monotonic change in Phase II. The performance of the proposed estimator is comprehensively investigated through some special cases, and compared with estimators under step change and drift. The results show that the proposed estimator has better performance in small and medium shifts under different increasing changes. Finally, the applicability of the proposed estimator is illustrated using a real case.     

SPC for short-run multivariate autocorrelated processes

This paper discusses the development of a multivariate control charting technique for short-run autocorrelated data manufacturing environment. The proposed approach is a combination of the multivariate residual charts for autocorrelated data and the multivariate transformation technique for i.i.d. process observations of short lengths. The proposed approach consists in fitting adequate multivariate time-series model of various process outputs and computes the residuals, transforming them into standard normal N(0, 1) data and then using standardized data as inputs to plot conventional univariate i.i.d. control charts. The objective for applying multivariate finite horizon techniques for autocorrelated processes is to allow continuous process monitoring, since all process outputs are controlled trough the use of a single control chart with constant control limits. Throughout simulated examples, it is shown that the proposed short-run process monitoring technique provides approximately similar shifts detection properties as VAR residual charts.     

Bootstrap-Based T 2 Multivariate Control Charts

Control charts have been used effectively for years to monitor processes and detect abnormal behaviors. However, most control charts require a specific distribution to establish their control limits. The bootstrap method is a nonparametric technique that does not rely on the assumption of a parametric distribution of the observed data. Although the bootstrap technique has been used to develop univariate control charts to monitor a single process, no effort has been made to integrate the effectiveness of the bootstrap technique with multivariate control charts. In the present study, we propose a bootstrap-based multivariate T ² control chart that can efficiently monitor a process when the distribution of observed data is nonnormal or unknown. A simulation study was conducted to evaluate the performance of the proposed control chart and compare it with a traditional Hotelling's T ² control chart and the kernel density estimation (KDE)-based T ² control chart. The results showed that the proposed chart performed better than the traditional T ² control chart and performed comparably with the KDE-based T ² control chart. Furthermore, we present a case study to demonstrate the applicability of the proposed control chart to real situations.     

A Double Multivariate Exponentially Weighted Moving Average (dMEWMA) Control Chart for a Process Location Monitoring

A new control scheme, dMEWMA, for detecting shifts in the mean vector of multivariately normally distributed quality characteristics is presented. It is shown that the ARL performance of dMEWMA depends on the mean and variance-covariance matricies only through the non-centrality parameter value. Through Monte Carlo simulations, the performance of dMEWMA for detecting various shifts is compared to the competing control schemes, MEWMA and Hotelling's χ². It is concluded that dMEWMA outperforms MEWMA and Hotelling's χ² control schemes for small and larger shifts. In comparison to MEWMA control schemes, dMEWMA schemes are optimal for larger values of the smoothing parameter λ and perform much better for very small shifts in the process mean. Finally, an example to illustrate the construction of the dMEWMA control scheme is introduced.     

A Comparative Study of Different Methodologies for Fault Diagnosis in Multivariate Quality Control

Different methodologies for fault diagnosis in multivariate quality control have been proposed in recent years. These methods work in the space of the original measured variables and have performed reasonably well when there is a reduced number of mildly correlated quality and/or process variables with a well-conditioned covariance matrix. These approaches have been introduced by emphasizing their positive or negative virtues, generally on an individual basis, so it is not clear for the practitioner the best method to be used. This article provides a comprehensive study of the performance of diverse methodological approaches when tested on a large number of distinct simulated scenarios. Our primary aim is to highlight key weaknesses and strengths in these methods as well as clarifying their relationships and the requirements for their implementation in practice.     

O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N³) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N²)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener–Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N³) operations could be eliminated, and a typical speedup of 5–9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.