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.     

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.     

A semiparametric nonlinear mixed model approach to phase I profile monitoring

When process data follow a particular curve in quality control, profile monitoring is suitable and appropriate for assessing process stability. Previous research in profile monitoring focusing on nonlinear parametric (P) modeling, involving both fixed and random-effects, was made under the assumption of an accurate nonlinear model specification. Lately, nonparametric (NP) methods have been used in the profile monitoring context in the absence of an obvious linear P model. This study introduces a novel technique in profile monitoring for any nonlinear and auto-correlated data. Referred to as the nonlinear mixed robust profile monitoring (NMRPM) method, it proposes a semiparametric (SP) approach that combines nonlinear P and NP profile fits for scenarios in which a nonlinear P model is adequate over part of the data but inadequate of the rest. These three methods (P, NP, and NMRPM) account for the auto-correlation within profiles and treats the collection of profiles as a random sample with a common population. During Phase I analysis, a version of Hotelling’s T² statistic is proposed for each approach to identify abnormal profiles based on the estimated random effects and obtain the corresponding control limits. The performance of the NMRPM method is then evaluated using a real data set. Results reveal that the NMRPM method is robust to model misspecification and performs adequately against a correctly specified nonlinear P model. Control charts with the NMRPM method have excellent capability of detecting changes in Phase I data with control limits that are easily computable.     

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.     

New control charts for simultaneous monitoring of the mean vector and covariance matrix of multivariate multiple linear profiles

Simultaneous monitoring of the mean vector and covariance matrix in multivariate processes allows practitioners to avoid the inflated false alarm rate that results from using two independent control charts. In this paper, we extend exponentially weighted moving average semicircle and generally weighted moving average semicircle control charts to monitor the mean vector and covariance matrix of multivariate multiple linear regression profiles in Phase II simultaneously. These new control charts are compared with the existing control charts in the literature in terms of the average run length criterion. Finally, a case is considered to show the application of the proposed charts.     

Modified Tukey's Control Chart

Phase I of control analysis requires large amount of data to fit a distribution and estimate the corresponding parameters of the process under study. However, when only individual observations are available, and no a priori knowledge exists, the presence of outliers can bias the analysis. A relatively recent and successful approach to address this situation is Tukey's Control Chart (TCC), a charting method that applies the Box Plot technique to estimate the control limits. This procedure has proven to be effective for symmetric distributions. However, when skewness is present the average run length performance diminishes significantly. This article proposes a modified version of TCC to consider skewness with minimum assumptions on the underlying distribution of observations. Using theoretical results and Monte Carlo simulation, the modified TCC is tested over several distributions proving a better representation of skewed populations, even in cases when only a limited number of observations are available.     

The Performance of Phase II Simple Linear Profile Approaches when Parameters Are Estimated

Previous studies of statistical performance of Phase II simple linear profile approaches were reported only for the case of known profile parameters assumption. The main objective of this article is to evaluate and compare the performance of these approaches when the profile parameters are estimated from an in-control Phase I profile data set. Simulations establish that the performance of these approaches is strongly affected when the parameters are estimated compared to the known parameters case. The in-control performance of the competing approaches significantly deteriorates if 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. The results show also that some profile monitoring approaches need much larger number of Phase I profiles than other approaches to achieve the expected statistical performance. They also show that the profile monitoring approach proposed by Mahmoud et al. (2010 Mahmoud , M. A. , Morgan , J. P. , Woodall , W. H. ( 2010 ). The monitoring of simple linear regression profiles with two observations per sample . Journal of Applied Statistics 37 : 1249 – 1263 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) has generally better out-of-control run length performance than the competing approaches when the estimated parameters are used in the charts design.     

Robust difference-based outlier detection

Abstract

In this paper, we propose an outlier-detection approach that uses the properties of an intercept estimator in a difference-based regression model (DBRM) that we first introduce. This DBRM uses multiple linear regression, and invented it to detect outliers in a multiple linear regression. Our outlier-detection approach uses only the intercept; it does not require estimates for the other parameters in the DBRM. In this paper, we first employed a difference-based intercept estimator to study the outlier-detection problem in a multiple regression model. We compared our approach with several existing methods in a simulation study and the results suggest that our approach outperformed the others. We also demonstrated the advantage of our approach using a real data application. Our approach can extend to nonparametric regression models for outliers detection.     

On the monitoring of mixture simple linear profiles

ABSTRACT

In some applications, the quality of a process or product is best characterized by a functional relationship between a response variable and one or more explanatory variables. Profile monitoring is used to understand and to check the stability of this relationship or curve over time. In the existing simple linear regression profile models, it is often assumed that the data follow a single mode distribution and consequently the noise of the functional relationship follows a normal distribution. However, in some applications, it is likely that the data may follow a multiple-modes distribution. In this case, it is more appropriate to assume that the data follow a mixture profile. In this study, we focus on a mixture simple linear profile model, and propose new control schemes for Phase II monitoring. The proposed methods are shown to have good performance in a simulation study.     

Penalized Spline Estimation for Varying-Coefficient Models

Varying-coefficient models are useful extensions of classical linear models. They arise from multivariate nonparametric regression, nonlinear time series modeling and forecasting, longitudinal data analysis, and others. This article proposes the penalized spline estimation for the varying-coefficient models. Assuming a fixed but potentially large number of knots, the penalized spline estimators are shown to be strong consistency and asymptotic normality. A systematic optimization algorithm for the selection of multiple smoothing parameters is developed. One of the advantages of the penalized spline estimation is that it can accommodate varying degrees of smoothness among coefficient functions due to multiple smoothing parameters being used. Some simulation studies are presented to illustrate the proposed methods.     

Modified Profile Likelihood for Fixed-Effects Panel Data Models

We show how modified profile likelihood methods, developed in the statistical literature, may be effectively applied to estimate the structural parameters of econometric models for panel data, with a remarkable reduction of bias with respect to ordinary likelihood methods. Initially, the implementation of these methods is illustrated for general models for panel data including individual-specific fixed effects and then, in more detail, for the truncated linear regression model and dynamic regression models for binary data formulated along with different specifications. Simulation studies show the good behavior of the inference based on the modified profile likelihood, even when compared to an ideal, although infeasible, procedure (in which the fixed effects are known) and also to alternative estimators existing in the econometric literature. The proposed estimation methods are implemented in an R package that we make available to the reader.     

Addressing the effect of parameter estimation on phase II monitoring of multivariate multiple linear profiles via a new cluster-based approach

Abstract

Profile monitoring is applied when the quality of a product or a process can be determined by the relationship between a response variable and one or more independent variables. In most Phase II monitoring approaches, it is assumed that the process parameters are known. However, it is obvious that this assumption is not valid in many real-world applications. In fact, the process parameters should be estimated based on the in-control Phase I samples. In this study, the effect of parameter estimation on the performance of four Phase II control charts for monitoring multivariate multiple linear profiles is evaluated. In addition, since the accuracy of the parameter estimation has a significant impact on the performance of Phase II control charts, a new cluster-based approach is developed to address this effect. Moreover, we evaluate and compare the performance of the proposed approach with a previous approach in terms of two metrics, average of average run length and its standard deviation, which are used for considering practitioner-to-practitioner variability. In this approach, it is not necessary to know the distribution of the chart statistic. Therefore, in addition to ease of use, the proposed approach can be applied to other type of profiles. The superior performance of the proposed method compared to the competing one is shown in terms of all metrics. Based on the results obtained, our method yields less bias with small-variance Phase I estimates compared to the competing approach.     

Phase I monitoring and change point estimation of autocorrelated poisson regression profiles

The objective of this paper is to study the Phase I monitoring and change point estimation of autocorrelated Poisson profiles where the response values within each profile are autocorrelated. Two charts, the SLRT and the Hotelling's T², are proposed along with an algorithm for parameter estimation. The detecting power of the proposed charts is compared using simulations in terms of the signal probability criterion. The performance of the SLRT method in estimating the change point in the regression parameters is also evaluated. Moreover, a real data example is presented to illustrate the application of the methods.     

A new look at the difference between the GEE and the GLMM when modeling longitudinal count responses

Poisson log-linear regression is a popular model for count responses. We examine two popular extensions of this model – the generalized estimating equations (GEE) and the generalized linear mixed-effects model (GLMM) – to longitudinal data analysis and complement the existing literature on characterizing the relationship between the two dueling paradigms in this setting. Unlike linear regression, the GEE and the GLMM carry significant conceptual and practical implications when applied to modeling count data. Our findings shed additional light on the differences between the two classes of models when used for count data. Our considerations are demonstrated by both real study and simulated data.     

Monitoring the parameter vector of regression models with time-to-event response in phase II processes

In most of the existing specialized literature, monitoring regression models are a special case of profile monitoring. However, not every regression model always represents appropriately a profile data structure. This is clearly the case of the Weibull regression model (WRM) with common shape parameter γ. Even though it might be thought that existing methodologies (especially likelihood-ratio (LRT)-based methods) for monitoring generalized linear profiles can also be successfully applied to monitoring regression models with time-to-event response, it will be shown in this paper that those methodologies work fairly acceptable just for data structures with 1000 observations at least approximately. It was found out that some corrections, often referred to as Bartlett's adjustments, are needed to be implemented in order to improve the accuracy of using the asymptotic distributional properties of the LRT statistic for carrying out the monitoring of WRM with relatively small and moderate dimensions of the available datasets. Simulation studies suggest that the use of the aforementioned corrections make the resulting charts work quite acceptable when available data structures contain 30 observations at least. Detection abilities of the proposed schemes improve as dataset dimension increases.     

Outlier detection using difference-based variance estimators in multiple regression

In this article, we propose an outlier detection approach in a multiple regression model using the properties of a difference-based variance estimator. This type of a difference-based variance estimator was originally used to estimate error variance in a non parametric regression model without estimating a non parametric function. This article first employed a difference-based error variance estimator to study the outlier detection problem in a multiple regression model. Our approach uses the leave-one-out type method based on difference-based error variance. The existing outlier detection approaches using the leave-one-out approach are highly affected by other outliers, while ours is not because our approach does not use the regression coefficient estimator. We compared our approach with several existing methods using a simulation study, suggesting the outperformance of our approach. The advantages of our approach are demonstrated using a real data application. Our approach can be extended to the non parametric regression model for outlier detection.     

Monitoring of autocorrelated general linear profiles

A collection of quality data represented by a functional relationship between response and explanatory variables is called a profile. In the literature, the errors of profiles are often assumed to be independent. However, quality data often exhibits time correlations in real applications. Therefore, in this paper, we investigate a general linear regression model with a between-profile autocorrelation. We propose a multivariate exponentially weighted moving average chart for monitoring shifts in the regression parameters, and an exponentially weighted moving average chart for monitoring shifts in the standard deviation. A simulation study reveals that our proposed schemes outperform competing existing schemes based on the average run length criterion. An example is used to illustrate the applicability of the proposed scheme.     

Bayesian Truncated Poisson Regression with Application to Dutch Illegal Immigrant Data

This article presents a Bayesian approach to the regression analysis of truncated data, with a focus on zero-truncated counts from the Poisson distribution. The approach provides inference not only on the regression coefficients but also on the total sample size and the parameters of the covariate distribution. The theory is applied to some illegal immigrant data from The Netherlands. Several models are fitted with the aid of Markov chain Monte Carlo methods and assessed via posterior predictive p-values. Inferences are compared with those obtained elsewhere using other approaches.     

Goodness-of-fit-based outlier detection for Phase I monitoring

ABSTRACT

Nonparametric charts are useful in statistical process control when there is a lack of or limited knowledge about the underlying process distribution. Most existing approaches in the literature of Phase I monitoring assume that outliers have the same distributions as the in-control sample but only differ in location or scale parameters, they may not be effective with distributional changes. This article develops a new procedure based on the integration of the classical Anderson–Darling goodness-of-fit test and the stepwise isolation method. Our proposed procedure is efficient in detecting potential shifts in location, scale, or shape, and thus it offers robust protection against variation in various underlying distributions. The finite sample performance of our method is evaluated through simulations and is compared with that of available outlier detection methods for Phase I monitoring.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.