Robust surface estimation in multi-response multistage statistical optimization problems

As the ordinary least squares (OLS) method is very sensitive to outliers as well as to correlated responses, a robust coefficient estimation method is proposed in this paper for multi-response surfaces in multistage processes based on M-estimators. In this approach, experimental designs are used in which the intermediate response variables may act as covariates in the next stages. The performances of both the ordinary multivariate OLS and the proposed robust multi-response surface approach are analyzed and compared through extensive simulation experiments. Sum of the squared errors in estimating the regression coefficients reveals the efficiency of the proposed robust approach.     

Robust estimation of complicated profiles using wavelets

Some quality characteristics are well defined when treated as the response variables and their relationships are identified to some independent variables. This relationship is called a profile. The parametric models, such as linear models, may be used to model the profiles. However, due to the complexity of many processes in practical applications, it is inappropriate to model the process using parametric models. In these cases non parametric methods are used to model the processes. One of the most applicable non parametric methods used to model complicated profiles is the wavelet. Many authors considered the use of the wavelet transformation only for monitoring the processes in phase II. The problem of estimating the in-control profile in phase I using wavelet transformation is not deeply addressed. Usually classical estimators are used in phase I to estimate the in-control profiles, even when the wavelet transformation is used. These estimators are suitable if the data do not contain outliers. However, when the outliers exist, these estimators cannot estimate the in-control profile properly. In this research, a robust method of estimating the in-control profiles is proposed, which is insensitive to the presence of outliers and could be applied when the wavelet transformation is used. The proposed estimator is the combination of the robust clustering and the S-estimator. This estimator is compared with the classical estimator of the in-control profile in the presence of outliers. The results from a large simulation study show that using the proposed method, one can estimate the in-control profile precisely when the data are contaminated either locally or globally.     

A robust posterior preference multi-response optimization approach in multistage processes

Nowadays, many manufacturing and service systems provide products and services to their customers in several consecutive stages of operations, in each of which one or more quality characteristics of interest are monitored. In these environments, the final quality in the last stage not only depends on the quality of the task performed in that stage but also is dependent on the quality of the products and services in intermediate stages as well as the design parameters in each stage. In this paper, a novel methodology based on the posterior preference approach is proposed to robustly optimize these multistage processes. In this methodology, a multi-response surface optimization problem is solved in order to find preferred solutions among different non dominated solutions (NDSs) according to decision maker's preference. In addition, as the intermediate response variables (quality characteristics) may act as covariates in the next stages, a robust multi-response estimation method is applied to extract the relationships between the outputs and inputs of each stage. NDSs are generated by the ?-constraint method. The robust preferred solutions are selected considering some newly defined conformance criteria. The applicability of the proposed approach is illustrated by a numerical example at the end.     

Robust multivariate control charts based on Birnbaum–Saunders distributions

Multivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

On designing new optimal synthetic Tukey’s control charts

Tukey’s control chart is generally used for monitoring the processes where the measurement process physically damages the product. It is based on single observation and robust to outliers. In this paper, two optimal synthetic Tukey’s control charts are proposed by integrating the conforming run length chart with the Tukey’s control chart and its modification. The performance comparison of the proposed charts with the existing Tukey’s control charts is made by using out-of-control average run length and extra quadratic loss as performance metrics. The proposed charts offer better protection against the process shifts as compare to the existing Tukey’s control charts when the underlying process distribution is symmetric or asymmetric. Simulation studies also establish the supremacy of the proposed control charts over the existing Tukey’s control charts. In the end, an illustrative example based on a real data set of the combined cycle power plant is provided for practical implementation.     

Robust Estimation of Parameters in Simple Linear Profiles Using M-Estimators

In many situations, the quality of a process or product may be better characterized and summarized by a relationship between the response variable and one or more explanatory variables. Parameter estimation is the first step in constructing control charts. Outliers may hamper proper classical estimators and lead to incorrect conclusions. To remedy the problem of outliers, robust methods have been developed recently. In this article, a robust method is introduced for estimating the parameters of simple linear profiles. Two weight functions, Huber and Bisquare, are applied in the estimation algorithm. In addition, a method for robust estimation of the error terms variance is proposed. Simulation studies are done to investigate and evaluate the performance of the proposed estimator, as well as the classical one, in the presence and absence of outliers under different scenarios by the means of MSE criterion. The results reveal that the robust estimators proposed in this research perform as well as classical estimators in the absence of outliers and even considerably better when outliers exist. The maximum value of variance estimate in one scenario obtained from classical estimator is 10.9, while this value is 1.66 and 1.27 from proposed robust estimators when its actual value is 1.     

基于稳健MM估计的统计数据质量评估方法

 政府统计数据质量是当前各界关注的热点问题,如何采用严谨的诊断方法,对我国统计数据进行科学的评估具有重要的现实意义。稳健回归方法可使求出的回归估计不受异常值的强烈影响,并且能更好的识别异常点。本文首次运用基于稳健MM估计的异常值诊断方法,在生产函数模型的框架下,分别使用两种不同的劳动投入数据,对改革以来我国GDP数据质量进行了评估。结果表明,基于稳健MM估计的异常值诊断方法可有效的解决传统方法容易出现的多个异常点的掩盖现象,改革以来我国的GDP数据是相对可靠的。     

Robust Coplot Analysis

CoPlot analysis is one of the multivariate data-visualizing techniques. It consists of two graphs: the first one represents the distribution of p-dimensional observations over two-dimensional space, whereas the second shows the relations of variables with the observations. At CoPlot analysis, multidimensional scaling (MDS) and Pearson’s correlation coefficient (PCC) are used to obtain a map that demonstrates observations and variables simultaneously. However, both MDS and PCC are sensitive to outliers. When multidimensional dataset contains outliers, interpretation of the map, which is obtained from classical CoPlot analysis, may result in wrong conclusions. At this study, a novel approach to classical CoPlot analysis is presented. By using robust MDS and median absolute deviation correlation coefficient (MADCC), robust CoPlot map is improved. Numerical examples are given to illustrate the merits of the proposed approach. Also, obtained results are compared with the classical CoPlot analysis to emphasize the superiority of introduced robust CoPlot approach.     

A clustering approach to detect multiple outliers in linear functional relationship model for circular data

Outlier detection has been used extensively in data analysis to detect anomalous observation in data. It has important applications such as in fraud detection and robust analysis, among others. In this paper, we propose a method in detecting multiple outliers in linear functional relationship model for circular variables. Using the residual values of the Caires and Wyatt model, we applied the hierarchical clustering approach. With the use of a tree diagram, we illustrate the detection of outliers graphically. A Monte Carlo simulation study is done to verify the accuracy of the proposed method. Low probability of masking and swamping effects indicate the validity of the proposed approach. Also, the illustrations to two sets of real data are given to show its practical applicability.     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

Shape bias of robust covariance estimators: an empirical study

Detecting outliers in a multivariate point cloud is not trivial, especially when dealing with a sizable fraction of contamination. Over time, it has increasingly been recognized that the safest and most feasible approach to exposing outliers starts by computing a highly robust estimator of location and scatter that can withstand a large proportion of contamination. Many such estimators have been proposed in recent years. We will compare the worst-case bias of several prominent robust multivariate estimators by means of simulation. We also propose a new tool to compare robust estimators on real data sets, and illustrate it.     

Robust PCA for high-dimensional data based on characteristic transformation

In this paper, we propose a novel robust principal component analysis (PCA) for high-dimensional data in the presence of various heterogeneities, in particular strong tailing and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. The suggested method has the distinct advantage of dealing with heavy-tail-distributed data, whose covariances may be non-existent (positively infinite, for instance), in addition to the usual outliers. The proposed approach is also a case of kernel principal component analysis (KPCA) and employs the robust and non-linear properties via a bounded and non-linear kernel function. The merits of the new method are illustrated by some statistical properties, including the upper bound of the excess error and the behaviour of the large eigenvalues under a spiked covariance model. Additionally, using a variety of simulations, we demonstrate the benefits of our approach over the classical PCA. Finally, using data on protein expression in mice of various genotypes in a biological study, we apply the novel robust PCA to categorise the mice and find that our approach is more effective at identifying abnormal mice than the classical PCA.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

A comparison of different procedures for principal component analysis in the presence of outliers

Principal component analysis (PCA) is a popular technique that is useful for dimensionality reduction but it is affected by the presence of outliers. The outlier sensitivity of classical PCA (CPCA) has caused the development of new approaches. Effects of using estimates obtained by expectation–maximization – EM and multiple imputation – MI instead of outliers were examined on the artificial and a real data set. Furthermore, robust PCA based on minimum covariance determinant (MCD), PCA based on estimates obtained by EM instead of outliers and PCA based on estimates obtained by MI instead of outliers were compared with the results of CPCA. In this study, we tried to show the effects of using estimates obtained by MI and EM instead of outliers, depending on the ratio of outliers in data set. Finally, when the ratio of outliers exceeds 20%, we suggest the use of estimates obtained by MI and EM instead of outliers as an alternative approach.     

Independent attributes control chart for Markov-dependent processes

Statistical process monitoring (SPM) is a very efficient tool to maintain and to improve the quality of a product. In many industrial processes, end product has two or more attribute-type quality characteristics. Some of them are independent, but the observations are Markovian dependent. It is essential to develop a control chart for such situations. In this article, we develop an Independent Attributes Control Chart for Markov Dependent Processes based on error probabilities criterion under the assumption of one-step Markov dependency. Implementation of the chart is similar to that of Shewhart-type chart. Performance of the chart has been studied using probability of detecting shift criterion. A procedure to identify the attribute(s) responsible for out-of-control status of the process is given.     

Optimal B-robust estimators for the parameters of the Burr XII distribution

The Burr XII distribution offers a flexible alternative to the distributions that play important role for modelling data in reliability, risk and process capability. However, estimating the shape parameters of the Burr XII distribution is a challenging problem. The classical estimation methods such as maximum likelihood and least squares are often used to estimate the parameters of the Burr XII distribution, but these methods are very sensitive to the outliers in the data. Thus, a robust estimation method alternative to the classical methods is needed to find robust estimators that are less sensitive to the outliers in the data. The purpose of this paper is to use the optimal B-robust estimation method [Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA. Robust statistics: the approach based on influence functions. New York: Wiley; 1986] to obtain robust estimators for the shape parameters of the Burr XII distribution. The simulation results show that the optimal B-robust estimators generally outperform the classical estimators in terms of the bias and root mean square errors when there are outliers in data.     

ARFIMA processes and outliers: a weighted likelihood approach

In this paper, we consider the problem of robust estimation of the fractional parameter, d, in long memory autoregressive fractionally integrated moving average processes, when two types of outliers, i.e. additive and innovation, are taken into account without knowing their number, position or intensity. The proposed method is a weighted likelihood estimation (WLE) approach for which needed definitions and algorithm are given. By an extensive Monte Carlo simulation study, we compare the performance of the WLE method with the performance of both the approximated maximum likelihood estimation (MLE) and the robust M-estimator proposed by Beran (Statistics for Long-Memory Processes, Chapman & Hall, London, 1994). We find that robustness against the two types of considered outliers can be achieved without loss of efficiency. Moreover, as a byproduct of the procedure, we can classify the suspicious observations in different kinds of outliers. Finally, we apply the proposed methodology to the Nile River annual minima time series.     

Weighting Method for a Linear Mixed Model

Maximum likelihood is a widely used estimation method in statistics. This method is model dependent and as such is criticized as being non robust. In this article, we consider using weighted likelihood method to make robust inferences for linear mixed models where weights are determined at both the subject level and the observation level. This approach is appropriate for problems where maximum likelihood is the basic fitting technique, but a subset of data points is discrepant with the model. It allows us to reduce the impact of outliers without complicating the basic linear mixed model with normally distributed random effects and errors. The weighted likelihood estimators are shown to be robust and asymptotically normal. Our simulation study demonstrates that the weighted estimates are much better than the unweighted ones when a subset of data points is far away from the rest. Its application to the analysis of deglutition apnea duration in normal swallows shows that the differences between the weighted and unweighted estimates are due to large amount of outliers in the data set.