Minimum Volume Ellipsoid Estimator Research Articles

Detecting outliers in the multivariate control charts for dispersion monitoring

AbstractOutlier detection is an important aspect of statistical process monitoring (SPM) because outliers affect the performance of control charts. SPM researchers study the negative impact of outliers on control charts for monitoring location parameters. However, there is little research on outlier detection in multivariate charts for monitoring process dispersion. This study aims to investigate the impact of outliers in multivariate control charts for monitoring covariance matrix of a process, and then to recommend techniques for detecting potential outliers present from Phase I samples. We propose a new multivariate dispersion chart that employs the determinant of logarithm of estimated covariance matrix as the monitoring statistic. Through Monte Carlo simulations, the results show how outliers from the first phase affect the overall performance of multivariate charts. The results also demonstrate that the minimum volume ellipsoid (MVE) estimator is effective in reducing the effect of outliers on the proposed control scheme than the other compared estimators.

Robust logistic regression in the presence of high leverage points

In this article we conceder the logistic regression model with high leverage points. For the logistic regression model with a binary response, we suggested a new robust approach called robust logistic regression (RLR) based on the robust mahalanobis distance (RMD) which depends on the minimum volume ellipsoid (MVE) estimators. The RMD is computed by using the algorithm of stochastic gradient descent (SGD). In order to assist the new suggested approach we compare it with some existing method such as maximum likelihood estimator and robust M-estimator in logistic regression model. The simulation study points that the RLR has supreme performances throw some measurement comparison.

Fast algorithms for the minimum volume estimator

The minimum volume ellipsoid (MVE) estimator is an important tool in robust regression and outlier detection in statistics. We develop fast and efficient algorithms for the MVE estimator problem and discuss how they can be implemented efficiently. The novelty of our approach stems from the recent developments in the first-order algorithms for solving the related minimum volume enclosing ellipsoid problem. Comparative computational results are provided which demonstrate the strength of the algorithms.

Least median of squares regression and minimum volume ellipsoid estimator for outliers detection in housing appraisal

In the real estate sector, regression analysis is the most used method for interpretative and predictive purposes. However, the presence of outliers in the estimative sample can lead to ordinary last squared regression models that do not represent the investigated market phenomenon, with the consequence of producing unreliable assessments. In the present research, the issues of the identification and the removal of outliers are discussed. The outliers identified by the least median of squares (LMS) regression and the minimum volume ellipsoid (MVE) estimator are compared in order to test the coincidence or the diversity. A complete diagnosis of the data of the initial estimative sample is carried out, combining the robust residuals obtained with LMS and the robust distances obtained with MVE. The data are classified into regular observations, vertical outliers, good leverage points and bad leverage points, and cases to delete and those to keep in the sample are identified.

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.

On the use of minimum volume ellipsoids and symplectic capacities for studying classicaluncertainties for joint position–momentum measurements

We study the minimum volume ellipsoid estimator associated with a cloud of points inphase space. Using as a natural measure of uncertainty the symplectic capacity of thecovariance ellipsoid we find that classical uncertainties obey relations similar to those foundin non-standard quantum mechanics.

A comparison of robust alternatives to Hotelling’s T 2 control chart

Control charts are one of the widest used techniques in statistical process control. In Phase I, historical observations are analysed in order to construct a control chart. Because of the existence of multiple outliers that are undetected by control charts such as Hotelling’s T 2 due to the masking effect, robust alternatives to Hotelling’s T 2 have been developed based on minimum volume ellipsoid (MVE) estimators, minimum covariance determinant (MCD) estimators, reweighted MCD estimators or trimmed estimators. In this paper, we use a simulation study to analyse the performance of each alternative in various situations and offer guidance for the correct use of each estimator.

Minimum volume ellipsoid

AbstractThe minimum volume ellipsoid (MVE) estimator is based on the smallest volume ellipsoid that covershof thenobservations. It is an affine equivariant, high‐breakdown robust estimator of multivariate location and scatter. The MVE can be computed by a resampling algorithm. Its low bias makes the MVE very useful for outlier detection in multivariate data, often through the use of MVE‐based robust distances.We review the basic MVE definition as well as some useful extensions such as the one‐step reweighted MVE. We discuss the main properties of the MVE including its breakdown value, affine equivariance, and efficiency. We discuss the basic resampling algorithm to calculate the MVE and illustrate its use on two examples. An overview of applications is given, as well as some related classes of robust estimators of multivariate location and scatter. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under:Statistical and Graphical Methods of Data Analysis > Robust Methods

Application and Simulation Study of the Hotelling's T2 Control Chart to Monitor a Wastewater Treatment Process

The purpose of this paper is to study the application of the Hotelling's T2 statistic to assess the performance of a wastewater treatment process. Standard control charts are widely used in engineering for quality improvement through the detection of special causes of variability. These methods assume that the observations are independent. In wastewater treatment processes the data are frequently autocorrelated. One possible way to apply control charts to autocorrelated observations is to monitor the one-step ahead forecast errors after estimating the process time series model. In this paper the application of control charts to individual forecast errors of the multivariate time series model from a wastewater process is discussed. Hotelling's T2 statistics are computed using the sample covariance matrix, the successive differences estimator and the minimum volume ellipsoid estimator. A Monte Carlo simulation study shows that the application of these charts to the time series model residuals is quite effective in detecting the mean changes. However, the multivariate T2 charts are more powerful when applied to independent observations. The simulation study also indicates that the autocorrelation affects the properties of the T2 control chart giving a large false alarm rate when using the successive difference estimator, or affecting its power to detect level shifts when using the sample covariance matrix or the minimum volume ellipsoid estimator. These two charts have essentially no power to detect the simulated shifts.

Two-Step Robust Diagnostic Method for Identification of Multiple High Leverage Points

Problem statement: High leverage points are extreme outliers in the X-direction. In regression analysis, the detection of these leverage points becomes important due to their arbitrary large effects on the estimations as well as multicollinearity problems. Mahalanobis Distance (MD) has been used as a diagnostic tool for identification of outliers in multivariate analysis where it finds the distance between normal and abnormal groups of the data. Since the computation of MD relies on non-robust classical estimations, the classical MD can hardly detect outliers accurately. As an alternative, Robust MD (RMD) methods such as Minimum Covariance Determinant (MCD) and Minimum Volume Ellipsoid (MVE) estimators had been used to identify the existence of high leverage points in the data set. However, these methods tended to swamp some low leverage points even though they can identify high leverage points correctly. Since, the detection of leverage points is one of the most important issues in regression analysis, it is imperative to introduce a novel detection method of high leverage points. Approach: In this study, we proposed a relatively new two-step method for detection of high leverage points by utilizing the RMD (MVE) and RMD (MCD) in the first step to identify the suspected outlier points. Then, in the second step the MD was used based on the mean and covariance of the clean data set. We called this method two-step Robust Diagnostic Mahalanobis Distance (RDMDTS) which could identify high leverage points correctly and also swamps less low leverage points. Results: The merit of the newly proposed method was investigated extensively by real data sets and Monte Carlo Simulations study. The results of this study indicated that, for small sample sizes, the best detection method is (RDMDTS) (MVE)-mad while there was not much difference between (RDMDTS) (MVE)-mad and (RDMDTS) (MCD)-mad for large sample sizes. Conclusion/Recommendations: In order to swamp less low leverage as high leverage point, the proposed robust diagnostic methods, (RDMDTS) (MVE)-mad and (RDMDTS) (MCD)-mad were recommended.

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix Σ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for Σ or another matrix Γ ∝ Σ . In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240–2247] in the context of local asymptotic normality theory.

Confidence Intervals Based on Robust Estimators

Classical estimation of confidence intervals based on the sample mean and variance is sensitive to outliers. Robust methods were proposed for reducing the influence of outliers. The Minimum Volume Ellipsoid estimator (MVE), having a high breakdown point, is one of the robust estimators for location and scale parameters. The robust confidence interval for location parameter is constructed based on the MVE, and compared with the proposed robust confidence interval estimation methods. The performance of the robust confidence interval based on MVE is illustrated with a simulation study. The lengths of 100(1-α)% confidence intervals were investigated.

Robust location estimation under dependence

We discuss the approximation of the mean of autocorrelated data under contaminations of different types. Many robust location estimators have been investigated carefully for independent data, but their properties have not been studied in detail under dependencies. We pay attention to estimators based on subranges like minimum volume ellipsoid and minimum covariance determinant estimators, mid-ranges and trimmed means, of which the sample mean and median are special cases, and also include the Hodges–Lehmann and the Bickel–Hodges estimators. Our interest is in small to moderate sample sizes.

High breakdown estimation methods for Phase I multivariate control charts

AbstractA goal of Phase I analysis of multivariate data is to identify multivariate outliers and step changes so that the Phase II estimated control limits are sufficiently accurate. High breakdown estimation methods based on the minimum volume ellipsoid (MVE) or the minimum covariance determinant (MCD) are well suited for detecting multivariate outliers in data. As a result of the inherent difficulties in their computation, many algorithms have been proposed to detect multivariate outliers. Due to their availability in standard software packages, we consider the subsampling algorithm to obtain the MVE estimators and the FAST‐MCD algorithm to obtain the MCD estimators. Previous studies have not clearly determined which of these two available estimation methods is best for control chart applications. The comprehensive simulation study presented in this paper gives guidance for the correct use of each estimator. Control limits are provided. High breakdown estimation methods based on the MCD and MVE approaches can be applied to a wide variety of multivariate quality control data. Copyright © 2006 John Wiley & Sons, Ltd.

Robust Estimation in Multivariate Control Charts for Individual Observations

Multivariate control charts, such as Hotelling's T2 control chart, are used for monitoring several quality characteristics measured simultaneously on a product or process. In the retrospective stage (Phase I) analysis of a historical data set, the presence of multiple outliers may go undetected by the conventional T2 chart, due to masking. In this paper we propose T2 control charts based on robust estimators of location and dispersion. Simulation studies show that the T2 control chart using the minimum volume ellipsoid (MVE) estimators is effective in detecting any reasonable number of outliers.

Nearest-Neighbor Variance Estimation (NNVE)

Robust covariance estimation commonly proceeds by downweighting outliers. In this article we measure the “outlyingness” of a data point by the standardized distance between the point and its Kth nearest neighbor. The appropriate weights for robust estimation are found by a model-based mixture modeling approach that follows from considering the data cloud as a realization of a high-dimensional point process. To correct a potential bias when there are no outliers, we introduce a boundary correction procedure that artificially adds in extra outlying points; the resulting methodology is called nearest-neighbor variance estimation (NNVE). The strength of NNVE is its robustness against a large proportion of noise points and against deviation from normality of the signal. A consistency result for the method is established. Under some reasonable assumptions, it is shown that the covariance estimate is bounded and that each point has only bounded influence on the final estimates. NNVE outperformed the popular minimum volume ellipsoid (MVE) estimator in simulation studies, with a big improvement when the proportion of outliers was very large (> 50%). In our simulation study, when the proportion of outliers was ≥ 50%, the mean squared error of the NNVE estimator of variance was at least 100 times smaller than that of the MVE estimator. The proposed estimator also outperformed MVE in cases where the underlying data distribution was not normal. Good performance of NNVE in several real examples is demonstrated. A potential drawback of NNVE is that data points condensed in moderate-sized clusters would be classified as signal. Even though we do not support approaches discarding moderate-sized clusters as outliers without checking, this feature of NNVE could be problematic, particularly when only the main data cloud is of interest. A simple diagnostic tool built on existing model-based clustering procedure is proposed. This procedure allows us to check whether there is more than one separate data cloud in the data after cleaning. It also supplies the central locations of the separated moderate-sized clusters, which allows further investigation. Finally, because NNVE reduces the problem of finding the robustness weights to a one-dimensional problem, it may be useful in high-dimensional problems, such as those encountered in data mining.

Location adjustment for the minimum volume ellipsoid estimator

Estimating multivariate location and scatter with both affine equivariance and positive breakdown has always been difficult. A well-known estimator which satisfies both properties is the Minimum Volume Ellipsoid Estimator (MVE). Computing the exact MVE is often not feasible, so one usually resorts to an approximate algorithm. In the regression setup, algorithms for positive-breakdown estimators like Least Median of Squares typically recompute the intercept at each step, to improve the result. This approach is called intercept adjustment. In this paper we show that a similar technique, called location adjustment, can be applied to the MVE. For this purpose we use the Minimum Volume Ball (MVB), in order to lower the MVE objective function. An exact algorithm for calculating the MVB is presented. As an alternative to MVB location adjustment we propose L1 location adjustment, which does not necessarily lower the MVE objective function but yields more efficient estimates for the location part. Simulations compare the two types of location adjustment. We also obtain the maxbias curves of L1 and the MVB in the multivariate setting, revealing the superiority of L1.

Using Angles to Identify Concentrated Multivariate Outliers

This article describes a procedure for the detection of multivariate outliers based on the analysis of certain angular properties of the observations. The method is simple, exploratory in nature, and particularly well suited for the detection of concentrated contamination patterns, in which the outliers appear to form a cluster, separated from the sample. It is shown that it presents good properties for the identification of contaminations on high-dimensional sample spaces and for high contamination levels, including some cases in which methods based on robust estimators (the minimum covariance determinant and minimum volume ellipsoid estimators, the Stahel–Donoho estimator, or other recent proposals) may fail. The use of the procedure is illustrated through several examples.

Small - sample correction factor of the minimum covariance determinant estimator

The Minimum Covariance Determinant (MCD) estimator proposed by Rousseeuw (1985) is a highly robust way to estimate location and scatter of multivariate and possibly contaminated data sets. Recently Rousseeuw and Van Driessen (1999) have proposed a fast algorithm which allows practical application of MCD even in very large and contaminated data sets. As an initial estimate of location and scatter, MCD could be a better alternative than the so-called minimum volume ellipsoid (MVE) estimator proposed by Rousseeuw (1985), because MCD is asymptotically normal and MVE has a lower convergence rate. Robust distances using MVE have been used for labeling multiple outliers, as it was suggested by Rousseeuw and van Zomeren (1990). For small - sample cases Rousseeuw and van Zomeren (1991) computed by simulation a correction factor in terms to be able to compare MVE robust distances against distribution quantiles. In accordance with its superior theoretical performance, MCD estimate might be used for the same purpose; however it also requires a correction factor for small samples. We present in this paper the results from a simulation study for small - sample cases, with two objectives: first, when both types of estimators are used to build tolerance regions, we compare them in relation to their actual coverage probabilities; and second, to compute required correction factors. We followed the methodology from Rousseeuw and van Zomeren (1991).

A Deterministic Method for Robust Estimation of Multivariate Location and Shape

The existence of outliers in a data set and how to deal with them is an important problem in statistics. The minimum volume ellipsoid (MVE) estimator is a robust estimator of location and covariate structure; however its use has been limited because there are few computationally attractive methods. Determining the MVE consists of two parts—finding the subset of points to be used in the estimate and finding the ellipsoid that covers this set. This article addresses the first problem. Our method will also allow us to compute the minimum covariance determinant (MCD) estimator. The proposed method of subset selection is called the effective independence distribution (EID) method, which chooses the subset by minimizing determinants of matrices containing the data. This method is deterministic, yielding reproducible estimates of location and scatter for a given data set. The EID method of finding the MVE is applied to several regression data sets where the true estimate is known. Results show that the EID method, when applied to these data sets, produces the subset of data more quickly than conventional procedures and that there is less than 6% relative error in the estimates. We also give timing results illustrating the feasibility of our method for larger data sets. For the case of 10,000 points in 10 dimensions, the compute time is under 25 minutes.