The feasible solution algorithm for the minimum covariance determinant estimator in multivariate data

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.

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 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.

The 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

Portfolio optimization relies in three main elements: the utility function, the use of multivariate dependence measure between assets and the returns of the assets. The first problem is well known in finance since one can use several different types of portfolio utility functions depending on her objective function and constraints. The two remaining problems are relevant topics of research: the measure and forecast of the dependence between assets and the forecast of the returns of assets. In this paper, we present a new method with artificial intelligence to estimate the variance and covariance matrices of financial time series data accounting for small sample size, large number of assets and presences of outliers. We compare results using different approaches, e.g., Minimum covariance determinant (MCD), Minimum Volume Ellipsoid (MVE), Minimum Regularized Covariance Determinant (MRCD) and Orthogonalized Gnanadesikan-Kettenring (OGK). The proposed approach is robust under several financial stylized facts and presents good performance with respect to the cumulative returns in the US markets (NASDAQ Stock Exchange) for 95 assets during 2015 to the beginning of 2019. These study contributes to fill in the literature gap in covariance estimation for small sample size, large number of assets and presences of outliers and contributes to investors to making better portfolio management.

This paper reviews the state of the art in the computation of robust estimates of multivariate location and shape using combinatorial estimators such as the minimum volume ellipsoid (MVE) and iterative M‐ and S‐estimators. We also present new results on the behavior of M‐ and S‐estimators in the presence of different types of outliers, and give the first computational evidence on compound estimators that use the MVE as a starting point for an S‐estimator. Problems with too many data points in too many dimensions cannot be handled by any available technology; however, the methods presented in this paper substantially extend the size of problem that can be successfully handled.

The minimum covariance determinant (MCD) estimator is a highly robust estimator of multivariate location and scatter. It can be computed efficiently with the FAST‐MCD algorithm of Rousseeuw and Van Driessen. Since estimating the covariance matrix is the cornerstone of many multivariate statistical methods, the MCD has also been used to develop robust and computationally efficient multivariate techniques.In this paper, we review the MCD estimator, along with its main properties such as affine equivariance, breakdown value, and influence function. We discuss its computation, and list applications and extensions of the MCD in theoretical and applied multivariate statistics. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Robust Methods

Among the most well known estimators of multivariate location and scatter is the Minimum Volume Ellipsoid (MVE). Many algorithms have been proposed to compute it. Most of these attempt merely to approximate as close as possible the exact MVE, but some of them led to the definition of new estimators which maintain the properties of robustness and affine equivariance that make the MVE so attractive. Rousseeuw and van Zomeren (1990) used the <$>(p+1)<$>- subset estimator which was modified by Croux and Haesbroeck (1997) to give rise to the averaged <$>(p+1)<$>- subset estimator . This note shows by means of simulations that the averaged <$>(p+1)<$>-subset estimator outperforms the exact estimator as far as finite-sample efficiency is concerned. We also present a new robust estimator for the MVE, closely related to the averaged <$>(p+1)<$>-subset estimator, but yielding a natural ranking of the data.

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.

Interest in generative models has grown tremendously in the past decade. However, their training performance can be adversely affected by contamination, where outliers are encoded in the representation of the model. This results in the generation of noisy data. In this paper, we introduce weighted conjugate feature duality in the framework of Restricted Kernel Machines (RKMs). The RKM formulation allows for an easy integration of methods from classical robust statistics. This formulation is used to fine-tune the latent space of generative RKMs using a weighting function based on the Minimum Covariance Determinant, which is a highly robust estimator of multivariate location and scatter. Experiments show that the weighted RKM is capable of generating clean images when contamination is present in the training data. We further show that the robust method also preserves uncorrelated feature learning through qualitative and quantitative experiments on standard datasets.

Robust principal component regression s-estimator is principal component regression that applies robust approach method at principal component analysis and s-estimator at principal component regression analysis. The aim of robust principal component regression s-estimator is to overcome multicollinearity problems in multiple linier regression Ordinary Least Square (OLS) and to overcome outlier problems in principal component regression so get the most effective model. Minimum Volume Ellipsoid (MVE) is one of the robust approach methods that applied when doing principal component analysis and S-Estimator is one of the estimation methods that applied when doing principal component regression analysis. The case in this study is the factors that influence the Number of Unemployment in Central Java in 2017. The model that provides the most effective result to handling multicolliniearity and ouliers in the case study Number of Unemployment in Central Java in 2017 is using robust principal component regression MVE-(S-Estimator) with Adjusted R2 value of 0.9615 and RSE value of 0.4073. Keywords: Robust Principal Component Regression S-Estimator, Multicollinearity, Outliers, Minimum Volume Ellipsoid (MVE), Number of Unemployment.

An easy way to increase the finite-sample efficiency of the resampled minimum volume ellipsoid estimator

Contagious ecthyma is a skin disease, caused by Orf virus, creating great economic threats to livestock farming worldwide. Zoonotic potential of this disease has gained recent attention owing to the re-emergence of disease in several parts of the world. Increased public health concern emphasizes the need for a predictive understanding of the geographic distributional potential of Orf virus. Here, we mapped the current distribution using occurrence records, and estimated the ecological niche in both geographical and environmental spaces. Twenty modeling experiments, resulting from two- and three-partition models, were performed to choose the candidate models that best represent the geographic distributional potential of Orf virus. For all of our models, it was possible to reject the null hypothesis of predictive performance no better than random expectations. However, statistical significance must be accompanied by sufficiently good predictive performance if a model is to be useful. In our case, omission of known distribution of the virus was noticed in all Maxent models, indicating inferior quality of our models. This conclusion was further confirmed by the independent final evaluation, using occurrence records sourced from the Centre for Agriculture and Bioscience International. Minimum volume ellipsoid (MVE) models indicated the broad range of environmental conditions under which Orf virus infections are found. The excluded climatic conditions from MVEs could not be considered as unsuitable owing to the broad distribution of Orf virus. These results suggest two possibilities: that the niche models fail to identify niche limits that constrain the virus, or that the virus has no detectable niche, as it can be found throughout the geographic distributions of its hosts. This potential limitation of component-based pathogen-only ENMs is discussed in detail.

A maxbias curve is a powerful tool to describe the robustness of an estimator. It tells us how much an estimator can change due to a given fraction of contamination. In this paper, maxbias curves are computed for some univariate location estimators based on subranges: midranges, trimmed means and the univariate Minimum Volume Ellipsoid (MVE) location estimators. These estimators are intuitively appealing and easy to calculate.

Detecting influential observations in principal components and common principal components