outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

axisymmetric vibration of a circular plate with double linear variable thickness abstract the rayleigh ritz method has been employed to study the axisymmetric transverse vibration of a circular plate with two different linear variations in thickness one in the central core and the other in the outer annular region the boundary has been taken either clamped or simply supported results for linearly varying thickness and uniform thickness have been obtained as special cases and are compared with those available in the literature the effects of the various parameters on the frequencies mode shapes and the nodal radii have been examined through extensive numerical results given in the form of graphs and tables

asymptotics for the minimum covariance determinant estimator consistency is shown for the minimum covariance determinant mcd estimators of multivariate location and scale and asymptotic normality is shown for the former the proofs are made possible by showing a separating ellipsoid property for the mcd subset of observations an analogous property is shown for the mcd subset computed from the population distribution

outlier detection and robust normal curvature estimation in mobile laser scanning 3d point cloud data this paper proposes two robust statistical techniques for outlier detection and robust saliency features such as surface normal and curvature estimation in laser scanning 3d point cloud data one is based on a robust z score and the other uses a mahalanobis type robust distance the methods couple the ideas of point to plane orthogonal distance and local surface point consistency to get maximum consistency with minimum distance mcmd the methods estimate the best fit plane based on most probable outlier free and most consistent points set in a local neighbourhood then the normal and curvature from the best fit plane will be highly robust to noise and outliers experiments are performed to show the performance of the algorithms compared to several existing well known methods from computer vision data mining machine learning and statistics using synthetic and real laser scanning datasets of complex planar and non planar objects results for plane fitting denoising sharp feature preserving and segmentation are significantly improved the algorithms are demonstrated to be significantly faster more accurate and robust quantitatively for a sample size of 50 with 20 outliers the proposed mcmd z is approximately 5 15 and 98 times faster than the existing methods ulsif ransac and rpca respectively the proposed mcmd md method can tolerate 75 clustered outliers whereas rpca and ransac can only tolerate 47 and 64 outliers respectively in terms of outlier detection for the same dataset mcmd z has an accuracy of 99 72 0 4 false positive rate and 0 false negative rate for rpca ransac and ulsif the accuracies are 97 05 47 06 and 94 54 respectively and they have misclassification rates higher than the proposed methods the new methods have potential for local surface reconstruction fitting and other point cloud processing tasks highlightstwo statistical techniques are proposed for outlier detection in point cloud data the proposed methods can fit robust plane in laser scanning data the proposed methods produce robust normal and curvature in point cloud processing they are faster and robust than ransac robust pca and other existing efficient methods they have potential for point cloud denoising segmentation and reconstruction

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

high breakdown linear discriminant analysis abstract the classification rules of linear discriminant analysis are defined by the true mean vectors and the common covariance matrix of the populations from which the data come because these true parameters are generally unknown they are commonly estimated by the sample mean vector and covariance matrix of the data in a training sample randomly drawn from each population however these sample statistics are notoriously susceptible to contamination by outliers a problem compounded by the fact that the outliers may be invisible to conventional diagnostics high breakdown estimation is a procedure designed to remove this cause for concern by producing estimates that are immune to serious distortion by a minority of outliers regardless of their severity in this article we motivate and develop a high breakdown criterion for linear discriminant analysis and give an algorithm for its implementation the procedure is intended to supplement rather than replace the usual sample moment methodology of discri

theoretical analysis of potential forces in markets abstract we show that random walks in a moving potential function with its center at the moving average of market prices are represented in the form of the self modulation model from this point of view we confirm the existence of non trivial autocorrelation in real market price changes by generalizing the formulation of potential function we prove that the arch model belongs to the special case of random walk in an asymmetric potential with randomly changing coefficient

a heuristic based fuzzy co clustering algorithm for categorization of high dimensional data fuzzy co clustering is a technique that performs simultaneous fuzzy clustering of objects and features it is known to be suitable for categorizing high dimensional data due to its dynamic dimensionality reduction mechanism achieved through simultaneous feature clustering we introduce a new fuzzy co clustering algorithm called heuristic fuzzy co clustering with the ruspini s condition hfcr which addresses several issues in some prominent existing fuzzy co clustering algorithms among these issues are the performance on data sets with overlapping feature clusters and the unnatural representation of feature clusters the key idea behind hfcr is the formulation of the dual partitioning approach for fuzzy co clustering replacing the existing partitioning ranking approach hfcr adopts an efficient and practical heuristic method that can be shown to be more robust than our earlier effort for the dual partitioning approach we explain the proposed algorithm in details and provide an analytical study on its advantages experimental results on 10 large benchmark document data sets confirm the effectiveness of the new algorithm

robust estimation in generalized partial linear models for clustered data in this article we consider robust generalized estimating equations for the analysis of semiparametric generalized partial linear models gplms for longitudinal data or clustered data in general we approximate the nonparametric function in the gplm by a regression spline and use bounded scores and leverage based weights in the estimating equation to achieve robustness against outliers we show that the regression spline approach avoids some of the intricacies associated with the profile kernel method and that robust estimation and inference can be carried out operationally as if a generalized linear model were used

robust linear model selection based on least angle regression in this article we consider the problem of building a linear prediction model when the number of candidate predictors is large and the data possibly contain anomalies that are difficult to visualize and clean we want to predict the nonoutlying cases therefore we need a method that is simultaneously robust and scalable we consider the stepwise least angle regression lars algorithm which is computationally very efficient but sensitive to outliers we introduce two different approaches to robustify lars the plug in approach replaces the classical correlations in lars by robust correlation estimates the cleaning approach first transforms the data set by shrinking the outliers toward the bulk of the data which we call multivariate winsorization and then applies lars to the transformed data we show that the plug in approach is time efficient and scalable and that the bootstrap can be used to stabilize its results we recommend using bootstrapped robustified lars to sequence a number of candidate pred

principal components and orthogonal regression based on robust scales both principal components analysis pca and orthogonal regression deal with finding a p dimensional linear manifold minimizing a scale of the orthogonal distances of the m dimensional data points to the manifold the main conceptual difference is that in pca p is estimated from the data to attain a small proportion of unexplained variability whereas in orthogonal regression p equals m 1 the two main approaches to robust pca are using the eigenvectors of a robust covariance matrix and searching for the projections that maximize or minimize a robust univariate dispersion measure this article is more akin to second approach but rather than finding the components one by one we directly undertake the problem of finding for a given p a p dimensional linear manifold minimizing a robust scale of the orthogonal distances of the data points to the manifold the scale may be either a smooth m scale or a trimmed scale an iterative algorithm is developed that is shown to converge to a local minimum a

breakdown points of affine equivariant estimators of multivariate location and covariance matrices finite sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices the role of various equivariance properties is illustrated the breakdown point is related to a measure of performance based on large deviations probabilities finally we show that one step reweighting preserves the breakdown point

detecting deviating data cells a multivariate dataset consists of n cases in d dimensions and is often stored in an n by d data matrix it is well known that real data may contain outliers depending on the situation outliers

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

breakdown and groups the concept of breakdown point was introduced by hampel ph d dissertation 1968 univ california berkeley ann math statist 42 1971 1887 1896 and developed further by among others huber robust statistics 1981 wiley new york and donoho and huber in a festschrift for erich l lehmann 1983 157 184 wadsworth belmont ca it has proved most successful in the context of location scale and regression problems attempts to extend the concept to other situations have not met with general acceptance in this paper we argue that this is connected to the fact that in the location scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one institute of mathematical statistics 2005

self modulation processes and resulting generic 1 f fluctuations we analyze high precision data of transaction intervals in a foreign exchange market and show that it is nicely approximated by a non stationary poisson process whose expectation value is given by a moving average of its own trace generalizing this result we introduce novel stochastic processes called the self modulation processes by the self modulation effect clustering occurs automatically resulting in fat tailed interval distributions including the zipf s law in an extreme case we prove rigorously that the corresponding power spectrum follows the 1 f spectrum

invariant co ordinate selection summary a general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented the method is based on the eigenvalue eigenvector decomposition of one scatter matrix relative to another in particular it is shown that the eigenvectors can be used to generate an affine invariant co ordinate system for the multivariate data consequently we view this method as a method for invariant co ordinate selection by plotting the data with respect to this new invariant co ordinate system various data structures can be revealed for example under certain independent components models it is shown that the invariant co ordinates correspond to the independent components another example pertains to mixtures of elliptical distributions in this case it is shown that a subset of the invariant co ordinates corresponds to fisher s linear discriminant subspace even though the class identifications of the data points are unknown some illustrative examples are given

an unusual exact closed form solution for axisymmetric vibration of inhomogeneous simply supported circular plates abstract apparently for the first time in the literature an exact closed form solution is derived for an axisymmetrically vibrating inhomogeneous circular plate that is simply supported at its boundary the solution is characterized here as an unusual one since for its counterpart the homogeneous plate transcendental functions are called for whereas here a solution is found in elementary functions namely polynomials the analysis is based on an inverse vibration problem given a candidate mode shape and density distribution we calculate the plate stiffness so that the governing equation for the plate mode shape is identically satisfied we also are able to obtain the expression for the corresponding natural frequency

a fast algorithm for the minimum covariance determinant estimator the minimum covariance determinant mcd method of rousseeuw is a highly robust estimator of multivariate location and scatter its objective is to find h observations out of n whose covariance matrix has the lowest determinant until now applications of the mcd were hampered by the computation time of existing algorithms which were limited to a few hundred objects in a few dimensions we discuss two important applications of larger size one about a production process at philips with n 677 objects and p 9 variables and a dataset from astronomy with n 137 256 objects and p 27 variables to deal with such problems we have developed a new algorithm for the mcd called fast mcd the basic ideas are an inequality involving order statistics and determinants and techniques which we call selective iteration and nested extensions for small datasets fast mcd typically finds the exact mcd whereas for larger datasets it gives more accurate results than existing algorithms and is faster by orders

fast and robust parameter estimation for statistical partial volume models in brain mri due to the finite spatial resolution of imaging devices a single voxel in a medical image may be composed of mixture of tissue types an effect known as partial volume effect pve partial volume estimation that is the estimation of the amount of each tissue type within each voxel has received considerable interest in recent years much of this work has been focused on the mixel model a statistical model of pve we propose a novel trimmed minimum covariance determinant tmcd method for the estimation of the parameters of the mixel pve model in this method each voxel is first labeled according to the most dominant tissue type voxels that are prone to pve are removed from this labeled set following which robust location estimators with high breakdown points are used to estimate the mean and the covariance of each tissue class comparisons between different methods for parameter estimation based on classified images as well as expectation maximization like em like procedure for simultaneous parameter and partial volume estimation are reported the robust estimators based on a pruned classification as presented here are shown to perform well even if the initial classification is of poor quality the results obtained are comparable to those obtained using the em like procedure but require considerably less computation time segmentation results of real data based on partial volume estimation are also reported in addition to considering the parameter estimation problem we discuss differences between different approximations to the complete mixel model in summary the proposed tmcd method allows for the accurate robust and efficient estimation of partial volume model parameters which is crucial to a variety of brain mri data analysis procedures such as the accurate estimation of tissue volumes and the accurate delineation of the cortical surface

nonparametric imputation by data depth abstractwe present single imputation method for missing values which borrows the idea of data depth a measure of centrality defined for an arbitrary point of a space with respect to a probability d

robust orthogonal complement principal component analysis abstractrecently the robustification of principal component analysis pca has attracted lots of attention from statisticians engineers and computer scientists in this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation based on a mathematical formulation of such transformed outliers a novel robust orthogonal complement principal component analysis roc pca is proposed the framework combines the popular sparsity enforcing and low rank regularization techniques to deal with row wise outliers as well as element wise outliers a nonasymptotic oracle inequality guarantees the accuracy and high breakdown performance of roc pca in finite samples to tackle the computational challenges an efficient algorithm is developed on the basis of stiefel manifold optimization and iterative thresholding furthermore a batch variant is proposed to significantly reduce the cost in ultra high dimensions th

three dimensional elasticity solutions for free vibrations of circular plates a polynomials ritz analysis this paper presents accurate three dimensional elasticity solutions for free vibration of circular plates the derivation of a linear frequency equation based on an exact three dimensional small strain linearly elastic principle is detailed the solution to this problem is made possible by using the ritz method with a set of orthogonal polynomial series to approximate the spatial displacements of the circular plate in cylindrical polar coordinates the perturbation of frequency responses due to the variations of boundary conditions and thickness is investigated first known frequency parameters and three dimensional deformed mode shapes are presented in vivid graphical forms the accuracy of these results are verified by appropriate convergence studies and when possible are checked with existing solutions

fuzziness in data analysis the first aim is to emphasize the use of fuzziness in data analysis to capture information that has been traditionally disregarded with a cost in the precision of the conclusions fuzziness can be considered in the data analysis process at various stages but the main target in this paper will be fuzziness in the data depending on the nature of the fuzzy data or the aim to which they are handled different approaches should be applied we attempt to contribute to the clarification of such a difference while focusing on the so called ontic approach in contrast to the epistemic approach the second aim is to underline the need of considering robust methods to reduce the misleading impact of outliers in fuzzy data analysis we propose trimming as a general and intuitive method to discard outliers we exemplify this approach with the case of the ontic fuzzy trimmed mean variance and highlight the differences with the epistemic case all the discussions and developments are illustrated by means of a case study concerning the perception of lengths of men and women

robpca a new approach to robust principal component analysis we introduce a new method for robust principal component analysis pca classical pca is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations two robust approaches have been developed to date the first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an s estimator and is limited to relatively low dimensional data the second approach is based on projection pursuit and can handle high dimensional data here we propose the robpca approach which combines projection pursuit ideas with robust scatter matrix estimation robpca yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data robpca can be computed rapidly and is able to detect exact fit situations as a by product robpca produces a diagnostic plot that displays and classifies the outliers we apply the algorithm to several datasets from chemometrics and engineering

computable robust estimation of multivariate location and shape in high dimension using compound estimators abstract estimation of multivariate shape and location in a fashion that is robust with respect to outliers and is affine equivariant represents a significant challenge the use of compound estimators that use a combinatorial estimator such as rousseeuw s minimum volume ellipsoid mve or minimum covariance determinant mcd to find good starting points for high efficiency robust estimators such as s estimators has been proposed in this article we indicate why this scheme will fail in high dimension due to combinatorial explosion in the space that must be searched for the mve or mcd we propose a meta algorithm based on partitioning the data that enables compound estimators to work in high dimension we show that even when the computational effort is restricted to a linear function of the number of data points the algorithm results in an estimator with good asymptotic properties extensive computational experiments are used to confirm that significant benefits accrue in finite samples as well we also g

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

alternatives to the median absolute deviation abstract in robust estimation one frequently needs an initial or auxiliary estimate of scale for this one usually takes the median absolute deviation mad n 1 4826 med xi med j x j because it has a simple explicit formula needs little computation time and is very robust as witnessed by its bounded influence function and its 50 breakdown point but there is still room for improvement in two areas the fact that mad n is aimed at symmetric distributions and its low 37 gaussian efficiency in this article we set out to construct explicit and 50 breakdown scale estimators that are more efficient we consider the estimator sn 1 1926 med med j xi xj and the estimator qn given by the 25 quantile of the distances xi x j i j note that sn and qn do not need any location estimate both sn and qn can be computed using o n log n time and o n storage the gaussian efficiency of sn is 58 whereas qn attains 82 we study sn and qn by means of their influence functions their b

robust ridge regression for high dimensional data ridge regression being based on the minimization of a quadratic loss function is sensitive to outliers current proposals for robust ridge regression estimators are sensitive to bad leverage observations cannot be employed when the number of predictors p is larger than the number of observations n and have a low robustness when the ratio p n is large in this article a ridge regression estimate based on repeated m estimation mm estimation is proposed it is a penalized regression mm estimator in which the quadratic loss is replaced by an average of where ri are the residuals and the residual scale from an initial estimator which is a penalized s estimator and is a bounded function the mm estimator can be computed for p n and is robust for large p n a fast algorithm is proposed the advantages of the proposed approach over its competitors are demonstrated through both simulated and real data supplemental materials are available online

robust multivariate regression we introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables as a robust estimator of location and scatter we use the minimum covariance determinant mcd estimator of rousseeuw based on simulations we investigate the finite sample performance and robustness of the estimator to increase the efficiency we propose a reweighted estimator selected from several possible reweighting schemes the resulting multivariate regression does not need much computation time and is applied to real datasets we show that the multivariate regression estimator has the appropriate equivariance properties has a bounded influence function and inherits the breakdown value of the mcd estimator these theoretical robustness properties confirm the good finite sample results obtained from the simulations

how to quantify deterministic and random influences on the statistics of the foreign exchange market it is shown that price changes of the u s dollar german mark exchange rates upon different delay times can be regarded as a stochastic marcovian process furthermore we show how kramers moyal coefficients can be estimated from the empirical data finally we present an explicit fokker planck equation which models very precisely the empirical probability distributions in particular their non gaussian heavy tails

fast robust correlation for high dimensional data the product moment covariance matrix is a cornerstone of multivariate data analysis from which one can derive correlations principal components mahalanobis distances and many other results unfo

robust nonparametric methods the terms nonparametric statistics and distributionfree methods have historically referred to a collection of statistical tests whose null distributions do not depend on the underlying distribution of the data the area has been expanded to include estimates and confidence intervals derived from the tests and includes asymptotically distribution free tests for complex models as well it is now generally recognized that these statistical methods are highly efficient over large sets of possible models and are robust as well this was not always the case the earliest work was by hotelling and pabst in 1936 references to works in this paragraph can be found in the books cited in the next paragraph on rank correlation and by friedman in 1937 on rank tests in a two way design followed in 1945 with the introduction of the signed rank and rank sum tests by wilcoxon in 1947 mann and whitney extended the ideas of wilcoxon all of these rank tests were considered quick and dirty but not competitive with the presumably more efficient t tests this perspective was about to change in 1948 pitman developed efficiency concepts in a set of lecture notes that although never published were widely disseminated during the 1950s and 1960s lehmann working with hodges and with his students at berkeley showed that rank tests are surprisingly efficient and robust in fact the wilcoxon rank tests are essentially efficient at the normal model and can be much more efficient than least squares methods when the underlying distribution of the data has heavy tails in the early 1960s hodges and lehmann derived estimates and confidence intervals from rank test statistics and also introduced aligned rank tests for use in the regression model in 1967 hajek and sid published their seminal book on the theory of rank tests it is impossible to mention all of the contributions to this early development hence we cite several key mono

potential force observed in market dynamics abstract as a model of market price we introduce a new type of random walk in a moving potential which is approximated by a quadratic function with its center given by the moving average of its own trace the properties of resulting random walks are similar to those of ordinary random walks for large time scales however their short time properties are approximated by abnormal diffusion with nontrivial exponents a new data analysis method based on this model enables us to observe temporal changes of potential forces from high precision market data directly

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

robust averaging of covariances for eeg recordings classification in motor imagery brain computer interfaces the estimation of covariance matrices is of prime importance to analyze the distribution of multivariate signals in motor imagery based brain computer interfaces mi bci covariance matrices play a central role in the extraction of features from recorded electroencephalograms eegs therefore correctly estimating covariance is crucial for eeg classification this letter discusses algorithms to average sample covariance matrices scms for the selection of the reference matrix in tangent space mapping tsm based mi bci tangent space mapping is a powerful method of feature extraction and strongly depends on the selection of a reference covariance matrix in general the observed signals may include outliers therefore taking the geometric mean of scms as the reference matrix may not be the best choice in order to deal with the effects of outliers robust estimators have to be used in particular we discuss and test the use of geometric medians and trimmed averages defined on the basis of several m

robust estimation of unbalanced mixture models on samples with outliers mixture models are often used to compactly represent samples from heterogeneous sources however in real world the samples generally contain an unknown fraction of outliers and the sources generate different or unbalanced numbers of observations such unbalanced and contaminated samples may for instance be obtained by high density data sensors such as imaging devices estimation of unbalanced mixture models from samples with outliers requires robust estimation methods in this paper we propose a novel robust mixture estimator incorporating trimming of the outliers based on component wise confidence level ordering of observations the proposed method is validated and compared to the state of the art fast tle method on two data sets one consisting of synthetic samples with a varying fraction of outliers and a varying balance between mixture weights while the other data set contained structural magnetic resonance images of the brain with tumors of varying volumes the results on both data sets clearly indicate that the proposed method is capable to robustly estimate unbalanced mixtures over a broad range of outlier fractions as such it is applicable to real world samples in which the outlier fraction cannot be estimated in advance

robust estimates of location and dispersion for high dimensional datasets the computing times of high breakdown point estimates of multivariate location and scatter increase rapidly with the number of variables which makes them impractical for high dimensional datasets such as those used in data mining we propose an estimator of location and scatter based on a modified version of the gnanadesikan kettenring robust covariance estimate we compare its behavior with that of the stahel donoho sd and rousseeuw and van driessen s fast mcd fmcd estimates in simulations with contaminated multivariate normal data our estimate is almost as good as sd and clearly better than fmcd it is much faster than both especially for large dimension we give examples with real data with dimensions between 5 and 93 in which the proposed estimate is as good as or better than sd and fmcd at detecting outliers and other structures with much shorter computing times

robust linear clustering structures are not well represented by points we present a method that searches for linear groups in the presence of outliers the method is based on the idea of impartial trimming we search for the best subsample containing a proportion 1 a of the data and the best k affine subspaces fitting to those non discarded observations by measuring discrepancies through orthogonal distances the population version of the sample problem is also considered we prove the existence of solutions for the sample and population problems together with their consistency a feasible algorithm for solving the sample problem is described as well finally some examples showing how the method proposed works in practice are provided

free vibration of polar orthotropic circular plates of variable thickness with elastically restrained edge abstract asymmetric vibrations of polar orthotropic circular plates of linearly varying thickness with elastic rigid support are discussed on the basis of the classical plate theory an approximate solution of the problem is obtained by the rayleigh ritz method using functions based on static deflection of polar orthotropic circular plates this type of approximating functions has a faster rate of convergence as compared to the polynomial co ordinate functions frequency parameters of the plate have been obtained for different values of taper parameter flexibility parameter and rigidity ratio in the first three modes of vibration a comparison of results for special cases of isotropy uniform thickness and of axisymmetric vibrations obtained by other techniques such as the finite element method and the receptance method shows excellent agreement

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

on one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

robust low rank approximation of data matrices with elementwise contamination we propose a robust method to approximate an n p data matrix with one of rank q the method is based on yohai s regression mm estimates it is intended to be resistant against the existence of both atypical rows and of scattered atypical cells and also to be able to cope with missing values we propose an algorithm based on alternating m regressions and a starting estimate based on successive rank one fits which involves o npq operations simulations show that our estimate outperforms competing estimates in terms of both efficiency and resistance three high dimensional real data sets are analyzed the running time of our estimate for large data sets is shown to be less than that of its competitors

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

robust reduced rank regression in high dimensional multivariate regression problems enforcing low rank in the coefficient matrix offers effective dimension reduction which greatly facilitates parameter estimation and model interpretation however commonly used reduced rank methods are sensitive to data corruption as the low rank dependence structure between response variables and predictors is easily distorted by outliers we propose a robust reduced rank regression approach for joint modelling and outlier detection the problem is formulated as a regularized multivariate regression with a sparse mean shift parameterization which generalizes and unifies some popular robust multivariate methods an efficient thresholding based iterative procedure is developed for optimization we show that the algorithm is guaranteed to converge and that the coordinatewise minimum point produced is statistically accurate under regularity conditions our theoretical investigations focus on non asymptotic robust analysis demonstrating that joint rank reduction and outlier detection leads to improved prediction accuracy in particular we show that redescending formula see text functions can essentially attain the minimax optimal error rate and in some less challenging problems convex regularization guarantees the same low error rate the performance of the proposed method is examined through simulation studies and real data examples

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

lekhnitskii s classic formula serving as an exact mode shape of simply supported polar orthotropic inhomogeneous circular plates the semi inverse problem of free vibration of simply supported inhomogeneous polar orthotropic plates is studied the mode shape is postulated in the form of the classic formula by lekhnitskii namely the static deflection of the associated uniform polar orthotropic circular plate under uniform loading the ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers so that the candidate mode shapes constitute the polynomial functions the flexural rigidities themselves are also represented by polynomials the semi inverse problem of identifying the coefficients of the flexural rigidities is solved analytically it appears remarkable that for numerous cases the simple method developed in this study provides novel closed form solutions for the design of the polar orthotropic circular plates with pre specified mode shapes

the asymptotics of rousseeuw s minimum volume ellipsoid estimator rousseeuw s minimum volume estimator for multivariate location and dispersion parameters has the highest possible breakdown point for an affine equivariant estimator in this paper we establish that it satisfies a local holder condition of order 1 2 and converges weakly at the rate of n 1 3 to a non gaussian distribution

a bounded influence high breakdown efficient regression estimator abstract we consider the multiple linear regression model y i x i e i i 1 2 n with random carriers and focus on the estimation of this article s main contribution is to present an estimator that is affine regression and scale equivariant has both a high breakdown point and a bounded influence function and has an asymptotic efficiency greater than 95 versus least squares under gaussian errors we give conditions under which the estimator a one step general m estimator that uses schweppe weights and is based on a high breakdown initial estimator satisfies these properties the major conditions necessary for the estimator are a it must be based on a n consistent initial estimator with a 50 breakdown point and b it must be based on a psi function that is odd bounded and strictly increasing the advantage of this estimator over previous approaches is that it does not downweight high leverage points without first considering how they fit the bulk of the data methods of computi

the complexity of computing the mcd estimator in modern statistics the robust estimation of parameters is a central problem i e an estimation that is not or only slightly affected by outliers in the data the minimum covariance determinant mcd estimator j amer statist assoc 79 1984 871 is probably one of the most important robust estimators of location and scatter the complexity of computing the mcd however was unknown and generally thought to be exponential even if the dimensionality of the data is fixed here we present a polynomial time algorithm for mcd for fixed dimension of the data in contrast we show that computing the mcd estimator is np hard if the dimension varies

finding an unknown number of multivariate outliers we use the forward search to provide robust mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic we also introduce several new robust distances with associated distributional results comparisons of our procedure with tests using other robust mahalanobis distances show the good size and high power of our procedure we also provide a unification of results on correction factors for estimation from truncated samples

a robust method for cluster analysis let there be given a contaminated list of n r d valued observations coming from g different normally distributed populations with a common covariance matrix we compute the ml estimator with respect to a certain statistical model with n r outliers for the parameters of the g populations it detects outliers and simultaneously partitions their complement into g clusters it turns out that the estimator unites both the minimum covariance determinant rejection method and the well known pooled determinant criterion of cluster analysis we also propose an efficient algorithm for approximating this estimator and study its breakdown points for mean values and pooled ssp matrix

inconsistency of resampling algorithms for high breakdown regression estimators and a new algorithm because high breakdown estimators hbes are impractical to compute exactly in large samples approximate algorithms are used the algorithm generally produces an estimator with a lower consistency rate and breakdown value than the exact theoretical estimator this discrepancy grows with the sample size with the implication that huge computations are needed for good approximations in large high dimension samples the workhorse for hbes has been the elemental set or basic resampling algorithm this turns out to be completely ineffective in high dimensions with high levels of contamination however enriching it with a concentration step turns it into a method that can handle even high levels of contamination provided that the regression outliers are located on random cases it remains ineffective if the regression outliers are concentrated on high leverage cases we focus on the multiple regression problem but several of the broad conclusions notably those of the inadequacy of fixed numbers of

applications of robust distances for regression the dd plot introduced by rousseeuw and van driessen 1999 is a plot of classical vs robust mahalanobis distances mdi vs rdi the dd plot can be used as a diagnostic for multivariate normality and elliptical symmetry and to assess the success of numerical transformations towards elliptical symmetry in the regression context many procedures can be adversely aected if strong nonlinearities are present in the predictors even if strong nonlinearities are present the robust distances can be used to help visualize important regression models such as generalized linear models

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates

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 mini

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

robust estimation of multivariate location and scatter in the presence of missing data two main issues regarding data quality are data contamination outliers and data completion missing data these two problems have attracted much attention and research but surprisingly they are seldom considered together popular robust methods such as s estimators of multivariate location and scatter offer protection against outliers but cannot deal with missing data except for the obviously inefficient approach of deleting all incomplete cases we generalize the definition of s estimators of multivariate location and scatter to simultaneously deal with missing data and outliers we show that the proposed estimators are strongly consistent under elliptical models when data are missing completely at random we derive an algorithm similar to the expectation maximization algorithm for computing the proposed estimators this algorithm is initialized by an extension for missing data of the minimum volume ellipsoid we assess the performance of our proposal by monte carlo simulation and give some real dat

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

multivariate outlier detection with high breakdown estimators in this paper we develop multivariate outlier tests based on the high breakdown minimum covariance determinant estimator the rules that we propose have good performance under the null hypothesis of no outliers in the data and also appreciable power properties for the purpose of individual outlier detection this achievement is made possible by two orders of improvement over the currently available methodology first we suggest an approximation to the exact distribution of robust distances from which cut off values can be obtained even in small samples our thresholds are accurate simple to implement and result in more powerful outlier identification rules than those obtained by calibrating the asymptotic distribution of distances the second power improvement comes from the addition of a new iteration step after one step reweighting of the estimator the proposed methodology is motivated by asymptotic distributional results its finite sample performance is evaluated through simulations and compared to

using robust dispersion estimation in support vector machines in this paper a novel support vector machine svm variant which makes use of robust statistics is proposed we investigate the use of statistically robust location and dispersion estimators in order to enhance the performance of svms and test it in two class and multi class classification problems moreover we propose a novel method for class specific multi class svm which makes use of the covariance matrix of only one class i e the class that we are interested in separating from the others while ignoring the dispersion of other classes we performed experiments in artificial data as well as in many real world publicly available databases used for classification the proposed approach performs better than other svm variants especially in cases where the training data contain outliers finally we applied the proposed method for facial expression recognition in three well known facial expression databases showing that it outperforms previously published attempts

robust regression methods for computer vision a review regression analysis fitting a model to noisy data is a basic technique in computer vision robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance we review several robust estimation techniques and describe in detail the least median of squares lmeds method the method yields the correct result even when half of the data is severely corrupted its efficiency in the presence of gaussian noise can be improved by complementing it with a weighted least squares based procedure the high time complexity of the lmeds algorithm can be reduced by a monte carlo type speed up technique we discuss the relationship of lmeds with the ransac paradigm and its limitations in the presence of noise corrupting all the data and we compare its performance with the class of robust m estimators references to published applications of robust techniques in computer vision are also given

statistical properties of the moving average price in dollar yen exchange rates we introduce a weighted moving average analysis for the tick by tick data of yen dollar exchange rates the weights are determined automatically for given data by applying the yule walker formula for autoregressive model although the data are non stationary the resulting moving average gives a quite nice property that the deviation around the moving average becomes a white noise the weights decay exponentially with time scale less than 2 min implying that dealers are watching only very recent market state

a general trimming approach to robust cluster analysis we introduce a new method for performing clustering with the aim of fitting clusters with different scatters and weights it is de signed by allowing to handle a proportionof contaminating data to guarantee the robustness of the method as a characteristic fea ture restrictions on the ratio between the maximum and the mini mum eigenvalues of the groups scatter matrices are introduced this makes the problem to be well defined and guarantees the consistency of the sample solutions to the population ones the method covers a wide range of clustering approaches depend ing on the strength of the chosen restrictions our proposal includes an algorithm for approximately solving the sample problem

outlier detection for high dimensional data outlier detection is an integral component of statistical modelling and estimation for high dimensional data classical methods based on the mahalanobis distance are usually not applicable we propose an outlier detection procedure that replaces the classical minimum covariance determinant estimator with a high breakdown minimum diagonal product estimator the cut off value is obtained from the asymptotic distribution of the distance which enables us to control the type i error and deliver robust outlier detection simulation studies show that the proposed method behaves well for high dimensional data

projection based outlier detection in functional data summarywe propose a procedure based on a high breakdown mean function estimator to detect outliers in functional data the robust estimator is obtained from a clean subset of observations excluding potential outliers by minimizing the least trimmed squares projection coefficients after functional principal component analysis a threshold rule based on the asymptotic distribution of the functional score based distance robustly controls the false positive rate and detects outliers effectively further improvement in power can be achieved by adding a one step reweighting procedure the finite sample performance of our method demonstrates satisfactory false positive and false negative rates compared with existing outlier detection methods for functional data

a cluster based outlier detection scheme for multivariate data detection power of the squared mahalanobis distance statistic is significantly reduced when several outliers exist within a multivariate dataset of interest to overcome this masking effect we propose a computer intensive cluster based approach that incorporates a reweighted version of rousseeuw s minimum covariance determinant method with a multi step cluster based algorithm that initially filters out potential masking points compared to the most robust procedures simulation studies show that our new method is better for outlier detection additional real data comparisons are given supplementary materials for this article are available online