computation of minimum volume covering ellipsoids we present a practical algorithm for computing the minimum volume n dimensional ellipsoid that must contain m given points a1 am n this convex constrained problem arises in a variety of applied computational settings particularly in data mining and robust statistics its structure makes it particularly amenable to solution by interior point methods and it has been the subject of much theoretical complexity analysis here we focus on computation we present a combined interior point and active set method for solving this problem our computational results demonstrate that our method solves very large problem instances m 30 000 and n 30 to a high degree of accuracy in under 30 seconds on a personal computer

on the homology of the hilbert scheme of points in the plane geir ellingsrud 1 and stein arild stromme 2 i matematisk institutt universitetet i oslo blindern n oslo 3 norway 2 matematisk institutt universitetet i bergen n 5014 bergen norway although several authors have been interested in the hilbert scheme hilba ip z parametrizing finite subschemes of length d in the projective plane i1 i2 f 1 f2 br among others not much is known about the topological properties of this space the picard group has been calculated i f2 and the homology groups of hilb3 ip 2 have been computed hi in this paper we give a precise description of the additive structure of the homology of hilba ip2 applying the results of birula bialynicki b1 b2 on the cellular decompositions de fined by a torus action to the natural action of a maximal torus of sl 3 on hilbe ipz a rather easy consequence of the fact that this action has finitely many fixpoints is that the cycle maps between the chow groups and the homology groups are isomorphisms in particular there is no odd homology and the homology groups are all free the main objective of this work is to compute their ranks the betti numbers of hilba ipz as a byproduct of our method we get similar results on the homology of the punctual hilbert scheme and of the hilbert scheme of points in the affine plane it seems natural to generalize our results to any toric smooth surface however we give the results only for the rational ruled surfaces if with an indication of the necessary changes in the proofs for simplicity we work over the field of complex numbers but with an appropriate interpretation of the word homology our results remain valid over any base field

three letters to walter feit on group representations and quaternions abstract representations of the quaternion group by 2 2 matrices with coefficients in the ring of integers of an imaginary quadratic field

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

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

towards the multiplicative behavior of the k theoretical mckay correspondence when the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the hilbert scheme of regular orbits of nakamura then there is a natural isomorphism between the grothendieck group of this resolution and the representation ring of the group given by the bridgeland king reid map however this isomorphism is not compatible with the ring structures for the hilbert scheme of points on the affine plane we study the multiplicative behavior of this map

poisson deformations of symplectic quotient singularities abstract we establish a connection between smooth symplectic resolutions and symplectic deformations of a possibly singular affine poisson variety in particular let v be a finite dimensional complex symplectic vector space and g sp v a finite subgroup our main result says that the so called calogero moser deformation of the orbifold v g is in an appropriate sense a versal poisson deformation that enables us to determine the algebra structure on the cohomology h x c of any smooth symplectic resolution x v g multiplicative mckay correspondence we prove further that if g gl h is an irreducible weyl group and v h h then no smooth symplectic resolution of v g exists unless g is of types a b c

on the mckay correspondences for the hilbert scheme of points on the affine plane the quotient of a finite dimensional vector space by the action of a finite subgroup of automorphisms is usually a singular variety under appropriate assumptions the mckay correspondence relates the geometry of nice resolutions of singularities and the representations of the group for the hilbert scheme of points on the affine plane we study how different correspondences mckay dual mckay and multiplicative mckay are related to each other

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

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

symmetric groups and the cup product on the cohomology of hilbert schemes let c sn be the z module of integer valued class functions on the symmetric group sn we introduce a graded version of the con volution product on c sn and show that there is a degree preserving ring isomorphism c sn h hilb n a 2 z to the cohomology of the hilbert scheme of points in the complex affine plane

trimming and likelihood robust location and dispersion estimation in the elliptical model robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated mahalanobis distance here we analyze some one or k step maximum likelihood estimators computed on a subsample obtained with such a procedure we introduce different models which arise naturally from the ways in which the discarded data can be treated leading to truncated or censored likelihoods as well as to a likelihood based on an only outliers gross errors model results on existence uniqueness robustness and asymptotic properties of the proposed estimators are included a remarkable fact is that the proposed estimators generally keep the breakdown point of the initial robust estimators but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate n 1 2 independently of the rate of convergence of the initial estimator

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

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

minimizing buffered probability of exceedance by progressive hedging stochastic programming problems have for a long time been posed in terms of minimizing the expected value of a random variable influenced by decision variables but alternative objectives can also be considered such as minimizing a measure of risk here something different is introduced minimizing the buffered probability of exceedance for a specified loss threshold the buffered version of the traditional concept of probability of exceedance has recently been developed with many attractive properties that are conducive to successful optimization in contrast to the usual concept which is often posed simply as the probability of failure the main contribution here is to demonstrate that in minimizing buffered probability of exceedance the underlying convexities in a stochastic programming problem can be maintained and the progressive hedging algorithm can be employed to compute a solution

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

semisimplicity in symmetric rigid tensor categories abstract let be a partition of a positive integer n let c be a symmetric rigid tensor category over a field k of characteristic 0 or char k n and let v be an object of c in our main result theorem 4 3 we introduce a finite set of integers f and prove that if the schur functor s v of v is semisimple and the dimension of v is not in f then v is semisimple moreover we prove that for each d f there exist a symmetric rigid tensor category c over k and a non semisimple object v c of dimension d such that s v is semisimple which shows that our result is the best possible in particular theorem 4 3 extends two theorems of serre for c rep g g is a group and s v is n v or sym n v and proves a conjecture of serre 1997 s2

asymptotics of reweighted estimators of multivariate location and scatter we investigate the asymptotic behavior ofa weighted sample mean and covariance where the weights are determined by the mahalanobis distances with respect to initial robust estimators we derive an explicit expansion for the weighted estimators from this expansion it can be seen that reweighting does not improve the rate ofconvergence ofthe initial estimators we also show that ifone uses smooth s estimators to determine the weights the weighted estimators are asymptotically normal finally we will compare the efficiency and local robustness of the reweighted s estimators with two other improvements of s estimators estimators and constrained m estimators

identifying multiple outliers in multivariate data summary we propose a procedure for the detection of multiple outliers in multivariate data let xbe an n x p data matrix representing n observations onp variates we first order the n observations using an appropriately chosen robust measure of outlyingness then divide the data set into two initial subsets a basic subset which containsp 1 good observations and a nonbasic subset which contains the remaining n p 1 observations second we compute the relative distance from each point in the data set to the centre of the basic subset relative to the possibly singular covariance matrix of the basic subset third we rearrange the n observations in ascending order accordingly then divide the data set into two subsets a basic subset which contains the first p 2 observations and a non basic subset which contains the remaining n p 2 observations this process is repeated until an appropriately chosen stopping criterion is met the final non basic subset of observations is declared an outlying subset the procedure proposed is illustrated and compared with existing methods by using several data sets the procedure is simple computationally inexpensive suitable for automation computable with widely available software packages effective in dealing with masking and swamping problems and most importantly successful in identifying multivariate outliers

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

maximization of auc and buffered auc in binary classification in binary classification performance metrics that are defined as the probability that some error exceeds a threshold are numerically difficult to optimize directly and also hide potentially important information about the magnitude of errors larger than the threshold defining similar metrics instead using buffered probability of exceedance bpoe generates counterpart metrics that resolve both of these issues we apply this approach to the case of auc the area under the roc curve and define buffered auc bauc we show that bauc can provide insights into classifier performance not revealed by auc while being closely related as the tightest concave lower bound and representable as the area under a modified roc curve additionally while auc is numerically difficult to optimize directly we show that bauc optimization often reduces to convex or linear programming extending these results we show that auc and bauc are special cases of generalized bauc and that popular support vector machine svm formulations for approximately maximizing auc are equivalent to direct maximization of generalized bauc we also prove bauc generalization bounds for these svm s as a central component to these results we provide an important novel formula for calculating bpoe the inverse of conditional value at risk using this formula we show that particular bpoe minimization problems reduce to convex and linear programming

robust diagnostic data analysis transformations in regression we introduce a very general forward method of data analysis that starts from a small robustly chosen subset of the data and shows the effect of adding observations by a forward search powerful diagnostic procedures result the observations are ordered by their agreement with the proposed transformation masking is overcome and the inferential effect of each observation is clear we apply the resulting method to the transformation of both univariate and multivariate data other applications of the forward search are mentioned

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

estimation and asymptotics for buffered probability of exceedance abstract this paper studies statistical properties of empirical sample estimates of the buffered probability of exceedance bpoe the estimation procedure is based on one dimensional minimization representation of the bpoe convergence rates and asymptotic properties of the suggested estimation procedures are investigated theoretical predictions are validated with numerical experiments including a special case of exponential distribution and a study proposing bpoe modification of minimum volume ellipsoid problem

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

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

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

the asymptotics of s estimators in the linear regression model we consider the consistency and weak convergence of s estimators in the linear regression model sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover for example polynomial trends and i i d carriers a weak convergence theorem for the hampel rousseeuw least median of squares estimator is obtained and it is shown under rather general conditions that the correct norming factor is n 1 3 finally the asymptotic normality of s estimators with a smooth rho function is obtained again under weak conditions on the carriers

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

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

location of several outliers in multiple regression data using elemental sets the outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model each such subset yields an elemental residual for the case in question and a suitable summary statistic of them can be used as an estimate of the case s outlying tendency we propose two such summary statistics an unweighted median which is of bounded influence and a weighted median which is more efficient but less robust the computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p as a byproduct the method yields useful information on the influence or leverage of cases and the mutual masking of high leverage points

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

complete reducibility and conjugacy classes of tuples in algebraic groups and lie algebras let h be a reductive subgroup of a reductive group g over an algebraically closed field k we consider the action of h on g n the n fold cartesian product of g with itself by simultaneous conjugation we give a purely algebraic characterization of the closed h orbits in g n generalizing work of richardson which treats the case h g this characterization turns out to be a natural generalization of serre s notion of g complete reducibility this concept appears to be new even in characteristic zero we discuss how to extend some key results on g complete reducibility in this framework we also consider some rationality questions

chern classes of tautological sheaves on hilbert schemes of points on surfaces we give an algorithmic description of the action of the chern classes of tautological bundles on the cohomology of hilbert schemes of points on a smooth surface within the framework of nakajima s oscillator algebra this leads to an identification of the cohomology ring of hilb n a2 with a ring of explicitly given differential operators on a fock space we end with the computation of the top segre classes of tautological bundles associated to line bundles on hilb n up to n 7 extending computations of severi lebarz tikhomirov and troshina and give a conjecture for the generating series

minimum volume ellipsoids containing compact sets application to parameter bounding abstract the problem of finding the minimum volume ellipsoid containing a compact set is shown to be equivalent to the determination of an optimal distribution of weights over several equivalent conditions for optimality of this distribution are obtained and used to construct algorithms guaranteed to converge to the optimum these algorithms can be used to approximate the posterior feasible set for the parameters of a model in the bounded error context linear and nonlinear illustrative examples are treated

unmasking multivariate outliers and leverage points comment 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 to avoid this masking effect we propose to compute distances based on very robust estimates of location and covariance in the case of regression data the outliers also may be unmasked by using a highly robust regression method a new display is proposed in which the robust regression residuals are plotted versus the robust distances

the brauer algebra and the symplectic schur algebra let k be an algebraically closed field of characteristic p 0 let m r be integers with m 1 r 0 and m r and let s0 2m r be the symplectic schur algebra over k as introduced by the first author we introduce the symplectic schur functor derive some basic properties of it and relate this to work of hartmann and paget we do the same for the orthogonal schur algebra we give a modified jantzen sum formula and a block result for the symplectic schur algebra under the assumption that r and the residue of 2m mod p are small relative to p from this we deduce a block result for the orthogonal schur algebra under similar assumptions we also show that in general the block relations of the brauer algebra and the symplectic and orthogonal schur algebra are the same finally we deduce from the previous results a new proof of the description of the blocks of the brauer algebra in characteristic 0 as obtained by cox de visscher and martin