discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

nonparametric variable transformation in sufficient dimension reduction sufficient dimension reduction sdr techniques have proven to be very useful data analysis tools in various applications underlying many sdr techniques is a critical assumption that the predictors are elliptically contoured when this assumption appears to be wrong practitioners usually try variable transformation such that the transformed predictors become nearly normal the transformation function is often chosen from the log and power transformation family as suggested in the celebrated box cox model however any parametric transformation can be too restrictive causing the danger of model misspecification we suggest a nonparametric variable transformation method after which the predictors become normal to demonstrate the main idea we combine this flexible transformation method with two well established sdr techniques sliced inverse regression sir and inverse regression estimator ire the resulting sdr techniques are referred to as tsir and tire respectively both simulation and real da

quantum isometry groups of 0 dimensional manifolds quantum isometry groups of spectral triples associated with ap proximately finite dimensional c algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated bratteli dia grams this is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middle third cantor set it is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural laplacians

existence and examples of quantum isometry groups for a class of compact metric spaces we formulate a definition of isometric action of a compact quantum group cqg on a compact metric space generalizing banica s definition for finite metric spaces for metric spaces x d which can be isometrically embedded in some euclidean space we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on x d in fact our existence theorem applies to a larger class namely for any compact metric space x d which admits a one to one continuous map f x rn for some n such that d0 f x f y d x y where d0 is the euclidean metric for some homeomorphism of r as concrete examples we obtain wang s quantum permutation group sn and also the free wreath product of z2 by sn as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in 13

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

discrete product systems and twisted crossed products by semigroups a product systemeover a semigrouppis a family of hilbert spaces es s p together with multiplicationses et est we vieweas a unitary valued cocycle onp and consider twisted crossed productsa e pinvolvingeand an action ofpby endomorphisms of ac algebraa whenpis quasi lattice ordered in the sense of nica we isolate a class of covariant representations ofe and consider a twisted crossed productbp e pwhich is universal for covariant representations ofewhenehas finite dimensional fibres and in general is slightly larger in particular whenp nand dim e1 our algebrabn e n is a new infinite analogue of the toeplitz cuntz algebras ton our main theorem is a characterisation of the faithful representations ofbp e p

identifying regression outliers and mixtures graphically regressions in practice can include outliers and other unknown subpopulation structures for example mixtures of regressions occur if there is an omitted categorical predictor like gender or location and different regressions occur within each category the theory of regression graphics based on central subspaces can be used to construct graphical solutions to long standing problems of this type it is argued that in practice the central subspace automatically expands to incorporate outliers and regression mixtures thus methods of estimating the central subspace can be used to identify these phenomena without specifying a model examples illustrating the power of the theory are presented

estimating central subspaces via inverse third moments summary modern graphical tools have enhanced our ability to learn many things from data directly in recent years dimension reduction has proven to be an effective tool for generating low dimensional summary plots without appreciable loss of information some well known inverse regression methods for dimension reduction such as sliced inverse regression li 1991 and sliced average variance estimation cook weisberg 1991 have been developed to estimate summary plots for regression and discriminant analysis in this paper we suggest a new method that makes use of inverse third moments this method can find structure beyond that found by sliced inverse regression and sliced average variance estimation particularly regression mixtures illustrative examples are presented

when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional evans d g and sims a when is the cuntz krieger algebra of a higher rank graph approximately finite dimensional preprint arxiv 1112 4549v1

transformations of the explanatory variables in the logistic regression model for binary data summary some results are presented on improving the fit of the logistic regression model for binary data by transforming the vector of explanatory variables the methods are based on consideration of the distributions of these variables conditional on outcome group the transformations required are the functions of the explanatory variables which appear in the log density ratio of the conditional distributions

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

dimension reduction for the conditional kth moment in regression the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model central subspaces are designed to capture all the information for the regression and to provide a population structure for dimension reduction here we introduce the central kth moment subspace to capture information from the mean variance and so on up to the kth conditional moment of the regression new methods are studied for estimating these subspaces connections with sliced inverse regression are established and examples illustrating the theory are presented

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

determining the dimensionality in sliced inverse regression abstract a general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term sliced inverse regression is a method for determining these linear combinations in this article we address the problem of determining how many linear combinations are involved procedures based on conditional means and conditional covariance matrices as well as a procedure combining the two approaches are considered in each case we develop a test that has an asymptotic chi squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution

graphical tools for quadratic discriminant analysis sufficient dimension reduction methods provide effective ways to visualize discriminant analysis problems for example cook and yin showed that the dimension reduction method of sliced average variance estimation save identifies variates that are equivalent to a quadratic discriminant analysis qda solution this article makes this connection explicit to motivate the use of save variates in exploratory graphics for discriminant analysis classification can then be based on the save variates using a suitable distance measure if the chosen measure is mahalanobis distance then classification is identical to qda using the original variables just as canonical variates provide a useful way to visualize linear discriminant analysis lda so do save variates help visualize qda this would appear to be particularly useful given the lack of graphical tools for qda in current software furthermore whereas lda and qda can be sensitive to nonnormality save is more robust

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

on distribution weighted partial least squares with diverging number of highly correlated predictors summary because highly correlated data arise from many scientific fields we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated for this we first develop a distribution weighted least squares estimator that can recover directions in the central subspace then use the distribution weighted least squares estimator as a seed vector and project it onto a krylov space by partial least squares to avoid computing the inverse of the covariance of predictors thus distrbution weighted partial least squares can handle the cases with high dimensional and highly correlated predictors furthermore we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares for theoretical investigation we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate o n1 2 log n and o n1 3 respectively where n is the sample size when there are no other constraints on the covariance of predictors the rates n1 2 and n1 3 are optimal we also propose a bayesian information criterion type of criterion to estimate the dimension of the krylov space in the partial least squares procedure illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non ellipticity and works well even in small n large p problems

the interplay between k graphs and the yang baxter equation in this paper we initiate the study of the interplay between k graphs and the yang baxter equation for this we provide two very different perspectives on one hand we show that the set of all set theoretic solutions of the yang baxter equation is a special class of single vertex k graphs as a consequence we construct an infinite family of large solutions of the yang baxter equation from an arbitrarily given one on the other hand we prove that all single vertex k graphs are yb semigroups of square free involutive solutions of the yang baxter equation other various connections are also investigated

on surrogate dimension reduction for measurement error regression an invariance law we consider a general nonlinear regression problem where the predictors contain measurement error it has been recently discovered that several well known dimension reduction methods such as ols sir and phd can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor in this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces which implies that at least at the population level the two dimension reduction problems are in fact equivalent consequently we can apply all existing dimension reduction methods to measurement error regression problems the equivalence holds exactly for multivariate normal predictors and approximately for arbitrary predictors we also characterize the rate of convergence for the surrogate dimension reduction estimators finally we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

non existence of genuine compact quantum symmetries of compact connected smooth manifolds abstract suppose that a compact quantum group q acts faithfully on a smooth compact connected manifold m i e has a c co action on c m such that c m c m q and the linear span of c m 1 q is dense in c m q with respect to the frechet topology it was conjectured by the author quite a few years ago that q must be commutative as a c algebra i e q c g for some compact group g acting smoothly on m the goal of this paper is to prove the truth of this conjecture a remarkable aspect of the proof is the use of probabilistic techniques involving brownian stopping time

topological automorphism groups of compact quantum groups we study the topological structure of the automorphism groups of compact quantum groups showing that in parallel to a classical result due to iwasawa the connected component of identity of the automorphism group and of the inner automorphism group coincide for compact matrix quantum groups which can be thought of as quantum analogues of compact lie groups we prove that the inner automorphism group is a compact lie group and the outer automorphism group is discrete applications of this to the study of group actions on compact quantum groups are highlighted we end by providing examples of compact matrix quantum groups with infinitely generated fusion rings in stark contrast with the classical situation along the way we study the invariant theory of finite group actions on free laurent rings and show that the rings of invariants are in general not finitely generated

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

sufficient dimension reduction for spatial point processes directed by gaussian random fields summary we develop a sufficient dimension reduction paradigm for inhomogeneous spatial point processes driven by gaussian random fields specifically we introduce the notion of the kth order central intensity subspace we show that a central subspace can be defined as the combination of all central intensity subspaces for many commonly used spatial point process models we find that the central subspace is equivalent to the first order central intensity subspace to estimate the latter we propose a flexible framework under which most existing benchmark inverse regression methods can be extended to the spatial point process setting we develop novel graphical and formal testing methods to determine the structural dimension of the central subspace these methods are extremely versatile in that they do not require any specific model assumption on the correlation structures of the covariates and the spatial point process to illustrate the practical use of the methods proposed we apply them to both simulated data and two real examples

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

metric transforms and euclidean embeddings it is proved that if 0 c 0 72 then for any point met ric space x d the metric space x dc is isometrically embeddable into a euclidean space for 6 point metric space c j log2 is the largest exponent that guarantees the existence of isometric embeddings into a euclidean space such largest exponent is also determined for all point graphs with truncated distance

assessing the number of linear components in a general regression problem abstract this article presents a method for testing the dimension of a general regression model without assuming any specific model for the distribution of the regressors examples of application and comparisons with existing methods are also studied

dimension reduction for non elliptically distributed predictors second order methods summary many classical dimension reduction methods especially those based on inverse conditional moments require the predictors to have elliptical distributions or at least to satisfy a linearity condition such conditions however are too strong for some applications li and dong 2009 introduced the notion of the central solution space and used it to modify first order methods such as sliced inverse regression so that they no longer rely on these conditions in this paper we generalize this idea to second order methods such as sliced average variance estimation and directional regression in doing so we demonstrate that the central solution space is a versatile framework we can use it to modify essentially all inverse conditional moment based methods to relax the distributional assumption on the predictors simulation studies and an application show a substantial improvement of the modified methods over their classical counterparts

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

estimating the structural dimension of regressions via parametric inverse regression a new estimation method for the dimension of a regression at the outset of an analysis is proposed a linear subspace spanned by projections of the regressor vector x which contains part or all of the modelling information for the regression of a vector y on x and its dimension are estimated via the means of parametric inverse regression smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model no restrictions are placed on the distribution of the regressors the estimate of the dimension of the regression is based on optimal estimation procedures a simulation study shows the method to be more powerful than sliced inverse regression in some situations

a two stage dimension reduction method for transformed responses and its applications researchers in the biological sciences nowadays often encounter the curse of dimensionality to tackle this sufficient dimension reduction aims to estimate the central subspace in which all the necessary information supplied by the covariates regarding the response of interest is contained subsequent statistical analysis can then be made in a lower dimensional space while preserving relevant information many studies are concerned with the transformed response rather than the original one but they may have different central subspaces when estimating the central subspace of the transformed response direct methods will be inefficient in this article we propose a more efficient two stage estimator of the central subspace of a transformed response this approach is extended to censored responses and is applied to combining multiple biomarkers simulation studies and data examples support the superiority of the procedure copyright 2012 oxford university press

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

sufficient dimension reduction through discretization expectation estimation in the context of sufficient dimension reduction the goal is to parsimoniously recover the central subspace of a regression model many inverse regression methods use slicing estimation to recover the central subspace the efficacy of slicing estimation depends heavily upon the number of slices however the selection of the number of slices is an open and long standing problem in this paper we propose a discretization expectation estimation method which avoids selecting the number of slices while preserving the integrity of the central subspace this generic method assures root n consistency and asymptotic normality of slicing estimators for many inverse regression methods and can be applied to regressions with multivariate responses a bic type criterion for the dimension of the central subspace is proposed comprehensive simulations and an illustrative application show that our method compares favourably with existing estimators

quantum group of isometries in classical and noncommutative geometry we formulate a quantum generalization of the notion of the group of riemannian isometries for a compact riemannian manifold by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples and then proving the existence of a universal object called the quantum isometry group in the category of compact quantum groups acting smoothly and isometrically on a given possibly noncommutative manifold satisfying certain regularity assumptions the idea of quantum families due to woronowicz and soltan are relevant to our construction a number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed

existence and rigidity of quantum isometry groups for compact metric spaces we prove the existence of a quantum isometry groups for new classes of metric spaces i geodesic metrics for compact connected riemannian manifolds possibly with boundary and ii metric spaces admitting a uniformly distributed probability measure in the former case it also follows from recent results of the second author that the quantum isometry group is classical i e the commutative c algebra of continuous functions on the riemannian isometry group

semisimple hopf actions on commutative domains abstract let h be a semisimple so finite dimensional hopf algebra over an algebraically closed field k of characteristic zero and let a be a commutative domain over k we show that if a arises as an h module algebra via an inner faithful h action then h must be a group algebra this answers a question of e kirkman and j kuzmanovich and partially answers a question of m cohen the main results of this article extend to working over k of positive characteristic on the other hand we obtain results on hopf actions on weyl algebras as a consequence of the main theorem

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

discrete product systems and their c algebras continuous product systems or product systems over r were recently introduced by arveson to develope an index theory for continuous semigroups of endomorphisms of b h in this paper we continue his work by studying discrete product systems or product systems over g where g is a countable dense subgroup of r we give examples of discrete product systems and compute their automorphism groups to each discrete product system e over g we associate a separable c algebra oe g whose non degenerate representations correspond bijectively to representations of e this family of c algebras may be regarded as analogues of cuntz algebras or car algebras in some degenerate cases these c algebras are the generalized toeplitz algebras we show that oe g is simple and study its gauge automorphisms quasi free automorphisms and fock state