a vectorization optimization method based type 2 fuzzy neural network for noisy data classification this paper proposes a vectorization optimization method vom based type 2 fuzzy neural network vom2fnn for noisy data classification in handling problems with uncertainties such as noisy data type 2 fuzzy systems usually outperform their type 1 counterparts hence type 2 fuzzy sets are adopted in the antecedent parts to model the uncertainty to consider the classification problems the discriminative capability is crucial to determine the performance therefore a vom is proposed in the consequent parts to increase the discriminability and reduce the parameters compared with other existing fuzzy neural networks the novelty of the proposed vom2fnn is its consideration of both uncertainty and discriminability the effectiveness of the proposed vom2fnn is demonstrated by three classification problems experimental results and theoretical analysis indicate that the proposed vom2fnn performs better than the other fuzzy neural networks

super hilbert spaces the basic mathematical framework for super hilbert spaces over a grasmann algebra with a grasmann number valued inner product is formulated super hilbert spaces over infinitely generated grasmann algebras arise in the functional schrodinger representation of spinor quantum field theory in a natural way

spherical harmonics and integration in superspace the study of spherical harmonics in superspace introduced in de bie and sommen 2007 j phys a math theor 40 7193 is further elaborated a detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces which also determines the irreducible pieces under the action of so m x sp 2n in the second part of the paper this decomposition is used to describe all possible integrations over the supersphere it is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees this is the so called pizzetti integral of which it was shown in de bie and sommen 2007 j phys a math theor 40 7193 that it leads to the berezin integral

visual assessment of clustering tendency for rectangular dissimilarity matrices we have an m times n matrix d and assume that its entries correspond to pair wise dissimilarities between m row objects or and n column objects oc which taken together as a union comprise a set o of n m n objects this paper develops a new visual approach that applies to four different cluster assessment problems associated with o the problems are the assessment of cluster tendency pi amongst the row objects or p2 amongst the column objects oc p3 amongst the union of the row and column objects or u oc and p4 amongst the union of the row and column objects that contain at least one object of each type co clusters the basis of the method is to regard d as a subset of known values that is part of a larger unknown n times n dissimilarity matrix and then impute the missing values from d this results in estimates for three square matrices dr dc druc that can be visually assessed for clustering tendency using the previous vat or svat algorithms the output from assessment of druc ultimately leads to a rectangular covat image which exhibits clustering tendencies in d five examples are given to illustrate the new method two important points i because vat is scalable by svat to data sets of arbitrary size and because covat depends explicitly and only on vat this new approach is immediately scalable to say the scovat model which works for even very large unloadable data sets without alteration and ii vat svat and covat are autonomous parameter free models no hidden values are needed to make them work

irreducible characters of general linear superalgebra and super duality we develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra including all the finite dimensional modules by directly relating the problem to the classical kazhdan lusztig theory furthermore we prove that certain parabolic bgg categories over the general linear algebra and over the general linear superalgebra are equivalent we also verify a parabolic version of a conjecture of brundan on the irreducible characters in the bgg category of the general linear superalgebra

character formula for infinite dimensional unitarizable modules of the general linear superalgebra the fock space of m p bosonic and n q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm p n q its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight glm p n q modules we obtain an explicit decomposition of any tensor power of this fock space into irreducibles and develop a character formula for the irreducible glm p n q modules arising in this way

hermite and gegenbauer polynomials in superspace using clifford analysis the clifford hermite and the clifford gegenbauer polynomials of standard clifford analysis are generalized to the new framework of clifford analysis in superspace in a merely symbolic way this means that one does not a priori need an integration theory in superspace furthermore a lot of basic properties such as orthogonality relations differential equations and recursion formulae are proven finally an interesting physical application of the super clifford hermite polynomials is discussed thus giving an interpretation to the super dimension

differential invariants and curved bernstein gelfand gelfand sequences we give a simple construction of the bernstein gelfand gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry this method permits us to define the additional structure of a bilinear differential cup product on this sequence satisfying a leibniz rule up to curvature terms it is not associative but is part of an a algebra of multilinear differential operators which we also obtain explicitly we illustrate the construction in the case of conformal differential geometry where the cup product provides a wide reaching generalization of helicity raising and lowering for conformally invariant field equations

robustness properties of k means and trimmed k means abstract the generalized k means method is based on the minimization of the discrepancy between a random variable or a sample of this random variable and a set with k points measured through a penalty function as in the m estimators setting k 1 a penalty function with unbounded derivative naturally leads to nonrobust generalized k means however surprisingly the lack of robustness extends also to the case of bounded that is generalized k means do not inherit the robustness properties of the m estimator from which they came attempting to robustify the generalized k means method the generalized trimmed k means method arises from combining k means idea with a so called impartial trimming procedure in this article study generalized k means and generalized trimmed k means performance from the viewpoint of hampel s robustness criteria that is we investigate the influence function breakdown point and qualitative robustness confirming the superiority provided by the trimming we inc

a central limit theorem for multivariate generalized trimmed k means a central limit theorem for generalized trimmed k means is obtained in a very general framework that covers the multivariate setting general penalty functions and general k 1 several applications including the location estimator case k 1 for elliptical distributions and the construction of multivariate not necessarily connected tolerance zones are also given

linear fuzzy clustering with selection of variables using graded possibilistic approach linear fuzzy clustering is a useful tool for knowledge discovery in databases kdd and several modifications have been proposed in order to analyze real world data this paper proposes a new approach for estimating local linear models in which linear fuzzy clustering is performed by selecting variables that are useful for extracting correlation structure in each cluster the new clustering model uses two types of memberships one is the conventional membership that represents the degree of membership of each sample in each cluster the other is the additional parameter that represents the relative responsibility of each variable for estimation of local linear models the additional membership takes large values when the variable has close relationship with local principal components and is calculated by using the graded possibilistic approach numerical experiments demonstrate that the proposed method is useful for identifying local linear model taking typicality of each variable into account

a robust em clustering algorithm for gaussian mixture models clustering is a useful tool for finding structure in a data set the mixture likelihood approach to clustering is a popular clustering method in which the em algorithm is the most used method however the em algorithm for gaussian mixture models is quite sensitive to initial values and the number of its components needs to be given a priori to resolve these drawbacks of the em we develop a robust em clustering algorithm for gaussian mixture models first creating a new way to solve these initialization problems we then construct a schema to automatically obtain an optimal number of clusters therefore the proposed robust em algorithm is robust to initialization and also different cluster volumes with automatically obtaining an optimal number of clusters some experimental examples are used to compare our robust em algorithm with existing clustering methods the results demonstrate the superiority and usefulness of our proposed method

a generalization of distance functions for fuzzy c means clustering with centroids of arithmetic means fuzzy c means fcm is a widely used fuzzy clustering method which allows an object to belong to two or more clusters with a membership grade between zero and one despite the considerable efforts made by the clustering community the common characteristics of distance functions suitable for fcm remain unclear to fill this crucial void in this paper we first provide a generalized definition of distance functions that fit fcm directly the goal is to provide more flexibility to fcm in the choice of distance functions while preserving the simplicity of fcm by using the centroids of arithmetic means indeed we show that any distance function that fits fcm directly can be derived by a continuously differentiable convex function and thus is an instance of the generalized point to centroid distance p2c d by definition in addition we prove that if the membership grade matrix is nondegenerate any instance of the p2c d fits fcm directly finally extensive experiments have been conducted to demonstrate that the p2c d leads to the global convergence of fcm and that the clustering performances are significantly affected by the choices of distance functions

kostant homology formulas for oscillator modules of lie superalgebras abstract we provide a systematic approach to obtain formulas for characters and kostant u homology groups of the oscillator modules of the finite dimensional general linear and ortho symplectic superalgebras via howe dualities for infinite dimensional lie algebras specializing these lie superalgebras to lie algebras we recover in a new way formulas for kostant homology groups of unitarizable highest weight representations of hermitian symmetric pairs in addition two new reductive dual pairs related to the above mentioned u homology computation are worked out

two characterizations of crooked functions in this correspondence we give two characterizations of crooked functions one based on the minimum distance of a preparata like code and the other based on the distance regularity of a crooked graph

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

classification of all star irreps of gl m n all the finite dimensional star irreps of gl m n are classified in terms of their highest weights thereby completing the classification of all finite dimensional star irreps of the basic classical lie superalgebras the lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl m n are star irreps of type 1 and 2 respectively this explains the applicability of the young diagram method to these two classes of representations however it is shown that one parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps

orthosymplectically invariant functions in superspace the notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced this leads to dimensional reduction theorems for differentiation and integration in superspace these spherically symmetric functions can be used to solve orthosymplectically invariant schrodinger equations in superspace such as the an harmonic oscillator or the kepler problem finally the obtained machinery is used to prove the funk hecke theorem and bochner s relations in superspace

molecular dynamics like data clustering approach based on the molecular kinetic theory a molecular dynamics like data clustering approach is proposed in this paper clusters are extracted after data points fuse in the iterating space by the dynamical mechanism that is similar to the interacting mechanism between molecules through molecular forces this approach is to find possible natural clusters without pre specifying the number of clusters compared with 3 other clustering methods trimmed k means jp algorithm and another gravitational model based method this approach found clusters better than the other 3 methods in the experiments

orthogonality of hermite polynomials in superspace and mehler type formulae in this paper hermite polynomials related to quantum systems with orthogonal o m symmetry finite reflection group symmetry g o m symplectic symmetry sp 2n and superspace symmetry o m x sp 2n are considered after an overview of the results for o m and g the orthogonality of the hermite polynomials related to sp 2n is obtained with respect to the berezin integral as a consequence an extension of the mehler formula for the classical hermite polynomials to grassmann algebras is proved next hermite polynomials in a full superspace with o m x sp 2n symmetry are considered it is shown that they are not orthogonal with respect to the canonically defined inner product however a new inner product is introduced which behaves correctly with respect to the structure of harmonic polynomials on superspace this inner product allows to restore the orthogonality of the hermite polynomials and also restores the hermiticity of a class of schrodinger operators in superspace subsequently a mehler formula for the full superspace is obtained thus yielding an eigenfunction decomposition of the super fourier transform finally the new results for the sp 2n and o m x sp 2n symmetry are compared with the results in the different types of symmetry

a new apn function which is not equivalent to a power mapping a new almost perfect nonlinear function apn on f 2 sup 10 which is not equivalent to any of the previously known apn mappings is constructed this is the first example of an apn mapping which is not equivalent to a power mapping

on the integrability of a representation of sl 2 r abstract the dunkl operators involve a multiplicity function k as parameter c f dunkl differential difference operators associated to reflection groups trans amer math soc 311 1989 167 183 for positive real values of this function we consider on the schwartz space s r n a representation k of sl 2 r defined in terms of the dunkl laplacian operator by means of a beautiful theorem due to e nelson we prove that k exponentiates to a unique unitary representation of the universal covering group of sl 2 r the representation theory is used to derive an identity of bochner type for the dunkl transform

classification of star and grade star representations of c n 1 the two types of and grade representations of the lie superalgebra c n 1 are classified a type 1 irreducible representation is characterized by the single condition s 0 and s being the highest weight and the odd simple root respectively while the type 2 representations are duals of type 1 representations for n 1 only the identity and vector representations of c n 1 are shown to be grade the superalgebra c 2 proves to be an exception and admits several classes of nontrivial irreducible grade representations

u cohomology formula for unitarizable modules over general linear superalgebras abstract the u cohomology groups with coefficients in infinite dimensional irreducible unitarizable highest weight modules are introduced for the general linear superalgebra the corresponding cohomology groups associated with a large class of infinite dimensional irreducible unitarizable representations known as the oscillator representations are computed explicitly results lead to a more detailed understanding of the character formula for such modules derived in an earlier publication using other methods

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

observer biased fuzzy clustering as generated by clustering algorithms clusterings or partitions are hypotheses on data explanation which are better evaluated by experts from the application domain in general clustering algorithms allow a limited usage of domain knowledge about the cluster formation process in this study we propose both a design technique and a new partitioning based clustering algorithm which can be used to assist the data analyst while looking for a set of meaningful clusters i e clusters that actually correspond to the underlying data structure following an observer metaphor according to which the perception of a group of objects depends on the observer position the closer an observer is from an image more details s he perceives we resort to shrinkage to incorporate a regularization term accounting for the observation point within the objective function of an otherwise unbiased clustering algorithm this technique allows our resulting biased algorithm to generate a set of reasonable partitions i e partitions validated by a given cluster validity index corresponding to views of data with different levels of granularity levels of detail in different regions of the data space for the illustration of the design technique we adopted the fuzzy c means fcm algorithm as the unbiased clustering algorithm and include a convergence theorem assuring that changing the point of observation in the corresponding biased algorithm fcm with focal point fcmfp does not jeopardize its convergence experimental studies on both synthetic and real data are included to illustrate the usefulness of the approach in addition and as a convenient side effect of using shrinkage the experimental results suggest that our biased algorithm fcmfp not only seems to scale better than the successive runs of the unbiased one fcm but on the average seems to produce clusters exhibiting higher validity index values as well in addition less sensitivity to initialization was observed for the biased algorithm when compared with the unbiased one

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

on almost perfect nonlinear functions over mmb f_2 n we investigate some open problems on almost perfect nonlinear apn functions over a finite field of characteristic 2 we provide new characterizations of apn functions and of apn permutations by means of their component functions we generalize some results of nyberg 1994 and strengthen a conjecture on the upper bound of nonlinearity of apn functions we also focus on the case of quadratic functions we contribute to the current works on apn quadratic functions by proving that a large class of quadratic functions cannot be apn

robust clustering methods a unified view clustering methods need to be robust if they are to be useful in practice in this paper we analyze several popular robust clustering methods and show that they have much in common we also establish a connection between fuzzy set theory and robust statistics and point out the similarities between robust clustering methods and statistical methods such as the weighted least squares technique the m estimator the minimum volume ellipsoid algorithm cooperative robust estimation minimization of probability of randomness and the epsilon contamination model by gleaning the common principles upon which the methods proposed in the literature are based we arrive at a unified view of robust clustering methods we define several general concepts that are useful in robust clustering state the robust clustering problem in terms of the defined concepts and propose generic algorithms and guidelines for clustering noisy data we also discuss why the generalized hough transform is a suboptimal solution to the robust clustering problem

frbc a fuzzy rule based clustering algorithm fuzzy clustering is superior to crisp clustering when the boundaries among the clusters are vague and ambiguous however the main limitation of both fuzzy and crisp clustering algorithms is their sensitivity to the number of potential clusters and or their initial positions moreover the comprehensibility of obtained clusters is not expertized whereupon in data mining applications the discovered knowledge is not understandable for human users to overcome these restrictions a novel fuzzy rule based clustering algorithm frbc is proposed in this paper like fuzzy rule based classifiers the frbc employs a supervised classification approach to do the unsupervised cluster analysis it tries to automatically explore the potential clusters in the data patterns and identify them with some interpretable fuzzy rules simultaneous classification of data patterns with these fuzzy rules can reveal the actual boundaries of the clusters to illustrate the capability of frbc to explore the clusters in data the experimental results on some benchmark datasets are obtained and compared with other fuzzy clustering algorithms the clusters specified by fuzzy rules are human understandable with acceptable accuracy

trimmed comparison of distributions this article introduces an analysis of similarity of distributions based on the l2 wasserstein distance between trimmed distributions our main innovation is the use of the impartial trimming methodology already considered in robust statistics which we adapt to this setup instead of simply removing data at the tails to provide some robustness to the similarity analysis we develop a data driven trimming method aimed at maximizing similarity between distributions dissimilarity is then measured in terms of the distance between the optimally trimmed distributions we provide illustrative examples showing the improvements over previous approaches and give the relevant asymptotic results to justify the use of this methodology in applications

robust clustering using outlier sparsity regularization notwithstanding the popularity of conventional clustering algorithms such as k means and probabilistic clustering their clustering results are sensitive to the presence of outliers in the data even a few outliers can compromise the ability of these algorithms to identify meaningful hidden structures rendering their outcome unreliable this paper develops robust clustering algorithms that not only aim to cluster the data but also to identify the outliers the novel approaches rely on the infrequent presence of outliers in the data which translates to sparsity in a judiciously chosen domain leveraging sparsity in the outlier domain outlier aware robust k means and probabilistic clustering approaches are proposed their novelty lies on identifying outliers while effecting sparsity in the outlier domain through carefully chosen regularization a block coordinate descent approach is developed to obtain iterative algorithms with convergence guarantees and small excess computational complexity with respect to their non robust counterparts kernelized versions of the robust clustering algorithms are also developed to efficiently handle high dimensional data identify nonlinearly separable clusters or even cluster objects that are not represented by vectors numerical tests on both synthetic and real datasets validate the performance and applicability of the novel algorithms

linear fuzzy clustering techniques with missing values and their application to local principal component analysis in this paper we propose two methods for partitioning an incomplete data set with missing values into several linear fuzzy clusters by extracting local principal components one is an extension of fuzzy c varieties clustering that can be regarded as the algorithm for the local principal component analysis of fuzzy covariance matrices the other is a simultaneous application of fuzzy clustering and principal component analysis of fuzzy correlation matrices both methods estimate prototypes ignoring only missing values and they need no preprocessing of data such as the elimination of samples with missing values or the imputation of missing elements numerical examples show that the methods provide useful tools for interpretation of the local structures of a database

hilbert space for quantum mechanics on superspace in superspace a realization of sl2 is generated by the super laplace operator and the generalized norm squared in this paper an inner product on superspace for which this representation is skew symmetric is considered this inner product was already defined for spaces of weighted polynomials see k coulembier h de bie and f sommen orthogonality of hermite polynomials in superspace and mehler type formulae proc london math soc accepted arxiv 1002 1118 in this article it is proven that this inner product can be extended to the super schwartz space but not to the space of square integrable functions subsequently the correct hilbert space corresponding to this inner product is defined and studied a complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this hilbert space then the integrability of the sl2 representation is proven finally the heisenberg uncertainty principle for the super fourier transform is constructed

fuzzy pca guided robust k means clustering this paper proposes a new approach to robust clustering in which a robust k means partition is derived by using a noise rejection mechanism based on the noise clustering approach the responsibility weight of each sample for the k means process is estimated by considering the noise degree of the sample and cluster indicators are calculated in a fuzzy principal component analysis pca guided manner where fuzzy pca guided robust k means is performed by considering responsibility weights of samples then the proposed method achieves cluster core estimation in a deterministic way the validity of the derived cluster cores is visually assessed through distance sensitive ordering which considers responsibility weights of samples numerical experiments demonstrate that the proposed method is useful for capturing cluster cores by rejecting noise samples and we can easily assess cluster validity by using cluster crossing curves

principal points and self consistent points of elliptical distributions the k principal points of a p variate random vector x are those points 1 k r p which approximate the distribution of x by minimizing the expected squared distance of x from the nearest of the j any set of k points y 1 y k partitions r p into domains of attraction d 1 d k according to minimal distance following hastie and stuetzle we call y 1 y k self consistent if e xx d j y j for j 1 k principal points are a special case of self consistent points in this paper we study principal points and self consistent points of p variate elliptical distributions the main results are the following 1 if k self consistent points of x span a subspace of dimension q p then this subspace is also spanned by q principal components that is self consistent points of elliptical distributions exist only in principal component subspaces 2 the subspace spanned by k principal points of x is identical with the subspace spanned by the principal components associated with the largest roots this proves a conjecture of flury we also discuss implications of our results for the computation and estimation of principal points

design of natural classification kernels using prior knowledge a new class of kernels has been designed to enhance the usability of prior knowledge prior knowledge is shown to improve the generalization ability of kernel algorithms for binary classification problems the prior knowledge is expressed in natural language via fuzzy rules first the concepts of a fuzzy rule base and prior confidence region are proposed to formulate the prior knowledge then the new kernels which are referred to as natural classification kernels ncks are represented by fuzzy equivalence relations based on the formulation of prior knowledge an nck is interpreted as a measure of similarities between samples it is proven that ncks have two desired properties 1 transitivity with respect to the triangular norms and 2 the ability to provide higher similarities to spatially closer samples from the same class using transitivity a large number of ncks may be directly obtained by means of triangular norms additionally the theoretical results show that the second property makes it possible for the support vector machine svm and convex hull separation algorithm to generalize from training samples to test samples in the prior confidence region finally some synthetic datasets and a benchmark dataset are employed to validate the proposed approach

new classes of almost bent and almost perfect nonlinear polynomials new infinite classes of almost bent and almost perfect nonlinear polynomials are constructed it is shown that they are affine inequivalent to any sum of a power function and an affine function

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

trimmed k means an attempt to robustify quantizers a class of procedures based on impartial trimming self determined by the data is introduced with the aim of robustifying k means hence the associated clustering analysis we include a detailed study of optimal regions showing that only nonpathological regions can arise from impartial trimming procedures the asymptotic results provided in the paper focus on strong consistency of the suggested methods under widely general conditions a section is devoted to exploring the performance of the procedure to detect anomalous data in simulated data sets

lda based clustering algorithm and its application to an unsupervised feature extraction research has shown fuzzy c means fcm clustering to be a powerful tool to partition samples into different categories however the objective function of fcm is based only on the sum of distances of samples to their cluster centers which is equal to the trace of the within cluster scatter matrix in this study we propose a clustering algorithm based on both within and between cluster scatter matrices extended from linear discriminant analysis lda and its application to an unsupervised feature extraction fe our proposed methods comprise between and within cluster scatter matrices modified from the between and within class scatter matrices of lda the scatter matrices of lda are special cases of our proposed unsupervised scatter matrices the results of experiments on both synthetic and real data show that the proposed clustering algorithm can generate similar or better clustering results than 11 popular clustering algorithms k means k medoid fcm the gustafson kessel gath geva possibilistic c means pcm fuzzy pcm possibilistic fcm fuzzy compactness and separation a fuzzy clustering algorithm based on a fuzzy treatment of finite mixtures of multivariate student s t distributions algorithms and a fuzzy mixture of the student s t factor analyzers model the results also show that the proposed fe outperforms principal component analysis and independent component analysis

strongly typical representations of the basic classical lie superalgebras 1 1 in ps1 i penkov and v serganova show that the category of representations of a basic classical lie superalgebra g of type i with a fixed typical central character is equivalent to the category of representations of the even part go with a suitable central character a similar result for type ii was proven by i penkov in p for generic central characters the aim of this paper is to understand for which central characters such an equivalence holds we also study a simplest example when such an equivalence fails to exist but the corresponding category of representations of a lie superalgebra still has a good description

super duality and irreducible characters of ortho symplectic lie superalgebras we formulate and establish a super duality which connects parabolic categories o for the ortho symplectic lie superalgebras and classical lie algebras of bcd types this provides a complete and conceptual solution of the irreducible character problem for the ortho symplectic lie superalgebras in a parabolic category o which includes all finite dimensional irreducible modules in terms of classical kazhdan lusztig polynomials

rederivation of the fuzzypossibilistic clustering objective function through bayesian inference unsupervised clustering of a set of datums into homogeneous groups is a primitive operation required in many signal and image processing applications in fact different incarnations and hybrids of fuzzy c means fcm and possibilistic c means pcm have been suggested which address additional requirements such as accepting weighted sets and being robust to the presence of outliers nevertheless arriving at a general framework which is independent of the datum model and the notion of homogeneity of a particular problem class is a challenge however this process has not been followed organically and clustering algorithms are generally based on exogenous objective functions which are heuristically engineered and are believed to lead to the satisfaction of a required behavior these techniques also commonly depend on regularization coefficients which are to be set prudently by the user or through separate processes in contrast in this work we utilize bayesian inference and derive a robustified objective function for a fuzzypossibilistic clustering algorithm by assuming a generic datum model and a generic notion of cluster homogeneity we utilize this model for the purpose of cluster validity assessment as well we emphasize the epistemological importance of the theoretical basis on which the developed methodology rests at the end of this paper we present experimental results to exhibit the utilization of the developed framework in the context of four different problem classes

fischer decomposition for polynomials on superspace recently the fischer decomposition for polynomials on superspace m 2n that is polynomials in m commuting and 2n anti commuting variables has been obtained unless the superdimension m m 2n is even and non positive in this case it turns out that the fischer decomposition of polynomials into spherical harmonics is quite analogous as in m and it is an irreducible decomposition under the natural action of lie superalgebra m 2n in this paper we describe explicitly the fischer decomposition in the exceptional case when m 2 0 in particular we show that under the action of m 2n the fischer decomposition is not in general a decomposition into irreducible but just indecomposable pieces

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

soft transition from probabilistic to possibilistic fuzzy clustering in the fuzzy clustering literature two main types of membership are usually considered a relative type termed probabilistic and an absolute or possibilistic type indicating the strength of the attribution to any cluster independent from the rest there are works addressing the unification of the two schemes here we focus on providing a model for the transition from one schema to the other to exploit the dual information given by the two schemes and to add flexibility for the interpretation of results we apply an uncertainty model based on interval values to memberships in the clustering framework obtaining a framework that we term graded possibility we outline a basic example of graded possibilistic clustering algorithm and add some practical remarks about its implementation the experimental demonstrations presented highlight the different properties attainable through appropriate implementation of a suitable graded possibilistic model an interesting application is found in automated segmentation of diagnostic medical images where the model provides an interactive visualization tool for this task

a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

bernstein gelfand gelfand resolutions for basic classical lie superalgebras abstract we study kostant cohomology and bernstein gelfand gelfand resolutions for finite dimensional representations of basic classical lie superalgebras for each choice of parabolic subalgebra and representation of such a lie superalgebra there is a natural definition of the boundary and coboundary operators which define co homology of the nilradical of the parabolic subalgebra we prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of generalised verma modules every such a resolution is then given by modules induced by these homology groups we also prove that if these homology groups are completely reducible a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the kostant quabla operator which is equivalent with disjointness of the boundary and coboundary operators then we use these results to derive very explicit conditions under which bgg resolutions exist which are particularly useful for the superalgebras of type i for the unitarisable representations of gl m n and osp 2 2 n we derive conditions on the parabolic subalgebra under which the bgg resolutions exist we also apply the obtained theory to construct specific examples of bgg resolutions for osp m 2 n finally we state some results for typical modules of gl m n and osp 2 2 n

the strong law of large numbers for k means and best possible nets of banach valued random variables let b be a uniformly convex banach space x a b valued random variable and k a given positive integer number a random sample of x is substituted by the set of k elements which minimizes a criterion we found conditions to assure that this set converges a s as the sample size increases to the set of k elements which minimizes the same criterion for x

higher symmetries of the laplacian we identify the symmetry algebra of the laplacian on euclidean space as an explicit quotient of the universal enveloping algebra of the lie algebra of conformal motions we construct analogues of these symmetries on a general conformal manifold

bernstein gelfand gelfand sequences the bernstein gelfand gelfand sequences extend the complexes of homogeneous vector bundles to curved cartan geometries

cohomology of generalized supergrassmannians and character formulae for basic classical lie superalgebras we use geometrical methods adapted from borel weil bott theory to compute the character of every finite dimensional simple module over a basic classical lie superalgebra the answer is given by a combinatorial algorithm in terms of weight and cap diagrams which were defined by brundan and stroppel for the general linear superalgebra ams subject classification 20g05 14f05

robust estimation in the normal mixture model based on robust clustering summary we introduce a robust estimation procedure that is based on the choice of a representative trimmed subsample through an initial robust clustering procedure and subsequent improvements based on maximum likelihood to obtain the initial trimming we resort to the trimmed k means a simple procedure designed for finding the core of the clusters under appropriate configurations by handling the trimmed data as censored maximum likelihood estimation provides in each step the location and shape of the next trimming data driven restrictions on the parameters requiring that every distribution in the mixture must be sufficiently represented in the initial clustered region allow singularities to be avoided and guarantee the existence of the estimator our analysis includes robustness properties and asymptotic results as well as worked examples

fourier transform and related integral transforms in superspace abstract in this paper extensions of the classical fourier fractional fourier and radon transforms to superspace are studied previously a fourier transform in superspace was already studied but with a different kernel in this work the fermionic part of the fourier kernel has a natural symplectic structure derived using a clifford analysis approach several basic properties of these three transforms are studied using suitable generalizations of the hermite polynomials to superspace see h de bie f sommen hermite and gegenbauer polynomials in superspace using clifford analysis j phys a 40 2007 10441 10456 an eigenfunction basis for the fourier transform is constructed

a possibilistic fuzzy c means clustering algorithm in 1997 we proposed the fuzzy possibilistic c means fpcm model and algorithm that generated both membership and typicality values when clustering unlabeled data fpcm constrains the typicality values so that the sum over all data points of typicalities to a cluster is one the row sum constraint produces unrealistic typicality values for large data sets in this paper we propose a new model called possibilistic fuzzy c means pfcm model pfcm produces memberships and possibilities simultaneously along with the usual point prototypes or cluster centers for each cluster pfcm is a hybridization of possibilistic c means pcm and fuzzy c means fcm that often avoids various problems of pcm fcm and fpcm pfcm solves the noise sensitivity defect of fcm overcomes the coincident clusters problem of pcm and eliminates the row sum constraints of fpcm we derive the first order necessary conditions for extrema of the pfcm objective function and use them as the basis for a standard alternating optimization approach to finding local minima of the pfcm objective functional several numerical examples are given that compare fcm and pcm to pfcm our examples show that pfcm compares favorably to both of the previous models since pfcm prototypes are less sensitive to outliers and can avoid coincident clusters pfcm is a strong candidate for fuzzy rule based system identification

integration in superspace using distribution theory in this paper a new class of cauchy integral formulae in superspace is obtained using formal expansions of distributions this allows us to solve five open problems in the study of harmonic and clifford analysis in superspace

strong consistency of the prototype based clustering in probabilistic space in this paper we formulate in general terms an approach to prove strong consistency of the empirical risk minimisation inductive principle applied to the prototype or distance based clustering this approach was motivated by the divisive information theoretic feature clustering model in probabilistic space with kullback leibler divergence which may be regarded as a special case within the clustering minimisation framework

modeling the manifolds of images of handwritten digits this paper describes two new methods for modeling the manifolds of digitized images of handwritten digits the models allow a priori information about the structure of the manifolds to be combined with empirical data accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models one of the methods is grounded in principal components analysis the other in factor analysis both methods are based on locally linear low dimensional approximations to the underlying data manifold links with other methods that model the manifold are discussed

regularized linear fuzzy clustering and probabilistic pca mixture models fuzzy c means fcm type fuzzy clustering approaches are closely related to gaussian mixture models gmms and em like algorithms have been used in fcm clustering with regularized objective functions especially fcm with regularization by kullback leibler information klfcm is a fuzzy counterpart of gmms in this paper we propose to apply probabilistic principal component analysis pca mixture models to linear clustering following a discussion on the relationship between local pca and linear fuzzy clustering although the proposed method is a kind of the constrained model of klfcm the algorithm includes the fuzzy c varieties fcv algorithm as a special case and the algorithm can be regarded as a modified fcv algorithm with regularization by k l information numerical experiments demonstrate that the proposed clustering algorithm is more flexible than the maximum likelihood approaches and is useful for capturing local substructures properly

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets