the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

dynamics stability and statistics on lattices and networks these lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution emerging from the spatio temporal organization of several coupled dynamical variables the first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from out of equilibrium statistical mechanics the second message is that the overall scenario is definitely reacher than the standard problems in these fields for instance systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior i e lyapunov chaos as it happens for dynamical models made of a few degrees of freedom in fact a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of lyapunov instability such a mechanism may emerge from the combination of spatial extension and non

comparison of linear and classical velocity update rules in particle swarm optimization notes on scale and frame invariance in this paper we investigate whether the particle swarm optimization pso algorithm is invariant of the scale and frame i e translation and rotation in which an objective function is posed to do so we study the linear and classical velocity update rules we will show that the linear velocity update rule is scale and frame invariant but that the classical velocity update rule lacks rotational invariance it is known that the linear velocity update rule lacks diversity resulting in particle trajectories that collapse to line searches in contrast the classical velocity update rule maintains diverse space filling particle trajectories to illustrate that diversity and invariance are not necessarily exclusive we propose a new velocity update rule this update rule which is just one example of many possible formulations is rotationally invariant and at the same time directionally diverse this is achieved through consistent perturbation of the search directions we quantify the in variance and performance of the three different implementations using a popular test set the test problems are evaluated in both unrotated and arbitrarily rotated reference frames copyright 2006 john wiley sons ltd

k means type algorithms a generalized convergence theorem and characterization of local optimality the k means algorithm is a commonly used technique in cluster analysis in this paper several questions about the algorithm are addressed the clustering problem is first cast as a nonconvex mathematical program then a rigorous proof of the finite convergence of the k means type algorithm is given for any metric it is shown that under certain conditions the algorithm may fail to converge to a local minimum and that it converges under differentiability conditions to a kuhn tucker point finally a method for obtaining a local minimum solution is given

discovery of time inconsecutive co movement patterns of foreign currencies using an evolutionary biclustering method abstract this paper proposes an evolutionary biclustering algorithm to discover inconsecutive co movement patterns of different foreign exchange rates the rows columns of a bicluster i e a submatrix with a subset of rows and a subset of columns are not necessarily consecutive a typical bicluster with constant values on rows and or columns is represented as a hyperplane in a high dimensional space and the coefficients of the hyperplane are determined using a genetic algorithm a detected bicluster demonstrates the co moving behaviors of a subset of currencies in inconsecutive time periods indicating that the currencies moved in different manners in some specific time periods in our experiments we relate these patterns to the geographically close economic connections and find out the correspondence between the nominal exchange rates and the economic conditions the findings are useful as a guide for investing foreign currencies

towards improving fuzzy clustering using support vector machine application to gene expression data recent advancement in microarray technology permits monitoring of the expression levels of a large set of genes across a number of time points simultaneously for extracting knowledge from such huge volume of microarray gene expression data computational analysis is required clustering is one of the important data mining tools for analyzing such microarray data to group similar genes into clusters researchers have proposed a number of clustering algorithms in this purpose in this article an attempt has been made in order to improve the performance of fuzzy clustering by combining it with support vector machine svm classifier a recently proposed real coded variable string length genetic algorithm based clustering technique and an iterated version of fuzzy c means clustering have been utilized in this purpose the performance of the proposed clustering scheme has been compared with that of some well known existing clustering algorithms and their svm boosted versions for one simulated and six real life gene expression data sets statistical significance test based on analysis of variance anova followed by posteriori tukey kramer multiple comparison test has been conducted to establish the statistical significance of the superior performance of the proposed clustering scheme moreover biological significance of the clustering solutions have been established

quadratic estimates and functional calculi of perturbed dirac operators we prove quadratic estimates for complex perturbations of dirac type operators and thereby show that such operators have a bounded functional calculus as an application we show that spectral projections of the hodge dirac operator on compact manifolds depend analytically on l changes in the metric we also recover a unified proof of many results in the calderon program including the kato square root problem and the boundedness of the cauchy operator on lipschitz curves and surfaces

convergence rates of the splitting scheme for parabolic linear stochastic cauchy problems we study the splitting scheme associated with the linear stochastic cauchy problem du t au t dt dw t t in 0 t u 0 x here a is the generator of an analytic c 0 semigroup s s t t geqslant0 on a banach space e and w w t t geqslant0 is a brownian motion with values in a fractional domain space e beta associated with a we prove that if alpha beta gamma theta geqslant0 are such that gamma theta 1 and alpha beta theta gamma frac 1 2 then the approximate solutions u n converge to the solution u in the space c gamma 0 t e alpha both in l p means and almost surely with rate 1 n theta

the impact of cluster representatives on the convergence of the k modes type clustering as a leading partitional clustering technique k modes is one of the most computationally efficient clustering methods for categorical data in the k modes a cluster is represented by a x201c mode x201d which is composed of the attribute value that occurs most frequently in each attribute domain of the cluster whereas in real applications using only one attribute value in each attribute to represent a cluster may not be adequate as it could in turn affect the accuracy of data analysis to get rid of this deficiency several modified clustering algorithms were developed by assigning appropriate weights to several attribute values in each attribute although these modified algorithms are quite effective their convergence proofs are lacking in this paper we analyze their convergence property and prove that they cannot guarantee to converge under their optimization frameworks unless they degrade to the original k modes type algorithms furthermore we propose two different modified algorithms with weighted cluster prototypes to overcome the shortcomings of these existing algorithms we rigorously derive updating formulas for the proposed algorithms and prove the convergence of the proposed algorithms the experimental studies show that the proposed algorithms are effective and efficient for large categorical datasets

pointwise convergence for semigroups in vector valued l p spaces suppose that tt t 0 is a symmetric diffusion semigroup on l2 x and denote by widetilde t t t geq0 its tensor product extension to the bochner space l p x mathcal b where mathcal b belongs to a certain broad class of umd spaces we prove a vector valued version of the hopf dunford schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of widetilde t t t geq 0 on l p x mathcal b as an application we show that such continuations exhibit pointwise convergence

clustering with evolution strategies tbe applicability of evolution strategies ess population based stochastic optimization techniques to optimize clustering objective functions is explored clustering objective functions are categorized into centroid and non centroid type of functions optimization of the centroid type of objective functions is accomplished by formulating them as functions of real valued parameters using ess both hard and fuzzy clustering objective functions are considered in this study applicability of ess to discrete optimization problems is extended to optimize the non centroid type of objective functions as ess are amenable to parallelization a parallel model master slave model is described in the context of the clustering problem results obtained for selected data sets substantiate the utility of ess in clustering

space structure and clustering of categorical data learning from categorical data plays a fundamental role in such areas as pattern recognition machine learning data mining and knowledge discovery to effectively discover the group structure inherent in a set of categorical objects many categorical clustering algorithms have been developed in the literature among which k modes type algorithms are very representative because of their good performance nevertheless there is still much room for improving their clustering performance in comparison with the clustering algorithms for the numeric data this may arise from the fact that the categorical data lack a clear space structure as that of the numeric data to address this issue we propose in this paper a novel data representation scheme for the categorical data which maps a set of categorical objects into a euclidean space based on the data representation scheme a general framework for space structure based categorical clustering algorithms sbc is designed this framework together with the applications of two kinds of dissimilarities leads two versions of the sbc type algorithms to verify the performance of the sbc type algorithms we employ as references four representative algorithms of the k modes type algorithms experiments show that the proposed sbc type algorithms significantly outperform the k modes type algorithms

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

a convergence theorem for the fuzzy subspace clustering fsc algorithm we establish the convergence of the fuzzy subspace clustering fsc algorithm by applying zangwill s convergence theorem we show that the iteration sequence produced by the fsc algorithm terminates at a point in the solution set s or there is a subsequence converging to a point in s in addition we present experimental results that illustrate the convergence properties of the fsc algorithm in various scenarios

tagged particle diffusion in 3d lattice gas cellular automata abstract we report simulations of tagged particle diffusion in three dimensional lattice gas cellular automata lgca in particular we looked at the decay of the velocity autocorrelation function vacf using a new technique that is about a million times more efficient than the conventional techniques for longer times the simulations clearly show the algebraic t 3 2 tail of the vacf we compare the observed long time tail with the predictions of mode coupling theory in three dimensions the amplitude of this tail is found to agree within the small statistical error with these predictions

performance evaluation of some clustering algorithms and validity indices in this article we evaluate the performance of three clustering algorithms hard k means single linkage and a simulated annealing sa based technique in conjunction with four cluster validity indices namely davies bouldin index dunn s index calinski harabasz index and a recently developed index i based on a relation between the index i and the dunn s index a lower bound of the value of the former is theoretically estimated in order to get unique hard k partition when the data set has distinct substructures the effectiveness of the different validity indices and clustering methods in automatically evolving the appropriate number of clusters is demonstrated experimentally for both artificial and real life data sets with the number of clusters varying from two to ten once the appropriate number of clusters is determined the sa based clustering technique is used for proper partitioning of the data into the said number of clusters

a fuzzy sv k modes algorithm for clustering categorical data with set valued attributes abstract in this paper we propose a fuzzy sv k modes algorithm that uses the fuzzy k modes clustering process to cluster categorical data with set valued attributes in the proposed algorithm we use jaccard coefficient to measure the dissimilarity between two objects and represent the center of a cluster with set valued modes a heuristic update way of cluster prototype is developed for the fuzzy partition matrix these extensions make the fuzzy sv k modes algorithm can cluster categorical data with single valued and set valued attributes together and the fuzzy k modes algorithm is its special case experimental results on the synthetic data sets and the three real data sets from different applications have shown the efficiency and effectiveness of the fuzzy sv k modes algorithm

integrated rough fuzzy clustering for categorical data analysis abstract in recent times advanced data mining research has been mostly focusing on clustering of categorical data where a natural ordering in attribute values is missing to address this fact the rough fuzzy k modes clustering technique has been recently developed in order to handle imperfect information i e indiscernibility coarseness and vagueness within the dataset however it has been observed that the said technique suffers from the problem of local optima due to the random choice of initial cluster modes hence in this paper we have proposed an integrated clustering technique using multi phase learning in this regard first simulated annealing based rough fuzzy k modes and genetic algorithm based rough fuzzy k modes are proposed in order to perform the clustering better by considering clustering as an underlying optimization problem these clustering methods individually produce clusters having set of central and peripheral points thereafter for each case final improved clustering results are obtained by assigning peripheral points to a particular crisp cluster using random forest where central points are used as training set second the varying cardinality of the training and testing sets produced by each clustering method further motivated us to propose a generalized technique called integrated rough fuzzy clustering using random forest where results of three aforementioned clustering techniques are used to compute the roughness measure based on this measure three different sets namely best central points semi best central points and pure peripheral points are determined thereafter using multi phase learning best central points are used to classify the semi best central points and then using both of them pure peripheral points are classified by random forest experimental results are reported quantitatively and visually to demonstrate the effectiveness of the proposed methods in comparison with well known state of the art methods for six synthetic and five real life datasets finally statistical significance tests are conducted to establish the superiority of the results produced by the proposed methods

comparison of linear and classical velocity update rules in particle swarm optimization notes on diversity in this study we investigate the significance of diversity in the particle swarm optimization pso algorithm to do so we study two different implementations of the pso being the so called linear and classical pso formulations while the behaviour of these two implementations is markedly different they only differ in the formulation of the velocity update rule in fact the differences are merely due to subtle differences in the introduction of randomness into the algorithm in this paper we show that in algorithms employing the linear pso velocity update rule particle trajectories collapse to line searches in n dimensional space the classical formulation does not suffer this drawback instead directional diverse stochastic search trajectories are retained which in turn helps to alleviate premature convergence the performance of the classical implementation is therefore superior for all test problems considered due to the presence of adequate diversity copyright 2006 john wiley sons ltd

gaps a clustering method using a new point symmetry based distance measure in this paper an evolutionary clustering technique is described that uses a new point symmetry based distance measure the algorithm is therefore able to detect both convex and non convex clusters kd tree based nearest neighbor search is used to reduce the complexity of finding the closest symmetric point adaptive mutation and crossover probabilities are used the proposed ga with point symmetry gaps distance based clustering algorithm is able to detect any type of clusters irrespective of their geometrical shape and overlapping nature as long as they possess the characteristic of symmetry gaps is compared with existing symmetry based clustering technique sbkm its modified version and the well known k means algorithm sixteen data sets with widely varying characteristics are used to demonstrate its superiority for real life data sets anova and manova statistical analyses are performed

rapid and brief communication a k populations algorithm for clustering categorical data in this paper the conventional k modes type algorithms for clustering categorical data are extended by representing the clusters of categorical data with k populations instead of the hard type centroids used in the conventional algorithms use of a population based centroid representation makes it possible to preserve the uncertainty inherent in data sets as long as possible before actual decisions are made the k populations algorithm was found to give markedly better clustering results through various experiments

a semi supervised regression model for mixed numerical and categorical variables in this paper we develop a semi supervised regression algorithm to analyze data sets which contain both categorical and numerical attributes this algorithm partitions the data sets into several clusters and at the same time fits a multivariate regression model to each cluster this framework allows one to incorporate both multivariate regression models for numerical variables supervised learning methods and k mode clustering algorithms for categorical variables unsupervised learning methods the estimates of regression models and k mode parameters can be obtained simultaneously by minimizing a function which is the weighted sum of the least square errors in the multivariate regression models and the dissimilarity measures among the categorical variables both synthetic and real data sets are presented to demonstrate the effectiveness of the proposed method

the fuzzy c means algorithm with fuzzy p mode prototypes for clustering objects having mixed features frequency based cluster prototypes have been used to cluster categorical objects based on the simple matching dissimilarity measure this paper introduces a new generalization called fuzzy p mode prototype of frequency based prototypes a fuzzy p mode cluster prototype at a categorical feature is expressed as a list of p labels that have larger frequencies than others in the cluster this paper also presents a new generalization of the fuzzy c means clustering algorithm for the objects of mixed features in the general fuzzy c means clustering algorithm any dissimilarity measures at the categorical feature level are assumed not like other clustering algorithms that use the simple matching dissimilarity the convergence of the general fuzzy c means clustering algorithm under the optimization framework is proved it is also explained through experiments over real object sets that the size of fuzzy p mode prototypes and the fuzzification coefficients affect clustering performance

stochastic maximal lp regularity in this article we prove a maximal lp regularity result for stochastic convolutions which extends krylov s basic mixed lp lq inequality for the laplace operator on d to large classes of elliptic operators both on d and on bounded domains in d with various boundary conditions our method of proof is based on mcintosh s h functional calculus r boundedness techniques and sharp lp lq square function estimates for stochastic integrals in lq spaces under an additional invertibility assumption on a a maximal space time lp regularity result is obtained as well

clustering ensemble selection for categorical data based on internal validity indices propose a clustering ensemble selection algorithm for categorical data sivid sivid measures the quality of base clusterings with internal validity indices sivd measures the diversity of base clusterings with nmi experimental results show the effectiveness and robustness of the proposed algorithm clustering ensemble selection is an effective technique for improving the quality of clustering results however traditional methods usually measure the quality and diversity based on the cluster labels of base clusterings while missing the information of the original data to solve this problem a new clustering ensemble selection algorithm for categorical data is presented in this algorithm five popular internal validity indices and the normalized mutual information are utilized to measure the quality and diversity of the base clusterings respectively according to the quality measure the partition with the highest value is firstly selected to participate in the ensemble then the base partitions with the highest clustering quality and diversity with respect to the selected base partitions in previous iterations are iteratively selected until the size of selected base clusterings is satisfied the effectiveness and robustness of the proposed algorithm are evaluated in comparison with full ensemble random selection ensemble and the state of the art ensemble selection algorithms experimental results on real categorical data sets show that the proposed algorithm is competitive with the existing ensemble selection algorithms in terms of clustering quality

a genetic algorithm using hyper quadtrees for low dimensional k means clustering the k means algorithm is widely used for clustering because of its computational efficiency given n points in d dimensional space and the number of desired clusters k k means seeks a set of k cluster centers so as to minimize the sum of the squared euclidean distance between each point and its nearest cluster center however the algorithm is very sensitive to the initial selection of centers and is likely to converge to partitions that are significantly inferior to the global optimum we present a genetic algorithm ga for evolving centers in the k means algorithm that simultaneously identifies good partitions for a range of values around a specified k the set of centers is represented using a hyper quadtree constructed on the data this representation is exploited in our ga to generate an initial population of good centers and to support a novel crossover operation that selectively passes good subsets of neighboring centers from parents to offspring by swapping subtrees experimental results indicate that our ga finds the global optimum for data sets with known optima and finds good solutions for large simulated data sets

modified fuzzy c means for ordinal valued attributes with particle swarm for optimization there are well established methods for fuzzy clustering especially for the cases where the feature values are numerical of ratio or interval scale not so well established are methods to be applied when the feature values are ordinal or nominal in that case there is no satisfactory method it seems this paper discusses a modified fuzzy c means clustering method where an ordinal to numeric mapping for the ordinal features is obtained as part of the clustering process part of minimizing the objective function for the clustering is to find this ordinal to numeric mapping having this mapping allows standard methods of fuzzy c means clustering to be used since then if there are no categorical features all the features are numeric the mapping is not of interest in itself and to obtain it is only a subsidiary objective of the clustering process the mapping allows for the degrees of freedom that is characteristic of the data the method involves solving a rather challenging optimization problem since the objective function has many local minima this makes the use of a global optimization method such as particle swarm optimization pso attractive for determining the membership matrix for the clustering to minimize computational effort a bayesian stopping criterion may be used in combination with a multi start strategy for the pso other clustering methods generally find local optimum of their objective function through simulations and experiments with real and artificial data the method proposed here is shown to be quite effective

genetic algorithm based clustering technique abstract a genetic algorithm based clustering technique called ga clustering is proposed in this article the searching capability of genetic algorithms is exploited in order to search for appropriate cluster centres in the feature space such that a similarity metric of the resulting clusters is optimized the chromosomes which are represented as strings of real numbers encode the centres of a fixed number of clusters the superiority of the ga clustering algorithm over the commonly used k means algorithm is extensively demonstrated for four artificial and three real life data sets

mode coupling theory and tails in ca fluids abstract after a summary of important effects of mode coupling theory in continuous fluids the theory is extended to ca fluids and the long time tails calculated for the velocity correlation function and for the stress stress correlation function including the spurious contributions from the staggered momentum densities at finite densities the fermi statistics strongly suppresses the long time singularities arising from two fluid modes but not those from tagged particle diffusion

a robust evolutionary algorithm for the recovery of rational gielis curves gielis curves gc can represent a wide range of shapes and patterns ranging from star shapes to symmetric and asymmetric polygons and even self intersecting curves such patterns appear in natural objects or phenomena such as flowers crystals pollen structures animals or even wave propagation gielis curves and surfaces are an extension of lame curves and surfaces superquadrics which have benefited in the last two decades of extensive researches to retrieve their parameters from various data types such as range images 2d and 3d point clouds etc unfortunately the most efficient techniques for superquadrics recovery based on deterministic methods cannot directly be adapted to gielis curves indeed the different nature of their parameters forbids the use of a unified gradient descent approach which requires initial pre processings such as the symmetry detection and a reliable pose and scale estimation furthermore even the most recent algorithms in the literature remain extremely sensitive to initialization and often fall into local minima in the presence of large missing data we present a simple evolutionary algorithm which overcomes most of these issues and unifies all of the required operations into a single though efficient approach the key ideas in this paper are the replacement of the potential fields used for the cost function closed form by the shortest euclidean distance sed iterative approach the construction of cost functions which minimize the shortest distance as well as the curve length using r functions and slight modifications of the evolutionary operators we show that the proposed cost function based on sed and r function offers the best compromise in terms of accuracy robustness to noise and missing data

adaptive learning of ordinal numerical mappings through fuzzy clustering for the objects of mixed features ordinal feature values are totally ordered labels that can be considered as fuzzy sets the formulation of proper fuzzy sets for ordinal labels is important for the systems that deal with the objects of mixed feature types when a proper ordinal numerical mapping for an ordinal feature of interest is given proper fuzzy sets for the labels of the ordinal feature can be easily formulated in this paper we propose an adaptive method to learn proper ordinal numerical mappings for ordinal features of interest from a given objects of mixed features including the ordinal features the method starts with uniform ordinal numerical mappings and performs two steps iteratively the first step computes a fuzzy partition over the given object set with the ordinal numerical mappings the second step learns new ordinal numerical mappings from the new fuzzy partition in the way that the new mappings make the similarity between two ordinal labels be similar to the average similarity between the objects having the two labels respectively through the alternate repetition of the two steps both of the ordinal numerical mappings and the clustering quality become gradually improved the validity of the proposed method is strongly supported through the experiments with a modified fuzzy c means clustering algorithm in which the proposed method is implemented

hydrodynamics and time correlation functions for cellular automata hydrodynamic excitations in lattice gas cellular automata are described in terms of equilibrium time correlation functions for the local conserved variables for large space and time scales the linearized hydrodynamic equations are obtained to navier stokes order exact expressions for the associated susceptibilities and transport coefficients are identified in terms of correlation functions the general form of the time correlation functions for conserved densities in the hydrodynamic limit is given and illustrated by some examples suitable for comparison with computer simulation the transport coefficients are related to time correlation functions for the conserved fluxes in a way analogous to the green kubo expressions for continuous fluids the general results are applied for a one component fluid and several types of binary diffusion also discussed are the effects of unphysical slow modes such as staggered particle or momentum densities

a joint functional calculus for sectorial operators with commuting resolvents 1 introduction in this paper we are concerned with h 1 functional calculus in the sense of the construction introduced by mcintosh 23 and developed recently by cowling doust mcintosh and yagi 8 given v in 0 p we will use the notation sv fz 2 c n f0g jarg z j vg to denote the open sector of angle 2v around the half line r let x be a complex banach space given a linear operator a on x we denote by d a n a and r a the domain the kernel and the range of a respectively we also denote by j a the spectrum of a and by r a the resolvent set of a

initialization of k modes clustering using outlier detection techniques we considered the initialization of k modes clustering from the view of outlier detection we proposed an initialization algorithm for k modes clustering via the distance based outlier detection technique we presented a partition entropy based outlier detection technique and designed an initialization algorithm via it we proposed a new distance metric weighted matching distance metric the effectiveness of our initialization algorithms was shown on several uci data sets the k modes clustering has received much attention since it works well for categorical data sets however the performance of k modes clustering is especially sensitive to the selection of initial cluster centers therefore choosing the proper initial cluster centers is a key step for k modes clustering in this paper we consider the initialization of k modes clustering from the view of outlier detection we present two different initialization algorithms for k modes clustering where the first is based on the traditional distance based outlier detection technique and the second is based on the partition entropy based outlier detection technique by using the above two outlier detection techniques to calculate the degree of outlierness of each object our algorithms can guarantee that the chosen initial cluster centers are not outliers moreover during the process of initialization we adopt a new distance metric weighted matching distance metric to calculate the distance between two objects described by categorical attributes experimental results on several uci data sets demonstrate the effectiveness of our initialization algorithms for k modes clustering

generalized ornstein uhlenbeck semigroups littlewood paley stein inequalities and the p a meyer equivalence of norms abstract let be a nondegenerate gaussian measure on a hilbert space h for an arbitrary selfadjoint nonnegative operator a we consider the semigroup e tl e ta on l p where stands for the second quantization operator we provide an explicit characterization of the domains of i l m 2 in l p in terms of gaussian sobolev spaces thus extending the p a meyer result on equivalence of norms the main tools are the littlewood paley stein inequalities which are proved under minimal assumptions by a purely analytic method following e stein 1970 topics in harmonic analysis princeton univ press princeton nj

fuzzy c means optimality of solutions and effective termination of the algorithm abstract in this paper the solutions produced by the fuzzy c means algorithm for a general class of problems are examined and a method to test for the local optimality of such solutions is established an equivalent mathematical program is defined for the c means problem utilizing a generalized norm then the properties of the resulting optimization problem are investigated it is shown that the gradient of the resulting objective function at the solution produced by the c means algorithm in this case takes a special structure which can be used in terminating the algorithm moreover the local optimality of the solution obtained is checked utilizing the hessian of the criterion function the solution is a local minimum point if the hessian matrix at this point is positive semidefinite simple rules are proposed to help in checking the definiteness of the matrix

local convergence of the fuzzy c means algorithms abstract much understanding has recently been gained concerning global convergence properties of the fuzzy c means fcm family of clustering algorithms these global convergence properties which hold for all iteration sequences guarantee that every fcm iteration sequence converges at least along a subsequence to a stationary point of an fcm objective function in this paper we prove a local convergence property that is a property pertaining to iteration sequences started near a solution specifically a simple result is proved which shows that whenever an fcm algorithm is started sufficiently near a minimizer of the corresponding objective function then the iteration sequence must converge to that particular minimizer the result guarantees that once captured by the local neighborhood of a minimizer the succeeding iterate sequence will not escape thus infinite oscillation of such a sequence cannot occur the rate of convergence of the sequence to such a point is also discussed

hierarchical clustering of mixed data based on distance hierarchy data clustering is an important data mining technique which partitions data according to some similarity criterion abundant algorithms have been proposed for clustering numerical data and some recent research tackles the problem of clustering categorical or mixed data unlike the subtraction scheme used for numerical attributes there is no standard for measuring distance between categorical values in this article we propose a distance representation scheme distance hierarchy which facilitates expressing the similarity between categorical values and also unifies distance measuring of numerical and categorical values we then apply the scheme to mixed data clustering in particular to integrate with a hierarchical clustering algorithm consequently this integrated approach can uniformly handle numerical data and categorical data and also enables one to take the similarity between categorical values into consideration experimental results show that the proposed approach produces better clustering results than conventional clustering algorithms when categorical attributes are present and their values have different degree of similarity

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

a novel fuzzy clustering algorithm with between cluster information for categorical data in this paper we present a new fuzzy clustering algorithm for categorical data in the algorithm the objective function of the fuzzy k modes algorithm is modified by adding the between cluster information so that we can simultaneously minimize the within cluster dispersion and enhance the between cluster separation for obtaining the local optimal solutions of the modified objective function the corresponding update formulas of the membership matrix and the cluster prototypes are strictly derived the convergence of the proposed algorithm under the optimization framework is proved on several real data sets from uci the performance of the proposed algorithm is studied the experimental results illustrate that the algorithm is effective and suitable for categorical data sets

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

use of a fuzzy granulation degranulation criterion for assessing cluster validity the identification of a suitable clustering algorithm to partition data and assessment of the validity of the resultant partitioning are ongoing quests in unsupervised learning in this study a fuzzy granulation degranulation criterion is proposed to evaluate the goodness of a fuzzy partitioning of the data this in turn is used to determine the appropriate clustering algorithm suitable for a particular data set in general the quality of a partitioning is measured by computing the variance within it which is a measure of compactness of the obtained partitioning here a new error function which reflects how well the computed cluster centers represent the whole data set is used as the goodness measure of the obtained partitioning thus a clustering algorithm providing a good set of cluster centers which approximate well the whole data set is considered to be the most suited thereafter this new fuzzy granulation degranulation criterion is used to develop six new cluster validity indices these indices mimic the definitions of the existing and well known cluster validity indices such as pbm index xb index ps index fs index k index and sv index but use the new fuzzy granulation degranulation based error function instead of cluster compactness in order to evaluate the effectiveness of the proposed error function in correctly identifying the appropriate clustering algorithm for a particular data set eight well known clustering algorithms k means fuzzy c means gak means genetic algorithm based k means algorithm a newly developed genetic point symmetry based clustering technique gaps clustering average linkage clustering algorithm expectation maximization em clustering algorithm self organizing map som and spectral clustering technique are evaluated on a set of six artificially generated and six real life data sets results show that gak means is the most appropriate for most of the data sets used for the experiments thereafter the effectiveness of the proposed cluster validity indices in identifying the appropriate number of clusters automatically from different data sets are shown for above mentioned 12 data sets for the purpose of comparison results obtained with the original versions of the proposed cluster validity indices and results obtained by a density based clustering technique are also presented

a fuzzy k modes algorithm for clustering categorical data this correspondence describes extensions to the fuzzy k means algorithm for clustering categorical data by using a simple matching dissimilarity measure for categorical objects and modes instead of means for clusters a new approach is developed which allows the use of the k means paradigm to efficiently cluster large categorical data sets a fuzzy k modes algorithm is presented and the effectiveness of the algorithm is demonstrated with experimental results

optimality tests for fixed points of the fuzzy c means algorithm abstract the fuzzy c means fcm clustering algorithms are known to converge to either local minima or saddle points of the objective function which defines the fcm method the object of this paper is to derive efficient numerical tests for local extrema of the fcm functional that enable one to identify each candidate as a local minimum or saddle point numerical examples of the theory derived illustrate that the tests proposed cover all possible cases

a dynamic niching clustering algorithm based on individual connectedness and its application to color image segmentation in this paper a dynamic niching clustering algorithm based on individual connectedness dnic is proposed for unsupervised classification with no prior knowledge it aims to automatically evolve the optimal number of clusters as well as the cluster centers of the data set based on the proposed adaptive compact k distance neighborhood algorithm more specifically with the adaptive selection of the number of the nearest neighbor and the individual connectedness algorithm dnic often achieves several sets of connecting individuals and each set composes an independent niche in practice each set of connecting individuals corresponds to a homogeneous cluster and this ensures the separability of an arbitrary data set theoretically an application of the dnic clustering algorithm in color image segmentation is also provided experimental results demonstrate that the dnic clustering algorithm has high performance and flexibility highlightsthe dnic aims to automatically evolve the number of clusters and the cluster centers a simpler representation is adopted that each individual represents a single center an algorithm is proposed to select the optimal number of the neighbor automatically an individual connectedness algorithm is proposed to dynamic identify the niches the dynamic niching is accomplished without assuming any a priori knowledge

a comparison of heuristic procedures for minimum within cluster sums of squares partitioning perhaps the most common criterion for partitioning a data set is the minimization of the within cluster sums of squared deviation from cluster centroids although optimal solution procedures for within cluster sums of squares wcss partitioning are computationally feasible for small data sets heuristic procedures are required for most practical applications in the behavioral sciences we compared the performances of nine prominent heuristic procedures for wcss partitioning across 324 simulated data sets representative of a broad spectrum of test conditions performance comparisons focused on both percentage deviation from the best found wcss values as well as recovery of true cluster structure a real coded genetic algorithm and variable neighborhood search heuristic were the most effective methods however a straightforward two stage heuristic algorithm hk means also yielded exceptional performance a follow up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data our findings have important implications for behavioral science researchers whose theoretical conclusions could be adversely affected by poor algorithmic performances

minimization by random search techniques we give two general convergence proofs for random search algorithms we review the literature and show how our results extend those available for specific variants of the conceptual algorithm studied here we then exploit the convergence results to examine convergence rates and to actually design implementable methods finally we report on some computational experience

an optimization algorithm for clustering using weighted dissimilarity measures one of the main problems in cluster analysis is the weighting of attributes so as to discover structures that may be present by using weighted dissimilarity measures for objects a new approach is developed which allows the use of the k means type paradigm to efficiently cluster large data sets the optimization algorithm is presented and the effectiveness of the algorithm is demonstrated with both synthetic and real data sets

application of honey bee mating optimization algorithm on clustering cluster analysis is one of attractive data mining technique that use in many fields one popular class of data clustering algorithms is the center based clustering algorithm k means used as a popular clustering method due to its simplicity and high speed in clustering large datasets however k means has two shortcomings dependency on the initial state and convergence to local optima and global solutions of large problems cannot found with reasonable amount of computation effort in order to overcome local optima problem lots of studies done in clustering over the last decade modeling the behavior of social insects such as ants and bees for the purpose of search and problem solving has been the context of the emerging area of swarm intelligence honey bees are among the most closely studied social insects honey bee mating may also be considered as a typical swarm based approach to optimization in which the search algorithm is inspired by the process of marriage in real honey bee honey bee has been used to model agent based systems in this paper we proposed application of honeybee mating optimization in clustering hbmk means we compared hbmk means with other heuristics algorithm in clustering such as ga sa ts and aco by implementing them on several well known datasets our finding shows that the proposed algorithm works than the best one

a fuzzy k partitions model for categorical data and its comparison to the gom model the grade of membership gom model uses fuzzy sets as memberships of each individual to extreme profiles or classes on the likelihood function of multivariate multinomial distributions the gom clustering algorithm derived from the gom model is used in cluster analysis for categorical data but it is iterated with complicated calculations in this paper we create another approach termed a fuzzy k partitions fkp model which is also based on the likelihood function of multivariate multinomial distributions however the calculations of the fkp algorithm for clustering categorical data derived from the proposed fkp model are simpler the proposed fkp clustering algorithm is not only easier in calculation than the gom but also has more accuracy and computation efficiency to verify it we employ real empirical data and also some simulation data we find that fkp has superior results to gom we then apply these two algorithms to classification of pathology the results show the superiority of the fkp clustering algorithm moreover the proposed fkp algorithm can be used as a fuzzy clustering algorithm for categorical data some comparisons between fkp and two popular algorithms fuzzy k modes and fuzzy centroids are made these results show that the fkp clustering algorithm can be another useful tool in analyzing categorical data

data clustering by minimizing disconnectivity identifying clusters of arbitrary shapes remains a challenge in the field of data clustering we propose a new measure of cluster quality based on minimizing the penalty of disconnection between objects that would be ideally clustered together this disconnectivity is based on analysis of nearest neighbors and the principle that an object should be in the same cluster as its nearest neighbors an algorithm called mindisconnect is proposed that heuristically minimizes disconnectivity and numerical results are presented that indicate that the new algorithm can effectively identify clusters of complex shapes and is robust in finding clusters of arbitrary shapes

genetic clustering for automatic evolution of clusters and application to image classification abstract in this article the searching capability of genetic algorithms has been exploited for automatically evolving the number of clusters as well as proper clustering of any data set a new string representation comprising both real numbers and the do not care symbol is used in order to encode a variable number of clusters the davies bouldin index is used as a measure of the validity of the clusters effectiveness of the genetic clustering scheme is demonstrated for both artificial and real life data sets utility of the genetic clustering technique is also demonstrated for a satellite image of a part of the city calcutta the proposed technique is able to distinguish some characteristic landcover types in the image

symbolic clustering using a new dissimilarity measure abstract a new dissimilarity measure based on position span and content of symbolic objects is proposed for symbolic clustering the dissimilarity measure is new in the sense that it is not just another aspect of a similarity measure in the proposed hierarchical agglomerative clustering methodology composite symbolic objects are formed using a cartesian join operator whenever a mutual pair of symbolic objects is selected for agglomeration based on minimum dissimilarity the minimum dissimilarity values of different merging levels are used to compute the cluster indicator values and hence to determine the number of clusters in the data the results of the application of the algorithm on numeric data of known number of classes are described first so as to show the efficacy of the method subsequently the results of the experiments on two data sets of assertion type of symbolic objects drawn from the domains of fat oil and microcomputers are presented

incremental learning based multiobjective fuzzy clustering for categorical data the problem of clustering categorical data where attribute values cannot be naturally ordered as numerical values has gained more importance in recent time due to the special properties of categorical attributes the clustering of categorical data seems to be more complicated than that of numerical data although a few clustering algorithms that optimize single clustering objective have been proposed it has found that such single measure may not be appropriate for all kind of datasets hence in this article an incremental learning based multiobjective fuzzy clustering for categorical data is proposed for this purpose a multiobjective modified differential evolution based fuzzy clustering algorithm is developed thereafter it integrates with the well known supervised classifier called random forest using incremental learning to propose the aforementioned technique here the multiobjective algorithm produces a set of optimal clustering solutions known as pareto optimal solutions by optimizing two conflicting objectives simultaneously subsequently through incremental learning using random forest classifier final solution is evolved from the ensemble pareto optimal solutions the results of the proposed method are demonstrated quantitatively and visually in comparison with widely used state of the art methods for six synthetic and four real life datasets finally statistical test is conducted to show the superiority of the results produced by the proposed method

clustering categorical data sets using tabu search techniques clustering methods partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some defined criteria the fuzzy k means type algorithm is best suited for implementing this clustering operation because of its effectiveness in clustering data sets however working only on numeric values limits its use because data sets often contain categorical values in this paper we present a tabu search based clustering algorithm to extend the k means paradigm to categorical domains and domains with both numeric and categorical values using tabu search based techniques our algorithm can explore the solution space beyond local optimality in order to aim at finding a global solution of the fuzzy clustering problem it is found that the clustering results produced by the proposed algorithm are very high in accuracy