statistical mechanics and phase transitions in clustering a new approach to clustering based on statistical physics is presented the problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance the corresponding lagrange multiplier is related to the temperature and motivates a deterministic annealing process where the free energy is minimized at each temperature critical temperatures are derived for phase transitions when existing clusters split it is a hierarchical clustering estimating the most probable cluster parameters at various average variances

on relational possibilistic clustering this paper initially describes the relational counterpart of possibilistic c means pcm algorithm called relational pcm or rpcm rpcm is then improved to better handle arbitrary dissimilarity data first a re scaling of the pcm membership function is proposed in order to obtain zero membership values when the distance to prototype equals the maximum value allowed in bounded dissimilarity measures second a heuristic method of reference distance initialisation is provided which diminishes the known pcm tendency of producing coincident clusters finally rpcm improved with our initialisation strategy is tested on both synthetic and real data sets with satisfactory results

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

fuzzy clustering for data time arrays with inlier and outlier time trajectories in many knowledge discovery and data mining tasks fuzzy clustering is one of the most common tools for data partitioning in this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories time series sequences are developed in particular by adopting an exploratory approach based on a geometric algebraic formulation of the data time array different kinds of dynamic fuzzy clustering models based on cross sectional and longitudinal aspects are suggested furthermore a modified version of the previous clustering models that can be seen as a generalization of these models is proposed by utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories trajectories with anomalous positions and slopes are present in the dataset in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure s domains and are characterized by strange slopes in these models the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given furthermore some remarks on dynamic three way extensions of a few robust fuzzy clustering models for two way data are suggested demonstrative examples are shown and a comparison assessment based on artificial multivariate time varying data is carried out

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

visual stability analysis for model selection in graded possibilistic clustering abstract when compared to standard clustering fuzzy clustering provides more flexible and powerful data representation most fuzzy methods require setting some parameters as is the case for the graded possibilistic c means clustering method which has two parameters in addition to number of centroids however for this model selection task there is no well established criterion available building on our own previous work on fuzzy clustering similarity indexes we introduce a technique to evaluate the stability of clusterings by using the fuzzy jaccard index and use this procedure to select the most suitable values of parameters the experiments indicate that the procedure is effective for fuzzy and possibilistic clustering

evaluation of stability of k means cluster ensembles with respect to random initialization many clustering algorithms including cluster ensembles rely on a random component stability of the results across different runs is considered to be an asset of the algorithm the cluster ensembles considered here are based on k means clusterers each clusterer is assigned a random target number of clusters k and is started from a random initialization here we use 10 artificial and 10 real data sets to study ensemble stability with respect to random k and random initialization the data sets were chosen to have a small number of clusters two to seven and a moderate number of data points up to a few hundred pairwise stability is defined as the adjusted rand index between pairs of clusterers in the ensemble averaged across all pairs nonpairwise stability is defined as the entropy of the consensus matrix of the ensemble an experimental comparison with the stability of the standard k means algorithm was carried out for k from 2 to 20 the results revealed that ensembles are generally more stable markedly so for larger k to establish whether stability can serve as a cluster validity index we first looked at the relationship between stability and accuracy with respect to the number of clusters k we found that such a relationship strongly depends on the data set varying from almost perfect positive correlation 0 97 for the glass data to almost perfect negative correlation 0 93 for the crabs data we propose a new combined stability index to be the sum of the pairwise individual and ensemble stabilities this index was found to correlate better with the ensemble accuracy following the hypothesis that a point of stability of a clustering algorithm corresponds to a structure found in the data we used the stability measures to pick the number of clusters the combined stability index gave best results

a possibilistic approach to latent component analysis for symmetric fuzzy data in many situations the available amount of data is huge and can be intractable when the data set is single valued latent component models are recognized techniques which provide a useful compression of the information this is done by considering a regression model between observed and unobserved latent variables in this paper an extension of latent component analysis to deal with fuzzy data is proposed our extension follows the possibilistic approach widely used both in the cluster and regression frameworks in this case the possibilistic approach involves the formulation of a latent component analysis for fuzzy data by optimization specifically a non linear programming problem in which the fuzziness of the model is minimized is introduced in order to show how our model works the results of two applications are proposed

possibilistic and probabilistic fuzzy clustering unification within the framework of the non extensive thermostatistics fuzzy clustering algorithms are becoming the major technique in cluster analysis in this paper we consider the fuzzy clustering based on objective functions they can be divided into two categories possibilistic and probabilistic approaches leading to two different function families depending on the conditions required to state that fuzzy clusters are a fuzzy c partition of the input data recently we have presented in menard and eboueya fuzzy sets and systems 27 to be published an axiomatic derivation of the possibilistic and maximum entropy inference mei clustering approaches based upon an unifying principle of physics that of extreme physical information epi defined by frieden physics from fisher information a unification cambridge university press cambridge 1999 here using the same formalism we explicitly give a new criterion in order to provide a theoretical justification of the objective functions constraint terms membership functions and weighting exponent m used in the probabilistic and possibilistic fuzzy clustering moreover we propose an unified framework including the two procedures this approach is inspired by the work of frieden and plastino and plastino and miller physics a 235 577 extending the principle of extremal information in the framework of the non extensive thermostatistics then we show how with the help of epi one can propose extensions of the fcm and possibilistic algorithms

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

analysis of aperiodicities in nearly periodic waveforms waveforms from the oscillations of sustained tone acoustic musical instruments are never precisely periodic the source of their aperiodicity is instrument dependent and typically small with the exception of some studies on the period fluctuations of bowed string instruments musical instrument aperiodicities have never been quantitatively characterized this paper introduces two techniques for the characterization of nearly periodic waveforms the norm difference method and the fourier transform of sonograms the former is appropriate for the study of cycle to cycle fluctuations and yields its best results for fluctuations that have a typical time scale of up to 16 periods of the nominally periodic oscillation the second method is appropriate for a frequency range of 2 hz to approximately 100 hz the ranges of applicability overlap for pitch frequencies in the treble region the techniques are described in detail and explored with various computer constructed waveforms with appropriate properties de

proximity based frameworks for generating embeddings from multi output data this paper is about supervised and semi supervised dimensionality reduction dr by generating spectral embeddings from multi output data based on the pairwise proximity information two flexible and generic frameworks are proposed to achieve supervised dr sdr for multilabel classification one is able to extend any existing single label sdr to multilabel via sample duplication referred to as mesd the other is a multilabel design framework that tackles the sdr problem by computing weight proximity matrices based on simultaneous feature and label information referred to as mope as a generalization of many current techniques a diverse set of different schemes for label based proximity calculation as well as a mechanism for combining label based and feature based weight information by considering information importance and prioritization are proposed for mope additionally we summarize many current spectral methods for unsupervised dr udr single multilabel sdr and semi supervised dr ssdr and express them under a common template representation as a general guide to researchers in the field we also propose a general framework for achieving ssdr by combining existing sdr and udr models and also a procedure of reducing the computational cost via learning with a target set of relation features the effectiveness of our proposed methodologies is demonstrated with experiments with document collections for multilabel text categorization from the natural language processing domain

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

dual fuzzy possibilistic coclustering for categorization of documents in this paper we develop a new soft model dual fuzzy possibilistic coclustering dfpc for document categorization the proposed model targets robustness to outliers and richer representations of coclusters dfpc is inspired by an existing algorithm called possibilistic fuzzy c means pfcm that hybridizes fuzzy and possibilistic clustering it has been shown that pfcm can perform effectively for low dimensional data clustering to achieve our goal we expand this existing idea by introducing a novel pfcm like coclustering model the new algorithm dfpc preserves the desired properties of pfcm in addition as a coclustering algorithm dfpc is more suitable for our intended high dimensional application document clustering besides the coclustering mechanism enables dfpc to generate together with document clusters fuzzy possibilistic word memberships these word memberships which are absent in the existing pfcm model can play an important role in generating useful descriptions of document clusters we detail the formulation of the proposed model and provide an extensive analytical study of the algorithm dfpc experiments on an artificial dataset and various benchmark document datasets demonstrate the effectiveness and potential of dfpc

possibilistic fuzzy co clustering of large document collections in this paper we propose a new co clustering algorithm called possibilistic fuzzy co clustering pfcc for automatic categorization of large document collections pfcc integrates a possibilistic document clustering technique and a combined formulation of fuzzy word ranking and partitioning into a fast iterative co clustering procedure this novel framework brings about simultaneously some benefits including robustness in the presence of document and word outliers rich representations of co clusters highly descriptive document clusters a good performance in a high dimensional space and a reduced sensitivity to the initialization in the possibilistic clustering we present the detailed formulation of pfcc together with the explanations of the motivations behind the advantages over other existing works and the algorithm s proof of convergence are provided experiments on several large document data sets demonstrate the effectiveness of pfcc

cluster ensembles a knowledge reuse framework for combining multiple partitions this paper introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings we first identify several application scenarios for the resultant knowledge reuse framework that we call cluster ensembles the cluster ensemble problem is then formalized as a combinatorial optimization problem in terms of shared mutual information in addition to a direct maximization approach we propose three effective and efficient techniques for obtaining high quality combiners consensus functions the first combiner induces a similarity measure from the partitionings and then reclusters the objects the second combiner is based on hypergraph partitioning the third one collapses groups of clusters into meta clusters which then compete for each object to determine the combined clustering due to the low computational costs of our techniques it is quite feasible to use a supra consensus function that evaluates all three approaches against the objective function and picks the best solution for a given situation we evaluate the effectiveness of cluster ensembles in three qualitatively different application scenarios i where the original clusters were formed based on non identical sets of features ii where the original clustering algorithms worked on non identical sets of objects and iii where a common data set is used and the main purpose of combining multiple clusterings is to improve the quality and robustness of the solution promising results are obtained in all three situations for synthetic as well as real data sets

mercer kernel based clustering in feature space the article presents a method for both the unsupervised partitioning of a sample of data and the estimation of the possible number of inherent clusters which generate the data this work exploits the notion that performing a nonlinear data transformation into some high dimensional feature space increases the probability of the linear separability of the patterns within the transformed space and therefore simplifies the associated data structure it is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data

a similarity based robust clustering method this paper presents an alternating optimization clustering procedure called a similarity based clustering method scm it is an effective and robust approach to clustering on the basis of a total similarity objective function related to the approximate density shape estimation we show that the data points in scm can self organize local optimal cluster number and volumes without using cluster validity functions or a variance covariance matrix the proposed clustering method is also robust to noise and outliers based on the influence function and gross error sensitivity analysis therefore scm exhibits three robust clustering characteristics 1 robust to the initialization cluster number and initial guesses 2 robust to cluster volumes ability to detect different volumes of clusters and 3 robust to noise and outliers several numerical data sets and actual data are used in the scm to show these good aspects the computational complexity of scm is also analyzed some experimental results of comparing the proposed scm with the existing methods show the superiority of the scm method

model transitions in descending flvq fuzzy learning vector quantization flvq also known as the fuzzy kohonen clustering network was developed to improve performance and usability of online hard competitive kohonen vq and soft competitive self organizing map som algorithms flvq s effectiveness seems to depend on the range of change of the weighting exponent m t extreme m t values 1 and spl infin are employed to investigate flvq asymptotic behaviors this analysis shows that when m t tends to either extreme flvq is affected by trivial vq which causes centroids to collapse in the grand mean of the input data set no analytical criterion has been found to improve the heuristic choice of the range of m t change two flvq and som classification experiments of remote sensed data are then presented the two nets are connected in cascade to a supervised second stage based on the delta rule the results confirm that flvq performance can be greatly affected by the user s definition of the range of change of the weighting exponent moreover flvq shows instability when its traditional termination criterion is applied empirical recommendations are proposed for the enhancement of flvq robustness both the analytical and the experimental data reported seem to indicate that the choice of the range of m t change is still open to discussion and that alternative clustering neural network approaches should be developed to pursue during training 1 monotone reduction of the neurons learning rate and 2 monotone reduction of the overlap among neuron receptive fields

applying the possibilistic c means algorithm in kernel induced spaces in this paper we study a kernel extension of the classic possibilistic c means in the proposed extension we implicitly map input patterns into a possibly high dimensional space by means of positive semidefinite kernels in this new space we model the mapped data by means of the possibilistic clustering algorithm we study in more detail the special case where we model the mapped data using a single cluster only since it turns out to have many interesting properties the modeled memberships in kernel induced spaces yield a modeling of generic shapes in the input space we analyze in detail the connections to one class support vector machines and kernel density estimation thus suggesting that the proposed algorithm can be used in many scenarios of unsupervised learning in the experimental part we analyze the stability and the accuracy of the proposed algorithm on some synthetic and real datasets the results show high stability and good performances in terms of accuracy

discrimination of musical instrument sounds resynthesized with simplified spectrotemporal parameters the perceptual salience of several outstanding features of quasiharmonic time variant spectra was investigated in musical instrument sounds spectral analyses of sounds from seven musical instruments clarinet flute oboe trumpet violin harpsichord and marimba produced time varying harmonic amplitude and frequency data six basic data simplifications and five combinations of them were applied to the reference tones amplitude variation smoothing coherent variation of amplitudes over time spectral envelope smoothing forced harmonic frequency variation frequency variation smoothing and harmonic frequency flattening listeners were asked to discriminate sounds resynthesized with simplified data from reference sounds resynthesized with the full data averaged over the seven instruments the discrimination was very good for spectral envelope smoothing and amplitude envelope coherence but was moderate to poor in decreasing order for forced harmonic frequency variation frequency variation smoothing frequency flattening and amplitude variation smoothing discrimination of combinations of simplifications was equivalent to that of the most potent constituent simplification objective measurements were made on the spectral data for harmonic amplitude harmonic frequency and spectral centroid changes resulting from simplifications these measures were found to correlate well with discrimination results indicating that listeners have access to a relatively fine grained sensory representation of musical instrument sounds

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

worst case analysis of weight inaccuracy effects in multilayer perceptrons we derive a method for the analysis of weight quantization effects in multilayer perceptrons based on the application of interval arithmetic differently from previous results we find worst case bounds on the errors due to weight quantization that are valid for every distribution of the input or weight values given a trained network our method allows us to easily compute the minimum number of bits needed to encode its weights

a new credibilistic clustering algorithm abstract this paper focuses on credibilistic clustering approach a data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition there are different models for objective function based fuzzy clustering such as fuzzy c means fcm possibilistic c mean pcm and their combinations credibilistic clustering is a new approach in this field in this paper a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering also in objective function the separation of clusters is considered in addition to the compactness within clusters the steps of clustering are designed based on this approach finally the main issues about model are discussed and the results of computational experiments are presented to show the efficiency of the proposed model

vector quantization in dct domain using fuzzy possibilistic c means based on penalized and compensated constraints in this paper fuzzy possibilistic c means fpcm approach based on penalized and compensated constraints are proposed to vector quantization vq in discrete cosine transform dct for image compression these approaches are named penalized fuzzy possibilistic c means pfpcm and compensated fuzzy possibilistic c means cfpcm the main purpose is to modify the fpcm strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to 4nd near global solution in optimal problem the information transformed by dct was separated into dc and ac coe5cients then the ac coe5cients are trained by using the proposed methods to generate better codebook based on vq the compression performances using the proposed approaches are compared with fpcm and conventional vq method from the experimental results the promising performances can be obtained using the proposed approaches 2002 pattern recognition society published by elsevier science ltd all rights reserved

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

relational generalizations of cluster validity indices numerous computational schemes have arisen over the years that attempt to learn information about objects based upon the similarity or dissimilarity of one object to another one such scheme clustering looks for self similar groups of objects to use clustering algorithms an investigator must often have a priori knowledge of the number of clusters i e c to search for in the data moreover it is often convenient to have ways to rank the returned results either for a single value of c a range of c s different clustering methods or any combination thereof however the task of assessing the quality of the results so that c may be determined objectively is currently ill defined for object object relationships to bridge this gap we generalize three well known validity indices the modified hubert s gamma xie beni and the generalized dunn s indices to relational data in doing so we develop a framework to convert many other validity indices to a relational form numerical examples on 12 datasets samples from four normal mixtures four real world object datasets and four real world pure relational datasets using the relational duals of the hard fuzzy and possibilistic c means cluster algorithms are offered to illustrate and evaluate the new indices

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

comments on a possibilistic approach to clustering in this comment we report a difficulty with the application of the possibilistic approach to fuzzy clustering pcm proposed by keller and krishnapuram 1993 in applying this algorithm we found that it has the undesirable tendency to produce coincidental clusters results illustrating this tendency are reported and a possible explanation for the pcm behavior is suggested

a method for comparing two hierarchical clusterings abstract this article concerns the derivation and use of a measure of similarity between two hierarchical clusterings the measure bk is derived from the matching matrix mij formed by cutting the two hierarchical trees and counting the number of matching entries in the k clusters in each tree the mean and variance of bk are determined under the assumption that the margins of mij are fixed thus bk represents a collection of measures for k 2 n 1 k bk plots are found to be useful in portraying the similarity of two clusterings bk is compared to other measures of similarity proposed respectively by baker 1974 and rand 1971 the use of k bk plots for studying clustering methods is explored by a series of monte carlo sampling experiments an example of the use of k bk on real data is given

unsupervised possibilistic clustering in fuzzy clustering the fuzzy c means fcm clustering algorithm is the best known and used method since the fcm memberships do not always explain the degrees of belonging for the data well krishnapuram and keller proposed a possibilistic approach to clustering to correct this weakness of fcm however the performance of krishnapuram and keller s approach depends heavily on the parameters in this paper we propose another possibilistic clustering algorithm pca which is based on the fcm objective function the partition coefficient pc and partition entropy pe validity indexes the resulting membership becomes the exponential function so that it is robust to noise and outliers the parameters in pca can be easily handled also the pca objective function can be considered as a potential function or a mountain function so that the prototypes of pca can be correspondent to the peaks of the estimated function to validate the clustering results obtained through a pca we generalized the validity indexes of fcm this generalization makes each validity index workable in both fuzzy and possibilistic clustering models by combining these generalized validity indexes an unsupervised possibilistic clustering is proposed some numerical examples and real data implementation on the basis of the proposed pca and generalized validity indexes show their effectiveness and accuracy

a robust automatic merging possibilistic clustering method krishnapuram and keller ieee trans fuzzy syst vol 1 no 2 pp 98 110 may 1993 proposed a possibilistic c means pcm clustering by relaxing the constraint in fuzzy c means fcms that the memberships of a data point across classes sum to 1 the pcm algorithm has a tendency to produce coincident clusters this can be a merit of pcm as a good mode seeking algorithm if initials and parameters are suitably chosen however the performance of pcm heavily depends on initializations and parameter selection in this paper we propose a mechanism of robust automatic merging we then create an automatic merging possibilistic clustering method am pcm where the proposed algorithm does not only solve these parameter selection and initialization problems but also obtains an optimal cluster number the proposed am pcm algorithm first uses all data points as initial cluster centers and then automatically merges these surrounding points around each cluster mode such that it can self organize data groups according to the original data structure the am pcm can exhibit the robustness to parameter noise cluster number different volumes and initializations the computational complexity of am pcm is also analyzed comparisons between am pcm and other clustering methods are made some numerical data and real datasets are used to show these good aspects of am pcm experimental results and comparisons actually demonstrate that the proposed am pcm is an effective and parameter free robust clustering algorithm

characterizing the clarinet tone measurements of lyapunov exponents correlation dimension and unsteadiness the clarinet tone is produced by a self sustained oscillation involving nonlinearity between the flow through the reed and the mechanical response of the reed and acoustic coupling via the air column response regimes of oscillation include periodic biperiodic and other quasi periodic signals yet even a nominally periodic tone has small but perceptually and musically important deviations that are obscured in a conventional power spectrum such deviations may be due to the nonlinear dynamics underlying sound production or to perturbations in the performer s control of the instrument via changes in lip embouchure blowing pressure and vocal tract configuration techniques based upon experimental nonlinear dynamics and short time signal processing are applied to the acoustic signal measured within the clarinet mouthpiece to assess the role of these additional deviations on sound production these include the lyapunov exponent correlation dimension and a normalized period synchronous energy variance

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

isometric decomposition operators function spaces ep qλ and applications to nonlinear evolution equations abstract by using the isometric decomposition to the frequency spaces we will introduce a new class of function spaces e p q which is a subspace of gevrey 1 class g 1 r n c r n for 0 and we will study the cauchy problem for the nonlinear schrodinger equation the complex ginzburg landau equation and the navier stokes equation some well posed results are obtained for the cauchy data in e 2 1 0 and the regularity behavior in e 2 1 ct g 1 r n for the complex ginzburg landau equation and the navier stokes equation is also obtained as time t 0

bounded fuzzy possibilistic method abstract this paper introduces the bounded fuzzy possibilistic method bfpm by addressing several issues that previous clustering classification methods have not considered in fuzzy clustering the data object s membership values should sum to 1 hence any data object may obtain full membership in at most one cluster possibilistic clustering methods remove this restriction however bfpm differs from previous fuzzy and possibilistic clustering approaches by allowing the membership function to take larger values with respect to all clusters furthermore in bfpm a data object can have full membership in multiple clusters or even in all clusters bfpm relaxes the boundary conditions restrictions in membership assignment the assigned boundaries satisfy the necessity of obtaining full memberships and overcome the issues with conventional methods overlapping the functionality of the proposed method has been proved in geometry set theory and other disciplines when learning methods should provide the important arithmetic operations on different domains weighted feature distance function wfd is also used to cover diversity validity and comparison indexes have been applied to evaluate the accuracy of bfpm conventional fuzzy and possibilistic methods are compared with bfpm and bfpm wfd in terms of accuracy fuzzification constant different norms and covering diversity and overlapping the results show that the proposed algorithms perform better than other fuzzy and possibilistic methods

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

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

improved possibilistic c means clustering algorithms a possibilistic approach was proposed in a previous paper for c means clustering and two algorithms realizing this approach were reported in two previous papers although the possibilistic approach is sound these two algorithms tend to find identical clusters in this paper we modify and improve these algorithms to overcome their shortcoming the numerical results demonstrate that the improved algorithms can determine proper clusters and they can realize the advantages of the possibilistic approach

a stability based validity method for fuzzy clustering an important goal in cluster analysis is the internal validation of results using an objective criterion of particular relevance in this respect is the estimation of the optimum number of clusters capturing the intrinsic structure of your data this paper proposes a method to determine this optimum number based on the evaluation of fuzzy partition stability under bootstrap resampling the method is first characterized on synthetic data with respect to hyper parameters like the fuzzifier and spatial clustering parameters such as feature space dimensionality clusters degree of overlap and number of clusters the method is then validated on experimental datasets furthermore the performance of the proposed method is compared to that obtained using a number of traditional fuzzy validity rules based on the cluster compactness to separation criteria the proposed method provides accurate and reliable results and offers better generalization capabilities than the classical approaches

generalized metaplectic operators and the schrodinger equation with a potential in the sjostrand class it is well known that the matrix of a metaplectic operator with respect to phase space shifts is concentrated along the graph of a linear symplectic map we show that the algebra generated by metaplectic operators and by pseudodifferential operators in a sjostrand class enjoys the same decay properties we study the behavior of these generalized metaplectic operators and represent them by fourier integral operators our main result shows that the one parameter group generated by a hamiltonian operator with a potential in the sjostrand class consists of generalized metaplectic operators as a consequence the schrodinger equation preserves the phase space concentration as measured by modulation space norms

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

a novel adaptive possibilistic clustering algorithm in this paper a novel possibilistic c means clustering algorithm called adaptive possibilistic c means is presented its main feature is that its parameters after their initialization are properly adapted during its execution provided that the algorithm starts with a reasonable overestimate of the number of physical clusters formed by the data it is capable in principle to unravel them a long standing issue in the clustering literature this is due to the fully adaptive nature of the proposed algorithm that enables the removal of the clusters that gradually become obsolete in addition the adaptation of all its parameters increases the flexibility of the algorithm in following the variations in the formation of the clusters that occur from iteration to iteration theoretical results that are indicative of the convergence behavior of the algorithm are also provided finally extensive simulation results on both synthetic and real data highlight the effectiveness of the proposed algorithm

a robust competitive clustering algorithm with applications in computer vision this paper addresses three major issues associated with conventional partitional clustering namely sensitivity to initialization difficulty in determining the number of clusters and sensitivity to noise and outliers the proposed robust competitive agglomeration rca algorithm starts with a large number of clusters to reduce the sensitivity to initialization and determines the actual number of clusters by a process of competitive agglomeration noise immunity is achieved by incorporating concepts from robust statistics into the algorithm rca assigns two different sets of weights for each data point the first set of constrained weights represents degrees of sharing and is used to create a competitive environment and to generate a fuzzy partition of the data set the second set corresponds to robust weights and is used to obtain robust estimates of the cluster prototypes by choosing an appropriate distance measure in the objective function rca can be used to find an unknown number of clusters of various shapes in noisy data sets as well as to fit an unknown number of parametric models simultaneously several examples such as clustering mixture decomposition line plane fitting segmentation of range images and estimation of motion parameters of multiple objects are shown

on possibilistic clustering with repulsion constraints for imprecise data in possibilistic clustering objects are assigned to clusters according to the so called membership degrees taking values in the unit interval differently from fuzzy clustering it is not required that the sum of the membership degrees of an object to all clusters is equal to one this is very helpful in the presence of outliers which are usually assigned to the clusters with membership degrees close to zero unfortunately a drawback of the possibilistic approach is the tendency to produce coincident clusters a remedy is to add a repulsion term among prototypes in the loss function forcing the prototypes to be far enough from each other here a possibilistic clustering algorithm with repulsion constraints for imprecise data managed in terms of fuzzy sets is introduced applications to synthetic and real fuzzy data are considered in order to analyze how the proposed clustering algorithm works in practice

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

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

a characterization of modulation spaces by symplectic rotations abstract this note contains a new characterization of modulation spaces m m p r n 1 p by symplectic rotations precisely instead to measure the time frequency content of a function by using translations and modulations of a fixed window as building blocks we use translations and metaplectic operators corresponding to symplectic rotations technically this amounts to replace in the computation of the m m p r n norm the integral in the time frequency plane with an integral on r n u 2 n r with respect to a suitable measure u 2 n r being the group of symplectic rotations more conceptually we are considering a sort of polar coordinates in the time frequency plane to have invariance under symplectic rotations we choose a gaussian as suitable window function we also provide a similar and easier characterization with the group u 2 n r being reduced to the n dimensional torus t n

comparing fuzzy probabilistic and possibilistic partitions when clustering produces more than one candidate to partition a finite set of objects o there are two approaches to validation i e selection of a best partition and implicitly a best value for c which is the number of clusters in o first we may use an internal index which evaluates each partition separately second we may compare pairs of candidates with each other or with a reference partition that purports to represent the true cluster structure in the objects this paper generalizes many of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for candidate partitions of any type i e crisp or soft with soft comprising the fuzzy probabilistic and possibilistic cases space prevents inclusion of all of the possible generalizations that can be realized this way here we concentrate on the rand index and its modifications we compare our fuzzy rand index with those of campello hullermeier and rifqi and brouwer and show that our extension of the rand index is o n while the other three are all o n2 numerical examples are given to illustrate various facets of the new indices in particular we show that our indices can be used even when the partitions are probabilistic or possibilistic and that our method of generalization is valid for any index that depends only on the entries of the classical i e four pair types contingency table for this problem