autocorrelation based fuzzy clustering of time series the traditional approaches to clustering a set of time series are generally applicable if there is a fixed underlying structure to the time series so that each will belong to one cluster or the other however time series often display dynamic behaviour in their evolution over time this dynamic behaviour should be taken into account when attempting to cluster time series for instance during a certain period a time series might belong to a certain cluster afterwards its dynamics might be closer to that of another cluster in this case the traditional clustering approaches are unlikely to find and represent the underlying structure in the given time series this switch from one time state to another which is typically vague can be naturally treated following a fuzzy approach this paper proposes a fuzzy clustering approach based on the autocorrelation functions of time series in which each time series is not assigned exclusively to only one cluster but it is allowed to belong to different clusters with various membership degrees

orthogonal polynomial approach to estimation of poles of rational functions from data on open curves this paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane for this problem nara and ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration the main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials it is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations moreover to cope with discretization errors caused by numerical integration we introduce new polynomials similar in spirit to discrete orthogonal polynomials which yield an algorithm free from discretization errors

modeling fuzziness measures for best wavelet selection uncertainty measures model different types of uncertainty that are inherent in complex information systems measures that model either fuzzy or probabilistic uncertainty types have been explored in the literature this paper shows that a combination of fuzzy and probabilistic uncertainty types combined with the generalized maximum uncertainty principle can be applied to time series sequence classification and analysis we present a novel algorithm that selects a wavelet from a wavelet library such that it best represents a time series sequence in a maximum uncertainty sense transformation coefficients are combined together in feature vectors that capture sequence temporal trends a neural network is trained and tested using extracted gait sequence temporal features results have shown that models that combine together fuzzy and probabilistic uncertainty types better classify time series gait sequences

numerical recovery of location and residue of poles of meromorphic functions we present a method able to recover location and residue of poles of functions meromorphic in a half plane from samples of the function on the real positive semi axis the function is assumed to satisfy appropriate asymptotic conditions including in particular that required by carlson s theorem the peculiar features of the present procedure are a it does not make use of the approximation of meromorphic functions by rational functions b it does not use the standard methods of regularization of ill posed problems the data required for the determination of the pole parameters i e location and residue are the approximate values of the meromorphic function on a finite set of equidistant points on the real positive semi axis we show that this method is numerically stable by proving that the algorithm is convergent as the number of data points tends to infinity and the noise on the input data goes to zero moreover we can also evaluate the degree of approximation of the estimates of pole location and residue which we obtain from the knowledge of a finite number of noisy samples

clustering spatiotemporal data an augmented fuzzy c means in spatiotemporal data commonly encountered in geographical systems biomedical signals and the like each datum is composed of features comprising a spatial component and a temporal part clustering of data of this nature poses challenges especially in terms of a suitable treatment of the spatial and temporal components of the data in this study proceeding with the objective function based clustering such as e g fuzzy c means we revisit and augment the algorithm to make it applicable to spatiotemporal data an augmented distance function is discussed and the resulting clustering algorithm is provided two optimization criteria i e a reconstruction error and a prediction error are introduced and used as a vehicle to optimize the performance of the clustering method experimental results obtained for synthetic and real world data are reported

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

a fuzzy approach for multitype relational data clustering mining interrelated data among multiple types of objects or entities is important in many real world applications despite extensive study on fuzzy clustering of vector space data very limited exploration has been made on fuzzy clustering of relational data that involve several object types in this paper we propose a new fuzzy clustering approach for multitype relational data fc mr in fc mr different types of objects are clustered simultaneously an object is assigned a large membership with respect to a cluster if its related objects in this cluster have high rankings in each cluster an object tends to have a high ranking if its related objects have large memberships in this cluster the fc mr approach is formulated to deal with multitype relational data with various structures the objective function of fc mr is locally optimized by an efficient iterative algorithm which updates the fuzzy membership matrix and the ranking matrix of one type at once while keeping those of other types constant we also discuss the simplified fc mr for multitype relational data with two special structures namely star structure and extended star structure experimental studies are conducted on benchmark document datasets to illustrate how the proposed approach can be applied flexibly under different scenarios in real world applications the experimental results demonstrate the feasibility and effectiveness of the new approach compared with existing ones

segmentation of m fish images for improved classification of chromosomes with an adaptive fuzzy c means clustering algorithm an adaptive fuzzy c means algorithm was developed and applied to the segmentation and classification of multicolor fluorescence in situ hybridization m fish images which can be used to detect chromosomal abnormalities for cancer and genetic disease diagnosis the algorithm improves the classical fuzzy c means algorithm fcm by the use of a gain field which models and corrects intensity inhomogeneities caused by a microscope imaging system flairs of targets chromosomes and uneven hybridization of dna other than directly simulating the inhomogeneousely distributed intensities over the image the gain field regulates centers of each intensity cluster the algorithm has been tested on an m fish database that we have established which demonstrates improved performance in both segmentation and classification when compared with other fcm clustering based algorithms and a recently reported region based segmentation and classification algorithm our method gave the lowest segmentation and classification error which will contribute to improved diagnosis of genetic diseases and cancers

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

pseudodifferential operators on supermanifolds and the atiyah singer index theorem fermionic quantization or clifford algebra is combined with pseudodifferential operators to simplify the proof of the atiyah singer index theorem for the dirac operator on a spin manifold

taiex forecasting based on fuzzy time series and fuzzy variation groups in this paper we present a new method to forecast the daily taiwan stock exchange capitalization weighted stock index taiex based on fuzzy time series and fuzzy variation groups where the main input factor is the previous day s taiex and the secondary factor is either the dow jones the nasdaq the m 1b or their combination first the proposed method fuzzifies the historical training data of the taiex into fuzzy sets to form fuzzy logical relationships second it groups the fuzzy logical relationships into fuzzy logical relationship groups flrgs based on the fuzzy variations of the secondary factor third it evaluates the leverage of the fuzzy variations between the main factor and the secondary factor to construct fuzzy variation groups fourth it gets the statistics of the fuzzy variations appearing in each fuzzy variation group fifth it calculates the weights of the statistics of the fuzzy variations appearing in each fuzzy variation group respectively finally based on the weights of the statistics of the fuzzy variations appearing in the fuzzy variation groups and the flrgs it performs the forecasting of the daily taiex because the proposed method uses both fuzzy variation groups and flrgs to analyze in detail the historical training data it gets higher forecasting accuracy rates to forecast the taiex than the existing methods

deterministic annealing gustafson kessel fuzzy clustering algorithm abstract the gustafson and kessel gk fuzzy clustering algorithm proposed by gustafson and kessel in 1979 was the first important extension to the fuzzy c means fcm algorithm up to now the gk algorithm had become one of the most commonly used fuzzy clustering algorithms where the mahalanobis distance is used as a dissimilarity measure to provide more effectiveness and robustness than the fcm algorithm recently chaomurilige et al 2015 proposed a theoretical analysis on the parameter selection for the gk algorithm in which they indicated that the parameter of the fuzziness index heavily influences the performance of the gk algorithm in this paper we propose a novel gk fuzzy clustering algorithm based on the deterministic annealing approach for decreasing the effect of parameters we first consider maximizing the shannon s entropy of membership functions to the gk objective function and then use deterministic annealing to adjust the annealing parameter we also mathematically provide a theoretical initialization lower bound for the annealing parameter of the proposed deterministic annealing gk da gk algorithm comparisons between the da gk algorithm and other methods are made the computational complexity of the proposed method is also provided experimental results and comparisons actually verify theoretical results and also indicate the superiority and effectiveness of the proposed da gk algorithm

fuzzy clustering with volume prototypes and adaptive cluster merging two extensions to objective function based fuzzy clustering are proposed first the point prototypes are extended to hypervolumes whose size can be fixed or can be determined automatically from the data being clustered it is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function second a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization starting with an overestimation of the number of clusters in the data similar clusters are merged in order to obtain a suitable partitioning an adaptive threshold for merging is proposed the extensions proposed are applied to gustafson kessel and fuzzy c means algorithms and the resulting extended algorithm is given the properties of the new algorithm are illustrated by various examples

enhancement of the classification and reconstruction performance of fuzzy c means with refinements of prototypes abstract owning to their abilities to reveal structural relationships in data fuzzy clustering plays a pivotal role in fuzzy modeling pattern recognition and data analysis as supporting an unsupervised mode of learning fuzzy clustering brings about unique opportunities to build a structural backbone of numerous constructs in the areas identified above a follow up phase is required when the structural findings developed in the form of fuzzy clusters need some refinements usually when the clustering results are used afterwards in the model being developed in a supervised mode following this general line of thought in this study we propose a novel approach to optimize the clustering and classification performance of the fuzzy c means fcm algorithm proceeding with a collection of clusters information granules produced by the fcm we carry out the refinements of the results in order to improve the representation or classification capabilities of fuzzy clusters by adjusting a location of the prototypes so that a certain performance index becomes optimized at this phase the optimization is carried out in a supervised mode with the aid of differential evolution de we propose five different strategies to adjust a location of the prototypes experimental studies completed on synthetic data and publicly available real world data quantify the improvement of the representation and classification abilities of the clustering method

clustering granular data and their characterization with information granules of higher type the study is devoted to the clustering of granular data and an evaluation of the results of such clustering a comprehensive and systematic approach is developed which is composed of three fundamental phases 1 representation of granular data 2 clustering carried out in the representation space of information granules and 3 evaluation of quality of clusters following the reconstruction criterion the reconstruction criterion formed originally for numeric data and leading to an idea of granular prototypes is revisited we show here an emergence of granular information of higher type which are used to implement granular interval prototypes we discuss a way of forming granular data in the context of representation of time series and present clustering of granular time series

robust learning fuzzy c means clustering algorithm with unknown number of clusters abstract in fuzzy clustering the fuzzy c means fcm algorithm is the most commonly used clustering method various extensions of fcm had been proposed in the literature however the fcm algorithm and its extensions are usually affected by initializations and parameter selection with a number of clusters to be given a priori although there were some works to solve these problems in fcm there is no work for fcm to be simultaneously robust to initializations and parameter selection under free of the fuzziness index without a given number of clusters in this paper we construct a robust learning based fcm framework called a robust learning fcm rl fcm algorithm so that it becomes free of the fuzziness index m and initializations without parameter selection and can also automatically find the best number of clusters we first use entropy type penalty terms for adjusting bias with free of the fuzziness index and then create a robust learning based schema for finding the best number of clusters the computational complexity of the proposed rl fcm algorithm is also analyzed comparisons between rl fcm and other existing methods are made experimental results and comparisons actually demonstrate these good aspects of the proposed rl fcm where it exhibits three robust characteristics 1 robust to initializations with free of the fuzziness index 2 robust to without parameter selection and 3 robust to number of clusters with unknown number of clusters

time series clustering with arma mixtures clustering problems are central to many knowledge discovery and data mining tasks however most existing clustering methods can only work with flxed dimensional representations of data patterns in this paper we study the clustering of data patterns that are represented as sequences or time series possibly of difierent lengths we propose a model based approach to this problem using mixtures of autoregressive moving average arma models we derive an expectation maximization em algorithm for learning the mixing coe cients as well as the parameters of the component models to address the model selection problem we use the bayesian information criterion bic to determine the number of clusters in the data experiments are conducted on a number of simulated and real datasets results from the experiments show that our method compares favorably with other methods proposed previously by others for similar time series clustering tasks

spatiotemporal extended fuzzy c means clustering algorithm for hotspots detection and prediction abstract we present a spatiotemporal clustering method namely sefcm which is a generalization of the extended fuzzy c means efcm method for detecting hotspots in spatial analysis each pattern is formed by three features the geographical coordinates and the period in which a certain event is occurred this method is applied to a spatial dataset formed by earthquake epicenters occurred in southern italy since 2001 till to 2014 for prediction of the hotspots obtained in a given year comparisons of the prediction results are also made with the ones obtained by applying the known st dbscan algorithm

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

proximity based clustering a search for structural consistency in data with semantic blocks of features a class of clustering problems that is studied here is concerned with the development of a structure of a global nature given a collection of structures clusters constructed locally for data that are represented by several collections blocks of features these blocks of features come with a well defined semantics for instance in spatiotemporal data a certain block of features concerns a spatial component of the data say x y or x y z coordinates while another one deals with the features that describe time series associated with the corresponding locations the results of clustering that are being produced locally are reconciled by minimizing a distance between the proximity matrices that are formed at the higher conceptual level and induced by the individual partition matrices the optimization problem is formulated and presented along with its iterative scheme

on the solution of a class of cauchy integral equations in a previous paper we have presented a new method for solving a class of cauchy integral equations in this work we discuss in detail how to manage this method numerically when only a finite and noisy data set is available particular attention is focused on the question of the numerical stability

a global averaging method for dynamic time warping with applications to clustering mining sequential data is an old topic that has been revived in the last decade due to the increasing availability of sequential datasets most works in this field are centred on the definition and use of a distance or at least a similarity measure between sequences of elements a measure called dynamic time warping dtw seems to be currently the most relevant for a large panel of applications this article is about the use of dtw in data mining algorithms and focuses on the computation of an average of a set of sequences averaging is an essential tool for the analysis of data for example the k means clustering algorithm repeatedly computes such an average and needs to provide a description of the clusters it forms averaging is here a crucial step which must be sound in order to make algorithms work accurately when dealing with sequences especially when sequences are compared with dtw averaging is not a trivial task starting with existing techniques developed around dtw the article suggests an analysis framework to classify averaging techniques it then proceeds to study the two major questions lifted by the framework first we develop a global technique for averaging a set of sequences this technique is original in that it avoids using iterative pairwise averaging it is thus insensitive to ordering effects second we describe a new strategy to reduce the length of the resulting average sequence this has a favourable impact on performance but also on the relevance of the results both aspects are evaluated on standard datasets and the evaluation shows that they compare favourably with existing methods the article ends by describing the use of averaging in clustering the last section also introduces a new application domain namely the analysis of satellite image time series where data mining techniques provide an original approach

the higher index theorem for manifolds with boundary abstract the global calculus of pseudodifferential operators on manifolds with cylindrical ends is developed to study the higher indices of elliptic boundary value problems the higher index theorem is then proved by employing the asymptotic symbolic calculus of the global pseudodifferential operators based on a scaling similar to that of getzler

cluster centric fuzzy modeling in this study we propose a cluster oriented development of fuzzy models an overall design process is focused on an efficient usage of fuzzy clustering fuzzy c means fcm in particular to form information granules clusters that are used in the construction of the fuzzy model fuzzy models are regarded as mappings from information granules expressed in the input and output spaces this position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules this study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages the augmented fcm method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized it is shown that the cluster oriented modeling gives rise to the mamdani like fuzzy rules and a zero order takagi sugeno model under a certain decoding scheme we identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding decoding or granulation degranulation encountered in processing fuzzy sets and granular computing in general furthermore refinements of zero order fuzzy models are investigated leading to first order fuzzy models with linear functions standing in the conclusions of the rules a series of experiments is reported where we used synthetic and real world data using which an issue of generalization capabilities is elaborated in detail

optimizing the prototypes with a novel data weighting algorithm for enhancing the classification performance of fuzzy clustering abstract fuzzy clustering is regarded as an unsupervised learning process that constitutes a prerequisite for many other data mining techniques deciding how to classify data efficiently and accurately has been one of the topics pursued by many researchers we anticipate that the classification performance of the clustering is strongly dependent on the boundary data viz data located at the boundaries of the clusters the boundary data hold some levels of uncertainties and as such contain more information than others usually the greater the uncertainty the more information contained in such data to improve the quality of clustering this study develops an augmented scheme of fuzzy clustering in which a novel weighted data based fuzzy clustering is proposed in the introduced scheme a dataset is composed of boundary data and non boundary data the partition matrix is used to determine the boundary data and the non boundary data to be next considered in the clustering process then we assign different weights to each datum to construct the weighted data during this process we make the weights for the boundary data and the non boundary data different which makes the contributions of the boundary data and the non boundary data to the prototypes being reduced and enhanced respectively furthermore we build a weighting function to determine the weights of the data the weighted data are used to optimize the prototypes with the optimized prototypes the partition matrix can be refined which ultimately makes the boundaries of the clusters optimized finally the classification performance of fuzzy clustering is enhanced we offer a thorough analysis of the developed scheme comprehensive experimental studies involving synthetic and publicly available datasets are reported to demonstrate the performance of the proposed approach

multivariable gaussian evolving fuzzy modeling system this paper introduces a class of evolving fuzzy rule based system as an approach for multivariable gaussian adaptive fuzzy modeling the system is an evolving takagi sugeno ets functional fuzzy model whose rule base can be continuously updated using a new recursive clustering algorithm based on participatory learning the fuzzy sets of the rule antecedents are multivariable gaussian membership functions which have been adopted to preserve information between input variable interactions the parameters of the membership functions are estimated by the clustering algorithm a weighted recursive least squares algorithm updates the parameters of the rule consequents experiments considering time series forecasting and nonlinear system identification are performed to evaluate the performance of the approach proposed the multivariable gaussian evolving fuzzy models are compared with alternative evolving fuzzy models and classic models with fixed structures the results suggest that multivariable gaussian evolving fuzzy modeling is a promising approach for adaptive system modeling

fuzzy clustering of time series in the frequency domain traditional and fuzzy cluster analyses are applicable to variables whose values are uncorrelated hence in order to cluster time series data which are usually serially correlated one needs to extract features from the time series the values of which are uncorrelated the periodogram which is an estimator of the spectral density function of a time series is a feature that can be used in the cluster analysis of time series because its ordinates are uncorrelated additionally the normalized periodogram and the logarithm of the normalized periodogram are also features that can be used in this paper we consider a fuzzy clustering approach for time series based on the estimated cepstrum the cepstrum is the spectrum of the logarithm of the spectral density function we show in our simulation studies for the typical generating processes that have been considered fuzzy clustering based on the cepstral coefficients performs very well compared to when it is based on other features

predictive modular fuzzy systems for time series classification we introduce the so called predictive modular fuzzy system premofs which performs time series classification a premofs consists of 1 a bank of prediction modules and 2 a fuzzy decision module it is assumed that the time series is generated by a source belonging to a finite search set universal set then the classification problem is to select the source that best represents the observed data classification is based on a membership function which is updated recursively according to the predictive accuracy of each model two algorithms are presented for updating the membership function the first is based on sum product fuzzy inference and the second on max min fuzzy inference in short premofs is a fuzzy modular system that classifies time series to one of a finite number of classes using the full set of past data without preprocessing to perform a recursive competitive computation of membership function based on predictive accuracy convergence proofs are given for both premofs algorithms in both cases the membership grade tends to one for the source that best predicts the observed data and to less than one for the remaining sources hence correct classification is guaranteed simulation results are also presented premofs are applied to signal detection system identification and phoneme classification tasks

takagi sugeno fuzzy modeling using mixed fuzzy clustering this paper proposes the use of mixed fuzzy clustering mfc algorithm to derive takagi sugeno t s fuzzy models fms mixed fuzzy clustering handles both time invariant and multivariate time variant features allowing the user to control the weight of each component in the clustering process two model designs based on mfc are investigated in the first the antecedent fuzzy sets of the t s model are obtained from the clusters obtained by the mfc algorithm in the second fms based on fuzzy c means fcm are constructed over the input space of the partition matrix generated by mfc the proposed fuzzy modeling approaches are used in health care classification problems where time series of unequal lengths are very common mfc based t s fms outperform fcm based t s fms in four out of five datasets and k nearest neighbors classifiers in five out of five datasets dynamic time warping performs better than the euclidean distance in one dataset and similarly in the remaining given the different nature of time variant and invariant data the choice of a clustering algorithm that treats data differently should be considered for model construction

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed