feature interaction in subspace clustering using the choquet integral subspace clustering has recently emerged as a popular approach to removing irrelevant and redundant features during the clustering process however most subspace clustering methods do not consider the interaction between the features this unawareness limits the analysis performance in many pattern recognition problems in this paper we propose a novel subspace clustering technique by introducing the feature interaction using the concepts of fuzzy measures and the choquet integral this new framework of subspace clustering can provide optimal subsets of interacted features chosen for each cluster and hence can improve clustering based pattern recognition tasks various experimental results illustrate the effective performance of the proposed method

collocation method for generalized abel s integral equations the purpose of this paper is to present an approximate method for solving the generalized abel s integral equations the approximate method is based on the collocation method for solving volterra integral equations generalized abel s integral equations could be considered as a more general form of volterra integral equations collocation method in sense of atkinson s approach atkinson 2016 is applied to get the approximate solution of generalized abel s integral equations the convergence analysis of the presented method is also established the different polynomials such as 1 jacobi polynomials 2 legendre polynomials 3 chebyshev polynomials and 4 gegenbauer polynomials are considered to get the numerical solution of the generalized abel s integral equations illustrative examples with different solutions are considered to show the validity and applicability of the proposed method numerical results show that the proposed method works well and achieve good accuracy even for less number of polynomials further the performance of the proposed method is compared under the effect of different polynomials we present polynomials based collocation methods for generalized abel s integral equations the convergence of the presented method is also established illustrative examples show the validity and applicability of the proposed method

a feature group weighting method for subspace clustering of high dimensional data this paper proposes a new method to weight subspaces in feature groups and individual features for clustering high dimensional data in this method the features of high dimensional data are divided into feature groups based on their natural characteristics two types of weights are introduced to the clustering process to simultaneously identify the importance of feature groups and individual features in each cluster a new optimization model is given to define the optimization process and a new clustering algorithm fg k means is proposed to optimize the optimization model the new algorithm is an extension to k means by adding two additional steps to automatically calculate the two types of subspace weights a new data generation method is presented to generate high dimensional data with clusters in subspaces of both feature groups and individual features experimental results on synthetic and real life data have shown that the fg k means algorithm significantly outperformed four k means type algorithms i e k means w k means lac and ewkm in almost all experiments the new algorithm is robust to noise and missing values which commonly exist in high dimensional data

subspace clustering of categorical and numerical data with an unknown number of clusters in clustering analysis data attributes may have different contributions to the detection of various clusters to solve this problem the subspace clustering technique has been developed which aims at grouping the data objects into clusters based on the subsets of attributes rather than the entire data space however the most existing subspace clustering methods are only applicable to either numerical or categorical data but not both this paper therefore studies the soft subspace clustering of data with both of the numerical and categorical attributes also simply called mixed data for short specifically an attribute weighted clustering model based on the definition of object cluster similarity is presented accordingly a unified weighting scheme for the numerical and categorical attributes is proposed which quantifies the attribute to cluster contribution by taking into account both of intercluster difference and intracluster similarity moreover a rival penalized competitive learning mechanism is further introduced into the proposed soft subspace clustering algorithm so that the subspace cluster structure as well as the most appropriate number of clusters can be learned simultaneously in a single learning paradigm in addition an initialization oriented method is also presented which can effectively improve the stability and accuracy of k means type clustering methods on numerical categorical and mixed data the experimental results on different benchmark data sets show the efficacy of the proposed approach

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

an extension of the spectral tau method for numerical solution of multi order fractional differential equations with convergence analysis the main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations fdes based on a spectral tau method an extension of the operational approach of the tau method with the orthogonal polynomial bases is proposed to convert fdes to its matrix vector multiplication representation the fractional derivatives are described in the caputo sense the spectral rate of convergence for the proposed method is established in the l 2 norm we tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

extensions of kmeans type algorithms a new clustering framework by integrating intracluster compactness and intercluster separation kmeans type clustering aims at partitioning a data set into clusters such that the objects in a cluster are compact and the objects in different clusters are well separated however most kmeans type clustering algorithms rely on only intracluster compactness while overlooking intercluster separation in this paper a series of new clustering algorithms by extending the existing kmeans type algorithms is proposed by integrating both intracluster compactness and intercluster separation first a set of new objective functions for clustering is developed based on these objective functions the corresponding updating rules for the algorithms are then derived analytically the properties and performances of these algorithms are investigated on several synthetic and real life data sets experimental studies demonstrate that our proposed algorithms outperform the state of the art kmeans type clustering algorithms with respect to four metrics accuracy randindex fscore and normal mutual information

analysis of ciarlet raviart mixed finite element methods for solving damped boussinesq equation abstract in this paper we consider the numerical solution of damped boussinesq equation using ciarlet raviart mixed finite element method an implicit finite difference scheme is used for the time discretization a priori error estimates are analyzed and stability analysis of the method is shown we obtain an optimal error estimate in l 2 norm with quadratic or higher order element for both semi and fully discrete finite element approximations finally numerical examples are given to verify the theoretical results

a fractional differential equation for a mems viscometer used in the oil industry a mathematical model is developed for a micro electro mechanical system mems instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration it is shown that in one mode of operation the displacement of the device satisfies a fractional differential equation fde the theory of fdes is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme it is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised it is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring where the resulting fde is nonlinear

application of taylor series in obtaining the orthogonal operational matrix in this research first we explicitly obtain the relation between the coefficients of the taylor series and jacobi polynomial expansions then we present a new method for computing classical operational matrices derivative integral and product for general jacobi orthogonal functions polynomial and rational this method can be used for many classes of orthogonal functions

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

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

soft subspace clustering of categorical data with probabilistic distance categorical data clustering is an important subject in pattern recognition currently subspace clustering of categorical data remains an open problem due to the difficulties in estimating attribute interestingness according to the statistics of categories in clusters in this paper a new algorithm is proposed for clustering categorical data with a novel soft feature selection scheme by which each categorical attribute is automatically assigned a weight that correlates with the smoothed dispersion of the categories in a cluster in the proposed algorithm dissimilarity between categorical data objects is measured using a probabilistic distance function based on kernel density estimation for categorical attributes we also make use of the probabilistic distances to define a cluster validity index for estimating the number of categorical clusters the suitability of the proposal is demonstrated in an empirical study done with some widely used real world data sets and synthetic data sets and the results show its outstanding performance highlightswe define the cluster scatter on object to cluster distances for categorical data we propose a probabilistic distance function using a kernel density estimation method categorical attributes are weighted based on the smoothed dispersion of categories two weighting schemes are offered depending on the attribute types significantly improve clustering performance compared to mode based algorithms

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

a tabu search based algorithm for the fuzzy clustering problem the fuzzy clustering problem fcp is a mathematical program which is difficult to solve since it is nonconvex which implies possession of many local minima the fuzzy c means heuristic is the widely known approach to this problem but it is guaranteed only to yield local minima in this paper we propose a new approach to this problem which is based on tabu search technique and aims at finding a global solution of fcp we compare the performance of the algorithm with the fuzzy c means algorithm

comparative study of three numerical schemes for fractional integro differential equations this paper presents a comparative study of three numerical schemes such as linear quadratic and quadratic linear scheme for the fractional integro differential equations defined in terms of the caputo fractional derivatives the error estimates of the respective approximations are also established numerical tests of the discussed schemes show that all schemes work well and when the number of terms approximating the solution is increased the desired solution is achieved the accuracy of the numerical schemes with respect to the step size h is analysed and illustrated through various tables finally comparative performances of the schemes are discussed comparative study of three schemes such as linear quadratic and quadratic linear schemes is presented for the fractional integro differential equations the convergences of the schemes are also established the performances of the schemes are compared through the test examples

collocation method with convergence for generalized fractional integro differential equations abstract in this paper we study a numerical approach for some class of generalized fractional integro differential equations gfides defined in terms of the b operators presented recently we develop collocation method for linear and nonlinear forms of gfides the numerical approach uses the idea of collocation methods for solving integral equations legendre polynomials are used to approximate the solution in finite dimensional space with convergence analysis the obtained approximate solution recovers the solution of the fractional integro differential equation fide defined using caputo derivatives in a special case fides containing convolution type kernels appear in diverse area of science and engineering applications therefore some test examples varying the kernel in the b operator are considered to perform the numerical investigations the numerical results validate the presented scheme and provide good accuracy using few legendre basis functions

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

error estimate of finite element finite difference technique for solution of two dimensional weakly singular integro partial differential equation with space and time fractional derivatives abstract in the current investigation an error estimate has been proposed to solve the two dimensional weakly singular integro partial differential equation with space and time fractional derivatives based on the finite element finite difference scheme the time and space derivatives are based on the riemann liouville and riesz fractional derivatives respectively at first the temporal variable has been discretized by a second order difference scheme and then the space variable has been approximated by the finite element method fem the analytical study shows that the presented scheme is unconditionally stable and convergent finally some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique

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

free vibration of regular polygonal plates with simply supported edges the free vibration of regular polygonal plates with simply supported edges is studied by the dynamical analogy with membranes a regular polygonal membrane is formed on a rectangular membrane by fixing several segments with the reaction forces acting on all edges of an actual polygonal membrane regarded as unknown harmonic loads the stationary response of the membrane to these loads is expressed by the eigenfunctions of the extended rectangular membrane without internal supports the force distributions along the edges are expanded into fourier sine series with unknown coefficients and the homogeneous equations for the coefficients are derived by restraint conditions on the edges the natural frequencies and the mode shapes of the actual membrane are determined by calculating the eigenvalues and eigenvectors of the equations the method is applied to an equilateral triangular through a regular decagonal membrane the natural frequencies and mode shapes are calculated numerically and the effect of the shape of membrane is discussed the numerical values obtained for polygonal membranes are immediately converted into those of simply supported polygonal plates

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

minkowski metric feature weighting and anomalous cluster initializing in k means clustering this paper represents another step in overcoming a drawback of k means its lack of defense against noisy features using feature weights in the criterion the weighted k means method by huang et al 2008 2004 2005 5 7 is extended to the corresponding minkowski metric for measuring distances under minkowski metric the feature weights become intuitively appealing feature rescaling factors in a conventional k means criterion to see how this can be used in addressing another issue of k means the initial setting a method to initialize k means with anomalous clusters is adapted the minkowski metric based method is experimentally validated on datasets from the uci machine learning repository and generated sets of gaussian clusters both as they are and with additional uniform random noise features and appears to be competitive in comparison with other k means based feature weighting algorithms

approximate solution of nonlinear fractional integro differential equations using fractional alternative legendre functions abstract in this paper a new set of functions called fractional alternative legendre is defined for solving nonlinear fractional integro differential equations the concept of the fractional derivative in this problem is in the caputo sense to solve the problem first we obtain the operational matrix of the riemann liouville fractional integration of these functions for the first time then this matrix and collocation method are utilized to reduce the solution of the nonlinear fractional integro differential equations to a system of algebraic equations also the convergence analysis of the proposed method is investigated finally some examples are included to demonstrate the validity and applicability of the approach

fuzzy partition based soft subspace clustering and its applications in high dimensional data as one of the most popular clustering techniques for high dimensional data soft subspace clustering ssc algorithms have been receiving a great deal of attention in recent years unfortunately most existing works do not cluster high dimensional sparse data and noisy data in an effective manner in this study a novel soft subspace clustering algorithm called pi ssc is proposed by introducing a partition index pi into the objective function a novel soft subspace clustering algorithm that combines the concepts of hard and fuzzy clustering is proposed furthermore the robust property of pi ssc is analyzed from the viewpoint of e insensitive distance a convergence theorem for pi ssc is also established by applying zangwill s convergence theorem the results of the experiment demonstrate the effectiveness of the proposed algorithm in high dimensional sparse text data and noisy texture data

enhanced soft subspace clustering integrating within cluster and between cluster information while within cluster information is commonly utilized in most soft subspace clustering approaches in order to develop the algorithms other important information such as between cluster information is seldom considered for soft subspace clustering in this study a novel clustering technique called enhanced soft subspace clustering essc is proposed by employing both within cluster and between class information first a new optimization objective function is developed by integrating the within class compactness and the between cluster separation in the subspace based on this objective function the corresponding update rules for clustering are then derived followed by the development of the novel essc algorithm the properties of this algorithm are investigated and the performance is evaluated experimentally using real and synthetic datasets including synthetic high dimensional datasets uci benchmarking datasets high dimensional cancer gene expression datasets and texture image datasets the experimental studies demonstrate that the accuracy of the proposed essc algorithm outperforms most existing state of the art soft subspace clustering algorithms

exact discretization of fractional laplacian abstract an exact discretization of fractional order laplacian for n dimensional space is suggested particular solutions of fractional order partial difference equations with the proposed discrete laplace operators are suggested

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

novel soft subspace clustering with multi objective evolutionary approach for high dimensional data many conventional soft subspace clustering techniques merge several criteria into a single objective to improve performance however the weighting parameters become important but difficult to set in this paper a novel soft subspace clustering with a multi objective evolutionary approach moeassc is proposed to this problem this clustering method considers two types of criteria as multiple objectives and optimizes them simultaneously by using a modified multi objective evolutionary algorithm with new encoding and operators an indicator called projection similarity validity index psvindex is designed to select the best solution and cluster number experiments on many datasets demonstrate the usefulness of moeassc and psvindex and show that our algorithm is insensitive to its parameters and is scalable to large datasets

distance metric learning for soft subspace clustering in composite kernel space soft subspace clustering algorithms have been successfully used for high dimensional data in recent years however the existing algorithms often utilize only one distance function to evaluate the distance between data items on each feature which cannot deal with datasets with complex inner structures in this paper a composite kernel space cks is constructed based on a set of basis kernels and a novel framework of soft subspace clustering is proposed by integrating distance metric learning in the cks two soft subspace clustering algorithms i e entropy weighting fuzzy clustering in cks for kernel space cks ewfc k and feature space cks ewfc f are thus developed in both algorithms the prototype in the feature space is mapped into the cks by multiple simultaneous mappings one mapping for each cluster which is distinct from existing kernel based clustering algorithms by evaluating the distance on each feature in the cks both cks ewfc k and cks ewfc f learn the distance function adaptively during the clustering process experimental results have demonstrated that the proposed algorithms in general outperform classical clustering algorithms and are immune to ineffective kernels and irrelevant features in soft subspace the composite kernel space is constructed based on a set of basis kernels the general form of soft subspace clustering in cks is presented cks ewfc k and cks ewfc f are proposed under the framework of cks ssc the properties of cks ewfc k and cks ewfc f are investigated both cks ewfc k and cks ewfc f are immune to ineffective kernels

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

fundamental frequency of clamped plates with circularly periodic boundaries abstract a boundary perturbation method is developed to determine the fundamental frequency of vibrating plates the method is then applied to wavy star shape and polygonal plates with clamped boundary conditions approximate analytical solutions of the fundamental frequency are obtained with an accuracy of o e4 where e is the deviation from the unit circle

a novel attribute weighting algorithm for clustering high dimensional categorical data due to data sparseness and attribute redundancy in high dimensional data clusters of objects often exist in subspaces rather than in the entire space to effectively address this issue this paper presents a new optimization algorithm for clustering high dimensional categorical data which is an extension of the k modes clustering algorithm in the proposed algorithm a novel weighting technique for categorical data is developed to calculate two weights for each attribute or dimension in each cluster and use the weight values to identify the subsets of important attributes that categorize different clusters the convergence of the algorithm under an optimization framework is proved the performance and scalability of the algorithm is evaluated experimentally on both synthetic and real data sets the experimental studies show that the proposed algorithm is effective in clustering categorical data sets and also scalable to large data sets owning to its linear time complexity with respect to the number of data objects attributes or clusters

a fast numerical algorithm based on the second kind chebyshev polynomials for fractional integro differential equations with weakly singular kernels a spectral method based on operational matrices of the second kind chebyshev polynomials skcps is employed for solving fractional integro differential equations with weakly singular kernels firstly properties of shifted skcps operational matrix of fractional integration and product operational matrix are introduced and then utilized to reduce the given equation to the solution of a system of linear algebraic equations this new approach provides a significant computational advantage by converting the given original problem to an equivalent linear volterra integral equation of the second kind with the same initial conditions approximate solution is achieved by expanding the functions in terms of skcps and employing operational matrices unknown coefficients are determined by solving final system of linear equations an estimation of the error is given finally illustrative examples are included to demonstrate the high precision fast computation and good performance of the new scheme

recovering the number of clusters in data sets with noise features using feature rescaling factors in this paper we introduce three methods for re scaling data sets aiming at improving the likelihood of clustering validity indexes to return the true number of spherical gaussian clusters with additional noise features our method obtains feature re scaling factors taking into account the structure of a given data set and the intuitive idea that different features may have different degrees of relevance at different clusters we experiment with the silhouette using squared euclidean manhattan and the pth power of the minkowski distance dunn s calinski harabasz and hartigan indexes on data sets with spherical gaussian clusters with and without noise features we conclude that our methods indeed increase the chances of estimating the true number of clusters in a data set

the minkowski central partition as a pointer to a suitable distance exponent and consensus partitioning we generate optimal minkowski partitions at various values of the exponent p we define the minkowski profile based on the average similarity between partitions minkowski profile is highly correlated with ari vectors related to the ground truth we define the central minkowski partition which can serve as a consensus partition the silhouette width should be used for selecting the optimal minkowski exponent the minkowski weighted k means mwk means is a recently developed clustering algorithm capable of computing feature weights the cluster specific weights in mwk means follow the intuitive idea that a feature with low variance should have a greater weight than a feature with high variance the final clustering found by this algorithm depends on the selection of the minkowski distance exponent this paper explores the possibility of using the central minkowski partition in the ensemble of all minkowski partitions for selecting an optimal value of the minkowski exponent the central minkowski partition appears to be also a good consensus partition furthermore we discovered some striking correlation results between the minkowski profile defined as a mapping of the minkowski exponent values into the average similarity values of the optimal minkowski partitions and the adjusted rand index vectors resulting from the comparison of the obtained partitions to the ground truth our findings were confirmed by a series of computational experiments involving synthetic gaussian clusters and real world data

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

a fully spectral collocation approximation for multi dimensional fractional schrodinger equations a shifted legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one and two dimensional time fractional schrodinger equations tfses subject to initial boundary and non local conditions the first step depends mainly on shifted legendre gauss lobatto collocation sl gl c method for spatial discretization an expansion in a series of shifted legendre polynomials for the approximate solution and its spatial derivatives occurring in the tfse is investigated in addition the legendre gauss lobatto quadrature rule is established to treat the nonlocal conservation conditions thereby the expansion coefficients are then determined by reducing the tfse with its nonlocal conditions to a system of fractional differential equations sfdes for these coefficients the second step is to propose a shifted legendre gauss radau collocation sl gr c scheme for temporal discretization to reduce such system into a system of algebraic equations which is far easier to be solved the proposed collocation scheme both in temporal and spatial discretizations is successfully extended to solve the two dimensional tfse numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms by selecting relatively limited legendre gauss lobatto and gauss radau collocation nodes we are able to get very accurate approximations demonstrating the utility and high accuracy of the new approach over other numerical methods

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

the k means type algorithms versus imbalanced data distributions k means is a partitional clustering technique that is well known and widely used for its low computational cost the representative algorithms include the hard k means and the fuzzy k means however the performance of these algorithms tends to be affected by skewed data distributions i e imbalanced data they often produce clusters of relatively uniform sizes even if input data have varied cluster sizes which is called the uniform effect in this paper we analyze the causes of this effect and illustrate that it probably occurs more in the fuzzy k means clustering process than the hard k means clustering process as the fuzzy index m increases the uniform effect becomes evident to prevent the effect of the uniform effect we propose a multicenter clustering algorithm in which multicenters are used to represent each cluster instead of one single center the proposed algorithm consists of the three subalgorithms the fast global fuzzy k means best m plot and grouping multicenter algorithms they will be respectively used to address the three important problems 1 how are the reliable cluster centers from a dataset obtained 2 how are the number of clusters which these obtained cluster centers represent determined 3 how is it judged as to which cluster centers represent the same clusters the experimental studies on both synthetic and real datasets illustrate the effectiveness of the proposed clustering algorithm in clustering balanced and imbalanced data

kernel based hard clustering methods with kernelization of the metric and automatic weighting of the variables this paper presents kernel based hard clustering methods with kernelization of the metric and automatic weighting of the variables the proposed methodology is supported by the fact that a kernel function can be written as a sum of kernels evaluated separately on each variable thus in the proposed algorithms dissimilarity measures are obtained as sums of euclidean distances between patterns and centroids computed individually for each variable by means of kernels the main advantage of this approach over the conventional approach is that it allows the use of kernelized adaptive distances which are suitable to learn the weights of the variables dynamically improving the performance of the algorithms moreover various partition and cluster interpretation tools are introduced experiments with a number of benchmark datasets corroborate the usefulness of the proposed algorithms and the merit of the partition and cluster interpretation tools highlightspresents kernel based clustering methods with automatic weighting of the variables kernelized local and global adaptive distances are introduced the proposed algorithms are suitable to learn the weights of the variables partition and cluster interpretation tools are given experiments with several benchmark data sets corroborate the proposed methods

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

a boundary perturbation method for circularly periodic plates with a core fundamental frequencies of vibrating circularly periodic plates with a circular core have been determined analytically a boundary perturbation method is developed to extract the fundamental eigenvalue of the governing biharmonic boundary value problem the method is then applied to wavy and polygonal plates with clamped and simply supported outer boundary conditions clamped simply supported and free circular cores are considered approximate analytical formulations of the fundamental frequency for such plates with core are obtained

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

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

flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables a set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the rayleigh ritz method the set of orthogonal polynomials is constructed by employing the gram schmidt orthogonalization procedure the first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted results are compared with those obtained previously by other methods