alternative c means clustering algorithms abstract in this paper we propose a new metric to replace the euclidean norm in c means clustering procedures on the basis of the robust statistic and the influence function we claim that the proposed new metric is more robust than the euclidean norm we then create two new clustering methods called the alternative hard c means ahcm and alternative fuzzy c means afcm clustering algorithms these alternative types of c means clustering have more robustness than c means clustering numerical results show that ahcm has better performance than hcm and afcm is better than fcm we recommend afcm for use in cluster analysis recently this afcm algorithm has successfully been used in segmenting the magnetic resonance image of ophthalmology to differentiate the abnormal tissues from the normal tissues

consistency of spectral clustering consistency is a key property of all statistical procedures analyzing randomly sampled data surprisingly despite decades of work little is known about consistency of most clustering algorithms in this paper we investigate consistency of the popular family of spectral clustering algorithms which clusters the data with the help of eigenvectors of graph laplacian matrices we develop new methods to establish that for increasing sample size those eigenvectors converge to the eigenvectors of certain limit operators as a result we can prove that one of the two major classes of spectral clustering normalized clustering converges under very general conditions while the other unnormalized clustering is only consistent under strong additional assumptions which are not always satisfied in real data we conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

estimation of a fuzzy regression model using fuzzy distances regression analysis is a powerful statistical tool that has many applications in different areas the problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input output data crisp or fuzzy however we realize that in general most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances minimizing a real error function and the interpretation of fuzzy data as possibility distributions the main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers the new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error the estimation process in general can be considered easier to apply in practice and it is not limited to triangular fuzzy numbers numerical examples are provided to illustrate the usefulness and applicability of these results and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory

a new approach to fuzzy regression models with application to business cycle analysis recently fuzzy regression analysis has been largely applied in the modeling of economic or financial data however those data often exhibit certain kinds of linguistic terms for instance very good a little reclining or stable in the business cycle or the growth rate of gdp etc the goal of this paper is to construct a fuzzy regression model by fuzzy parameters estimation using the fuzzy samples it deals with imprecise measurement of observed variables fuzzy least square estimation and nonparametric methods this is different from the assumptions as well as the estimation techniques of the classical analysis empirical results demonstrate that our new approach is efficient and more realistic than the traditional regression analysis

on a new methodology for ranking fuzzy numbers and its application to real economic data abstract ranking fuzzy numbers plays an important role in the decision process several strategies can be found in the abundant literature existing on this topic which can be basically classified into two main groups approaches based on the so called ranking indices and based on fuzzy binary relations ranking indices are functions from fuzzy numbers to real values these defuzzification methods are critiqued by its implicit information loss which leads to results that might not be consistent with human intuition in certain cases in a fuzzy context approaches based on binary relations can be considered more suitable solutions following this strategy this paper introduces a new technique for ranking fuzzy numbers which is applicable to the whole set of fuzzy numbers it is proved that many desired properties in particular we give several new properties are satisfied and numerical examples show reasonable results the developed technique is very easy to compute and interpret in practice and it overcomes certain shortcomings that appear when applying other more complex algorithms in the case of triangular or trapezoidal fuzzy numbers the procedure is particularly simple and intuitive finally a real life example involving data of chinese economy is also carried out

trapezoidal approximations of fuzzy numbers the problem of the trapezoidal approximation of fuzzy numbers is discussed a set of criteria for approximation operators is formulated these constraints can be used for direct operator derivation a new nearest trapezoidal approximation operator preserving expected interval is suggested

symmetric submodular system contractions and gomory hu tree abstract let s v f be a symmetric submodular system for two distinct elements s and l of v let s l denote the set of all subsets of v which separate s from l by using any gomory hu tree of s we can obtain an element of s l which has minimum value among all the elements of s l this tree can be constructed iteratively by solving v 1 minimum s l separator problem pendant pairs of a symmetric submodular system play a key role in finding a minimizer of this system in this paper we obtain a gomory hu tree of a contraction of s with respect to some subsets of v only by using contraction in a gomory hu tree of s furthermore we obtain some pendant pairs of s and of its contractions by using a gomory hu tree of s

on the complexity of computing the k restricted edge connectivity of a graph the k restricted edge connectivity of a graph g denoted by k g is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices this graph invariant which can be seen as a generalization of a minimum edge cut has been extensively studied from a combinatorial point of view however very little is known about the complexity of computing k g very recently in the parameterized complexity community the notion of good edge separation of a graph has been defined which happens to be essentially the same as the k restricted edge connectivity motivated by the relevance of this invariant from both combinatorial and algorithmic points of view in this article we initiate a systematic study of its computational complexity with special emphasis on its parameterized complexity for several choices of the parameters we provide a number of np hardness and w 1 hardness results as well as fpt algorithms

a robust varying coefficient approach to fuzzy multiple regression model abstract the varying coefficient models are powerful tools for exploring the dynamic pattern between a response and a group of predictors in multiple regression models in addition robust regression is another solid approach in the regression analyses for cases whose data are contaminated with outliers or influential observations this paper proposed a novel varying coefficient model with exact predictors and fuzzy responses which can be used in cases where outliers occur in the data set for this purpose a locally weighted approximation idea and a popular m estimator were combined to estimate unknown fuzzy nonparametric varying coefficients some common goodness of fit criteria including an outlier detection criterion were also applied to examine the performance of the proposed method the effectiveness of the presented method was then illustrated through two numerical examples including a simulation study it was also compared with several common fuzzy multiple regression models the numerical results clearly indicate that the proposed method is not sensitive to the outliers moreover compared to the available fuzzy multiple regressions with constant coefficients the proposed fuzzy varying coefficient model managed to provide more accurate results

linear regression analysis for fuzzy input and output data using the extension principle abstract the method for obtaining the fuzzy least squares estimators with the help of the extension principle in fuzzy sets theory is proposed the membership functions of fuzzy least squares estimators will be constructed according to the usual least squares estimators in order to obtain the membership value of any given value taken from the fuzzy least squares estimator optimization problems have to be solved we also provide the methodology for evaluating the predicted fuzzy output from the given fuzzy input data

a new approach for ranking fuzzy numbers by distance method abstract many ranking methods have been proposed so far however there is yet no method that can always give a satisfactory solution to every situation some are counterintuitive not discriminating some use only the local information of fuzzy values some produce different rankings for the same situation for overcoming the above problems we propose a new method for ranking fuzzy numbers by distance method our method is based on calculating the centroid point where the distance means from original point to the centroid point x 0 y 0 and the x 0 index is the same as murakami et al s x 0 however the y 0 index is integrated from the inverse functions of an lr type fuzzy number thus we use ranking function r a x 2 y 2 distance index as the order quantities in a vague environment our method can rank more than two fuzzy numbers simultaneously and the fuzzy numbers need not be normal furthermore we also propose the coefficient of variation cv index to improve lee and li s method comput math appl 15 1988 887 896 lee and li rank fuzzy numbers based on two different criteria namely the fuzzy mean and the fuzzy spread of the fuzzy numbers and they pointed out that human intuition would favor a fuzzy number with the following characteristics higher mean value and at the same time lower spread however when higher mean value and at the same time higher spread or lower mean value and at the same time lower spread exists it is not easy to compare its orderings clearly our cv index is defined as cv standard error mean which can overcome lee and li s problem efficiently in this way our proposed method can also be easily calculated by the mathematica package to solve problems of ranking fuzzy numbers at last we present three numerical examples to illustrate our proposed method and compare with other ranking methods

hierarchical inference of unicast network topologies based on end to end measurements in this paper we address the problem of topology discovery in unicast logical tree networks using end to end measurements without any cooperation from the internal routers topology estimation can be formulated as hierarchical clustering of the leaf nodes based on pairwise correlations as similarity metrics unlike previous work that first assumes the network topology is a binary tree and then tries to generalize to a nonbinary tree we provide a framework that directly deals with general logical tree topologies a hierarchical algorithm to estimate the topology is developed in a recursive manner by finding the best partitions of the leaf nodes level by level our simulations show that the algorithm is more robust than binary tree based methods

i o efficient algorithms for the minimum cut problem on unweighted undirected graphs the problem of finding the minimum cut of an undirected unweighted graph is studied on the external memory model first a lower bound of e v sort v on the number of i os is shown for the problem where v is the number of vertices and e is the number of edges then the following are presented for m b 2 1 a minimum cut algorithm that uses o c log e rm msf v e frac v b rm sort v i os here msf v e is the number of i os needed to compute a minimum spanning tree of the graph and c is the value of the minimum cut the algorithm performs better on dense graphs than the algorithm of 7 which requires o e c 2 v log v c i os when executed on the external memory model for a fat graph for 0 the maximum tree packing of the graph is at least 1 c 2 our algorithm computes a minimum cut in o c loge msf v e sort e i os 2 a randomized algorithm that computes minimum cut with high probability in o c log e cdot rm msf v e rm sort e log 2 v frac v b rm sort v log v i os 3 a 2 e minimum cut algorithm that requires o e v msf v e i os and performs better on sparse graphs than our exact minimum cut algorithm

weight reduction problems with certain bottleneck objectives abstract this paper is concerned with bottleneck weight reduction problems wrps stated as follows we are given a finite set e a class f of nonempty subsets of e a weight w e r and a cost c e r for each e e c e stands for the cost of reducing weight w e by one unit for each subset f f the bottleneck weight of f is w f min e f w e the weight of the family f is the maximum of w f for all f in f the problem is to determine new weights x e w e such that the weight of f is minimized under the constraint that the overall reduction cost does not exceed a given budget b similarly to capacity expansion problems wrps include np hard problems a wrp can be formulated as a parametric optimization problem over all transversal sets t of the class f this leads to strongly polynomial solution procedures for special systems f in particular we outline a polynomial algorithm in the case when f is the class of all spanning trees in an undirected graph

fuzzy regression models using the least squares method based on the concept of distance fuzzy regression models are developed to construct the relationship between explanatory variables and responses in a fuzzy environment in order to increase the explanatory performance of the model the least squares method is applied to determine the numeric coefficients based on the concept of distance unlike most existing approaches the numeric coefficients in the proposed model can have negative values the proposed model minimizes total estimation error in terms of the sum of the average squared distance between the observed and estimated responses based on a few alpha cuts the proposed approach is not limited to triangular fuzzy numbers it can be used to carry out a large number of fuzzy observations efficiently because the model is based on traditional statistical methods comparisons with existing methods show that based on the total estimation error using the mean squared error and kim and bishu s criterion the explanatory performance of the proposed model is satisfactory

a partial robust ridge based regression model with fuzzy predictors responses abstract this paper extends the conventional semi parametric partial linear regression model with fuzzy predictors and fuzzy response in cases where 1 outliers occur in data set 2 multicollinearity exists in predictors for this purpose the classical kernel fitting method is first extended for estimating unknown fuzzy smooth function then inspired by the conventional local linear smoothing method a kernel based weighted criterion is suggested for determining the exact coefficients on the basis of ridge methodology some common goodness of fit criteria are also used to examine the performance of the proposed method the effectiveness of the proposed method is then illustrated through two numerical examples including a simulation study the proposed method is also compared with some common fuzzy multiple regression models with fuzzy predictors and fuzzy responses the numerical results clearly indicate that 1 the proposed method is not sensitive to the outliers 2 the proposed method provides sufficiently accurate results in cases where multicollinearity happens among predictors and 3 comparing to existing fuzzy multiple regressions the proposed model provides more efficient predictions

least squares estimates in fuzzy regression analysis abstract regression is a very powerful methodology for forecasting which is considered as an essential component of successful or applications in this paper an idea stemmed from the classical least squares is proposed to handle fuzzy observations in regression analysis based on the extension principle the membership function of the sum of squared errors is constructed the fuzzy sum of squared errors is a function of the regression coefficients to be determined which can be minimized via a nonlinear program formulated under the structure of the chen klein method for ranking fuzzy numbers to illustrate how the proposed method is applied three cases one crisp input fuzzy output one fuzzy input fuzzy output and one non triangular fuzzy observations are exemplified the results show that the least squares method of this paper is able to determine the regression coefficients with better explanatory power most important it works for all types of fuzzy observations not restricted to the triangular one

graph theory applied to noise and vibration control in statistical energy analysis models a fundamental aspect of noise and vibration control in statistical energy analysis sea models consists in first identifying and then reducing the energy flow paths between subsystems in this work it is proposed to make use of some results from graph theory to address both issues on the one hand linear and path algebras applied to adjacency matrices of sea graphs are used to determine the existence of any order paths between subsystems counting and labeling them finding extremal paths or determining the power flow contributions from groups of paths on the other hand a strategy is presented that makes use of graph cut algorithms to reduce the energy flow from a source subsystem to a receiver one modifying as few internal and coupling loss factors as possible

a mathematical programming approach to fuzzy linear regression analysis most of previous studies on fuzzy regression analysis have a common characteristic of increasing spreads for the estimated fuzzy responses as the independent variable increases its magnitude which is not suitable for general cases in this paper fuzzy linear regression models with fuzzy crisp output fuzzy crisp input are considered and an estimated method along with a mathematical programming based approach is proposed the advantages of the proposed approach are simplicity in programming and computation and minimum difference of total spread between observed and estimated values

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

fuzzy linear regression models with least square errors to estimate the parameters of fuzzy linear regression models with fuzzy output and crisp inputs we develop a mathematical programming model in this paper the method is constructed on the basis of minimizing the square of the total difference between observed and estimated spread values or in other words minimizing the least square errors the advantage of the proposed approach is its simplicity in programming and computation as well as its performance to compare the performance of the proposed approach with the other methods two examples are presented

some applications of the study of the image of a fuzzy number in this paper we present a characterization of the image of a fuzzy number in terms of the extremes of its level sets then we deduce some consequences we show some relationships between the image of two fuzzy numbers and the images of their usual operations and it is proved that the continuity of some fuzzy numbers is preserved under the usual operations as applications we introduce a fuzzy regression model involving a parametric family of fuzzy semidistances between fuzzy numbers with finite image and an ordering process of fuzzy numbers with compact support that satisfies specificity type properties

possibilistic linear regression analysis for fuzzy data abstract fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined recently it has become important to deal with fuzzy data in connection with expert knowledge three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data since our formulations can be reduced to linear programming problems the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters this approach can be regarded as a fuzzy interval analysis in a fuzzy environment

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

aggregation of subjective evaluations based on discrete fuzzy numbers in this paper aggregation functions defined on the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers are introduced and the particular cases of uninorms and nullnorms are studied in detail these aggregation functions are constructed from discrete aggregation functions defined on a finite chain and they are applied to the aggregation of subjective evaluations

constrained clustering by constraint programming cluster analysis is an important task in data mining with hundreds of different approaches in the literature since the last decade the cluster analysis has been extended to constrained clustering also called semi supervised clustering so as to integrate previous knowledge on data to clustering algorithms in this dissertation we explore constraint programming cp for solving the task of constrained clustering the main principles in cp are 1 users specify declaratively the problem in a constraint satisfaction problem 2 solvers search for solutions by constraint propagation and search relying on cp has two main advantages the declarativity which enables to easily add new constraints and the ability to find an optimal solution satisfying all the constraints when there exists one we propose two models based on cp to address constrained clustering tasks the models are flexible and general and supports instance level constraints and different cluster level constraints it also allows the users to choose among different optimization criteria in order to improve the efficiency different aspects have been studied in the dissertation experiments on various classical datasets show that our models are competitive with other exact approaches we show that our models can easily be embedded in a more general process and we illustrate this on the problem of finding the pareto front of a bi criterion optimization process

time series represented by means of fuzzy piecewise lineal segments the aim of this paper is to represent time series in a fuzzy way by means of a piecewise linear segment method that represents the time series in an efficient way but in addition this representation collects the uncertainty generated in the process of generation of the segments this representation can be used for instance to compare time series or to query information from these that is why in this approach we also propose a method to compare different parts of the output obtaining as result a dissimilarity value representing how close their outputs are several tests have been done in order to know if input time series are represented correctly with this method and to know if the comparison process obtains the expected results the input data selected for the tests is the real data obtained from the electric consumption of three buildings from the campus in the city of toledo of our university

diffuse interface models on graphs for classification of high dimensional data this paper is a republication of an mms paper a l bertozzi and a flenner multiscale model simul 10 2012 pp 1090 1118 describing a new class of algorithms for classification of high dimensional data these methods combine ideas from spectral methods on graphs with nonlinear edge region detection methods traditionally used in the pde based imaging community the algorithms use the ginzburg landau functional extended to a graphical framework which has classical pde connections to total variation minimization convex splitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graph theoretic problems we review the diverse computational examples presented in the original paper involving both basic clustering and semisupervised learning for different applications case studies include feature identification in images segmentation in social networks and segmentation of shapes in high dimensional datasets since the pape

random matrix elements and eigenfunctions in chaotic systems the expected root mean square value of a matrix element a ensuremath alpha ensuremath beta in a classically chaotic system where a is a smooth ensuremath elzxh independent function of the coordinates and momenta and ensuremath alpha and ensuremath beta label different energy eigenstates has been evaluated in the literature in two different ways by treating the energy eigenfunctions as gaussian random variables and averaging a ensuremath alpha ensuremath beta 2 over them and by relating a ensuremath alpha ensuremath beta 2 to the classical time correlation function of a we show that these two methods give the same answer only if berry s formula for the spatial correlations in the energy eigenfunctions which is based on a microcanonical density in phase space is modified at large separations in a manner that we previously proposed

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs we consider multiobjective and parametric versions of the global minimum cut problem in undirected graphs and bounded rank hypergraphs with multiple edge cost functions for a fixed number of edge cost functions we show that the total number of supported non dominated snd cuts is bounded by a polynomial in the numbers of nodes and edges i e is strongly polynomial this bound also applies to the combinatorial facet complexity of the problem i e the maximum number of facets linear pieces of the parametric curve for the parametrized linear combination objective over the set of all parameter vectors such that the parametrized edge costs are nonnegative and the parametrized cut costs are positive we sharpen this bound in the case of two objectives the bicriteria problem for which we also derive a strongly polynomial upper bound on the total number of non dominated pareto optimal cuts in particular the bicriteria global minimum cut problem in an n node graph admits o n 3 log n o n3logn snd cuts and o n 5 log n o n5logn pareto optimal cuts these results significantly improve on earlier graph cut results by mulmuley siam j comput 28 4 1460 1509 1999 and armon and zwick algorithmica 46 1 15 26 2006 they also imply that the parametric curve and all snd cuts and for the bicriteria problems all pareto optimal cuts can be computed in strongly polynomial time when the number of objectives is fixed

the median of a random fuzzy number the 1 norm distance approach in quantifying the central tendency of the distribution of a random fuzzy number or fuzzy random variable in puri and ralescu s sense the most usual measure is the aumann type mean which extends the mean of a real valued random variable and preserves its main properties and behavior although such a behavior has very valuable and convenient implications extreme values or changes of data entail too much influence on the aumann type mean of a random fuzzy number this strong influence motivates the search for a more robust central tendency measure in this respect this paper aims to explore the extension of the median to random fuzzy numbers this extension is based on the 1 norm distance and its adequacy will be shown by analyzing its properties and comparing its robustness with that of the mean both theoretically and empirically

sparse spectral clustering method based on the incomplete cholesky decomposition a novel sparse spectral clustering method using linear algebra techniques is proposed spectral clustering methods solve an eigenvalue problem containing a graph laplacian the proposed method exploits the structure of the laplacian to construct an approximation not in terms of a low rank approximation but in terms of capturing the structure of the matrix with this approximation the size of the eigenvalue problem can be reduced to obtain the indicator vectors from the eigenvectors the method proposed by zha et al 2002 26 which computes a pivoted lq factorization of the eigenvector matrix is adapted this formulation also gives the possibility to extend the method to out of sample points

canonical representations of discrete fuzzy numbers in this paper we obtain two canonical representations of discrete fuzzy numbers in particular it is shown that any discrete fuzzy number can be represented by a triangular or a block discrete fuzzy number having the same value ambiguity and fuzziness as the original number

multi criteria decision making involving uncertain information via fuzzy ranking and fuzzy aggregation functions abstract many advances in artificial intelligence and machine learning are based on decision making especially in uncertain settings due to its possible applications decision making is currently a broad field of study in many areas like computation economics and business management the first techniques appeared in scenarios where information was modeled by real numbers in all cases one of the key steps in such processes was the summarization of the available information into a few values that helped the decision maker to complete this task in this paper we introduce a novel multi criteria decision making methodology in the fuzzy context in which weights and experts opinions may be translated by linguistic labels are stated as triangular fuzzy numbers to do that we take advantage of a recently presented fuzzy binary relation whose properties are according to human intuition and we carry out a study of the main properties that an aggregation function a mapping to sum up information must satisfy in the fuzzy framework the presented procedure makes a final decision based on parabolic fuzzy numbers not triangular and this will be shown in an illustrative example

evaluation of fuzzy linear regression models by comparing membership functions fuzzy linear regression models can provide an estimated fuzzy number that has a fuzzy membership function if a point that has the highest membership value from the estimated fuzzy number is not within the support of the observed fuzzy membership function a decision maker can have high risk from the estimate in this study a modification of fuzzy linear regression analysis based on a criterion of minimizing the difference of the fuzzy membership values between the observed and estimated fuzzy numbers is proposed two numerical examples are used to evaluate the fuzzy regression models

asymptotic rate of quantum ergodicity in chaotic euclidean billiards the quantum unique ergodicity que conjecture of rudnick and sarnak is that every eigenfunction n of the laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit eigenvalue en that is strong scars are absent we study numerically the rate of equidistribution for a uniformly hyperbolic sinai type planar euclidean billiard with dirichlet boundary condition the drum problem at unprecedented high e and statistical accuracy via the matrix elements n m of a piecewise constant test function a by collecting 30 000 diagonal elements up to level n 7 105 we find that their variance decays with eigenvalue as a power 0 48 0 01 close to the semiclassical estimate of feingold and peres this contrasts with the results of existing studies which have been limited to en a factor 102 smaller we find strong evidence for que in this system we also compare off diagonal variance as a function of distance from the diagonal against feingold peres or spectral measure at the highest accuracy 0 7 thus far in any chaotic system we outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions 2006 wiley periodicals inc

deterministic edge connectivity in near linear time we present a deterministic algorithm that computes the edge connectivity of a graph in near linear time this is for a simple undirected unweighted graph g with n vertices and m edges this is the first o mn time deterministic algorithm for the problem our algorithm is easily extended to find a concrete minimum edge cut in fact we can construct the classic cactus representation of all minimum cuts in near linear time the previous fastest deterministic algorithm by gabow from stoc 91 took o m 2 n where is the edge connectivity but can be as big as n 1 karger presented a randomized near linear time monte carlo algorithm for the minimum cut problem at stoc 96 but the returned cut is only minimum with high probability our main technical contribution is a near linear time algorithm that contracts vertex sets of a simple input graph g with minimum degree producing a multigraph with o m edges which preserves all minimum cuts of g with at least two vertices on each side in our deterministic near linear time algorithm we will decompose the problem via low conductance cuts found using pagerank a la brin and page 1998 as analyzed by andersson chung and lang at focs 06 normally such algorithms for low conductance cuts are randomized monte carlo algorithms because they rely on guessing a good start vertex however in our case we have so much structure that no guessing is needed

efficient implementation of graph algorithms using contraction the component merging problem is a new graph problem versions of this problem appear as bottlenecks in various graph algorithms a new data structure solves this problem efficiently and two special cases of the problem have even more efficient solutions based on other data structures the performance of the data structures is sped up by introducing a new algorithmic tool called packets the algorithms that use these solutions to the component merging problem also exploit new properties of two existing data structures specifically trees can be used simultaneously as a priority queue and a concatenable queue similarly f heaps support some kinds of split operations with no loss of efficiency an immediate application of the solution to the simplest version of the merging problem is an o t m n algorithm for finding minimum spanning trees in undirected graphs without using f heaps where t m n m log 2 log 2 log d n the graph has n vertices and m edges and d max m n 2 packets also improve the f heap minimum spanning tree algorithm giving the fastest algorithm currently known for this problem the efficient solutions to the merging problem and the new observation about f heaps lead to an o n t m n n log n algorithm for finding a maximum weighted matching in general graphs this settles an open problem posed by tarjan 15 p 123 where the weaker bound of o nm log n 2 m was conjectured

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

comparison of fuzzy numbers using a fuzzy distance measure a new approach for ranking fuzzy numbers based on a distance measure is introduced a new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first and then it is used to formulate the distance measure for fuzzy numbers the approach is illustrated by numerical examples showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches

a simple method for computation of fuzzy linear regression we propose a new method for computation of fuzzy regression that is simple and gives good solutions we consider two cases first when only the dependent variable is fuzzy our approach is given and is compared with those suggested in the literature secondly when both dependent and independent variables are fuzzy our approach is extended and compared with those given in the literature in each case a simple example is used to compare the competing approaches

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

a fuzzy regression model based on finite fuzzy numbers and its application to real world financial data different sources of imprecision and uncertainty are encountered in practical problems and thus many elements may need to be imprecisely observed defined or treated in this setting one of the most successfully applied techniques to describe possible relationships between fuzzy variables is the regression methodology in this paper we introduce a fuzzy regression procedure involving a class a fuzzy numbers defined by some level sets called finite fuzzy numbers we give a characterization of the image of a finite fuzzy number in terms of the extremes of its level sets and we present a parametric family of fuzzy semidistances between them that let us to consider a total fuzzy error of estimation described as a fuzzy sum of squares of residuals in particular cases the estimation process consists in finding a regression model that minimizes in a fuzzy sense such fuzzy error although spreads of finite fuzzy numbers can take some values very close to zero which complicate the task of finding nonnegative models the presented algorithm is able to guarantee that the predicted response is a fuzzy variable finally a numerical example based on fuzzy economic data of china is given to illustrate the use of the proposed method

fuzzy spline univariate regression with exact predictors and fuzzy responses abstract spline smoothing is a form of nonlinear regression when there is reason to believe that relationship between the predictor and the response is curvilinear in such cases the spline smoothing is an effective method to improve the performances of the conventional polynomial based regression models this paper proposed a fuzzy spline method based on a weighted absolute error distance measure with exact predictors and fuzzy responses unknown fuzzy coefficients tuning parameter were selected according to a hybrid optimization algorithm the effectiveness of the proposed method was also examined and compared with some well established fuzzy nonlinear regression models through some numerical examples including a simulation study for this purpose several common goodness of fit criteria were employed the numerical results clearly indicated that our prediction estimation gives more accurate results compared to other methods

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

clustering by minimum cut hyperplanes minimum normalised graph cuts are highly effective ways of partitioning unlabeled data having been made popular by the success of spectral clustering this work presents a novel method for learning hyperplane separators which minimise this graph cut objective when data are embedded in euclidean space the optimisation problem associated with the proposed method can be formulated as a sequence of univariate subproblems in which the optimal hyperplane orthogonal to a given vector is determined these subproblems can be solved in log linear time by exploiting the trivial factorisation of the exponential function experimentation suggests that the empirical runtime of the overall algorithm is also log linear in the number of data asymptotic properties of the minimum cut hyperplane both for a finite sample and for an increasing sample assumed to arise from an underlying probability distribution are discussed in the finite sample case the minimum cut hyperplane converges to the maximum margin hyperplane as the scaling parameter is reduced to zero applying the proposed methodology both for fixed scaling and the large margin asymptotes is shown to produce high quality clustering models in comparison with state of the art clustering algorithms in experiments using a large collection of benchmark datasets

consensus and products of random stochastic matrices exact rate for convergence in probability we find the exact rate for convergence in probability of products of independent identically distributed symmetric stochastic matrices it is well known that if the matrices have positive diagonals almost surely and the support graph of the mean or expected value of the random matrices is connected the products of the matrices converge almost surely to the average consensus matrix and thus in probability in this paper we show that the convergence in probability is exponentially fast and we explicitly characterize the exponential rate of this convergence our analysis reveals that the exponential rate of convergence in probability depends only on the statistics of the support graphs of the random matrices further we show how to compute this rate for commonly used random models gossip and link failure with these models the rate is found by solving a min cut problem and hence it is easily computable finally as an illustration we apply our results to solving power allocation among networked sensors in a consensus innovations distributed detection problem