on finity countability cardinalities and cylindric extensions of type 2 fuzzy sets in linguistic summarization of databases the discussion in this paper is closely related to our idea of a type 2 linguistic summary of a database presented by niewiadomski a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008 which is intended to be an efficient tool of knowledge discovery from large databases in that approach we put emphasis on algorithms and applications of type 2 fuzzy sets in summarizing databases hence we have implicitly assumed and considered linguistic expressions represented by sets in discrete or even finite universes of discourse now in this paper we discuss more thoroughly some of the properties and formal aspects of both discrete and continuous type 2 fuzzy sets that represent the elements of linguistic summaries i e quantifiers summarizers and or qualifiers we underline differences between type 2 fuzzy sets in finite and in infinite usually continuous universes of discourse and henceforth between their cardinalities and similar measures which are used in evaluating the goodness quality of type 2 linguistic summaries in addition we define new imprecision measures of linguistic expressions represented by type 2 fuzzy sets and propose a definition of the cylindric extension of a type 2 fuzzy set finally we apply these new concepts to generalize algorithms of type 2 linguistic summarization and propose some new features in comparison to our previous approach a niewiadomski a type 2 fuzzy approach to linguistic summarization of data ieee trans fuzzy syst vol 16 no 1 pp 198 213 feb 2008

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

structured invariant spaces of vector valued rational functions hermitian matrices and a generalization of the lohvidov laws abstract finite dimensional indefinite inner product spaces of vector valued rational functions which are 1 invariant under the generalized backward shift and 2 subject to a structural identity and subspaces and superspaces thereof are studied the theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of hermitian toeplitz and hermitian hankel matrices the connecting link rests on the identification of a hermitian matrix as the gram matrix of a space of vector valued functions of the type considered in the first part of the paper corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non hermitian teoplitz and non hermitian hankel matrices are also stated but the proofs will be presented elsewhere

on a new class of reproducing kernel spaces and a new generalization of the iohvidov laws abstract a new class of finite dimensional reproducing kernel spaces of m 1 vector valued analytic functions on a fairly general domain is introduced the reproducing kernels of these spaces have a special form which is based on an m m matrix valued function which is j unitary on the boundary of every invertible hermitian matrix can be interpreted as the gram matrix of a suitably chosen basis in such a space where m is equal to the appropriately defined displacement rank of the given matrix orthogonal decompositions of the space are developed in terms of schur complements of the matrix and factorizations of the much as in the classical cases wherein is either the unit disc or the half plane finally a new generalization of the iohvidov laws which extends earlier generalizations by the same authors is deduced as an application of the theory

mpeg vbr video traffic modeling and classification using fuzzy technique we present an approach for mpeg variable bit rate vbr video modeling and classification using fuzzy techniques we demonstrate that a type 2 fuzzy membership function i e a gaussian mf with uncertain variance is most appropriate to model the log value of i p b frame sizes in mpeg vbr video the fuzzy c means fcm method is used to obtain the mean and standard deviation std of t p b frame sizes when the frame category is unknown we propose to use type 2 fuzzy logic classifiers flcs to classify video traffic using compressed data five fuzzy classifiers and a bayesian classifier are designed for video traffic classification and the fuzzy classifiers are compared against the bayesian classifier simulation results show that a type 2 fuzzy classifier in which the input is modeled as a type 2 fuzzy set and antecedent membership functions are modeled as type 2 fuzzy sets performs the best of the five classifiers when the testing video product is not included in the training products and a steepest descent algorithm is used to tune its parameters

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

rational matrix functions with coisometric values on the imaginary line abstract given signature matrices j1 and j2 we obtain a necessary and sufficient condition for a rational matrix function w analytic at infinity to satisfy equation j1 w z j2w z on the imaginary axis the condition is based on a lyapunov equation involving matrices in an observable realization of w and generalizes the fact well known in the case where j1 j2 if the condition is satisfied for every observable realization a b c d of w there exists a unique possibly singular hermitian matrix g such that g satisfies the lyapunov equation and cg dj2b we call g the hermitian matrix associated with the realization the minimal factorizations w w1w2 where w1 and w2 satisfy equations j1 w1 z j1w1 z and j1 w2 z j2w2 z for z on the imaginary line can be characterized in terms of decompositions of the state space into subspaces determined by a possibly indefinite inner product induced by g as a corollary we obtain a sufficient condition for existence of a minimal factorization w w1w2 with w1 the multiplicative inverse of a blaschke potapov factor

enhanced interval approach for encoding words into interval type 2 fuzzy sets and its convergence analysis construction of interval type 2 fuzzy set models is the first step in the perceptual computer which is an implementation of computing with words the interval approach ia has so far been the only systematic method to construct such models from data intervals that are collected from a survey however as pointed out in this paper it has some limitations and its performance can be further improved this paper proposes an enhanced interval approach eia and demonstrates its performance on data that are collected from a web survey the data part of the eia has more strict and reasonable tests than the ia and the fuzzy set part of the eia has an improved procedure to compute the lower membership function we also perform a convergence analysis to answer two important questions 1 does the output interval type 2 fuzzy set from the eia converge to a stable model as increasingly more data intervals are collected and 2 if it converges then how many data intervals are needed before the resulting interval type 2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals we show that the eia converges in a mean square sense and generally 30 data intervals seem to be a good compromise between cost and accuracy

on a new class of structured reproducing kernel spaces abstract a class of reproducing kernel spaces with reproducing kernels of the form k j j with p a a is characterized in terms of invariance under a pair of generalized shift operators and a structural identity this incorporates a characterization of de branges for the line case and a later analogue due to ball for the circle case as well as many other possibilities by specializing the choice of these results also permit the extension of some earlier characterizations by the authors of finite dimensional spaces with reproducing kernels of the form given above to the infinite dimensional case the non hermitian case is also considered

a new version of 2 tuple fuzzy linguistic representation model for computing with words in this paper we provide a new proportional 2 tuple fuzzy linguistic representation model for computing with words cw which is based on the concept of symbolic proportion this concept motivates us to represent the linguistic information by means of 2 tuples which are composed by two proportional linguistic terms for clarity and generality we first study proportional 2 tuples under ordinal contexts then under linguistic contexts and based on canonical characteristic values ccvs of linguistic labels we define many aggregation operators to handle proportional 2 tuple linguistic information in a computational stage for cw without any loss of information our approach for this proportional 2 tuple fuzzy linguistic representation model deals with linguistic labels which do not have to be symmetrically distributed around a medium label and without the traditional requirement of having equal distance between them moreover this new model not only provides a space to allow a continuous interpolation of a sequence of ordered linguistic labels but also provides an opportunity to describe the initial linguistic information by members of a continuous linguistic scale domain which does not necessarily require the ordered linguistic terms of a linguistic variable being equidistant meanwhile under the assumption of equally informative which is defined by a condition based on the concept of ccv we show that our model reduces to herrera and mart spl inodot spl acute nez s translational 2 tuple fuzzy linguistic representation model

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

ensuring the centroid of an interval type 2 fuzzy set in computing with words establishing a fuzzy set fs model for a word to capture its uncertainties is an important issue an interval type 2 it2 fs can be used to model a word how to establish an it2 fs from the collected data about a word has been a challenging problem it has been reported that one way is to extract the centroid of an it2 fs from the collected data and to obtain geometric parameters of its footprint of uncertainty fou such that its centroid matches the extracted one how to extract the centroid of an it2 fs from the collected data about a word has thoroughly been studied however there exists no method for obtaining fou parameters for an it2 fs such that its centroid matches the desired one to fill this gap this paper presents an approach to obtaining fou parameters for an it2 fs by establishing equations using the centroid requirement to propose this approach a sufficient and necessary condition for ensuring the centroid of an it2 fs is developed using this sufficient and necessary condition two equations about all of the fou parameters are established to obtain the fou parameters all of them except two are predetermined so that the established equations can be simplified to two single variable equations the other two fou parameters can then be determined by solving these two single variable equations using existing root finding algorithms among existing root finding algorithms the false position algorithm is recommended the overall merits of the proposed approach are its simplicity in implementation and its applicability to it2 fss with arbitrary fou shapes in addition numerical examples are provided to further illustrate how to apply the proposed approach to obtain fou parameters for an it2 fs

linguistic summarization using if then rules and interval type 2 fuzzy sets linguistic summarization ls is a data mining or knowledge discovery approach to extract patterns from databases many authors have used this technique to generate summaries like most senior workers have high salary which can be used to better understand and communicate about data however few of them have used it to generate if then rules like if x is large and y is medium then z is small which not only facilitate understanding and communication of data but can also be used in decision making in this paper an ls approach to generate if then rules for causal databases is proposed both type 1 and interval type 2 fuzzy sets are considered five quality measures the degrees of truth sufficient coverage reliability outlier and simplicity are defined among them the degree of reliability is especially valuable for finding the most reliable and representative rules and the degree of outlier can be used to identify outlier rules and data for close up investigation an improved parallel coordinates approach for visualizing the if then rules is also proposed experiments on two datasets demonstrate our ls and rule visualization approaches finally the relationships between our ls approach and the wang mendel wm method perceptual reasoning and granular computing are pointed out

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

from interval valued data to general type 2 fuzzy sets in this paper a new approach is presented to model interval based data using fuzzy sets fss specifically we show how both crisp and uncertain intervals where there is uncertainty about the endpoints of intervals collected from individual or multiple survey participants over single or repeated surveys can be modeled using type 1 interval type 2 or general type 2 fss based on zslices the proposed approach is designed to minimize any loss of information when transferring the interval based data into fs models and to avoid as much as possible assumptions about the distribution of the data furthermore our approach does not rely on data preprocessing or outlier removal which can lead to the elimination of important information different types of uncertainty contained within the data namely intra and inter source uncertainty are identified and modeled using the different degrees of freedom of type 2 fss thus providing a clear representation and separation of these individual types of uncertainty present in the data we provide full details of the proposed approach as well as a series of detailed examples based on both real world and synthetic data we perform comparisons with analogue techniques to derive fss from intervals namely the interval approach and the enhanced interval approach and highlight the practical applicability of the proposed approach

spectral factorization of rectangular rational matrix functions with application to discrete wiener hopf equations the properties of a discrete wiener hopf equation are closely related to the factorization of the symbol of the equation we give a necessary and sufficient condition for existence of a canonical wiener hopf factorization of a possibly nonregular rational matrix function w relative to a contour which is a positively oriented boundary of a region in the finite complex plane the condition involves decomposition of the state space in a minimal realization of w and if it is satisfied we give explicit formulas for the factors the results are generalized by means of centered realizations to arbitrary rational matrix functions the proposed approach can be used to solve discrete wiener hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle

multivariate modeling and type 2 fuzzy sets this paper explores the link between type 2 fuzzy sets and multivariate modeling elements of a space x are treated as observations fuzzily associated with values in a multivariate feature space a category or class is likewise treated as a fuzzy allocation of feature values possibly dependent on values in x we observe that a type 2 fuzzy set on x generated by these two fuzzy allocations captures imprecision in the class definition and imprecision in the observations in practice many type 2 fuzzy sets are in fact generated in this way and can therefore be interpreted as the output of a classification task we then show that an arbitrary type 2 fuzzy set can be so constructed by taking as a feature space a set of membership functions on x this construction presents a new perspective on the representation theorem of mendel and john the multivariate modeling underpinning the type 2 fuzzy sets can also constrain realizable forms of membership functions because averaging operators such as centroid and subsethood on type 2 fuzzy sets involve a search for optima over membership functions constraining this search can make computation easier and tighten the results we demonstrate how the construction can be used to combine representations of concepts and how it therefore provides an additional tool alongside standard operations such as intersection and subsethood for concept fusion and computing with words

data informed fuzzy measures for fuzzy integration of intervals and fuzzy numbers the fuzzy integral fi with respect to a fuzzy measure fm is a powerful means of aggregating information the most popular fis are the choquet and sugeno and most research focuses on these two variants the arena of the fm is much more populated including numerically derived fms such as the sugeno lambda measure and decomposable measure expert defined fms and data informed fms the drawback of numerically derived and expert defined fms is that one must know something about the relative values of the input sources however there are many problems where this information is unavailable such as crowdsourcing this paper focuses on data informed fms or those fms that are computed by an algorithm that analyzes some property of the input data itself gleaning the importance of each input source by the data they provide the original instantiation of a data informed fm is the agreement fm which assigns high confidence to combinations of sources that numerically agree with one another this paper extends upon our previous work in data informed fms by proposing the uniqueness measure and additive measure of agreement for interval valued evidence we then extend data informed fms to fuzzy number fn valued inputs we demonstrate the proposed fms by aggregating interval and fn evidence with the choquet and sugeno fis for both synthetic and real world data

type 2 fuzzy hidden markov models and their application to speech recognition this paper presents an extension of hidden markov models hmms based on the type 2 t2 fuzzy set fs referred to as type 2 fuzzy hmms t2 fhmms membership functions mfs of t2 fss are three dimensional and this new third dimension offers additional degrees of freedom to evaluate the hmms fuzziness therefore t2 fhmms are able to handle both random and fuzzy uncertainties existing universally in the sequential data we derive the t2 fuzzy forward backward algorithm and viterbi algorithm using t2 fs operations in order to investigate the effectiveness of t2 fhmms we apply them to phoneme classification and recognition on the timit speech database experimental results show that t2 fhmms can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical hmms

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

zero cancellation for general rational matrix functions the problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints firstly the class of all factors that dislocate the zeros and feature minimal mcmillan degree are derived further necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being j unitary or j inner either with respect to the imaginary axis or to the unit circle the main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial proper or improper rank deficient with arbitrary poles and zeros including at infinity a particular attention is paid to the numerically sound construction of solutions by employing at each stage unitary transformations reliable numerical algorithms for eigenvalue assignment and efficient lyapunov equation solvers

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

on a new class of realization formulas and their application abstract a new set of realization formulas is derived for a class of matrix valued functions w these include the standard realization formulas for rational w observability controllability and minimality are defined and characterized conditions for w to be j 1 j 2 isometric and or j 1 j 2 coisometric with respect to a pair of signature matrices j 1 and j 2 are given in terms of the realizations minimal factorizations for square w are considered and formulas for the factors are deduced associated reproducing kernel spaces of pairs too are discussed

beurling lax type theorems in the complex and quaternionic setting abstract we give a generalization of the beurling lax theorem both in the complex and quaternionic settings we consider in the first case functions meromorphic in the right complex half plane and functions slice hypermeromorphic in the right quaternionic half space in the second case in both settings we also discuss a unified framework which includes both the disk and the half plane for the complex case and the open unit ball and the half space in the quaternionic setting

perceptual reasoning for perceptual computing a similarity based approach perceptual reasoning pr is an approximate reasoning method that can be used as a computing with words cww engine in perceptual computing there can be different approaches to implement pr e g firing interval based pr fi pr which has been proposed in j m mendel and d wu ieee trans fuzzy syst vol 16 no 6 pp 1550 1564 dec 2008 and similarity based pr s pr which is proposed in this paper both approaches satisfy the requirement on a cww engine that the result of combining fired rules should lead to a footprint of uncertainty fou that resembles the three kinds of fous in a cww codebook a comparative study shows that s pr leads to output fous that resemble word fous which are obtained from subject data much more closely than fi pr hence s pr is a better choice for a cww engine than fi pr

network coding a computational perspective in this work we study the computational perspective of network coding focusing on two issues first we address the computational complexity of finding a network code for acyclic multicast networks second we address the issue of reducing the amount of computation performed by network nodes in particular we consider the problem of finding a network code with the minimum possible number of encoding nodes i e nodes that generate new packets by performing algebraic operations on packets received over incoming links we present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph g v e in time 0 e kh v k2h2 h4k3 k h where k is the number of destinations and h is the number of packets our algorithm improves the best known running time for network code construction in addition our algorithm guarantees that the number of encoding nodes in the obtained network code is upper bounded by 0 h3k2 next we address the problem of finding integral and fractional network codes with the minimum number of encoding nodes we prove that in the majority of settings this problem is np hard however we show that if h o 1 k o 1 and the underlying communication graph is acyclic then there exists an algorithm that solves this problem in polynomial time

on advanced computing with words using the generalized extension principle for type 1 fuzzy sets in this paper we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve advanced computing with words acww problems such problems involve implicit assignments of linguistic truth probability and possibility to begin we establish the vocabularies of the words involved in the problems and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the interval approach ia or the enhanced interval approach eia next the solutions of the acww problems which involve the fuzzy set models of the words are formulated using the generalized extension principle because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically we then develop a numerical method for their solution finally the resulting fuzzy set solutions are decoded into natural language words using jaccard s similarity measure we explain how acww problems can solve some potential prototype engineering problems and connect the methodology of this paper with perceptual computing

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

a probabilistic framework for interval type 2 fuzzy linguistic summarization many studies on linguistic summarization have addressed the use of ordinary fuzzy set type 1 fuzzy set t1fs for modeling words however few of them have exploited interval type 2 fuzzy set it2fs although it2fs is better able to deal with the uncertainty associated with words the existing studies on linguistic summarization using it2fs have focused on scalar cardinality based degree of truth in this paper for the first time we propose a probabilistic framework based on the idea of interval mass assignment to evaluate interval type 2 fuzzy linguistic summaries based on the type ii quantified sentences and the semi fuzzy quantifiers as an alternative method to the scalar cardinality based methods we implement a real case study on linguistic summarization of time series data of europe brent spot price as well as a comparison of the results obtained with our approach and those of the existing approaches

minimizing adjusted simple terms in the consensus reaching process with hesitant linguistic assessments in group decision making in some real world decision processes decision makers may prefer to provide their opinions using linguistic expressions instead of a single linguistic term particularly they may hesitate between several linguistic terms in this paper we deal with the consensus issue in the hesitant linguistic group decision making gdm problem firstly a novel distance based consensus measure is proposed then using this consensus measure we develop an optimization based consensus model in the hesitant linguistic gdm which minimizes the number of adjusted simple terms in the consensus building furthermore a two stage model is displayed to further optimize the solutions to the proposed consensus model through which we obtain the unique optimal adjustment suggestion to support the consensus reaching process in the hesitant linguistic gdm finally several desirable properties are proposed to justify the proposal and two examples are used to demonstrate the validity of the models