a sketched finite element method for elliptic models abstract we consider a sketched implementation of the finite element method for elliptic partial differential equations on high dimensional models motivated by applications in real time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low dimensional subspace and sketching the reduced equations using randomised sampling we show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete laplacian operator can achieve nearly optimal performance and a significant speedup we derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability and an upper bound for the distance between the sketched and the high dimensional solutions our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction

factorization of saddle point matrices in dynamical systems optimization reusing pivots abstract in this paper we consider the application of direct methods for solving a sequence of saddle point systems our goal is to design a method that reuses information from one factorization and applies it to the next one in more detail when we compute the pivoted l d l t factorization we speed up computation by reusing already computed pivots and permutations we develop our method in the frame of dynamical systems optimization experiments show that the method improves efficiency over bunch parlett and bunch kaufman while delivering the same results

reduction to least squares estimates in multiple fuzzy regression analysis in this paper we deal with the problem of least squares multiple regression with fuzzy data the regression coefficients are assumed to be real crisp a formula for solving the regression coefficients in one variable models is derived if each independent variable is effective i e its corresponding regression coefficient is nonzero the multiple regression problem can be replaced with a 0 1 programming problem its optimal solution is easily computed finally we also propose effective algorithms to compute the regression coefficients in a general case

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

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

development of fuzzy and control charts using α cuts statistical process control spc is an approach that uses statistical techniques to monitor the process shewhart introduced the control charts that are one of the most important techniques of quality control to detect if assignable causes exist the widely used control charts are and these are called traditional variable control charts in the traditional variable control charts center line upper control limit and lower control limit are represented by numeric values a process is either in control or out of control depending on numeric observation values for many problems control limits could not be so precise uncertainty comes from the measurement system including operators and gauges and environmental conditions in this context fuzzy set theory is a useful tool to handle this uncertainty numeric control limits can be transformed to fuzzy control limits by using membership functions if a sample mean is too close to the control limits and the used measurement system is not so sensitive the decision may be faulty fuzzy control limits provide a more accurate and flexible evaluation this study constructs the fuzzy and control charts with a cuts an application is presented for fuzzy control charts by using fuzzy and control charts the flexibility of traditional control limits is increased

fuzzy nonparametric regression based on local linear smoothing technique in a great deal of literature on fuzzy regression analysis most of research has focused on some predefined parametric forms of fuzzy regression relationships especially on the fuzzy linear regression models in many practical situations it may be unrealistic to predetermine a fuzzy parametric regression relationship in this paper a fuzzy nonparametric model with crisp input and lr fuzzy output is considered and based on the distance measure for fuzzy numbers suggested by diamond p diamond fuzzy least squares information sciences 46 1988 141 157 the local linear smoothing technique in statistics with the cross validation procedure for selecting the optimal value of the smoothing parameter is fuzzified to fit this model some simulation experiments are conducted to examine the performance of the proposed method and three real world datasets are analyzed to illustrate the application of the proposed method the results demonstrate that the proposed method works quite well not only in producing satisfactory estimate of the fuzzy regression function but also in reducing the boundary effect significantly

a possibilistic regression based on gradual interval b splines application for hyperspectral imaging lake sediments abstract according to an epistemic view of intervals this paper proposes a possibilistic regression based on gradual interval b splines to represent the input output data mapping in this context an improvement of the parametric fuzzy regression through the notion of gradual interval b splines is proposed the proposed gradual regression based on b splines can be regarded as an extension of the nonparametric interval based regression where an uncertain dimension is integrated the proposed method is validated through illustrative and comparative examples moreover it is applied for modeling the input output behavior between reflectance and wavelengths in an application for hyperspectral imaging for lake sediments

on lagerstrom s model of slow incompressible viscous flow the model discussed is a nonlinear boundary value problem which contains a parameter varepsilon that models the reynolds number the matched asymptotic expansions an inner stokes expansion valid near the inner boundary and an outer oseen expansion valid away from it that describe the solutions of the model problem for varepsilon small are extended numerical calculations show that these matched expansions have only a small range of usefulness with the addition of further terms generally causing a worse rather than better approximation at moderate values of varepsilon far better results are achieved when a single expansion the outer expansion is used throughout the additional terms that have been calculated then consistently give improved approximations for all varepsilon it is also rigorously proved that an iterative method of solution of the model equation based on the outer oseen approximation converges for all varepsilon to a unique solution the results presented here for lagerstrom s model suggest that iterative improvement of the oseen expansion may be an effective method of approximation of viscous flows at moderate reynolds number

a fuzzy varying coefficient model and its estimation the fuzzy linear regression model has been a useful tool for analyzing relationships between a set of variables in a fuzzy environment and has been extensively studied in the literature however this model may fail to reflect the more complicated regression relationships that are usually found in practice because of its simple and predefined linear structure in order to enhance the feasibility and adaptability of the fuzzy linear models we propose in this paper a fuzzy varying coefficient model in which the fuzzy coefficients in the fuzzy linear models are allowed to vary with a covariate a restricted weighted least squares estimation is suggested for locally fitting the model furthermore some real world datasets are analyzed in order to evaluate the performance of the proposed method and the results show that the proposed model with its estimation approach performs satisfactorily in predicting the fuzzy response even in the case where the regression relationship is complicated

fuzzy least absolutes regression using shape preserving operations the purpose of this study is to introduce a new fuzzy regression model based on the least absolutes method the fuzzy simple and fuzzy multivariate regression models with fuzzy input fuzzy output are considered in which the coefficients of the models are themselves fuzzy the proposed method is based on a new metric on the space of lr fuzzy numbers fuzzy arithmetic operations are based on the weakest triangular norm t w which is the unique triangular norm that preserves the shape of fuzzy numbers during multiplication the results of comparative studies and numerical examples indicate that using the similarity measure criterion as well as a predictive ability index the proposed method has a better performance than the least squares method especially when the data set includes some outlier data point s

on bounds for scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a positive definite diagonal matrix in a recent paper stewart considered the weighted pseudoinverse x d x t dx 1 x t d and the associated oblique projection p d xx d and gave bounds independent of d for the norms of these matrices in this note we answer a question he raised by showing that the bounds are computable

random fuzzy variables of second order and applications to statistical inference abstract this paper summarizes some results on random fuzzy variables with existing expectation and variance called random fuzzy variables of second order using the frechet principle and via support functions the embedding of convex fuzzy sets into a banach space of functions it especially presents a unified view on expectation and variance of random fuzzy variables these notions are applied in developing linear statistical inference with fuzzy data detailed investigations are presented concerning best linear unbiased estimation in linear regression models with fuzzy observations

falscal a fuzzy multidimensional scaling algorithm the conventionally adopted alternating least squares scaling alscal procedure of multidimensional scaling mds is a valuable mathematical scheme for analyzing data in areas where organized concepts and underlying dimensions are inadequately defined or developed fuzzy set theory fst attempts to formulate human reasoning and perceptions therefore targeting problems in areas where human factors significantly impact the result of decision making to our knowledge the fst and alscal approaches have not yet been integrated this study integrates and modifies the fst and alscal procedures fuzzy data collected from fuzzy questionnaires are adopted as the input of the mds ensuring that the uncertainty of input data can be incorporated into the analysis the conventionally adopted alscal procedure is then modified to cope with fuzzy input data by adopting the notion of fuzzy distances fuzzy disparities and fuzzy ranking to represent the similarities between fuzzy data related approximation operations of the triangular fuzzy number are also introduced to facilitate computation in fuzzy alscal

fuzzy least squares linear regression analysis for fuzzy input output data a fuzzy regression model is used in evaluating the functional relationship between the dependent and independent variables in a fuzzy environment most fuzzy regression models are considered to be fuzzy outputs and parameters but non fuzzy crisp inputs in general there are two approaches in the analysis of fuzzy regression models linear programming based methods and fuzzy least squares methods in 1992 sakawa and yano considered fuzzy linear regression models with fuzzy outputs fuzzy parameters and also fuzzy inputs they formulated multiobjective programming methods for the model estimation along with a linear programming based approach in this paper two estimation methods along with a fuzzy least squares approach are proposed these proposed methods can be effectively used for the parameter estimation comparisons are also made between them

a parametric model for fusing heterogeneous fuzzy data presented is a model that integrates three data types numbers intervals and linguistic assessments data of these three types come from a variety of sensors one objective of sensor fusion models is to provide a common framework for data integration processing and interpretation that is what our model does we use a small set of artificial data to illustrate how problems as diverse as feature analysis clustering cluster validity and prototype classifier design can all be formulated and attacked with standard methods once the data are converted to the generalized coordinates of our model the effects of reparameterization on computational outputs are discussed numerical examples illustrate that the proposed model affords a natural way to approach problems which involve mixed data types

fuzzy least squares algorithms for interactive fuzzy linear regression models fuzzy regression analysis can be thought of as a fuzzy variation of classical regression analysis it has been widely studied and applied in diverse areas in general the analysis of fuzzy regression models can be roughly divided into two categories the first is based on tanaka s linear programming approach the second category is based on the fuzzy least squares approach in this paper new types of fuzzy least squares algorithms with a noise cluster for interactive fuzzy linear regression models are proposed these algorithms are robust for the estimation of fuzzy linear regression models especially when outliers are present numerical examples are given to detail the effectiveness of this approach

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

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

fuzzy prediction based on regression models abstract we use a linguistic variable to represent imprecise information to be inserted in regression models used for prediction we show how one can obtain probabilistic statements about the forecasted variables

s curve regression model in fuzzy environment abstract the purpose of this paper is to discuss the problem for least squares fitting of fuzzy valued data which are expressed as fuzzy numbers and to develop an s shaped curve regression model for fitting this type of data it is shown that the solution of the s curve regression model is equivalent to the solution of the corresponding linear equations and furthermore the solution can be explicitly obtained by solving the linear equations

granular box regression granular computing grc has gained increasing attention in the past decade although not uniquely defined its basic idea is to approximate detailed machine like information by a coarser presentation on a human like level within granular computing the mapping of continuous variables into intervals plays an important role these intervals are often prerequisites for the formulation of linguistic variables in this paper we suggest a piecewise interval approximation and propose granular box regression its objective is to establish relationships between independent and dependent variables by multidimensional boxes we interpret granular box regression as interval regression and show its potential for the extraction of fuzzy rules from data in two experiments we apply granular box regression to an artificial as well as to a real dataset in the field of finance and evaluate its properties

fuzzy least absolute deviations regression and the conflicting trends in fuzzy parameters a general fuzzy linear model for fuzzy regression analysis was first formulated based on both this general model and a fuzzy difference ranking method 1 2 approaches for fuzzy least squares regression and fuzzy least absolute value deviations regression were proposed the former approach resulted in a nonlinear programming problem while the latter resulted linear numerical examples were solved by using the absolute deviations approach to illustrate the problems of conflicting trends and ways to at least partially overcome these problems furthermore these examples showed that the absolute deviations formulation forms an effective computational tool

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

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

development of a cost function for wastewater treatment systems with fuzzy regression abstract a fuzzy linear regression analysis is performed to study the cost function of wastewater that is developed by using a pool of data from 26 municipal wastewater treatment systems in taiwan the choice of a primary or more higher wastewater treatment process is intimately related to financial land availability effluent standards and managerial factors especially financial consideration plays an important role in the planning stage in addition cost estimation is difficult to precisely evaluate in an uncertain environment in the conventional regression model deviations between the observed values and the estimated values are supposed to be due to measurement errors here taking a different perspective these deviations are regarded as the fuzziness of the system s parameters thus these deviations are reflected in a linear function with fuzzy parameters using linear programming algorithm this fuzzy linear regression model might be very convenient and useful for finding a fuzzy structure in an evaluation system in this paper the details of the fuzzy linear regression concept and its applications to the cost function in an uncertain environment are shown and discussed

from fuzzy regression to gradual regression interval based analysis and extensions abstract this paper proposes an analysis of parametric interval based regression methodologies according to ontic and epistemic visions of intervals when assuming an epistemic point of view a new interpretation of fuzzy regression through the notion of gradual intervals is developed which leads to gradual regression gradual regression is viewed as an extension of the imprecise interval based regression which is obtained by integrating an uncertain dimension gradual intervals can yield improved specificity compared to conventional intervals and jointly consider the concepts of imprecision and uncertainty through a single and coherent formalism the formulation of the gradual regression problem its resolution and the propagation of the information through the obtained regressive models are carried out via gradual interval arithmetic the proposed method allows not only the extension of the interval vision to the gradual case but also interesting interpretations according to non additive confidence measure theories possibility and belief functions

quality mesh generation in higher dimensions we consider the problem of triangulating a d dimensional region our mesh generation algorithm called qmg is a quadtree based algorithm that can triangulate any polyhedral region including nonconvex regions with holes furthermore our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles finally our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by qmg is bounded above by a factor times the number produced by any competing algorithm where the factor depends on the aspect ratio bound satisfied by the competing algorithm the qmg algorithm has been implemented in c and is used as a mesh generator for the finite element method

theoretical choice of the optimal threshold for possibilistic linear model with noisy input based on possibility concepts various possibilistic linear models plms have been proposed and their pivotal role in fuzzy modeling and associated applications has been established when adopting plms one has to adopt an appropriate threshold lambda value however choosing such a value is by no means trivial and is still an open theoretical issue in this paper we propose a solution by first extending the plm to its regularized version i e a regularized plm rplm such that its generalization capability can be enhanced the rplm is then formulated as a maximum a posteriori map framework which facilitates the determination of the theoretically optimal threshold value for the rplm with noisy input our mathematical derivations reveal the approximately inversely proportional relationship between the threshold and the standard deviation of gaussian noisy input this is also confirmed by the simulation results this finding is very helpful for the practical applications of both plms and rplms

a geometric analysis of the lagerstrom model problem abstract lagerstrom s model problem is a classical singular perturbation problem which was introduced to illustrate the ideas and subtleties involved in the analysis of viscous flow past a solid at low reynolds number by the method of matched asymptotic expansions in this paper the corresponding boundary value problem is analyzed geometrically by using methods from the theory of dynamical systems in particular invariant manifold theory as an essential part of the dynamics takes place near a line of non hyperbolic equilibria a blow up transformation is introduced to resolve these singularities this approach leads to a constructive proof of existence and local uniqueness of solutions and to a better understanding of the singular perturbation nature of the problem in particular the source of the logarithmic switchback phenomenon is identified

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

a fuzzy linear regression model with better explanatory power previous studies on fuzzy linear 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 this paper proposes a two stage approach to construct the fuzzy linear regression model in the first stage the fuzzy observations are defuzzified so that the traditional least squares method can be applied to find a crisp regression line showing the general trend of the data in the second stage the error term of the fuzzy regression model which represents the fuzziness of the data in a general sense is determined to give the regression model the best explanatory power for the data the results from two examples one with crisp data and the other with fuzzy data for the independent variable indicate that the two stage method proposed in this paper has better performance than the previous studies

finknn a fuzzy interval number k nearest neighbor classifier for prediction of sugar production from populations of samples this work introduces finknn a k nearest neighbor classifier operating over the metric lattice of conventional interval supported convex fuzzy sets we show that for problems involving populations of measurements data can be represented by fuzzy interval numbers fins and we present an algorithm for constructing fins from such populations we then present a lattice theoretic metric distance between fins with arbitrary shaped membership functions which forms the basis for finknn s similarity measurements we apply finknn to the task of predicting annual sugar production based on populations of measurements supplied by hellenic sugar industry we show that finknn improves prediction accuracy on this task and discuss the broader scope and potential utility of these techniques

fuzzy linear regression analysis a multi objective programming approach fuzzy regression analysis is an extension of the classical regression analysis that is used in evaluating the functional relationship between the dependent and independent variables in a fuzzy environment fuzzy regression has been criticized because it is sensitive to outliers it does not allow all data points to influence the estimated parameters and the spread of the estimated values become wider as more data are included in the model in this paper we develop a multi objective fuzzy linear regression model to overcome the above shortcomings

note on the lagerstrom singular perturbation models by a change of dependent variable the lagerstrom model problem for compressible low reynolds number flow is converted to the simpler ncompressible model form and new uniformly valid expansions are obtained

generalized finite element methods their performance and their relation to mixed methods the notion of a generalized finite element method is introduced this class of methods is analyzed and their relation to mixed methods is discussed the class of generalized finite element methods offers a wide variety of computational procedures from which particular procedures can be selected for particular problems a particular generalized finite element method which is very effective for problems with rough coefficients is discussed in detail

interval regression analysis by quadratic programming approach when we use linear programming in possibilistic regression analysis some coefficients tend to become crisp because of the characteristic of linear programming on the other hand a quadratic programming approach gives more diverse spread coefficients than a linear programming one therefore to overcome the crisp characteristic of linear programming we propose interval regression analysis based on a quadratic programming approach another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression by changing the weights of the quadratic function we can analyze the given data from different viewpoints for data with crisp inputs and interval outputs the possibility and necessity models can be considered therefore the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed even though there always exist possibility estimation models the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model thus we consider polynomials as regression models since any curve can be represented by the polynomial approximation using polynomials we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models furthermore from the obtained possibility and necessity regression models a trapezoidal fuzzy output can be constructed

an efficient vector finite element method for nonlinear electromagnetic modeling we have developed a mixed vector finite element method vfem for maxwell s equations with a nonlinear polarization term the method allows for discretization of complicated geometries with arbitrary order representations of the b and e fields in this paper we will describe the method and a series of optimizations that significantly reduce the computational cost additionally a series of test simulations will be presented to validate the method finally a nonlinear waveguide mode mixing example is presented and discussed

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

higher level fuzzy numbers arising from fuzzy regression consider a fuzzy random variable y with expectation symbol b x where b is an unknown fuzzy number and an unknown real number for n observations yi xi there is a model yi b xi ei i 1 2 n where ei are fuzzy valued errors independently and identically distributed in some sense the aim is to obtain estimates of b when all fuzzy numbers are triangular and the ei are uniformly distributed maximum likelihood estimators emerge naturally as a special form of higher level fuzzy number thus mle can be interpreted as an estimation procedure which transforms randomness into higher levels of fuzziness

fuzzy nonlinear regression analysis of dose response relationships abstract a general methodology for fuzzy regression is developed and illustrated by analyzing a dose response relationship fuzzy regression is used as an alternative to statistical regression analysis because the relationship between variables is imprecise data are uncertain and sample sizes are insufficient three goodness of fit criteria are used namely the maximum the average vagueness and the prediction vagueness criterion in the case of a nonlinear relationship the calculation of fuzzy regression parameters leads to a nonlinear programming problem this technique is illustrated by the relationship between the dose of a n nitroso compound and the probability of tumor development in rats this relationship is nonlinear imprecise and based on only six data points in the present case three common dose response models are fitted using nonlinear fuzzy regression techniques with the six data points regression parameters have been obtained by minimizing a nonlinear objective function under a set of specific nonlinear constraints the methodology is relatively simple and offers a viable alternative to statistical regression techniques

fuzzy linear regression using rank transform method in regression analysis the rank transform rt method is known to be neither dependent on the shape of the error distribution nor sensitive to outliers in this paper we construct a so called level fuzzy regression model based on the resolution identity theorem and apply rt method to this model fuzzy regression models with crisp input fuzzy output and fuzzy input fuzzy output are investigated to show the effectiveness of the proposed method to compare its effectiveness with existing methods we introduce a new performance measure in addition we propose a method to obtain a predicted output with respect to a specific target value and show that our model is more robust compared with other methods when the data contain some outliers

fuzzy nonlinear regression analysis using a random weight network modeling a fuzzy in fuzzy out system where both inputs and outputs are uncertain is of practical and theoretical importance fuzzy nonlinear regression fnr is one of the approaches used most widely to model such systems in this study we propose the use of a random weight network rwn to develop a fnr model called fnrrwn where both the inputs and outputs are triangular fuzzy numbers unlike existing fnr models based on back propagation bp and radial basis function rbf networks fnrrwn does not require iterative adjustment of the network weights and biases instead the input layer weights and hidden layer biases of fnrrwn are selected randomly the output layer weights for fnrrwn are calculated analytically based on a derived updating rule which aims to minimize the integrated squared error between cut sets that correspond to the predicted fuzzy outputs and target fuzzy outputs respectively in fnrrwn the integrated squared error is solved approximately by riemann integral theory the experimental results show that the proposed fnrrwn method can effectively approximate a fuzzy in fuzzy out system fnrrwn obtains better prediction accuracy in a lower computational time compared with existing fnr models based on bp and rbf networks

the annihilator of fuzzy subgroups abstract we implement the notion of annihilator into fuzzy subgroups of an abelian group there are different uses for the term annihilator in algebraic contexts that have been used fuzzy systems in this paper we refer to another type of annihilator which is essential in classical duality theory and extends the widely applied notion of orthogonal complement in euclidean spaces we find that in the natural algebraic duality of a group a fuzzy subgroup can be recovered after taking the inverse annihilator of its annihilator we also study the behavior of annihilators with respect to unions and intersections some illustrative examples of annihilators of fuzzy subgroups are shown both with finite and infinite rank

asymptotic properties of least squares estimation with fuzzy observations this paper deals with the asymptotic properties of the least squares estimators for fuzzy linear regression models with fuzzy triangular input output and random error terms the asymptotic normality and strong consistency of the fuzzy least squares estimator flse are investigated a confidence region based on a class of flses is proposed the asymptotic relative efficiency of flses with respect to the crisp least squares estimators is also provided and a numerical example is given some simulation results are also presented to illustrate the behavior of flses

robust fuzzy regression analysis in this paper we propose a robust fuzzy linear regression model based on the least median squares weighted least squares lms wls estimation procedure the proposed model is general enough to deal with data contaminated by outliers due to measurement errors or extracted from highly skewed or heavy tailed distributions we also define suitable goodness of fit indices useful to evaluate the performances of the proposed model the effectiveness of our model in reducing the outliers influence is shown by using applicative examples based both on simulated and real data and by a simulation study

dual possibilistic regression analysis using support vector networks abstract this study proposes a novel and efficient dual regression model for possibilistic regression analysis that incorporates the principles of support vector machine svm theory the dual regression model which comprises an upper model and a lower model approximates the observed fuzzy phenomena from the outside and inside directions respectively such that the inclusion relationship between those two models holds the proposed dual regression model better explains the inherent vagueness that exists in a given dataset it provides the outer and inner bounds for the estimated vagueness region and allows an estimation of the degree of confidence in the predicted fuzzy output using the principles for a twin support vector machine tsvm the upper and lower models are estimated by solving two smaller svm type quadratic programming problems qpps instead of a single larger qpp this strategy significantly increases the learning speed for the proposed algorithm the structural risk minimization principle of svm makes the proposed method to yield better generalization ability the kernel function method offers a model independent framework for the proposed dual regression model this paper focuses on the class of radial kernels which enables the proposed method to conquer the problem of increasing spreads the radial kernel also gives the proposed method a unified framework that allows both crisp and fuzzy inputs the experimental results verify the effectiveness and efficiency of the proposed method in comparison with previous svm based dual regression model the proposed approach significantly improves the sparsity prediction speed and training speed

identification of possibilistic linear systems by quadratic membership functions of fuzzy parameters abstract we have already discussed several models of the possibilities linear regression analysis under the assumption that probabilitioes parameters in the models are non interactive i e the possibilistic distribution of parameters is defined by minimum operators in this paper quadratic membership functions as defined by a celmi s are considered to propose an identification method of interactive fuzzy parameters in possibilistic linear systems our method can be reduced to linear programming so that it is very easy to obtain the posssibilistic distribution of parameters

ridge estimation for regression models with crisp inputs and gaussian fuzzy output this paper deals with ridge estimation of fuzzy multiple linear and nonlinear regression models with crisp inputs and gaussian fuzzy output using ridge regression learning algorithm in the lagrangian dual space we describe a ridge estimation of fuzzy multiple linear regression model of xu and li fuzzy sets and systems 119 2001 215 it allows us to perform nonlinear regression for xu and li s model by constructing a fuzzy linear regression function in a high dimensional feature space experimental results are then presented which indicate the performance of this algorithm

solving elliptic finite element systems in near linear time with support preconditioners we consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems our main result is that these linear systems which are symmetric and positive semidefinite are well approximated by symmetric diagonally dominant matrices our framework for defining matrix approximation is support theory significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case and in particular it is known that such systems can be solved with iterative methods in nearly linear time thus our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time we show that the support number bounds which control the number of iterations in the preconditioned iterative solver depend on mesh quality measures but not on the problem size or shape of the domain

fuzzy linear regression models for assessing risks of cumulative trauma disorders abstract cumulative trauma disorders ctds are a major problem in industry and today s office environments especially for keyboard users since comprehensive knowledge about individual or a combination of risk factors contributing to ctds of the hand and forearm is still lacking developing methodologies to quantify relationships of risk factors is greatly needed the purpose of this research was to build fuzzy linear regression models to reveal the relationship of ctd risk factors to predict the injuries and to evaluate risk levels of individuals twenty seven keyboard users twenty two for model building and five for model validation and three ctd experts participated in the model building four fuzzy models were built corresponding to four risk categories and a final fuzzy linear model was established using ahp pairwise comparisons multicolinearity effects were addressed and a partial standard deviation scaling method was used to eliminate the effects from a methodological point of view fuzzy linear regression models provide useful insight into risk factors ctd relationship

additions of interactive fuzzy numbers this paper provides an account of an approach to modeling unknown data by means of fuzzy set theory and addresses the problem of deriving the uncertainty on a sum of variables whose values lie within fuzzy intervals the first part is an extensive presentation of the theoretical background of the approach the extension principle is stated in terms of possibility of an event the concept of variable interaction is investigated at length section ii gives new results regarding the effective practical computation of additions of fuzzy numbers its originality lies in the introduction of interaction which enables to control the growth of uncertainty in calculations moreover the problem of computing mathematical expectations with fuzzy probabilities is solved the results derived in this paper can easily be used in decision problems where values of parameters or decision variables are not yet precisely fixed or assessed typical applications could be multicriteria optimization and decision making under uncertainty where fuzzy expected utilities can be obtained out of uncompletely assessed probabilities more generally fuzzy arithmetic can be an important tool for sophisticated computationally tractable sensitivity analysis in systems modeling computer aided design and operations research

robust support vector regression for uncertain input and output data in this paper a robust support vector regression rsvr method with uncertain input and output data is studied first the data uncertainties are investigated under a stochastic framework and two linear robust formulations are derived linear formulations robust to ellipsoidal uncertainties are also considered from a geometric perspective second kernelized rsvr formulations are established for nonlinear regression problems both linear and nonlinear formulations are converted to second order cone programming problems which can be solved efficiently by the interior point method simulation demonstrates that the proposed method outperforms existing rsvrs in the presence of both input and output data uncertainties

a dantzig wolfe like variant of karmarkar s interior point linear programming algorithm we show that a variant of karmarkar s projective algorithm for linear programming can be viewed as following the approach of dantzig wolfe decomposition at each iteration the current primal feasible solution generates prices which are used to form a simple subproblem the solution to the subproblem is then incorporated into the current feasible solution with a suitable choice of stepsize a constant reduction in potential function is achieved at each iteration we also use our analysis to motivate a new primal simplex pivot rule that is closely related to rules used by e klotz and l schrage

an orderwise polynomial regression procedure for fuzzy data a procedure for polynomial fit with fuzzy data is presented as in real case studies there is often dependence between modes and spreads we propose a regression polynomial model capable to take into account possible interactions between the estimated spreads and modes this method finds applications in several fields such as reliability quality control psychometrics marketing image processing etc an application to a software reliability problem is also presented

block jacobi two stage method with gauss sidel inner iterations for fuzzy system of linear equations in this paper we study block two stage iterative method for finding the approximate solution of fuzzy system of linear equation fsle with convergence conditions the application of this method is solving large linear system on parallel computer algorithm is illustrated by solving some numerical examples

on a class of fuzzy c numbers clustering procedures for fuzzy data this paper describes a class of fuzzy clustering procedures for fuzzy data most fuzzy clustering techniques are designed for handling crisp data with their class memberships using the idea of fuzzy set theory here we derive new types of fuzzy clustering procedures in dealing with fuzzy data these procedures are called fuzzy c numbers fcn clusterings specially we construct these fcns for u type triangular trapezoidal and normal fuzzy numbers k jwoy s cluster analysis fuzzy clustering fuzzy data analysis fuzzy numbers fuzzy intervals fuzzy c means clustering fuzzy c numbers clustering

a set arithmetic based linear regression model for modelling interval valued responses through real valued variables a new linear regression model for an interval valued response and a real valued explanatory variable is presented the approach is based on the interval arithmetic comparisons with previous methods are discussed the new linear model is theoretically analyzed and the regression parameters are estimated some properties of the regression estimators are investigated finally the performance of the procedure is illustrated using both a real life application and simulation studies

extended fuzzy regression models using regularization method regularization approach to regression can be easily found in statistics and information science literatures the technique of regularization was introduced as a way of controlling the smoothness properties of regression function the main purpose of this paper is to extend the models of diamond inform sci 46 1988 141 to fuzzy multiple linear regression models and to extend the model f3 with numerical input of diamond inform sci 46 1988 141 to fuzzy nonlinear model here all extensions are based on regularization technique

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

an enhanced fuzzy linear regression model with more flexible spreads one of the deficiencies of previous fuzzy linear regression models is that with the increase of the magnitudes of independent variables the spreads of estimated fuzzy dependent variables are increasing even though the spreads of observed dependent variables actually decrease or remain unchanged some solutions have been proposed to solve this spreads increasing problem however those solutions still cannot model a decreasing trend in the spreads of the observed dependent variables as the magnitudes of the independent variables increase in this paper we propose an enhanced fuzzy linear regression model model flr f s in which the spreads of the estimated dependent variables are able to fit the spreads of the observed dependent variables no matter the spreads of the observed dependent variables are increased decreased or unchanged as the magnitudes and spreads of the independent variables change four numerical examples are used to demonstrate the effectiveness of model flr f s

on scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a diagonal matrix with positive diagonal elements the weighted pseudoinverse defined by x d x t dx 1 x t d and the associated oblique projection p d xx d arise in many applications in this paper we show that the norms of both matrices are bounded by numbers that are independent of d

exponential possibility regression analysis by identification method of possibilistic coefficients abstract this paper introduces exponential possibility regression analysis by the proposed identification method to identify the exponential possibility distribution first we solve all possible different sets of the input output linear systems from the given data and obtain the corresponding coefficient vectors then we select minimum number of coefficient vectors which represent all the given data using the selected coefficient vectors we apply the identification method to determine a possibility distribution of coefficients in the fuzzy linear system as a criterion to select minimum coefficient vectors we use the inner product by the proposed approach the smaller area of possibility distribution can be obtained this is the main characteristic of our proposed regression comparing with the other regression analyses

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

fuzzy linear least squares for the identification of possibilistic regression models abstract the stochastic approach to regression e g the least squares method is a commonly used technique in a wide range of disciplines as it offers an intuitive and well founded solution to the problem of fitting a set of parameters to existing data specifically in the case where the parameters of meta models are identified the linear least squares formulation receives wide appreciation among engineers due to its simple yet powerful formulation and solution usually this approach provides a reasonable fit depending largely on the quality of the model that is employed however it has two main drawbacks from a statistical point of view the premises justifying its use cannot always be assumed in good faith and most importantly the resulting model will almost certainly fail at correctly predicting the output without falling victim to the effect of over fitting the presented fuzzy regression algorithm originates from a set of different axioms for the solution of the linear least squares problem it is able to provide the worst case optimal parameter variation necessary to cover all of the crisp data points while encoding meaningful information in the membership function this is accomplished by means of a sensible fuzzification procedure of the crisp data points and the application of an exact inverse fuzzy arithmetic for estimating the fuzzy valued parameters of the model in question

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

a unified approach to asymptotic behaviors for the autoregressive model with fuzzy data we propose a unified estimator for the autoregressive model with fuzzy input output variables based on the least squares method the least squares estimation is investigated in presence of a unified r distance defined on the space of fuzzy numbers we investigate asymptotic properties of the unified estimator under some simple conditions as well as a generalization which reduces to the asymptotic properties under those distances when the distances are the special cases of r distance some simulation studies are included to compare the asymptotic properties of estimators formed under several distances being the special cases of r distance

fuzzy neural networks a survey abstract in this paper a fuzzy neural network will be a layered feedforward neural net that has fuzzy signals and or fuzzy weights we survey recent results on learning algorithms and applications for fuzzy neural networks