on the variance of fuzzy random variables abstract this paper deals with an expectation and a real valued variance of fuzzy random variables the expectation and the variance of a fuzzy random variable is characterized by frechet s principle in a metric space we study properties of the variance of a fuzzy random variable and compare it with the common variance of real valued random variables using the expectation and the variance of fuzzy random variables we consider a linear regression problem and limit theorems

a new gradient descent approach for local learning of fuzzy neural models the majority of reported learning methods for takagi sugeno kang tsk fuzzy neural models to date mainly focus on improvement of their accuracy however one of the key design requirements in building an interpretable fuzzy model is that each obtained rule consequent must match well with the system local behavior when all the rules are aggregated to produce the overall system output this is one of the distinctive characteristics from black box models such as neural networks therefore how to find a desirable set of fuzzy partitions and hence identify the corresponding consequent models which can be directly explained in terms of system behavior presents a critical step in fuzzy neural modeling in this paper a new learning approach considering both nonlinear parameters in the rule premises and linear parameters in the rule consequents is proposed unlike the conventional two stage optimization procedure widely practiced in the field where the two sets of parameters are optimized separately the consequent parameters are transformed into a dependent set on the premise parameters thereby enabling the introduction of a new integrated gradient descent learning approach thus a new jacobian matrix is proposed and efficiently computed to achieve a more accurate approximation of the cost function by using the second order levenberg marquardt optimization method several other interpretability issues regarding the fuzzy neural model are also discussed and integrated into this new learning approach numerical examples are presented to illustrate the resultant structure of the fuzzy neural models and the effectiveness of the proposed new algorithm and compared with the results from some well known methods

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

generalized linear array models with applications to multidimensional smoothing summary data with an array structure are common in statistics and the design or regression matrix for analysis of such data can often be written as a kronecker product factorial designs contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models in such a setting we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array we show how this arithmetic leads to low storage high speed computation in the scoring algorithm of the generalized linear model we refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays we illustrate our procedure with the analysis of three data sets mortality data indexed by age at death and year of death spatially varying microarray background data and disease incidence data indexed by age at death year of death and month of death

fuzzy linear regression with fuzzy intervals abstract in this paper we introduce a new class of fuzzy linear regression models based on tanaka s approach unlike in the tanaka model here all training data influence the estimated interval the crisp lp of tanaka s approach is generalized to a fuzzy linear programming problem with the proposed model an adaptation of the fuzzy regression equation to new data becomes possible

general fuzzy least squares abstract a model for least squares fitting of fuzzy valued data is described a previous process obtained for triangular fuzzy numbers is generalized and improved to include all fuzzy numbers represented by single maxima piecewise continuous functions with compact support the new model is compared with the previous procedure and its applicability is demonstrated

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

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

the hybrid fuzzy least squares regression approach to modeling manufacturing processes uncertainty in manufacturing processes is caused both by randomness as in material properties and by fuzziness as in the inexact knowledge previous research has seldom considered these two types of uncertainty when modeling manufacturing processes in this paper a hybrid fuzzy least squares regression hflsr approach to modeling manufacturing processes which does take into consideration these two types of uncertainty is proposed and described and a new form of weighted fuzzy arithmetic is introduced to develop the hybrid fuzzy least squares regression method the proposed hflsr approach not only features the capability of dealing with the two types of uncertainty but also addresses the consideration of replication of responses in experiments to investigate the effectiveness of the proposed approach to process modeling it was applied to the modeling solder paste dispensing process modeling results were compared with those based on statistical regression and fuzzy linear regression it was found that the accuracy of prediction based on the hflsr is slightly better than that based on statistical regression and much better than that based on the peters fuzzy regression

possibilistic linear systems and their application to the linear regression model abstract it is the purpose of this paper to illustrate possibilistic linear systems based on possibility measure and to formulate a linear regression analysis by possibilistic linear models linear regression by a possibilistic model is called possibilistic linear regression this is a new interpretation of fuzzy linear regression and also includes a new method by which interval analysis can be done in fuzzy numbers

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

a linear regression model using triangular fuzzy number coefficients fuzzy regression analysis using fuzzy linear models with symmetric triangular fuzzy number coefficients has been formulated earlier the goal of this regression is to find the coefficients of a proposed model for all given input output data sets in this paper we extend the results of a fuzzy linear regression model that uses symmetric triangular coefficient to one with non symmetric fuzzy triangular coefficients this work eradicates the inflexibility of existing fuzzy linear regression models

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

a generalized fuzzy weighted least squares regression abstract a fairly general fuzzy regression technique is proposed based on the least squares approach the main concept is to estimate the modal value and the spreads separately in order to do this the interactions between the modal value and the spreads are first analyzed in detail the advantages of this new fuzzy weighted least squares regression fwlsr approach are 1 the estimation of both non interactive and interactive fuzzy parameters can be performed by the same method 2 the decision makers confidence in the gathered data and in the established model can be incorporated into the process and 3 suspicious outliers or fuzzy outliers that is data points that are obviously and suspiciously lying outside the usual range can be treated and their effects can be reduced a numerical example is provided to show that the proposed method can be an effective computational tool in fuzzy regression analysis

a tabu harmony search based approach to fuzzy linear regression we propose an unconstrained global continuous optimization method based on tabu search and harmony search to support the design of fuzzy linear regression flr models tabu and harmony search strategies are used for diversification and intensification of flr respectively the proposed approach offers the flexibility to use any kind of an objective function based on client s requirements or requests and the nature of the dataset and then attains its minimum error moreover we elaborate on the error produced by this method and compare it with the errors resulting from the other known estimation methods to study the performance of the method three categories of datasets are considered numeric inputs symmetric fuzzy outputs symmetric fuzzy inputs symmetric fuzzy outputs and numeric inputs asymmetric fuzzy outputs through a series of experiments we demonstrate that in terms of the produced error with different model fitting measurements the proposed method outperforms or is pareto equivalent to the existing methods reported in the literature

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

fuzzy regression analysis by support vector learning approach support vector machines svms have been very successful in pattern classification and function approximation problems for crisp data in this paper we incorporate the concept of fuzzy set theory into the support vector regression machine the parameters to be estimated in the svm regression such as the components within the weight vector and the bias term are set to be the fuzzy numbers this integration preserves the benefits of svm regression model and fuzzy regression model and has been attempted to treat fuzzy nonlinear regression analysis in contrast to previous fuzzy nonlinear regression models the proposed algorithm is a model free method in the sense that we do not have to assume the underlying model function by using different kernel functions we can construct different learning machines with arbitrary types of nonlinear regression functions moreover the proposed method can achieve automatic accuracy control in the fuzzy regression analysis task the upper bound on number of errors is controlled by the user predefined parameters experimental results are then presented that indicate the performance of the proposed approach

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

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

finite sample confidence envelopes for shape restricted densities a conservative finite sample simultaneous confidence envelope for a density can be found by solving a finite set of finite dimensional linear programming problems if the density is known to be monotonic or to have at most k modes relative to a positive weight function the dimension of the problems is at most n log n 1 3 where n is the number of observations the linear programs find densities attaining the largest and smallest values at a point among cumulative distribution functions in a confidence set defined using the assumed shape restriction and differences between the empirical cumulative distribution function evaluated at a subset of the observed points bounds at any finite set of points can be extrapolated conservatively using the shape restriction the optima are attained by densities piecewise proportional to the weight function with discontinuities at a subset of the observations and at most five other points if the weight function is constant and the density satisfies a local lipschitz condition with exponent the width of the bounds converges to zero at the optimal rate log n n 1 2 outside every neighborhood of the set of modes if a bandwidth parameter is chosen correctly the integrated width of the bounds converges at the same rate on intervals where the density satisfies a lipschitz condition if the intervals are strictly within the support of the density the approach also gives algorithms to compute confidence intervals for the support of monotonic densities and for the mode of unimodal densities lower confidence intervals on the number of modes of distribution and conservative tests of the hypothesis of k modality we use the method to compute confidence bounds for the probability density of aftershocks of the 1984 morgan hill ca earthquake assuming aftershock times are an inhomogeneous poisson point process with decreasing intensity

extended fuzzy linear models and least squares estimates least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads 1 the idea is to introduce best fit difference models which exploit both the hukuhara difference and the l2 metric distance the fuzzy models use lr fuzzy numbers fitted models are compared by using the coefficient of determination in a similar way to its use in classical statistical least squares fitting the non lr fuzzy case is also considered

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

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

set valued approximations and newton s methods to rm under the assumption of semi smoothness of the mapping we prove that the approximation can be obtained through the clarke generalized jacobian ioffe ralph generalized jacobian b subdifferential and their approximations as an application we propose a generalized newton s method based on the point based set valued approximation for solving nonsmooth equations we show that the proposed method converges locally superlinearly without the assumption of semi smoothness finally we include some well known generalized newton s methods in our method and consolidate the convergence results of these methods

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

optimal adaptive estimation of a quadratic functional minimax mean squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes in contrast to many nonparametric function estimation problems there are both regular and irregular cases in the regular cases the minimax mean squared error converges at a rate proportional to the inverse of the sample size whereas in the irregular case much slower rates are the rule we investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known it is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case a lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case on the other hand we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case moreover we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small

penalized clustering of large scale functional data with multiple covariates in this article we propose a penalized clustering method for large scale data with multiple covariates through a functional data approach in our proposed method responses and covariates are linked together through nonparametric multivariate functions fixed effects which have great flexibility in modeling various function features such as jump points branching and periodicity functional anova is used to further decompose multivariate functions in a reproducing kernel hilbert space and provide associated notions of main effect and interaction parsimonious random effects are used to capture various correlation structures the mixed effects models are nested under a general mixture model in which the heterogeneity of functional data is characterized we propose a penalized henderson s likelihood approach for model fitting and design a rejection controlled em algorithm for the estimation our method selects smoothing parameters through generalized cross validation furthermore bayesian confidence in

interval regression analysis using quadratic loss support vector machine support vector machines svms have been very successful in pattern recognition and function estimation problems for crisp data this paper proposes a new method to evaluate interval linear and nonlinear regression models combining the possibility and necessity estimation formulation with the principle of quadratic loss svm this version of svm utilizes quadratic loss function unlike the traditional svm for data sets with crisp inputs and interval outputs the possibility and necessity models have been recently utilized which are based on quadratic programming approach giving more diverse spread coefficients than a linear programming one the quadratic loss svm also uses quadratic programming approach whose another advantage in interval regression analysis is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression however this is not a computationally expensive way the quadratic loss svm allows us to perform interval nonlinear regression analysis by constructing an interval linear regression function in a high dimensional feature space the proposed algorithm is a very attractive approach to modeling nonlinear interval data and is model free method in the sense that we do not have to assume the underlying model function for interval nonlinear regression model with crisp inputs and interval output experimental results are then presented which indicate the performance of this algorithm

sufficient conditions for the generalized problem of bolza this paper presents sufficient conditions for strong local optimality in the generalized problem of bolza these conditions represent a unification in the sense that they can be applied to both the calculus of variations and to optimal control problems as well as problems with nonsmooth data also this paper brings to light a new point of view concerning the jacobi condition in the classical calculus of variations showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original hamiltonian to a locally concave convex hamiltonian

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

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

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

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

linear regression with random fuzzy variables extended classical estimates best linear estimates least squares estimates this paper deals with problems which arise if for well justified statistical models like linear regression only fuzzy data are available three approaches are discussed the first one is an application of zadeh s extension principle to optimal classical estimators here the result is that they in general do not keep their optimality properties the second one is the attempt to develop a certain kind of linear theory for fuzzy random variables w r t extended addition and scalar multiplication the main problem in this connection is that fuzzy sets with these extended operations do not constitute a linear space therefore in the third approach the least squares approximation principle for fuzzy data is investigated which leads to most acceptable results

efficient computation of smoothing splines via adaptive basis sampling summary smoothing splines provide flexible nonparametric regression estimators however the high computational cost of smoothing splines for large datasets has hindered their wide application in this article we develop a new method named adaptive basis sampling for efficient computation of smoothing splines in super large samples except for the univariate case where the reinsch algorithm is applicable a smoothing spline for a regression problem with sample size n can be expressed as a linear combination of n basis functions and its computational complexity is generally o n 3 we achieve a more scalable computation in the multivariate case by evaluating the smoothing spline using a smaller set of basis functions obtained by an adaptive sampling scheme that uses values of the response variable our asymptotic analysis shows that smoothing splines computed via adaptive basis sampling converge to the true function at the same rate as full basis smoothing splines using simulation studies and a large scale deep earth core mantle boundary imaging study we show that the proposed method outperforms a sampling method that does not use the values of response variables

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

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

fuzzy data analysis by possibilistic linear models abstract since fuzzy data can be regarded as distribution of possibility fuzzy data analysis by possibilistic linear models is proposed in this paper possibilistic linear systems are defined by the extension principle fuzzy parameter estimations are discussed in possibilistic linear systems and possibilistic linear models are employed for fuzzy data analysis with non fuzzy inputs and fuzzy outputs defined by fuzzy numbers the estimated possibilistic linear system can be obtained by solving a linear programming problem this approach can be regarded as fuzzy interval analysis

adaptive confidence bands for nonparametric regression functions this article proposes a new formulation for the construction of adaptive confidence bands cbs in nonparametric function estimation problems cbs which have size that adapts to the smoothness of the function while guaranteeing that both the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band are small it is shown that the bands adapt over a maximum range of lipschitz classes the adaptive cb can be easily implemented in standard statistical software with wavelet support we investigate the numerical performance of the procedure using both simulated and real datasets the numerical results agree well with the theoretical analysis the procedure can be easily modified and used for other nonparametric function estimation models supplementary materials for this article are available online

multiobjective fuzzy linear regression analysis for fuzzy input output data abstract fuzzy linear regression models where both input data and output data are fuzzy numbers are introduced by using three indices for equalities between fuzzy numbers by considering the conflict between the fuzzy threshold for the three indices and the fuzziness of the fuzzy linear regression model three types of multiobjective programming problems for obtaining fuzzy linear regression models are formulated corresponding to the three indices then a linear programming based interactive decision making method to derive the satisficing solution of the decision maker for the formulated multiobjective programming problems is developed a numerical example demonstrates the appropriateness and efficiency of the proposed method

a probabilistic approach to pattern matching in the continuous domain the goal of this paper is to solve the following basic problem given discrete noisy samples from a continuous signal compute the probability distribution of its distance from a fixed template as opposed to the typical restoration problem which considers a single optimal signal the computation of the entire probability distribution necessitates integrating over the entire signal space to achieve this we apply path integration techniques the problem is studied in one and two dimensions and an accurate solution as well as an efficient approximation scheme are provided

multiple regression with fuzzy data in this paper we propose an iterative algorithm for multiple regression with fuzzy variables while using the standard least squares criterion as a performance index we pose the regression problem as a gradient descent optimisation the separation of the evaluation of the gradient and the update of the regression variables makes it possible to avoid undue complication of analytical formulae for multiple regression with fuzzy data the origins of fuzzy input data are traced back to the fundamental concept of information granulation and an example fcm based granulation method is proposed and illustrated by some numerical examples the proposed multiple regression algorithm is applied to one three and nine dimensional synthetic data sets as well as the 13 dimensional boston housing dataset from the machine learning repository the algorithm s performance is illustrated by the corresponding plots of convergence of regression parameters and the values of the prediction error of the resulting regression model general comments on the numerical complexity of the proposed algorithm are also provided

confidence bands in density estimation given a sample from some unknown continuous density f we construct adaptive confidence bands that are honest for all densities in a generic subset of the union of t holder balls 0 t r where r is a fixed but arbitrary integer the exceptional nongeneric set of densities for which our results do not hold is shown to be nowhere dense in the relevant holder norm topologies in the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators which are of independent interest

adaptive nonparametric confidence sets we construct honest confidence regions for a hilbert space valued parameter in various statistical models the confidence sets can be centered at arbitrary adaptive estimators and have diameter which adapts optimally to a given selection of models the latter adaptation is necessarily limited in scope we review the notion of adaptive confidence regions and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation applications include the finite normal mean model the white noise model density estimation and regression with random design institute of mathematical statistics 2006

nonparametric regression in exponential families most results in nonparametric regression theory are developed only for the case of additive noise in such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive in this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function which include for example poisson regression binomial regression and gamma regression we propose a unified approach of using a mean matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic gaussian regression problem then in principle any good nonparametric gaussian regression procedure can be applied to the transformed data to illustrate our general methodology in this paper we use wavelet block thresholding to construct the final estimators of the regression function the procedures are easily implementable both theoretical and numerical properties of the estimators are investigated the estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near optimal asymptotic performance over a wide range of besov spaces the estimators also perform well numerically

an approximate approach for ranking fuzzy numbers based on left and right dominance this study presents an approximate approach for ranking fuzzy numbers based on the left and right dominance the proposed approach only requires a few left and right spreads at some levels of fuzzy numbers to determine the respective dominance of one fuzzy number over the other the total dominance is then determined by combining the left and right dominance based on a decision maker s optimistic perspectives such a dominance is useful in ranking the fuzzy numbers when membership functions cannot be acquired the approach proposed herein is relatively simple in terms of computational efforts and is efficient when ranking a large quantity of fuzzy numbers by using a few left and right spreads two groups of examples demonstrate the accuracy and applicability of the proposed approach

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

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

a note on smoothness priors and nonlinear regression abstract a generalization of a nonparametric estimator proposed by shiller 1984 is considered the estimator is shown to be directly related to penalized least squares estimation and spline smoothing some simplifications concerning the computation of shiller s estimator and corresponding posterior covariances are indicated

evaluation of fuzzy linear regression models abstract fuzzy linear regression provides means for tackling regression problems lacking a significant amount of data for determining regression models and with vague relationships between the dependent variable and independent variables in the paper a two stage construction of a linear regression model is proposed using an enhancement of a minimal vagueness criterion already discussed in fuzzy regression analysis two different experiments are set up and used for evaluating both models the choice of the fuzzy regression model should be made in the context of a practical application

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

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

adaptive confidence balls adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level the confidence balls are centered on adaptive estimators based on special local block thresholding rules the radius is derived from an analysis of the loss of this adaptive estimator in addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of r n and expected squared radius adapting over a maximum range of besov bodies

adaptive confidence bands we show that there do not exist adaptive confidence bands for curve estimation except under very restrictive assumptions we propose instead to construct adaptive bands that cover a surrogate function f which is close to but simpler than f the surrogate captures the significant features in f we establish lower bounds on the width for any confidence band for f and construct a procedure that comes within a small constant factor of attaining the lower bound for finite samples

on p values for smooth components of an extended generalized additive model the problem of testing smooth components of an extended generalized additive model for equality to zero is considered confidence intervals for such components exhibit good across the function coverage probabilities if based on the approximate result hat f i n f i v f i i where f is the vector of evaluated values for the smooth component of interest and v f is the covariance matrix for f according to the bayesian view of the smoothing process based on this result a wald type test of f 0 is proposed it is shown that care must be taken in selecting the rank used in the test statistic the method complements previous work by extending applicability beyond the gaussian case while considering tests of zero effect rather than testing the parametric hypothesis given by the null space of the component s smoothing penalty the proposed p values are routine and efficient to compute from a fitted model without requiring extra model fits or null distribution simulation

properties of certain fuzzy linear regression methods abstract fuzzy linear regression was originally introduced by tanaka uejima and asai ieee trans systems man cybern 12 1982 903 907 in subsequent years several different approaches to fuzzy linear regression have been proposed the purpose of this paper is to review and examine some of these formulations to discuss their strengths and weaknesses relative to each other and to suggest possible improvements in addition we compare and contrast these methods to the method of ordinary least squares regression

constructing a sequence of discrete hessian matrices of an sc 1 function uniformly convergent to the generalized hessian matrix we construct a uniform approximation for generalized hessian matrix of an sc 1 function using the discrete gradient and the extended second order derivative we define the discrete hessian matrix we construct a sequence of sets where each set is composed of discrete hessian matrices we first show some new properties of sc 1 functions then we prove that for sc 1 functions the sequence of the set of discrete hessian matrices is uniformly convergent to the generalized hessian matrix

asymptotic equivalence and adaptive estimation for robust nonparametric regression asymptotic equivalence theory developed in the literature so far are only for bounded loss functions this limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded in this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions the results imply that all the gaussian nonparametric regression procedures can be robustified in a unified way a key step in our equivalence argument is to bin the data and then take the median of each bin the asymptotic equivalence results have significant practical implications to illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems robust estimation of the regression function and the estimation of a quadratic functional in both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion

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

robust nonparametric estimation via wavelet median regression in this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions including those that are heavy tailed and may not even possess variances or means our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point a key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable this median coupling inequality may be of independent interest

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

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

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

generalized hessian for c 1 1 functions in infinite dimensional normed spaces the subject of this paper is the systematic study of second order notions concerning differentiable functions with lipschitz derivative the results and notions are motivated by recent papers of cominetti correa and hiriart urruty the first goal of this paper is the comparison of several known second order directional derivatives the second goal is the introduction of a generalized hessian which is a set of certain symmetric bilinear forms the relation of this generalized hessian to other existing second order derivatives is also described

a signal extraction approach to modeling hormone time series with pulses and a changing baseline abstract hormones serve as regulating signals for many biological processes in recent years it was determined that many hormones are secreted in a pulsatile manner and that the pulsatile secretion pattern in addition to the absolute concentration level is important in regulating biological processes consequently it is necessary to characterize the latent secretion patterns from measurements of concentration levels the characterization is complicated by the presence of a biological circadian rhythm when hormone concentrations are plotted over time the resultant time series usually exhibits occasional short rises superimposed on a slowly changing baseline this is a result of a mixture of pulsatile secretions and a circadian rhythm in this article we present a signal extraction approach to model simultaneously a slowly changing component and a pulsatile component of a time series a smoothing spline is used to model the baseline and a multiprocess dynamic linear model is used to model the pulsati