on the minimum of the christoffel function abstract we consider the christoffel function n d associated with a measure d on a compact interval and the question where the minimum of the christoffel function is attained sufficient conditions on the coefficients appearing in the continued fraction expansion of the stieltjes transform are obtained for the n th christoffel function to attain its minimum at the boundary points of the underlying interval in contrary to several monotonicity results for christoffel functions in the literature our approach does not require the absolute continuity of the measure with respect to the lebesgue measure

m estimates of location for the robust central tendency of fuzzy data the aumann type mean has been shown to possess valuable properties as a measure of the location or central tendency of fuzzy data associated with a random experiment however concerning robustness its behavior is not appropriate the aumann type mean is highly affected by slight changes in the fuzzy data or when outliers arise in the sample robust estimators of location on the other hand avoid such adverse effects for this purpose this paper considers the m estimation approach and discusses conditions under which this alternative yields valid fuzzy valued m estimators the resulting m estimators are applied to a real life example finally some simulation studies show empirically the suitability of the introduced estimators

a note on trapezoidal approximations of fuzzy numbers in this paper we propose some generalized and new properties of the trapezoidal approximations of fuzzy numbers first we claim that the trapezoidal approximation operator is linear with respect to the operations of fuzzy numbers second we show a new property concerning the centroid points of fuzzy numbers furthermore a comment for computing the trapezoidal approximations is provided finally some properties about the distance between a fuzzy number and its trapezoidal approximation are proved

optimal discrimination designs for multifactor experiments in this paper efficient designs are determined when anderson s procedure is applied in order to identify the degree of a multivariate polynomial regression model it is shown that the optimal designs are very closely related to model robust designs which maximize a weighted p mean of d efficiencies as a consequence we obtain designs with high efficiency for model discrimination and for the statistical analysis in the identified model

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

fuzzy number approximation via shadowed sets interval and trapezoidal approximation methods for fuzzy numbers have been intensively developed and examined recently in this paper another approach to simplify fuzzy numbers called shadowed set approximation is suggested some basic properties of the proposed method is also discussed it seems that the proposed approximation might be useful for granular computing as a tool that removes excessive precision in describing imprecise phenomena

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

optimal designs for comparing curves we consider the optimal design problem for a comparison of two regression curves which is used to establish the similarity between the dose response relationships of two groups an optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions optimal design theory equivalence theorems efficiency bounds is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly the results are illustrated in several examples modeling dose response relationships it is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models in particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used non optimal designs yields a reduction of the width of the confidence band by more than 50

experimental designs for model discrimination abstract we present designs that perform well for several objectives simultaneously three different approaches are discussed to augment a given design in an optimal way to evaluate a mixture of the various criteria and to optimize one objective subject to achieving a prescribed efficiency level for the others our sample designs are for the situation of discriminating between a second and third degree polynomial fit under the d criterion and geometric mixtures of d criteria

trapezoidal approximations of fuzzy numbers revisited fuzzy number approximation by trapezoidal fuzzy numbers which preserve expected interval is discussed the previously proposed approximation operator is improved so as to always produce a well formed trapezoidal fuzzy number

on g efficiency calculation for polynomial models we study properties of the variance function of the least squares estimator for the response surface for polynomial models we identify a class of approximate designs for which their variance functions are maximized at the extreme points of the design space as an application we examine robustness properties of d optimal designs and d n r optimal designs under various polynomial model assumptions analytic formulas for the g efficiencies of these designs are derived along with their d efficiencies

trapezoidal and triangular approximations preserving the expected interval in this paper an improvement of the nearest trapezoidal approximation operator preserving the expected interval is studied which is proposed by grzegorzewski and mrowka a formula for computing the improved approximation is presented moreover the nearest triangular approximation operator preserving the expected interval is also investigated

bayesian d optimal designs on a fixed number of design points for heteroscedastic polynomial models summary we consider design issues in a polynomial regression model where the variance of the response depends on the independent variable exponentially however this dependence is not known precisely and additional parameters are required in the model our design criteria permit various subsets of the parameters to be estimated with different emphasis bayesian d optimal designs on a compact interval with the number of support points restricted to be one more than the degree of the polynomial are found analytically for a large class of priors these designs may or may not be optimal within the class of all designs depending on the prior distribution

nearest interval approximation of a fuzzy number the problem of the interval approximation of fuzzy numbers is discussed a new interval approximation operator which is the best one with respect to a certain measure of distance between fuzzy numbers is suggested

trapezoidal approximations of fuzzy numbers preserving the expected interval algorithms and properties fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is discussed algorithms for calculating the proper approximations are proposed and some properties of the approximation operators are discussed it is shown that an adequate approximation operator might be chosen through the comparisons of some characteristics of the fuzzy number like its ambiguity width its value and weighted expected value

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

optimal designs for identifying the degree of a polynomial regression if an experimenter wants to determine the degree of a polynomial regression on the basis of a sample of observations anderson showed that the following method is optimal starting with the highest specified degree the procedure is to test in sequence whether the coefficients are 0 in this paper optimal designs for anderson s procedure are determined explicitly the optimal design maximizes the minimum power of a given set of alternatives

why triangular membership functions abstract in this note we look at a certain theoretically sound motivation behind the common use of triangular and trapezoidal membership functions the studies are completed within a conceptual framework of fuzzy modelling whose structure comprises of input and output interfaces linked with a single transformation module aimed at processing linguistic information it is shown that under some additional mild assumptions these triangular fuzzy sets comply with a request for a uniformly excited codebook in the case of the input interfaces and a satisfaction of a zero error reconstruction criterion being formulated for the output interface

robustly stabilizing controllers for dissipative infinite dimensional systems with collocated actuators and sensors we derive formulas for normalized doubly coprime factorizations over h for a class of nonexponentially stabilizable systems using these results we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of infinite dimensional systems s a b b d with finite rank inputs and outputs and dissipative operator a these systems are not exponentially stabilizable we give a parameterization of controllers that achieve a given robustness margin

metrics and orders in space of fuzzy numbers abstract two classes of metrics p q and p where 1 p 0 q 1 in space of fuzzy numbers are introduced then a method of ranking fuzzy numbers based on these metrics is proposed and investigated not only fuzzy numbers but also the left sided and right sided fuzzy numbers are considered applications in making decisions and statistics are suggested

optimal designs for a class of polynomials of odd or even degree prior y y1 v y d where the numbers yj correspond to the models of degree 2j 1 2j for j 1 r for a special class of priors optimal designs of a very simple structure are calculated generalizing the d1 optimal design for polynomial regression of degree 2r 1 2r the support of these designs splits up in three sets and the masses of the optimal design at the support points of every set are all equal the results are derived in a general context using the theory of canonical moments and continued fractions some applications are given to the d optimal design problem for polynomial regression with vanishing coefficients of odd or even powers

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

fuzzy entropy a more comprehensible perspective for interval shadowed sets of fuzzy sets shadowed sets proposed by pedrycz provide a three way approximation scheme for transforming the universe of a fuzzy set into three disjoint areas i e elevated area reduced area and shadow area to calculate a pair of decision making thresholds an analytic method was proposed by solving a minimization problem of the uncertainty arising from the three areas however some uncertainty will be lost in the process of constructing the shadowed set model using pedrycz s method moreover few references on how to measure the uncertainty of shadow sets exist in this article a comprehensible method for measuring the fuzzy entropy of a shadowed set i e interval fuzzy entropy is defined based on the interval fuzzy entropy a new shadowed set model namely interval shadowed sets is proposed compared with pedrycz s model the main difference is that the range of the shadow area in this model is beta alpha 0 leq beta while not 0 1 by solving a fuzzy entropy loss minimization problem a pair of optimal thresholds alpha and beta can be obtained finally the results of the instance analysis of different types of representative membership functions and many experiments show that the fuzzy entropy loss of the interval shadowed set is lower than that of the traditional shadowed set of a fuzzy set these results enrich shadowed set theory from a new perspective

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

on the nearest parametric approximation of a fuzzy number many nearest parametric approximation methods of fuzzy sets are proposed in the literature it is clear that the specific approximations may lead to the loss of information about fuzziness to overcome this problem most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation but these approximations mostly are not flexible to the decision maker s choice hence in this paper we offer a parametric fuzzy approximation method based on the decision maker s strategy as an extension of trapezoidal approximation of a fuzzy number this method comprises the selection of the form of the parametric membership function and its evaluation

the nearest approximation of a fuzzy quantity in parametric form applications of fuzzy logic and fuzzy mathematics are increasing widely all around the world thus working with fuzzy numbers are very important in many applications of fuzzy mathematics we need or it is better to work with the same fuzzy numbers in this work we approximate parametric fuzzy numbers with polynomial parametric fuzzy numbers

decision theoretic three way approximations of fuzzy sets abstract a three way three valued or three region approximation of a fuzzy set is constructed from a pair of thresholds on the fuzzy membership function an element whose membership grade equals to or is greater than is put into the positive region an element whose membership grade equals to or is less than is put into the negative region and an element whose membership grade is between and is put into the boundary region a fundamental issue is the determination and interpretation of the required pair of thresholds in the framework of shadowed sets i e an example of three way approximations of fuzzy sets pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions to gain further insights into three way approximations of fuzzy sets we introduce an alternative decision theoretic formulation in which the required thresholds are computed by minimizing decision cost

approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval the problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well known metric which preserves the expected interval of the fuzzy number is completely solved the previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number properties of this new trapezoidal approximation operator are studied

the mean value of a fuzzy number abstract the notion of expectation is defined for fuzzy numbers of intervals viewing them as consonant random sets this expectation is interval valued and remains additive in the sense of fuzzy interval addition as a by product some light is shed on the debate between probability and possibility theory and the problem of how to carry measure theoretic notions into the field of possibility theory

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

on a canonical representation of fuzzy numbers abstract fuzzy numbers and more generally linguistic values are approximate assessments given by experts and accepted by decision makers when obtaining more accurate values is impossible or unnecessary to simplify the task of representing and handling fuzzy numbers several authors have introduced real indices in order to capture the information contained in a fuzzy number in this paper we propose two parameters value and ambiguity for this purpose we use these parameters to obtain canonical representations and to deal with fuzzy numbers in decision making problems several examples illustrate these ideas

the expected value of a fuzzy number abstract the notions of expected interval and expected value of fuzzy number are introduced the expected interval is defined as the expected value of an interval random set generated by the fuzzy number and the expected value of this number is defined as the centre of the expected interval the linearity of the expected interval is studied

a general model of decision theoretic three way approximations of fuzzy sets based on a heuristic algorithm abstract a model of decision theoretic three way approximations of fuzzy sets that can calculate a pair of interpretable thresholds by combining shadowed sets with three way decisions was established by deng in that model the significant contribution is that the value 0 5 which denotes a membership grade with the highest uncertainty is used to replace the uncertain region with the interval 0 1 in shadowed sets from the principle of the minimum decision cost although the lowest overall cost can be achieved precisely in some data distributions based on the value 0 5 there is a difference between the obtained overall cost and the least cost in some cases therefore in this paper based on deng s model the concept of a general three way approximation of a fuzzy set is proposed to replace 0 5 with a variable value c 0

on the interval approximation of a fuzzy number the notion of an approximation interval of a fuzzy number is introduced it is the interval which fulfills two conditions in the first its width is equal to the width of a fuzzy number being approximated in the second the hamming distance between this interval and the approximated number is minimal the introduced notion is compared with the notion of expected interval of a fuzzy number known in literature