performance comparison of numerical inversion methods for laplace and hankel integral transforms in engineering problems abstract different methods for the numerical evaluations of the inverse laplace and inverse of joint laplace hankel integral transforms are applied to solve a wide range of initial boundary value problems often arising in engineering and applied mathematics the aim of the paper is to present a performance comparison among different numerical methods when they are applied to transformed functions related to actual engineering problems found in the literature most of our selected test functions have been found in the solution of boundary value problems of applied mechanics such as those related to transient responses of isotropic and transversely isotropic half space to concentrated impulse or those related to viscoelastic wave motion in layered media these classes of test functions are frequently encountered in similar problems such as those in boundary element or boundary integral equations theoretical seismology soil structure interaction in time domain and so on therefore their behavior with different numerical inversion algorithms could make a useful guide to a precise choice of more suitable inversion method to be used in similar problems some different methods are also investigated in detail and compared for the inversion of the joint hankel laplace transforms where more sophisticated integrand functions are encountered it is shown that durbin crump d amore fixed talbot gaver whyn rho gwr and direct integration methods have excellent performance and produce good results when applied to the same problems on the contrary gaver stehfest and piessens methods furnish results not very reliable for almost all classes of transformed functions and they seem good only for simple transformed functions particularly the performance of gwr algorithm is very good even for transformed functions with infinite number of singularities where the other methods fail in addition in case of double integral transforms only the fixed talbot durbin and weeks methods are recommended

comprehensive analysis of a new fuzzy rule interpolation method the first published result in fuzzy rule interpolation was the spl alpha cut based fuzzy rule interpolation termed as kh fuzzy rule interpolation originally devoted for complexity reduction a modified version of the kh approach has been presented by yam et al 1999 which eliminates the subnormality problem while at the same time intending to maintain the advantageous computational properties of the original method this paper presents a comprehensive analysis of the new method which includes detailed comparison with the original kh fuzzy rule interpolation method concerning the explicit functions of the methods preservation of piecewise linearity and stability the fuzziness of the conclusion with respect to the fuzziness of the observation is also investigated in comparison with several interpolation techniques all these comparisons shows that the new method preserves the advantageous properties of the kh method and alleviates its most significant disadvantage the problem of subnormality

interpolativity of at least and at most models of monotone fuzzy rule bases with multiple antecedent variables of the many desirable properties of fuzzy inference systems not all are known to co exist for instance a fuzzy inference system should be interpolative i e if a fuzzy set equivalent to one of the antecedent fuzzy sets appears on the input of such a system the inferred output should be equivalent to the respective consequent fuzzy set similarly the defuzzified output obtained from a system based on a monotone fuzzy rule base should be monotone i e if x x are crisp inputs to the system that are ordered x x then the corresponding defuzzified outputs from the system y y should also be ordered accordingly i e y y however the particular setting that ensures monotonicity need not simultaneously ensure interpolativity recently st pnicka and de baets have investigated and demonstrated the co existence of the above two properties in the case of fuzzy relational inference systems and single input single output siso rule bases an extension of these results to the multiple input single output miso case is not straightforward due to the lack of a natural ordering in higher dimensions in this work we study the miso case and show that similar results can be obtained when the monotone rule base is modeled based on at most and at least modifiers

weierstrass formula and zero finding methods classical weierstrass formula 29 has been often the subject of investigation of many authors in this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions we are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros conditions for the safe convergence of weierstrass method depending only on initial approximations are given in particular we study polynomials with interval coefficients using an interval version of weierstrass method enclosures in the form of disks for the complex valued set containing all zeros of a polynomial with varying coefficients are obtained we also present weierstrass like algorithm for approximating simultaneously all zeros of a class of analytic functions in a given closed region to demonstrate the proposed algorithms three numerical examples are included

fuzzy interpolative reasoning via scale and move transformations interpolative reasoning does not only help reduce the complexity of fuzzy models but also makes inference in sparse rule based systems possible this paper presents an interpolative reasoning method by means of scale and move transformations it can be used to interpolate fuzzy rules involving complex polygon gaussian or other bell shaped fuzzy membership functions the method works by first constructing a new inference rule via manipulating two given adjacent rules and then by using scale and move transformations to convert the intermediate inference results into the final derived conclusions this method has three advantages thanks to the proposed transformations 1 it can handle interpolation of multiple antecedent variables with simple computation 2 it guarantees the uniqueness as well as normality and convexity of the resulting interpolated fuzzy sets and 3 it suggests a variety of definitions for representative values providing a degree of freedom to meet different requirements comparative experimental studies are provided to demonstrate the potential of this method

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

interpolation of fuzzy data analytical approach and overview we propose a general framework for the interpolation problem our framework stems from the classical elaboration of the problem we introduce the notion of an interpolating fuzzy function and show how this function can be characterized we examine and analyze previously published fuzzy interpolation algorithms to choose those algorithms that can be represented analytically we also propose an analytic solution of the interpolation problem that unifies various algorithmic approaches

an accurate numerical inversionof laplace transforms based on the location of their poles we introduce an efficient and easily implemented numerical method for the inversion of laplace transforms using the analytic continuation of integrands of bromwich s integrals after deforming the bromwich s contours so that it consists of the union of small circles around singular points we evaluate the bromwich s integrals by quadrature rules we prove that the error bound of our method has spectral accuracy of type n eps p 0 1 and provide several numerical examples

improved fft based numerical inversion of laplace transforms via fast hartley transform algorithm the disadvantages of numerical inversion of the laplace transform via the conventional fast fourier transform fft are identified and an improved method is presented to remedy them the improved method is based on introducing a new integration step length mt for trapezoidal rule approximation of the bromwich integral in which a new parameter m is introduced for controlling the accuracy of the numerical integration naturally this method leads to multiple sets of complex fft computations a new inversion formula is derived such that n equally spaced samples of the inverse laplace transform function can be obtained by m2 1 sets of n point complex fft computations or by m sets of real fast hartley transform fht computations

approximate reasoning based on linguistic truth value with α operator abstract in this paper we discuss the approximate reasoning based on linguistic truth value rlv with the operator a new method of rlv is proposed it is equivalent to the original rlv method but without using the linguistic truth value fuzzy set under given conditions we convert the problem of calculating sup a a b into the problem of solving the equation of a ta b this significantly simplifies the computations of rlv some properties of the rlv are studied with examples such as the fuzziness of the rlv based on deferent operators a graphical method of rlv is also given to provide a convenient tool to implement the proposed method

practical inference with systems of gradual implicative rules a general approach to practical inference with gradual implicative rules and fuzzy inputs is presented gradual rules represent constraints restricting outputs of a fuzzy system for each input they are tailored for interpolative reasoning our approach to inference relies on the use of inferential independence it is based on fuzzy output computation under an interval valued input a double decomposition of fuzzy inputs is done in terms of alpha cuts and in terms of a partitioning of these cuts according to areas where only a few rules apply the case of 1 d and 2 d inputs is considered as well as higher dimensional cases an application to a cheese making process illustrates the approach

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

closed form fuzzy interpolation fuzzy interpolation enhances the robustness of fuzzy systems and helps to reduce systems complexity although a number of important fuzzy rule interpolation approaches have been proposed in the literature most of these approaches cannot be expressed in a closed form this is usually caused by the effort to avoid possible invalid interpolated results this paper proposes a different fuzzy rule interpolation approach it not only can be represented in a closed form but also guarantees that the interpolated results are valid fuzzy sets this approach is based on a direct use of the extension principle which has been widely utilised for the development of a variety of fuzzy systems the mathematical properties of the proposed approach are analysed by taking the advantage of the closed form representation this approach has been applied to a practical problem of predicting diarrhoeal disease rates in remote villages the results demonstrate the potential of the proposed work in enhancing the robustness of fuzzy interpolation

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

more on the weeks method for the numerical inversion of the laplace transform most of the numerical methods for the inversion of the laplace transform require the values of several incidental parameters generally these parameters are related to the properties of the algorithm and to the analytical properties of the laplace transform functionf s

the generalized tp model transformation for t s fuzzy model manipulation and generalized stability verification this paper integrates various ideas about the tensor product tp model transformation into one conceptual framework and formulates it in terms of the takagi sugeno t s fuzzy model manipulation and control design framework several new extensions of the tp model transformation are proposed such as the quasi and full compact and rank reduced higher order singular value decomposition based canonical form of t s fuzzy models and the bilinear multi pseudo convex partial tp model transformations all of these extensions together form the generalized tp model transformation which provides an effective tool to freely and readily manipulate the antecedent sets and rules of t s fuzzy models and also provides main fuzzy rule component analysis as well as a means for complexity and accuracy tradeoffs it is demonstrated in this paper that the proposed manipulation forms a new effective and necessary optimization step of t s fuzzy or polytopic models and linear matrix inequality based control design and can also decrease conservativeness identification techniques are typically constructed according to the available data and measurement set as well as the type of system to be identified as a result they may not always provide good representations for control design frameworks this paper demonstrates that the proposed tp model transformation is unique in that it bridges between various soft computing based identification techniques and t s fuzzy model based approaches finally this paper proposes the multi tp model transformation which is a tractable and nonheuristic framework to verify the stability of the result of fuzzy or various soft computing based control designs the multi tp model transformation could provide an answer to the frequently emerging criticisms regarding the lack of mathematical stability verification techniques in the soft computing based control design control examples are provided in this paper

a laplace transorm potential theoretic method for acoustic propagation in subsonic flows this paper introduces a competitive computational approach for determining time dependent far field sound generated by subsonic flows around lifting airfoils the procedure assumes the linearity of the sound field away from a bounded region surrounding the airfoil it is assumed that the sound pressure on the boundary of this enclosed region referred to as the kirchhoff surface is specified possibly by another procedure such as solving the full euler equations away from the kirchhoff surface the euler equations are linearized about a uniform mean flow it is well known that linearized euler equations can be uncoupled into a scalar convective wave equation however due to the anisotropy present in the convective wave equation it is difficult to compute solutions in this context direct numerical simulation of the convective wave equation requires proper numerical descriptions of far field boundary conditions which is a non trivial task moreover if accurate far field conditions can be formulated the computational cost of direct simulation can be prohibitive even in a modest computational domain in this paper we present an alternate solution procedure first the problem is transformed via the laplace transform with appropriate initial conditions into a reduced wave equation the convective term in the reduced wave equation is removed using a dependent variable transformation then we use gothert s rule to obtain a helmholtz like equation with complex wave number which is subsequently solved using double layer potential theory finally upon application of numerical inverse laplace transform techniques far field acoustic pressure is obtained as a function of space and time

an analysis of bilinear transform polynomial methods of inversion of laplace transforms methods for the numerical inversion of a laplace transform f s which use a special bilinear transformation of s are particularly effective in many cases and are widely used the main purpose of this paper is to analyze the convergence and conditioning properties of a special class of such methods characterized by the use of lagrange interpolation the results derived apply both to complex and real inversion and show that some known inversion methods are in fact in this class

a note on reasoning conditions of ko czy s interpolative reasoning method abstract in this note we analyze continually the interpolative reasoning method by koczy and hirota and prove that the reasoning conditions given by the authors are also necessary conditions to guarantee the fuzzy inference consequence to be of triangular type if the fuzzy rules and an observation are defined by triangular type membership functions

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

fuzzy interpolative reasoning for sparse fuzzy rule based systems based on bm alpha cuts and transformations techniques in sparse fuzzy rule based systems the fuzzy rule bases are usually incomplete in this situation the system may not properly perform fuzzy reasoning to get reasonable consequences in order to overcome the drawback of sparse fuzzy rule based systems there is an increasing demand to develop fuzzy interpolative reasoning techniques in sparse fuzzy rule based systems in this paper we present a new fuzzy interpolative reasoning method via cutting and transformation techniques for sparse fuzzy rule based systems it can produce more reasonable results than the existing methods the proposed method provides a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy rule based systems

fuzzy interpolation and extrapolation a practical approach fuzzy interpolation does not only help to reduce the complexity of fuzzy models but also makes inference in sparse rule based systems possible it has been successfully applied to systems control but limited work exists for its applications to tasks like prediction and classification almost all fuzzy interpolation techniques in the literature make strong assumptions that there are two closest adjacent rules available to the observation and that such rules must flank the observation for each attribute also some interpolation approaches cannot handle fuzzy sets whose membership functions involve vertical slopes to avoid such limitations and develop a more practical approach this paper extends the work of huang and shen the result enables both interpolation and extrapolation which involve multiple fuzzy rules with each rule consisting of multiple antecedents two realistic applications namely truck backer upper control and computer activity prediction are provided in this paper to demonstrate the utility of the extended approach experiment based comparisons to the most commonly used mamdani fuzzy reasoning mechanism and to other existing fuzzy interpolation techniques are given to show the significance and potential of this research

adaptability interpretability and rule weights in fuzzy rule based systems this paper discusses interpretability in two main categories of fuzzy systems fuzzy rule based classifiers and interpolative fuzzy systems our goal is to show that the aspect of high level interpretability is more relevant to fuzzy classifiers whereas fuzzy systems employed in modeling and control benefit more from low level interpretability we also discuss the interpretability accuracy tradeoff and observe why various rule weighting schemes that have been brought into play to increase adaptability of fuzzy systems rather just increase computational overhead and seriously compromise interpretability of fuzzy systems

the guaranteed convergence of laguerre like method abstract the construction of initial conditions that provide a guaranteed convergence of zero finding methods has attracted a great deal of attention for many years in this paper we consider convergent properties of the laguerre like method of the fourth order for the simultaneous approximation of polynomial zeros using a procedure based on smale s point estimation theory and some recent results concerned with localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of practical importance

a generalized concept for fuzzy rule interpolation the concept of fuzzy rule interpolation in sparse rule bases was introduced in 1993 it has become a widely researched topic in recent years because of its unique merits in the topic of fuzzy rule base complexity reduction the first implemented technique of fuzzy rule interpolation was termed as spl alpha cut distance based fuzzy rule base interpolation despite its advantageous properties in various approximation aspects and in complexity reduction it was shown that it has some essential deficiencies for instance it does not always result in immediately interpretable fuzzy membership functions this fact inspired researchers to develop various kinds of fuzzy rule interpolation techniques in order to alleviate these deficiencies this paper is an attempt into this direction it proposes an interpolation methodology whose key idea is based on the interpolation of relations instead of interpolating spl alpha cut distances and which offers a way to derive a family of interpolation methods capable of eliminating some typical deficiencies of fuzzy rule interpolation techniques the proposed concept of interpolating relations is elaborated here using fuzzy and semantic relations this paper presents numerical examples in comparison with former approaches to show the effectiveness of the proposed interpolation methodology

fuzzy interpolative reasoning for sparse fuzzy rule based systems based on the areas of fuzzy sets fuzzy interpolative reasoning is an inference technique for dealing with the sparse rules problem in sparse fuzzy rule based systems in this paper we present a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the areas of fuzzy sets the proposed method uses the weighted average method to infer the fuzzy interpolative reasoning results and has the following advantages 1 it holds the normality and the convexity of the fuzzy interpolative reasoning result 2 it can deal with fuzzy interpolative reasoning with complicated membership functions 3 it can deal with fuzzy interpolative reasoning when the fuzzy sets of the antecedents and the consequents of the fuzzy rules have different kinds of membership functions 4 it can handle fuzzy interpolative reasoning with multiple antecedent variables 5 it can handle fuzzy interpolative reasoning with multiple fuzzy rules and 6 it can handle fuzzy interpolative reasoning with logically consistent properties with respect to the ratios of fuzziness we use some examples to compare the fuzzy interpolative reasoning results of the proposed method with those of the existing fuzzy interpolative reasoning methods in terms of the six evaluation indices the experimental results show that the proposed method performs more reasonably than the existing methods the proposed method provides us a useful way to deal with fuzzy interpolative reasoning in sparse fuzzy rule based systems

closeness in similarity based reasoning with an interpolation condition this study considers an approximate reasoning scheme where the knowledge base is a set of fuzzy if then rules and the inference mechanism is characterized in terms of closeness we propose the interpretation of this scheme which uses ruspini s theory of conditional consistency and implication measures and its adaptation to fuzzy sets in the proposed interpretation we find the necessary and sufficient conditions to ensure that the computation of the conclusion fulfills the interpolation condition we show that the inference mechanism is equivalent to the compositional rule of inference in the form of inf composition finally we construct a particular interpretation where all of the considered requirements are satisfied and the inference operator reduces to a piecewise linear interpolation function

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

error analysis of a laplace transform inversion procedure this paper establishes new results for attaining accuracy and efficiency in a numerical method for inverting laplace transforms the method developed by d l jagerman bell system tech j 57 1978 pp 669 710 op cit 61 1982 pp 1995 2002 defines a sequence of positive operators from widder s inversion formula that generate a sequence of functions called the approximation sequence that converge uniformly to the inverse of a transform some internal parameters introduced by the method must be properly specified for efficient and accurate computation until now computational instability and slow convergence have been the major drawbacks of the method the new results change this state of affairs by giving explicit formulas for determining the parameters and controlling instability thus permitting computations with high order terms of the approximation sequence that ensure convergence

construction of a neurofuzzy network capable of extrapolating and interpolating with respect to the convex hull of a set of input samples in bb r n the problem of regression estimation is considered with a specific regard for the distinction between interpolation and extrapolation a neurofuzzy network named nfech is proposed that is capable of extrapolating and interpolating with respect to the convex hull of a finite set of input samples x sub ropfn the geometrical construction of the proposed network is explained both mathematically and graphically the illustrations explain how the particular parts of the construction work and also show the final surfaces of the obtained models the method is tested on artificial datasets generated from mathematical functions according to various statistical distributions also comparisons to the commonly used radial basis function rbf multilayered perceptron mlp neural networks and to fuzzy rule interpretation fri fuzzy rule extrapolation fre approach are presented

a modification of the weeks method for numerical inversion of the laplace transform the weeks method for calculating numerical values of f t from its laplace transform f s involves the numerical evaluation of a set of contour integrals in this paper we describe a powerful modification by means of which results of comparable accuracy can be obtained using approximately half the number of complex function values this and other possible minor modifications are discussed at any values of s we like we wish to construct an algorithm which obtains or uses this information to evaluate f t at a given selection of values of t real to a specified accuracy one may assume that f t is real in which case f s f in almost all methods one needs to know or one has to find out first the numerical value of the laplace convergence abscissa ao this may be defined as the limit of the set of values of re s for which this integral converges or alternatively as the maximum of the real parts of the singular points sj of f s a good brief description of the formulas on which the weeks method 2 is based is given in piessens and branders 1 in order to provide a proper background we repeat some of these here briefly we expand f t in terms of laguerre functions thus we assume that

application of limit fuzzy controllers to stability analysis abstract we propose a method to examine the stability of fuzzy control systems it is based on the limit law of fuzzy controllers constructed from regular fuzzy rules the main idea is to use lyapunov functions for the system with limit laws to examine the stability of the system with fuzzy controllers this method is particularly useful when the system and the limit law are both linear it is applied to an inverted pendulum system two cases with and without friction are considered simulation results and the analysis confirm certain experimental and theoretical results already reported

interpolative reasoning with insufficient evidence in sparse fuzzy rule bases abstract rule based fuzzy approximate reasoning uses various techniques of modified modus ponens the observation is in most cases not identical with any of the antecedents in the rules however a conclusion still can be computed by using some combination of all consequents where an overlapping of observation and antecedent is present if the rule base is sparse i e it contains insufficient information on the total state space it might occur that an observation has absolutely no overlapping with any of the antecedents and so not even a single rule is fired i e no conclusion can be computed on the basis of modus ponens in such a case interpolative reasoning in the strict sense can be applied some kind of weighted average of the flanking rules is calculated this technique can be extended to a form of extrapolation when the observation is not flanked from both sides linear interpolation and extrapolation is presented and then the idea is extended to arbitrary approximation

weighted fuzzy interpolative reasoning for sparse fuzzy rule based systems based on piecewise fuzzy entropies of fuzzy sets in this paper we propose a new method for weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems based on piecewise fuzzy entropies of fuzzy sets first the proposed method uses the representative values of antecedent fuzzy sets the representative values of observation fuzzy sets and the representative values of consequence fuzzy sets of fuzzy rules to get the characteristic points of the fuzzy interpolative result represented by a fuzzy set then it calculates the piecewise fuzzy entropies between any two characteristic points of the antecedent fuzzy sets the piecewise fuzzy entropies between any two characteristic points of the observation fuzzy sets and the piecewise fuzzy entropies between any two characteristic points of the consequence fuzzy sets of the fuzzy rules respectively then it calculates the weights of the antecedent fuzzy sets of each fuzzy rule respectively and calculates the weight of each fuzzy rule then it calculates the piecewise fuzzy entropies between any two characteristic points of the fuzzy interpolative result finally it uses the secant method to calculate the degree of membership of each obtained characteristic point of the fuzzy interpolative result the experimental results show that the proposed method outperforms the existing methods for dealing with the multivariate regression problems the mackey glass chaotic time series prediction problem and the time series prediction problems

a family of root finding methods with accelerated convergence a parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered the convergence of the basic method of the fourth order is accelerated using newton s and halley s corrections thus generating total step methods of orders five and six further improvements are obtained by applying the gauss seidel approach accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations detailed convergence analysis and two numerical examples are given

adaptive fuzzy interpolation fuzzy interpolative reasoning strengthens the power of fuzzy inference by the enhancement of the robustness of fuzzy systems and the reduction of the systems complexity however after a series of interpolations it is possible that multiple object values for a common variable are inferred leading to inconsistency in interpolated results such inconsistencies may result from defective interpolated rules or incorrect interpolative transformations this paper presents a novel approach for identification and correction of defective rules in interpolative transformations thereby removing the inconsistencies in particular an assumption based truth maintenance system atms is used to record dependences between interpolations and the underlying technique that the classical general diagnostic engine gde employs for fault localization is adapted to isolate possible faulty interpolated rules and their associated interpolative transformations from this an algorithm is introduced to allow for the modification of the original linear interpolation to become first order piecewise linear the approach is applied to a realistic problem which predicates the diarrheal disease rates in remote villages to demonstrate the potential of this study

reasoning conditions on ko czy s interpolative reasoning method in sparse fuzzy rule bases abstract in our pre work the two sufficient and necessary conditions have been given on koczy s interpolative reasoning method in sparse fuzzy rule bases to guarantee that the reasoning consequence is of triangular type if the fuzzy rules and an observation are defined by triangular type membership functions however the two conditions are too strong to use the reasoning method on practical applications in this paper we analyze the properties of the reasoning method in detail and give several applicable sufficient conditions on it in order to make the reasoning consequence always a normal and convex fuzzy set

fuzzy rules interpolation for sparse fuzzy rule based systems based on interval type 2 gaussian fuzzy sets and genetic algorithms in this paper we present a new method for fuzzy rules interpolation for sparse fuzzy rule based systems based on interval type 2 gaussian fuzzy sets and genetic algorithms first we present a method to deal with the interpolation of fuzzy rules based on interval type 2 gaussian fuzzy sets we also prove that the proposed method guarantees to produce normal interval type 2 gaussian fuzzy sets then we present a method to learn optimal interval type 2 gaussian fuzzy sets for sparse fuzzy rule based systems based on genetic algorithms we also apply the proposed fuzzy rules interpolation method and the proposed learning method to deal with multivariate regression problems and time series prediction problems the experimental results show that the proposed fuzzy rules interpolation method using the optimally learned interval type 2 gaussian fuzzy sets gets higher average accuracy rates than the existing methods

the numerical performance of tricomi s formula for inverting the laplace transform a method for inverting the laplace transform which uses an expansion into laguerre polynomials is considered by means of a recently established generalization of the euler knopp transformation the rate of convergence of the series of laguerre polynomials is accelerated for computing the transformed series a recursive algorithm is given results of theoretical and practical nature make the usefulness of the new transformation evident

interpolation with function space representation of membership functions this paper generalizes a previous cartesian approach for interpolating fuzzy rules comprised of membership functions with finite number of characteristic points instead of representing membership functions as points in cartesian spaces they now become elements in the space of square integrable function interpolation is thus conducted between the antecedent and consequent function spaces the generalized representation allows an extended class of membership functions satisfying two monotonicity conditions to be accommodated in the interpolation process they include the popular bell shaped membership functions which were not possible before with the cartesian representation the work also extends the similarity triangle based interpolation technique from the previous cartesian representation to the new representation ensuing issues on computational complexity and nonunique conclusion are discussed other concepts such as spanning set and extensibility functions are also presented under the generalized framework examples to illustrate the extended approach and to compare with the cartesian approach are given

the theory of smale s point estimation and its applications abstract the main result of this paper is that we exact smale s point estimation theory i e without assuming k p z 1 p k z k k 2 being bounded by the point estimation convergence theorem of the ne method is set up by making use of the majorizing method the proof of the theorem is simple and precise while the required point estimation conditions are weaker than all those of known point estimation convergence theorems another result of this paper is an application of the above new theory to the durand kerner method we compare the point estimation conditions for the durand kerner method with other known point estimation conditions numerical results show that our results have evident advantages

gradual inference rules in approximate reasoning abstract gradual inference rules of the form the more x is f the more y is g or of similar forms with the less instead of the more which express a progressive change of the degree to which the entity y satisfies the gradual property g when the degree to which the entity x satisfies the gradual property f is modified are often encountered in commonsense reasoning a representation of such rules by means of fuzzy sets is proposed and discussed this representation turns out to be based on a special implication function already considered in multiple valued logic patterns of reasoning involving gradual inference rules are formalized their links with interpolation mechanisms are pointed out

improved structure optimization for fuzzy neural networks fuzzy neural network based inference systems are well known universal approximators which can produce linguistically interpretable results unfortunately their dimensionality can be extremely high due to an excessive number of inputs and rules which raises the need for overall structure optimization in the literature various input selection methods are available but they are applied separately from rule selection often without considering the fuzzy structure this paper proposes an integrated framework to optimize the number of inputs and the number of rules simultaneously first a method is developed to select the most significant rules along with a refinement stage to remove unnecessary correlations an improved information criterion is then proposed to find an appropriate number of inputs and rules to include in the model leading to a balanced tradeoff between interpretability and accuracy simulation results confirm the efficacy of the proposed method

weighted fuzzy interpolative reasoning systems based on interval type 2 fuzzy sets abstract in this paper we present a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on interval type 2 fuzzy sets we also apply the proposed weighted fuzzy interpolative reasoning method to deal with the truck backer upper control problem the proposed method satisfies the seven evaluation indices for fuzzy interpolative reasoning the experimental results show that the proposed method outperforms the existing methods it provides us with a useful way for dealing with fuzzy interpolative reasoning in sparse fuzzy rule based systems

rough fuzzy rule interpolation abstract fuzzy rule interpolation forms an important approach for performing inference with systems comprising sparse rule bases even when a given observation has no overlap with the antecedent values of any existing rules fuzzy rule interpolation may still derive a useful conclusion unfortunately very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one form of uncertainty in the rules or observations in particular the difficulty in defining the required precise valued membership functions for the fuzzy sets that are used in conventional fuzzy rule interpolation techniques significantly restricts their application in this paper a novel rough fuzzy approach is proposed in an attempt to address such difficulties the proposed approach allows the representation handling and utilisation of different levels of uncertainty in knowledge this allows transformation based fuzzy rule interpolation techniques to model and harness additional uncertain information in order to implement an effective fuzzy interpolative reasoning system final conclusions are derived by performing rough fuzzy interpolation over this representation the effectiveness of the approach is illustrated by a practical application to the prediction of diarrhoeal disease rates in remote villages it is further evaluated against a range of other benchmark case studies the experimental results confirm the efficacy of the proposed work

overlap indices construction of and application to interpolative fuzzy systems in this paper we analyze the minimal set of axioms that overlap indices between fuzzy sets must fulfill we also present a method to build overlap indices using overlap functions and we carry on a theoretical study about the continuity the invariance and some general properties of such indices we present a generalization of the inference algorithm for interpolative fuzzy systems using overlap indices we also propose an algorithm for choosing among a set of overlap indices the best one for a considered problem

fuzzy interpolative reasoning based on ranking values of polygonal fuzzy sets and automatically generated weights of fuzzy rules in this paper we propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on ranking values of polygonal fuzzy sets and automatically generated weights of fuzzy rules first the proposed method gets the characteristic points of the observation polygonal fuzzy sets and the characteristic points of the antecedent polygonal fuzzy sets of the fuzzy rules respectively by using the cut operations respectively where 0 1 then it calculates the ranking value of each observation polygonal fuzzy set and the ranking value of each antecedent polygonal fuzzy set respectively then it calculates the difference between the ranking value of each observation polygonal fuzzy set and the ranking value of each antecedent polygonal fuzzy set then it calculates the weight of each fuzzy rule based on the obtained differences between the ranking values of the observation polygonal fuzzy sets and the ranking values of the antecedent polygonal fuzzy sets then based on the obtained characteristic points of the observation polygonal fuzzy sets the obtained characteristic points of the antecedent polygonal fuzzy sets and the obtained weights of the fuzzy rules it gets the characteristic points of the fuzzy interpolative reasoning result represented by a polygonal fuzzy set the experimental results show that the proposed method can overcome the drawbacks of the existing methods for fuzzy interpolative reasoning in sparse fuzzy rule based systems

fuzzy interpolative reasoning based on the ratio of fuzziness of rough fuzzy sets in this paper we propose a method to construct a polygonal rough fuzzy set from a set of polygonal fuzzy sets representing the aggregation of multiple experts opinions and propose a new fuzzy interpolative reasoning method for sparse fuzzy rule based systems based on the ratio of fuzziness of the constructed polygonal rough fuzzy sets where the values of the antecedent variables and the consequence variable appearing in the fuzzy rules are represented by the constructed polygonal rough fuzzy sets the proposed fuzzy interpolative reasoning method can overcome the drawbacks of the existing method due to the fact that it can deal with fuzzy interpolative reasoning using polygonal rough fuzzy sets and it gets more reasonable fuzzy interpolative reasoning results than the existing method

reduction of fuzzy rule base via singular value decomposition introduces a singular value based method for reducing a given fuzzy rule set the method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set the present work characterizes membership functions by the conditions of sum normalization sn nonnegativeness nn and normality no algorithms to preserve the sn and nn conditions in the new membership functions are presented preservation of the no condition relates to a high dimensional convex hull problem and is not always feasible in which case a closed to no solution may be sought the proposed method is applicable regardless of the adopted inference paradigms with product sum gravity inference and singleton support fuzzy rule base output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values the work discusses three specific applications of fuzzy reduction fuzzy rule base with singleton support fuzzy rule base with nonsingleton support which includes the case of missing rules and the takagi sugeno kang tsk model numerical examples are presented to illustrate the reduction process

laguerre like inclusion method for polynomial zeros a new method of the fourth order for the simultaneous inclusion of simple polynomial zeros is proposed the presented method is based on the fixed point relation of laguerre s type and realized in circular complex interval arithmetic the established initial conditions guarantee the convergence of the considered method and the fourth order of convergence these conditions are computationally verifiable which is of practical importance the presented iteration formula is suitable for the construction of modified interval methods with improved convergence rate with negligible additional operations convergence analysis and numerical results are given

generalized adaptive fuzzy rule interpolation as a substantial extension to fuzzy rule interpolation that works based on two neighboring rules flanking an observation adaptive fuzzy rule interpolation is able to restore system consistency when contradictory results are reached during interpolation the approach first identifies the exhaustive sets of candidates with each candidate consisting of a set of interpolation procedures which may jointly be responsible for the system inconsistency then individual candidates are modified such that all contradictions are removed and thus interpolation consistency is restored it has been developed on the assumption that contradictions may only be resulted from the underlying interpolation mechanism and that all the identified candidates are not distinguishable in terms of their likelihood to be the real culprit however this assumption may not hold for real world situations this paper therefore further develops the adaptive method by taking into account observations rules and interpolation procedures all as diagnosable and modifiable system components in addition given the common practice in fuzzy systems that observations and rules are often associated with certainty degrees the identified candidates are ranked by examining the certainty degrees of its components and their derivatives from this the candidate modification is carried out based on such ranking this study significantly improves the efficacy of the existing adaptive system by exploiting more information during both the diagnosis and modification processes

genetic feature selection in a fuzzy rule based classification system learning process for high dimensional problems abstract the inductive learning of a fuzzy rule based classification system frbcs is made difficult by the presence of a large number of features that increases the dimensionality of the problem being solved the difficulty comes from the exponential growth of the fuzzy rule search space with the increase in the number of features considered in the learning process in this work we present a genetic feature selection process that can be integrated in a multistage genetic learning method to obtain in a more efficient way frbcss composed of a set of comprehensible fuzzy rules with high classification ability the proposed process fixes a priori the number of selected features and therefore the size of the search space of candidate fuzzy rules the experimentation carried out using sonar example base shows a significant improvement on simplicity precision and efficiency achieved by adding the proposed feature selection processes to the multistage genetic learning method or to other learning methods

representing membership functions as points in high dimensional spaces for fuzzy interpolation and extrapolation this paper introduces a new approach for fuzzy interpolation and extrapolation of sparse rule base comprising of membership functions with finite number of characteristic points the approach calls for representing membership functions as points in high dimensional cartesian spaces using the locations of their characteristic points as coordinates hence a fuzzy rule base can be viewed as a set of mappings between the antecedent and consequent spaces and the interpolation and extrapolation problem becomes searching for an image in the consequent space upon given an antecedent observation the present approach divides observations into two groups 1 observations within the antecedent spanning set contain the same geometric properties as the given antecedents and 2 observations lying outside the antecedent spanning set contain new geometric properties beyond those of the given rules heuristic reasoning must therefore be applied in this case a two step approach with certain flexibility to accommodate additional criteria and design objectives is formulated

weighted fuzzy interpolative reasoning based on weighted increment transformation and weighted ratio transformation techniques in this paper we present a new weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems the proposed method uses weighted increment transformation and weighted ratio transformation techniques to handle weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems it allows each variable that appears in the antecedent parts of fuzzy rules to associate with a weight between zero and one moreover we also propose an algorithm that automatically tunes the optimal weights of the antecedent variables appearing in the antecedent parts of fuzzy rules we also apply the proposed weighted fuzzy interpolative reasoning method to handle the truck backer upper control problem the proposed weighted fuzzy interpolative reasoning method performs better than the ones obtained by the traditional fuzzy inference system 2000 huang and shen s method 2008 and chen and ko s method 2008 the proposed method provides us with a useful way to deal with weighted fuzzy interpolative reasoning in sparse fuzzy rule based systems

similarity rules and gradual rules for analogical and interpolative reasoning with imprecise data similarity rules and gradual rules in the form of functional mappings of imprecise data are proposed to implement analogical and interpolative reasoning the rules are deduced from the property of the horizontal graduality of fuzzy sets and the horizontal graduality is represented by introducing a horizontal difference function as an alternative to the proposed extensions of distance between fuzzy concepts a thorough discussion of the main characteristics of fuzzy sets and horizontal difference function is given an analogical reasoning algorithm based on these characteristics and a set of intuitively sound axioms are proposed the algorithm is consistent with the intuitive concepts of analogical reasoning expressed by the axioms it guarantees a convex and normal output regardless of the shape of the precondition and action patterns and input data when some axiomatic conditions are not satisfied the algorithm relaxes the less important ones and produces an approximate solution the fuzziness cardinality and shape of an output fuzzy set are easily controlled some steps of the algorithm can be carried out in parallel numerical examples confirm the theoretical analysis of the properties of the algorithm some extensions for interpolative reasoning are discussed the proposed rules are particularly well suited for systems which support qualitative reasoning and qualitative modelling

fuzzy systems and approximation abstract the basic motivation of using fuzzy rule based systems especially for control purposes is to deduce simple and fast approximations of the unknown or too complicated models fuzzy rule based systems have become very popuar because of their transparency and easiness of tuning and modification recently some results concerning the explicit functions implemented by realistic fuzzy controllers presented the class of functions that could be implemented in this way some parallel results on the other hand attempted to prove that the main advantage of using fuzzy systems was the suitability for approximation with arbitrary accuracy in their universality the explicit formulas and some very recent theoretical results made it clear however that fuzzy systems were not really good approximators as realistic fuzzy controllers could generate only very rough approximations of given transference functions in connection with approximation the question can be asked whether there is an optimal fineness roughness of a fuzzy rule base that controls a certain action with roughness gives minimal time complexity as an example a target tracking problem was chosen cat and mouse or hawk and sparrow problem where the antagonistic criteria of minimizing inference time by the given rule base and minimizing action time search for the target with given uncertainty provided by the rule model were examined under certain assumptions the solution of this optimization problem leads to nontrivial rule base sizes these results have also practical applicability since if a fine enough model of the system is known it is always possible to generate a rougher version of the same by applying the model transformation technique offered by rule interpolation with levels

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

weighted fuzzy rule interpolation based on ga based weight learning techniques in this paper we propose a weighted fuzzy interpolative reasoning method for sparse fuzzy rule based systems it is based on a genetic algorithm ga based weight learning technique the proposed method can deal with fuzzy rule interpolation with weighted antecedent variables it also can deal with fuzzy rule interpolation based on polygonal membership functions and bell shaped membership functions we also propose a ga based weight learning algorithm to automatically learn the optimal weights of the antecedent variables of the fuzzy rules furthermore we apply the proposed weighted fuzzy interpolative reasoning method and the proposed ga based weight learning algorithm to deal with the truck backer upper control problem the computer activity prediction problem multivariate regression problems and time series prediction problems based on statistical analysis techniques the experimental results show that the proposed weighted fuzzy interpolative reasoning method by the use of the optimally learned weights that were obtained by the proposed ga based weight learning algorithm has statistically significantly smaller error rates than the existing methods

fuzzy rule interpolation for multidimensional input spaces with applications a case study fuzzy rule based systems have been very popular in many engineering applications however when generating fuzzy rules from the available information this may result in a sparse fuzzy rule base fuzzy rule interpolation techniques have been established to solve the problems encountered in processing sparse fuzzy rule bases in most engineering applications the use of more than one input variable is common however the majority of the fuzzy rule interpolation techniques only present detailed analysis to one input variable case this paper investigates characteristics of two selected fuzzy rule interpolation techniques for multidimensional input spaces and proposes an improved fuzzy rule interpolation technique to handle multidimensional input spaces the three methods are compared by means of application examples in the field of petroleum engineering and mineral processing the results show that the proposed fuzzy rule interpolation technique for multidimensional input spaces can be used in engineering applications

fuzzy reasoning method by optimizing the similarity of truth tables this paper presents a new fuzzy reasoning method by optimizing the similarity of truth tables os method for short its basic idea is to find a fuzzy set such that the truth tables generated by the antecedent rule and the consequent rule are as similar as possible based on this idea the principle of os method and the fuzzy reasoning with os method are given and discussed and then the os methods with certain similarity measure and several fuzzy implications are investigated finally numerical examples are analyzed to compare the proposed method with compositional rule of inference cri method

ordering distance and closeness of fuzzy sets abstract in dense rule bases where the observation usually overlaps with several antecedents in the rule base various algorithms are used for approximate reasoning and control if the antecedents are located sparsely and the observation does not overlap as a rule with any of the antecedents function approximation techniques combined with the resolution principle lead to applicable conclusions this kind of approximation is possible only if a new concept of ordering and distance i e a metric in the state space and a partial ordering among convex and normal fuzzy sets cnf sets is introduced so the fuzzy distance of two cnf sets can be defined and by this distance closeness and similarity of cnf sets as well

fuzziness measure on complete hedge algebras and quantifying semantics of terms in linear hedge algebras in the paper we shall examine the fuzziness measure fm of terms or of complete and linear hedge algebras of a linguistic variable the notion of semantically quantifying mappings sqms previously examined by the first author will be redefined more generally and a closed relation between the fm of linguistic terms and a family of sqms with the parameters to be the fm of primary terms and linguistic hedges will be established a semantics based topology of hedge algebras and a closed and interesting relation between this topology the fm and the above family of sqms will be discovered and examined an applicability of the fm and sqms will be shown by an examination of some application examples

joint block structure sparse representation for multi input multi output mimo t s fuzzy system identification takagi sugeno t s fuzzy system identification provides a reasonable framework for modeling and approximation by decomposition of a nonlinear system into a collection of local linear models in this paper we extend the multidimensional output fuzzy rule from the corresponding single output fuzzy rule with the important difference in the variable definition of output namely the output of each fuzzy rule is represented by a multidimensional vector instead of a scalar value in this case the consequent of the fuzzy rule with the multidimensional output variable share a common antecedent part different from the traditional methods that separate the multi input multi output fuzzy system into a group of multi input single output fuzzy models in this paper we take into account the block structure information in the t s fuzzy system and cast the problem of multidimensional output fuzzy model identification as a joint structure sparse optimization problem where the consequent parameters are estimated with a common block structured sparsity pattern over all dimensions of the output variable furthermore we exploit a joint block sparse orthogonal match pursuit algorithm to reduce the number of fuzzy rules in terms of all dimensions of the output variable and prove the sufficient conditions in consideration of the multidimensional output together with the block structure in the t s fuzzy model this method is efficient and shows good performance in well known benchmark datasets and real world problems