adaptive fuzzy rule based classification systems this paper proposes an adaptive method to construct a fuzzy rule based classification system with high performance for pattern classification problems the proposed method consists of two procedures an error correction based learning procedure and an additional learning procedure the error correction based learning procedure adjusts the grade of certainty of each fuzzy rule by its classification performance that is when a pattern is misclassified by a particular fuzzy rule the grade of certainty of that rule is decreased on the contrary when a pattern is correctly classified the grade of certainty is increased because the error correction based learning procedure is not meaningful after all the given patterns are correctly classified we cannot adjust a classification boundary in such a case to acquire a more intuitively acceptable boundary we propose an additional learning procedure we also propose a method for selecting significant fuzzy rules by pruning unnecessary fuzzy rules which consists of the error correction based learning procedure and the concept of forgetting we can construct a compact fuzzy rule based classification system with high performance

induction of fuzzy rules and membership functions from training examples abstract most fuzzy controllers and fuzzy expert systems must predefine membership functions and fuzzy inference rules to map numeric data into linguistic variable terms and to make fuzzy reasoning work in this paper we propose a general learning method as a framework for automatically deriving membership functions and fuzzy if then rules from a set of given training examples to rapidly build a prototype fuzzy expert system based on the membership functions and the fuzzy rules derived a corresponding fuzzy inference procedure to process inputs is also developed

julia sets and mandelbrot sets in noor orbit in recent literature researchers have generated julia sets and mandelbrot sets in mann and ishikawa orbits that are examples of two step and three step feedback processes respectively this paper presents further generalization of julia and mandelbrot sets for complex valued polynomials such as quadratic cubic and higher degree polynomials using a noor orbit which is a four step iterative procedure the graphical images of julia and mandelbrot sets have been visualized and certain patterns in mandelbrot sets have been recognized it is fascinating to see that a few mandelbrot sets are akin to a butterfly or a coupled urn or a coupled trident

implementing dempster s rule for hierarchial evidence this article gives an algorithm for the exact implementation of dempster s rule in the case of hierarchical evidence this algorithm is computationally efficient and it makes the approximation suggested by gordon and shortliffe unnecessary the algorithm itself is simple but its derivation depends on a detailed understanding of the interaction of hierarchical evidence

accurate computation of component centers in the degree n bifurcation set a highly accurate computation of component centers in the degree n bifurcation set is presented via newton s method applied to the transformed polynomial equation that governs component centers the transformed polynomial as well as some properties of the degree n bifurcation set are investigated and efficiently used in newton s method since the initial values are taken from the approximated roots found by fsolve routine of maple v the newton s method shows a great accuracy with a confirmed convergence of order two and with at least 48 significant digits after the decimal point in both real and imaginary parts of the computed component centers although many cases are studied for 2 n 25 and 1 k 10 the limited space allows us to list only typical cases for n 3 4 12 and 25 with 2 k 5 reflecting highly numerical accuracy our study extends the results given by peitgen and richter 1

data driven fuzzy modeling for takagi sugeno kang fuzzy system this paper presents a systematic approach to design first order tagaki sugeno kang tsk fuzzy systems this approach attempts to obtain the fuzzy rules without any assumption about the structure of the data the structure identification and parameter optimization steps in this approach are carried out automatically and are capable of finding the optimal number of the rules with an acceptable accuracy starting with an initial structure the system first tries to improve the structure and then as soon as an improved structure is found it fine tunes its rules parameters then it goes back to improve the structure again to find a better structure and re fine tune the rules parameters this loop continues until a satisfactory solution tsk model is found the proposed approach has successfully been applied to well known benchmark datasets and real world problems the obtained results are compared with those obtained with other methods from the literature experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules finally as a case study the proposed approach is applied to the desulfurization process of a real steel industry comparing the proposed approach with some other fuzzy systems and neural networks it is shown that the developed tsk fuzzy system exhibits better results with higher accuracy and smaller size of architecture

infinite differentiability for one dimensional spin system with long range random interaction we consider one dimensional spin systems with hamiltonian h left sigma lambda right sum limits t t in lambda frac varepsilon tt left t t right alpha sigma t sigma t h sum limits t in lambda sigma t where tt are independent random variables and using decimation and the cluster expansion we show that when 3 2 ande tt 0 for any magnetic fieldh and inverse temperature the correlation functions and the free energy arec both inh and

the thermodynamic limit for long range random systems long range spin systems with random interactions are considered a simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape provided there is no external field the argument is valid for both classical and quantum spin systems and can be applied to a spins randomly distributed on a lattice and interacting via dipolar interactions and b spin systems with potentials of the formj x 1 x 2 x 1 x 2 d where thej x 1 x 2 are independent random variables with mean zero d is the dimension and 1 2 the key to the proof is a multidimensional subadditive ergodic theorem as a corollary we show that for random ferromagnets the correlation length is a nonrandom quantity

renormalization group and analyticity in one dimension a proof of dobrushin s theorem we consider one dimensional systems described by many body potentials with finite first moment and prove that the correlation functions are analytic in the interaction parameters this result is not new dobrushin 1973 but our proof is simpler and physically more transparent we show that by introducing suitable blocks and averaging over the variables associated to a subset of the blocks decimation procedure the resulting effective interaction is such that the system can always be dealt with as a high temperature system

elicitation and fine tuning of fuzzy control rules using symbiotic evolution this paper exploits the ability of symbiotic evolution se as a generic methodology to elicit a fuzzy rule base of the mamdani type almost all fuzzy rule base generation algorithms produce rule bases with redundant and overlapped membership functions that limit their interpretability elegance in their application we address this problem by applying an algorithm to merge any similar membership functions it is shown that our proposed algorithm leads generally to a more transparent and more interpretable rule base with a minimum number of membership functions and a reduced number of rules in addition a new post processing approach is proposed for recovering any probable performance lost after membership functions merging the proposed methodology has been applied successfully for the design of an active control suspension system using a non linear bond graphs bg based half car model with parameters that relate to a ford fiesta mk2

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

a novel approach to fuzzy clustering based on a dissimilarity relation extracted from data using a ts system clustering refers to the process of unsupervised partitioning of a data set based on a dissimilarity measure which determines the cluster shape considering that cluster shapes may change from one cluster to another it would be of the utmost importance to extract the dissimilarity measure directly from the data by means of a data model on the other hand a model construction requires some kind of supervision of the data structure which is exactly what we look for during clustering so the lower the supervision degree used to build the data model the more it makes sense to resort to a data model for clustering purposes conscious of this we propose to exploit very few pairs of patterns with known dissimilarity to build a ts system which models the dissimilarity relation among other things the rules of the ts system provide an intuitive description of the dissimilarity relation itself then we use the ts system to build a dissimilarity matrix which is fed as input to an unsupervised fuzzy relational clustering algorithm denoted any relation clustering algorithm arca which partitions the data set based on the proximity of the vectors containing the dissimilarity values between each pattern and all the other patterns in the data set we show that combining the ts system and the arca algorithm allows us to achieve high classification performance on a synthetic data set and on two real data sets further we discuss how the rules of the ts system represent a sort of linguistic description of the dissimilarity relation

adaptive fuzzy inference neural network an adaptive fuzzy inference neural network afinn is proposed in this paper it has self construction ability parameter estimation ability and rule extraction ability the structure of afinn is formed by the following four phases 1 initial rule creation 2 selection of important input elements 3 identification of the network structure and 4 parameter estimation using lms least mean square algorithm when the number of input dimension is large the conventional fuzzy systems often cannot handle the task correctly because the degree of each rule becomes too small afinn solves such a problem by modification of the learning and inference algorithm

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

interval type 2 credibilistic clustering for pattern recognition this paper presents a new approach to interval type 2 fuzzy clustering in order to consider compactness within the clusters and separation of them simultaneously the objective function of this paper is designed such that it generates both degrees of membership and non membership of each data in each cluster and integrates them using credibility concept also a new approach to separation of clusters is proposed and utilized in designing the objective function in this approach the borders of clusters and therefore their compactness contribute in attaining their separation so the separation of clusters is not assessed only by the distance of their centers the credibility degrees are transformed to interval type 2 form to handle different sources of uncertainty finally a new validity index to characterize the number of clusters based on proposed approach to separation of clusters and choquet integrals are proposed the advantages of paper contributions are illustrated using several examples a new objective function based fuzzy clustering is introduced using credibility concept a new approach to separation of clusters is presented a new validity index is designed by proposed approach to separation and choquet integral the obtained results show that proposed model can handle different types of clusters

parameter conditions for monotonic takagi sugeno kang fuzzy system this paper presents sufficient conditions on the parameters of the takagi sugeno kang tsk fuzzy system under which the output of the tsk fuzzy system becomes monotonic with respect to its input the output of the tsk fuzzy system is first differentiated with respect to its input and the parameter conditions are derived so as to make the derivative nonnegative the parameter conditions are developed for both single input and multi input tsk fuzzy systems where the involved fuzzy membership functions are differentiable everywhere or bar some finite points the derived monotonicity conditions consist of two parts the conditions on the consequent parts and the conditions on the fuzzy membership functions they are further characterized for the most important types of fuzzy membership functions

ten years of genetic fuzzy systems current framework and new trends fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains currently there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities two of the most successful approaches to hybridise fuzzy systems with learning and adaptation methods have been made in the realm of soft computing neural fuzzy systems and genetic fuzzy systems hybridise the approximate reasoning method of fuzzy systems with the learning capabilities of neural networks and evolutionary algorithms the objective of this paper is to provide an account of genetic fuzzy systems with special attention to genetic fuzzy rule based systems after a brief introduction to models and applications of genetic fuzzy systems the field is overviewed new trends are identified a critical evaluation of genetic fuzzy systems for fuzzy knowledge extraction is elaborated and open questions that remain to be addressed in the future are raised the paper also includes some of the key references required to quickly access implementation details of genetic fuzzy systems

from approximative to descriptive fuzzy classifiers this paper presents an effective and efficient approach for translating fuzzy classification rules that use approximative sets to rules that use descriptive sets and linguistic hedges of predefined meaning it works by first generating rules that use approximative sets from training data and then translating the resulting approximative rules into descriptive ones hedges that are useful for supporting such translations are provided the translated rules are functionally equivalent to the original approximative ones or a close equivalent given search time restrictions while reflecting their underlying preconceived meaning thus fuzzy descriptive classifiers can be obtained by taking advantage of any existing approach to approximative modeling which is generally efficient and accurate while employing rules that are comprehensible to human users experimental results are provided and comparisons to alternative approaches given

rspop rough set based pseudo outer product fuzzy rule identification algorithm system modeling with neuro fuzzy systems involves two contradictory requirements interpretability verses accuracy the pseudo outer product pop rule identification algorithm used in the family of pseudo outer product based fuzzy neural networks popfnn suffered from an exponential increase in the number of identified fuzzy rules and computational complexity arising from high dimensional data this decreases the interpretability of the popfnn in linguistic fuzzy modeling this article proposes a novel rough set based pseudo outer product rspop algorithm that integrates the sound concept of knowledge reduction from rough set theory with the pop algorithm the proposed algorithm not only performs feature selection through the reduction of attributes but also extends the reduction to rules without redundant attributes as many possible reducts exist in a given rule set an objective measure is developed for popfnn to correctly identify the reducts that improve the inferred consequence experimental results are presented using published data sets and real world application involving highway traffic flow prediction to evaluate the effectiveness of using the proposed algorithm to identify fuzzy rules in the popfnn using compositional rule of inference and singleton fuzzifier popfnn cri s architecture results showed that the proposed rough set based pseudo outer product algorithm reduces computational complexity improves the interpretability of neuro fuzzy systems by identifying significantly fewer fuzzy rules and improves the accuracy of the popfnn

a neural fuzzy modelling framework based on granular computing concepts and applications fuzzy and neural fuzzy systems have successfully and extensively applied to solve problems in many research areas such as those associated with industrial medical and academic applications however recent trends reveal a demand for a workflow with a particular emphasis on transparency simplicity system interpretability as well as on a strong link with human cognition such requirement is mainly driven by research areas where expert knowledge is of very high importance and any new proposed modelling system falls under the interpretability scrutiny of experts in order to confirm the system s validity the relatively recent paradigm of granular computing grc offers an ideal opportunity for a transparent knowledge discovery methodology to be combined with fuzzy logic thereby towards a systematic modelling framework with a focus on the overall transparency of the system such transparency in the workflow allows for better interaction between the expert process knowledge and the system design which translates into a better performing system in this paper a systematic modelling approach using granular computing grc and neural fuzzy modelling is presented in this research study a grc algorithm is used to extract relational information and data characteristics out of an initial database the extracted knowledge and granular features are then translated into a linguistic rule base of a fuzzy system this rule base is finally elicited and optimised via a neural fuzzy modelling structure during the various steps of this methodology the transparency features are highlighted and it is shown here how the system designer can take advantage of such features to enhance the system the proposed modelling framework is applied to a multi dimensional and complex data set consisting of measurements of mechanical properties of heat treated steel the data set is collected from a real industrial process and the measurements are dictated by customer production demands and the data set is very sparse with many discontinuities the proposed framework successfully models the mechanical properties of heat treated steel and it further improves upon the performance of previously established modelling structures

stability analysis and design for a class of continuous time fuzzy control systems this paper presents a design method for a class of fuzzy control systems the class of fuzzy systems considered can be represented by the takagi sugeno fuzzy model which is a type of dynamic fuzzy model a constructive algorithm is developed to obtain the stabilizing feedback control law for the system the main contribution of this paper is the development of an equivalent principle that is the design of a fuzzy control system is equivalent to the design of a set of linear time invariant extreme systems thus any design method in linear control system theory can be used to design a fuzzy control system an example is given to illustrate the application of the method

the generalized mandelbrot set perturbed by composing noise of additive and multiplicative adopting the experimental mathematics method combining complex variable function theory with computer aided drawing this paper researches on the structural characteristic and the fission evolution law of the generalized mandelbrot set generalized m set in short perturbed by composing noise of additive and multiplicative analyzes the effect of random perturbation to the generalized m set

a constrained takagi sugeno fuzzy system that allows for better interpretation and analysis abstract in this paper we present a modification of the generalized takagi sugeno gts model fiordaliso proc eufit96 conf vol 1 1996 pp 581 586 the key idea is to constrain the conclusions of each rule to perform a convex combination of the input patterns this constraint allows to interpret each rule as an input patterns mixer and therefore contributes to a better understanding of the fuzzy system inference we illustrate the functioning of the constrained system which is next applied for purposes of combining forecasts

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

a characteristic point based fuzzy inference system aimed to minimize the number of fuzzy rules this paper presents a characteristic point based fuzzy inference system cpfis for fuzzy modeling from training data the aim of the cpfis is not only satisfactory precision performance but also to employ as few purely linguistic fuzzy rules as possible by using a minimization based systematic training method characteristic points cps are defined as the few data points among the original training data which when they are directly mapped to fuzzy rules and thus form the entire rule base allow the underlying system to be effectively modeled three minimization based algorithms in a sequence are proposed to train the cpfis a gradient projection method a gauss jordan elimination based column elimination and back propagation the cps are determined by iterative computations of the first two minimization algorithms after which the resulting fuzzy sets are further fine tuned by the third algorithm experiments conducted on three benchmark problems showed that the cpfis used one of the smallest number of fuzzy rules among the reported results for other methods the gaussian membership functions in both the input and output fuzzy sets and the small number of fuzzy rules make the rule interpretation of the cpfis much easier than that of other methods thus enhancing human computer cooperation in knowledge discovery

one dimensional random ising systems with interaction decay r 1 epsilon a convergent cluster expansion we consider a one dimensional random ising model with hamiltonian h sum limits i ddag j frac j ij left i j right 1 varepsilon s i s j h sum limits i s i where e 0 andj ij are independent identically distributed random variables with distributiondf x such that int xdf left x right 0 i int e tx df left x right infty forall t in mathbb r ii we construct a cluster expansion for the free energy and the gibbs expectations of local observables this expansion is convergent almost surely at every temperature in this way we obtain that the free energy and the gibbs expectations of local observables arec functions of the temperature and of the magnetic fieldh moreover we can estimate the decay of truncated correlation functions in particular for every e 0 there exists a random variablec m finite almost everywhere such that left left langle s 0 s j right rangle h left langle s 0 right rangle h left langle s j right rangle h right leqq frac c left omega right left j right 1 varepsilon varepsilon where h denotes the gibbs average with respect to the hamiltonianh

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

on a topological closeness of perturbed mandelbrot sets in the present work we propose a numerical and visual tool for the study of the deformation of the mandelbrot sets of perturbed mandelbrot maps by noise in comparison with the original mandelbrot set further by employing these numerical tools we support the invariance of the mandelbrot set of a noise perturbed mandelbrot map under different noise realizations finally we provide evidence for the non fractal structure of the mandelbrot set of a noise perturbed mandelbrot map

genetically optimized fuzzy polynomial neural networks with fuzzy set based polynomial neurons in this paper we propose and investigate a new category of neurofuzzy networks fuzzy polynomial neural networks fpnn endowed with fuzzy set based polynomial neurons fspns we develop a comprehensive design methodology involving mechanisms of genetic optimization and genetic algorithms gas in particular the conventional fpnns developed so far are based on the mechanisms of self organization fuzzy neurocomputing and evolutionary optimization the design of the network exploits the fspns as well as the extended group method of data handling gmdh let us stress that in the previous development strategies some essential parameters of the networks such as the number of input variables the order of the polynomial the number of membership functions and a collection of the specific subset of input variables being available within the network are provided by the designer in advance and kept fixed throughout the overall development process this restriction may hamper a possibility of developing an optimal architecture of the model the design proposed in this study addresses this issue the augmented and genetically developed fpnn gfpnn results in a structurally optimized structure and comes with a higher level of flexibility in comparison to the one we encounter in the conventional fpnns the ga based design procedure being applied at each layer of the fpnn leads to the selection of the most suitable nodes or fspns available within the fpnn in the sequel two general optimization mechanisms are explored first the structural optimization is realized via gas whereas the ensuing detailed parametric optimization is carried out in the setting of a standard least square method based learning the performance of the gfpnn is quantified through experimentation in which we use a number of modeling benchmarks synthetic and experimental data being commonly used in fuzzy or neurofuzzy modeling the obtained results demonstrate the superiority of the proposed networks over the models existing in the references

finding relevant attributes and membership functions fuzzy systems that automatically derive fuzzy if then rules from numeric data have been developed most have to predefine membership functions in order to learn hong and lee proposed a general learning method that automatically derives fuzzy if then rules and membership functions from a set of given training examples using a decision table all available attributes were included in the decision table and the initial membership functions for each attribute were built according to the predefined smallest unit although hong and lee s method accurately derives the fuzzy if then rules and final membership functions the decision table and the initial membership functions are complex if there are many attributes or if the predefined unit is small we improve hong and lee s method by first selecting relevant attributes and building appropriate initial membership functions these attributes and membership functions are then used in a decision table to derive final fuzzy if then rules and membership functions experimental results on iris data show that the proposed method effectively induces membership functions and fuzzy if then rules

bootstrap rank ordered conditional mutual information brocmi a nonlinear input variable selection method for water resources modeling the input variable selection problem has recently garnered much interest in the time series modeling community especially within water resources applications demonstrating that information theoretic nonlinear based input variable selection algorithms such as partial mutual information pmi selection pmis provide an improved representation of the modeled process when compared to linear alternatives such as partial correlation input selection pcis pmis is a popular algorithm for water resources modeling problems considering nonlinear input variable selection however this method requires the specification of two nonlinear regression models each with parametric settings that greatly influence the selected input variables other attempts to develop input variable selection methods using conditional mutual information cmi an analog to pmi have been formulated under different parametric pretenses such as k nearest neighbor knn statistics or kernel density estimates kde in this paper we introduce a new input variable selection method based on cmi that uses a nonparametric multivariate continuous probability estimator based on edgeworth approximations ea we improve the ea method by considering the uncertainty in the input variable selection procedure by introducing a bootstrap resampling procedure that uses rank statistics to order the selected input sets we name our proposed method bootstrap rank ordered cmi brocmi we demonstrate the superior performance of brocmi when compared to cmi based alternatives ea kde and knn pmis and pcis input variable selection algorithms on a set of seven synthetic test problems and a real world urban water demand uwd forecasting experiment in ottawa canada

a new credibilistic clustering algorithm abstract this paper focuses on credibilistic clustering approach a data clustering method partitions unlabeled data sets into clusters and labels them for various goals such as computer vision and pattern recognition there are different models for objective function based fuzzy clustering such as fuzzy c means fcm possibilistic c mean pcm and their combinations credibilistic clustering is a new approach in this field in this paper a new credibilistic clustering model is introduced in which credibility measure is applied instead of possibility measure in possibilistic clustering also in objective function the separation of clusters is considered in addition to the compactness within clusters the steps of clustering are designed based on this approach finally the main issues about model are discussed and the results of computational experiments are presented to show the efficiency of the proposed model

research on fractal structure of generalized m j sets utilized lyapunov exponents and periodic scanning techniques abstract in order to show details of fractal structure of mandelbrot set precisely lyapunov exponents and periodic scanning techniques have been brought forward by shirriff and welstead this paper generalizes these two techniques and puts forward periodicity orbit search and comparison technique which can be used to discuss the relationship of the generalized mandelbrot julia sets the generalized m j sets adopting the techniques mentioned above and the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing this paper researches on the structure topological inflexibility and the discontinuity evolution law of the generalized m j sets generated from the complex mapping z z c r and explores structure and distributing of periodicity petal and topological law of periodicity orbits of the generalized m sets and finds that the generalized m set contains abundant information of structure of the generalized j sets by founding the whole portray of the generalized j sets based on the generalized m set qualitatively furthermore the physical meaning of the generalized m j sets have been expounded

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

t s fuzzy model identification with a gravitational search based hyperplane clustering algorithm in order to improve the performance of the fuzzy clustering algorithm in fuzzy space partition in the identification of the takagi sugeno t s fuzzy model a hyperplane prototype fuzzy clustering model is proposed to solve the clustering objective function which could not be handled by the gradient method as the traditional clustering method fuzzy c means does a newly developed excellent global search method which is the gravitational search algorithm gsa is employed then the gsa based hyperplane clustering algorithm gshpc is proposed and illuminated gshpc is used to partition the fuzzy space and identify premise parameters of the t s fuzzy model and orthogonal least squares is exploited to identify the consequent parameters comparative experiments are designed to verify the validity of the proposed clustering algorithm and the t s fuzzy model identification method and the results show that the new method is effective in describing a complicated nonlinear system with significantly high accuracies compared with approaches in the literature

a recursive rule base adjustment algorithm for a fuzzy logic controller this paper introduces a recursive rule base adjustment to enhance the performance of fuzzy logic controllers here the fuzzy controller is constructed on the basis of a decision table dt relying on membership functions and fuzzy rules that incorporate heuristic knowledge and operator experience if the controller performance is not satisfactory it has previously been suggested that the rule base be altered by combined tuning of membership functions and controller scaling factors the alternative approach proposed here entails alteration of the fuzzy rule base the recursive rule base adjustment algorithm proposed in this paper has the benefit that it is computationally more efficient for the generation of a dt and advantage for online realization simulation results are presented to support this thesis

quadratic stability analysis of the takagi sugeno fuzzy model the nonlinear dynamic takagi sugeno fuzzy model with offset terms is analyzed as a perturbed linear system a sufficient criterion for the robust stability of this nominal system against nonlinear perturbations guarantees quadratic stability of the fuzzy model the criterion accepts a convex programming formulation of reduced computational cost compared to the common lyapunov matrix approach parametric robust control techniques suggest synthesis tools for stabilization of the fuzzy system application examples on fuzzy models of nonlinear plants advocate the efficiency of the method the examples demonstrate reduced conservatism compared to norm based criteria

input selection for nonlinear regression models a simple and effective method for the selection of significant inputs in nonlinear regression models is proposed given a set of input output data and an initial superset of potential inputs the relevant inputs are selected by checking whether after deleting a particular input the data set is still consistent with the basic property of a function in order to be able to handle real valued and noisy data in a sensible manner fuzzy clustering is first applied the obtained clusters are compared by using a similarity measure in order to find inconsistencies within the data several examples using simulated and real world data sets are presented to demonstrate the effectiveness of the algorithm

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

fuzzy data mining for interesting generalized association rules due to the increasing use of very large databases and data warehouses mining useful information and helpful knowledge from transactions is evolving into an important research area most conventional data mining algorithms identify the relationships among transactions using binary values and find rules at a single concept level transactions with quantitative values and items with hierarchy relation are however commonly seen in real world applications in this paper we thus introduce the problem of mining fuzzy generalized association rules from quantitative data a fuzzy mining algorithm based on srikant and agrawal s method is proposed for extracting implicit generalized knowledge from transactions stored as quantitative values it integrates fuzzy set concepts and generalized data mining technologies to achieve this purpose items in rules may be from any level of the given taxonomy the effect of numbers of fuzzy regions on the performance of the proposed algorithm is also discussed

processing individual fuzzy attributes for fuzzy rule induction fuzzy systems that can automatically derive fuzzy if then rules and membership functions from numeric data have recently been developed in this paper we propose two new fuzzy learning methods for automatically deriving membership functions and fuzzy if then rules from a set of given training examples the proposed methods first select relevant attributes and build appropriate initial membership functions they then simplify the intervals and the membership functions of each attribute before forming a decision table these attributes and membership functions are then used in a decision table to derive the final fuzzy if then rules and membership functions experimental results for the iris data show that our methods can achieve a high degree of accuracy the proposed methods are thus useful in constructing membership functions and in managing uncertainty and vagueness they can also reduce the time and effort needed to develop a fuzzy knowledge base

knowledge discovery by a neuro fuzzy modeling framework in this paper a neuro fuzzy modeling framework is proposed which is devoted to discover knowledge from data and represent it in the form of fuzzy rules the core of the framework is a knowledge extraction procedure that is aimed to identify the structure and the parameters of a fuzzy rule base through a two phase learning of a neuro fuzzy network in order to obtain reliable and readable knowledge two further stages are integrated with the knowledge extraction procedure a pre processing stage performing variable selection on the available data to obtain simpler and more reliable fuzzy rules and a post processing stage that granulates outputs of the extracted fuzzy rules so as to provide a validity range of estimated outputs moreover the framework can address complex multi input multi output problems in such case two distinct modeling strategies can be followed with the opportunity of producing both a single mimo model or a collection of miso models the proposed framework is verified on a real world case study involving prediction of chemical composition of ashes produced by combustion processes carried out in thermo electric generators located in italy

a neuro fuzzy system modeling with self constructing rule generationand hybrid svd based learning we propose an approach for neuro fuzzy system modeling a neuro fuzzy system for a given set of input output data is obtained in two steps first the data set is partitioned automatically into a set of clusters based on input similarity and output similarity tests membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster then a fuzzy if then rule is extracted from each cluster to form a fuzzy rule base second a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule base to decrease the size of the search space and to speed up the convergence we develop a hybrid learning algorithm which combines a recursive singular value decomposition based least squares estimator and the gradient descent method the proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points besides it learns faster consumes less memory and produces lower approximation errors than other methods

on the symmetries of the julia sets for the process z zp c the self replicating properties of the julia sets jc p for the iterative processes z implies zp c are examined for integers p 1 it is shown that the corresponding mandelbrot sets m p contain p 1 fold symmetries which results in jc p having p fold symmetries

subtractive clustering based modeling of job sequencing with parametric search in this paper an extended subtractive clustering based fuzzy system identification method and the sugeno type reasoning mechanism are used for modeling job sequencing problems this approach can be used to build a fuzzy model of the sequencing system from an existing sequence output data and possible job attributes input data the single machine weighted flowtime problem is used as an example to demonstrate the proposed methodology the effects of data scarcity on the modeling performance is studied by using three data sets with varying degrees of available data furthermore a parametric search on various clustering parameters is performed to identify the best model as a result of parametric search ranges of clustering parameters that provide best models are also identified

genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction tuning fuzzy rule based systems for linguistic fuzzy modeling is an interesting and widely developed task it involves adjusting some of the components of the knowledge base without completely redefining it this contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions to adjust the former component we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability to alter the latter component two different approaches changing their basic parameters and using nonlinear scaling factors are proposed as the accomplished experimental study shows the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance the paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process a good interpretability accuracy tradeoff is obtained combining both processes with a sequential scheme first reducing the rule set and subsequently tuning the model

induction of fuzzy decision trees abstract most decision tree induction methods used for extracting knowledge in classification problems do not deal with cognitive uncertainties such as vagueness and ambiguity associated with human thinking and perception in this paper cognitive uncertainties involved in classification problems are explicitly represented measured and incorporated into the knowledge induction process a fuzzy decision tree induction method which is based on the reduction of classification ambiguity with fuzzy evidence is developed fuzzy decision trees represent classification knowledge more naturally to the way of human thinking and are more robust in tolerating imprecise conflict and missing information

on numerical approximations of the area of the generalized mandelbrot sets in the present work the area of the generalized mandelbrot sets is defined as the double limit of the areas of the plotted generalized mandelbrot sets in a given square lattice using the finite escape algorithm while the lattice resolution and the number of iteration counts used to plot them tends to infinity the asymptotic behavior of the areas of the generalized mandelbrot sets in terms of their degree growth is investigated finally numerical approximations of the area of the mandelbrot set are proposed by using tools from regression analysis

improvements and critique on sugeno s and yasukawa s qualitative modeling investigates sugeno s and yasukawa s 1993 qualitative fuzzy modeling approach we propose some easily implementable solutions for the unclear details of the original paper such as trapezoid approximation of membership functions rule creation from sample data points and selection of important variables we further suggest an improved parameter identification algorithm to be applied instead of the original one these details are crucial concerning the method s performance as it is shown in a comparative analysis and helps to improve the accuracy of the built up model finally we propose a possible further rule base reduction which can be applied successfully in certain cases this improvement reduces the time requirement of the method by up to 16 in our experiments

fuzzy modeling and control of multilayer incinerator abstract a method to design a fuzzy controller based on a fuzzy model is suggested the identification method of a fuzzy model is also discussed the suggested method is applied for the modelling and control of a multilayer incinerator the performance of the designed fuzzy controller is tested by computer simulation using the identified process model the results are satisfactory

regranulation a granular algorithm enabling communication between granular worlds abstract in this paper we describe a granular algorithm for translating information between two granular worlds represented as fuzzy rulebases these granular worlds are defined on the same universe of discourse but employ different granulations of this universe in order to translate information from one granular world to the other we must regranulate the information so that it matches the information granularity of the target world this is accomplished through the use of a first order interpolation algorithm implemented using linguistic arithmetic a set of elementary granular computing operations we first demonstrate this algorithm by studying the common fuzzy pd rulebase at several different granularities and conclude that the 3 3 granulation may be too coarse for this objective we then examine the question of what the natural granularity of a system might be this is studied through a 10 fold cross validation experiment involving three different granulations of the same underlying mapping for the problem under consideration we find that a 7 7 granulation appears to be the minimum necessary precision

design of a new adaptive neuro fuzzy inference system based on a solution for clustering in a data potential field in this study we propose a new method for building adaptive neuro fuzzy inference systems anfis via datasets in order to improve the performance of conventional anfis to handle noisy data we focus on ameliorating the cluster data space established from a given dataset to achieve this we propose a weighted clustering process in the joint input output data space thus during the clustering process the cluster with the smallest potential distance which is a combination of the euclidean distance and the size of the clusters has priority when obtaining the surveyed sample based on this principle we formulate a new algorithm for synthesizing an anfis via the proposed data potential field called anfis pf which has the following features it establishes a data potential field that covers the entire initial data space a cluster data space is built based on the generated data potential field and the anfis is synthesized using this cluster data space finally we performed experiments using datasets with and without noise to demonstrate the effectiveness of the proposed method in several applications including dynamic response noisy datasets obtained from a magnetorheological damper

a fast genetic method for inducting descriptive fuzzy models under certain inference mechanisms fuzzy rule bases can be regarded as extended additive models this relationship can be applied to extend some statistical techniques to learn fuzzy models from data the interest in this parallelism is twofold theoretical and practical first extended additive models can be estimated by means of the matching pursuit algorithm which has been related to support vector machines boosting and radial basis neural networks learning this connection can be exploited to better understand the learning of fuzzy models in particular the technique we propose here can be regarded as the counterpart to boosting fuzzy classifiers in the field of fuzzy modeling second since matching pursuit is very efficient in time we can expect to obtain faster algorithms to learn fuzzy rules from data we show that the combination of a genetic algorithm and the backfitting process learns faster than ad hoc methods in certain datasets

a fuzzy based customer classification method for demand responsive logistical distribution operations in some cases customer classification is important for the development of advanced logistical distribution strategies in response to the growing complexity in business logistical markets this paper presents a new approach that can be employed to cluster customers before executing fleet routing in logistical operations the proposed approach is developed on the basis of fuzzy clustering techniques and involves three sequential mechanisms including 1 binary transformation 2 generation of a fuzzy correlation matrix and 3 customer clustering such a customer clustering method should be performed prior to vehicle dispatching and routing in the process of goods distribution the proposed methodology clusters customers on the basis of their demand attributes rather than the static geographic property which is considered extensively in most published vehicle routing algorithms in addition to methodology development a case study was conducted to demonstrate the potential advantages of the proposed fuzzy clustering based method it is expected that this study can stimulate more research on time based logistics control and management

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

finding useful fuzzy concepts for pattern classification using genetic algorithm in this paper a fuzzy classifier is treated as a fuzzy information retrieval system to construct such a system a fuzzy data mining method is employed to determine useful fuzzy concepts subsequently each of the classes and patterns can be represented by a fuzzy set of useful fuzzy concepts from the viewpoint of fuzzy information retrieval a pattern can be categorized into one class if there exists a maximum degree of similarity between them the genetic algorithm ga whose objective is to find a compact set consisting of useful fuzzy concepts with high classification capability is further employed to automatically determine parameter specifications that are not easily specified by users to evaluate classification performance of the proposed method computer simulations are performed on some well known classification problems demonstrating that the generalization ability of the proposed method is comparable to other fuzzy classification methods

a novel generic hebbian ordering based fuzzy rule base reduction approach to mamdani neuro fuzzy system there are two important issues in neuro fuzzy modeling 1 interpretability the ability to describe the behavior of the system in an interpretable way and 2 accuracy the ability to approximate the outcome of the system accurately as these two objectives usually exert contradictory requirements on the neuro fuzzy model certain compromise has to be undertaken this letter proposes a novel rule reduction algorithm namely hebb rule reduction and an iterative tuning process to balance interpretability and accuracy the hebb rule reduction algorithm uses hebbian ordering which represents the degree of coverage of the samples by the rule as an importance measure of each rule to merge the membership functions and hence reduces the number of the rules similar membership functions mfs are merged by a specified similarity measure in an order of hebbian importance and the resultant equivalent rules are deleted from the rule base the rule with a higher hebbian importance will be retained among a set of rules the mfs are tuned through the least mean square lms algorithm to reduce the modeling error the tuning of the mfs and the reduction of the rules proceed iteratively to achieve a balance between interpretability and accuracy three published data sets by nakanishi nakanishi turksen sugeno 1993 the pat synthetic data set pal mitra mitra 2003 and the traffic flow density prediction data set are used as benchmarks to demonstrate the effectiveness of the proposed method good interpretability as well as high modeling accuracy are derivable simultaneously and are suitably benchmarked against other well established neuro fuzzy models

identification of key variables using fuzzy average with fuzzy cluster distribution identification of the significance of input variables is very important for complex systems with high dimensional input space in this paper a method using fuzzy average with fuzzy cluster distribution is proposed to avoid the interference of different distributions of the sampling data the distribution of fuzzy clusters in the sampling data is considered instead of the original data set to discover the input output relationship the methods of fuzzy rules and fuzzy c means are first used to partition the original sampling data set into fuzzy clusters a new data set with the same distribution of the fuzzy clusters is produced the fuzzy average method is then applied to the new data set by doing so the interference of distribution of the original sampling data is removed this method is straightforward and computationally easy the performance is tested on both benchmark data and real world data

on a topological closeness of perturbed julia sets in the present work we expand our previous work in 1 by introducing the julia deviation distance and the julia deviation plot in order to study the stability of the julia sets of noise perturbed mandelbrot maps we observe a power law behaviour of the julia deviation distance of the julia sets of a family of additive dynamic noise mandelbrot maps from the julia set of the mandelbrot map as a function of the noise level additionally using the above tools we support the invariance of the julia set of a noise perturbed mandelbrot map under different noise realizations

absence of phase transitions in certain one dimensional long range random systems an ising chain is considered with a potential of the formj i j i j where thej i j are independent random variables with mean zero the chain contains both randomness and frustration and serves to model a spin glass a simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if 1 this is to be contrasted with a ferromagnetic interaction which requires 2 the basic idea is to prove that the surface free energy between two half lines is finite although the surface energy may be unbounded ford dimensional systems it is shown that the free energy does not depend on the specific boundary conditions if 1 2 d

finite range spin glasses in the kac limit free energy and local observables we study a finite range spin glass model in arbitrary dimension where the intensity of the coupling between spins decays to zero over some distance 1 we prove that under a positivity condition for the interaction potential the infinite volume free energy of the system converges to that of the sherrington kirkpatrick model in the kac limit 0 we study the implication of this convergence for the local order parameter i e the local overlap distribution function and a family of susceptibilities associated with it and we show that locally the system behaves like its mean field analogue similar results are obtained for models with p spin interactions finally we discuss a possible approach to the problem of the existence of long range order for finite based on a large deviation functional for overlap profiles this will be developed in future work

cluster centric fuzzy modeling in this study we propose a cluster oriented development of fuzzy models an overall design process is focused on an efficient usage of fuzzy clustering fuzzy c means fcm in particular to form information granules clusters that are used in the construction of the fuzzy model fuzzy models are regarded as mappings from information granules expressed in the input and output spaces this position motivates us to look at the development of the models through the perspective of the construction and efficient usage of information granules this study directly associates fuzzy clustering with fuzzy modeling both in terms of conceptual and algorithmic linkages the augmented fcm method is formed predominantly for modeling purposes so that a balance between the structural content present in the input and output spaces is achieved and this way the performance of the resulting fuzzy model is optimized it is shown that the cluster oriented modeling gives rise to the mamdani like fuzzy rules and a zero order takagi sugeno model under a certain decoding scheme we identify an interesting and direct linkage between the developed fuzzy models and a fundamental idea of encoding decoding or granulation degranulation encountered in processing fuzzy sets and granular computing in general furthermore refinements of zero order fuzzy models are investigated leading to first order fuzzy models with linear functions standing in the conclusions of the rules a series of experiments is reported where we used synthetic and real world data using which an issue of generalization capabilities is elaborated in detail

an identification algorithm in fuzzy relational systems abstract the paper presents an effective identification method in fuzzy relational systems we propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques the usefulness of the method provided is demonstrated by means of two numerical examples also a possible way of generating a linguistic decision making algorithm is discussed

input selection and partition validation for fuzzy modelling using neural network a simple and effective method for selecting significant input variables and determining optimal number of fuzzy rules when building a fuzzy model from data is proposed in contrast to the existing clustering based methods in this approach both input selecting and partition validating are determined on the basis of a class of sub clusters created by a self organising network instead of on the data the important input variables which independently and significantly influence the system output can be extracted by a fuzzy neural network on the other hand the optimal number of fuzzy rules can be determined separately via the fuzzy c means algorithm with a modified fuzzy entropy as the criterion of cluster validation the simulation results show that the proposed method can provide good model structures for fuzzy modelling and has high computing efficiency

type i and type ii fuzzy system modeling abstract two system modeling approaches are proposed with a unification of fuzzy methodologies type i and type ii knowledge representation are reviewed together with their proposed approximate reasoning schemes type i representation and inference are based on boolean normal forms type i models are myopic because in these models either the disjunctive or the conjunctive boolean normal forms are selected rather arbitrarily for knowledge representation and inference and then they are fuzzified whereas in type ii models representation and inference are based on fuzzy disjunctive and conjunctive normal forms which are derived from the fuzzy truth tables

a simple but powerful heuristic method for generating fuzzy rules from numerical data abstract in this paper we propose a simple but powerful heuristic method for automatically generating fuzzy if then rules from numerical data fuzzy if then rules with nonfuzzy singletons i e real numbers in the consequent parts are generated by the proposed heuristic method the main advantage of the proposed heuristic method is its simplicity i e it involves neither time consuming iterative learning procedures nor complicated rule generation mechanisms we also suggest a linguistic representation method for deriving linguistic rules from fuzzy if then rules with consequent real numbers the proposed linguistic approximation method consists of two linguistic rule tables which can realize exactly the same nonlinear mapping as an original system based on fuzzy if then rules with consequent real numbers using computer simulations on rice taste data we demonstrate the high performance of the proposed heuristic method and illustrate the proposed linguistic representation method

multi central general type 2 fuzzy clustering approach for pattern recognitions this paper presents a new approach to general type 2 fuzzy clustering called multi central general type 2 fuzzy clustering model this model mainly focuses on uncertainty associated with the cluster centers in this model a set of points is considered as the center for each cluster the membership values to the clusters are defined as general type 2 fuzzy sets including primary and secondary variables primary variable indicates the degree of belonging to the central objects and the secondary variables indicate the degree of belonging of the central objects to the center of the cluster there is not any type reduction or defuzzification for updating cluster prototypes in the proposed clustering algorithm the compatible indexes with the proposed model are defined for validation and verification of the clustering process and results several experimental results are given to evaluate the performance of the proposed multi central general type 2 fuzzy clustering the results also compared with the results of the type 1 fuzzy clustering model

additive perturbed generalized mandelbrot julia sets abstract adopting the experimental mathematics method combining complex variable function theory with computer aided drawing this paper researches on the structural characteristic and the fission evolution law of additive perturbed generalized mandelbrot julia sets generalized m j sets in short the corresponding relationship between point coordinates in generalized m set and the general structure of generalized j sets has been founded qualitatively and the physical meaning of the generalized m j sets has been expounded the following conclusions are deduced 1 chaotic patterns of fractal structure of generalized j sets may emerge out of double periodic bifurcation which shows that brownian movement can be chaotic 2 experimental evidence of li yorke theorem is given out 3 the additive perturbed generalized m set contains abundant information on the construction of generalized j sets 4 resemble logistic map in the process of a series of double periodic bifurcation coming into chaos generalized j sets also present self similarity in parameter space

construction of fuzzy models through clustering techniques abstract in this note we will discuss fuzzy models designed with the aid of fuzzy clustering the clustering method utilizes a generalized objective function involving a collection of linear varieties in this way the model is distributed and consists of a series of local linear type models based on the revealed clusters in comparison to the algorithms existing in the literature and producing function like models the proposed one is of a relational character allowing for multidirectional accessibility a detailed identification algorithm is given and illustrated with the aid of numerical examples

handling the nonlinearity of a fuzzy logic controller at the transition between rules abstract a fuzzy logic controller approximates a desired control surface by using the outputs of its fuzzy control rules the shape of this control surface is mainly influenced by the linguistic variables of the rule base the basic operators of the fuzzy sets the implication the inference and the defuzzification method the linearity nonlinearity of this control surface is discussed in many studies however none of those studies has explicitly considered the linearity nonlinearity at the transition between fuzzy logic rules in this paper a simple approach to control the linearity nonlinearity at this transition by a modified center of gravity defuzzification method is proposed namely by introducing the so called defuzzification weights to the overlapping areas of the consequent those defuzzification weights can be determined heuristically or automatically both cases will be demonstrated for the latter a feedforward neural network is employed