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

ensembles of fuzzy linear model trees for the identification of multioutput systems we address the task of discrete time modeling of nonlinear dynamic systems with multiple outputs using measured data in the area of control engineering this task is typically converted into a set of classical regression problems one for each output which can then be solved with any nonlinear regression approach fuzzy models in the takagi sugeno form are popular in this context we use lolimot which is a tree learning method to build fuzzy linear model trees in this paper we propose implement and empirically evaluate three extensions of fuzzy linear model trees first we consider and evaluate multioutput models second we propose to use ensembles of such models third we investigate the use of a search heuristic based on simulation error as opposed to one step ahead prediction error specific to the context of modeling dynamic systems finally we perform an empirical evaluation and compare these approaches on six multioutput case studies using both measured and simulated data with noise the case studies include modeling of the inverse dynamics of a robot arm as well as five additional process industry systems ensembles improve the performance of both single and multioutput trees while the heuristic specific to modeling dynamic systems only improves performance very slightly multioutput model trees exhibit comparable or worse predictive performance to a set of single output models while providing a more compact model overall we can recommend the use of bagging of single output lolimot models learned by using the simulation error as a search heuristic

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

constructing narmax models using armax models this paper outlines how it is possible to decompose a complex non linear modelling problem into a set of simpler linear modelling problems local armax models valid within certain operating regimes are interpolated to construct a global narmax non linear narmax model knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models hence it should not be considered as a pure black box approach but as an approach that utilizes a limited amount of a priori system knowledge it is shown that a large class of non linear systems can be modelled in this way and indicated how to decompose the systems range of operation into operating regimes standard system identification algorithms can be used to identify the narmax model and several aspects of the system identification problem are discussed and illustrated by a simulation example

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

top down induction of first order logical decision trees a first order framework for top down induction of logical decision trees is introduced the expressivity of these trees is shown to be larger than that of the flat logic programs which are typically induced by classical ilp systems and equal to that of first order decision lists these results are related to predicate invention and mixed variable quantification finally an implementation of this framework the tilde system is presented and empirically evaluated

adaptive control of mimo time varying systems with indicator function based parametrization abstract this paper studies a new solution framework for adaptive control of a class of mimo time varying systems with indicator function based parametrization motivated by a general discrete time mimo takagi sugeno t s fuzzy system model in an input output form with unknown parameters an indicator membership function based parametrization has some favorable capacity to deal with certain large parameter variations a new discrete time mimo system prediction model is derived for approximating a nonlinear dynamic system and its system properties are clarified an adaptive control scheme is developed with desired controller parametrization and stable parameter estimation for control of such uncertain mimo time varying systems a control singularity problem is addressed and the closed loop stability and output tracking properties are analyzed this work provides a new method for multivariable t s fuzzy system modeling and adaptive control an illustrative example and simulation results are presented to demonstrate the proposed novel concepts and to verify the desired adaptive control system performance

theories for mutagenicity a study in first order and feature based induction abstract a classic problem from chemistry is used to test a conjecture that in domains for which data are most naturally represented by graphs theories constructed with inductive logic programming ilp will significantly outperform those using simpler feature based methods one area that has long been associated with graph based or structural representation and reasoning is organic chemistry in this field we consider the problem of predicting the mutagenic activity of small molecules a property that is related to carcinogenicity and an important consideration in developing less hazardous drugs by providing an ilp system with progressively more structural information concerning the molecules we compare the predictive power of the logical theories constructed against benchmarks set by regression neural and tree based methods

interval valued membership function estimation for fuzzy modeling abstract fuzzy modeling is an important topic in fuzzy sets theory and applications a powerful method for constructing an interval valued takagi sugeno fuzzy model ivfm based on input output data of the identified system is presented in this investigation a takagi sugeno fuzzy model is automatically generated in three steps 1 structure identification 2 envelope detection and 3 parameters identification in the structure identification phase a clustering method based on gustafson kessel algorithm is used in order to detect the linear subsystems of the whole nonlinear system local linearization then an envelope detection algorithm eda based on derivative concept is proposed to estimate both the upper and lower functions of the interval valued membership function defined point wise in the parameter identification step the least squares algorithm is applied to compute the best parameter values of the premises gaussians and the kalman filter algorithm to compute the consequences straight lines parameters the effectiveness of this approach is demonstrated on approximating some nonlinear static functions real world data and dynamical systems

the tase nf model for function approximation problems optimization of the local sub models when using neuro fuzzy systems to model input output data can be spoiled during the global optimization process therefore special care has to be taken to avoid this loss some approaches involving grid input space partitioning that optimize local sub models whilst still obtaining a good global system performance have appeared however this is a more complex task for radial basis function networks scatter partitioning fuzzy systems and neuro fuzzy models in general this work presents a modified neuro fuzzy model known as the tase nf model which is a special case of a scatter partitioned takagi sugeno kang fuzzy system or normalized radial basis function neural network the tase nf model maintains the optimization properties of the local sub models fuzzy rules or rbf nodes when the model is globally optimized thanks to the modified calculation of the final normalized activation of the rules which provides an appropriate input space partitioning to achieve these objectives to show the characteristics of the proposed model a learning methodology for this model which consists of a clustering algorithm especially suited to function approximation problems a local search technique and a membership function merging approach with the objective of improving the transparency i e ease of interpretability of the extracted fuzzy rule set is also proposed

fuzzy modeling of high dimensional systems complexity reduction and interpretability improvement fuzzy modeling of high dimensional systems is a challenging topic this paper proposes an effective approach to data based fuzzy modeling of high dimensional systems an initial fuzzy rule system is generated based on the conclusion that optimal fuzzy rules cover extrema redundant rules are removed based on a fuzzy similarity measure then the structure and parameters of the fuzzy system are optimized using a genetic algorithm and the gradient method during optimization rules that have a very low firing strength are deleted finally interpretability of the fuzzy system is improved by fine training the fuzzy rules with regularization the resulting fuzzy system generated by this method has the following distinct features 1 the fuzzy system is quite simplified 2 the fuzzy system is interpretable and 3 the dependencies between the inputs and the output are clearly shown this method has successfully been applied to a system that has 11 inputs and one output with 20 000 training data and 80 000 test data

supervised hierarchical clustering in fuzzy model identification this paper presents a new supervised hierarchical clustering algorithm suhiclust for fuzzy model identification the presented algorithm solves the problem of global model accuracy together with the interpretability of local models as valid linearizations of the modeled nonlinear system the algorithm combines the advantages of supervised hierarchical algorithms which are based on heuristic tree construction algorithms together with the advantages of fuzzy product space clustering the high flexibility of the validity functions that is obtained by fuzzy clustering combined with supervised learning results in an efficient partitioning algorithm which is independent of initialization and results in a parsimonious fuzzy model furthermore the usability of suhiclust is very undemanding because it delivers in contrast with many other methods reproducible results in order to get reasonable results the user only has to set either a threshold for the maximum number of local models or a value for the maximum allowed global model error as a termination criterion for fine tuning the interpolation smoothness controls the degree of regularization the performance is illustrated on both analytical examples and benchmark problems from the literature

identifying fuzzy models utilizing genetic programming abstract fuzzy models offer a convenient way to describe complex nonlinear systems moreover they permit the user to deal with uncertainty and vagueness due to these advantages fuzzy models are employed in various fields of applications e g control forecasting and pattern recognition nevertheless it has to be emphasized that the identification of a fuzzy model is a complex optimization task with many local minima genetic programming provides a way to solve such complex optimization problems in this work the use of genetic programming to identify the input variables the rule base and the involved membership functions of a fuzzy model is proposed for this purpose several new reproduction operators are introduced

forced vertical vibrations of an elastic elliptic plate on an elastic half space a direct approach using orthogonal polynomials the forced vibration of an elastic half space produced by a rigid elliptic indenter oscillating about an axis perpendicular to the plane face of the half space is considered the boundary conditions lead to a two dimensional dual integral equation in terms of the unknown normal stress by appropriate substitution the dual integral equation is first reduced to a two dimensional fredholm integral equation this is transformed to an infinite set of equations using abelian transformations next the abel transformed variable of the unknown normal stress is expanded in terms of orthogonal jacobi polynomials and by solving the system of linear equations orthogonal polynomial solutions are obtained the method used to obtain the orthogonal polynomial solutions of this problem is new and the major advantage of this expansion technique is that it is valid for all frequencies detailed numerical work has been given for the total load on the disc for different values of frequencies

singular foliations with trivial canonical class this paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds

improving the interpretability of tsk fuzzy models by combining global learning and local learning the fuzzy inference system proposed by takagi sugeno and kang known as the tsk model in fuzzy system literature provides a powerful tool for modeling complex nonlinear systems unlike conventional modeling where a single model is used to describe the global behavior of a system tsk modeling is essentially a multimodel approach in which simple submodels typically linear models are combined to describe the global behavior of the system most existing learning algorithms for identifying the tsk model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model although these algorithms can generate a tsk model with good global performance i e the model is capable of approximating the given system with arbitrary accuracy provided that sufficient rules are used and sufficient training data are available they cannot guarantee the resulting model to have a good local performance often the submodels in the tsk model may exhibit an erratic local behavior which is difficult to interpret since one of the important motivations of using the tsk model also other fuzzy models is to gain insights into the model it is important to investigate the interpretability issue of the tsk model we propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework this algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics but remains the component of global fitting in the existing learning algorithms the algorithm is capable of adjusting its parameters based on the user s preference generating models with good tradeoff in terms of global fitting and local interpretation we illustrate the performance of the proposed algorithm using a motorcycle crash modeling example

hypersurfaces quasi invariant by codimension one foliations we present a variant of the classical darboux jouanolou theorem our main result provides a characterization of foliations which are pull backs of foliations on surfaces by rational maps as an application we provide a structure theorem for foliations on 3 folds admitting an infinite number of extremal rays

flexfis a robust incremental learning approach for evolving takagi sugeno fuzzy models in this paper we introduce a new algorithm for incremental learning of a specific form of takagi sugeno fuzzy systems proposed by wang and mendel in 1992 the new data driven online learning approach includes not only the adaptation of linear parameters appearing in the rule consequents but also the incremental learning of premise parameters appearing in the membership functions fuzzy sets together with a rule learning strategy in sample mode a modified version of vector quantization is exploited for rule evolution and an incremental learning of the rules premise parts the modifications include an automatic generation of new clusters based on the nature distribution and quality of new data and an alternative strategy for selecting the winning cluster rule in each incremental learning step antecedent and consequent learning are connected in a stable manner meaning that a convergence toward the optimal parameter set in the least squares sense can be achieved an evaluation and a comparison to conventional batch methods based on static and dynamic process models are presented for high dimensional data recorded at engine test benches and at rolling mills for the latter the obtained data driven fuzzy models are even compared with an analytical physical model furthermore a comparison with other evolving fuzzy systems approaches is carried out based on nonlinear dynamic system identification tasks and a three input nonlinear function approximation example

on the optimization of fuzzy decision trees the induction of fuzzy decision trees is an important way of acquiring imprecise knowledge automatically fuzzy id3 and its variants are popular and efficient methods of making fuzzy decision trees from a group of training examples this paper points out the inherent defect of the likes of fuzzy id3 presents two optimization principles of fuzzy decision trees proves that the algorithm complexity of constructing a kind of minimum fuzzy decision tree is np hard and gives a new algorithm which is applied to three practical problems the experimental results show that with regard to the size of trees and the classification accuracy for unknown cases the new algorithm is superior to the likes of fuzzy id3

a cloned linguistic decision tree controller for real time path planning in hostile environments abstract the idea of a cloned controller to approximate optimised control algorithms in a real time environment is introduced a cloned controller is demonstrated using linguistic decision trees ldts to clone a model predictive controller mpc based on mixed integer linear programming milp for unmanned aerial vehicle uav path planning through a hostile environment modifications to the ldt algorithm are proposed to account for attributes with circular domains such as bearings and discontinuous output functions the cloned controller is shown to produce near optimal paths whilst significantly reducing the decision period further investigation shows that the cloned controller generalises to the multi obstacle case although this can lead to situations far outside of the training dataset and consequently result in decisions with a high level of uncertainty a modification to the algorithm to improve the performance in regions of high uncertainty is proposed and shown to further enhance generalisation the resulting controller combines the high performance of mpc milp with the rapid response of an ldt while providing a degree of transparency interpretability of the decision making

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

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

fault tolerant measurement system based on takagi sugeno fuzzy models for a gas turbine in a combined cycle power plant a fault tolerant measurement system for a gas turbine in a combined cycle power plant based on dynamic models principal component analysis pca and q test is presented the proposed scheme makes use of a model based symptom generator which delivers fault signals obtained by using direct identification of parity relations and structured residuals symptoms are then analyzed in a statistical module achieving fault diagnosis and reconstruction of the faulty signals the scheme presents as main advantage the ability of detecting faults in both input and output sensors due to its particular structure tests carried out on the gas turbine of the san isidro combined cycle power plant in the v region chile show that takagi sugeno fuzzy models present the best fitting performance and an acceptable computational cost in comparison with autoregressive exogenous state space and neural models real time software based on this scheme has been developed and connected to osisoft pi system tm the software is running at endesa monitoring and diagnosis center in santiago chile

fuzzy modelling and identification with genetic algorithm based learning abstract a ga based learning algorithm is proposed in this paper for the identification of tsk models the algorithm consists of four blocks partition block ga block tuning block and termination block the partition block is to determine an estimated partition of input variables the ga block is to optimise the structure of a tsk model the tuning block is to fine tune the parameters of the tsk model using the gradient descent based approach and the termination block checks that the resultant tsk model is satisfactory the proposed gabl algorithm has the advantage of simplicity flexibility high accuracy and automation the presented numerical examples indicate that the gabl algorithm is effective in constructing a good tsk model for complex nonlinear systems

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

forced oscillations of an axisymmetric structure in contact with an elastic half space by a version of global local finite elements abstract a version of the global local finite element method glfem is presented for determining the steady state forced response of an axisymmetric structure in contact with an elastic homogeneous isotropic half space in this glfem version conventional finite elements are used to model the structure and some portion of the sorrounding medium and global functions in the form of a complete set of outgoing spherical harmonic waves for the entire space are used to capture the behavior in the half space region beyond the finite element mesh an arbitrary distribution of steady state normal loads may be applied to the structure full traction and displacement continuity are enforced at the finite element mesh interface with the outer region on the half space surface of the outer region traction free surface conditions are met by requiring a sequence of weighted average integrals of the tractions to vanish comparison of the present results for the problem of a rigid plate in frictionless contact with the half space with those obtained by krenk and schmidt 1 show excellent agreement over a wide frequency range other examples on circular plates under various contact conditions with the half space are presented as illustrations of the forced response

disjunctive normal random forests we develop a novel supervised learning classification method called disjunctive normal random forest dnrf a dnrf is an ensemble of randomly trained disjunctive normal decision trees dndt to construct a dndt we formulate each decision tree in the random forest as a disjunction of rules which are conjunctions of boolean functions we then approximate this disjunction of conjunctions with a differentiable function and approach the learning process as a risk minimization problem that incorporates the classification error into a single global objective function the minimization problem is solved using gradient descent dnrfs are able to learn complex decision boundaries and achieve low generalization error we present experimental results demonstrating the improved performance of dndts and dnrfs over conventional decision trees and random forests we also show the superior performance of dnrfs over state of the art classification methods on benchmark datasets

identification of non linear system structure and parameters using regime decomposition abstract an off line algorithm for empirical modeling and identification of non linear dynamic systems is presented the minimal input to the algorithm is a sequence of empirical data and the model order using this information the algorithm searches for an optimal model structure and parameters within a rich non linear model set the model representation is based on the interpolation of a number of simple local models where each local model has a limited range of validity but the local models yield a complete global model when interpolated the method is illustrated using simulated data

on constructing a fuzzy inference framework using crisp decision trees this paper proposes a framework which consists of a novel fuzzy inference algorithm to generate fuzzy decision trees from induced crisp decision trees fuzzy theoretical techniques are used to fuzzify crisp decision trees in order to soften sharp decision boundaries at decision nodes inherent in this type of trees a framework for the investigation of various types of membership functions and fuzzy inference techniques is proposed once the decision tree has been fuzzified all branches throughout the tree will fire resulting in a membership grade being generated at each branch five different fuzzy inference mechanisms are used to investigate the degree of interaction between membership grades on each path in the decision tree which ultimately leads to a final crisp classification a genetic algorithm is used to optimize and automatically determine the set of fuzzy regions for all branches and simultaneously the degree in which the inference parameters will be applied comparisons between crisp trees and the fuzzified trees suggest that the later fuzzy tree is significantly more robust and produces a more balanced classification in addition the results obtained from five real world data sets show that there is a significant improvement in the accuracy of the fuzzy trees when compared with crisp trees

forced asymmetric vibrations of an axisymmetric body in contact with an elastic half space a global local finite element approach forced asymmetric vibrations of an axisymmetric structure in contact with an elastic half space are investigated by means of an extended version of the global local finite element method glfem in glfem the structure and part of the surrounding medium are modeled by conventional finite elements the behavior beyond this finite element model i e in the remainder of the half space region is represented by analytical functions called global functions in the form of spherical harmonics the solution process requires that both displacement and traction continuity at the finite element interface with the outer field be satisfield also to satisfy traction free surface conditions the non vanishing spherical harmonic tractions on the free surface beyond the finite element mesh are constrained in the form of weighted average integrals that express zero net work due to these stress components the results for rocking vibration of a rigid circular plate in frictionless contact with the half space compare well with existing analytical data response results that illustrate the effects of various embedment conditions of a rigid circular plate completely bonded to the half space are presented lastly an example on flexible transducers fully bonded to a half space is considered data describing the transducers contact displacement distribution near resonance are given

geno fuzzy classification trees making the non terminal nodes of a binary tree classifier fuzzy can mitigate tree brittleness using a genetic algorithm two optimization techniques are explored in one case each generation minimizes classification error by optimizing a common fuzzy percent pt used to determine parameters at every node in the other case each generation yields a sequence of minimized node specific parameters the output value is determined through defuzzification after input vectors in general take both paths at each node with a weighting factor determined by the node membership functions experiments conducted using this geno fuzzy approach yield an improvement compared with other classical algorithms

designing a fuzzy model by adaptive macroevolution genetic algorithms abstract in this paper the adaptive macroevolution genetic algorithms are proposed to identify three different types of fuzzy models several newly established techniques such as adaptive choice function and macroevolution are adopted into the simple genetic algorithms to improve the optimization capability the genetic algorithms used here are controlled to retain the best solution in the population until a better one is found in the next generation the procedures of applying the proposed genetic algorithms to optimize the fuzzy models are also investigated we also compare whether the instantaneous or batch model is more suitable to adjust the fuzzy models the refined fuzzy models are then used to remedy the prediction output from a grey system the superiority of the adaptive macroevolution genetic algorithms to the simple ones is discussed and an example is given to verify our viewpoints simulation results from difference equation and prediction filter are presented to illustrate the effectiveness of the proposed genetic based fuzzy grey model

lmi based design of constrained fuzzy predictive control predictive control of nonlinear systems subject to output and input constraints is considered a fuzzy model is used to predict the future behavior two new ideas are proposed here first an added constraint on the applied control action is used to ensure the decrease of a quadratic lyapunov function and so guarantee lyapunov exponential stability of the closed loop system second the feasibility of the finite horizon optimization problem with the added constraints is ensured based on an off line solution of a set of lmis the novel stability method is compared to the existing methods such as the techniques based on the end point constraints terminal constraint set and the robust stability techniques based on the small gain theory the proposed method ensures lyapunov exponential stability does not need an auxiliary controller and can be used with any feasible controller parameters illustrative examples including the predictive control of a highly nonlinear chemical reactor cstr are discussed

decision tree based fuzzy classifier of h1 magnetic resonance spectra from cerebrospinal fluid samples this paper presents a method for classifying cerebrospinal fluid csf samples studied by proton magnetic resonance spectroscopy h1 mrs into clinical subgroups by means of a fuzzy classifier the method focuses on the analysis of a low signal to noise region of the spectra and is designed to use a small number of samples because sampling can only be done through an invasive technique the proposed method involves the fusion of classifiers based on decision trees designed using fuzzy techniques the fusion step was carried out by ordered weighted averaging owa operators the quality of the proposed classifier was evaluated by efficiency and robustness quality indexes using a method based on a cross validation technique results show excellent classification levels and satisfactory robustness in both training and test sets

globally optimal fuzzy decision trees for classification and regression a fuzzy decision tree is constructed by allowing the possibility of partial membership of a point in the nodes that make up the tree structure this extension of its expressive capabilities transforms the decision tree into a powerful functional approximant that incorporates features of connectionist methods while remaining easily interpretable fuzzification is achieved by superimposing a fuzzy structure over the skeleton of a cart decision tree a training rule for fuzzy trees similar to backpropagation in neural networks is designed this rule corresponds to a global optimization algorithm that fixes the parameters of the fuzzy splits the method developed for the automatic generation of fuzzy decision trees is applied to both classification and regression problems in regression problems it is seen that the continuity constraint imposed by the function representation of the fuzzy tree leads to substantial improvements in the quality of the regression and limits the tendency to overfitting in classification fuzzification provides a means of uncovering the structure of the probability distribution for the classification errors in attribute space this allows the identification of regions for which the error rate of the tree is significantly lower than the average error rate sometimes even below the bayes misclassification rate

top down induction of fuzzy pattern trees fuzzy pattern tree induction was recently introduced as a novel machine learning method for classification roughly speaking a pattern tree is a hierarchical tree like structure whose inner nodes are marked with generalized fuzzy logical operators and whose leaf nodes are associated with fuzzy predicates on input attributes a pattern tree classifier is composed of an ensemble of such pattern trees one for each class label this type of classifier is interesting for several reasons for example since a single pattern tree can be considered as a kind of logical description of a class it is quite appealing from an interpretation point of view moreover in terms of classification accuracy the method has shown promising performance in first experimental studies yet as will be argued in this paper the algorithm that has originally been proposed for learning fuzzy pattern trees from data offers scope for improvement here we propose a new method that modifies the original proposal in several ways notably our learning algorithm reverses the direction of pattern tree construction from bottom up to top down additionally a different termination criterion is proposed that is more adapted to the learning problem at hand experimentally it will be shown that our approach is indeed able to outperform the original learning method in terms of predictive accuracy

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

identification and simplification of t s fuzzy neural networks based on incremental structure learning and similarity analysis abstract this paper proposes a novel identification method and a simplification scheme for t s fuzzy neural networks which consists of two steps the first step refers to the structure design based on an incremental clustering approach whose basic ideas are that the structure identification of fuzzy neural networks is guided by the attenuation of output approximation error in each cluster and processed by a recursive refined clustering iteration with the input space clustering and sub clustering as the main steps once the structure of a t s fuzzy neural network is identified by the incremental clustering approach its parameters are further learned and refined by the levenberg marquardt optimization algorithm the second step refers to the structure simplification including removing redundant fuzzy rules and merging highly similar fuzzy rules furthermore the performances of several similarity calculating methods are analyzed and discussed which provides a basis for the selection of the appropriate similarity analysis and effective calculating method for the merging of fuzzy rules and system simplification that is the given performance analysis provides the methodology basis and design guide for structure simplification based on similarity analysis and merger of fuzzy rules several experiments are implemented to illustrate the feasibility and effectiveness of the proposed approach

first order jk clausal theories are pac learnable abstract we present positive pac learning results for the nonmonotonic inductive logic programming setting in particular we show that first order range restricted clausal theories that consist of clauses with up to k literals of size at most j each are polynomial sample polynomial time pac learnable with one sided error from positive examples only in our framework concepts are clausal theories and examples are finite interpretations we discuss the problems encountered when learning theories which only have infinite nontrivial models and propose a way to avoid these problems using a representation change called flattening finally we compare our results to pac learnability results for the normal inductive logic programming setting

author s reply to comments on a proposal to improve the accuracy of linguistic modelling in our opinion there are two main concerns in roubos and babugka s note that are summarized as follows 1 the kinds of problems used in our paper to test the algorithm proposed in a proposal to improve the accuracy of linguistic modeling and other studies the authors claim that they are very simple to be considered as benchmarks for nonlinear modeling techniques 2 the interpretability of the different kinds of models considered roubos and babuska think that there is no difference between the interpretability of fuzzy linguistic models takagi sugeno kang tsk fuzzy models and mathematical formulations linear models in this case we agree with some of the opinions of the authors of the note but not with some others

some new qualitative insights into quality of fuzzy rule based models rules in fuzzy rule based models convey essential knowledge about the system under discussion as such they capture the essence of relationships occurring among input and output variables while the quality of such fuzzy models is predominantly related with the accuracy and eventual interpretability of rules although to a limited extent the quality of rules being regarded as generic pieces of knowledge has not been studied so far in this study we formulate and investigate this problem by looking at the quality of rules including aspects of a stability b generalizability and c conflict we identify a concept of rule multiplicity conflict and study an emergence of rule generalization a number of pertinent performance indices are developed and their usage is presented through a series of experimental studies

why fuzzy decision trees are good rankers several fuzzy extensions of decision tree induction which is an established machine learning method have already been proposed in the literature so far however fuzzy decision trees have almost exclusively been used for the performance task of classification in this paper we show that a fuzzy extension of decision trees is arguably more useful for another performance task namely ranking roughly the goal of ranking is to order a set of instances from most likely positive to most likely negative the motivation for applying fuzzy decision trees to this problem originates from recent investigations of the ranking performance of conventional decision trees these investigations will be continued and complemented in this paper our results reveal some properties that seem to be crucial for a good ranking performance properties that are better and more naturally offered by fuzzy than by conventional decision trees most notably a fuzzy decision tree produces scores in terms of membership degrees on a fine granular scale using these membership degrees as a ranking criterion a key problem of conventional decision trees is solved in an elegant way namely the question of how to break ties between instances in the same leaf or more generally between equally scored instances

on the interpretation of local models in blended multiple model structures the construction of non linear dynamics by means of interpolating the behaviour of locally valid models offers an attractive and intuitively pleasing method of modelling non linear systems the approach is used in fuzzy logic modelling operating regime based models and nonlinear statistical models the model structure suggests that the composite local models can be used to interpret in some appropriate manner the overall non linear dynamics in this paper we demonstrate that the interpretation of these local models in the context of multiple model structures is not as straightforward as it might initially appear we argue that the blended multiple model system can be interpreted in two ways as an interpolation of linearisations or as a full parameterisation of the system the choice of interpretation affects experiment design parameter identification and model validation we then show that in some cases the local models give insight into full model behaviour only in a very small region of state space more alarmingly we demonstrate that for off equilibrium behaviour subject to some approximation error a non unique parameterisation of the model dynamics exists hence qualitative conclusions drawn from the behaviour of an identified local model e g regarding stable unstable nodal or complex behaviour must be treated with extreme caution the example of muscle modelling is used to illustrate these points clearly

multiobjective identification of takagi sugeno fuzzy models the problem of identifying the parameters of the constituent local linear models of takagi sugeno fuzzy models is considered in order to address the tradeoff between global model accuracy and interpretability of the local models as linearizations of a nonlinear system two multiobjective identification algorithms are studied particular attention is paid to the analysis of conflicts between objectives and we show that such information can be easily computed from the solution of the multiobjective optimization this information is useful to diagnose the model and tune the weighting priorities of the multiobjective optimization moreover the result of the conflict analysis can be used as a constructive tool to modify the fuzzy model structure including membership functions in order to meet the multiple objectives simple illustrative examples as well as experimental results show the usefulness of the method

arcing classifier with discussion and a rejoinder by the author recent work has shown that combining multiple versions of unstable classifiers such as trees or neural nets results in reduced test set error one of the more effective is bagging here modified training sets are formed by resampling from the original training set classifiers constructed using these training sets and then combined by voting freund and schapire propose an algorithm the basis of which is to adaptively resample and combine hence the acronym arcing so that the weights in the resampling are increased for those cases most often misclassified and the combining is done by weighted voting arcing is more successful than bagging in test set error reduction we explore two arcing algorithms compare them to each other and to bagging and try to understand how arcing works we introduce the definitions of bias and variance for a classifier as components of the test set error unstable classifiers can have low bias on a large range of data sets their problem is high variance combining multiple versions either through bagging or arcing reduces variance significantly

posterior probability based ensemble strategy using optimizing decision directed acyclic graph for multi class classification ensemble strategy is important to develop a decomposition and ensemble method for multi class classification problems most existing ensemble strategies use predetermined and heuristic decision rules in this work we build up the decision rules by optimizing decision directed acyclic graph oddag with classical and fuzzy decision trees to ensemble the posterior probabilities of binary classifiers from one vs one ovo or one vs all ova decomposition strategies for multi class classification problems four widely used extensible algorithms and ten decomposition and ensemble methods incorporating four binary classifiers bcs have been tested on 25 data sets the empirical results show that the methods based on ensemble strategies using oddag are among the top methods that achieve the best performance in terms of two different measures

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

asymmetric vibrations of a circular elastic plate on an elastic half space abstract the asymmetric problem of a vibrating circular elastic plate in frictionless contact with an elastic half space is solved by an integral equation method where the contact stress appears as the unknown function by a trigonometric expansion the problem is reduced to a number of uncoupled two dimensional problems the radial variations of contact stresses and surface displacements are represented by polynomials the coefficients of which are directly related by an infinite matrix that is a function of the vibration frequency the results include a parametric study of the power input as a function of the vibration frequency of various plate stiffnesses and the normal component of the surface displacement field for simple excitation of the plate and passage of a plane rayleigh wave

function approximation with decomposed fuzzy systems if the number of input signals increases a fuzzy system gets increasingly intractable for two reasons on the one hand the knowledge acquisition suffers increasingly from the knowledge engineering bottleneck on the other hand computational and memory demands of fuzzy systems increase strongly thus suffering from the curse of dimensionality the first problem is classically addressed by learning techniques the second by decomposing the fuzzy system the netfan approach network of fuzzy adaptive nodes combines decomposition with learning in order to apply fuzzy techniques to more complex applications different ways to decompose a fuzzy system are discussed in this paper for the favorite decomposition principle it is shown to be an universal function approximator despite decomposition the real world example of controlling a pneumatic positioning system finally demonstrates the applicability and benefits of this type of hierarchical decomposition namely a significantly reduced number of rules

solving the multiple instance problem with axis parallel rectangles the multiple instance problem arises in tasks where the training examples are ambiguous a single example object may have many alternative feature vectors instances that describe it and yet only one of those feature vectors may be responsible for the observed classification of the object this paper describes and compares three kinds of algorithms that learn axis parallel rectangles to solve the multiple instance problem algorithms that ignore the multiple instance problem perform very poorly an algorithm that directly confronts the multiple instance problem by attempting to identify which feature vectors are responsible for the observed classifications performs best giving 89 correct predictions on a musk odor prediction task the paper also illustrates the use of artificial data to debug and compare these algorithms

a complete fuzzy decision tree technique in this paper a new method of fuzzy decision trees called soft decision trees sdt is presented this method combines tree growing and pruning to determine the structure of the soft decision tree with refitting and backfitting to improve its generalization capabilities the method is explained and motivated and its behavior is first analyzed empirically on 3 large databases in terms of classification error rate model complexity and cpu time a comparative study on 11 standard uci repository databases then shows that the soft decision trees produced by this method are significantly more accurate than standard decision trees moreover a global model variance study shows a much lower variance for soft decision trees than for standard trees as a direct cause of the improved accuracy

incremental evolutionary design of tsk fuzzy controllers this paper proposes a novel method for the incremental design and optimization of first order tagaki sugeno kang tsk fuzzy controllers by means of an evolutionary algorithm starting with a single linear control law the controller structure is gradually refined during the evolution structural augmentation is intertwined with evolutionary adaptation of the additional parameters with the objective not only to improve the control performance but also to maximize the stability region of the nonlinear system from the viewpoint of optimization the proposed method follows a divide and conquer approach additional rules and their parameters are introduced into the controller structure in a neutral fashion such that the adaptations of the less complex controller in the previous stage are initially preserved the proposed scheme is evaluated at the task of tsk fuzzy controller design for the upswing and stabilization of a rotational inverted pendulum in the first case the objective is a time optimal controller that upswings the pendulum in to the upper equilibrium point in shortest time the stabilizing controller is designed as a state optimal controller in a second application the optimization method is applied to the design of a fuzzy controller for vision based mobile robot navigation the results demonstrate that the incremental scheme generates solutions that are similar in control performance to pure parameter optimization of only the gains of a tsk system even more important whereas direct optimization of control systems with more than 35 rules fails to identify a stabilizing control law the incremental scheme optimizes fuzzy state space partitions and gains for hundreds of rules

pattern trees induction a new machine learning method fuzzy classification is one of the most important applications in fuzzy set and fuzzy logic related research its goal is to find a set of fuzzy rules that form a classification model most of the existing fuzzy rule induction methods e g the fuzzy decision trees fdts induction method focus on searching rules consisting of triangular norms t norms i e and only but not triangular conorms t conorms or explicitly this may lead to the omission of generating important rules that involve t conorms explicitly this paper proposes a type of tree termed pattern trees pts that makes use of different aggregations including both t norms and t conorms like decision trees pts are an effective tool for classification applications this paper discusses the difference between decision trees and pts and also shows that the subsethood based method sbm and the weighted subsethood based method wsbm are two specific cases of pt induction a novel pt induction method is proposed using similarity measure and fuzzy aggregations the comparison to other classification methods including sbm wsbm c4 5 nearest neighbor support vector machine and fdt induction shows that 1 pts can obtain high accuracy rates in classifications 2 pts are robust to overfltting and 3 pts especially simple pattern trees spts maintain compact tree structures

greedy regression ensemble selection theory and an application to water quality prediction this paper studies the greedy ensemble selection family of algorithms for ensembles of regression models these algorithms search for the globally best subset of regressors by making local greedy decisions for changing the current subset we abstract the key points of the greedy ensemble selection algorithms and present a general framework which is applied to an application domain with important social and commercial value water quality prediction

learning pairwise image similarities for multi classification using kernel regression trees we are often faced with the problem of distinguishing between visually similar objects that share the same general appearance characteristics as opposed to object categorization this task is focused on capturing fine image differences in a pose dependent fashion our research addresses this particular family of problems and is centered around the concept of learning from example pairs formally we construct a parameterized visual similarity function optimally separating the pairs of images that depict the objects of the same class or identity from the pairs representing different object classes identities it combines various image distances that are quantified by comparing local descriptor responses at the corresponding locations in both paired images to find the best combinations we train ensembles of so called kernel regression trees which model the desired similarity function as a hierarchy of fuzzy decision stumps the obtained function is then used within a k nn like framework to address complex multi classification problems through the experiments with several image datasets we demonstrate the numerous advantages of the proposed approach high classification accuracy scalability ease of interpretation and the independence of the feature representation

method for higher order polynomial sugeno fuzzy inference systems a method for higher order polynomial sugeno fuzzy inference systems formation is presented compared to other existing higher order sugeno implementations it uses fewer parameters and compared to zero and 1st order sugeno fuzzy systems it has overall improved model performance while best models are not always obtained via a higher order representation in our proposed method it is possible to choose the polynomial order which best fits the desired data its input is a previously established model found by a clustering algorithm subtractive algorithm in this case afterward parameters of all higher order polynomials are adjusted using recursive least square algorithm for experimental validation multiple benchmark datasets are tested using hold out and k fold validation as well as data forecasting various performance measures are used although akaike information criterion is used as a primary measure to demonstrate that our proposed higher order polynomials have overall better model performance over zero and 1st order polynomials

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

a methodology for automated fuzzy model generation in this paper we propose a generic methodology for the automated generation of fuzzy models the methodology is realized in three stages initially a crisp model is created and in the second stage it is transformed to a fuzzy one in the third stage all parameters entering the fuzzy model are optimized the proposed methodology is novel and generic since it can integrate alternative techniques in each of its stages a specific realization of this methodology is implemented using decision trees for the creation of the crisp model the sigmoid function the min max operators and the maximum defuzzifier for the transformation of the crisp model into a fuzzy one and four different optimization strategies including global and local optimization techniques as well as hybrid approaches the proposed methodology presents several advantages and novelties the transformation of the crisp model to the respective fuzzy one is straightforward ensuring its full automated nature and it introduces a set of parameters expressing the importance of each fuzzy rule the above realization is extensively evaluated using several benchmark data sets from the uci machine learning repository and the obtained results indicate its high efficiency

tase a taylor series based fuzzy system model that combines interpretability and accuracy typically takagi sugeno kang tsk fuzzy rules have been used as a powerful tool for function approximation problems since they have the capability of explaining complex relations among variables using rule consequents that are functions of the input variables but they present the great drawback of the lack of interpretability which makes them not to be so suitable for a wide range of problems where interpretability of the obtained model is a fundamental key in this paper we present a novel approach that extends the work by bikdash ieee trans fuzzy systems 7 6 1999 686 696 in order to obtain an interpretable and accurate model for function approximation from a set of i o data samples which make use of the taylor series expansion of a function around a point to approximate the function using a low number of rules our approach also provides an automatic methodology for obtaining the optimum structure of our taylor series based tase fuzzy system as well as its pseudo optimal rule parameters both antecedents and consequents

fuzzy classification boundaries against adversarial network attacks abstract adversarial machine learning copes with the development of methods to prevent machine learning algorithms from being misled by malicious users this field is especially relevant for applications where machine learning lies in the core of security systems in the field of network security adversarial samples are actually novel network attacks or old attacks with tuned properties this paper proposes to blur classification boundaries in order to enhance machine learning robustness and improve the detection of adversarial samples that exploit learning weaknesses we test this concept by an experimental setup with network traffic in which linear decision trees are wrapped by a one class membership scoring algorithm we benchmark our proposal with plain linear decision trees and fuzzy decision trees results show that evasive attacks i e false negatives tend to be ranked with low class membership levels meaning that they are located in zones close to classification thresholds in addition classification performances improve when membership scores are added as new features using fuzzy class boundaries is highly consistent with the interpretation of many network traffic features used for malware detection moreover it prevents network attackers from exploiting classification boundaries as attack objectives

on codimension 1 del pezzo foliations on varieties with mild singularities in this paper we extend to the singular setting the theory of fano foliations developed in our previous paper araujo and druel adv math 238 70 118 2013 a mathbb q fano foliation on a complex projective variety x is a foliation fancyscript f subsetneq t x whose anti canonical class k fancyscript f is an ample mathbb q cartier divisor in the spirit of kobayashi ochiai theorem we prove that under some conditions the index i fancyscript f of a mathbb q fano foliation is bounded by the rank r of fancyscript f and classify the cases in which i fancyscript f r next we consider mathbb q fano foliations fancyscript f for which i fancyscript f r 1 these are called del pezzo foliations we classify codimension 1 del pezzo foliations on mildly singular varieties

fuzzy system identification using an adaptive learning rule with terminal attractors a fuzzy model with an adaptive learning rule is proposed to approximate a class of nonlinear systems the proposed adaptive learning rule uses the concept of terminal attractors and guarantees that the identification is stable and fast convergence in finite time another subject of this paper is to determine the appropriate rule number of the fuzzy model required for a desired modeling error a learning algorithm is also introduced to identify the centers of membership functions of fuzzy outputs

genetic programming for model selection of tsk fuzzy systems this paper compares a genetic programming gp approach with a greedy partition algorithm lolimot for structure identification of local linear neuro fuzzy models the crisp linear conclusion part of a takagi sugeno kang tsk fuzzy rule describes the underlying model in the local region specified in the premise the objective of structure identification is to identify an optimal partition of the input space into gaussian axis orthogonal fuzzy sets the linear parameters in the rule consequent are then estimated by means of a local weighted least squares algorithm lolimot is an incremental tree construction algorithm that partitions the input space by axis orthogonal splits in each iteration it greedily adds the new model that minimizes the classification error gp performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub optimal intermediate split decisions we compare the performance of both methods for function approximation of a highly non linear two dimensional test function and an engine characteristic map

neuro fuzzy id3 a method of inducing fuzzy decision trees with linear programming for maximizing entropy and an algebraic method for incremental learning abstract id3 is a popular and efficient method of making a decision tree for classification from symbolic data without much computation fuzzy reasoning rules in the form of a decision tree which can be viewed as a fuzzy partition are obtained by fuzzy id3 the aims of this paper are 1 not only the learning from examples but also the interview with domain specialists are needed for knowledge acquisition in expert systems in order to avoid dangerous simplification of the tree by discarding partial knowledge of the experts a measure of uncertainty with maximizing entropy is applied to fuzzy id3 2 basically the tree based learning is nonincremental or single step an algebraic method to facilitate incremental learning like the neural nets is adapted and the fuzzy decision tree which consists of b spline membership function is regarded as three layered neural network 3 prototypes of expert system for estimation of wheel wear characteristics and surface roughness in abrasive cut off are developed

multi target regression with rule ensembles methods for learning decision rules are being successfully applied to many problem domains in particular when understanding and interpretation of the learned model is necessary in many real life problems we would like to predict multiple related nominal or numeric target attributes simultaneously while several methods for learning rules that predict multiple targets at once exist they are all based on the covering algorithm which does not work well for regression problems a better solution for regression is the rule ensemble approach that transcribes an ensemble of decision trees into a large collection of rules an optimization procedure is then used to select the best and much smaller subset of these rules and to determine their respective weights we introduce the fire algorithm for solving multi target regression problems which employs the rule ensembles approach we improve the accuracy of the algorithm by adding simple linear functions to the ensemble we also extensively evaluate the algorithm with and without linear functions the results show that the accuracy of multi target regression rule ensembles is high they are more accurate than for instance multi target regression trees but not quite as accurate as multi target random forests the rule ensembles are significantly more concise than random forests and it is also possible to create compact rule sets that are smaller than a single regression tree but still comparable in accuracy