Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques

In this paper, we present a new method for multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques. First, the proposed method constructs training samples based on the variation rates of the training data set and then uses the training samples to construct fuzzy rules by making use of the fuzzy C-means clustering algorithm, where each fuzzy rule corresponds to a given cluster. Then, we determine the weight of each fuzzy rule with respect to the input observations and use such weights to determine the predicted output, based on the multiple fuzzy rules interpolation scheme. We apply the proposed method to the temperature prediction problem and the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) data. The experimental results show that the proposed method produces better forecasting results than several existing methods.     

Fuzzy rule interpolation for multidimensional input spaces with applications: a case study

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

Fuzzy control on the basis of equality relations with an examplefrom idle speed control

The way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications. In most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment, which reflects the indistinguishability of measurements or control values. In this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function. As a special case of our approach the authors obtain Mamdani's model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations, and they can explain why typical fuzzy partitions are preferred. The authors also obtain a criterion for reasonable defuzzification strategies. The fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the Volkswagen Golf GTI     

Rainfall prediction model using soft computing technique

 Rainfall prediction in this paper is a spatial interpolation problem that makes use of the daily rainfall information to predict volume of rainfall at unknown locations within area covered by existing observations. This paper proposed the use of self-organising map (SOM), backpropagation neural networks (BPNN) and fuzzy rule systems to perform rainfall spatial interpolation based on local method. The SOM is first used to separate the whole data space into some local surface automatically without any knowledge from the analyst. In each sub-surface, the complexity of the whole data space is reduced to something more homogeneous. After classification, BPNNs are then use to learn the generalization characteristics from the data within each cluster. Fuzzy rules for each cluster are then extracted. The fuzzy rule base is then used for rainfall prediction. This method is used to compare with an established method, which uses radial basis function networks and orographic effect. Results show that this method could provide similar results from the established method. However, this method has the advantage of allowing analyst to understand and interact with the model using fuzzy rules.     

Time series forecasting with a neuro-fuzzy modeling scheme

Time series forecasting concerns the prediction of future values based on the observations previously taken at equally spaced time points. Statistical methods have been extensively applied in the forecasting community for the past decades. Recently, machine learning techniques have drawn attention and useful forecasting systems based on these techniques have been developed. In this paper, we propose an approach based on neuro-fuzzy modeling for time series prediction. Given a predicting sequence, the local context of the sequence is located in the series of the observed data. Proper lags of relevant variables are selected and training patterns are extracted. Based on the extracted training patterns, a set of TSK fuzzy rules are constructed and the parameters involved in the rules are refined by a hybrid learning algorithm. The refined fuzzy rules are then used for prediction. Our approach has several advantages. It can produce adaptive forecasting models. It works for univariate and multivariate prediction. It also works for one-step as well as multi-step prediction. Several experiments are conducted to demonstrate the effectiveness of the proposed approach.     

Stability analysis and design of Takagi-Sugeno fuzzy systems

This work presents stable composite control criteria for multivariable Takagi-Sugeno (T-S) fuzzy systems. On the basis of the linear matrix inequality (LMI) control strategy and parametric optimization, the composite fuzzy control algorithms are derived. Unlike earlier studies of fuzzy control systems on an LMI framework, this investigation develops a supervisory control approach, such that a fuzzy controller can be synthesized more efficiently. Moreover, a robust control scheme is applied to the T-S fuzzy model with parametric uncertainties. The sufficient conditions are deduced in the form of reduced LMIs and adaptive tuning rules. Finally, numeric simulations are given to validate the proposed approach.     

Feature space mapping as a universal adaptive system

The most popular realizations of adaptive systems are based on the neural network type of algorithms, in particular feedforward multilayered perceptrons trained by backpropagation of error procedures. In this paper an alternative approach based on multidimensional separable localized functions centered at the data clusters is proposed. In comparison with the neural networks that use delocalized transfer functions this approach allows for full control of the basins of attractors of all stationary points. Slow learning procedures are replaced by the explicit construction of the landscape function followed by the optimization of adjustable parameters using gradient techniques or genetic algorithms. Retrieving information does not require searches in multidimensional subspaces but it is factorized into a series of one-dimensional searches. Feature Space Mapping is applicable to learning not only from facts but also from general laws and may be treated as a fuzzy expert system (neurofuzzy system). The number of nodes (fuzzy rules) is growing as the network creates new nodes for novel data but the search time is sublinear in the number of rules or data clusters stored. Such a system may work as a universal classificator, approximator and reasoning system. Examples of applications for the identification of spectra (classification), intelligent databases (association) and for the analysis of simple electrical circuits (expert system type) are given.     

Online elicitation of Mamdani-type fuzzy rules via TSK-based generalized predictive control

Many synergies have been proposed between soft-computing techniques, such as neural networks (NNs), fuzzy logic (FL), and genetic algorithms (GAs), which have shown that such hybrid structures can work well and also add more robustness to the control system design. In this paper, a new control architecture is proposed whereby the on-line generated fuzzy rules relating to the self-organizing fuzzy logic controller (SOFLC) are obtained via integration with the popular generalized predictive control (GPC) algorithm using a Takagi-Sugeno-Kang (TSK)-based controlled autoregressive integrated moving average (CARIMA) model structure. In this approach, GPC replaces the performance index (PI) table which, as an incremental model, is traditionally used to discover, amend, and delete the rules. Because the GPC sequence is computed using predicted future outputs, the new hybrid approach rewards the time-delay very well. The new generic approach, named generalized predictive self-organizing fuzzy logic control (GPSOFLC), is simulated on a well-known nonlinear chemical process, the distillation column, and is shown to produce an effective fuzzy rule-base in both qualitative (minimum number of generated rules) and quantitative (good rules) terms.     

A parallel genetic local search algorithm for intrusion detection in computer networks

The security of networked computers plays a strategic role in modern computer systems. This task is so complicated because the determination of normal and abnormal behaviors in computer networks is hard, as the boundaries cannot be well defined. One of the difficulties in such a prediction process is the generation of false alarms in many anomaly based intrusion detection systems. However, fuzzy logic is an important solution to reduce the false alarm rate in determining intrusive activities. This paper proposes a parallel genetic local search algorithm (PAGELS) to generate fuzzy rules capable of detecting intrusive behaviors in computer networks. The system uses the Michigan's approach, where each individual represents a fuzzy rule which has the form “if condition then prediction.” In the presented algorithm the global population is divided into some subpopulations, each assigned to a distinct processor. Each subpopulation consists of the same class fuzzy rules. These rules evolve independently in the proposed parallel manner. Experimental results show that the presented algorithm produces fuzzy rules, which can be used to construct a reliable intrusion detection system.     

模糊推理协处理器芯片

模糊推理协处理器VLSI芯片F200采用0.8μm CMOS工艺,作为一种模糊 控制器,主要用于实时过程控制和其它适合的应用场合,例如机器人控制、分类器、专家系 统等.F200芯片支持多个模糊知识库工作,支持最常用的两种模糊模型,Mamdani和 Trakagi-Sugeno模型.芯片精度12位,主频20MHz,推理速度约为每秒1.2M条模糊规则.     

Interpolation and extrapolation of fuzzy quantities revisited – an axiomatic approach

 This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. For this purpose the well-known fuzzy inference mechanisms have to be extended or replaced by more general ones. The methods proposed so far in the literature for rule interpolation are mainly conceived for the application to fuzzy control and miss certain logical characteristics of an inference. First, a set of axioms is proposed in this paper. With this, a definition is given for the notion of interpolation, extrapolation, linear interpolation and linear extrapolation of fuzzy rules. The axioms include all the conditions that have been of interest in the previous attempts and others which either have logical characteristics or try to capture the linearity of the interpolation. A new method for linear interpolation and extrapolation of compact fuzzy quantities of the real line is suggested and analyzed in the spirit of the given definition. The method is extended to non-linear interpolation and extrapolation as well.     

Multiobjective genetic fuzzy rule selection of single granularity-based fuzzy classification rules and its interaction with the lateral tuning of membership functions

Multiobjective genetic fuzzy rule selection is based on the generation of a set of candidate fuzzy classification rules using a preestablished granularity or multiple fuzzy partitions with different granularities for each attribute. Then, a multiobjective evolutionary algorithm is applied to perform fuzzy rule selection. Since using multiple granularities for the same attribute has been sometimes pointed out as to involve a potential interpretability loss, a mechanism to specify appropriate single granularities at the rule extraction stage has been proposed to avoid it but maintaining or even improving the classification performance. In this work, we perform a statistical study on this proposal and we extend it by combining the single granularity-based approach with a lateral tuning of the membership functions, i.e., complete contexts learning. In this way, we analyze in depth the importance of determining the appropriate contexts for learning fuzzy classifiers. To this end, we will compare the single granularity-based approach with the use of multiple granularities with and without tuning. The results show that the performance of the obtained classifiers can be even improved by obtaining the appropriate variable contexts, i.e., appropriate granularities and membership function parameters.     

Genetic tuning of fuzzy inference within fuzzy classifier systems

Abstract: In generating a suitable fuzzy classifier system, significant effort is often placed on the determination and the fine tuning of the fuzzy sets. However, in such systems little thought is given to the way in which membership functions are combined within the fuzzy rules. Often traditional fuzzy inference strategies are used which consequently provide no control over how strongly or weakly the inference is applied within these rules. Furthermore such strategies will allow no interaction between grades of membership. A number of theoretical fuzzy inference operators have been proposed for both regression and classification problems but they have not been investigated in the context of real-world applications. In this paper we propose a novel genetic algorithm framework for optimizing the strength of fuzzy inference operators concurrently with the tuning of membership functions for a given fuzzy classifier system. Each fuzzy system is generated using two well-established decision tree algorithms: C4.5 and CHAID. This will enable both classification and regression problems to be addressed within the framework. Each solution generated by the genetic algorithm will produce a set of fuzzy membership functions and also determine how strongly the inference will be applied within each fuzzy rule. We investigate several theoretical proven fuzzy inference techniques (T-norms) in the context of both classification and regression problems. The methodology proposed is applied to a number of real-world data sets in order to determine the effects of the simultaneous tuning of membership functions and inference parameters on the accuracy and robustness of fuzzy classifiers.     

Evolving Takagi–Sugeno fuzzy model based on switching to neighboring models

In this paper, we propose a new online identification approach for evolving Takagi–Sugeno (TS) fuzzy models. Here, for a TS model, a certain number of models as neighboring models are defined and then the TS model switches to one of them at each stage of evolving. We define neighboring models for an in-progress (current) TS model as its fairly evolved versions, which are different with it just in two fuzzy rules. To generate neighboring models for the current model, we apply specially designed split and merge operations. By each split operation, a fuzzy rule is replaced with two rules; while by each merge operation, two fuzzy rules combine to one rule. Among neighboring models, the one with the minimum sum of squared errors – on certain time intervals – replaces the current model.To reduce the computational load of the proposed evolving TS model, straightforward relations between outputs of neighboring models and that of current model are established. Also, to reduce the number of rules, we define and use first-order TS fuzzy models whose generated local linear models can be localized in flexible fuzzy subspaces. To demonstrate the improved performance of the proposed identification approach, the efficiency of the evolving TS model is studied in prediction of monthly sunspot number and forecast of daily electrical power consumption. The prediction and modeling results are compared with that of some important existing evolving fuzzy systems.     

模糊推理的函数变换观点

基于函数论的立场 ,指出模糊推理过程是一个函数变换过程 ,模糊规则蕴涵了一个从函数空间到函数空间的映射 ,现存的种种模糊推理方法都是对这种映射的估计 ,进而指出插值和回归的方法都适用于这种估计。系统地提出了用回归的方法处理模糊推理的思想 ,并结合线性回归模型进行了示范 ,证明了基于线性回归模型的模糊推理系统 (FIS)同样是一个万能函数逼近器。     

Simplification of a fuzzy mechanism with rule combination

 In this paper, a systematic approach to reduce the complexity of a fuzzy controller with the rule combination of a fuzzy rule base is presented. The complexity of a fuzzy controller is defined to be the computation load in this work. The proposed rule combination approach can be applied to the fuzzy mechanisms with product–sum and min–max inferences. With the input membership functions indexed in sequence for each input variable, the n-dimensional fuzzy rule table is represented as vectors so that the combination of the fuzzy rule base is realizable. Then the adjacent fuzzy rules with the same output consequent are combined to have smaller size of fuzzy rule base. The fuzzy mechanism with the combined rule table is shown to have the same output with the original fuzzy mechanism (without rule combination). Thus, in many applications, the rule combination approach presented in this paper can be used to reduce the complexity of the fuzzy mechanism without degrading the performances. Moreover, the Don't Care fuzzy rules are defined and it is indicated that the number of the necessary fuzzy rules might be decreased when the Don't Care fuzzy rules are taken into consideration. Further, the properties of the simplification approach for the fuzzy rule base of the fuzzy mechanism are discussed.     

Application of interval type-2 fuzzy neural networks in non-linear identification and time series prediction

Neural networks (NNs), type-1 fuzzy logic systems and interval type-2 fuzzy logic systems (IT2FLSs) have been shown to be important methods in real world applications, which range from pattern recognition, time series prediction, to intelligent control. Recent research shows that embedding an IT2FLS on an NN can be very effective for a wide number of non-linear complex systems, especially when handling imperfect or incomplete information. In this paper we are presenting several models of interval type-2 fuzzy neural networks (IT2FNNs) that use a set of rules and interval type-2 membership functions for that purpose. Simulation results of non-linear function identification using the IT2FNN for one and three variables and for the Mackey–Glass chaotic time series prediction are presented to illustrate that the proposed models have potential for real world applications.     

Stability analysis of nonlinear multivariable Takagi-Sugeno fuzzycontrol systems

Points out how the nonlinearities involved in multivariable Takagi-Sugeno (T-S) fuzzy control systems could originate complex behavior phenomena, such as multiple equilibrium points or limit cycles, that cannot be detected using conventional stability analysis techniques. In the paper, the application of MIMO frequency-domain methods to predict the existence of multiple equilibria and of limit cycles are presented. The proposed method is based on the formulation of a Lur'e problem from the original structure of a T-S fuzzy system with a fuzzy controller. Furthermore, this technique makes straightforward the application of input-output stability techniques such as the multivariable circle criterion, also called the conicity criterion, and the harmonic balance method. Moreover, in the paper, the application of the harmonic balance method has been generalized to the case of a MIMO fuzzy system with asymmetric nonlinearities and improved by the decreasing conservatism. A new and more general stability index which could be used to perform a bifurcation analysis of fuzzy control systems is presented. The paper includes a collection of examples where the advantages of the proposed approach are made explicit comparing it to the input-output conicity criterion and the Lyapunov direct method     

Towards a theory of fuzzy causality for diagnosis

Parsimonious covering offers an alternative to rules for building diagnostic expert systems. Abductive paradigms, such as parsimonious covering, are a departure from the forward-chaining, rule-based approach, which is based on deduction. Parsimonious covering addresses weaknesses of rule-based systems where the diagnosis may contain multiple faults or disorders, or where the need to include all the necessary context for each rule's application in the antecedent clauses of each rule would make the representation of the knowledge base too large or overly complex.

In this paper, we compare the notions of deterministic covering and the probabilistic causal model with two fuzzy analogies: fuzzy subsethood and fuzzy similarity. Monotonic upper and lower bounds for fuzzy similarity are derived, and pruning opportunities are identified for search through the power set of disorders, given a measured, crisp manifestation set.     

两类模糊系统具有插值性的充要条件

当模糊系统具有插值性时,它必具有泛逼近性.因此,由插值性可以分析模糊系统的逼近能力.本文讨论了由“交”和“并”的方式聚合推理规则所生成的两类模糊系统的插值性问题.首先,通过分析由“单点”模糊化方法、CRI(com positional ru le of inference)算法以及“重心法”构造的模糊系统,指出模糊系统是否具有插值性关键取决于模糊蕴含算子的第二个变量为0和1时的表达式或取值.在此基础上,得到两类模糊系统具有插值性的充要条件.最后给出了满足这两个充要条件的一些常用的蕴涵算子.