Gaussian kernel-based fuzzy inference systems for high dimensional regression

We propose a novel architecture for a higher order fuzzy inference system (FIS) and develop a learning algorithm to build the FIS. The consequent part of the proposed FIS is expressed as a nonlinear combination of the input variables, which can be obtained by introducing an implicit mapping from the input space to a high dimensional feature space. The proposed learning algorithm consists of two phases. In the first phase, the antecedent fuzzy sets are estimated by the kernel-based fuzzy c-means clustering. In the second phase, the consequent parameters are identified by support vector machine whose kernel function is constructed by fuzzy membership functions and the Gaussian kernel. The performance of the proposed model is verified through several numerical examples generally used in fuzzy modeling. Comparative analysis shows that, compared with the zero-order fuzzy model, first-order fuzzy model, and polynomial fuzzy model, the proposed model exhibits higher accuracy, better generalization performance, and satisfactory robustness.     

A novel and fast MIMO fuzzy inference system based on a class of fuzzy clustering algorithms with interpretability and complexity analysis

A data driven Fuzzy Inference System (FIS) employs Membership Functions (MFs) with adjustable parameters in its IF part to fuzzify the input data. The input space is partitioned simply by dividing universe of discourse of each input variable into some fuzzy subspaces. The MFs are then defined on the fuzzy subspaces of the input variables. Parameters of the MFs are tuned for maximum accuracy of the system (which demands high runtime) without considering the data structure which impairs interpretability of the FIS and degenerates the system into a black-box tool. Such a FIS does not represent actual structure of the data and its MFs are not necessarily in accord with the data distribution in the input space. In addition, the FIS suffers from exponential complexity of order O(T^r) where T is number of linguistic terms (number of subspaces on the universe of discourse of input variables) and r is number of input variables. This article presents a novel Multiple-Input and Multiple-Output Clustering based Fuzzy Inference System (MIMO CFIS) which is made directly from a class of fuzzy clustering algorithms to overcome these shortcomings. CFIS identifies dense regions of the input data using fuzzy clustering and then places a cluster on each of these regions. These fuzzy clusters represent actual structure of the data and serve as fuzzy rules in the rule base of CFIS and provide MFs that exactly fit the dense regions of the data that makes the system more interpretable and avoids redundant rules. These MFs are normal, convex, and continuous and have no parameter to be tuned (which makes CFIS much faster than other FISs) and fuzzify the input data according to their membership in the clusters. THEN part of CFIS is generalized form of THEN part of Takagi-Sugeno (TS) fuzzy system which accommodates any function of input variables. Despite less number of adjustable parameters, testing error of CFIS is less than that of TS system and its modified versions. Moreover, number of fuzzy rules in CFIS rule base is the same as the number of linguistic terms (or fuzzy clusters) and consequently its complexity is of orderO(T). Also, CFIS is a MIMO system and avoids inconsistent (contradictory) rules by generating well-separated fuzzy clusters whereas TS system is MISO and never guarantees generation of consistent rules. In addition, CFIS satisfies most of the interpretability criteria of FISs.     

Special issue on interpretable fuzzy systems

Interpretability is acknowledged as one of the most appreciated advantages of fuzzy systems in many applications, especially in those with high human interaction where it actually becomes a strong requirement. However, it is important to remark that there is a somehow misleading but widely extended belief, even in part of the fuzzy community, regarding fuzzy systems as interpretable no matter how they were designed. Of course, we are aware the use of fuzzy logic favors the interpretability of designed models. Thanks to their semantic expressivity, close to natural language, fuzzy variables and rules can be used to formalize linguistic propositions which are likely to be easily understandood by human beings. Obviously, this fact makes easier the knowledge extraction and representation tasks carried out when modeling real-world complex systems. Notwithstanding, fuzzy logic is not enough by itself to guarantee the interpretability of the final model. As it is thoroughly illustrated in this special issue, achieving interpretable fuzzy systems is a matter of careful design because fuzzy systems cannot be deemed as interpretable per se. Thus, several constraints have to be imposed along the whole design process with the aim of producing really interpretable fuzzy systems, in the sense that every element of the whole system may be checked and understood by a human being. Otherwise, fuzzy systems may even become black-boxes.     

Hypoglycaemia detection using fuzzy inference system with intelligent optimiser

Hypoglycaemia is a medical term for a body state with a low level of blood glucose. It is a common and serious side effect of insulin therapy in patients with diabetes. In this paper, we propose a system model to measure physiological parameters continuously to provide hypoglycaemia detection for Type 1 diabetes mellitus (TIDM) patients. The resulting model is a fuzzy inference system (FIS). The heart rate (HR), corrected QT interval of the electrocardiogram (ECG) signal (QT_c), change of HR, and change of QT_c are used as the input of the FIS to detect the hypoglycaemic episodes. An intelligent optimiser is designed to optimise the FIS parameters that govern the membership functions and the fuzzy rules. The intelligent optimiser has an implementation framework that incorporates two wavelet mutated differential evolution optimisers to enhance the training performance. A multi-objective optimisation approach is used to perform the training of the FIS in order to meet the medical standards on sensitivity and specificity. Experiments with real data of 16 children (569 data points) with TIDM are studied in this paper. The data are randomly separated into a training set with 5 patients (l99 data points), a validation set with 5 patients (177 data points) and a testing set with 5 patients (193 data points). Experiment results show that the proposed FIS tuned by the proposed intelligent optimiser can offer good performance of classification.     

GSETSK: a generic self-evolving TSK fuzzy neural network with a novel Hebbian-based rule reduction approach

Takagi–Sugeno–Kang (TSK) fuzzy systems have been widely applied for solving function approximation and regression-centric problems. Existing dynamic TSK models proposed in the literature can be broadly classified into two classes. Class I TSK models are essentially fuzzy systems that are limited to time-invariant environments. Class II TSK models are generally evolving systems that can learn in time-variant environments. This paper attempts to address the issues of achieving compact, up-to-date fuzzy rule bases and interpretable knowledge bases in TSK models. It proposes a novel rule pruning method which is simple, computationally efficient and biologically plausible. This rule pruning algorithm applies a gradual forgetting approach and adopts the Hebbian learning mechanism behind the long-term potentiation phenomenon in the brain. It also proposes a merging approach which is used to improve the interpretability of the knowledge bases. This approach can prevent derived fuzzy sets from expanding too many times to protect their semantic meanings. These two approaches are incorporated into a generic self-evolving Takagi–Sugeno–Kang fuzzy framework (GSETSK) which adopts an online data-driven incremental-learning-based approach.Extensive experiments were conducted to evaluate the performance of the proposed GSETSK against other established evolving TSK systems. GSETSK has also been tested on real world dataset using the high-way traffic flow density and Dow Jones index time series. The results are encouraging. GSETSK demonstrates its fast learning ability in time-variant environments. In addition, GSETSK derives an up-to-date and better interpretable fuzzy rule base while maintaining a high level of modeling accuracy at the same time.     

On employing fuzzy modeling algorithms for the valuation of residential premises

In this paper, we investigate fuzzy modeling techniques for predicting the prices of residential premises, based on some main drivers such as usable area of premises, age of a building, number of rooms in a flat, floor on which a flat is located, number of storeys in a building as well as the distance from the city center. Our proposed modeling techniques rely on two aspects: the first one (called SparseFIS) is a batch off-line modeling method and tries to out-sparse an initial dense rule population by optimizing the rule weights within an iterative optimization procedure subject to constrain the number of important rules; the second one (called FLEXFIS) is a single-pass incremental method which is able to build up fuzzy models in an on-line sample-wise learning context. As such, it is able to adapt former generated prediction models with new data recordings on demand and also to cope with on-line data streams. The final obtained fuzzy models provide some interpretable insight into the relations between the various features and residential prices in form of linguistically readable rules (IF-THEN conditions). Both methods will be compared with a state-of-the-art premise estimation method usually conducted by many experts and exploiting heuristic concepts such as sliding time window, nearest neighbors and averaging. The comparison is based on a two real-world data set including prices for residential premises within the years 1998-2008.     

Combining fuzzy weight average with fuzzy inference system for material substitution selection in electric industry

Material selection is a very important issue for an electronics company as it includes many qualitative or quantification factors. The material selection problem is associated with design and manufacturing problems which have been widely investigated. This study develops a hybrid fuzzy decision-making model which combines the fuzzy weight average (FWA) with the fuzzy inference system (FIS) for material substitution selection in the electronics industry. FWA is employed to select a substitute material in an uncertain environment, while FIS is used for reasoning purposes. FWA with α-cuts arithmetic (FWA_α-cut) is a popularly technology in decision-making problems. However, FWA_α-cut may result in the following unanticipated situations: (1) unclear decision situations; (2) undecided results expressed by fuzzy membership functions; and (3) high computational complexity. Therefore, a fuzzy weight average with the weakest t-norm (FWA_Tω) is designed as an alternative method for group decision making. In contrast to traditional FWA methods, FWA_Tω obtains more visible fuzzy results for decision makers with lower computational complexity, and can provide exacter estimation by the weakest t-norm operations in uncertain environment. Thus, the proposed hybrid fuzzy decision-making model imitates an expert’s experiences and can estimate substitution purchasing in various statuses. A real material substitution selection case is employed to examine the feasibility of the proposed model; experimental results reveal that the proposed model performs better than the traditional FWA model in coping with material substitution selection problems.     

Fuzzy inference systems: An integrated modeling environment for collaboration between expert knowledge and data using FisPro

The present paper aims to demonstrate the interest of fuzzy inference systems in system modeling when human interaction is important. It discusses the originality of FIS and their capability to integrate expertise and rule learning from data into a single framework, analyzing their place relatively to concurrent approaches. An open source software implementation is presented, with a focus on the useful features for modeling. Two real world case studies are presented to illustrate the approach and the software utility.     

Designing fuzzy inference systems from data: Aninterpretability-oriented review

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

Design of fuzzy rule-based classifiers with semantic cointension

In computing with words (CWW), knowledge is linguistically represented and has an explicit semantics defined through fuzzy information granules. The linguistic representation, in turn, naturally bears an implicit semantics that belongs to users reading the knowledge base; hence a necessary condition for achieving interpretability requires that implicit and explicit semantics are cointensive. Interpretability is definitely stringent when knowledge must be acquired from data through inductive learning. Therefore, in this paper we propose a methodology for designing interpretable fuzzy models through semantic cointension. We focus our analysis on fuzzy rule-based classifiers (FRBCs), where we observe that rules resemble logical propositions, thus semantic cointension can be partially regarded as the fulfillment of the “logical view”, i.e. the set of basic logical laws that are required in any logical system. The proposed approach is grounded on the employment of a couple of tools: DCf, which extracts interpretable classification rules from data, and Espresso, that is capable of fast minimization of Boolean propositions. Our research demonstrates that it is possible to design models that exhibit good classification accuracy combined with high interpretability in the sense of semantic cointension. Also, structural parameters that quantify model complexity show that the derived models are also simple enough to be read and understood.     

Fault isolation in analog circuits using a fuzzy inference system

This paper introduces a fuzzy inference system (FIS) for single analog fault diagnosis. The ability of fuzzy logic to encode structured knowledge in a numerical framework is exploited in isolating faults in analog circuits. A training set that simulates the behaviour of the circuit due to a set of anticipated single faults as well as the fault-free situation is first constructed. For each anticipated fault, this set relates the circuit measurements to the corresponding deviation in the faulty circuit element from its nominal. These measurements and the deviations in circuit elements are both fuzzified into appropriate linguistic fuzzy values. A fuzzy rule base for each fault that characterizes the circuit response by linking symptoms to causes is built. The outputs of the fuzzy rule bases are then defuzzified to recover crisp values for the deviations in circuit elements. A fault diagnosis procedure that utilizes the proposed FIS is also presented along with a brief analysis and comparison with a number of existing artificial intelligence-based techniques. A test example that demonstrates the potential of this procedure in fault isolation is illustrated.     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

Medical diagnosis of cardiovascular diseases using an interval-valued fuzzy rule-based classification system

ObjectiveTo develop a classifier that tackles the problem of determining the risk of a patient of suffering from a cardiovascular disease within the next 10 years. The system has to provide both a diagnosis and an interpretable model explaining the decision. In this way, doctors are able to analyse the usefulness of the information given by the system.MethodsLinguistic fuzzy rule-based classification systems are used, since they provide a good classification rate and a highly interpretable model. More specifically, a new methodology to combine fuzzy rule-based classification systems with interval-valued fuzzy sets is proposed, which is composed of three steps: (1) the modelling of the linguistic labels of the classifier using interval-valued fuzzy sets; (2) the use of the K_α operator in the inference process and (3) the application of a genetic tuning to find the best ignorance degree that each interval-valued fuzzy set represents as well as the best value for the parameter α of the K_α operator in each rule.ResultsThe suitability of the new proposal to deal with this medical diagnosis classification problem is shown by comparing its performance with respect to the one provided by two classical fuzzy classifiers and a previous interval-valued fuzzy rule-based classification system. The performance of the new method is statistically better than the ones obtained with the methods considered in the comparison. The new proposal enhances both the total number of correctly diagnosed patients, around 3% with respect the classical fuzzy classifiers and around 1% vs. the previous interval-valued fuzzy classifier, and the classifier ability to correctly differentiate patients of the different risk categories.ConclusionThe proposed methodology is a suitable tool to face the medical diagnosis of cardiovascular diseases, since it obtains a good classification rate and it also provides an interpretable model that can be easily understood by the doctors.     

Position tracking of a 3-PSP parallel robot using dynamic growing interval type-2 fuzzy neural control

Parallel robots have complicated structures as well as complex dynamic and kinematic equations, rendering model-based control approaches as ineffective due to their high computational cost and low accuracy. Here, we propose a model-free dynamic-growing control architecture for parallel robots that combines the merits of self-organizing systems with those of interval type-2 fuzzy neural systems. The proposed approach is then applied experimentally to position control of a 3-PSP (Prismatic–Spherical–Prismatic) parallel robot. The proposed rule-base construction is different from most conventional self-organizing approaches by omitting the node pruning process while adding nodes more conservatively. This helps preserve valuable historical rules for when they are needed. The use of interval type-2 fuzzy logic structure also better enables coping with uncertainties in parameters, dynamics of the robot model and uncertainties in rule space. Finally, the adaptation structure allows learning and further adapts the rule base to changing environment. Multiple simulation and experimental studies confirm that the proposed approach leads to fewer rules, lower computational cost and higher accuracy when compared with two competing type-1 and type-2 fuzzy neural controllers.     

支持向量机和一类模糊推理系统的等效性及其应用

支持向量机（ＳＶＭ）和模糊推理系统（ＦＩＳ）分别源于统计学习理论（ＳＬＴ）和认知学两个不同的领域．在一定约束条件下,提出并证明了ＳＶＭ 和一类基于规则的ＦＩＳ的函数等效性定理．在此基础上,提出基于ＳＶＭ 学习过程的ＦＩＳ（ＭＢＦＩＳ）的设计方法．ＭＢＦＩＳ继承了ＳＶＭ 良好的泛化能力和对“维数灾难”的避免能力,也继承了基于规则的ＦＩＳ的显式推理能力．Ｂｅｎｃｈｍａｒｋ数据实验表明,ＭＢＦＩＳ具有良好的分类性能．     

Interval type-2 fuzzy expert system for prediction of carbon monoxide concentration in mega-cities

This paper presents an indirect approach to interval type-2 fuzzy logic system modeling to forecaste the level of air pollutants. The type-2 fuzzy logic system permits us to model the uncertainties among rules and the parameters related to data analysis. In this paper, we propose an indirect method to create an interval type-2 fuzzy logic system from a historical data, where Footprint of Uncertainties of fuzzy sets are extracted by implementation of an interval type-2 FCM algorithm and based on an upper and lower value for the level of fuzziness m in FCM. Finally, the proposed model is applied for prediction of carbon monoxide concentration in Tehran air pollution. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to type-1 fuzzy logic systems in terms of two performance indices.     

模糊竞争神经网络在线辨识及在CSTR中的应用

基于改进的T-S模糊模型构造了一种自适应模糊竞争神经网络模型(FCNN),给出了网络的连接结构和学习算法。它依据模糊竞争学习算法确定系统的模糊空间和模糊规则数,得出每个样本对每条规则的适用程度,并利用卡尔曼滤波算法在线辨识FCNN的后件参数。将其应用于化工过程连续搅拌反应器(CSTR)的建模中,仿真结果表明,FCNN具有结构简洁、收敛速度快、辨识精度高等特点,可当作复杂系统建模的一种有效手段。     

Adaptive Rule Weights in Neuro-Fuzzy Systems

Neuro-fuzzy systems have recently gained a lot of interest in research and application. They are approaches that use learning techniques derived from neural networks to learn fuzzy systems from data. A very simple ad hoc approach to apply a learning algorithm to a fuzzy system is to use adaptive rule weights. In this paper, we argue that rule weights have a negative effect on the linguistic interpretation of a fuzzy system, and thus remove one of the key advantages for applying fuzzy systems. We show how rule weights can be equivalently replaced by modifying the fuzzy sets of a fuzzy system. If this is done, the actual effects that rule weights have on a fuzzy rule base become visible. We demonstrate at a simple example the problems of using rule weights. We suggest that neuro-fuzzy learning should be better implemented by algorithms that modify the fuzzy sets directly without using rule weights.     

Comparison of BRDF models with a fuzzy inference system for correction of bidirectional effects

With the advances in computing and imaging technology, the field of precision agriculture is rapidly becoming a practical means for farm management. An important step in the delivery of highly accurate images for farm managers is the within-image correction for viewing geometry effects. Reflected light on an imaging sensor is influenced by properties of view zenith angle, solar zenith angle, and relative azimuth. There are a number of models that describe this effect termed the bidirectional reflectance distribution function (BRDF) or more generically “viewing geometry effects.” In this paper, we compared three BRDF models (Roujean, Shibayama-Wiegand, and Dymond-Qi) with a fuzzy inference system (FIS) for three data sets for correction of geometric effects. One data set consisted of ground data collected at different viewing angles of a cotton crop. Another data set included six aircraft images of a corn plot in a different part of each image. The final data set was an aerial image of a planting density experiment of cotton. All the models performed reasonably well, but the FIS was the most consistent predictor of BRDF for all three data sets. For the ground data set, R² statistics for predicting the reflectance based on the trained models ranged from 0.53 to 0.93 for the BRDF models and from 0.94 to 0.97 for the FIS.     

Interpretable fuzzy partitioning of classified data with variable granularity

Fuzzy rule-based systems are effective tools for acquiring knowledge from data and represent it in a linguistically interpretable form. To achieve interpretability, input features are granulated in fuzzy partitions. A critical design decision is the selection of the granularity level for each input feature. This paper presents an approach, called DC^* (Double Clustering with A^*), for automatically designing interpretable fuzzy partitions with optimal granularity. DC^* is specific for classification problems and is mainly based on a two-stage process: the first stage identifies clusters of multidimensional samples in order to derive class-labeled prototypes; in the second stage the one-dimensional projections of such prototypes are further clustered along each dimension simultaneously, thus minimizing the number of clusters for each feature. Moreover, the resulting one-dimensional clusters provide information to define fuzzy partitions that satisfy a number of interpretability constraints and exhibit variable granularity levels. The fuzzy sets in each partition can be labeled by meaningful linguistic terms and used to represent knowledge in a natural language form. Experimental results on both synthetic and real data show that the derived fuzzy partitions can be exploited to define very compact fuzzy rule-based systems that exhibit high linguistic interpretability and good classification accuracy.