Fuzzy If-then Rules Research Articles

On approximate reasoning with graded rules

This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IF–THEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.

System of fuzzy relation equations as a continuous model of IF–THEN rules

We claim that a fuzzy relation which models a system of fuzzy IF–THEN rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable. In view of this connection, it is important to consider the problem of solvability of a system of fuzzy relation equations – with respect to an unknown fuzzy relation. This problem is analyzed in the present paper. We propose several new necessary and sufficient criteria for the so-called Mamdani relation to be a solution to the system. We prove that in the general case, these same criteria are sufficient (but not always necessary) for the solvability of the system. A new, easy-to-check criterion is proposed for the solvability of the system with special fuzzy parameters.

FCMAC-EWS: A bank failure early warning system based on a novel localized pattern learning and semantically associative fuzzy neural network

In the banking industry, it is highly desirable to identify potential bank failure or high-risk banks. Successful early warning systems (EWS) would provide capabilities to avoid adverse financial repercussions and a massive bail out costs for the failing banks. Very often, these failures are due to financial distress. Various traditional statistical models have been used to study failures of financial institutions (Sinkey, J., Jr. (1975). A multivariate statistical analysis of the characteristics of problem banks. Journal of Finance, 1, 21-36; Martin, D. (1977). Early warning of bank failure: A logit regression approach. Journal of Banking and Finance, 1(3), 249-276; Lane, W., Looney, S., & Wansley, J. (1986). An application of the Cox proportional hazards model to bank failure. Journal of Banking and Finance, 10, 511-531; Cole, R., & Gunther, J. (1995). Separating the likelihood and timing of bank failure. Journal of Banking and Finance, 19, 1073-1089.). However, these models do not have the capability to identify the characteristics of financial distress and thus function as black boxes. This paper proposes a novel fuzzy CMAC (cerebellar model articulation controller) model based on compositional rule of inference, named FCMAC-CRI(S), as a new approach to tackle the problem using localized learning. The CRI-based FCMAC network, based on localized training, is able to identify the inherent traits and patterns of financial distress based on financial covariates derived from publicly available financial statements. The use of localized learning is akin to the neocortex semantic associative memory which is superior to the hippocampal form of global learning. The reason is that the hippocampal memory system rapidly learns arbitrary patterns of activity, whereas the neocortical system learns slowly. The slow learning of the neocortex is a requirement for any system that is able to eventually extract and model the similarity structure in its environment. The rapid learning of the hippocampal system, in contrast, sacrifices the ability to generalize. When both systems are intact, the hippocampal memory system trains the neocortical learning system through a process of repeated patterns, allowing for the gradual extraction of the similarity structure that is central to generalization. In FCMAC-CRI(S), its interactive relations among the selected pattern features are captured in the form of highly intuitive fuzzy IF-THEN rules, which form the knowledge base of the early warning system and provide insights into the characteristics of financial distress. The performance of the FCMAC-CRI(S) is benchmarked against that of the Cox's proportional hazard model and GenSoFNN-CRI(S) network, a functional hippocampal fuzzy semantic learning memory structure, in predicting bank failures based on a population of 3635 US banks observed over 21 years. The localized models and learning yield superior results and interpretation to fuzzy neural network such as GenSoFNN-CRI(S) that are based on global learning. The performance of the new approach as a bank failure classification and early warning system is highly encouraging.

Inferring operating rules for reservoir operations using fuzzy regression and ANFIS

The methods of ordinary least-squares regression (OLSR), fuzzy regression (FR), and adaptive network-based fuzzy inference system (ANFIS) are compared in inferring operating rules for a reservoir operations optimization problem. Dynamic programming (DP) is used as an example optimization tool to provide the input–output data set to be used by OLSR, FR, and ANFIS models. The coefficients of an FR model are found by solving a linear programming (LP) problem. The objective function of the LP is to minimize the total fuzziness of the FR model, which is related to the width of fuzzy coefficients in the regression model. Before applying FR to the reservoir operations problem, two FR formulations and interval regression (IR) are first examined in a simple tutorial example. ANFIS is also used to derive the reservoir operating rules as fuzzy IF-THEN rules. The OLSR, FR, and ANFIS based rules are then simulated and compared based on their performance in simulation. The methods are applied to a long-term planning problem as well as to a medium-term implicit stochastic optimization model. The results indicate that FR is useful to derive operating rules for a long-term planning model, where imperfect and partial information is available. ANFIS is beneficial in medium-term implicit stochastic optimization as it is able to extract important features of the system from the generated input–output set and represent those features as general operating rules.

Knowledge Representation Using High-Level Fuzzy Petri Nets

This correspondence presents a high-level fuzzy Petri net (HLFPN) model to represent the fuzzy production rules of a knowledge-based system, where a fuzzy production rule is the one that describes the fuzzy relation between the antecedent and the consequent. The HLFPN can be used to model fuzzy IF-THEN rules and IF-THEN-ELSE rules, where the fuzzy truth values of the propositions are restricted to [0, 1]. Based on the HLFPN model, an efficient algorithm is proposed to automatically reason about imprecise and fuzzy information. In this correspondence, a novel model to represent fuzzy knowledge is developed. When compared with other related models, the HLFPN model preserves several significant advantages. Finally, main results are presented in the form of eight properties and are supported by a comparison with other existing algorithms

Biological Systems Drug Infusion Controller Using FREN With Sliding Bounds

In this paper, a direct adaptive control for drug infusion of biological systems is presented. The proposed controller is accomplished using our adaptive network called Fuzzy Rules Emulated Network (FREN). The structure of FREN resembles the human knowledge in the form of fuzzy IF-THEN rules. After selecting the initial value of network's parameters, an on-line adaptive process based on Lyapunov's criteria is performed to improve the controller performance. The control signal from FREN is designed to keep in the region which is calculated by the modified Sliding Mode Control (SMC). The simulation results indicate that the proposed algorithm can satisfy the setting point and the robust performance.

A knowledge acquisition method for determining utilities of linguistic values for product factors

This paper proposes a fuzzy knowledge acquisition method to discover simplified fuzzy if-then rules, where the antecedent and consequent parts of a fuzzy if-then rule are referred to as a combination of linguistic values and the corresponding utility, respectively, from questionnaire data regarding the consumers’ subjective evaluation for a product or service. The main aim of the proposed method is to support decision makers in making appropriate marketing strategies, by identifying factors of concern to consumers through the analysis of the combinations of linguistic values with higher or lower utilities. To demonstrate the usefulness of the proposed method, computer simulations and possible marketing strategy analysis are performed on the rice taste data and the questionnaire data that evaluates the service quality of fast food stores.

Correct models of fuzzy IF–THEN rules are continuous

We show that a system of fuzzy IF–THEN rules being modeled correctly works as a partially given (fuzzy) function. Its behavior is determined by a chosen structure for IF–THEN rules which assigns meaning to fuzzy sets in the IF and THEN parts as well as to basic connectives. This partial function can be extended to a whole domain of fuzzy sets with the help of the Compositional Rule of Inference which plays a role of a mechanism for computing the dependent functional values. We investigate the question whether this function is continuous. Two new notions are introduced: a correct model of fuzzy IF–THEN rules in a structure and a continuous model of fuzzy IF–THEN rules with respect to given data. We prove that correctness of models of fuzzy IF–THEN rules is connected with the problem of solvability of the respective system of fuzzy relation equations. In this and only this case a model of fuzzy IF–THEN rules is continuous. As a side result we have established a new criterion of a solvability of a system of fuzzy relation equations with sup - * -composition.

DIRECT ADAPTIVE TYPE-2 FUZZY CONTROL FOR NONLINEAR SYSTEMS

A new direct adaptive type-2 fuzzy controller for a nonlinear dynamical system is developed in this paper. The parameters of the membership functions characterizing the linguistic terms in the type-2 fuzzy IF–THEN rules change according to some adaptive law for the purpose of controlling a plant to track a reference trajectory. A supervisory controller is appended to the type-2 fuzzy controller to force the state to be within the constraint set. Stability of this adaptive scheme is established using Lyapunov stability tools, where we guarantee the global stability of the resulting closed-loop system, in the sense that all signals involved are uniformly bounded. The simulation results for a Duffing forced-oscillation system show better performances, i.e. tracking error and control effort can be made smaller.

Prediction of wave parameters by using fuzzy logic approach

The purpose of this study is to investigate the relationship between wind speed, previous and current wave characteristics. It is expected that such a non-linear relationship includes some uncertainties. A fuzzy inference system employing fuzzy IF–THEN rules has an ability to deal with ill-defined and uncertain systems. Compared with traditional approaches, fuzzy logic is more efficient in linking the multiple inputs to a single output in a non-linear domain. In this paper, a sophisticated intelligent model, based on Takagi–Sugeno (TS) fuzzy modeling principles, was developed to predict the changes in wave characteristics such as significant wave height and zero up-crossing period due to the wind speed. Past measurements of significant wave height values and wind speed variables are used for training the adaptive model and it is then employed to predict the significant wave height amounts for future time intervals such as 1, 3, 6 and 12 h. The verification of the proposed model is achieved through the wave characteristics time series plots and various numerical error criterias. Also the model results were compared with classical Auto Regressive Moving Average with exogenous input (ARMAX) models. For the application of the proposed approach the offshore station located in the Pacific Ocean was used.

SVM-based fuzzy modeling for the arc welding process

This paper proposes a fuzzy modeling method based on support vector machine (SVM) for the arc welding process. SVM provides a mechanism to extract support vectors for generating fuzzy IF-THEN rules from training data. In the proposed SVM-based fuzzy system (SVM-FS), SVM is used to extract IF-THEN rules; the fuzzy basis function inference system is adopted as the fuzzy inference system. So the approach possesses good comprehensibility as well as satisfactory generalization capability. Modeling is one of the key techniques in the automatic control of the arc welding process, and is still a very difficult problem. We give the main steps of modeling for the process using SVM-FS, including selecting input/output variables, acquiring raw data, extracting fuzzy rules and fuzzy inference. Experimental results show that the proposed approach outperforms the rough set method.

Fast generation method of fuzzy rules and its application to flux optimization in process of matter converting

A fast generation method of fuzzy rules for flux optimization decision-making was proposed in order to extract the linguistic knowledge from numerical data in the process of matter converting. The fuzzy if-then rules with consequent real number were extracted from numerical data, and a linguistic representation method for deriving linguistic rules from fuzzy if-then rules with consequent real numbers was developed. The linguistic representation consisted of two linguistic variables with the degree of certainty and the storage structure of rule base was described. The simulation results show that the method involves neither the time-consuming iterative learning procedure nor the complicated rule generation mechanisms, and can approximate complex system. The method was applied to determine the flux amount of copper converting furnace in the process of matter converting. The real result shows that the mass fraction of Cu in slag is reduced by 0.5%.

Water bath temperature control by a recurrent fuzzy controller and its FPGA implementation

A hardware implementation of the Takagi-Sugeno-Kan (TSK)-type recurrent fuzzy network (TRFN-H) for water bath temperature control is proposed in this paper. The TRFN-H is constructed by a series of recurrent fuzzy if-then rules built on-line through concurrent structure and parameter learning. To design TRFN-H for temperature control, the direct inverse control configuration is adopted, and owing to the structure of TRFN-H, no a priori knowledge of the plant order is required, which eases the design process. Due to the powerful learning ability of TRFN-H, a small network is generated, which significantly reduces the hardware implementation cost. After the network is designed, it is realized on a field-programmable gate array (FPGA) chip. Because both the rule and input variable numbers in TRFN-H are small, it is implemented by combinational circuits directly without using any memory. The good performance of the TRFN-H chip is verified from comparisons with computer-based proportional-integral fuzzy (PI) and neural network controllers for different sets of experiments on water bath temperature control.

Hierarchical fuzzy relational models: linguistic interpretation and universal approximation

Hierarchical fuzzy structures composed of a series of sub-models connected in cascade have been found to be effective tools for dealing with the dimensionality problem in fuzzy systems. This paper addresses both the problems of universal approximation and linguistic interpretation of complex systems using hierarchical fuzzy models. Fuzzy relational equations are used to implement the sub-models of a hierarchical structure which has two very important properties: It is a universal approximator and it can be converted into a completely equivalent non-hierarchical model which in turn allows the extraction of linguistic knowledge in the form of consistent fuzzy IF-THEN rules. These properties are analytically derived and their application is illustrated by means of an example.

THREE-PARAMETER FUZZY ARITHMETIC APPROXIMATION OF L-R FUZZY NUMBERS FOR FUZZY NEURAL NETWORKS

In this study, we proposed an alternative operation of fuzzy arithmetic on L-R fuzzy numbers by three parameters of mode, left spread and right spread. Then, based on this approximation method, a new learning algorithm of a fully fuzzified neural network was developed in which the L-R fuzzy numbers were considered as the fuzzy signals. While the forward operations of fuzzy signals were based on the proposed three-parameter fuzzy arithmetic approximation method, the backward learning adopted a back-propagation learning procedure with a measurable error function. The learning algorithm was illustrated by an example of the recognition of fuzzy IF-THEN rules. The simulation result showed that the proposed approximation method used in such learning model was efficient and accurate.

COMPLETING FUZZY IF-THEN RULE BASES BY MEANS OF SMOOTHING SPLINES

A fuzzy if-then rule base may be viewed as a partial function between universes of fuzzy sets. For the construction of a fuzzy inference module, this partial function needs to be extended to a total one. Here, we propose a new method how to do so, making use of the method of smoothing splines. To this end, we identify the fuzzy sets with elements of a finite-dimensional real parameter space in an approximate way, using Perfilieva's fuzzy transforms. We then determine a function between two such parameter spaces by requiring that it reproduces the rule base as precise as possible and that it minimizes a parameter depending on its smoothness.

Fuzzy decision support system for demand forecasting with a learning mechanism

In this paper, a new decision support system for demand forecasting DSS_DF is presented. A demand forecast is generated in DSS_DF by combining four forecasts values. Two of them are obtained independently, one by a customer and the other by a market expert. They represent subjective judgments on future demand, given as linguistic values, such as “demand is around a certain value” or “demand is not lower than a certain value”, etc. Two additional forecasts are crisp values, obtained using conventional statistical methods, one using time-series analysis based on decomposition (TSAD), and the other using an auto regressive integrated moving average (ARMA) model. The combination of these four forecast values into one improved forecast is made by applying fuzzy IF-THEN rules. A modified Mamdani-style inference is used, which enables reasoning with fuzzy inputs. A new learning mechanism is developed and incorporated into the DSS_DF to adapt the rule bases that combine the individual forecasted values. The rule bases are adapted taking into consideration the performance of each of the forecast methods recorded in the past. The application of DSS_DF is demonstrated by an illustrative example. The forecasts obtained by DSS_DF are compared with results procured by applying the conventional TSAD and ARMA methods separately. The results obtained are encouraging and indicate that combining forecasts obtained by different methods may be beneficial.

Logical structure of fuzzy IF-THEN rules

This paper provides a logical basis for manipulation with fuzzy IF-THEN rules. Our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to Lotfi A. Zadeh but also other kinds of approximate reasoning methods, e.g., perception-based deduction, provided that there exists a possibility to characterize them within a formal logical system. In contrast with other approaches employing variants of multiple-valued first-order logic, the approach presented here employs fuzzy type theory of V. Novák which has sufficient expressive power to present the essential concepts and results in a compact, elegant and justifiable form. Within the effectively formalized representation developed here, based on a complete logical system, it is possible to reconstruct numerous well-known properties of CRI-related fuzzy inference methods, albeit not from the analytic point of view as usually presented, but as formal derivations of the logical system employed. The authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy IF-THEN rule bases will be represented, formalized and backed up by proof within the well-founded logical representation presented here. An immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy IF-THEN rules are ‘first class citizens´ of the representation: there are clear, logically founded definitions for fuzzy IF-THEN rule bases to be consistent, complete, or independent.

Design of hierarchical fuzzy model for classification problem using GAs

This paper proposes a new hierarchical fuzzy model (HFM) to solve the classification problem. The developed classification model comprises of two stages; one is to generate the fuzzy IF-THEN rules for each subsystem and the other is to determine the classification unit. For the classification problem, number of rules and the correct classification rate are the fundamental requirements. In this paper, we also advance two genetic algorithms (GAs) to tune the HFM. One is used to determine the combination of the input features for each subsystem on the HFM and the other is to reduce the number of rules in each fuzzy subsystem. The performance has been tested by simulations on the well known Wine and Iris databases. Simulations demonstrate that the proposed HFM under a few rules can provide sufficiently high classification rate even with higher feature dimensions.

Statistical fuzzy interval neural networks for currency exchange rate time series prediction

In this paper, the statistical fuzzy interval neural network with statistical interval input and output values is proposed to perform statistical fuzzy knowledge discovery and the currency exchange rate prediction. Time series data sets are grouped into time series data granules with statistical intervals. The statistical interval data sets including week-based averages, maximum errors of estimate and standard deviations are used to train the fuzzy interval neural network to discover fuzzy IF-THEN rules. The output of the fuzzy interval neural network is an interval value with certain percent confidence. Simulations are completed in terms of the exchange rates between US Dollar and other three currencies (Japanese Yen, British Pound and Hong Kong Dollar). The simulation results show that the fuzzy interval neural network can provide more tolerant prediction results.