Parameter optimization for Takagi-Sugeno fuzzy models-lessons learnt

Development of a new fuzzy exposure model

Adjusting the membership functions to satisfy one pattern may deteriorate the inference outcomes of the others. This incompatible issue can be retarded by the extension theory. A novel extension-based fuzzy modeling method, which differs from the traditional fuzzy inference, is proposed. Instead of directly applying the given data to building the fuzzy model, the given data are transformed to another domain by a sigmoidal function to obtain a better fuzzy model. We also define the extended correlation functions to relate the data with the fuzzy sets. During the refining process, the extended fuzzy model, which considers the positive and negative sets simultaneously, is adjusted by the gradient descent method. Simulation results from both single-input-single-output and double-input-single-output systems verified that better results than the conventional methods can be obtained.

Ozone forecast models using nonlinear regression (NLR) have been successfully applied to daily ozone forecast for seven metro areas in Kentucky, including Ashland, Bowling Green, Covington, Lexington, Louisville, Owensboro, and Paducah. In this study, the updated 2005 NLR ozone forecast models for these metro areas were evaluated on both the calibration data sets and independent data sets. These NLR ozone forecast models explained at least 72% of the variance of the daily peak ozone. Using the models to predict the ozone concentrations during the 2005 ozone season, the metro area mean absolute errors (MAEs) of the model hindcasts ranged from 5.90 ppb to 7.20 ppb. For the model raw forecasts, the metro area MAEs ranged from 7.90 ppb to 9.80 ppb. Based on previously developed NLR ozone forecast models for those areas, Takagi-Sugeno fuzzy system models were developed for the seven metro areas. The fuzzy "c-means" clustering technique coupled with an optimal output predefuzzification approach (least square method) was used to train the Takagi-Sugeno fuzzy system. Two types of fuzzy models, basic fuzzy and NLR-fuzzy system models, were developed. The basic fuzzy and NLR-fuzzy models exhibited essentially equivalent performance to the existing NLR models on 2004 ozone season hindcasts and forecasts. Both types of fuzzy models had, on average, slightly lower metro area averaged MAEs than the NLR models. Among the seven Kentucky metro areas Ashland, Covington, and Louisville are currently designated nonattainment areas for both ground level O 3 and PM 2.5 . In this study, summer PM 2.5 forecast models were developed for providing daily average PM 2.5 forecasts for the seven metro areas. The performance of the PM 2.5 forecast models was generally not as good as that of the ozone forecast models. For the summer 2004 model hindcasts, the metro-area average MAE was 5.33ìg/m 3 . Exploratory research was conducted to find the relationship between the winter PM 2.5 concentrations and the meteorological parameters and other derived prediction parameters. Winter PM 2.5 forecast models were developed for seven selected metro areas in Kentucky. For the model fits, the MAE for the seven forecast models ranged from 3.23 ìg/m 3 to 4.61 ìg/m 3 (~26-28% NMAE). The fuzzy technique was also applied on PM 2.5 forecast models to seek more accurate PM 2.5 prediction. The NLR-fuzzy PM 2.5 had slightly better performance than the NLR models.

The term “fuzzy modeling” was used in [30]. After that, pioneer works in the field of fuzzy modeling were done in [26, 31]. In 80’s, several fuzzy modeling techniques were developed (e.g., [24], [25], and [27]). In particular, Takagi and Sugeno [11] proposed a new type of fuzzy model. The model is called “Takagi-Sugeno fuzzy model (T-S fuzzy model)”. Furthermore, they proposed a procedure to identify the T-S fuzzy model from input-output data of systems in [11]. This work has been referred in many papers on fuzzy modeling for a long time. Sugeno and Kang [9, 10] extended the T-S procedure. It consists of parameter identification and structure identification. They stated in [10] that the structure identification is important and the method is better than GMDH1 in prediction problems for complex systems.

In this chapter we consider the identification of a Takagi-Sugeno fuzzy model (TS fuzzy model) [2]. This type of fuzzy model is especially useful in the area of fuzzy model-based control [10]. The TS fuzzy model is a nonlinear system model represented by fuzzy rules of the type $$ {R^i}:If{x_1}isA_1^iand...and{x_m}isA_m^ithen{y^i} = a_0^i + a_1^i{x_1} + ... + a_m^i{x_m}$$ (1.1) where R i ( i = 1, 2, …,n denotes that the i-th fuzzy rule, x j (j = 1,2,…, m) are input variables and y i is an output. Furthermore, a j i are the parameters contained in the consequent (then-part) of the i-th rule, and the A 1 i ,A 2 i ,…,A m i are the linguistic values taken by the input variables in the antecedent (if-part) of the i-th rule. The meaning of these linguistic values is defined by corresponding membership functions. As shown in (1.1) and (1.2), this fuzzy model describes a nonlinear input-output relation.

This paper reviews some frequently used methods to initialize an radial basis function (RBF) network and presents systematic design procedures for pre-processing unit(s) to initialize RBF network from available input–output data sets. The pre-processing units are computationally hybrid two-step training algorithms that can be named as (1) construction of initial structure and (2) coarse-tuning of free parameters. The first step, the number, and the locations of the initial centers of RBF network can be determined. Thus, an orthogonal least squares algorithm and a modified counter propagation network can be employed for this purpose. In the second step, a coarse-tuning of free parameters is achieved by using clustering procedures. Thus, the Gustafson–Kessel and the fuzzy C-means clustering methods are evaluated for the coarse-tuning. The first two-step behaves like a pre-processing unit for the last stage (or fine-tuning stage—a gradient descent algorithm). The initialization ability of the proposed four pre-processing units (modular combination of the existing methods) is compared with three non-linear benchmarks in terms of root mean square errors. Finally, the proposed hybrid pre-processing units may initialize a fairly accurate, IF–THEN-wise readable initial model automatically and efficiently with a minimum user inference.

A new approach is presented to enhance fuzzy modeling using multidimensional fuzzy sets directly in the fuzzy model in place of decomposed fuzzy sets by projection. The antecedent fuzzy sets in the form of multivariable functions are represented in continuous form. This is accomplished by using a multi input and multi output neural network, which is in particular radial basis functions (RBF) network providing the required multivariable function approximation properties in a fuzzy model. The inputs of the network are fuzzy model inputs, outputs are the membership function values as to the multivariable fuzzy sets involved in the model. Thus, each output is restricted to one multivariable fuzzy set. The RBF network is trained according to the multidimensional fuzzy sets which are identified after fuzzy clustering of the data. Based on this, the linear model parameters are determined by the method of least squares in a straightforward manner. The transparency issues can be handled by means of projection process to have information about the shape and locations of the membership functions pertinent to each variable at the input. The accurate fuzzy modeling having been thus guaranteed the information on the linear model parameters are used to establish the multivariable fuzzy rules with their respective validity regions. The redundancy of the number of fuzzy sets can be circumvented by classical cluster merging algorithms available. In this approach, the fuzzy rules are determined from the local linear models, however in contrast with the conventional fuzzy modeling, the model outputs are determined by the multivariable fuzzy sets. To obtain regular sets, RBF network is trained by means of orthogonal least squares (OLS) method so that at the same time the convergence issues of general neural network training is eliminated.

The kinetics of bioreactions often involve some uncertainties and the dynamics of the process vary during the course of fermentation. For such processes, conventional control schemes may not provide satisfactory control performance and demands extra effort to design advanced control schemes. In this study, a dynamic fuzzy model based predictive controller (DFMBPC) is presented for the control of a biochemical reactor. The DFMBPC incorporates an adaptive fuzzy modeling framework into a model based predictive control scheme to derive analytical controller output. The DFMBPC has the flexibility to opt with various types of fuzzy models whose choice also lead to improve the control performance. The performance of DFMBPC is evaluated by comparing with a fuzzy model based predictive controller (FMBPC) with no model adaptation and a conventional PI controller. The results show that DFMBPC provides better performance for tracking setpoint changes and rejecting unmeasured disturbances in the biochemical reactor.

The optimization of structural parameters of controllable reactor of transformer type (CRT) is of great significance for improving its economic benefits. This paper takes the new-type magnetic integration structure CRT as the research object. Qualitative analysis is employed to explore the relationship between loss, cost and structural parameters of CRT. The optimization model of structural parameters is established with the objective of minimizing cost and loss. The genetic algorithm is used to optimize the structural parameters of CRT. Based on this, a new economic calculation method of CRT's structural parameters is proposed. The results show that the optimization of CRT's structural parameters aiming at minimizing cost and loss is essentially to optimize the cross-sectional area of iron core and the length of winding coil. A new method for calculating CRT's structural parameters based on genetic algorithm optimization has more reasonable calculation results. With the increase of CRT operating time, the proportion of CRT use costs caused by loss will gradually increase. The methods and conclusions used in this paper provide some theoretical guidance for the structural design and optimization of CRT.

In this paper, the design problem of delayed output feedback control scheme using two-layer interval fuzzy observers for a class of nonlinear systems with state and output delays is investigated. The Takagi-Sugeno type fuzzy linear model with an on-line update law is used to approximate the nonlinear system. Based on the fuzzy model, a two-layer interval fuzzy observer is used to reconstruct the system states according to equal interval output time delay slices. Subsequently, a delayed output feedback adaptive fuzzy controller is developed to override the nonlinearities, time delays, and external disturbances such that the H∞ tracking performance is achieved. The linguistic information is developped by setting the membership functions of the fuzzy logic system and the adaptation parameters to estimate the model uncertainties directly for using linear analytical results instead of estimating nonlinear system functions. The filtered tracking error dynamics are designed to satisfy the Strictly Positive Realness (SPR) condition. Based on the Lyapunov stability criterion and linear matrix inequalities (LMIs), some sufficient conditions are derived so that all states of the system are uniformly ultimately bounded and the effect of the external disturbances on the tracking error can be attenuated to any prescribed level and consequently an H∞ tracking control is achieved. Finally, a numerical example of a two-link robot manipulator is given to illustrate the effectiveness of the proposed control scheme.

Clickbait has been considered a problem in the modern age of technology, especially in Indonesia. Various attempts have been made to research clickbait detection, however, researches which use Indonesian data are relatively scarce compared to other languages such as English because the availability of Indonesian datasets are lacking. Backpropagation Neural Network was used in clickbait detection on article's title and has achieved quite good accuracy result, however there is still a chance to improve the accuracy. This paper shows the results of using a modified backpropagation neural network algorithm to detect clickbait using article titles when compared to the standard algorithm. The research compares the results of standard stochastic gradient descent algorithm, mini-batch gradient descent algorithm, and a version of stochastic gradient descent with Adam optimizer and three hidden layers. The results show that using Adam optimizer and three hidden layers in stochastic gradient descent algorithm significantly improves the results compared to the standard architecture. The modified algorithm shows a precision score of 78% and a recall and F1 score of 76%, where the standard algorithm has a precision score of 67% and a recall and F1 score of 66%. The resulting algorithm is then implemented to a desktop application, which is considered easy to use.

С.А. Глушенко - кандидат экономических наук, старший преподаватель кафедры информационных систем и прикладной информатики, Ростовский государственный экономический университет (РИНХ)Адрес: 344002, г. Ростов-на-Дону, ул. Большая Садовая, д. 69E-mail: gs-gears@yandex.ru В статье обосновывается важность применения оценки рисков при реализации системы обеспечения информационной безопасности. Рассматриваются наиболее распространенные методики оценки риска и предлагается использовать для этих целей теорию нечеткой логики. Описывается предложенная нечеткая продукционная модель (НПМ), в которой определены семь входных лингвистических переменных, характеризующих факторы риска, четыре выходных лингвистических переменных, характеризующих риски различных областей информационной безопасности, а также четыре базы правил. Отмечается, что НПМ является первым приближением для рассматриваемой предметной области и требует оптимизации с целью минимизации ошибки выходов модели. Рассматриваются наиболее распространенные методы оптимизации параметров нечетких моделей и обосновываются преимущества применения методов, основанных на использовании нейро-нечетких сетей (ННС). Описывается процесс преобразования элементов нечеткой модели, таких как блок фаззификации, блок базы правил и блок дефаззификации во фрагменты нейронной сети. Результатом данного процесса является нейро-нечеткая сеть, соответствующая нечеткой модели. Построение разработанной ННС осуществляется на основе системы нейро-нечеткого вывода (adaptive neuro-fuzzy inference system, ANFIS) посредством применения специализированного пакета Neuro-Fuzzy Designer программного средства MATLAB. Обучение модели было выполнено гибридным методом, который представляет собой комбинацию методов наименьших квадратов и обратного распространения ошибки. Результатом данного процесса является оптимизация (настройка) параметров функций принадлежности входных лингвистических переменных. Использованный подход нейро-нечеткого моделирования позволил получить более адекватную нечеткую продукционную модель, которая позволяет проводить лингвистический анализ рисков информационной безопасности организации. Полученные с ее помощью сведения позволяют ИТ-менеджерам определять приоритеты рисков и разрабатывать эффективные планы мероприятий по снижению влияния наиболее опасных угроз. [1] Исследование выполнено при финансовой поддержке РФФИ, в рамках научного проекта № 16-31-00285 «Методы и модели нечеткой логики в системах принятия решений управления рисками»

Fuzzy models and relational equations

A novel hybrid algorithm for creating self-organizing fuzzy neural networks

Rule extraction from data is one of the key technologies for solving the bottlenecks in artificial intelligence. Artificial neural networks are well suited for representing any knowledge in given data. Extraction of logical/fuzzy rules from the trained artificial neural network is of great importance to researchers in the fields of artificial intelligence and soft computing. Fuzzy rule sets are capable of approximating any nonlinear mapping relationships. Extraction of rules from data has been discussed in terms of fuzzy modeling, fuzzy clustering, and classification with fuzzy rule sets. This special issue entitled"Rule Extraction from Data" is aimed at providing the readers with good insights into the advanced studies in the field of rule extraction from data using neural networks/fuzzy rule sets. I invited seven research papers best suited for the theme of this special issue. All the papers were reviewed rigorously by two reviewers each. The first paper proposes an interesting rule extraction method from data using neural networks. Ishikawa presents a combination of learning with an immediate critic and a structural learning with forgetting. This method is capable of generating skeletal networks for logical rule extraction from data with correct and wrong answers. The proposed method is applied to rule extraction from lense data. The second paper presents a new methodology for logical rule extraction based on transformation of MLP (multilayered perceptron) to a logical network. Duck et al. applied their C-MLP2LN to the Iris benchmark classification problem as well as real-world medical data with very good results. In the third paper, Geczy and Usui propose fuzzy rule extraction from trained artificial neural networks. The proposed algorithm is implied from their theoretical study, not from heuristics. Their study enables to initially consider derivation of crisp rules from trained artificial neural network, and in case of conflict, application of fuzzy rules. The proposed algorithm is experimentally demonstrated with the Iris benchmark classification problem. The fourth paper presents a new framework for fuzzy modeling using genetic algorithm. The authors have broken new ground of fuzzy rule extraction from neural networks. For the fuzzy modeling, they have proposed a particular type of neural networks containing nodes representing membership functions. In this fourth paper, the authors discuss input variable selection for the fuzzy modeling under multiple criteria with different importance. A target system with a strong nonlinearity is used for demonstrating the proposed method. Kasabov, et al. present, in the fifth paper, a method for extraction of fuzzy rules that have different level of abstraction depending on several modifiable thresholds. Explanation quality becomes better with higher threshold values. They apply the proposed method to the Iris benchmark classification problem and to a real world problem. J. Yen and W. Gillespie address interpretability issue of Takagi-Sugeno-Kang model, one of the most popular fuzzy mdoels, in the fifth paper. They propose a new approach of fuzzy modeling that ensures not only a high approximation of the input-output relationship in the data, but also good insights about the local behavior of the model. The proposed method is applied to fuzzy modeling of sinc function and Mackey-Glass chaotic time series data. The last paper discusses fuzzy rule extraction from numerical data for high-dimensional classification problems. H.Ishibuchi, et al. have been pioneering methods for classification of data using fuzzy rules and genetic algorithm. In this last paper, they introduced a new criterion, simplicity of each rule, together with the conventional ones, compactness of rule base and classification ability, for high-dimensional problem. The Iris data is used for demonstrating their new classification method. They applied it also to wine data and credit data. I hope that the readers will be encouraged to explore the frontier to establish a new paradigm in the field of knowledge representation and rule extraction.