Box-Jenkins Gas Furnace Research Articles

MFRFNN: Multi-Functional Recurrent Fuzzy Neural Network for Chaotic Time Series Prediction

Chaotic time series prediction, a challenging research topic in dynamic system modeling, has drawn great attention from researchers around the world. In recent years extensive researches have been done on developing chaotic time series prediction methods, and various models have been proposed. Among them, recurrent fuzzy neural networks (RFNNs) have shown significant potential in this area. Most of the proposed RFNNs learn a single function, but when dealing with chaotic time series, different outputs may be generated for a specific input based on the system’s state. So, a network is required that can learn multiple functions simultaneously. Based on this concept, a novel multi-functional recurrent fuzzy neural network (MFRFNN) is proposed in this paper. MFRFNN consists of two fuzzy neural networks with Takagi-Sugeno-Kang fuzzy rules, one is used to produce the output, and the other to determine the system’s state. There is a feedback loop between these two networks, which makes MFRFNN capable of learning and memorizing historical information of past observations. Employing the states allows the proposed network to learn multiple functions simultaneously. Moreover, a new learning algorithm, which employs the particle swarm optimization algorithm, is developed to train the networks’ weights. The effectiveness of MFRFNN is validated using the Lorenz and Rossler chaotic time series and four real-world datasets, including Box–Jenkins gas furnace, wind speed prediction, Google stock price prediction, and air quality index prediction. Based on the root mean square error, the proposed method shows a decrease of 35.12%,13.95%, and 49.62% from the second best methods in the Lorenz time series, Box–Jenkins gas furnace, and wind speed prediction dataset, respectively.

(2004-5811) Improved fuzzy clustering algorithm using adaptive particle swarm optimization for nonlinear system modeling and identification

In this paper, an improved Type2-PCM clustering algorithm based on improved adaptive particle swarm optimization called Type2-PCM-IAPSO is proposed. Firstly, a new clustering algorithm called Type2-PCM is proposed. The Type2-PCM algorithm can solve the problems encountered by fuzzy c-means algorithm (FCM), Gustafson-Kessel algorithm (G-K), possibilistic c-means algorithm (PCM) and NPCM (sensitivity to noise or aberrant points and local minimal sensitivity). . . etc. Secondly, we combined our Type2-PCM algorithm with the improved adaptive particle swarm optimization algorithm (IAPSO) to ensure proper convergence to a local minimum of the objective function. The effectiveness of the two proposed algorithms Type2-PCM and Type2-PCM-IAPSO was tested on a system described by a different equation, Box-Jenkins gas furnace, dryer system and the convection system. The validation tests used showed good performance of these algorithms. However, their average square error test (MSE) shows a better behaviour of the Type2-PCM-IAPSO algorithm compared to the FCM, G-K, PCM, FCM-PSO, Type2-PCM-PSO, RKPFCM and RKPFCM-PSO algorithms.

MULTIPLICATIVE NEURON MODEL BASED ON SINE COSINE ALGORITHM FOR TIME SERIES PREDICTION

Time series prediction is a method to predict the system behavior in the future based on current given data. Neural Networks (NNs) approach is a well-known technique that is useful for time series prediction. In the literature many NN models such Multilayer Perceptron (MLP), Pi-Sigma NN (PSNN), Recurrent NN etc. are proposed for solving time series prediction. In this paper, we use Multiplicative Neuron Model (MNM) to predict time series. For training this model, we propose use newly developed evolutionary optimization algorithm called Sine Cosine algorithm (SCA), and this algorithm has not been used as far as we know in training the MNM. The proposed SCA-MNM model is employed for the most known time series problems. In this paper, the application of the SCA-MNM on time prediction is illustrated using two mostly used datasets Mackey-Glass time series dataset, Box-Jenkins gas furnace dataset. To investigate the effect of the proposed SCA-MNM model, comparisons were made with some of the results given in the literature.

Forecasting the Behavior of Gas Furnace Multivariate Time Series Using Ridge Polynomial Based Neural Network Models

In this paper, a new application of ridge polynomial based neural network models in multivariate time series forecasting is presented. The existing ridge polynomial based neural network models can be grouped into two groups. Group A consists of models that use only autoregressive inputs, whereas Group B consists of models that use autoregressive and moving-average (i.e., error feedback) inputs. The well-known Box-Jenkins gas furnace multivariate time series was used in the forecasting comparison between the two groups. Simulation results show that the models in Group B achieve significant forecasting performance as compared to the models in Group A. Therefore, the Box-Jenkins gas furnace data can be modeled better using neural networks when error feedback is used.

Stability of Evolving Fuzzy Systems Based on Data Clouds

Evolving fuzzy systems (EFSs) are now well developed and widely used, thanks to their ability to self-adapt both their structures and parameters online. Since the concept was first introduced two decades ago, many different types of EFSs have been successfully implemented. However, there are only very few works considering the stability of the EFSs, and these studies were limited to certain types of membership functions with specifically predefined parameters, which largely increases the complexity of the learning process. At the same time, stability analysis is of paramount importance for control applications and provides the theoretical guarantees for the convergence of the learning algorithms. In this paper, we introduce the stability proof of a class of EFSs based on data clouds, which are grounded at the AnYa type fuzzy systems and the recently introduced empirical data analytics (EDA) methodological framework. By employing data clouds, the class of EFSs of AnYa type considered in this paper avoids the traditional way of defining membership functions for each input variable in an explicit manner and its learning process is entirely data driven. The stability of the considered EFS of AnYa type is proven through the Lyapunov theory, and the proof of stability shows that the average identification error converges to a small neighborhood of zero. Although, the stability proof presented in this paper is specially elaborated for the considered EFS, it is also applicable to general EFSs. The proposed method is illustrated with Box–Jenkins gas furnace problem, one nonlinear system identification problem, Mackey–Glass time series prediction problem, eight real-world benchmark regression problems as well as a high-frequency trading prediction problem. Compared with other EFSs, the numerical examples show that the considered EFS in this paper provides guaranteed stability as well as a better approximation accuracy.

System Identification Using Gray-Based Adaptive Heterogeneous Multi-Swarm PSO Algorithm: Application to an Irrigation Station

In this paper, a modified methodology to find an optimal T-S fuzzy model based on an extended heterogeneous Multi-swarm PSO (MsPSO) algorithm is developed in order to enhance search ability of the classical MsPSO algorithm. However, this simple MsPSO algorithm search behavior is not always optimal to find the potential solution to a special problem, and it may trap the individuals into local regions leading to premature convergence. To overcome this drawback, two parameter automation strategies (inertia weight and acceleration coefficients) are introduced based on gray relational analysis in MsPSO to guarantee a highly accurate model. The performance of the proposed algorithm is evaluated by adopting standard tests and indicators which are reported in the specialized literature. The numerical results demonstrate that the proposed algorithm is significantly better than the original MsPSO algorithm and the rest of compared algorithms according to mean and standard deviation (std) tests. Next, to further validate the generalization ability of the Improved OptiFel approach, the proposed algorithm is secondly applied on the BoxJenkins Gas Furnace system. Then, the improved OptiFel method is applied to an irrigation station process in order to provide an optimal T-S fuzzy model. Compared to the other existing methods, we achieve the result that the improved OptiFel can generate good fuzzy model with high accuracy and strong generalization ability.

An improved heterogeneous multi-swarm PSO algorithm to generate an optimal T-S fuzzy model of a hydraulic process

In this paper, a new methodology to develop an Optimal Fuzzy model (OptiFel) using an improved Multi-swarm Particle Swarm Optimization (MsPSO) algorithm is proposed with a new adaptive inertia weight based on Grey relational analysis. Since the classical MsPSO suffers from premature convergence and can be trapped into local optima, which significantly affects the model accuracy, a modified MsPSO algorithm is presented here. The most important advantage of the proposed algorithm is the adjustment of fewer parameters in which the main parameter is the inertia weight. In fact, the control of this parameter could facilitate the convergence and prevent an explosion of the swarm. The performance of the proposed algorithm is evaluated by adopting standard tests and indicators which are reported in the specialized literature. The proposed Grey MsPSO is first applied to solve the optimization problems of six benchmark functions and then, compared with the other nine variants of particle swarm optimization. In order to demonstrate the higher search performance of the proposed algorithm, the comparison is then made via two performance tests such as the standard deviation and central processing unit time. To further validate the generalization ability of the Improved OptiFel approach, the proposed algorithm is secondly applied on the Box–Jenkins Gas Furnace system and on a irrigation station prototype. A comparative study based on Mean Square Error is then performed between the proposed approach and other existing methods. As a result, the improved Grey MsPSO is well adopted to find an optimal model for the real processes with high accuracy and strong generalization ability.

A New Approach of Takagi–Sugeno Fuzzy Modeling Using an Improved Genetic Algorithm Optimization for Oxygen Content in a Coke Furnace

The oxygen content modeling of the coke furnace is important for advanced control design but not an easy job because of various disturbances and nonlinearity. A novel approach is proposed by using an improved genetic algorithm (IGA) combined with the dynamic autoregressive with exogenous input (ARX) Takagi–Sugeno (T-S) fuzzy model. The IGA algorithm automatically generates the input variable, the appropriate fuzzy if–then rules, and the ARX structure to characterize the dynamic nonlinear feature of the oxygen content by processing the operation data from the industrial coke furnace. And a more comprehensive objective function is constructed considering both the modeling precision and structure simplicity. Hybrid encoding, modified genetic operators, particularly the maintain operator, are designed to obtain the satisfactory optimization performance. The modeling accuracy and system structure of the T-S fuzzy model are compared with a benchmark Box–Jenkins gas furnace and the complex industrial coke furnac...

A new modeling algorithm based on ANFIS and GMDH

System modeling is one of the most important tasks of dynamic analysis and prediction systems, and imprecise model may lead to high bias. The presence of noise in sample data can make it more difficult to obtain precise system models. A new modeling algorithm called ANFIS-GMDH is presented in this paper, which builds upon the traditional ANFIS structure and utilizes the self-organizing mechanisms of GMDH. The aim of ANFIS-GMDH is to improve upon the traditional ANFIS method and prevent overfitting of noisy data. The well-studied Box-Jenkins gas furnace data is utilized to validate the algorithm, with results showing that the proposed algorithm performs better than traditional ANFIS, GMDH and subtractive clustering for both noisy and noiseless data, without any significant increase in execution time.

A multilayer-perceptron based method for variable selection in soft sensor design

The paper proposes a new method for variable selection for prediction settings and soft sensors applications. The new variable selection method is based on the multi-layer perceptron (MLP) neural network model, where the network is trained a single time, maintaining low computational cost. The proposed method was successfully applied, and compared with four state-of-the-art methods in one artificial dataset and three real-world datasets, two publicly available datasets (Box–Jenkins gas furnace and gas mileage), and a dataset of a problem where the objective is to estimate the fluoride concentration in the effluent of a real urban water treatment plant (WTP). The proposed method presents similar or better approximation performance when compared to the other four methods. In the experiments, among all the five methods, the proposed method selects the lowest number of variables and variables-delays pairs to achieve the best solution. In soft sensors applications having a lower number of variables is a positive factor for decreasing implementation costs, or even making the soft sensor feasible at all.

On the inference and approximation properties of belief rule based systems

Belief rule based (BRB) system provides a generic inference framework for approximating complicated nonlinear causal relationships between antecedent inputs and output. It has been successfully applied to a wide range of areas, such as fault diagnosis, system identification and decision analysis. In this paper, we provide analytical and theoretical analyses on the inference and approximation properties of BRB systems. We first investigate the unified multi-model decomposition structure of BRB systems, under which the input space is partitioned into different local regions. Then we analyse the distributed approximation process of BRB systems. These analysis results unveil the underlying inference mechanisms that enable BRB systems to have superior approximation performances. Furthermore, by using the Stone–Weierstrass theorem, we constructively prove that BRB systems can approximate any continuous function on a compact set with arbitrary accuracy. This result provides a theoretical foundation for using and training BRB systems in practical applications. Finally, a numerical simulation study on the well-known benchmark nonlinear system identification problem of Box–Jenkins gas furnace is conducted to illustrate the validity of a BRB system and show its inference and approximation capability.

A GMDH-based fuzzy modeling approach for constructing TS model

In this paper, a new learning algorithm based on group method of data handling (GMDH) is proposed for the identification of Takagi–Sugeno fuzzy model. Different from existing methods, the new approach, called TS-GMDH, starts from simple elementary TS fuzzy models, and then uses the mechanism of GMDH to produce candidate fuzzy models of growing complexity until the TS model of optimal complexity has been created. The main characteristic of the new approach is its ability to identify the structure of TS model automatically. Experiments on Box–Jenkins gas furnace data and UCI datasets have shown that the proposed method can achieve satisfactory results and is more robust to noise in comparison with other TS modeling techniques such as ANFIS.

Extended fuzzy function model with stable learning methods for online system identification

AbstractThe aim of the online nonlinear system identification is the accurate modeling of the current local input‐output behavior of the plant without using any prior knowledge and offline modeling phase. It is a challenging task for many intelligent systems when used for real‐time control applications. In this paper, we propose a novel computationally efficient extended fuzzy functions (EFF) model for system identification of unknown nonlinear discrete‐time systems. The main contributions are to introduce an effective quasi‐nonlinear model (EFF) and propose adaptive learning rates (ALR) for recursive least squares (RLS) and gradient‐descent (GD) methods. The asymptotic convergence of the modeling errors and boundedness of the parameters are proved by using the input‐to‐state stability (ISS) approach. Numerical simulations are performed for Box–Jenkins gas furnace system and a nonlinear dynamic system. The benefits of its accuracy, stability and simple implementation in practice indicate that EFF model is a promising technique for online identification of nonlinear systems. Copyright © 2010 John Wiley & Sons, Ltd.

A simplified linguistic information feedback-based dynamical fuzzy system

Inspired by the linguistic information feedback-based dynamical fuzzy system (LIFDFS) recently proposed by the authors, we present a simplified LIFDFS (S-LIFDFS) model in this paper, which has a simpler linguistic information feedback structure. Compared with the LIFDFS, the S-LIFDFS can offer us with a considerably reduced computational complexity. We first give a detailed description of its underlying principle. Based on the gradient descent method, an adaptive learning algorithm for the feedback parameters is next derived. We also discuss the application of this S-LIFDFS in time series prediction. Three evaluation examples including prediction of two artificial time sequences and the well-known Box–Jenkins gas furnace data are demonstrated here. Simulation results illustrate that with a compact structure, our S-LIFDFS can still retain the advantage of inherent dynamics of linguistic information feedback and is, therefore, well suited for handling temporal problems like prediction, modeling, and control.

Nonlinear system identification via Laguerre network based fuzzy systems

In this study, identification of nonlinear systems via Laguerre network based fuzzy model is introduced. We first describe the proposed modeling approach in detail and suggest a fast learning scheme for its training. The proposed approach is applied in three dynamic system modeling problems including Box–Jenkins gas furnace data and forced Van der Pol oscillator. When we compare the performance of the proposed approach against the classical Sugeno and adaptive network based fuzzy inference system modeling, our approach is found to have superior modeling performance and generalization capability.

A fuzzy modeling method via Enhanced Objective Cluster Analysis for designing TSK model

This paper proposes a fuzzy modeling method via Enhanced Objective Cluster Analysis to obtain the compact and robust approximate TSK fuzzy model. In our approach, the Objective Cluster Analysis algorithm is introduced. In order to obtain more compact and more robust fuzzy rule prototypes, this algorithm is enhanced by introducing the Relative Dissimilarity Measure and the new consistency criterion to represent the similarity degree between the clusters. By these additional criteria, the redundant clusters caused by iterations are avoided; the subjective influence from human judgment for clustering is weakened. Moreover the clustering results including the number of clusters and the cluster centers are considered as the initial condition of the premise parameters identification. Thus the traditional iteration modeling procedure for determining the number of rules and identifying parameters is changed into one-off modeling, which significantly reduces the burden of computation. Furthermore the decomposition errors and the approximation errors resulted from premise parameters identification by Fuzzy c-Means clustering are decreased. For the consequence parameters identification, the Stable Kalman Filter algorithm is adopted. The performance of the proposed modeling method is evaluated by the example of Box–Jenkins gas furnace. The simulation results demonstrate the power of our model.

Design of hybrid differential evolution and group method of data handling networks for modeling and prediction

This paper proposes a hybrid modeling approach based on two familiar non-linear methods of mathematical modeling; the group method of data handling (GMDH) and differential evolution (DE) population-based algorithm. The proposed method constructs a GMDH self-organizing network model of a population of promising DE solutions. The new hybrid implementation is then applied to modeling tool wear in milling operations and also applied to two representative time series prediction problems of exchange rates of three international currencies and the well-studied Box-Jenkins gas furnace process data. The results of the proposed DE–GMDH approach are compared with the results obtained by the standard GMDH algorithm and its variants. Results presented show that the proposed DE–GMDH algorithm appears to perform better than the standard GMDH algorithm and the polynomial neural network (PNN) model for the tool wear problem. For the exchange rate problem, the results of the proposed DE–GMDH algorithm are competitive with all other approaches except in one case. For the Box-Jenkins gas furnace data, the experimental results clearly demonstrates that the proposed DE–GMDH-type network outperforms the existing models both in terms of better approximation capabilities as well as generalization abilities. Consequently, this self-organizing modeling approach may be useful in modeling advanced manufacturing systems where it is necessary to model tool wear during machining operations, and in time series applications such as in prediction of time series exchange rate and industrial gas furnace problems.

The Application of an Ensemble of Boosted Elman Networks to Time Series Prediction: A Benchmark Study

In this paper, the application of multiple Elman neural networks to time series data regression problems is studied. An ensemble of Elman networks is formed by boosting to enhance the performance of the individual networks. A modified version of the AdaBoost algorithm is employed to integrate the predictions from multiple networks. Two benchmark time series data sets, i.e., the Sunspot and Box-Jenkins gas furnace problems, are used to assess the effectiveness of the proposed system. The simulation results reveal that an ensemble of boosted Elman networks can achieve a higher degree of generalization as well as performance than that of the individual networks. The results are compared with those from other learning systems, and implications of the performance are discussed.

Linguistic information feedforward-based dynamical fuzzy systems

In this paper, we first propose a linguistic information feedforward-based dynamical fuzzy system (LIFFDFS) in which the past fuzzy inference output represented by a membership function is fed forward locally with trainable parameters. Our LIFFDFS can overcome the common static mapping drawback of conventional fuzzy systems. We also give a detailed description of its underlying principle and general structure. Next, based on the gradient descent method, an adaptive learning algorithm for the feedforward parameters is derived. The proposed LIFFDFS is further employed in the prediction of time series. The well-known Box-Jenkins gas furnace data are used here as an evaluation example. Simulation results demonstrate that this new dynamical fuzzy system has the advantage of inherent dynamics and is, therefore, well suited for handling temporal problems, such as process modeling and control

Constraint-Based Fuzzy Models for an Environment with Heterogeneous Information-Granules

A novel framework for fuzzy modeling and model-based control design is described. Based on the theory of fuzzy constraint processing, the fuzzy model can be viewed as a generalized Takagi-Sugeno (TS) fuzzy model with fuzzy functional consequences. It uses multivariate antecedent membership functions obtained by granular-prototype fuzzy clustering methods and consequent fuzzy equations obtained by fuzzy regression techniques. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. The fuzzy-constraint-based approach provides the following features. 1) The knowledge base of a constraint-based fuzzy model can incorporate information with various types of fuzzy predicates. Consequently, it is easy to provide a fusion of different types of knowledge. The knowledge can be from data-driven approaches and/or from control-relevant physical models. 2) A corresponding inference mechanism for the proposed model can deal with heterogeneous information granules. 3) Both numerical and linguistic inputs can be accepted for predicting new outputs.The proposed techniques are demonstrated by means of two examples: a nonlinear function-fitting problem and the well-known Box-Jenkins gas furnace process. The first example shows that the proposed model uses fewer fuzzy predicates achieving similar results with the traditional rule-based approach, while the second shows the performance can be significantly improved when the control-relevant constraints are considered.