Neurofuzzy models to automate the grading of old‐age depression

Manual grading of depression is sometimes difficult due to the subjective signs‐symptoms. The aim of this paper is to automate the process of depression grading using a neurofuzzy model (NFM). Two hundred and seventy real‐world depression cases are considered in this work. Each case has seven symptoms, which are obtained according to DSM‐IV‐TR. Each case is graded as ‘mild’ or ‘moderate’. However, in practice, the boundaries of ‘mild’ and ‘moderate’ grading are fuzzy in nature. The paper attempts to solve this fuzzy overlapping zone of these grades. To reduce the number of symptoms, significantly correlated symptoms are mined using a paired t‐test. Then, two NFMs have been developed. NFM‐1 has been developed with all seven symptoms, while only significantly correlated symptoms have been used to construct the NFM‐2 model. Two fuzzy membership functions, such as triangular membership function (TRMF) and Gaussian membership function (GMF) have been considered to note with which better fuzzification could be achieved. The paper concludes that NFM‐1 with GMF is the best model with average predicting accuracy of 94.4% and robustness.     

A polynomial fuzzy neural network for modeling and control

This paper introduces a novel neurofuzzy system based on polynomial fuzzy neural network (PFNN) architecture. A PFNN consists of a set of if-then rules with appropriate membership functions (MFs) whose parameters are optimized via a hybrid genetic algorithm. A polynomial neural network is employed in the defuzzification scheme to improve output performance and to select appropriate rules. A performance criterion for model selection is defined to overcome the overfitting problem in the modeling procedure. For a performance assessment of the PFNN inference system, two well-known problems are employed for a comparison with other methods. The results of these comparisons show that the PFNN inference system out-performs the other methods and exhibits robustness characteristics. This work was presented in part at the Fourth International Symposium on Artificial Life and Robotics, Oita, Japan, January 19–22, 1999     

Generalized neurofuzzy network modeling algorithms usingBezier-Bernstein polynomial functions and additive decomposition

This paper introduces a new neurofuzzy model construction algorithm for nonlinear dynamic systems based upon basis functions that are Bezier-Bernstein polynomial functions. This paper is generalized in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n. This new construction algorithm also introduces univariate Bezier-Bernstein polynomial functions for the completeness of the generalized procedure. Like the B-spline expansion based neurofuzzy systems, Bezier-Bernstein polynomial function based neurofuzzy networks hold desirable properties such as nonnegativity of the basis functions, unity of support, and interpretability of basis function as fuzzy membership functions, moreover with the additional advantages of structural parsimony and Delaunay input space partition, essentially overcoming the curse of dimensionality associated with conventional fuzzy and RBF networks. This new modeling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier-Bernstein polynomial functions used in model construction. The overall network weights are then learnt using conventional least squares methods. Numerical examples are included to demonstrate the effectiveness of this new data based modeling approach.     

Fully Evolvable Optimal Neurofuzzy Controller Using Adaptive Critic Designs

A near-optimal neurofuzzy external controller is designed in this paper for a static compensator (STATCOM) in a multimachine power system. The controller provides an auxiliary reference signal for the STATCOM in such a way that it improves the damping of the rotor speed deviations of its neighboring generators. A zero-order Takagi–Sugeno fuzzy rule base constitutes the core of the controller. A heuristic dynamic programming (HDP) based approach is used to further train the controller and enable it to provide nonlinear near-optimal control at different operating conditions of the power system. Based on the connectionist systems theory, the parameters of the neurofuzzy controller, including the membership functions, undergo training. Simulation results are provided that compare the performance of the neurofuzzy controller with and without updating the fuzzy set parameters. Simulation results indicate that updating the membership functions can noticeably improve the performance of the controller and reduce the size of the STATCOM, which leads to lower capital investment.      

Fuzzy functions based ARX model and new fuzzy basis function models for nonlinear system identification

In this study, auto regressive with exogenous input (ARX) modeling is improved with fuzzy functions concept (FF-ARX). Fuzzy function with least squares estimation (FF-LSE) method has been recently developed and widely used with a small improvement with respect to least squares estimation method (LSE). FF-LSE is structured with only inputs and their membership values. This proposed model aims to increase the capability of the FF-LSE by widening the regression matrix with lagged input–output values. In addition, by using same idea, we proposed also two new fuzzy basis function models. In the first, basis of the fuzzy system and lagged input–output values are structured together in the regression matrix and named as “L-FBF”. Secondly, instead of using basis function, the membership values of the lagged input–output values are used in the regression matrix by using Gaussian membership functions, called “M-FBF”. Therefore, the power of the fuzzy basis function is also enhanced. For the corresponding models, antecedent part parameters for the input vectors are determined with fuzzy c-means (FCM) clustering algorithm. The consequent parameters of the all models are estimated with the LSE. The proposed models are utilized and compared for the identification of nonlinear benchmark problems.     

Type-2 fuzzy modeling for desulphurization of steel process

This paper presents a new type-2 fuzzy logic system model for desulphurization process of a real steel industry in Canada. The type-2 fuzzy logic system permits us to model rule uncertainties where every membership value of an element has a second order membership value of its own. In this paper, we propose an indirect method to create second order membership grades that are amplitudes of type-2 secondary membership functions, where the primary memberships are extracted by implementation of fuzzy clustering approach. In this research, Gaussian Mixture Model (GMM) is used for the creation of second order membership grades. Furthermore, a reduction scheme is implemented which results in type-1 membership grades. In turn, this leads to a reduction of the complexity of the system. Two methods are used for the estimation of the membership functions: indirect and direct methods. In the indirect method, the system uses an interpolation scheme for the estimation of the most appropriate membership functions. In the direct method, the system is tuned by an inference algorithm for the optimization of the main parametric system. In this case, the parameters are: Schweizer and Sklar t-norm and s-norm, combination of FATI and FITA inference approaches, and Yager defuzzification. Finally, the system model is applied to the desulphurization process of a real steel industry in Canada. It is shown that the proposed type-2 fuzzy logic system is superior in comparison to multiple regression and type-1 fuzzy logic systems in terms of robustness, and error reduction.     

From a Gaussian Mixture Model to Nonadditive Fuzzy Systems

This paper presents the formulation of nonadditive generalized fuzzy model (GFM) by using the framework of the Gaussian mixture model, which provides the membership functions for the input fuzzy sets. By treating the consequent part as a function of fuzzy measures, we derive its coefficients. The defuzzified output constructed from both the premise and consequent parts of the GFM rules takes the form of Choquet integral. The computational burden involved with the solution of lambda-measure is mitigated using Q-measure. This nonadditive fuzzy model is applied on two benchmark applications, and the results are found to be better than those obtained from the additive fuzzy models.     

Fuzzy polynomial neurons as neurofuzzy processing units

In this study, we introduce and study fuzzy polynomial neurons (FPNs) being regarded as generic processing units in neurofuzzy computing. The underlying topology of FPNs is formed through fuzzy rules, fuzzy inference and polynomials. Each polynomial offers a nonlinear mapping and is centred around a modal value of the corresponding membership functions defined in the input space of the neuron. The adjustable order of the polynomial is essential when addressing the level of nonlinearity to be handled in the approximation problem. We demonstrate that fuzzy polynomial neurons form a certain class of functional neurons and afterwards discuss their properties and an overall design process. Furthermore, these neurons are discussed in the context of universal approximation and universal approximators     

Wireless Sensor Network Lifetime Analysis Using Interval Type-2 Fuzzy Logic Systems

Extending the lifetime of the energy constrained wireless sensor networks is a crucial challenge in sensor network research. In this paper, we present a novel approach based on fuzzy logic systems to analyze the lifetime of a wireless sensor network. We demonstrate that a type-2 fuzzy membership function (MF), i.e., a Gaussian MF with uncertain standard deviation (std) is most appropriate to model a single node lifetime in wireless sensor networks. In our research, we study two basic sensor placement schemes: square-grid and hex-grid. Two fuzzy logic systems (FLSs): a singleton type-1 FLS and an interval type-2 FLS are designed to perform lifetime estimation of the sensor network. We compare our fuzzy approach with other nonfuzzy schemes in previous papers. Simulation results show that FLS offers a feasible method to analyze and estimate the sensor network lifetime and the interval type-2 FLS in which the antecedent and the consequent membership functions are modeled as Gaussian with uncertain std outperforms the singleton type-1 FLS and the nonfuzzy schemes.     

An Experimental Study on Nonlinear Function Computation for Neural/Fuzzy Hardware Design

An experimental study on the influence of the computation of basic nodal nonlinear functions on the performance of (NFSs) is described in this paper. Systems' architecture size, their approximation capability, and the smoothness of provided mappings are used as performance indexes for this comparative paper. Two widely used kernel functions, the sigmoid-logistic function and the Gaussian function, are analyzed by their computation through an accuracy-controllable approximation algorithm designed for hardware implementation. Two artificial neural network (ANN) paradigms are selected for the analysis: backpropagation neural networks (BPNNs) with one hidden layer and radial basis function (RBF) networks. Extensive simulation of simple benchmark approximation problems is used in order to achieve generalizable conclusions. For the performance analysis of fuzzy systems, a functional equivalence theorem is used to extend obtained results to fuzzy inference systems (FISs). Finally, the adaptive neurofuzzy inference system (ANFIS) paradigm is used to observe the behavior of neurofuzzy systems with learning capabilities     

Adaptive non-additive generalized fuzzy systems

An adaptive non-additive generalized fuzzy model (GFM) is presented in this paper using the framework of Gaussian mixture model (GMM) which provides the membership functions for the input fuzzy sets. By replacing the consequent part of the additive GFM rule by a non-additive function, we obtain the non-additive GFM. The coefficients of the non-additive function then become the fuzzy measures. The defuzzified output constructed from both the premise and consequent parts of the modified GFM rules in the wake of non-additiveness takes the form of Choquet fuzzy integral. The parameters of the premise and the consequent parts of the non-additive fuzzy rules are updated based on the estimation error on the arrival of each online data to make the system adaptive. The resulting adaptive non-additive fuzzy model is applied on two benchmark applications and the results demonstrate the advantage of the adaptive feature.     

A TSK-type neurofuzzy network approach to system modeling problems.

We develop a neurofuzzy network technique to extract TSK-type fuzzy rules from a given set of input-output data for system modeling problems. Fuzzy clusters are generated incrementally from the training dataset, and similar clusters are merged dynamically together through input-similarity, output-similarity, and output-variance tests. The associated membership functions are defined with statistical means and deviations. Each cluster corresponds to a fuzzy IF-THEN rule, and the obtained rules can be further refined by a fuzzy neural network with a hybrid learning algorithm which combines a recursive singular value decomposition-based least squares estimator and the gradient descent method. The proposed technique has several advantages. The information about input and output data subspaces is considered simultaneously for cluster generation and merging. Membership functions match closely with and describe properly the real distribution of the training data points. Redundant clusters are combined, and the sensitivity to the input order of training data is reduced. Besides, generation of the whole set of clusters from the scratch can be avoided when new training data are considered.     

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

An approach to Nonlinear Output Error (NOE) modelling using Takagi–Sugeno (TS) fuzzy model for a class of nonlinear dynamic systems having variability in their outputs is presented. Furthermore, the approach is compared and graphically illustrated with other alternate approaches on the basis of interval data and interval membership functions. Assuming the identification method can be repeated offline a number of times under similar conditions, multiple input–output time series can be obtained from the underlying system. These time series are pre-processed using the techniques of statistics and probability theory to generate the envelopes of response (curves outlining the upper and lower extremes of response) at each time instant. Two types of envelopes are described in this research: the max–min envelopes and the envelopes based on the confidence intervals provided by extended Chebyshev's inequality. By incorporating interval data in fuzzy modelling and using the theory of symbolic interval-valued data, a TS fuzzy model with interval antecedent and consequent parameters is obtained. This algorithm provides a model for predicting the expected response as well as envelopes. In order to validate the presented model, a simulation case study is devised in this paper. Moreover, it is demonstrated on the real data obtained from an electro-mechanical throttle valve.     

Generalised influential rule search scheme for fuzzy function approximation

The present paper is a humble attempt to develop a fuzzy function approximator which can completely self-generate its fuzzy rule base and input-output membership functions from an input-output data set. The fuzzy system can be further adapted to modify its rule base and output membership functions to provide satisfactory performance. This proposed scheme, called generalised influential rule search scheme, has been successfully implemented to develop pure fuzzy function approximators as well as fuzzy logic controllers. The satisfactory performance of the proposed scheme is amply demonstrated by implementing it to develop different major components in a process control loop. The versatility of the algorithm is further proved by implementing it for a benchmark nonlinear function approximation problem.     

Performance Enhancement for T–S Fuzzy Control Using Neural Networks

A new control scheme is proposed to improve the system performance for Takagi–Sugeno (T–S) fuzzy system using control grade functions tuned by neural networks. First, systematic modeling method is introduced to construct the exact T–S fuzzy model for a nonlinear control system. For the T–S fuzzy model, the system uncertainty affects only the membership functions. To cope with this problem, the grade functions, resulting from the membership functions of the control rules, are tuned by a back-propagation network. On the other hand, the feedback gains of the control rules are determined by solving a set of linear matrix inequalities (LMIs) which satisfy sufficient conditions of the closed-loop stability. As a result, both stability guarantee and better performance are concluded. The scheme is applied to a ball-and-beam system example verified by numerical simulations.     

Fine Tuning Of Membership Functions For Fuzzy Neural Systems

This paper presents a new method for fine‐tuning the Gaussian membership functions of a fuzzy neural network ( FNN ) to improve approximation accuracy. This method results in special shape membership functions without the convex property. We first recall that any continuous function can be represented by a linear combination of Gaussian functions with any standard deviation. Therefore, the Gaussian membership function in the second layer of the FNN can be replaced by several small Gaussian functions; the weighting vectors of this new network (called FNN₅ ) can then be updated using the backpropagation algorithm. The proposed method can adapt proper membership functions for any nonlinear input/output mapping to achieve highly accurate approximation. Convergence analysis shows that the weighting vectors of the FNN₅ eventually converge to the optimal values. Simulation results indicate that (a) this approach improves approximation accuracy, and (b) that the number of rules can be reduced for any given level of accuracy. For the purpose of illustrating the proposed method, the FNN₅ is also applied to tune PI controllers such that gain and phase margins of the closed‐loop system achieve the desired specifications.     

Multivariable gain-scheduled fuzzy logic control of an exothermic reactor

Fuzzy logic control frequently exhibits superior performance to classical linear controllers even for ‘hard’, mathematically well defined plants, as described in this paper. The case-study of a highly nonlinear exothermic continuous stirred tank reactor, which poses a multivariable control problem with two interacting loops and open-loop instability, is used. The behaviour of the fuzzy logic controller is compared with that of a PID controller. A smooth, easily tuneable gain-schedule is designed to handle offset-like problems with a fuzzy controller. It is analytically shown that such a gain-schedule is the simpler, intuitive equivalent of a manipulation of the corresponding fuzzy membership functions. The fuzzy controller structure chosen is a parsimonious one, with the choice of Gaussian bell-shaped membership functions generating a smooth input/output surface with nontrivial inferencing spanning the entire input space. This provides a clear, non-heuristic reason to select Gaussian over triangular shapes for membership functions. The gain-scheduled fuzzy controller shows excellent control performance, significantly outperforming the PID controllers in both servo and regulatory modes. The disturbance rejection behaviour of the modified fuzzy controller is observed to be particularly good.     

Influential rule search scheme (IRSS) - a new fuzzy pattern classifier

Automatic generation of fuzzy rule base and membership functions from an input-output data set, for reliable construction of an adaptive fuzzy inference system, has become an important area of research interest. We propose a new robust, fast acting adaptive fuzzy pattern classification scheme, named influential rule search scheme (IRSS). In IRSS, rules which are most influential in contributing to the error produced by the adaptive fuzzy system are identified at the end of each epoch and subsequently modified for satisfactory performance. This fuzzy rule base adjustment scheme is accompanied by an output membership function adaptation scheme for fine tuning the fuzzy system architecture. This iterative method has shown a relatively high speed of convergence. Performance of the proposed IRSS is compared with other existing pattern classification schemes by implementing it for Fisher's iris data problem and Wisconsin breast cancer data problems.     

A Fuzzy Inverse Model Construction Method for General Monotonic Multi-Input-- Single-Output (MISO) Systems

This paper presents a novel method of systematically constructing a fuzzy inverse model for general multi-input--single-output (MISO) systems represented with triangular input membership functions, singleton output membership function, and fuzzy-mean defuzzification. The fuzzy inverse model construction method has the ability of uniquely determining the inverse relationship for each input–output pair. It is derived in a straightforward way and the required input variables can be simultaneously obtained by the fuzzy inferencing calculation to realize the desired output value. Simulation examples are provided to demonstrate the effectiveness of the proposed method to find the inverse kinematics solutions for complex multiple degree-of-freedom industrial robot manipulators.      

Neurofuzzy identification of an autonomous underwater vehicle

Neurofuzzy modelling is ideally suited to many nonlinear system identification and data modelling applications. By combining the attractive attributes of fuzzy systems and neural networks transparent models of ill-defined systems can be identified. Available expert a priori knowledge is used to construct an initial model. Data modelling techniques from the neural network, statistical and conventional system identification communities are then used to adapt these models. As a result accurate parsimonious models which are transparent and easy to validate are identified. Recent advances in the datadriven identification algorithms have now made neurofuzzy modelling appropriate for high-dimensional problems for which the expert knowledge and data may be of a poor quality. In this paper neurofuzzy modelling techniques are presented. This powerful approach to system identification is demonstrated by its application to the identification of an Autonomous Underwater Vehicle (AUV).