A neurofuzzy network knowledge extraction and extended gram-Schmidt algorithm for model subspace decomposition

This paper introduces a new neurofuzzy model construction and parameter estimation algorithm from observed finite data sets, based on a Takagi and Sugeno (T-S) inference mechanism and a new extended Gram-Schmidt orthogonal decomposition algorithm, for the modeling of a priori unknown dynamical systems in the form of a set of fuzzy rules. The first contribution of the paper is the introduction of a one to one mapping between a fuzzy rule-base and a model matrix feature subspace using the T-S inference mechanism. This link enables the numerical properties associated with a rule-based matrix subspace, the relationships amongst these matrix subspaces, and the correlation between the output vector and a rule-base matrix subspace, to be investigated and extracted as rule-based knowledge to enhance model transparency. The matrix subspace spanned by a fuzzy rule is initially derived as the input regression matrix multiplied by a weighting matrix that consists of the corresponding fuzzy membership functions over the training data set. Model transparency is explored by the derivation of an equivalence between an A-optimality experimental design criterion of the weighting matrix and the average model output sensitivity to the fuzzy rule, so that rule-bases can be effectively measured by their identifiability via the A-optimality experimental design criterion. The A-optimality experimental design criterion of the weighting matrices of fuzzy rules is used to construct an initial model rule-base. An extended Gram-Schmidt algorithm is then developed to estimate the parameter vector for each rule. This new algorithm decomposes the model rule-bases via an orthogonal subspace decomposition approach, so as to enhance model transparency with the capability of interpreting the derived rule-base energy level. This new approach is computationally simpler than the conventional Gram-Schmidt algorithm for resolving high dimensional regression problems, whereby it is computationally desirable to decompose complex models into a few submodels rather than a single model with large number of input variables and the associated curse of dimensionality problem. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

On the use of fuzzy rule interpolation techniques for monotonic multi-input fuzzy rule base models

Constructing a monotonicity relating function is important, as many engineering problems revolve around a monotonicity relationship between input(s) and output(s). In this paper, we investigate the use of fuzzy rule interpolation techniques for monotonicity relating fuzzy inference system (FIS). A mathematical derivation on the conditions of an FIS to be monotone is provided. From the derivation, two conditions are necessary. The derivation suggests that the mapped consequence fuzzy set of an FIS to be of a monotonicity order. We further evaluate the use of fuzzy rule interpolation techniques in predicting a consequent associated with an observation according to the monotonicity order. There are several findings in this article. We point out the importance of an ordering criterion in rule selection for a multi-input FIS before the interpolation process; and hence, the practice of choosing the nearest rules may not be true in this case. To fulfill the monotonicity order, we argue with an example that conventional fuzzy rule interpolation techniques that predict each consequence separately is not suitable in this case. We further suggest another class of interpolation techniques that predicts the consequence of a set of observations simultaneously, instead of separately. This can be accomplished with the use of a search algorithm, such as the brute force, genetic algorithm or etc.

Fuzzy rule base assessment models : theoretical analyses and a case study

An assessment model is usually a mathematical model that produces a measuring index, in the form of a numerical score to a situation/object, with respect to the subject of measure. To allow a valid and useful comparison among various situations/objects according to their associated numerical scores to be made, two important properties, i.e., the monotone output property and output resolution properties, are essential in fuzzy inference-based assessment problems. In this paper, the conditions for a fuzzy assessment model to fulfill the monotone output property is investigated using a derivative approach. A guideline on how the input membership functions should be tuned is also provided. Besides, the output resolution property is defined as the derivative of the output of the assessment model with respect to the input, whereby the derivative should be greater than a minimum resolution. Based on the derivative, improvements to the output resolution property by refining the fuzzy production rules are suggested. A case study on the Bowles fuzzy RPN model to demonstrate the effectiveness of the properties is also included.

On the use of fuzzy inference techniques in assessment models : part I — theoretical properties

An assessment model is a mathematical model that produces a measuring index, either in the form of a numerical score or a category to a situation/object, with respect to the subject of measure. From the numerical score, decision can be made and action can be taken. To allow valid and useful comparisons among various situations/objects according to their associated numerical scores to be made, the monotone output property and the output resolution property are essential in fuzzy inference-based assessment problems. We investigate the conditions for a fuzzy assessment model to fulfill the monotone output property using a derivative approach. A guideline on how the input membership functions should be tuned is also provided. Besides, the output resolution property is defined as the derivative of the output of the assessment model with respect to its input. This derivative should be greater than the minimum resolution required. From the derivative, we suggest improvements to the output resolution property by refining the fuzzy production rules.

On the use of fuzzy inference techniques in assessment models : part II : industrial applications

In this paper, we study the applicability of the monotone output property and the output resolution property in fuzzy assessment models to two industrial Failure Mode and Effect Analysis (FMEA) problems. First, the effectiveness of the monotone output property in a single-input fuzzy assessment model is demonstrated with a proposed fuzzy occurrence model. Then, the usefulness of the two properties to a multi-input fuzzy assessment model, i.e., the Bowles fuzzy Risk Priority Number (RPN) model, is assessed. The experimental results indicate that both the fuzzy occurrence model and Bowles fuzzy RPN model are able to fulfill the monotone output property, with the derived conditions (in Part I) satisfied. In addition, the proposed rule refinement technique is able to improve the output resolution property of the Bowles fuzzy RPN model.

A new monotonicity index for fuzzy rule-based systems

A search in the literature reveals that mathematical conditions (usually sufficient conditions) for the Fuzzy Inference System (FIS) models to satisfy the monotonicity property have been developed. A monotonically-ordered fuzzy rule base is important to maintain the monotonicity property of an FIS. However, it may difficult to obtain a monotonically-ordered fuzzy rule base in practice. We have previously introduced the idea of fuzzy rule relabeling to tackle this problem. In this paper, we further propose a monotonicity index for the FIS system, which serves as a metric to indicate the degree of a fuzzy rule base fulfilling the monotonicity property. The index is useful to provide an indication whether a fuzzy rule base should (or should not) be used in practice, even with fuzzy rule relabeling. To illustrate the idea, the zero-order Sugeno FIS model is exemplified. We add noise as errors into the fuzzy rule base to formulate a set of non-monotone fuzzy rules. As such, the metric also acts as a measure of noise in the fuzzy rule base. The results show that the proposed metric is useful to indicate the degree of a fuzzy rule base fulfilling the monotonicity property.

Fuzzy portfolio allocation models through a new risk measure and fuzzy sharpe ratio

A new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper. Beyond the well-known Sharpe ratio (i.e., the reward-to-variability ratio) in modern portfolio theory, we initiate the so-called fuzzy Sharpe ratio in the fuzzy modeling context. In addition to the introduction of the new risk measure, we also put forward the reward-to-uncertainty ratio to assess the portfolio performance in fuzzy modeling. Corresponding to two approaches based on TM and TW fuzzy arithmetic, two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized, while the fuzzy Sharpe ratio is maximized. These models are solved by the fuzzy approach or by the genetic algorithm (GA). Solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean-variance optimization (MVO) model used prevalently in the financial literature. In terms of portfolio performance evaluated by the fuzzy Sharpe ratio and the reward-to-uncertainty ratio, the model using TW fuzzy arithmetic results in higher performance portfolios than those obtained by both the MVO and the fuzzy model, which employs TM fuzzy arithmetic. We also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using GA, although GA can offer a series of well-diversified portfolio solutions diagrammed in a Pareto frontier.

A proposal for reliability evaluation of components on electric power distribution system integrating probabilistic models and fuzzy inference systems

The system reliability depends on the reliability of its components itself. Therefore, it is necessary a methodology capable of inferring the state of functionality of these components to establish reliable indices of quality. Allocation models for maintenance and protective devices, among others, have been used in order to improve the quality and availability of services on electric power distribution systems. This paper proposes a methodology for assessing the reliability of distribution system components in an integrated way, using probabilistic models and fuzzy inference systems to infer about the operation probability of each component. © 2012 IEEE.

Modelos computacionais fuzzy e neuro-fuzzy para avaliarem os efeitos da poluição do ar

Pós-graduação em Engenharia Mecânica - FEG

Epidemiological Models of Directly Transmitted Diseases: An Approach via Fuzzy Sets Theory

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A Fuzzy Decision Strategy for Topic Identification and Dynamic Selection of Language Models

International audience

Constrained fuzzy power flow models with correlation between nodal injections

This paper extends the symmetric/constrained fuzzy powerflow models by including the potential correlations between nodal injections. Therefore, the extension of the model allows the specification of fuzzy generation and load values and of potential correlations between nodal injections. The enhanced version of the symmetric/constrained fuzzy powerflow model is applied to the 30-bus IEEE test system. The results prove the importance of the inclusion of data correlations in the analysis of transmission system adequacy.

Probabilistic Security Constrained Fuzzy Power Flow Models

In restructured power systems, generation and commercialization activities became market activities, while transmission and distribution activities continue as regulated monopolies. As a result, the adequacy of transmission network should be evaluated independent of generation system. After introducing the constrained fuzzy power flow (CFPF) as a suitable tool to quantify the adequacy of transmission network to satisfy 'reasonable demands for the transmission of electricity' (as stated, for instance, at European Directive 2009/72/EC), the aim is now showing how this approach can be used in conjunction with probabilistic criteria in security analysis. In classical security analysis models of power systems are considered the composite system (generation plus transmission). The state of system components is usually modeled with probabilities and loads (and generation) are modeled by crisp numbers, probability distributions or fuzzy numbers. In the case of CFPF the component’s failure of the transmission network have been investigated. In this framework, probabilistic methods are used for failures modeling of the transmission system components and possibility models are used to deal with 'reasonable demands'. The enhanced version of the CFPF model is applied to an illustrative case.

A prioritised fuzzy constraint satisfaction approach to model agent negotiation for railway scheduling

With the recent regulatory reforms in a number of countries, railways resources are no longer managed by a single party but are distributed among different stakeholders. To facilitate the operation of train services, a train service provider (SP) has to negotiate with the infrastructure provider (IP) for a train schedule and the associated track access charge. This paper models the SP and IP as software agents and the negotiation as a prioritized fuzzy constraint satisfaction (PFCS) problem. Computer simulations have been conducted to demonstrate the effects on the train schedule when the SP has different optimization criteria. The results show that by assigning different priorities on the fuzzy constraints, agents can represent SPs with different operational objectives.