Linguistic models and linguistic modeling

The study is concerned with a linguistic approach to the design of a new category of fuzzy (granular) models. In contrast to numerically driven identification techniques, we concentrate on budding meaningful linguistic labels (granules) in the space of experimental data and forming the ensuing model as a web of associations between such granules. As such models are designed at the level of information granules and generate results in the same granular rather than pure numeric format, we refer to them as linguistic models. Furthermore, as there are no detailed numeric estimation procedures involved in the construction of the linguistic models carried out in this way, their design mode can be viewed as that of a rapid prototyping. The underlying algorithm used in the development of the models utilizes an augmented version of the clustering technique (context-based clustering) that is centered around a notion of linguistic contexts-a collection of fuzzy sets or fuzzy relations defined in the data space (more precisely a space of input variables). The detailed design algorithm is provided and contrasted with the standard modeling approaches commonly encountered in the literature. The usefulness of the linguistic mode of system modeling is discussed and illustrated with the aid of numeric studies including both synthetic data as well as some time series dealing with modeling traffic intensity over a broadband telecommunication network.     

Linguistic models as a framework of user-centric system modeling

In this paper, the fundamental idea of linguistic models introduced by Pedrycz and Vasilakos (1999) is followed and their comprehensive design framework is developed. The paradigm of linguistic modeling is concerned with constructing models that: 1) are user centric and 2) inherently dwell upon collections of highly interpretable and user-oriented entities such as information granules. The objective of this paper is to investigate and compare alternative design options, present an organization of the overall optimization process, and come up with a specification of several evaluation mechanisms of the performance of the models. The underlying design tool guiding the development of linguistic models revolves around the augmented version of fuzzy clustering known as a context-based or conditional fuzzy C-means (C-FCM). The design process comprises several main phases such as: 1) defining and further refining context fuzzy sets; 2) completing conditional fuzzy clustering; and 3) optimizing parameters (connections) linking information granules in the input and output spaces. An iterative process of forming information granules in the input and output spaces is discussed. Their membership functions are adjusted by the gradient-based learning guided by the minimization of some performance index. The paper comes with a comprehensive suite of experiments that lead to some design guidelines of the models. Furthermore, the performance of linguistic models is contrasted with that of other fuzzy models, especially radial basis function neural networks (RBFNNs) and related constructs that are based on concepts of fuzzy clustering.     

Risk evaluation through decision-support architectures in threat assessment and countering terrorism

Owing to their inherent nature, terrorist activities could be highly diversified. The risk assessment becomes a crucial component as it helps us weigh pros and cons versus possible actions or some planning pursuits. The recognition of threats and their relevance/seriousness is an integral part of the overall process of classification, recognition, and assessing eventual actions undertaken in presence of acts of chem.-bio terrorism. In this study, we introduce an overall scheme of risk assessment realized on a basis of classification results produced for some experimental data capturing the history of previous threat cases. The structural relationships in these experimental data are first revealed with the help of information granulation – fuzzy clustering. We introduce two criteria using which information granules are evaluated, that is (a) representation capabilities which are concerned with the quality of representation of numeric data by abstract constructs such as information granules, and (b) interpretation aspects which are essential in the process of risk evaluation. In case of representation facet of information granules, we demonstrate how a reconstruction criterion quantifies their quality. Three ways in which interpretability is enhanced are studied. First, we show how to construct the information granules with extended cores (where the uncertainty associated with risk evaluation could be reduced) and shadowed sets, which provide a three-valued logic perspective of information granules given in the form of fuzzy sets. Subsequently, we show a way of interpreting fuzzy sets via an optimized set of its α-cuts.     

SoHyFIS-Yager: A self-organizing Yager based Hybrid neural Fuzzy Inference System

The Hybrid neural Fuzzy Inference System (HyFIS) is a multilayer adaptive neural fuzzy system for building and optimizing fuzzy models using neural networks. In this paper, the fuzzy Yager inference scheme, which is able to emulate the human deductive reasoning logic, is integrated into the HyFIS model to provide it with a firm and intuitive logical reasoning and decision-making framework. In addition, a self-organizing gaussian Discrete Incremental Clustering (gDIC) technique is implemented in the network to automatically form fuzzy sets in the fuzzification phase. This clustering technique is no longer limited by the need to have prior knowledge about the number of clusters present in each input and output dimensions. The proposed self-organizing Yager based Hybrid neural Fuzzy Inference System (SoHyFIS-Yager) introduces the learning power of neural networks to fuzzy logic systems, while providing linguistic explanations of the fuzzy logic systems to the connectionist networks. Extensive simulations were conducted using the proposed model and its performance demonstrates its superiority as an effective neuro-fuzzy modeling technique.     

The Development of Incremental Models

In this study, we introduce and discuss a concept of an incremental granular model. In contrast to typical rule-based systems encountered in fuzzy modeling, the underlying principle exploited here is to consider a two-phase development of fuzzy models. First, we build a standard regression model which could be treated as a preliminary construct capturing the linear part of the data and in this way forming a backbone of the entire construct. Next, all modeling discrepancies are compensated by a collection of rules that become attached to the regions of the input space where the error is localized. The incremental model is constructed by building a collection of information granules through some specialized fuzzy clustering, called context-based (conditional) fuzzy C-means that is guided by the distribution of error of the linear part of the model. The architecture of the model is discussed along with the major algorithmic phases of its development. In particular, the issue of granularity of fuzzy sets of context and induced clusters is discussed vis-a-vis the performance of the model. Numeric studies concern some low-dimensional synthetic data and several datasets coming from the machine learning repository.     

模糊树模型及其在复杂系统辨识中的应用

基于二叉树和模糊逻辑理论,提出了一种用于复杂系统建模的模糊树模型.将线性 模型和模糊集组织在树结构上,并给出了更新线性模型系数和模糊集隶属度函数的混合算 法.与其他建模方法相比,如ANFIS,模糊树模型计算量小,精度高,尤其在高维数据建模中 更为明显.仿真结果描述了这种方法的性能.     

Abstraction and specialization of information granules

In his paper, we introduce a model of generalization and specialization of information granules. The information granules themselves are modeled as fuzzy sets or fuzzy relations. The generalization is realized by or-ing fuzzy sets while the specialization is completed through logic and operation. These two logic operators are realized using triangular norms (that is t- and a-norms). We elaborate on two (top-down and bottom-up) strategies of constructing information granules that arise as results of generalization and specialization. Various triangular norms are experimented with and some conclusions based on numeric studies are derived.     

Structural developments of fuzzy systems with the aid of information granulation

We introduce a design procedure for fuzzy systems using the concept of information granulation and genetic optimization. Information granulation and resulting information granules themselves become an important design aspect of fuzzy models. By accommodating the formalism of fuzzy sets, the model is geared towards capturing relationship between information granules (fuzzy sets) rather than concentrating on plain numeric data. Information granulation realized with the use of the standard C-Means clustering helps determine the initial values of the parameters of the fuzzy models. This in particular concerns such essential components of the rules as the initial apexes of the membership functions standing in the premise part of the fuzzy rules and the initial values of the polynomial functions standing in the consequence part. The initial parameters are afterwards tuned with the aid of the genetic algorithms (GAs) and the least square method (LSM). The overall design methodology arises as a hybrid development process involving structural and parametric optimization. Especially, genetic algorithms and C-Means are used to generate the structurally as well as parametrically optimized fuzzy model. To identify the structure and estimate parameters of the fuzzy model we exploit the methodologies such as joint and successive method realized by means of genetic algorithms. The proposed model is evaluated using experimental data and its performance is contrasted with the behavior of the fuzzy models available in the literature.     

Rule-based multi-FNN identification with the aid of evolutionary fuzzy granulation

In this paper, we introduce a category of Multi-Fuzzy-Neural Networks (Multi-FNNs) models, analyze the underlying architectures and propose a comprehensive identification framework. The proposed Multi-FNNs are based on a concept of fuzzy rule-based FNNs that use H ard C-M eans (HCM) clustering and evolutionary fuzzy granulation and exploit linear inference being treated as a generic inference mechanism of approximate reasoning. By this nature, this FNN model is geared toward capturing relationships between information granules–fuzzy sets. The form of the information granules themselves (in particular their distribution and a type of membership functions) becomes an important design feature of the FNN model that contributes to its structural and parametric optimization. The genetically guided global optimization is then augmented by more refined gradient-based learning mechanisms such as a standard back-propagation. The HCM algorithm, whose role is to carry out preprocessing of the experimental data, is utilized to determine the structure of Multi-FNNs. The detailed parameters of the Multi-FNN (such as apexes of membership functions, learning rates, and momentum coefficients) are adjusted using genetic algorithms. The proposed aggregate performance index helps achieve a sound balance between approximation and generalization (predictive) abilities of the model. To evaluate an effectiveness of the introduced model, several numeric data sets are experimented with. Those include a time-series data of gas furnace, NOx emission process of gas turbine power plant and some synthetic data.     

模糊元图及其特性分析

模糊图已被广泛地应用于系统的分析与建模。然而现有的结构不适合于分析元素集之间的有向模糊关系。文中讨论了一种新的模糊图论结论－－模糊元图,它描述的是集合而非单个元素之间的有向关系,具有很强的图形化描述功能和形式化分析能力。文中着重分析了模糊元图的特性,尤其是其邻近矩阵的特性,这是模糊元图研究与应用的基础,模糊元图在系统建模和模糊规则管理方面有着广阔的应用前景。     

模糊规则模型的粒度性能指标评估方法

模糊规则模型广泛应用于许多领域,而现有的模糊规则模型主要使用基于数值形式的性能评估指标,忽略了对于模糊集合本身的评价,因此提出了一种模糊规则模型性能评估的新方法。该方法可以有效地评估模糊规则模型输出结果的非数值（粒度）性质。不同于通常使用的数值型性能指标（比如均方误差（MSE））,该方法通过信息粒的特征来表征模型输出的粒度结果的质量,并将该指标使用在模糊模型的性能优化中。信息粒性能采用（数据的）覆盖率和（信息粒自身的）特异性两个基本指标得以量化,并通过使用粒子群优化实现了粒度输出质量（表示为覆盖率和特异性的乘积）的最大化。此外,该方法还优化了模糊聚类形成的信息粒的分布。实验结果表明该指标对于模糊规则模型性能评估的有效性。     

The granular extension of Sugeno-type fuzzy models based on optimal allocation of information granularity and its application to forecasting of time series

The Sugeno-type fuzzy models are used frequently in system modeling. The idea of information granulation inherently arises in the design process of Sugeno-type fuzzy model, whereas information granulation is closely related with the developed information granules. In this paper, the design method of Sugeno-type granular model is proposed on a basis of an optimal allocation of information granularity. The overall design process initiates with a well-established Sugeno-type numeric fuzzy model (the original Sugeno-type model). Through assigning soundly information granularity to the related parameters of the antecedents and the conclusions of fuzzy rules of the original Sugeno-type model (i.e. granulate these parameters in the way of optimal allocation of information granularity becomes realized), the original Sugeno-type model is extended to its granular counterpart (granular model). Several protocols of optimal allocation of information granularity are also discussed. The obtained granular model is applied to forecast three real-world time series. The experimental results show that the method of designing Sugeno-type granular model offers some advantages yielding models of good prediction capabilities. Furthermore, those also show merits of the Sugeno-type granular model: (1) the output of the model is an information granule (interval granule) rather than the specific numeric entity, which facilitates further interpretation; (2) the model can provide much more flexibility than the original Sugeno-type model; (3) the constructing approach of the model is of general nature as it could be applied to various fuzzy models and realized by invoking different formalisms of information granules.     

A Hybrid Learning Method for the Data-Driven Design of Linguistic Dynamic Systems

In lots of data based prediction or modeling applications, uncertainties and/or noises in the observed data cannot be avoided. In such cases, it is more preferable and reasonable to provide linguistic (fuzzy) predicted results described by fuzzy memberships or fuzzy sets instead of the crisp estimates depicted by numbers. Linguistic dynamic system (LDS) provides a powerful tool for yielding linguistic (fuzzy) results. However, it is still difficult to construct LDS models from observed data. To solve this issue, this paper first presents a simplified LDS whose inputoutput mapping can be determined by closed-form formulas. Then, a hybrid learning method is proposed to construct the data-driven LDS model. The proposed hybrid learning method firstly generates fuzzy rules by the subtractive clustering method, then carries out further optimization of centers of the consequent triangular fuzzy sets in the fuzzy rules, and finally adopts multiobjective optimization algorithm to determine the left and right end-points of the consequent triangular fuzzy sets. The proposed approach is successfully applied to three real-world prediction applications which are: prediction of energy consumption of a building, forecasting of the traffic flow, and prediction of the wind speed. Simulation results show that the uncertainties in the data can be effectively captured by the linguistic (fuzzy) estimates. It can also be extended to some other prediction or modeling problems, in which observed data have high levels of uncertainties.      

The modeling of time series based on fuzzy information granules

A lot of research has resulted in many time series models with high precision forecasting realized at the numerical level. However, in the real world, higher numerical precision may not be necessary for the perception, reasoning and decision-making of human. Model of time series with an ability of humans to perceive and process abstract entities (rather than numeric entities) is more adaptable for some problems of decision-making. With this regard, information granules and granular computing play a primordial role. Fox example, if change range (intervals) of stock prices for a certain period in the future is regarded as information granule, constructing model that can forecast change ranges (intervals) of stock prices for a period in the future is better able to help stock investors make reasonable decisions in comparison with those based upon specific forecasting numerical value of stock price. In this paper, we propose a new modeling approach to realize interval prediction, in which the idea of information granules and granular computing is integrated with the classical Chen’s method. The proposed method is to segment an original numeric time series into a collection of time windows first, and then build fuzzy granules expressed as a certain fuzzy set over each time windows by exploiting the principle of justifiable granularity. Finally, fuzzy granular model can be constructed by mining fuzzy logical relationships of adjacent granules. The constructed model can carry out interval prediction by degranulation operation. Two benchmark time series are used to validate the feasibility and effectiveness of the proposed approach. The obtained results demonstrate the effectiveness of the approach. Besides, for modeling and prediction of large-scale time series, the proposed approach exhibit a clear advantage of reducing computation overhead of modeling and simplifying forecasting.     

Data validation and missing data reconstruction using self-organizing map for water treatment

Applications in the water treatment domain generally rely on complex sensors located at remote sites. The processing of the corresponding measurements for generating higher-level information such as optimization of coagulation dosing must therefore account for possible sensor failures and imperfect input data. In this paper, self-organizing map (SOM)-based methods are applied to multiparameter data validation and missing data reconstruction in a drinking water treatment. The SOM is a special kind of artificial neural networks that can be used for analysis and visualization of large high-dimensional data sets. It performs both in a nonlinear mapping from a high-dimensional data space to a low-dimensional space aiming to preserve the most important topological and metric relationships of the original data elements and, thus, inherently clusters the data. Combining the SOM results with those obtained by a fuzzy technique that uses marginal adequacy concept to identify the functional states (normal or abnormal), the SOM performances of validation and reconstruction process are tested successfully on the experimental data stemming from a coagulation process involved in drinking water treatment.     

Morphological perceptrons with competitive learning: Lattice-theoretical framework and constructive learning algorithm

A morphological neural network is generally defined as a type of artificial neural network that performs an elementary operation of mathematical morphology at every node, possibly followed by the application of an activation function. The underlying framework of mathematical morphology can be found in lattice theory.With the advent of granular computing, lattice-based neurocomputing models such as morphological neural networks and fuzzy lattice neurocomputing models are becoming increasingly important since many information granules such as fuzzy sets and their extensions, intervals, and rough sets are lattice ordered. In this paper, we present the lattice-theoretical background and the learning algorithms for morphological perceptrons with competitive learning which arise by incorporating a winner-take-all output layer into the original morphological perceptron model. Several well-known classification problems that are available on the internet are used to compare our new model with a range of classifiers such as conventional multi-layer perceptrons, fuzzy lattice neurocomputing models, k-nearest neighbors, and decision trees.     

NBSOM: The naive Bayes self-organizing map

The naive Bayes model has proven to be a simple yet effective model, which is very popular for pattern recognition applications such as data classification and clustering. This paper explores the possibility of using this model for multidimensional data visualization. To achieve this, a new learning algorithm called naive Bayes self-organizing map (NBSOM) is proposed to enable the naive Bayes model to perform topographic mappings. The training is carried out by means of an online expectation maximization algorithm with a self-organizing principle. The proposed method is compared with principal component analysis, self-organizing maps, and generative topographic mapping on two benchmark data sets and a real-world image processing application. Overall, the results show the effectiveness of NBSOM for multidimensional data visualization.     

虚拟环境中装配设计语义的表达、传递与转化研究

研究了虚拟环境产品装配建模过程中装配语义的表达、装配语义与装配约束的转化技术,提出了采用语义－约束图对装配设计语义与约束进行维护。通过语义－约束图可以从语义层次和约束层次对产品装配模型进行编辑和维护。提出了采用模糊参量表达和处理产品装配设计中的模糊语义。基于模糊参量的语义表达不仅为设计信息输入提供了更大的自由度,而且扩展了计算机对模糊信息的处理能力,有利于产品设计意图的维护。通过产品装配信息从抽象到具体、从模糊到精确的转化,实现了虚拟装配设计系统对抽象、模糊设计信息的表达、传递与处理。