Entropy-based shadowed set approximation of intuitionistic fuzzy sets

We propose a method to approximate Intuitionistic Fuzzy Sets (IFSs) with Shadowed Sets that could be used, in decision making or similar tasks, when the full information about membership values is not necessary, is difficult to process or to interpret. Our approach is based on an information-theoretic perspective and aims at preserving the uncertainty, represented through an entropy measure, in the original IFS by minimizing the difference between the entropy in the input IFS and the output Shadowed Set. We propose three different efficient optimization algorithms that retain Fuzziness, Lack of Knowledge, or both, and illustrate their computation through an illustrative example. We also evaluate the application of the proposed approximation methods in the Machine Learning setting by showing that the approximation, through the proposed methods, of IFS k-Nearest Neighbors is able to outperform, in terms of running time, the standard algorithm.     

On worst-case approximation of feasible system sets via orthonormal basis functions

This note deals with the approximation of sets of linear time-invariant systems via orthonormal basis functions. This problem is relevant to conditional set membership identification, where a set of feasible systems is available from observed data, and a reduced-complexity model must be estimated. The basis of the model class is made of impulse responses of linear filters. The objective of the note is to select the basis function poles according to a worst-case optimality criterion. Suboptimal conditional identification algorithms are introduced and tight bounds are provided on the associated identification errors.     

不确定性工艺知识的模糊粗糙隶属度函数表示方法*

分析了工艺知识不确定性和模糊性的特点,提出了基于模糊粗糙隶属度函数的工艺不确定性知识的表示与应用模型。实现了对各类用数值和符号描述的工艺不确定知识的表示,通过隶属度值代替粗糙集中的实数值作为模糊粗糙集的属性,依照属性的依赖度和属性表的核进行了工艺知识属性的约简,得到了工艺知识规则集,并以LED芯片制造领域的工艺知识表示为实例,验证了该方法的可行性与有效性。     

An approximation method for estimation in linear systems with parameter uncertainty

Estimation of the state variables of a linear system with parameter uncertainties is performed using an asymptotically unbiased linear minimum-variance recursive estimator in continuous time. Estimates of the parameters can be obtained simultaneously, but are found to be biased. By augmenting additional linear dynamic equations which represent an asymptotic expansion in the unknown parameters, a linear structure is formed which approximates the original nonlinear system. However, the initial conditions and additive process noise are not Gaussian. The convergence properties of the state variance for this expansion are illustrated analytically by a scalar dynamic system. The numerical aspects of this example illustrate the behavior of the actual variance of the error in the state estimate and the predicted error variance as the order of the approximation increases. For the vector state problem, only the multidimensional dynamic system in canonical form with a single output is developed. For ann-dimensional system withnunknown constant parameters, a first-order approximation requiresnadditional linear equations. This approach can be extended to correlated parameter processes.     

Identification and model approximation for continuous-time systems on finite parameter sets

Almost-sure convergence of the maximum likelihood and the maximum a posteriori probability estimates of unknown parameters of continuous-time stochastic dynamical linear time-invariant systems is investigated. The unknown parameter set is assumed to be finite. The situation where the ture parameter does not belong to the unknown parameter set is considered, as well as the situation where the true model is included in the unknown parameter set.     

Mid-frequency structural dynamics with parameter uncertainty

In this paper, a number of frequency-domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range are integrated into the stochastic finite element method. In all cases, frequency-domain model reduction strategies are used to minimize the computational effort in the mid-frequency range. Firstly, an energy operator approach (EOA) is investigated. In this procedure, an energy operator adapted to a fixed medium frequency band is defined whose dominant eigensubspace is used to construct a reduced model using a Ritz–Galerkin method. Secondly, the proper orthogonal decomposition method is used to extract the spatial dominant coherent structures in the vibration wave field in the mid-frequency band. The coherent structures are the eigenvectors corresponding to the dominant eigenvalues of the spatial autocorrelation function of the system response. Consequently, a close relationship between the energy operator approach and the proper orthogonal decomposition method is identified, although these two approaches are not mathematically identical. The proper orthogonal decomposition method appears to be more straightforward compared to the energy operator method from the viewpoint of numerical implementation. Thirdly, another approach, namely the dynamic element method based on frequency-dependent finite element shape functions, is considered. A stochastic reduction method is then utilized to represent the uncertain parameters, modelled as stochastic processes, in terms of their dominant scales of fluctuation. The Karhunen–Loeve and the polynomial chaos decompositions are used to that effect, in the context of a stochastic finite element formalism. The methodology adopted in the paper thus integrates efficient dynamical reduction techniques with a reduction scheme of stochastic processes for the mid-frequency vibration of linear random systems. The formulation is exemplified by its application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation. The example is also used to highlight some of the relative features of the three dynamic reduction strategies.     

From set membership to group membership: a separation of concerns

We revisit the well-known group membership problem and show how it can be considered a special case of a simple problem, the set membership problem. In the set membership problem, processes maintain a set whose elements are drawn from an arbitrary universe: They can request the addition or removal of elements to/from that set, and they agree on the current value of the set. Group membership corresponds to the special case where the elements of the set happen to be processes. We exploit this new way of looking at group membership to give a simple and succinct specification of this problem and to outline a simple implementation approach based on the state machine paradigm. This treatment of group membership separates several issues that are often mixed in existing specifications and/or implementations of group membership. We believe that this separation of concerns greatly simplifies the understanding of this problem.     

Optimality, approximation, and complexity in set membership H/sub /spl infin// identification

Investigates the set membership identification of time-invariant, discrete-time, exponentially stable, possibly infinite-dimensional, linear systems from time or frequency-domain data, corrupted by deterministic noise. The aim is to deliver not a single model, but a set of models whose size in H/sub /spl infin// norm measures the uncertainty in the identification. The main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets. A method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency. Such approximations allow one to compute, within any desired accuracy, the identification error of any identified model, and to evaluate an optimal model at any given number of frequencies. By suitably approximating these values, model sets with nominal models in RH/sub /spl infin// are then derived, whose order is selected by trading off between model set complexity and identification accuracy degradation. This degradation is evaluated by computing the optimality level, defined as the ratio between the reduced model identification error and the optimal one. A numerical example demonstrates the effectiveness of the presented results.     

Fuzzy rule interpolation based on principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets

Fuzzy rule interpolation is an important research topic in sparse fuzzy rule-based systems. In this paper, we present a new method for dealing with fuzzy rule interpolation in sparse fuzzy rule-based systems based on the principle membership functions and uncertainty grade functions of interval type-2 fuzzy sets. The proposed method deals with fuzzy rule interpolation based on the principle membership functions and the uncertainty grade functions of interval type-2 fuzzy sets. It can deal with fuzzy rule interpolation with polygonal interval type-2 fuzzy sets and can handle fuzzy rule interpolation with multiple antecedent variables. We also use some examples to compare the fuzzy interpolative reasoning results of the proposed method with the ones of an existing method. The experimental result shows that the proposed method gets more reasonable results than the existing method for fuzzy rule interpolation based on interval type-2 fuzzy sets.     

Asymptotic properties of set membership identification algorithms

This paper addresses the asymptotic worst-case properties of set membership identification (SMID) algorithms. We first present a set membership identification algorithm which can be used with a model structure consisting of parametric and nonparametric uncertainty, as well as output additive disturbances. This algorithm is then studied in the context of asymptotic worst-case behavior. We derive lower bounds on the worst-case achievable identification error measured by the volume, as well as the sum-of-sidelengths of the identified ellipsoidal uncertainty sets. We then show that there exist inputs which can shrink the uncertainty sets to these lower bounds.     

具有广义线性隶属函数的典型模糊系统的通用逼近性*

设计了一种将三角形和梯形隶属函数作为特例的广义线性隶属函数,推导了输入采用广义线性隶属函数的典型Mamdani模糊系统的解析结构,证明了典型模糊系统是单调、递减的有界连续函数;在此基础上证明了该类模糊系统能以任意精度逼近任意连续实函数,最后仿真实例证明了本设计的有效性。     

模型参数失配有界下的扩展集员估计方法

在非线性模型参数失配下,直接采用滤波算法很难获到理想的估计状态.本文基于扩展集员估计方法,在状态估计中引入参数的不确定信息,提出一种参数失配有界下的状态估计方法.该方法应用区间或集合运算的法则,计算由参数失配引起的偏差范围,并将其用椭球集外包.在状态估计的预测步,通过该偏差椭球集与先验椭球区间的并运算,得到预测椭球区间;在状态估计的更新步,利用观测椭球集对预测椭球区间进行更新,从而得到后验椭球集合以及状态估计值.最后,在数值仿真和发酵模型中的仿真应用验证了算法的有效性.     

Sequential pattern mining in databases with temporal uncertainty

Temporally uncertain data widely exist in many real-world applications. Temporal uncertainty can be caused by various reasons such as conflicting or missing event timestamps, network latency, granularity mismatch, synchronization problems, device precision limitations, data aggregation. In this paper, we propose an efficient algorithm to mine sequential patterns from data with temporal uncertainty. We propose an uncertain model in which timestamps are modeled by random variables and then design a new approach to manage temporal uncertainty. We integrate it into the pattern-growth sequential pattern mining algorithm to discover probabilistic frequent sequential patterns. Extensive experiments on both synthetic and real datasets prove that the proposed algorithm is both efficient and scalable.     

Optimal control of linear systems with parameter uncertainty

A class of infinite-horizon regulator problems is formulated for families of time-invariant linear systems with parameter uncertainty. Under a regularity assumption, the optimal linear, state-feedback control is shown to exist and is defined via a positive-definite solution of a family of Riccati-type, algebraic equations. The solvability of these equations is equivalent to the stabilizability of the family of linear systems by a constant, linear feedback.     

Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets

In this study, a new approach for the formation of type-2 membership functions is introduced. The footprint of uncertainty is formed by using rectangular type-2 fuzzy granules and the resulting membership function is named as granular type-2 membership function. This new approach provides more degrees of freedom and design flexibility in type-2 fuzzy logic systems. Uncertainties on the grades of membership functions can be represented independently for any region in the universe of discourse and free of any functional form. So, the designer could produce nonlinear, discontinuous or hybrid membership functions in granular formation and therefore could model any desired discontinuity and nonlinearity. The effectiveness of the proposed granular type-2 membership functions is firstly demonstrated by simulations done on noise corrupted Mackey–Glass time series prediction. Secondly, flexible design feature of granular type-2 membership functions is illustrated by modeling a nonlinear system having dead zone with uncertain system parameters. The simulation results show that type-2 fuzzy logic systems formed by granular type-2 membership functions have more modeling capabilities than the systems using conventional type-2 membership functions and they are more robust to system parameter changes and noisy inputs.     

具有任意形状隶属函数的分层模糊系统逼近性能研究

首先证明了对任意给定的矩阵A和正数c, 一定存在向量b, 使得方程Ax=b有非负解, 且b和解的范数均小于c. 在此基础上证明了具有任意形状隶属函数的分层模糊系统对紧集上连续函数的逼近性质, 为使用分层模糊系统进行辨识或控制以避免模糊规则数目随系统变量个数呈指数增长提供了理论依据.     

R–fuzzy隶属度粗糙近似集的优势测度理论

给出了一种R–fuzzy集隶属度的权重比较与量化方法,提出了优势测度概念,解决了粗糙近似集中隶属度的重要性难以确定的问题.首先给出优势测度的定义,然后,研究了优势测度的性质,指出了优势测度1型模糊集的本质属性.优势测度不仅实现了R-fuzzy集隶属度的量化,而且成为R-fuzzy集与2型模糊集联系的纽带.通过隶属度的优势测度,实现了人类感知领域中的群体共识与个性认识的区分,反过来通过优势测度的可视化,可以对人类不同感知下的隶属度值给出合理的推断与比较.最后,通过声音感知实验研究给出了优势测度可视化的特点及操作方法,讨论了不同职业测试组对于同一声音的理解在隶属度数值上的差异.对于涉及人类感知与模式辨识的应用领域具有较易的操作性与较强的实用性.