Type-2 Fuzzistics for Symmetric Interval Type-2 Fuzzy Sets: Part 1, Forward Problems

Interval type-2 fuzzy sets (T2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). In this two-part paper, we focus on symmetric interval T2 FSs for which the centroid (which is an interval type-1 FS) provides a measure of its uncertainty. Intuitively, we anticipate that geometric properties about the FOU, such as its area and the center of gravities (centroids) of its upper and lower MFs, will be associated with the amount of uncertainty in such a T2 FS. The main purpose of this paper (Part 1) is to demonstrate that our intuition is correct and to quantify the centroid of a symmetric interval T2 FS, and consequently its uncertainty, with respect to such geometric properties. It is then possible, for the first time, to formulate and solve forward problems, i.e., to go from parametric interval T2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems. These solutions are used in Part 2 to solve some inverse problems, i.e., to go from uncertain data to parametric interval T2 FS models (T2 fuzzistics)     

Type-2 Fuzzistics for Nonsymmetric Interval Type-2 Fuzzy Sets: Forward Problems

Interval type-2 fuzzy sets (IT2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). The centroid of an IT2 FS, which is an IT1 FS, provides a measure of the uncertainty in the IT2 FS. The main purpose of this paper is to quantify the centroid of a non-symmetric IT2 FS with respect to geometric properties of its FOU. This is very important because interval data collected from subjects about words suggests that the FOUs of most words are non-symmetrical. Using the results in this paper, it is possible to formulate and solve forward problems, i.e., to go from parametric non-symmetric IT2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems for non-symmetrical triangular, trapezoidal, Gaussian and shoulder FOUs.     

Encoding Words Into Interval Type-2 Fuzzy Sets Using an Interval Approach

This paper presents a very practical type-2-fuzzistics methodology for obtaining interval type-2 fuzzy set (IT2 FS) models for words, one that is called an interval approach (IA). The basic idea of the IA is to collect interval endpoint data for a word from a group of subjects, map each subject's data interval into a prespecified type-1 (T1) person membership function, interpret the latter as an embedded T1 FS of an IT2 FS, and obtain a mathematical model for the footprint of uncertainty (FOU) for the word from these T1 FSs. The IA consists of two parts: the data part and the FS part. In the data part, the interval endpoint data are preprocessed, after which data statistics are computed for the surviving data intervals. In the FS part, the data are used to decide whether the word should be modeled as an interior, left-shoulder, or right-shoulder FOU. Then, the parameters of the respective embedded T1 MFs are determined using the data statistics and uncertainty measures for the T1 FS models. The derived T1 MFs are aggregated using union leading to an FOU for a word, and finally, a mathematical model is obtained for the FOU. In order that all researchers can either duplicate our results or use them in their research, the raw data used for our codebook examples, as well as a MATLAB M-file for the IA, have been put on the Internet at: http://sipi.usc.edu/$sim$mendel.      

Type-2 fuzzy sets and systems: an overview

This paper provides an introduction to and an overview of type-2 fuzzy sets (T2 FS) and systems. It does this by answering the following questions: What is a T2 FS and how is it different from a T1 FS? Is there new terminology for a T2 FS? Are there important representations of a T2 FS and, if so, why are they important? How and why are T2 FSs used in a rule-based system? What are the detailed computations for an interval T2 fuzzy logic system (IT2 FLS) and are they easy to understand? Is it possible to have an IT2 FLS without type reduction? How do we wrap this up and where can we go to learn more?     

Type-2 fuzzy Gaussian mixture models

This paper presents a new extension of Gaussian mixture models (GMMs) based on type-2 fuzzy sets (T2 FSs) referred to as T2 FGMMs. The estimated parameters of the GMM may not accurately reflect the underlying distributions of the observations because of insufficient and noisy data in real-world problems. By three-dimensional membership functions of T2 FSs, T2 FGMMs use footprint of uncertainty (FOU) as well as interval secondary membership functions to handle GMMs uncertain mean vector or uncertain covariance matrix, and thus GMMs parameters vary anywhere in an interval with uniform possibilities. As a result, the likelihood of the T2 FGMM becomes an interval rather than a precise real number to account for GMMs uncertainty. These interval likelihoods are then processed by the generalized linear model (GLM) for classification decision-making. In this paper we focus on the role of the FOU in pattern classification. Multi-category classification on different data sets from UCI repository shows that T2 FGMMs are consistently as good as or better than GMMs in case of insufficient training data, and are also insensitive to different areas of the FOU. Based on T2 FGMMs, we extend hidden Markov models (HMMs) to type-2 fuzzy HMMs (T2 FHMMs). Phoneme classification in the babble noise shows that T2 FHMMs outperform classical HMMs in terms of the robustness and classification rate. We also find that the larger area of the FOU in T2 FHMMs with uncertain mean vectors performs better in classification when the signal-to-noise ratio is lower.     

Interval Type-2 Fuzzy Logic Systems Made Simple

To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way     

区间二型模糊集和模糊系统: 综述与展望

一型模糊集可以建模单个用户的语义概念中的不确定性, 即个体内不确定性. 一型模糊系统在控制和机器学习中得到了大量成功应用. 区间二型模糊集能同时建模个体内不确定性和个体间不确定性, 因而在很多应用中显示了比一型模糊系统更好的性能, 是近年来的研究热点. 本文首先介绍了区间二型模糊集的重要概念和理论研究进展, 总结了其在决策和机器学习中的成功应用, 然后介绍了区间二型模糊系统的基本操作和理论研究进展, 并回顾了其在控制和机器学习中的典型应用. 最后, 对区间二型模糊集和模糊系统未来的研究方向进行了展望.     

Modeling data uncertainty on electric load forecasting based on Type-2 fuzzy logic set theory

Real applications based on type-2 (T2) fuzzy sets are rare. The main reason is that the T2 fuzzy set theory requires massive computation and complex determination of secondary membership function. Thus most real-world applications are based on one simplified method, i.e. interval type-2 (IT2) fuzzy sets in which the secondary membership function is defined as interval sets. Consequently all computations in three-dimensional space are degenerated into calculations in two-dimensional plane, computing complexity is reduced greatly. However, ability on modeling information uncertainty is also reduced. In this paper, a novel methodology based on T2 fuzzy sets is proposed i.e. T2SDSA-FNN (Type-2 Self-Developing and Self-Adaptive Fuzzy Neural Networks). Our novelty is that (1) proposed system is based on T2 fuzzy sets, not IT2 ones; (2) it tackles one difficult problem in T2 fuzzy logic systems (FLS), i.e. massive computing time of inference so as not to be applicable to solve real world problem; and (3) membership grades on third dimensional space can be automatically determined from mining input data. The proposed method is validated in a real data set collected from Macao electric utility. Simulation and test results reveal that it has superior accuracy performance on electric forecasting problem than other techniques shown in existing literatures.     

广义区间二型模糊集合的词计算

普通的模糊集合是点值为二维的一型模糊集合,二型模糊集合(Type-2 fuzzy sets, T2 FS)是点值为三维的模糊集合, T2 FS比相应的一型难以理解和计算. 为了让人们更好地理解T2 FS并推广其应用, 本文提出了广义区间二型模糊集合(Generalized interval type-2 fuzzy sets, GIT2 FS)的定义, 并将其分成三类:离散型、半离散型及连续型,分别给出相应的数学表达式与扩展原理公式,并得到了GIT2 FS在两种不同的模糊逻辑算子下的词计算.     

New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule

The centroid of an interval type-2 fuzzy set (IT2 FS) provides a measure of the uncertainty of such a FS. Its calculation is very widely used in interval type-2 fuzzy logic systems. In this paper, we present properties about the centroid of an IT2 FS. We also illustrate many of the general results for a T2 fuzzy granule (FG) in order to develop some understanding about the uncertainty of the FG in terms of its vertical and horizontal dimensions. At present, the T2 FG is the only IT2 FS for which it is possible to obtain closed-form formulas for the centroid, and those formulas are in this paper.     

Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to C-Means

In many pattern recognition applications, it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set. Uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms. Therefore, various types of uncertainty may be taken into account when performing several pattern recognition methods. When one performs clustering with fuzzy sets, fuzzy membership values express assignment availability of patterns for clusters. However, when one assigns fuzzy memberships to a pattern set, imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment. When one encounters fuzzy clustering, fuzzy membership design includes various uncertainties (e.g., distance measure, fuzzifier, prototypes, etc.). In this paper, we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final C-partition in the fuzzy C-means (FCM) algorithm. To design and manage uncertainty for fuzzifier m, we extend a pattern set to interval type-2 fuzzy sets using two fuzzifiers m₁ and m₂ which creates a footprint of uncertainty (FOU) for the fuzzifier m. Then, we incorporate this interval type-2 fuzzy set into FCM to observe the effect of managing uncertainty from the two fuzzifiers. We also provide some solutions to type-reduction and defuzzification (i.e., cluster center updating and hard-partitioning) in FCM. Several experimental results are given to show the validity of our method     

Type-2 fuzzy hidden Markov models and their application to speech recognition

This paper presents an extension of hidden Markov models (HMMs) based on the type-2 (T2) fuzzy set (FS) referred to as type-2 fuzzy HMMs (T2 FHMMs). Membership functions (MFs) of T2 FSs are three-dimensional, and this new third dimension offers additional degrees of freedom to evaluate the HMMs fuzziness. Therefore, T2 FHMMs are able to handle both random and fuzzy uncertainties existing universally in the sequential data. We derive the T2 fuzzy forward-backward algorithm and Viterbi algorithm using T2 FS operations. In order to investigate the effectiveness of T2 FHMMs, we apply them to phoneme classification and recognition on the TIMIT speech database. Experimental results show that T2 FHMMs can effectively handle noise and dialect uncertainties in speech signals besides a better classification performance than the classical HMMs.     

Parallel Interval Type-2 Subsethood Neural Fuzzy Inference System

Neuro-fuzzy models are being increasingly employed in the domains like weather forecasting, stock market prediction, computational finance, control, planning, physics, economics and management, to name a few. These models enable one to predict system behavior in a more human-like manner than their crisp counterparts. In the present work, an interval type-2 neuro-fuzzy evolutionary subsethood based model has been proposed for its use in finding solutions to some well-known problems reported in the literature such as regression analysis, data mining and research problems relevant to expert and intelligent systems. A novel subsethood based interval type-2 fuzzy inference system, named as Interval Type-2 Subsethood Neural Fuzzy Inference System (IT2SuNFIS) is proposed in the present work. Mathematical modeling and empirical studies clearly bring out the efficacy of this model in a wide variety of practical problems such as Truck backer-upper control, Mackey–Glass time-series prediction, Narazaki–Ralescu and bell function approximation. The simulation results demonstrate intelligent decision making capability of the proposed system based on the available data. The major contribution of this work lies in identifying subsethood as an efficient measure for finding correlation in interval type-2 fuzzy sets and applying this concept to a wide variety of problems pertaining to expert and intelligent systems. Subsethood between two type-2 fuzzy sets is different from the commonly used sup-star methods. In the proposed model, this measure assists in providing better contrast between dissimilar objects. This method, coupled with the uncertainty handling capacity of type-2 fuzzy logic system, results in better trainability and improved performance of the system. The integration of subsethood with type-2 fuzzy logic system is a novel idea with several advantages, which is reported for the first time in this paper.     

关于二型模糊集合的一些基本问题

采用集合论的方法给出了单位模糊集合和二型模糊集合及其在一点的限制等定义,使得二型模糊集合更易于理解.通过定义嵌入单位模糊集合来描述一般二型模糊集合,并给出离散、半连通二型模糊集合的表达式.根据论域、主隶属度及隶属函数的特性将二型模糊集合分为四种类型：离散、半连通、连通及复合型,并根据连通的特点将连通二型模糊集合分为单连通及多连通两类.利用支集的闭包（Closure of support,CoS）划分法表述主隶属度及区间二型模糊集合.提出了CoS二、三次划分法分别来表述单、复连通二型模糊集合,并使每一个子区域的上下边界及次隶属函数在该子区域上的限制分别具有相同的解析表述式.最后,探讨了二型模糊集合在一点的限制、主隶属度、支集、嵌入单位模糊集合之间的关系.     

Reinforcement Interval Type-2 Fuzzy Controller Design by Online Rule Generation and Q-Value-Aided Ant Colony Optimization

This paper proposes a new reinforcement-learning method using online rule generation and Q-value-aided ant colony optimization (ORGQACO) for fuzzy controller design. The fuzzy controller is based on an interval type-2 fuzzy system (IT2FS). The antecedent part in the designed IT2FS uses interval type-2 fuzzy sets to improve controller robustness to noise. There are initially no fuzzy rules in the IT2FS. The ORGQACO concurrently designs both the structure and parameters of an IT2FS. We propose an online interval type-2 rule generation method for the evolution of system structure and flexible partitioning of the input space. Consequent part parameters in an IT2FS are designed using Q-values and the reinforcement local-global ant colony optimization algorithm. This algorithm selects the consequent part from a set of candidate actions according to ant pheromone trails and Q-values, both of which are updated using reinforcement signals. The ORGQACO design method is applied to the following three control problems: (1) truck-backing control; (2) magnetic-levitation control; and (3) chaotic-system control. The ORGQACO is compared with other reinforcement-learning methods to verify its efficiency and effectiveness. Comparisons with type-1 fuzzy systems verify the noise robustness property of using an IT2FS.     

二型模糊决策理论与方法研究综述

二型模糊集(type-2 fuzzy set, T2FS)是将模糊集中的隶属函数拓展为一型模糊集而产生的集合,其具有表示更深层次不确定性的优势,能够极大程度地增强对客观世界不确定性的刻画能力.因此,近年来围绕二型模糊环境下的决策理论与方法研究得到了蓬勃发展.鉴于此,对二型模糊决策理论与方法进行系统性综述,梳理该领域的发展脉络,阐明现有工作的研究态势,总结二型模糊信息集成与决策的主要研究成果.首先,介绍二型模糊集的发展历程和基础理论研究现状;然后,分别针对基于二型模糊信息的决策基础理论(信息融合理论、偏好关系理论和测度理论)以及决策方法的研究现状进行概述;最后,对二型模糊决策理论与方法的未来研究方向进行展望.     

区间二型模糊综合评判下的语言动力学分析

为了解决多人对事物的多因素动态评估问题,提出区间二型模糊综合评判下的语言动力学分析方法,给出半连通区间二型模糊集合的表述与运算.综合二型模糊集合与模糊综合评判,探讨二型模糊综合评判方法.结合不同时段的数据,形成多因素动态评估的语言动力学轨迹.最后将区间二型模糊综合评判下的语言动力学系统应用于旅游景区的动态评估中.     

A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process

The fuzzy analytic hierarchy process (FAHP) has been used to solve various multi-criteria decision-making problems where trapezoidal type-1 fuzzy sets are utilized in defining decision-makers’ linguistic judgment. Previous theories have suggested that interval type-2 fuzzy sets (IT2 FS) can offer an alternative that can handle vagueness and uncertainty. This paper proposes a new FAHP characterized by IT2 FS for linguistic variables. Differently from the typical FAHP, which directly utilizes trapezoidal type-1 fuzzy numbers, this method introduces IT2 FS to enhance judgment in the fuzzy decision-making environment. This new model includes linguistic variables in IT2 FS and a rank value method for normalizing upper and lower memberships of IT2 FS. The proposed model is illustrated by a numerical example of work safety evaluation. Comparable results are also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is consistent with the other two methods despite difference in weight priorities.     

A Type-2 Fuzzy Approach to Linguistic Summarization of Data

This paper introduces an application of type-2 fuzzy sets in data linguistic summarization. The original approach by Yager (1982) based on representing natural language statements via type-1, i.e., the Zadeh fuzzy sets, is generalized with type-2 fuzzy sets applied as models of linguistically expressed quantities and/or properties of objects. Type-2 sets extend the known summarization procedures by handling fuzzy values stored in databases, and allow to represent a linguistic term via a few different membership functions (e.g., provided by different experts), which makes the method more general and human-consistent. Furthermore, quality measures for type-2 summaries are discussed in order to evaluate the informativeness of the messages generated. Finally, two prototype applications are presented and the success of the new method is discussed.     

$alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications

This paper 1) reviews the alpha-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the alpha-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using alpha-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each alpha-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 alpha -planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.