Uncertainty measures for general Type-2 fuzzy sets

Five uncertainty measures have previously been defined for interval Type-2 fuzzy sets (IT2 FSs), namely centroid, cardinality, fuzziness, variance and skewness. Based on a recently developed α-plane representation for a general T2 FS, this paper generalizes these definitions to such T2 FSs and, more importantly, derives a unified strategy for computing all different uncertainty measures with low complexity. The uncertainty measures of T2 FSs with different shaped Footprints of Uncertainty and different kinds of secondary membership functions (MFs) are computed and are given as examples. Observations and summaries are made for these examples, and a Summary Interval Uncertainty Measure for a general T2 FS is proposed to simplify the interpretations. Comparative studies of uncertainty measures for Quasi-Type-2 (QT2), IT2 and T2 FSs are also performed to examine the feasibility of approximating T2 FSs using QT2 or even IT2 FSs.     

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.     

On answering the question “Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?”

This paper, which is tutorial in nature, demonstrates how the Embedded Sets Representation Theorem (RT) for a general type-2 fuzzy set (T2 FS), when specialized to an interval (I)T2 FS, can be used as the starting point to solve many diverse problems that involve IT2 FSs. The problems considered are: set theoretic operations, centroid, uncertainty measures, similarity, inference engine computations for Mamdani IT2 fuzzy logic systems, linguistic weighted average, person membership function approach to type-2 fuzzistics, and Interval Approach to type-2 fuzzistics. Each solution obtained from the RT is a structural solution but is not a practical computational solution, however, the latter are always found from the former. It is this author’s recommendation that one should use the RT as a starting point whenever solving a new problem involving IT2 FSs because it has had such great success in solving so many such problems in the past, and it answers the question “Where do I start in order to solve a new problem involving IT2 FSs?”     

Parameter Optimization of Interval Type-2 Fuzzy Neural Networks Based on PSO and BBBC Methods

Interval type-2 fuzzy neural networks (IT2FNNs) can be seen as the hybridization of interval type-2 fuzzy systems (IT2FSs) and neural networks (NNs). Thus, they naturally inherit the merits of both IT2FSs and NNs. Although IT2FNNs have more advantages in processing uncertain, incomplete, or imprecise information compared to their type-1 counterparts, a large number of parameters need to be tuned in the IT2FNNs, which increases the difficulties of their design. In this paper, big bang-big crunch (BBBC) optimization and particle swarm optimization (PSO) are applied in the parameter optimization for Takagi-Sugeno-Kang (TSK) type IT2FNNs. The employment of the BBBC and PSO strategies can eliminate the need of backpropagation computation. The computing problem is converted to a simple feed-forward IT2FNNs learning. The adoption of the BBBC or the PSO will not only simplify the design of the IT2FNNs, but will also increase identification accuracy when compared with present methods. The proposed optimization based strategies are tested with three types of interval type-2 fuzzy membership functions (IT2FMFs) and deployed on three typical identification models. Simulation results certify the effectiveness of the proposed parameter optimization methods for the IT2FNNs.      

基于区间二型模糊决策粗糙集的三支决策方法

考虑到区间二型模糊数在描述高度不确定性信息方面的优势,将区间二型模糊数拓展到决策粗糙集中,提出两种区间二型模糊三支决策方法.在没有类标签的区间二型模糊信息系统中,解释损失函数与确定条件概率是需要解决的两个关键问题.首先,根据区间二型模糊数的性质,将其引入决策粗糙集中,为损失函数提供一种新的解释.其次,基于贝叶斯决策过程...     

Data-driven modeling and optimization of thermal comfort and energy consumption using type-2 fuzzy method

In the research domain of intelligent buildings and smart home, modeling and optimization of the thermal comfort and energy consumption are important issues. This paper presents a type-2 fuzzy method based data-driven strategy for the modeling and optimization of thermal comfort words and energy consumption. First, we propose a methodology to convert the interval survey data on thermal comfort words to the interval type-2 fuzzy sets (IT2 FSs) which can reflect the inter-personal and intra-personal uncertainties contained in the intervals. This data-driven strategy includes three steps: survey data collection and pre-processing, ambiguity-preserved conversion of the survey intervals to their representative type-1 fuzzy sets (T1 FSs), IT2 FS modeling. Then, using the IT2 FS models of thermal comfort words as antecedent parts, an evolving type-2 fuzzy model is constructed to reflect the online observed energy consumption data. Finally, a multiobjective optimization model is presented to recommend a reasonable temperature range that can give comfortable feeling while reducing energy consumption. The proposed method can be used to realize comfortable but energy-saving environment in smart home or intelligent buildings.     

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.     

Uncertainty measures for interval type-2 fuzzy sets

Fuzziness (entropy) is a commonly used measure of uncertainty for type-1 fuzzy sets. For interval type-2 fuzzy sets (IT2 FSs), centroid, cardinality, fuzziness, variance and skewness are all measures of uncertainties. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions use a Representation Theorem for IT2 FSs. Formulas for computing the cardinality, fuzziness, variance and skewness of an IT2 FS are derived. These definitions should be useful in IT2 fuzzy logic systems design using the principles of uncertainty, and in measuring the similarity between two IT2 FSs.     

An interval number distance- and ranking-based method for remotely sensed image fuzzy clustering

ABSTRACT

Fuzzy c-means clustering is an important non-supervised classification method for remote-sensing images and is based on type-1 fuzzy set theory. Type-1 fuzzy sets use singleton values to express the membership grade; therefore, such sets cannot describe the uncertainty of the membership grade. Interval type-2 fuzzy c-means (IT2FCM) clustering and relevant methods are based on interval type-2 fuzzy sets. Real vectors are used to describe the clustering centres, and the average values of the upper and lower membership grades are used to determine the classification of each pixel. Thus, the width information for interval clustering centres and interval membership grades are ignored. The main contribution of this article is to propose an improved IT2FCM* algorithm by adopting interval number distance (IND) and ranking methods, which use the width information of interval clustering centres and interval membership grades, thus distinguishing this method from existing fuzzy clustering methods. Three different IND definitions are tested, and the distance definition proposed by Li shows the best performance. The second contribution of this work is that two fuzzy cluster validity indices, FS- and XB-, are improved using the IND. Three types of multi/hyperspectral remote-sensing data sets are used to test this algorithm, and the experimental results show that the IT2FCM* algorithm based on the IND proposed by Li performs better than the IT2FCM algorithm using four cluster validity indices, the confusion matrix, and the kappa coefficient (κ). Additionally, the improved FS- index has more indicative ability than the original FS- index.     

Bearing condition prediction considering uncertainty: An interval type-2 fuzzy neural network approach

Rolling-element bearings are critical components of rotating machinery. It is important to accurately predict in real-time the health condition of bearings so that maintenance practices can be scheduled to avoid malfunctions or even catastrophic failures. In this paper, an Interval Type-2 Fuzzy Neural Network (IT2FNN) is proposed to perform multi-step-ahead condition prediction of faulty bearings. Since the IT2FNN defines an interval type-2 fuzzy logic system in the form of a multi-layer neural network, it can integrate the merits of each, such as fuzzy reasoning to handle uncertainties and neural networks to learn from data. The interval type-2 fuzzy linguistic process in the IT2FNN enables the system to handle prediction uncertainties, since the type-2 fuzzy sets are such sets whose membership grades are type-1 fuzzy sets that can be used in failure prediction due to the difficult determination of an exact membership function for a fuzzy set. Noisy data of faulty bearings are used to validate the proposed predictor, whose performance is compared with that of a prevalent type-1 condition predictor called Adaptive Neuro-Fuzzy Inference System (ANFIS). The results show that better prediction accuracy can be achieved via the IT2FNN.     

On type-reduction of type-2 fuzzy sets: A review

As an undetachable module of type-2 (T2) fuzzy computations and reasoning, type-reduction methods play an important role in various fuzzy disciplines including fuzzy logic systems and fuzzy clustering. Importance of type-reduction techniques lies in the fact that they are the main tools for collecting the entire inherent vagueness of the data. Therefore, type-reduction methods form the output of type-2 fuzzy sets (T2 FSs) as the representative of the entire uncertainty in a given space. Hence, their accuracy, precision, and performance speed is of much interest. This paper, presents a comprehensive review on various type-reduction and defuzzification strategies for general and interval type-2 fuzzy sets and systems. It is tried to analyze the existing approaches from different point of views accompanied by extensive comparisons on different features of type-reduction methods to facilitate further research studies by the fuzzy community.     

Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets

In this paper, we present a new method to deal with fuzzy multiple attributes group decision-making problems based on ranking interval type-2 fuzzy sets. First, we propose a new method for ranking interval type-2 fuzzy sets. Then, we propose a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of interval type-2 fuzzy sets. We also use some examples to illustrate the fuzzy multiple attributes group decision-making process of the proposed method. The proposed method is simpler than the methods presented in [Chen and Lee, 2010a] and [Lee and Chen, 2010b] for fuzzy multiple attributes group decision-making based on interval type-2 fuzzy sets. It provides us with a useful way for dealing with fuzzy multiple attributes group decision-making problems based on interval type-2 fuzzy sets.     

Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets

The focus of this paper is the linguistic weighted average (LWA), where the weights are always words modeled as interval type-2 fuzzy sets (IT2 FSs), and the attributes may also (but do not have to) be words modeled as IT2 FSs; consequently, the output of the LWA is an IT2 FS. The LWA can be viewed as a generalization of the fuzzy weighted average (FWA) where the type-1 fuzzy inputs are replaced by IT2 FSs. This paper presents the theory, algorithms, and an application of the LWA. It is shown that finding the LWA can be decomposed into finding two FWAs. Since the LWA can model more uncertainties, it should have wide applications in distributed and hierarchical decision-making.     

区间二型模糊相似度与包含度

相似度与包含度是模糊集合理论中的两个重要概念,但对于二型模糊集合的研究还较为少见．鉴于此,提出了新的区间二型模糊相似度与包含度．首先选择了二者的公理化定义;然后基于公理化定义提出了新的计算公式,并讨论了二者的相互转换关系;最后通过实例来验证二者的性能,并将区间二型模糊相似度与Yang-Shih聚类方法相结合,用于高斯区间二型模糊集合的聚类分析,得到了合理的层次聚类树．仿真实例表明新测度具有一定的实用价值．     

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.     

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.     

A type-2 neural fuzzy system learned through type-1 fuzzy rules and its FPGA-based hardware implementation

This paper first proposes a type-2 neural fuzzy system (NFS) learned through its type-1 counterpart (T2NFS-T1) and then implements the built IT2NFS-T1 in a field-programmable gate array (FPGA) chip. The antecedent part of each fuzzy rule in the T2NFS-T1 uses interval type-2 fuzzy sets, while the consequent part uses a Takagi-Sugeno-Kang (TSK) type with interval combination weights. The T2NFS-T1 uses a simplified type-reduction operation to reduce system training time and hardware implementation cost. Given a training data set, a TSK type-1 NFS is first learned through structure and parameter learning. The built type-1 fuzzy logic system (FLS) is then extended to a type-2 FLS, where highly overlapped type-1 fuzzy sets are merged into interval type-2 fuzzy sets to reduce the total number of fuzzy sets. Finally, the rule consequent and antecedent parameters in the T2NFS-T1 are tuned using a hybrid of the gradient descent and rule-ordered recursive least square (RLS) algorithms. Simulation results and comparisons with various type-1 and type-2 FLSs verify the effectiveness and efficiency of the T2NFS-T1 for system modeling and prediction problems. A new hardware circuit using both parallel-processing and pipeline techniques is proposed to implement the learned T2NFS-T1 in an FPGA chip. The T2NFS-T1 chip reduces the hardware implementation cost in comparison to other type-2 fuzzy chips.     

$alpha$-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.     

Corrections to “Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets”

In the previous paper, we have proposed linguistic weighted average (LWA) algorithms that can be used in distributed and hierarchical decision making. The original LWA algorithms were completely based on the representation theorem for interval type-2 fuzzy sets (IT2 FSs). In later usage, we found that when the lower membership functions (LMFs) of the inputs and weights are of different heights, the LMF of the output IT2 FS may be nonconvex and discontinuous. In this letter, a correction to the original LWA algorithms is proposed. The new LWA algorithms are simpler and easier to understand; so, it should facilitate the applications of the LWAs.      

A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets

Ranking methods, similarity measures and uncertainty measures are very important concepts for interval type-2 fuzzy sets (IT2 FSs). So far, there is only one ranking method for such sets, whereas there are many similarity and uncertainty measures. A new ranking method and a new similarity measure for IT2 FSs are proposed in this paper. All these ranking methods, similarity measures and uncertainty measures are compared based on real survey data and then the most suitable ranking method, similarity measure and uncertainty measure that can be used in the computing with words paradigm are suggested. The results are useful in understanding the uncertainties associated with linguistic terms and hence how to use them effectively in survey design and linguistic information processing.