Dongrui Wu

Enhanced Karnik--Mendel Algorithms

The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and superexponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that, on average, the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time. An additional (at least) 23% computational cost can be saved if no sorting of the inputs is needed.

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.

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.

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.

A type-2 fuzzy logic controller for the liquid-level process

This paper focuses on evolving type-2 fuzzy logic controllers (FLCs) genetically and examining whether they are better able to handle modelling uncertainties. The study is conducted by utilizing a type-2 FLC, evolved by a genetic algorithm (GA), to control a liquid-level process. A two stage strategy is employed to design the type-2 FLC. First, the parameters of a type-1 FLC are optimized using GA. Next, the footprint of uncertainty is evolved by blurring the fuzzy input set. Experimental results show that the type-2 FLC copes well with the complexity of the plant, and can handle the modelling uncertainty better than its type-1 counterpart.

On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers

Interval type-2 fuzzy logic controllers (IT2 FLCs) have recently been attracting a lot of research attention. Many reported results have shown that IT2 FLCs are better able to handle uncertainties than their type-1 (T1) counterparts. A challenging question is the following: What are the fundamental differences between IT2 and T1 FLCs? Once the fundamental differences are clear, we can better understand the advantages of IT2 FLCs and, hence, make better use of them. This paper explains two fundamental differences between IT2 and T1 FLCs: 1) Adaptiveness, meaning that the embedded T1 fuzzy sets used to compute the bounds of the type-reduced interval change as input changes; and 2) Novelty, meaning that the upper and lower membership functions of the same IT2 fuzzy set may be used simultaneously in computing each bound of the type-reduced interval. T1 FLCs do not have these properties; thus, a T1 FLC cannot implement the complex control surface of an IT2 FLC given the same rulebase. We also present several methods to visualize and analyze the effects of these two fundamental differences, including the control surface, the P-map, the equivalent generalized T1 fuzzy sets, and the equivalent PI gains. Finally, we examine five alternative type reducers for IT2 FLCs and explain why they do not capture the fundamentals of IT2 FLCs.

Approaches for Reducing the Computational Cost of Interval Type-2 Fuzzy Logic Systems: Overview and Comparisons

Interval type-2 fuzzy logic systems (IT2 FLSs) have demonstrated better abilities to handle uncertainties than their type-1 (T1) counterparts in many applications; however, the high computational cost of the iterative Karnik-Mendel (KM) algorithms in type-reduction means that it is more expensive to deploy IT2 FLSs, which may hinder them from certain cost-sensitive real-world applications. This paper provides a comprehensive overview and comparison of three categories of methods to reduce their computational cost. The first category consists of five enhancements to the KM algorithms, which are the most popular type-reduction algorithms to date. The second category consists of 11 alternative type-reducers, which have closed-form representations and, hence, are more convenient for analysis. The third category consists of a simplified structure for IT2 FLSs, which can be combined with any algorithms in the first or second category for further computational cost reduction. Experiments demonstrate that almost all methods in these three categories are faster than the KM algorithms. This overview and comparison will help researchers and practitioners on IT2 FLSs choose the most suitable structure and type-reduction algorithms, from a computational cost perspective. A recommendation is given in the conclusion.

Computing With Words for Hierarchical Decision Making Applied to Evaluating a Weapon System

The perceptual computer (Per-C) is an architecture that makes subjective judgments by computing with words (CWWs). This paper applies the Per-C to hierarchical decision making, which means decision making based on comparing the performance of competing alternatives, where each alternative is first evaluated based on hierarchical criteria and subcriteria, and then, these alternatives are compared to arrive at either a single winner or a subset of winners. What can make this challenging is that the inputs to the subcriteria and criteria can be numbers, intervals, type-1 fuzzy sets, or even words modeled by interval type-2 fuzzy sets. Novel weighted averages are proposed in this paper as a CWW engine in the Per-C to aggregate these diverse inputs. A missile-evaluation problem is used to illustrate it. The main advantages of our approaches are that diverse inputs can be aggregated, and uncertainties associated with these inputs can be preserved and are propagated into the final evaluation.

Enhanced Interval Approach for Encoding Words Into Interval Type-2 Fuzzy Sets and Its Convergence Analysis

Construction of interval type-2 fuzzy set models is the first step in the perceptual computer, which is an implementation of computing with words. The interval approach (IA) has, so far, been the only systematic method to construct such models from data intervals that are collected from a survey. However, as pointed out in this paper, it has some limitations, and its performance can be further improved. This paper proposes an enhanced interval approach (EIA) and demonstrates its performance on data that are collected from a web survey. The data part of the EIA has more strict and reasonable tests than the IA, and the fuzzy set part of the EIA has an improved procedure to compute the lower membership function. We also perform a convergence analysis to answer two important questions: 1) Does the output interval type-2 fuzzy set from the EIA converge to a stable model as increasingly more data intervals are collected, and 2) if it converges, then how many data intervals are needed before the resulting interval type-2 fuzzy set is sufficiently similar to the model obtained from infinitely many data intervals? We show that the EIA converges in a mean-square sense, and generally, 30 data intervals seem to be a good compromise between cost and accuracy.

Linguistic Summarization Using IF–THEN Rules and Interval Type-2 Fuzzy Sets

Linguistic summarization (LS) is a data mining or knowledge discovery approach to extract patterns from databases. Many authors have used this technique to generate summaries like “Most senior workers have high salary,” which can be used to better understand and communicate about data; however, few of them have used it to generate IF-THEN rules like “IF X is large and Y is medium, THEN Z is small,” which not only facilitate understanding and communication of data but can also be used in decision-making. In this paper, an LS approach to generate IF-THEN rules for causal databases is proposed. Both type-1 and interval type-2 fuzzy sets are considered. Five quality measures-the degrees of truth, sufficient coverage, reliability, outlier, and simplicity-are defined. Among them, the degree of reliability is especially valuable for finding the most reliable and representative rules, and the degree of outlier can be used to identify outlier rules and data for close-up investigation. An improved parallel coordinates approach for visualizing the IF-THEN rules is also proposed. Experiments on two datasets demonstrate our LS and rule visualization approaches. Finally, the relationships between our LS approach and the Wang-Mendel (WM) method, perceptual reasoning, and granular computing are pointed out.