The performance of fuzzy systems has been shown both theoretically and empirically to improve with increasing fuzziness, e.g., when fuzzy values are modeled using interval type 2 (IT2) or interval type 3 (IT3) fuzzy sets. This performance gain comes at a computational cost. We describe a new class of higher order fuzzy sets constructed from IT2 fuzzy sets that supports efficient type reduction. Our focus is on representations that use input from experts (SMEs) in the subject matter of interest. Differing interpretations of a concept by different subject matter experts (SMEs) leads to imprecision, which is captured in an IT2 fuzzy set defined on the relevant domain by approaches such as the Hao–Mendel approach. A second source of imprecision that has largely been ignored is SME confidence in their own judgments. This can also be captured as an IT2 fuzzy set, defined on the unit interval. We present a novel and intuitive construction to combine these representations into an IT3 fuzzy set. We show more generally how imprecision in hierarchical estimates of knowledge can be incorporated into a type-n fuzzy set representation.

Fuzzy cognitive maps (FCMs) play an important role in high-level reasoning but are limited in their ability to model complex systems with singularities. We are interested in systems that exhibit discontinuous behaviors as one or more of their internal node states approach a threshold. In a new approach to FCM dynamics, we define general classes of aggregation functions which “jump” to a boundary value when any input crosses a threshold, or when all inputs do. The threshold value is a context-dependent parameter which can be readily understood by subject matter experts. Aggregation functions are applied separately to positively and negatively causal antecedents to each node then combined to form the nodal state. This modeling is applied in Computing with Words (CWW) settings, in which link strengths and activation levels are elicited using vocabulary words represented by interval type-2 fuzzy membership functions. We illustrate the behaviors of these novel FCM systems in comparison with their nonsingularity versions.

We introduce a new structure for fuzzy cognitive maps (FCM) where the traditional fan-in structure involving an inner product followed by a squashing function to describe the causal influences of antecedent nodes to a particular consequent node is replaced with a weighted mean type operator. In this paper, we employ the weighted power mean (WPM). Through appropriate selection of the weights and exponents in the WPM operators, we can both account for the relative importance of different antecedent nodes in the dynamics of a particular node, as well as take a perspective ranging continuously from the most pessimistic (minimum) to the most optimistic (maximum) on the normalized aggregation of antecedents for each node. We consider this FCM structure to be more intuitive than the traditional one, as the nonlinearity involved in the WPM is more scrutable with regard to the aggregation of its inputs. We provide examples of this new FCM structure to illustrate its behavior, including its convergence, and compare it with a traditional FCM architecture on a scenario presented in previous works.

Centroids are practically important in type-1 and type-2 fuzzy logic systems as a method of defuzzification and type reduction. However, computational problems arise when membership functions (MF) have singleton spikes. The novel thresholding aggregation operators that were described in our companion paper “ New Classes of Threshold Aggregation Functions Based Upon the Tsallis q-Exponential with Applications to Perceptual Computing” produce such MFs with spikes. Such spikes may occur when modeling concepts defined on a real-valued domain, and they are also formed in unions of fuzzy sets in which some have MFs with discrete support and others have support defined on an interval. This paper presents a modified definition of the centroid of a fuzzy set that avoids the computational problems associated with the usual definition and reduces to this definition when MFs are continuous and normal (i.e., of unit height) on some interval. We also present an enhanced Karnik-Mendel-type algorithm to compute the modified centroid of interval type-2 fuzzy sets whose MFs have spikes.

We introduce two new classes of single-parameter aggregation functions based upon the Tsallis q-exponential (QE) function of nonextensive statistical mechanics. These aggregation functions (denoted QE aggregation) facilitate simple modeling of the common human reasoning trait of “threshold” inference, where either 1) at least one input must exceed a threshold in order to achieve a nonzero aggregation output; or 2) if any one of the inputs exceeds a different threshold, the aggregation output takes its maximum value. We illustrate the thresholding behavior of these functions on interval type-2 fuzzy inputs using an example known in the literature as the Investment Judgment Advisor. We believe that the new QE class of aggregation operators will prove useful in extending the range of options available for the design of perceptual computing systems.

Certain classes of fuzzy rule-based systems have been shown to be universal approximators, capable of approximating any continuous mapping on a compact subset of the domain. In previous cases, this property has been proven for fuzzy systems employing t-norms to determine the firing level of each rule. In this paper, we prove that use of the weighted power mean to determine rule firing levels also results in a universal approximator. While it is only a t-norm in certain special cases, the weighted power mean is a more general aggregation operator, capable of providing greater logical flexibility in a rule. This will be of particular interest in computing with words (CWW) applications, where such flexibility is needed the better to mimic human reasoning.

We have introduced the linguistic weighted power mean and its computation via a generalization of the Karnick Mendel algorithm. This new family of aggregation operators admits interval type-2 fuzzy membership functions for both its inputs and its weights to account for imprecise knowledge of these quantities. The simplest member of this family, the linguistic weighted average, is the primary instantiation of a "perceptual computing" engine in Mendel and Wu's recent book by this title. This paper presents a comparison of the linguistic weighted power mean and the linguistic weighted average on one of the examples they describe in detail-the "investment judgment advisor". Our results illustrate the flexibility and range of logical inference provided by this very versatile aggregation operator for computing with words applications.

This paper recursively extends the fundamental set theory operations of intersection and union to Type-3 and higher level fuzzy sets. A generalized form of convexity and normality is assumed.

We show how Type-2 fuzzy set modeling strengthens the theory of conceptual spaces first presented by Gärdenfors and developed by various authors. In fuzzy conceptual spaces, domains are defined as sets with a measure, properties as Type-2 fuzzy sets on domains, contexts as sets of properties, and concepts as Type-2 fuzzy sets on pairs of properties in a context. An observation is a family of fuzzy sets on the domains that are relevant to a context. Membership of an observation in a property is a Type-2 fuzzy set on the collection of observations. A geographical application illustrates how improved modeling of imprecision can affect decisions.

For many readers and potential authors, type-2 (T2) fuzzy sets might be more readily understood if expressed by the use of standard mathematical notation and terminology. This paper, therefore, translates constructs associated with T2 fuzzy sets to the language of functions on spaces. Such translations may encourage researchers in different disciplines to investigate T2 fuzzy sets, thereby potentially broadening their application and strengthening the underlying theory.

In this paper, we use Zadeh's extension principle to extend Kosko's definition of the fuzzy subsethood measure S(G, H) to type-2 fuzzy sets defined on any set X equipped with a measure. Subsethood is itself a fuzzy set that is a crisp interval when G and H are interval type-2 sets. We show how to compute this interval and then use the result to compute subsethood for general type-2 fuzzy sets. A definition of subsethood for arbitrary fuzzy sets of type-n > 2 is then developed. This subsethood is a type-(n - 1) fuzzy set, and we provide a procedure to compute subsethood of interval type-3 fuzzy sets.

This paper discusses issues raised in the design of a dynamically generated Web-based natural history encyclopedia for children. The novelty of our proposal is the dynamic creation of encyclopedia entries using domain-specific middleware to access biological and ecological readers (users) with tailored reports summarizing information drawn from distributed datasets. A prerequisite, however, is a domain-dependent standardized data labeling system. Our project will piggyback on the host of projects currently developing metalabeling frameworks to dynamically create stories which are peopled by creatures from biological collections, are set in locations of local or other interest to the site visitor, and use scenarios based on relevant ecosystems. The underlying scientific databases provide a much broader range of subject matter (species and locations) that would not otherwise be feasible and alleviate content maintenance problems that beset community Web sites or professional encyclopedias. To incorporate such information in the sorts of format expected in a children's encyclopedia involves developing sets of “story” templates, rules for selection of a template, and rules for design of the multimedia presentation of the story. It also requires comprehensive labeling systems for data in the scientific databases.

The information in data depends on the subjective value system that the receiver of the data uses to interpret them. This paper looks at the information in a theory of first order logic (a knowledge base) from the perspective of a decision maker for whom the validation of formulae (facts and rules) have varying importance. The decision maker's preferences and prior knowledge are both incorporated into the information measure. The value of information is determined by what it conveys about the formulae of importance to the decision maker. The information measure is applied as a heuristic in commonsense reasoning; in relevance assessment ; and as a preference function in belief revision.

Concerns the estimation of the motion parameters of a rigid body from noisy point matches made on perspective views. A minimization problem which permits relatively robust parameter estimation and helps overcome poor zoom estimation when the field-of-view is small is formulated. It is assumed that the motion has a small rotary component, interframe differences are small, and the errors in the system are due to independently distributed errors in the components of the displacement vectors. A fast procedure for minimization is exposited, in which the parameter set is partitioned and conditional generalized least-squares formulas identified. Recursive application of these provides the search space for the minimization problem. Comparative results are presented using simulated data and displacement vectors obtained from an intensity-based matching algorithm.>

Matching algorithms which use image structure such as edges and junctions cannot perform well on unstructured images. For such images, global gradient-based methods may be more appropriate. Some of the assumptions involved in applying discretized conservation equations such as the optical flow equation are discussed and an intensity-weighted method of estimating dense displacement fields is presented, which attempts to avoid some of the problems of the standard methods. The only derivative required is the intensity gradient, and the method includes procedures for automatic parameter evaluation. The algorithm is expected to perform better than conventional algorithms on images without strong texture, as is demonstrated on the test set of images.>