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.

This paper presents a case study of what we believe to be the first commercial product to use interval type-2 fuzzy logic in a computing with words application to the digital economy. The Discovery Investing Scoreboard is a free online service for evaluating the investment potential of companies in a manner distinct from more traditional styles of investment analysis. It is particularly applicable to companies having small market capitalization, where the latter methods often are of little use. Prudent selection of such companies may offer exceptional investment returns through their discovery of world class assets, IP or technology, and their ability to execute on a business plan leading to production or acquisition at a premium valuation.

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.

A majority of artificial spiking neural network models are used for modeling and/or understanding biological processes, while only in a few instances have they been used for solving engineering tasks. When spiking neural networks are used for an engineering task often the inputs are fed directly into the network and it is up to the network to evolve the connections from a randomly connected pool of neurons. This approach may generate a network capable of producing the desired output (mapping), however, the resulting network is typically undecipherable. In modern digital systems a microprocessor is used to perform a task. Specifically, an analog to digital converter is used to allow the microprocessor to accurately interpret analog signals coming from sensors monitoring the task. In this paper, a spiking neural network is used to measure the intensity of an input analog signal using a coincidence detection approach, where a pool of intensity detecting neurons is responsible for identifying specific intensities. At any instant in time the active neuron in the pool represents current input intensity of the signal. We first show the circuit for a single intensity case. Next, the entire intensity detection circuit is examined over a large range of intensities. Finally, the entire network is tested when the input signal is mixed with Gaussian noise. The resulting analog coincidence intensity detection network has a rapid and accurate response while being noise-resistant. The modularity of the design allows the network to be easily incorporated as a preprocessing system in other and/or larger simulations.

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 introduce a novel system for recognition of partially occluded and rotated images. The system is based on a hierarchical network of integrate-and-fire spiking neurons with random synaptic connections and a novel organization process. The network generates integrated output sequences that are used for image classification. The proposed network is shown to provide satisfactory predictive performance given that the number of the recognition neurons and synaptic connections are adjusted to the size of the input image. Comparison of synaptic plasticity activity rule (SAPR) and spike timing dependant plasticity rules, which are used to learn connections between the spiking neurons, indicates that the former gives better results and thus the SAPR rule is used. Test results show that the proposed network performs better than a recognition system based on support vector machines.

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.

In this work, we consider a relatively new representation used in cognitive theory to describe how people understand concepts. This representation is called Conceptual Spaces, and is a geometrical way to represent human thought. Our work relates Conceptual Spaces to Data Fusion, first at Level 1, and later to be extended to Level 2 (as defined by JDL [1]). In this paper, we focus on modeling these Conceptual Spaces as a mathematical program. We then discuss uses and methodologies for Conceptual Spaces in the light of Data Fusion.

We describe a novel hypercube graph representation for labeled graphs with arbitrary edge weights in the interval [0, 1]. This representation admits graphical models for weighted adjacency matrices, which are useful in a number of real world applications wherein the strength of connections between graph nodes is important. It enables us to bring to bear a full arsenal of fuzzy set theoretic measures such as fuzzy subsethood, entropy, completeness, and mutual subsethood to the description of graphs. Our hypercube representation also provides a direct similarity metric between pairs of graphs, which is particularly useful for external comparisons among sets of graphs. The unitary complement of this similarity metric in turn provides a distance metric between two graphs, thus enabling us to perform vector processing operations on graphs, e.g., clustering, change detection, hypothesis testing as to the independence of two graphs, feature extraction for neural network and/or statistical classifiers, and antecedent specification for fuzzy mappings. We derive the probability mass function of this metric for two independent random graphs. The hypercube graph representation finds applications in problems where we are dealing with labeled graphs, e.g., computer networks, social networks, graphical information retrieval, and data fusion problems involving virtual networks of events. Of special interest are labeled graphs with fixed vertices whose edges and their corresponding weights vary over time, as well as graphs that evolve in time by the addition of new vertices and edges.

This paper presents a novel approach to data fusion knowledge representation using conceptual spaces. Conceptual spaces represent knowledge geometrically in multiple domains, each domain consisting of multiple dimensions with an associated distance metric and corresponding similarity measure. Complex concepts such as those required for level 2/3 fusion are described by multiple property regions within these domains, along with the property correlations and salience weights. These concepts are mapped into points in the unit hypercube that capture all of their essential elements. Observations are also mapped into points in the same unit hypercube. The relative similarity of observations to concepts can then be calculated using the fuzzy subsethood measure

This work analyzes the detection performance of autocoherence (i.e., the magnitude-squared complex autocorrelation coefficient estimate) as a nonparametric detector of narrowband signals in broadband noise. The distribution theory of this statistic is derived along with approximations that permit easy analysis of the detection performance. It is shown in particular case studies that only 1 to 2.6 dB additional signal-to-noise ratio is required by autocoherence to achieve performance equivalent to the optimal parametric energy detector. Unlike the mean-level detector, autocoherence does not require a separate pool of noise reference samples and is therefore easier to implement in cluttered environments.

In this paper, we derive the detection performance of an optimally beamformed 3-D random array system, assuming known Gaussian signal, noise, and interference statistics, and compare it with the performance of a conventional beamformed system. The receiver operating characteristics (ROC) are straightforward, nonlinear functions of the particular array realization and the second-order signal, noise, and interference statistics. Thus, a determination of the average ROCs (over the array ensemble) appears intractable. However, by making a reasonable approximation, we are able to derive an analytical expression for the average ROC for both conventional and optimal processing.

A probabilistic approach to the source encoding problem requires the specification of a fidelity criterion which measures the degradation of the source output produced by the encoding transformation. Historically, the fidelity criteria generated by difference-type distortion measures have received by far the widest acceptance, due in part to their analytical tractability. Two new fidelity criteria for discrete.time sources are introduced. These are designated as the "mean incoherence" (MIC) and the "mean lng-coherence" (MLC). Both criteria are functionals of the magnitude-squared coherence between the source output and its reproduction, and thus do not admit a recognizable time-domain distortion measure representation. Interesting features of these fidelity criteria are indicated, and comparisons are made with the classical mean-squared error (MSE) fidelity criterion. The rate-distortion functions for a stationary Gaussian source subject to constraints on either the MIC or MLC are derived, and the corresponding optimum encoder behavior is described. Finally, the applications where these new fidelity criteria may he advantageous over the MSE criterion are discussed.