What is a Fuzzy Bi-implication?

In order to make this document self-contained, we first present all the necessary theory as a background. Then we study several definitions that extended the classic bi-implication in to the domain of well stablished fuzzy logics, namely, into the [0; 1] interval. Those approaches of the fuzzy bi-implication can be summarized as follows: two axiomatized definitions, which we proved that represent the same class of functions, four defining standard (two of them proposed by us), which varied by the number of different compound operators and what restrictions they had to satisfy. We proved that those defining standard represent only two classes of functions, having one as a proper subclass of the other, yet being both a subclass of the class represented by the axiomatized definitions. Since those three clases satisfy some contraints that we judge unnecessary, we proposed a new defining standard free of those restrictions and that represents a class of functions that intersects with the class represented by the axiomatized definitions. By this dissertation we are aiming to settle the groundwork for future research on this operator.

Functional encryption for threshold functions (or fuzzy IBE) from lattices

Cryptosystems based on the hardness of lattice problems have recently acquired much importance due to their average-case to worst-case equivalence, their conjectured resistance to quantum cryptanalysis, their ease of implementation and increasing practicality, and, lately, their promising potential as a platform for constructing advanced functionalities. In this work, we construct “Fuzzy” Identity Based Encryption from the hardness of the Learning With Errors (LWE) problem. We note that for our parameters, the underlying lattice problems (such as gapSVP or SIVP) are assumed to be hard to approximate within supexponential factors for adversaries running in subexponential time. We give CPA and CCA secure variants of our construction, for small and large universes of attributes. All our constructions are secure against selective-identity attacks in the standard model. Our construction is made possible by observing certain special properties that secret sharing schemes need to satisfy in order to be useful for Fuzzy IBE. We also discuss some obstacles towards realizing lattice-based attribute-based encryption (ABE).

Vector valued similarity measures for Atanassov's intuitionistic fuzzy sets

We present a new approach for defining similarity measures for Atanassov's intuitionistic fuzzy sets (AIFS), in which a similarity measure has two components indicating the similarity and hesitancy aspects. We justify that there are at least two facets of uncertainty of an AIFS, one of which is related to fuzziness while other is related to lack of knowledge or non-specificity. We propose a set of axioms and build families of similarity measures that avoid counterintuitive examples that are used to justify one similarity measure over another. We also investigate a relation to entropies of AIFS, and outline possible application of our method in decision making and image segmentation. © 2014 Elsevier Inc. All rights reserved.

Segmentation of gray scale image based on intuitionistic fuzzy sets constructed from several membership functions

Segmentation is the process of extraction of objects from an image. This paper proposes a new algorithm to construct intuitionistic fuzzy set (IFS) from multiple fuzzy sets as an application to image segmentation. Hesitation degree in IFS is formulated as the degree of ignorance (due to the lack of knowledge) to determine whether the chosen membership function is best for image segmentation. By minimizing entropy of IFS generated from various fuzzy sets, an image is thresholded. Experimental results are provided to show the effectiveness of the proposed method.

A systematic design of interval type-2 fuzzy logic system using extreme learning machine for electricity load demand forecasting

This paper presents a novel design of interval type-2 fuzzy logic systems (IT2FLS) by utilizing the theory of extreme learning machine (ELM) for electricity load demand forecasting. ELM has become a popular learning algorithm for single hidden layer feed-forward neural networks (SLFN). From the functional equivalence between the SLFN and fuzzy inference system, a hybrid of fuzzy-ELM has gained attention of the researchers. This paper extends the concept of fuzzy-ELM to an IT2FLS based on ELM (IT2FELM). In the proposed design the antecedent membership function parameters of the IT2FLS are generated randomly, whereas the consequent part parameters are determined analytically by the Moore-Penrose pseudo inverse. The ELM strategy ensures fast learning of the IT2FLS as well as optimality of the parameters. Effectiveness of the proposed design of IT2FLS is demonstrated with the application of forecasting nonlinear and chaotic data sets. Nonlinear data of electricity load from the Australian National Electricity Market for the Victoria region and from the Ontario Electricity Market are considered here. The proposed model is also applied to forecast Mackey-glass chaotic time series data. Comparative analysis of the proposed model is conducted with some traditional models such as neural networks (NN) and adaptive neuro fuzzy inference system (ANFIS). In order to verify the structure of the proposed design of IT2FLS an alternate design of IT2FLS based on Kalman filter (KF) is also utilized for the comparison purposes.

Training of interval type-2 fuzzy logic system using extreme learning machine for load forecasting

Extreme learning machine (ELM) is originally proposed for single- hidden layer feed-forward neural networks (SLFN). From the functional equivalence of fuzzy logic systems and SLFN, the fuzzy logic systems can be interpreted as a special case of SLFN under some mild conditions. Hence the fuzzy logic systems can be trained using SLFN's learning algorithms. Considering the same equivalence, ELM is utilized here to train interval type-2 fuzzy logic systems (IT2FLSs). Based on the working principle of the ELM, the parameters of the antecedent of IT2FLSs are randomly generated while the consequent part of IT2FLSs is optimized using Moore-Penrose generalized inverse of ELM. Application of the developed model to electricity load forecasting is another novelty of the research work. Experimental results shows better forecasting performance of the proposed model over the two frequently used forecasting models.