Abstract - In this paper, we address the issue of type reduction of multi-dimensional interval type-2 (IT2) fuzzy sets (FSs). We utilize the Karnik- Mendel (KM) algorithm to estimate the centroid boundary of a multi-dimensional footprint of uncertainty (FOU). We deal with two-dimensional (2-D) fuzzy sets as we can visualize the FOU using 3-D plots, thus making the illustration of the methods simple. However, the basic idea can be extended to multiple dimensions. We give a formal definition of the centroid boundary of a 2-D IT2 fuzzy membership function (FMF) and propose two methods for its estimation. The first method computes embedded type-1 (T1) FSs whose centroids constitute the centroid boundary. We obtain the embedded sets by producing slices of the domain using different sets of parallel planes and then apply the KM algorithm over each slice, to obtain "embedded-curves." For the second method, we approximate our first method by restricting embedded-curves to be "embedded-lines" thus enhancing computational speed. These type reduction techniques can be applied to applications involving multi- dimensional centroid estimation such as, clustering, support vector estimation for dimensionality reduction, fuzzy logic controllers, to mention a few.
Abstract - Processing type-2 fuzzy sets is more demanding than processing interval type-2 fuzzy sets or type-1 fuzzy sets. In this paper we propose a method for calculating union and intersection operations using min t-norm and max t-conorm on general type-2 fuzzy sets. The proposed method is based on Zadeh's separation theorem and though is straightforward. The important feature of the algorithm is its simplicity and applicability on the type-2 fuzzy sets with convex membership grades.
Abstract - Multi-criteria decision making (MCDM) problems are a well known category of decision making problem that has received much attention in the literature, with a key approach being the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). While TOPSIS has been developed towards the use of Type-2 Fuzzy Sets (T2FS), to date, the additional information provided by T2FSs in TOPSIS has been largely ignored since the final output, the Closeness Coefficient (CC), has remained a crisp value. In this paper, we develop an alternative approach to T2 fuzzy TOPSIS, where the final CC values adopt an interval-valued form. We show in a series of systematically designed experiments, how increasing uncertainty in the T2 membership functions affects the interval-valued CC outputs. Specifically, we highlight the complex behaviour in terms of the relationship of the uncertainty levels and the outputs, including non-symmetric and non-linear growth in the CC intervals in response to linearly growing levels of uncertainty. As the first TOPSIS approach which provides an interval-valued output to capture output uncertainty, the proposed method is designed to reduce the loss of information and to maximize the benefit of using T2FSs. The initial results indicate substantial potential in the further development and exploration of the proposed and similar approaches and the paper highlights promising next steps.
Abstract - Restricted Boltzmann Machine (RBM) is a generative, stochastic neural network with two separate layers of hidden and visible units. Training data samples in RBM are usually corrupted by noise. RBM is not robust enough to handle such noises, which leads to uncertainty. In the literature, Fuzzy RBM (FRBM) has already been proposed for enhancing deep learning. In FRBM, the parameters of RBM are modelled as type-1 fuzzy numbers. However, there can be multiple sources of uncertainties such as noises in the data measurements, variations in the environment where they are deployed, etc. Such uncertainties cannot be modelled by type-1 fuzzy sets (T1 FS), since their membership values are themselves crisp in nature. On the other side, IT2 FS can model higher order uncertainties with their fuzzy membership grades. So, we propose to use interval type-2 fuzzy sets (IT2 FS) to model uncertain parameters of RBMs in the learning stage. Since deep neural network (DNN) is pre-trained using stacked RBMs, modeling noises using IT2 FS would demonstrate high performance and low root mean square error (RMSE) while learning. Thus, we propose a new algorithm viz. Interval type-2 fuzzy set based approach for enhanced deep learning (IT2 FS-EDL) in which RBM parameters are modeled as type-2 fuzzy sets in the learning process. Numerical examples and experimentations have been demonstrated to present the suitability of our proposed approach.
Abstract - This paper proposes a modified particle swarm optimization (PSO) algorithm that can be used to solve a variety of fuzzy nonlinear equations, i.e. fuzzy polynomials and exponential equations. Fuzzy nonlinear equations are reduced to a number of interval nonlinear equations using alpha cuts. These equations are then sequentially solved using the proposed methodology. Finally, the membership functions of the fuzzy solutions are constructed using the interval results at each alpha cut. Unlike existing methods, the proposed algorithm does not impose any restriction on the fuzzy variables in the problem. It is designed to work for equations containing both positive and negative fuzzy sets and even for the cases when the support of the fuzzy sets extends across 0, which is a particularly problematic case.
Abstract - Fuzzy Similarity Measure (FSM) is one of the most used techniques for classification, pattern recognition or knowledge reduction. While many type-1 FSMs exist, few ones exist for type-2 fuzzy sets. In this paper, we introduce three similarity measures between Interval Type-2 Fuzzy Sets (IT-2 FSs) as an extension of some distance measures between Intuitionistic Fuzzy Sets (IFSs). Many definitions and properties are exposed in order to prove that the formulas presented are indeed similarity measures. Experimental results are presented, comparison with other existing type-2 FSMs is done and interpretation is given in order to satisfy FSM properties.