In this study, we introduce a concept of hierarchical granular clustering and establish its algorithmic framework. We show that the proposed model naturally gives rise to information granules that are both of higher order and higher type, offering a compelling justification behind their emergence. In a concise way, we can capture the overall architecture of information granules as a hierarchy exhibiting conceptual layers of increasing abstraction: numeric data → information granules → information granules of type-2, order-2 → ... information granules of higher type/order. The elevated type of information granules is reflective of the visible hierarchical facet of processing and the inherent diversity of the individual locally revealed structures in data. While the concept and the methodology deliver some general settings, the detailed algorithmic aspects are discussed in detail when using fuzzy clustering realized by means of fuzzy c-means. Furthermore, for illustrative purposes, we mainly focus on interval-valued fuzzy sets and granular interval fuzzy sets arising at the higher level of the hierarchy. Higher type fuzzy sets are formed with the help of the principle of justifiable granularity. The conceptually sound hierarchy is established in a general way, which makes it equally applicable to various formalisms of representation of information granules. Experiments are reported for synthetic and publicly available datasets.

The plethora of ways of data representation and their applications to system modeling is inherently associated with dimensionality reduction. In a nutshell, the result of dimensionality reduction should support efficient ways of constructing ensuing models (classifiers, predictors) as well as an interpretation of the data themselves. Furthermore, there should be a suitable measure quantifying the quality of data positioned in the reduced space. We advocate that what makes the reduced data interpretable, goes hand in hand with revealing a logic fabric of the data, suppressing redundancy, and finally arriving at a logic description of data. The anticipation is that the reduced data can be described in the form of logic expressions formed over the original highly dimensional data. Evidently, having these above stated points in mind, the aim of this study is two-fold: (i) to develop a logic-oriented data representation, and (ii) to quantify the quality of results of dimensionality reduction by incorporating a facet of information granularity. In other words, we argue that the result of dimensionality reduction gives rise to information granules whose level of granularity associates with the quality of processing completed by the autoencoder. In light of the recent surge of architectures of deep learning, the study is focused on the construction and analysis of logic-oriented autoencoders. We propose a two-level architecture composed of the logic-oriented processing units (and processing carried out at the first layer of the autoencoder) followed by the or processing completed at the second layer. As data representation provided by the autoencoder is not ideal, we augment the original architecture by granular parameters which give rise to granular logic-oriented autoencoders. A suite of experiments is also reported.

In this paper, a fixed-time terminal synergetic observer for synchronization of fractional-order nonlinear chaotic systems is proposed. First, fixed-time terminal attractors for fractional-order nonlinear systems are introduced on the basis of fixed-time stability of integer-order nonlinear differential equations and on defining particular fractional-order macro-variables. Second, a new synergetic observer dedicated to the synchronization of fractional-order chaotic systems is developed. The proposed observer converges in a predefined fixed-time uniformly bounded with respect to initial conditions. Thanks to the step-by-step procedure, only one communication channel is used to achieve the synchronization. Third, a fixed-time synergetic extended observer with unknown input is constructed to simultaneously estimate the state variables and to recover the unknown input. Finally, computer simulations are performed to illustrate the efficiency of the proposed synchronization method and its application in a secure communication scheme.