Development of information granules of higher type and their applications to granular models of time series

The study is devoted to the design of information granules of higher type (especially type-2) with the use of the principle of justifiable granularity. The development of granules is realized in two key phases: first, information granules of type-1 are formed and then they are extended to type-2 constructs. Following the principle, information granules are designed by establishing a sound balance between their experimental justification (legitimacy) and specificity (associated with their underlying semantics). The definitions of coverage and specificity of type-2 information granules are revised to capture the essence of these constructs. Detailed formulas are derived for several main categories of membership functions (namely, triangular, parabolic, and square root) as well as intervals. The study delivers detailed results for interval-valued fuzzy sets described by membership functions coming from the main classes listed above. Illustrative studies include synthetic data exhibiting some probabilistic properties. The direct application of information granules of type-1 and type-2 is demonstrated in the description and prediction of time series realized in the setting of information granules (with the resulting models referred to as granular models of time series).