Infinite-Level Interpolation for Inference with Sparse Fuzzy Rules: Fundamental Analysis Toward Practical Use

Infinite-level interpolation is proposed for inference with sparse fuzzy rules. It is based on multi-level interpolation where fuzzy rule interpolation is performed at a number of multi-level points. Multi-level points are defined by the bounds of alpha-cuts of each given fact. As a feasibility study, fundamental analysis is focused on in order to theoretically derive convergent consequences in increasing the number of the levels of alpha for the alpha-cuts. By increasing the number of the levels, nonlinear mapping by the inference is made more precise in reflecting the distribution forms of sparse fuzzy rules to consequences. The convergent consequences make it unnecessary to examine the number of the levels for improving the mapping accuracy. It is confirmed that each of the consequences deduced with simulations converges to one theoretically derived with an infinite number of the levels of alpha. It is thereby proved that the fundamental analysis has its validity. Toward the practical use of the convergent consequences, further discussions may be possible to extend the fundamental analysis, considering practical conditions.