Theoretical choice of the optimal threshold for possibilistic linear model with noisy input

Based on possibility concepts, various possibilistic linear models (PLMs) have been proposed, and their pivotal role in fuzzy modeling and associated applications has been established. When adopting PLMs, one has to adopt an appropriate threshold (λ) value. However, choosing such a value is by no means trivial, and is still an open theoretical issue. In this paper, we propose a solution by first extending the PLM to its regularized version, i.e., a regularized PLM (RPLM), such that its generalization capability can be enhanced. The RPLM is then formulated as a maximum a posteriori (MAP) framework, which facilitates the determination of the theoretically optimal threshold value for the RPLM with noisy input. Our mathematical derivations reveal the approximately inversely proportional relationship between the threshold λ and the standard deviation of Gaussian noisy input. This is also confirmed by the simulation results. This finding is very helpful for the practical applications of both PLMs and RPLMs.