How Probabilistic Methods for Data Fitting Deal with Interval Uncertainty: A More Realistic Analysis

In our previous paper, we showed that a simplified probabilistic approach to interval uncertainty leads to the known notion of a united solution set. In this paper, we show that a more realistic probabilistic analysis of data fitting under interval uncertainty leads to another known notion -- the notion of a tolerable solution set. Thus, the notion of a tolerance solution set also has a clear probabilistic interpretation. Good news is that, in contrast to the united solution set whose computation is, in general, NP-hard, the tolerable solution set can be computed by a feasible algorithm

Interval Methods for Data Fitting under Uncertainty: A Probabilistic Treatment

How to estimate parameters from observations subject to errors and uncertainty? Very often, the measurement errors are random quantities that can be adequately described by the probability theory. When we know that the measurement errors are normally distributed with zero mean, then the (asymptotically optimal) Maximum Likelihood Method leads to the popular least squares estimates. In many situations, however, we do not know the shape of the error distribution, we only know that the measurement errors are located on a certain interval. Then the maximum entropy approach leads to a uniform distribution on this interval, and the Maximum Likelihood Method results in the so-called minimax estimates. We analyse specificity and drawbacks of the minimax estimation under essential interval uncertainty in data and discuss possible ways to solve the difficulties. Finally, we show that, for the linear functional dependency, the minimax estimates motivated by the Maximum Likelihood Method coincide with those produced by the Maximum Consistency Method that originate from interval analysis

Why Color Optical Computing

In this paper, we show that requirements that computations be fast and noise-resistant naturally lead to what we call color-based optical computing

Interval-Valued and Set-Valued Extensions of Discrete Fuzzy Logics, Belnap Logic, and Color Optical Computing

It has been recently shown that in some applications, e.g., in ship navigation near a harbor, it is convenient to use combinations of basic colors -- red, green, and blue -- to represent different fuzzy degrees. In this paper, we provide a natural explanation for the efficiency of this empirical fact: namely, we show that it is reasonable to consider discrete fuzzy logics, it is reasonable to consider their interval-valued and set-valued extensions, and that a set-valued extension of the 3-values logic is naturally equivalent to the use of color combinations