Interval or moments: which carry more information?

In many practical situations, we do not have enough observations to uniquely determine the corresponding probability distribution, we only have enough observations to estimate two parameters of this distribution. In such cases, the traditional statistical approach is to estimate the mean and the standard deviation. Alternatively, we can estimate the two bounds that form the range of the corresponding variable and thus, generate an interval. Which of these two approaches should we select? A natural idea is to select the most informative approach, i.e., an approach in which we need the smallest amount of additional information (in Shannon’s sense) to obtain the full information about the situation. In this paper, we follow this idea and come up with the following conclusion: in practical situations in which a 95 % confidence level is sufficient, interval bounds are more informative; however, in situations in which we need higher confidence, the moments approach is more informative.     

Roundoff-Free Number Fields for Interval Computations

Roundoff errors are inevitable if the exact result of some operation is not representable in a computer, and has, therefore, to be approximated. To avoid roundoff errors, it is hence necessary to choose a set of computer-representable numbers in such a way that the results of all basic operations will be still in this set. In this paper, we prove that if we include arithmetic operations and computing the interval range into this operations list, then the set F of numbers will be roundoff-free iff F is a real closed field; therefore, the smallest such set is the set of all real algebraic numbers (i.e., solutions of polynomial equations with rational coefficients).     

A Special Session on Granular Computing and Interval Computations at the 19th International Conference of the North American Fuzzy Information Processing Society (NAFIPS) Atlanta,Georgia, July 13–15, 2000

Reliable Computing -     

Unimodality,Independence Lead to NP-Hardness of Interval Probability Problems

In many real-life situations, we only have partial information about probabilities. This information is usually described by bounds on moments, on probabilities of certain events, etc. –i.e., by characteristics c(p) which are linear in terms of the unknown probabilities p _j. If we know interval bounds on some such characteristics , and we are interested in a characteristic c(p), then we can find the bounds on c(p) by solving a linear programming problem. In some situations, we also have additional conditions on the probability distribution –e.g., we may know that the two variables x ₁ and x ₂ are independent, or that the joint distribution of x ₁ and x ₂ is unimodal. We show that adding each of these conditions makes the corresponding interval probability problem NP-hard.     

Canonical-parameter estimation for instrument complex frequency response

Formulas are given for evaluating the time- constants T_i, T_k, T_p, and T_s together with the damping coefficients _k and _s and the factor K for rational- fraction complex frequency characteristics CFC in which the numerator and denominator are polynomials represented as products of two- term and three- term expressions. Formulas are also given of the variances of the error estimators for these parameters. All the formulas are given without deviation, which is fairly lengthy and hardly necessary here.Translated from Izmeritel'naya Tekhnika, No. 9, pp. 11–14, September, 1993.     

Uniqueness of reconstruction for Yager's t‐norm combination of probabilistic and possibilistic knowledge

Often, about the same real‐life system, we have both measurement‐related probabilistic information expressed by a probability measure P(S) and expert‐related possibilistic information expressed by a possibility measure M(S). To get the most adequate idea about the system, we must combine these two pieces of information. For this combination, R. Yager—borrowing an idea from fuzzy logic—proposed to use a t‐norm f_&(a,b) such as the product f_&(a,b)=a· b, i.e., to consider a set function f(S)=f_&(P(S),M(S)). A natural question is: can we uniquely reconstruct the two parts of knowledge from this function f(S)? In our previous paper, we showed that such a unique reconstruction is possible for the product t‐norm; in this paper, we extend this result to a general class of t‐norms. © 2011 Wiley Periodicals, Inc.     

Foreword

Reliable Computing -