Engineering Commons^™

Towards Interval Techniques For Model Validation, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

Most physical models are approximate. It is therefore important to find out how accurate are the predictions of a given model. This can be done by validating the model, i.e., by comparing its predictions with the experimental data. In some practical situations, it is difficult to directly compare the predictions with the experimental data, since models usually contain (physically meaningful) parameters, and the exact values of these parameters are often not known. One way to overcome this difficulty is to get a statistical distribution of the corresponding parameters. Once we substitute these distributions into a model, we get statistical predictions …

How To Tell When A Product Of Two Partially Ordered Spaces Has A Certain Property?, Francisco Zapata, Olga Kosheleva, Karen Villaverde

Departmental Technical Reports (CS)

In this paper, we describe how checking whether a givenproperty F is true for a product A₁ X A₂ of partiallyordered spaces can be reduced to checking several relatedproperties of the original spaces A_i.

This result can be useful in the analysis of propertiesof intervals [a,b] = {x: a <= x <= b}over general partially ordered spaces -- such as the spaceof all vectors with component-wise order or the set of allfunctions with component-wise ordering f <= g <-->for all x (f(x) <= g(x)). When we consider sets of pairs ofsuch objects A₁ X A₂, it is natural to define the orderon this set in terms of orders in A₁ and A₂ -- this is, e.g.,how ordering and intervals are defined on the set R² of all2-D vectors.

This result …

Theoretical Explanation Of Bernstein Polynomials' Efficiency: They Are Optimal Combination Of Optimal Endpoint-Related Functions, Jaime Nava, Vladik Kreinovich

Departmental Technical Reports (CS)

In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval [x_-, x₊] as linear combinations of Bernstein polynomials (x- x _- )^k * (x₊ - x)^n-k. In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the requirement that they are optimal combinations of optimal polynomials corresponding to the interval's endpoints.

Towards Fast And Accurate Algorithms For Processing Fuzzy Data: Interval Computations Revisited, Gang Xiang, Vladik Kreinovich

Departmental Technical Reports (CS)

Is It Possible To Have A Feasible Enclosure-Computing Method Which Is Independent Of The Equivalent Form?, Marcin Michalak, Vladik Kreinovich

Departmental Technical Reports (CS)

The problem of computing the range [y] of a given function f(x1, ..., xn) over given intervals [xi] -- often called the main problem of interval computations -- is, in general, NP-hard. This means that unless P = NP, it is not possible to have a feasible (= polynomial time) algorithm that always computes the desired range. Instead, interval computations algorithms compute an enclosure [Y] for the desired range. For all known feasible enclosure-computing methods -- starting with straightforward interval computations -- there exist two expressions f(x1, ..., xn) and g(x1, ..., xn) for computing the same function that lead …

Towards Faster Estimation Of Statistics And Odes Under Interval, P-Box, And Fuzzy Uncertainty: From Interval Computations To Rough Set-Related Computations, Vladik Kreinovich

Departmental Technical Reports (CS)

Interval computations estimate the uncertainty of the result of data processing in situations in which we only know the upper bounds $\Delta$ on the measurement errors. In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. As a result, we often have bounds with excess width. One way to remedy this problem is to extend interval technique to rough-set computations, where on each stage, in addition to intervals of possible values of the quantities, we also keep rough sets representing possible values of pairs (triples, etc.).