# From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval, p-Box, and Fuzzy Uncertainty

Martine Ceberio, Vladik Kreinovich, Andrzej Pownuk, Barnabás Bede
DOI: 10.4018/978-1-60566-324-1.ch006
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## Abstract

One of the important components of granular computing is interval computations. In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. In our previous papers, we proposed an extension of this technique to set computations, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time.
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## 1. Formulation Of The Problem

### 1.1. Need for Data Processing

In many real-life situations, we are interested in the value of a physical quantity that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure directly, a natural idea is to measure indirectly. Specifically, we find some easier-to-measure quantities which are related to by a known relation ; this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to a partial differential equation). Then, to estimate , we first measure or estimate the values of the quantities , and then we use the results of these measurements (estimations) to compute an estimate for as

Figure 1.

Computing an estimate for based on the results of direct measurements is called data processing; data processing is the main reason why computers were invented in the first place, and data processing is still one of the main uses of computers as number crunching devices.

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