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From Intervals to â€“?: Towards a General Description of Validated Uncertainty

By (author) Vladik Kreinovich, By (author) GraÃ§aliz Pereira Dimuro, By (author) AntÃ´nio Carlos da Rocha Costa

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This book is about methodological aspects of uncertainty propagation in data processing. Uncertainty propagation is an important problem: while computer algorithms efficiently process data related to many aspects of their lives, most of these algorithms implicitly assume that the numbers they process are exact. In reality, these numbers come from measurements, and measurements are never 100% exact. Because of this, it makes no sense to translate 61 kg into pounds and get the resultâ€”as computers doâ€”with 13 digit accuracy. In many casesâ€”e.g., in celestial mechanicsâ€”the state of a system can be described by a few numbers: the values of the corresponding physical quantities. In such cases, for each of these quantities, we know (at least) the upper bound on the measurement error. This bound is either provided by the manufacturer of the measuring instrumentâ€”or is estimated by the user who calibrates this instrument. However, in many other cases, the description of the system is more complex than a few numbers: we need a function to describe a physical field (e.g., electromagnetic field); we need a vector in Hilbert space to describe a quantum state; we need a pseudo-Riemannian space to describe the physical space-time, etc. To describe and process uncertainty in all such cases, this book proposes a general methodologyâ€”a methodology that includes intervals as a particular case. The book is recommended to students and researchers interested in challenging aspects of uncertainty analysis and to practitioners who need to handle uncertainty in such unusual situations.

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