Measurement, Information Channels, and Discretization: Exploring the Links

Abstract

The goal of this paper is to present a unified algebraic-analytic framework for (static and dynamic) deterministic measurement theory, which we find to be fully adequate in engineering and natural science applications. The starting point of this paradigm is the notion of a quantity algebra of a measured system and that of a measuring instrument, underlying the causal linkages in classical 'system + instrument' interactions. This approach is then further enriched by providing a superimposed data lattice of measurement outcomes, intended to handle the information flow from the measured system to its measurand's designated instrument. We argue that the language of Banach and von Neumann algebras is ideally suited for the treatment of quantities, encountered in theoretical and experimental science. These algebras and convex spaces of expectation functionals thereon together with information (co)channels between them provide a comprehensive information-theoretic framework for measurement theory. Concrete examples and applications to length and position measurements are also discussed and rigorously framed within the proposed quantity algebra and associated information channel paradigms. In modeling physical systems, investigators routinely rely on the assumption that state spaces and time domains form a continuum (locally homeomorphic to the real line or its Cartesian powers). But in sharp contrast, measurement and prediction outcomes pertaining to physical systems under consideration tend to be presented in terms of small discrete sets of rational numbers. We investigate this conceptual gap between theoretical and finitary data models from the perspectives of temporal, spatial and algebraic discretization schemes. The principal innovation in our approach to classical measurement theory is the representation of interactive instrument-based measurement processes in terms of channel-cochannel pairs constructed between dynamical quantity algebras of a target system and its measurand's measuring instrument.

Mathematical Models in Philosophy of Science

“…In sum, quantity algebras and associated convex spaces of statistical states allow model builders to construct mathematical models in an admittedly abstract albeit highly unified manner. More details on applications of quantity algebras to measurement theory may be found in Domotor and Batitsky (2009).…”

Algebraic Frameworks for Measurement in the Natural Sciences

“…At that point the scale mark that is closest to the flagpole's other end is read and added to the millimeters obtained from the total number of collinear ruler steppings along the flagpole's entire length. An earlier version of the length measurement structures presented here is discussed in [3]. 31 31 Today's technology is blessed with many types of non-contact length measurements, including ultrasound, laser and radar-based instruments.…”

“…As a by-product of classical systems and signal theories, STM is built around temporally (and spatially) varying smooth physical signals 3 and their transformations thereon by man-made physical systems, including but not limited to sensors, transducers, filters, amplifiers, analog-digital converters 4 , and signal reconstruction (inversion) modules. At the heart of STM rests the reductionist idea that measurement processes are merely special cases of other processes studied in mathematical systems and signal theories.…”