Checking If There Exists a Monotonic Function That Is Consistent with the Measurements: An Efficient Algorithm

In many problems in science and engineering ranging from astrophysics to geosciences to financial analysis, we know that a physical quantity y depends on the physical quantity x, i.e., y = f(x) for some function f(x), and we want to check whether this dependence is monotonic. Specifically, finitely many measurements of xi and yi = f(xi) have been made, and we want to check whether the results of these measurements are consistent with the monotonicity of f(x). An efficient parallelizable algorithm is known for solving this problem when the values xi are known precisely, while the values yi are known with interval uncertainty. In this paper, we extend this algorithm to a more general (and more realistic) situation when both xi and yi are known with interval uncertainty. 1 Formulation of the Problem Related problems – of checking monotonicity and locating local extrema – are important. In many problems in science and engineering, we know that a physical quantity y depends on the physical quantity x, i.e., y = f(x) for some function f(x), and we want to check whether this dependence is monotonic. In spectral analysis, chemical species are identified by locating local maxima of the spectra; see, e.g., [30, 31, 32]. Thus, to identify the chemical species, we must identify intervals between local extrema, i.e., intervals of monotonicity. In radioastronomy, sources of celestial radio emission and their subcomponents are identified by locating local maxima of the measured brightness of the