Bivariate dependency tracking in interval arithmetic

We propose a correlated bivariate interval arithmetic which allows for an initial dependence to be propagated, as well as the tracking of complicated non-linear dependencies arising from a computer program's execution. For this task, we extend several familiar concepts from probability theory to intervals, including bivariate copulas, conditioning, inference, and vine copulas. The interval copulas, which we call interval relations, may take any shape, and are represented by Boolean matrices defining where two intervals jointly exist or not. We use set conditioning to define an efficient correlated interval arithmetic, which may be used to find the input–output relations of operations. A key component of the presented arithmetic are interval relation networks, interval analogues to vine copulas, which store the interval relations throughout a program's execution, and use set inference to determine any unknown relations. The presented network inference can give a robust outer approximation to the exact multivariate interval dependency, which is found by projecting each pairwise bivariate relation into higher dimensions. Although some higher dimensional information is lost in this process, the bivariate projections are often sufficient to stop interval bounds becoming excessively wide. This extension allows for intervals to be rigorously and tightly propagated in deterministic engineering codes in an automatic fashion, and we apply the arithmetic on several engineering dynamics problems, including a non-linear ordinary differential equation.