Uncertainty propagation and quantification for digital twins

Probability has an inadequate model of ignorance, which creates practical problems in risk analysis. The mathematics used in risk assessments must be generalised to handle the epistemic uncertainty that commonly arises in imperfect measurements, unstudied dependencies, and incomplete scientific understanding generally.

Current methods force scientists and engineers to make untenable assumptions in order to get any quantitative answer at all, rather than developing technologies for making calculations that require only the assumptions they feel comfortable making. These methods can result in tests that get asymptotically more reliable with increasing sample size, without referencing any gold standard results. They can also result in probability dilation were, counterintuitively, more information can necessarily lead to higher uncertainty.

Confidence structures (c-boxes) generalise confidence distributions and provide an interpretation by which confidence intervals at any confidence level can be specified for a parameter of interest (Balch, 2012). C-boxes can be used in calculations using the methods of probability bounds analysis and yield results that also admit this confidence interpretation. The project will analyse ways in which c-boxes can be used to improve digital twin decision making methodology. Similar problems, including satellite conjunction analysis and medical diagnosis will be examined.

The project will also look at ways in which uncertainty quantification can be included in computer simulations, a fundamental problem in the development of digital twins. Uncertainty analysis is half the story in any computation. It is better to compute with information that we do know rather than making pretend calculations with infinitely precise numbers.

Consider the situation where an analyst has fifty thousand lines of 'uncertainty-naïve' source code but is unwilling, either due to mathematical complexity or the time required, to rewrite their code so that it contains full account of the uncertainty involved. A compiler will be developed that handles the specifications of uncertainties, either automatically or with end-user input, and inserts calls to an object-oriented library of 'intrusive' uncertainty quantification (UQ) algorithms, including probability distributions, confidence boxes, probability boxes and intervals. ANTLR, a parser/lexer generator, will be used along with Python to translate original code into UQ code in the same language. In theory, the approach could work with any computer language. Initially the compiler will be developed for Python but possibly extended to be able to handle MATLAB, C and FORTRAN languages. Ancillary strategies supporting automated simplification of mathematical expressions involving repeated variables will be developed to help reduce uncertainty.

Balch, M. S. (2012) 'Mathematical foundations for a theory of confidence structures', International Journal of Approximate Reasoning, 53(7), pp. 1003-1019. doi: 10.1016/j.ijar.2012.05.006.