  Most of statistics and AI draw insights through modelling discord or variance
between sources of information (i.e., inter-source uncertainty). Increasingly,
however, research is focusing upon uncertainty arising at the level of
individual measurements (i.e., within- or intra-source), such as for a given
sensor output or human response. Here, adopting intervals rather than numbers
as the fundamental data-type provides an efficient, powerful, yet challenging
way forward -- offering systematic capture of uncertainty-at-source, increasing
informational capacity, and ultimately potential for insight. Following recent
progress in the capture of interval-valued data, including from human
participants, conducting machine learning directly upon intervals is a crucial
next step. This paper focuses on linear regression for interval-valued data as
a recent growth area, providing an essential foundation for broader use of
intervals in AI. We conduct an in-depth analysis of state-of-the-art methods,
elucidating their behaviour, advantages, and pitfalls when applied to datasets
with different properties. Specific emphasis is given to the challenge of
preserving mathematical coherence -- i.e., ensuring that models maintain
fundamental mathematical properties of intervals throughout -- and the paper
puts forward extensions to an existing approach to guarantee this. Carefully
designed experiments, using both synthetic and real-world data, are conducted
-- with findings presented alongside novel visualizations for interval-valued
regression outputs, designed to maximise model interpretability. Finally, the
paper makes recommendations concerning method suitability for data sets with
specific properties and highlights remaining challenges and important next
steps for developing AI with the capacity to handle uncertainty-at-source.
