  Ex-post harmonisation is one of many data preprocessing processes used to
combine the increasingly vast and diverse sources of data available for
research and analysis. Documenting provenance and ensuring the quality of
multi-source datasets is vital for ensuring trustworthy scientific research and
encouraging reuse of existing harmonisation efforts. However, capturing and
communicating statistically relevant properties of harmonised datasets is
difficult without a universal standard for describing harmonisation operations.
Our paper combines mathematical and computer science perspectives to address
this need. The Crossmaps Framework defines a new approach for transforming
existing variables collected under a specific measurement or classification
standard to an imputed counterfactual variable indexed by some target standard.
It uses computational graphs to separate intended transformation logic from
actual data transformations, and avoid the risk of syntactically valid data
manipulation scripts resulting in statistically questionable data. In this
paper, we introduce the Crossmaps Framework through the example of ex-post
harmonisation of aggregated statistics in the social sciences. We define a new
provenance task abstraction, the crossmap transform, and formalise two
associated objects, the shared mass array and the crossmap. We further define
graph, matrix and list encodings of crossmaps and discuss resulting
implications for understanding statistical properties of ex-post harmonisation
and designing error minimising workflows.
