  The superposition of data sets with internal parametric self-similarity is a
longstanding and widespread technique for the analysis of many types of
experimental data across the physical sciences. Typically, this superposition
is performed manually, or recently by one of a few automated algorithms.
However, these methods are often heuristic in nature, are prone to user bias
via manual data shifting or parameterization, and lack a native framework for
handling uncertainty in both the data and the resulting model of the superposed
data. In this work, we develop a data-driven, non-parametric method for
superposing experimental data with arbitrary coordinate transformations, which
employs Gaussian process regression to learn statistical models that describe
the data, and then uses maximum a posteriori estimation to optimally superpose
the data sets. This statistical framework is robust to experimental noise, and
automatically produces uncertainty estimates for the learned coordinate
transformations. Moreover, it is distinguished from black-box machine learning
in its interpretability -- specifically, it produces a model that may itself be
interrogated to gain insight into the system under study. We demonstrate these
salient features of our method through its application to four representative
data sets characterizing the mechanics of soft materials. In every case, our
method replicates results obtained using other approaches, but with reduced
bias and the addition of uncertainty estimates. This method enables a
standardized, statistical treatment of self-similar data across many fields,
producing interpretable data-driven models that may inform applications such as
materials classification, design, and discovery.
