Interaction Decomposition of prediction function

This paper discusses the foundation of methods for accurately grasping the interaction effects. Among the existing methods that capture the interaction effects as terms, PD and ALE are known as global modelagnostic methods in the IML field. ALE, among the two, can theoretically provide a functional decomposition of the prediction function, and this study focuses on functional decomposition. Specifically, we mathematically formalize what we consider to be the requirements that must always be met by a decomposition (interaction decomposition, hereafter, ID) that decomposes the prediction function into main and interaction effect terms. We also present a theorem about how to produce a decomposition that meets these requirements. Furthermore, we confirm that while ALE is ID, PD is not, and we present examples of decomposition that meet the requirements of ID using methods other than existing ones (i.e., new methods).