On Invariant Within Equivalence Coordinate System (IWECS) Transformations

In exploratory data analysis and data mining in the very common setting of a data set $$\mathbb{X}$$ of vectors from $$\mathbb{R}^{d}$$ , the search for important features and artifacts of a geometrical nature is a leading focus. Here one must insist that such discoveries be invariant under selected changes of coordinates, at least within some specified equivalence relation on geometric structures. Otherwise, interesting findings could be merely artifacts of the coordinate system. To avoid such pitfalls, it is desirable to transform the data $$\mathbb{X}$$ to an associated data cloud $$\mathbb{X}^{{\ast}}$$ whose geometric structure may be viewed as intrinsic to the given data $$\mathbb{X}$$ but also invariant in the desired sense. General treatments of such “invariant coordinate system” transformations have been developed from various perspectives. As a timely step, here we formulate a more structured and unifying framework for the relevant concepts. With this in hand, we develop results that clarify the roles of the so-called transformation-retransformation transformations. We illustrate by treating invariance properties of some outlyingness functions. Finally, we examine productive connections with maximal invariants.