No More, No Less than Sum of Its Parts: Groups, Monoids, and the Algebra of Graphics, Statistics, and Interaction

Interactive data visualization has become a staple of modern data presentation. Yet, despite its growing popularity, we still lack a general framework for turning raw data into summary statistics that can be displayed by interactive graphics. This gap may stem from a subtle yet profound issue: while we would often like to treat graphics, statistics, and interaction in our plots as independent, they are in fact deeply connected. This article examines this interdependence in light of two fundamental concepts from category theory: groups and monoids. We argue that the knowledge of these algebraic structures can help us design sensible interactive graphics. Specifically, if we want our graphics to support interactive features which split our data into parts and then combine these parts back together (such as linked selection), then the statistics underlying our plots need to possess certain properties. By grounding our thinking in these algebraic concepts, we may be able to build more flexible and expressive interactive data visualization systems. Supplementary materials for this article are available online.