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# The Mathematical Foundation of Analytic Visualizations

The Grammar of Graphics (GoG) is the title of a book that lays out the mathematical foundation of analytic visualizations: statistical, cartographic, and other quantitative graphics designed to represent observed or abstract data. Analytic visualizations are distinguished from other graphics by their mathematical formalism. Informal diagrams, by contrast, are designed to communicate ideological, artistic, religious, or other metaphorical information.

The GoG foundation is based on the conventional definition of the graph of a function: a collection of ordered pairs (x, f(x)). A graphic is a visual representation of the graph of a function. In analytic visualizations, this function operates on observed or abstract data.

GoG decomposes the global visualization function into seven orthogonal classes that comprise a totally ordered function chain. Each class has a collection of member functions that are composable with functions in adjacent classes of the function chain. The ﬁrst class (Variables) maps data to an object called a varset (a set of variables). The next two classes (Algebra, Scales) are transformations on varsets. The next class (Statistics) takes a varset and creates a statistical graph (a statistical summary). The next class (Geometry) maps a statistical graph to a geometric graph. The next (Coordinates) embeds a graph in a coordinate space. And the last class (Aesthetics) maps a graph to a visible or perceivable display called a graphic.

A consequence of this class-orthogonality is a high degree of expressiveness: the product set of these seven function classes produces a huge variety of graphical forms or chart types. In fact, it is claimed that virtually the entire corpus of known statistical charts can be generated by this relatively parsimonious system, and perhaps a great number of meaningful but undiscovered chart types as well.

The second principal claim of GoG is that this function chain encapsulates the meaning of what we do when we construct formal statistical graphics, charts, and visualizations. It is not a taxonomy. It is a computational system based on the underlying mathematics of

representing functions of data. A consequence of this claim is to say that charts not definable within the GoG chain should be carefully examined for the possibility that they are ill-formed (meaningless).

This talk will include concrete examples to illustrate distinguishing characteristics of visualization languages based on GoG: simplicity, expressiveness, coherence, and meaningfulness. I will also survey software systems based on GoG that have been developed since the book was first published in 1999.