Subsections

• 11.1.1 Architecture

11.1 Introduction

The Grammar of Graphics, or GOG, denotes a system with seven orthogonal components ([43]). By orthogonal, we mean there are seven graphical component sets whose elements are aspects of the general system and that every combination of aspects in the product of all these sets is meaningful. This sense of the word orthogonality, a term used by computer designers to describe a combinatoric system of components or building blocks, is in some sense similar to the orthogonal factorial analysis of variance (ANOVA), where factors have levels and all possible combinations of levels exist in the ANOVA design. If we interpret each combination of features in a GOG system as a point in a network, then the world described by GOG is represented in a seven-dimensional rectangular lattice.

A consequence of the orthogonality of such a graphic system is a high degree of expressiveness. That is, it comprises a system that can produce a huge variety of graphical forms (chart types). In fact, it is claimed that virtually the entire corpus of known 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 system describes the meaning of what we do when we construct statistical graphs or charts. It is more than a taxonomy. It is a computational system based on the classical mathematics of representing functions and relations in Cartesian and other spaces. Because of this mathematical foundation, GOG specifications can serve as parsimonious and natural descriptions of famous statistical charts devised by Playfair, Minard, Jevons, Pearson, Bertin, Tukey, and other significant figures in the history of statistical graphics.

11.1.1 Architecture

Figure 11.1 shows a dataflow diagram that contains the seven GOG components. This dataflow is a chain that describes the sequence of mappings needed to produce a statistical graphic from a set of data. The first component (Variables) maps data to an object called a varset (a set of variables). The next three components (Algebra, Scales, Statistics) are transformations on varsets. The next component (Geometry) maps a varset to a graph and the next (Coordinates) embeds a graph in a coordinate space. The last component (Aesthetics) maps a graph to a visible or perceivable display called a graphic.

Figure 11.1: Dataflow

The dataflow architecture implies that the subtasks needed to produce a graphic from data must be done in this specified order. Imposing an order would appear to be unnecessarily restrictive, but changes of this ordering can produce meaningless graphics. For example, if we compute certain statistics on variables (e.g., sums) before scaling them (e.g., log scales), we can produce statistically questionable results because the log of a sum is not the sum of the logs.

The dataflow in Fig. 11.1 has many paths through it. We can choose different designs (factorial, nested, $\ldots$), scales (log, probability, $\ldots$), statistical methods (means, medians, modes, smoothers, $\ldots$), geometric objects (points, lines, bars, $\ldots$), coordinate systems (rectangular, polar, $\ldots$), and aesthetics (size, shape, color, $\ldots$). These paths reveal the richness of the system. The remainder of this article will summarize the seven GOG components, delineate these paths, and then briefly introduce sample applications.