Visualizing multivariate functions, data, and distributions

The problem of visualizing a scalar-dependent variable that is a function of many independent variables is addressed, focusing on cases with three or more independent variables. A hierarchical axis using different metrics for each independent variable is used, as are hierarchical data symbols. The technique is described for the case in which each independent variable is sampled in a regular grid or lattice-like fashion (that is, in equal increments), but it can be generalized to a variety of less restrictive domains. Rather than presenting a formal mathematical description, the authors use a visual means of describing the technique for a simple three-dimensional data case, and then demonstrate by example how to extend it to higher dimensions. It is demonstrated that these techniques for plotting scalar fields on an N-dimensional lattice work for such data visualization tasks as the location of maxima, minima, saddle points, and other features, as well as for visually fitting multivariate data and the visual determination of dominant and weak or irrelevant variables.<>