Visualizing Densities1

This paper focuses on visualizing densities. We first give a small generalization of kernel density estimators which is appropriate for smoothing general point masses including statistical data, but also more general data forms. We give a heuristic discussion to show that our smoother has some desirable approximation properties. We also show that for this class of kernel smoother the 12or 3-dimensional marginal densities of a high-dimensional kernel density approximator have the same formula as the 12or 3-dimensional kernel density approximator. We conclude that, for visualization purposes, it is unnecessary to compute kernel density approximators in higher than three dimensions. We also develop the formula for partial derivatives of the kernel density approximator. We develop the relationship between the isopleths, the gradient and the surface normals for a density with two-dimensional support. We show that these form a trihedron. We also develop the algorithm for computing the surface normal for the isopleths of a density with three-dimensional support. With this information in hand, we discuss rendering and lighting models, contouring algorithms, and stereoscopic display algorithms. We conclude with some examples and a discussion of our experiences in using rendering and lighting, transparency, stereoscopy, dynamic rotation and dynamic thresholding techniques to visualize densities.