Multivariate Density Estimation and Visualization

This chapter examines the use of flexible methods to approximate an unknown density function, and techniques appropriate for visualization of densities in up to four dimensions. The statistical analysis of data is a multilayered endeavor. Data must be carefully examined and cleaned to avoid spurious findings. A preliminary examination of data by graphical means is useful for this purpose. Graphical exploration of data was popularized by Tukey (1977) in his book on exploratory data analysis (EDA). Modern data mining packages also include an array of graphical tools such as the histogram, which is the simplest example of a density estimator. Exploring data is particularly challenging when the sample size is massive or if the number of variables exceeds a handful. In either situation, the use of nonparametric density estimation can aid in the fundamental goal of understanding the important features hidden in the data. In the following sections, the algorithms and theory of nonparametric density estimation will be described, as well as descriptions of the visualization of multivariate data and density estimates. For simplicity, the discussion will assume the data and functions are continuous. Extensions to discrete and mixed data are straightforward.