The Mode Tree: A Tool for Visualization of Nonparametric Density Features

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.Recognition and extraction of features in a nonparametric density estimate are highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one that is compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating factors such as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens's chondrite data and van Winkle's Hidalgo stamp data. Finally, the bivariate mode tree is introduced, together with an example using Scott's lipid data.

Multivariate Density Estimation and Visualization

“…For many exploratory purposes, this property alone is reason to use only the normal kernel. Minnotte and Scott (1993) proposed graphing the locations of all modes at all bandwidths in the "mode tree." Minnotte (1997) proposed an extension of Silverman's bootstrap test (Silverman, 1981) for the number of modes to test individual modes.…”

Multiresolution Jet Reconstruction with FFTJet

“…This method was originally inspired by Refs. [8] and [9] and initially proposed in [10]. 1 The study by Cheng [8] establishes an important connection between the iterative cone algorithm 2 and kernel density estimation (KDE) [16].…”

Compositional Gene Landscapes in Vertebrates

“…Our operations are similar, but require fewer adjustable parameters and no 'pixelization': we consider effectively a square scatterplot (0%-100% GC2/GC3) in order to achieve radial symmetry (isotropy), and all exact distances between the original data points are taken into account. Although such kernel smoothing is not fully equivalent to binning and contouring operations, quantitative results are available in this context, notably significance tests for multimodality and an algorithm for constructing a plausible null landscape (Silverman 1981(Silverman , 1986Minnotte and Scott 1993).…”