Densities as Functional Data

• Activity Number: 203
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: SSC
• Author(s): Hans-Georg Müller*+ and Alexander Petersen
• Companies: University of California, Davis and University of California, Davis
• Keywords: Functional Data Analysis ; Functional Mode of Variation ; Density Function ; Transformation

Abstract

Functional data which are non-negative and have a constrained integral can be considered as samples of one-dimensional density functions or derivatives of distributions. Due to their inherent constraints, densities do not live in a vector space and therefore common Hilbert space based methods of functional data analysis are not applicable. To address this problem, we introduce a transformation approach, mapping probability densities to a linear functional space through a continuous and invertible map. Common methods of functional data analysis, such as functional modes of variation or functional regression and classification, are then implemented in this linear space. Representations in density space are obtained by an application of the inverse map from the linear functional space to the density space. Transformations of interest include log quantile based transformations, among others. Theoretical support is provided for a general class of transformations that satisfy certain structural properties. The methods are illustrated by various examples, including an analysis of rise and fall over time in popularity for a sample of baby names.