Functional Data Analysis in Electronic Commerce Research

This paper describes opportunities and challenges of using functional data analysis (FDA) for the exploration and analysis of data originating from electronic commerce (eCommerce). We discuss the special data structures that arise in the online environment and why FDA is a natural approach for representing and analyzing such data. The paper reviews several FDA methods and motivates their usefulness in eCommerce research by providing a glimpse into new domain insights that they allow. We argue that the wedding of eCommerce with FDA leads to innovations both in statistical methodology, due to the challenges and complications that arise in eCommerce data, and in online research, by being able to ask (and subsequently answer) new research questions that classical statistical methods are not able to address, and also by expanding on research questions beyond the ones traditionally asked in the offline environment. We describe several applications originating from online transactions which are new to the statistics literature, and point out statistical challenges accompanied by some solutions. We also discuss some promising future directions for joint research efforts between researchers in eCommerce and statistics.Comment: Published at http://dx.doi.org/10.1214/088342306000000132 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian Functional Data Analysis Using WinBUGS

We provide user friendly software for Bayesian analysis of functional data models using \pkg{WinBUGS}~1.4. The excellent properties of Bayesian analysis in this context are due to: (1) dimensionality reduction, which leads to low dimensional projection bases; (2) mixed model representation of functional models, which provides a modular approach to model extension; and (3) orthogonality of the principal component bases, which contributes to excellent chain convergence and mixing properties. Our paper provides one more, essential, reason for using Bayesian analysis for functional models: the existence of software.

Functional Data Analysis with Increasing Number of Projections

Functional principal components (FPC's) provide the most important and most extensively used tool for dimension reduction and inference for functional data. The selection of the number, d, of the FPC's to be used in a specific procedure has attracted a fair amount of attention, and a number of reasonably effective approaches exist. Intuitively, they assume that the functional data can be sufficiently well approximated by a projection onto a finite-dimensional subspace, and the error resulting from such an approximation does not impact the conclusions. This has been shown to be a very effective approach, but it is desirable to understand the behavior of many inferential procedures by considering the projections on subspaces spanned by an increasing number of the FPC's. Such an approach reflects more fully the infinite-dimensional nature of functional data, and allows to derive procedures which are fairly insensitive to the selection of d. This is accomplished by considering limits as d tends to infinity with the sample size. We propose a specific framework in which we let d tend to infinity by deriving a normal approximation for the two-parameter partial sum process of the scores \xi_{i,j} of the i-th function with respect to the j-th FPC. Our approximation can be used to derive statistics that use segments of observations and segments of the FPC's. We apply our general results to derive two inferential procedures for the mean function: a change-point test and a two-sample test. In addition to the asymptotic theory, the tests are assessed through a small simulation study and a data example