Search

Two-way functional data consist of a data matrix whose row and column domains arc both structured, for example, temporally or spatially, as when the data are time series collected at different locations in space. We extend one-way functional principal component analysis (PCA) to two-way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition (SVD) of the data matrix. We focus on a penalization approach and solve the nontrivial problem of constructing proper two-way penalties from one way regression penalties. We introduce conditional cross-validated smoothing parameter selection whereby left-singular vectors are cross-validated conditional on right-singular vectors, and vice versa. The concept can be realized as part of an alternating optimization algorithm. In addition to the penalization approach, we briefly consider two-way regularization with basis expansion. The proposed methods are illustrated with one simulated and two real data examples. Supplemental materials available online show that several 'natural' approaches to penalized SVDs are flawed and explain why so. KEY WORDS: Basis expansion; Functional data analysis; Penalization; Regularization; Spatial-temporal modeling.

In the past decade there has been a resurgence of interest in nonlinear dimension reduction. Among new proposals are 'Local Linear Embedding,' 'Isomap,' and Kernel Principal Components Analysis which all construct global low-dimensional embeddings from local affine or metric information. We introduce a competing method called 'Local Multidimensional Scaling' (LMDS). Like LLE, Isomap, and KPCA, LMDS constructs its global embedding from local information, but it uses instead a combination of MDS and 'force-directed' graph drawing. We apply the force paradigm to create localized versions of MDS stress functions with a tuning parameter to adjust the strength of nonlocal repulsive forces. We solve the problem of tuning parameter selection with a meta-criterion that measures how well the sets of K-nearest neighbors agree between the data and the embedding. Tuned LMDS seems to be able to outperform MDS, PCA, LLE, Isomap, and KPCA, as illustrated with two well-known image datasets. The meta-criterion can also be used in a pointwise version as a diagnostic tool for measuring the local adequacy of embeddings and thereby detect local problems in dimension reductions. KEY WORDS: Cluster analysis; Energy functions; Force-directed layout; Isomap; Local linear embedding; Multidimensional scaling: Principal components analysis; Unsupervised learning.

A methodology is described for multidimensional scaling (MDS) and its implementation in two software systems, namely, GGvis and XGvis. The stability and multiplicity of MDS solutions, analysis of structure and junction between classical and distance MDS is also discussed.

A brief overview of Tukey's paper, 'The Future of Data Analysis' after 40 years by various authors is discussed considering the debates regarding whether statistics is a science and ways to attract bright students by showing excitement and rewards of applied work. It is argued that beyond the ideas of data analysis, one should look at how data is analyzed by others and the key concept should be statistical thinking.

Gelman's article is a thought-provoking mix of opinions and creative methodology. I agree with Gelman that the disjunction of models and exploratory data analysis (EDA) in mainstream statistics is unsound. [...]

This article describes how a mixture of two densities, [f.sub.0] and [f.sub.1], may be decomposed into a different mixture consisting of three densities. These new densities, [f.sub.+], [f.sub.-], and [f.sub.=], [...]

This article illustrates a technique for visualizing nonlinear mappings f : [R.sup.k] [right arrow] [R.sup.m] that arise frequently in engineering applications. The idea is based on viewing sections [f.sup.-1] (B) [...]

ANDREAS BUJA* Fisher's linear discriminant analysis is a valuable tool for multigroup classification. With a large number of predictors, one can find a reduced number of discriminant coordinate functions that [...]

Dedicated to the memory of the late John W. Tukey, one of the founders of computational and graphical statistics. Published research in the field of statistics is skewed to mathematics [...]