Aloke Phatak

The geometry of partial least squares

Our objective in this article is to clarify partial least squares (PLS) regression by illustrating the geometry of NIPALS and SIMPLS, two algorithms for carrying out PLS, in both object and variable space. We introduce the notion of the tangent rotation of a vector on an ellipsoid and show how it is intimately related to the power method of finding the eigenvalues and eigenvectors of a symmetric matrix. We also show that the PLS estimate of the vector of coefficients in the linear model turns out to be an oblique projection of the ordinary least squares estimate. With two simple building blocks—tangent rotations and orthogonal and oblique projections—it becomes possible to visualize precisely how PLS functions.

Short communication: Statistical downscaling of rainfall data using sparse variable selection methods

In many statistical downscaling methods, atmospheric variables are chosen by using a combination of expert knowledge with empirical measures such as correlations and partial correlations. In this short communication, we describe the use of a fast, sparse variable selection method, known as RaVE, for selecting atmospheric predictors, and illustrate its use on rainfall occurrence at stations in South Australia. We show that RaVE generates parsimonious models that are both sensible and interpretable, and whose results compare favourably to those obtained by a non-homogeneous hidden Markov model (Hughes et al., 1999).

Multivariate Process Monitoring: A New Use for the Biplot

Abstract In this article, we present a method for monitoring multivariate process data based on the Gabriel biplot. In contrast to existing methods that are based on some form of dimension reduction, we use reduction to two dimensions for displaying the state of the process butallthe data for determining whether it is in a state of statistical control. This approach allows us to detect changes in location, variation, and correlational structure accurately yet display a large amount of information concisely. We illustrate the use of the biplot on some examples of industrial data and also discuss some of the many issues related to a practical implementation of the method.