Data compression and regression through local principal curves and surfaces

We consider principal curves and surfaces in the context of multivariate regression modelling. For predictor spaces featuring complex dependency patterns between the involved variables, the intrinsic dimensionality of the data tends to be very small due to the high redundancy induced by the dependencies. In situations of this type, it is useful to approximate the high-dimensional predictor space through a low-dimensional manifold (i.e., a curve or a surface), and use the projections onto the manifold as compressed predictors in the regression problem. In the case that the intrinsic dimensionality of the predictor space equals one, we use the local principal curve algorithm for the the compression step. We provide a novel algorithm which extends this idea to local principal surfaces, thus covering cases of an intrinsic dimensionality equal to two, which is in principle extendible to manifolds of arbitrary dimension. We motivate and apply the novel techniques using astrophysical and oceanographic data examples

Data compression and regression based on local principal curves.

Frequently the predictor space of a multivariate regression problem of the type y = m(x_1, …, x_p ) + ε is intrinsically one-dimensional, or at least of far lower dimension than p. Usual modeling attempts such as the additive model y = m_1(x_1) + … + m_p (x_p ) + ε, which try to reduce the complexity of the regression problem by making additional structural assumptions, are then inefficient as they ignore the inherent structure of the predictor space and involve complicated model and variable selection stages. In a fundamentally different approach, one may consider first approximating the predictor space by a (usually nonlinear) curve passing through it, and then regressing the response only against the one-dimensional projections onto this curve. This entails the reduction from a p- to a one-dimensional regression problem. As a tool for the compression of the predictor space we apply local principal curves. Taking things on from the results presented in Einbeck et al. (Classification – The Ubiquitous Challenge. Springer, Heidelberg, 2005, pp. 256–263), we show how local principal curves can be parametrized and how the projections are obtained. The regression step can then be carried out using any nonparametric smoother. We illustrate the technique using data from the physical sciences

Localized Regression on Principal Manifolds

No abstract available

Mixtures of Spatial Spline Regressions

We present an extension of the functional data analysis framework for univariate functions to the analysis of surfaces: functions of two variables. The spatial spline regression (SSR) approach developed can be used to model surfaces that are sampled over a rectangular domain. Furthermore, combining SSR with linear mixed effects models (LMM) allows for the analysis of populations of surfaces, and combining the joint SSR-LMM method with finite mixture models allows for the analysis of populations of surfaces with sub-family structures. Through the mixtures of spatial splines regressions (MSSR) approach developed, we present methodologies for clustering surfaces into sub-families, and for performing surface-based discriminant analysis. The effectiveness of our methodologies, as well as the modeling capabilities of the SSR model are assessed through an application to handwritten character recognition

Fast Determination of Soil Behavior in the Capillary Zone Using Simple Laboratory Tests

INE/AUTC 13.1

Representing complex data using localized principal components with application to astronomical data

Often the relation between the variables constituting a multivariate data space might be characterized by one or more of the terms: ``nonlinear'', ``branched'', ``disconnected'', ``bended'', ``curved'', ``heterogeneous'', or, more general, ``complex''. In these cases, simple principal component analysis (PCA) as a tool for dimension reduction can fail badly. Of the many alternative approaches proposed so far, local approximations of PCA are among the most promising. This paper will give a short review of localized versions of PCA, focusing on local principal curves and local partitioning algorithms. Furthermore we discuss projections other than the local principal components. When performing local dimension reduction for regression or classification problems it is important to focus not only on the manifold structure of the covariates, but also on the response variable(s). Local principal components only achieve the former, whereas localized regression approaches concentrate on the latter. Local projection directions derived from the partial least squares (PLS) algorithm offer an interesting trade-off between these two objectives. We apply these methods to several real data sets. In particular, we consider simulated astrophysical data from the future Galactic survey mission Gaia.Comment: 25 pages. In "Principal Manifolds for Data Visualization and Dimension Reduction", A. Gorban, B. Kegl, D. Wunsch, and A. Zinovyev (eds), Lecture Notes in Computational Science and Engineering, Springer, 2007, pp. 180--204, http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-173750210-

Space environmental effects on graphite-epoxy compressive properties and epoxy tensile properties

This study characterizes the effects of electron radiation and temperature on a graphite-epoxy composite material. Compressive properties of the T300/934 material system were obtained at -250 F (-157 C), room temperature, and 250 F (121 C). Tensile specimens of the Fiberite 934 epoxy resin were fabricated and tested at room temperature and 250 F (121 C). Testing was conducted in the baseline (nonirradiated) and irradiated conditions. The radiation exposure was designed to simulate 30 year, worst-case exposure in geosynchronous Earth orbit. Mechanical properties tended to degrade at elevated temperature and improve at cryogenic temperature. Irradiation generally degraded properties at all temperatures