Exploring multivariate data structures with local principal curves.

A new approach to find the underlying structure of a multidimensional data cloud is proposed, which is based on a localized version of principal components analysis. More specifically, we calculate a series of local centers of mass and move through the data in directions given by the first local principal axis. One obtains a smooth ``local principal curve'' passing through the "middle" of a multivariate data cloud. The concept adopts to branched curves by considering the second local principal axis. Since the algorithm is based on a simple eigendecomposition, computation is fast and easy

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

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-

tourr: An R Package for Exploring Multivariate Data with Projections

This paper describes an R package which produces tours of multivariate data. The package includes functions for creating different types of tours, including grand, guided, and little tours, which project multivariate data (p-D) down to 1, 2, 3, or, more generally, d (⤠p) dimensions. The projected data can be rendered as densities or histograms, scatterplots, anaglyphs, glyphs, scatterplot matrices, parallel coordinate plots, time series or images, and viewed using an R graphics device, passed to GGobi, or saved to disk. A tour path can be stored for visualisation or replay. With this package it is possible to quickly experiment with different, and new, approaches to tours of data. This paper contains animations that can be viewed using the Adobe Acrobat PDF viewer.

Functional factor analysis for periodic remote sensing data

We present a new approach to factor rotation for functional data. This is achieved by rotating the functional principal components toward a predefined space of periodic functions designed to decompose the total variation into components that are nearly-periodic and nearly-aperiodic with a predefined period. We show that the factor rotation can be obtained by calculation of canonical correlations between appropriate spaces which make the methodology computationally efficient. Moreover, we demonstrate that our proposed rotations provide stable and interpretable results in the presence of highly complex covariance. This work is motivated by the goal of finding interpretable sources of variability in gridded time series of vegetation index measurements obtained from remote sensing, and we demonstrate our methodology through an application of factor rotation of this data.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS518 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Circular RNAs in Clear Cell Renal Cell Carcinoma: Their Microarray-Based Identification, Analytical Validation, and Potential Use in a Clinico-Genomic Model to Improve Prognostic Accuracy

Circular RNAs (circRNAs) may act as novel cancer biomarkers. However, a genome-wide evaluation of circRNAs in clear cell renal cell carcinoma (ccRCC) has yet to be conducted. Therefore, the objective of this study was to identify and validate circRNAs in ccRCC tissue with a focus to evaluate their potential as prognostic biomarkers. A genome-wide identification of circRNAs in total RNA extracted from ccRCC tissue samples was performed using microarray analysis. Three relevant differentially expressed circRNAs were selected (circEGLN3, circNOX4, and circRHOBTB3), their circular nature was experimentally confirmed, and their expression-along with that of their linear counterparts-was measured in 99 malignant and 85 adjacent normal tissue samples using specifically established RT-qPCR assays. The capacity of circRNAs to discriminate between malignant and adjacent normal tissue samples and their prognostic potential (with the endpoints cancer-specific, recurrence-free, and overall survival) after surgery were estimated by C-statistics, Kaplan-Meier method, univariate and multivariate Cox regression analysis, decision curve analysis, and Akaike and Bayesian information criteria. CircEGLN3 discriminated malignant from normal tissue with 97% accuracy. We generated a prognostic for the three endpoints by multivariate Cox regression analysis that included circEGLN3, circRHOBT3 and linRHOBTB3. The predictive outcome accuracy of the clinical models based on clinicopathological factors was improved in combination with this circRNA-based signature. Bootstrapping as well as Akaike and Bayesian information criteria confirmed the statistical significance and robustness of the combined models. Limitations of this study include its retrospective nature and the lack of external validation. The study demonstrated the promising potential of circRNAs as diagnostic and particularly prognostic biomarkers in ccRCC patients