Einbeck, J. and Evers, L. and Hinchliff, K. (2010) 'Data compression and regression based on local principal curves.', in Advances in data analysis, data handling and business intelligence. Berlin: Springer, pp. 701-712. Studies in classification, data analysis, and knowledge organization.
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.
|Item Type:||Book chapter|
|Additional Information:||Proceedings of the 32nd Annual Conference of the Gesellschaft für Klassifikation e.V., Joint Conference with the British Classification Society (BCS) and the Dutch/Flemish Classification Society (VOC), Helmut-Schmidt-University, Hamburg, July 16–18, 2008.|
|Keywords:||Dimension reduction, Principal component regression, Principal curves, Smoothing,|
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||http://dx.doi.org/10.1007/978-3-642-01044-6_64|
|Publisher statement:||The original publication is available at www.springerlink.com|
|Date accepted:||No date available|
|Date deposited:||31 January 2011|
|Date of first online publication:||2010|
|Date first made open access:||No date available|
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