Data Compression and Regression Through Local Principal Curves and Surfaces

Abstract

(2010) 'Data compression and regression through local principal curves and surfaces.', International journal of neural systems., 20 (3). pp. 177-192. Further information on publisher's website:http://dx.doi.org/10.1142/S0129065710002346Publisher's copyright statement:Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. 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 lowdimensional 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.

Bandwidth Selection for Mean-shift based Unsupervised Learning Techniques: a Unified Approach via Self-coverage

“…For instance, the local principal surface algorithm [7] does make use of the mean shift as the essential tool of estimation too, so that the methods proposed here should straightforwardly extend to this case, and preliminary investigations brought encouraging results.…”

“…Each data point x i can be projected onto the curve and represented ("compressed") through a univariate projection index λ i . Details on these techniques are irrelevant for the presentation of this paper [7]. If H = h 2 I p , one can show that, asymptotically [9],…”

Incorporating the geometry of dispersal and migration to understand spatial patterns of species distributions

“…The extension of PCA to curvilinear spaces is called 'manifold learning', a type of unsupervised machine learning technique , Einbeck et al 2010; it includes two types of method, the nonlinear PCA (Scholz et al 2008) and principal manifold techniques (Gorban and Zinovyev 2008). Some have been used to represent objects in space ranging from molecules (Gorban and Zinovyev 2008) to anatomical structures (Failmezger et al 2013).…”

Parameterization of White Matter Manifold-Like Structures Using Principal Surfaces

“…It highlights that the principal curve minimizes the sum of orthogonal distances, while a spline model fit tries to minimize the sum of distances parallel to the y axis. The principal surface algorithm and its extensions (Dong and McAvoy 1996; Einbeck et al 2010; Gerber et al 2009; Goldsmith et al 2011a; Jung et al 2011; Leblanc and Tibshirani 1994; Ozertem and Erdogmus 2011), do not provide a consistent parametrization in the 2D space (see Section 3). ISOMAP (Tenenbaum et al 2000) and Maximum Variance Unfolding (MVU) (Weinberger and Saul 2006) are methods for dimension reduction.…”