Mahalanobis distance

About: Mahalanobis distance is a research topic. Over the lifetime, 4616 publications have been published within this topic receiving 95294 citations.

Principal polynomial analysis

Abstract: This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

Classifying real-world data with the DDα-procedure

Abstract: The $${ DD}\alpha $$DD?-classifier, a nonparametric fast and very robust procedure, is described and applied to fifty classification problems regarding a broad spectrum of real-world data. The procedure first transforms the data from their original property space into a depth space, which is a low-dimensional unit cube, and then separates them by a projective invariant procedure, called $$\alpha $$?-procedure. To each data point the transformation assigns its depth values with respect to the given classes. Several alternative depth notions (spatial depth, Mahalanobis depth, projection depth, and Tukey depth, the latter two being approximated by univariate projections) are used in the procedure, and compared regarding their average error rates. With the Tukey depth, which fits the distributions' shape best and is most robust, `outsiders', that is data points having zero depth in all classes, appear. They need an additional treatment for classification. Evidence is also given about the dimension of the extended feature space needed for linear separation. The $${ DD}\alpha $$DD?-procedure is available as an R-package.

Mid-Infrared Fiber-Optics Reflectance Spectroscopy: A Noninvasive Technique for Remote Analysis of Painted Layers. Part II: Statistical Analysis of Spectra

Abstract: Mid-infrared fiber-optics reflectance spectroscopy supported by classification procedures based on the Mahalanobis distance in the principal component space was applied to investigate laboratory samples simulating actual paintings. The spectral data obtained were analyzed by means of principal component analysis (PCA). The application of PCA to first-derivative spectra resulted as a robust method of processing spectral data and made it possible to distinguish the binding medium and/or the pigment/dye, and to classify test samples by means of the Mahalanobis distance discrimination method.