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In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on exploiting such approximations in clustering, data compression, and prediction. Most of the literature relies on linear or locally linear approximations. In this article, we propose a simple and general alternative, which instead uses spheres, an approach we refer to as spherelets. We develop spherical principal components analysis (SPCA), and provide theory on the convergence rate for global and local SPCA, while showing that spherelets can provide lower covering numbers and mean squared errors for many manifolds. Results relative to state‐of‐the‐art competitors show gains in ability to accurately approximate manifolds with fewer components. Unlike most competitors, which simply output lower‐dimensional features, our approach projects data onto the estimated manifold to produce fitted values that can be used for model assessment and cross validation. The methods are illustrated with applications to multiple data sets. [ABSTRACT FROM AUTHOR]

The coupling of semiconductors with organic molecules results in a class of sensors whose chemoresistive properties are dictated by the nature of dyes. Organic molecules generally reduce conductivity, but in the case of optically active dyes, such as porphyrinoids, the conductivity is restored by illumination with visible light. In this paper, we investigated the gas sensing properties of ZnO nanoparticles coated with porphyrins and corroles. Under light illumination, the resistance of these materials increases with the adsorption of volatile compounds but decreases when these are strong electron donors. The behavior of these sensors can be explicated on the basis of the structural difference between free-base porphyrin and corrole, the influence of coordinated metal, and the corresponding electronic structures. These sensors are promising electronic noses that combine the selectivity to strong electron donors with the broad selectivity toward the other classes of chemicals. An efficient representation of the data of this peculiar array can be obtained by replacing the Euclidean distance with the angular distance. To this end, a recently introduced spherical principal component analysis algorithm is applied for the first time to gas sensor array data. Results show that a minimal gas sensor array (four elements) can produce a sort of chemotopic map, which enables us to cluster a very large class of pure chemical vapors. Furthermore, this map provides information about the composition of complex odor matrices, such as the headspaces of beef meat and their evolution over the time. [ABSTRACT FROM AUTHOR]