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AbstractFunctional data analysis (FDA) is an important modern paradigm for handling infinite-dimensional data. An important task in FDA is clustering, which identifies subgroups based on the shapes of measured curves. Considering that derivatives can provide additional useful information about the shapes of functionals, we propose a novel L2 distance between two random functions by incorporating the functions and their derivative information to determine the dissimilarity of curves under a unified scheme for dense observations. The Karhunen–Loève expansion is used to approximate the curves and their derivatives. Cluster membership prediction for each curve intends to minimize the new distances between the observed and predicted curves through subspace projection among all possible clusters. We provide consistent estimators for the curves, curve derivatives, and the proposed distance. Identifiability issues of the clustering procedure are also discussed. The utility of the proposed method is illustrated via simulation studies and applications to two real datasets. The proposed method can considerably improve cluster performance compared with existing functional clustering methods. Supplementary materials for the article are available online. [ABSTRACT FROM AUTHOR]

We report on nucleus‐independent magnetic shielding (NICS) scans over the centers of six‐ and five‐membered rings in selected metal phthalocyanines (MPc) and fullerene C60 for more accurate characterization of local aromaticity in these compounds. Detailed tests were conducted on model aromatic molecules including benzene, pyrrole, indole, isoindole, and carbazole and subsequently applied to H2Pc, ZnPc, Al(OH)Pc, and CuPc. Similar behavior of three selected magnetic probes, Bq, 3He, and 7Li+, approaching perpendicularly the ring centers, was observed. For better visualization of shielding zone over the centers of aromatic rings, we introduced a simple mathematical procedure: the first and second derivatives of scan curves with respect to magnetic probe position enabled their additional examination. It allowed an easier localization of curve minimum and discrimination between areas in space varying by the magnetic field magnitude and to illustrate local aromaticity of two different kinds of rings in MPc with better resolution. Our results supported earlier reports on very low aromaticity indexes of pyrrole ring incorporated into MPc and significant aromaticity of the central macrocycle. No direct dependence between harmonic oscillator model of aromaticity and NICS was observed. Instead, a correlation between position of scan curve minimum and its magnitude were observed. In addition, the NICS values and 3He chemical shifts in the middle of neutral C60 and C606− anion agreed well with the reported experimental NMR values for He@C60 and He@C606−. [ABSTRACT FROM AUTHOR]