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A general formalism, based upon tensor representation of multidimensional data blocks, is presented to express relationships between dependent properties and independent molecular feature measures. The solutions to these data set problems are three-dimensional quantitative structure-property relationships, 3D-QSPRs. The molecular features are partitioned into the intrinsic molecular shape tensor, the molecular field tensor, a nonshape/field feature tensor, and an experimental feature tensor. The intrinsic molecular shape tensor contains information on the shape of a molecule within the contact surface while the molecular field tensor contains information outside of the contact surface. Molecular features not directly related to molecular shape are put into the nonshape/field tensor. Experimental measures not being used as dependent variables can be considered as independent molecular features in the experimental feature tensor. The 3D-QSPR is realized by constructing the transformation tensor which optimizes the statistical significance between the dependent and independent variables. Use of partial least squares (PLS) regression permits the unfolding of the composite feature tensor and the identification of the optimum transformation tensor. It is pointed out that a variety of fragment, whole-molecule, two-dimensional, and/or three-dimensional features can be placed into a nonshape/field tensor.
Hopfinger et al. (Sat,) studied this question.