Profiling Psychological Traits in Kernel Space: from the 2D Plane to the Reproducing Kernel Hilbert Space (RKHS)

The article presents a new methodology of psychological profiling in the Hilbert space with a reproducing kernel (RKHS), in which individuals — rather than variables — become the basis for verifying psychological models. A transformation of classical trait profiles into geometric points in relational space is presented, and the coefficient~$\eta^\star$ is introduced as a new measure of fit in kernel space. The psychological model is analyzed as a spatial structure of individuals, which allows for its verification through the localization of persons relative to the theoretical trajectory.In contrast to classical methods based on variable covariation, the proposed approach reconfigures the very measurement space itself, introducing the radial kernel as a tool transforming data into a relational topology of higher dimensions. The RKHS space makes it possible to reveal the deep structure of the model in the geometric arrangement of individuals — through their mutual similarities rather than raw trait values.The proposed coefficient~$\eta^\star$, being an extension of Aranowska’s classical formula, makes it possible to assess whether a given psychological profile realizes the theoretical structure or rather fits into the empirical distribution. This method opens the possibility of verifying models not at the level of a correlation matrix, but in a space defined by the individuals themselves as vectors.The article thus constitutes a proposal for a new epistemology of psychological measurement — from variable space to person space, from classification to localization, from trait analysis to analysis of geometric relations.