Beyond principal component analysis: A trilinear decomposition model and least squares estimation (Q1205775)

Principal components and factor analysis for data \(X_{it}\) over subjects \(i=1,\dots,I\) and variables \(t=1,\dots,T\), are based on the bilinear decomposition \(X_{it}=\sum^ K_{k=1} A_{ki} C_{kt}+E_{it}\), where \(\underset\widetilde{} C=(C_{kt})\) is the \(K\times T\) ``loadings'' matrix, \(\underset\widetilde{} A=(A_{ki})\) is the \(K\times I\) ``scores'' matrix, and \(E_{it}\) is the error term. In the case of three-mode data \(X_{i\ell t}\) (for example, \(\ell\) for electrodes and \(t\) for times in evoked brain potential data), the analysis can be based on the trilinear decomposition \(X_{i\ell t}=\sum^ K_{k=1} A_{ki} B_{k\ell} C_{kt}+E_{i\ell t}\). Least squares estimators have been proposed for this model, and their properties are investigated in this paper. Sufficient conditions for consistency and asymptotic normality of the estimators are given. Detailed proofs, connections with multidimensional scaling, and references to the literature are provided.