Modeling multi-way data with linearly dependent loadings†

摘要: A generalization/specialization of the PARAFAC model is developed that improves its properties when applied to multi-way problems involving linearly dependent factors. This called PARALIND (PARAllel profiles with LINear Dependences). Linear dependences can arise empirical sources variation being modeled by factors are causally or logically linked during data generation, circumstantially collection. For example, this occur in a chemical context end products related precursor psychological single stimulus generates two incompatible feelings at once. such cases, most theoretically appropriate has loading vectors least one mode, and collinear, nonunique others. However, standard analysis fallible will have neither these features. Instead, latent linear become high surface correlations any nonuniqueness replaced meaningless surface-level ‘unique orientation’ optimally fits particular random noise sample. To avoid problems, set components theory should be rank deficient re-expressed as product matrices, explicitly represents their dependency relationships another, fewer columns, captures patterns variation. demonstrate approach, we apply it first fluorescence spectroscopy (excitation-emission EEM) which concentration values for analytes covary exactly, then flow injection (FIA) subsets columns constrained sum constant, but differently each modes. In solutions EEM data, all ‘unique’ only meaningful also unique level. contrast, directly display extent nature partial present level exhibiting corresponding uniqueness recovered loadings. FIA constraints restore estimates. Comparison shows more accurately recovers structure, presumably because uses parameters hence less error. Copyright © 2009 John Wiley & Sons, Ltd.