Shifted factor analysis—Part I: Models and properties

Abstract The factor model is modified to deal with the problem of factor shifts. This problem arises with sequential data (e.g. time series, spectra, digitized images) if the profiles of the latent factors shift position up or down the sequence of measurements: such shifts disturb multilinearity and so standard factor/component models no longer apply. To deal with this, we modify the model(s) to include explicit mathematical representation of any factor shifts present in a data set; in this way the model can both adjust for the shifts and describe/recover their patterns. Shifted factor versions of both two‐ and three (or higher)‐way factor models are developed. The results of applying them to synthetic data support the theoretical argument that these models have stronger uniqueness properties; they can provide unique solutions in both two‐way and three‐way cases where equivalent non‐shifted versions are under‐identified. For uniqueness to hold, however, the factors must shift independently; two or more factors that show the same pattern of shifts will not be uniquely resolved if not already uniquely determined. Another important restriction is that the models, in their current form, do not work well when the shifts are accompanied by substantial changes in factor profile shape. Three‐way factor models such as Parafac, and shifted factor models such as described here, may be just two of many ways that factor analysis can incorporate additional information to make the parameters identifiable. Copyright © 2003 John Wiley & Sons, Ltd.