Warped factor analysis

Abstract

Shifted factor analysis (SFA) is designed to fit overall position shifts of sequential factors as well as the variation of factor weights. SFA is one kind of nonlinear generalization of linear factor models such as the two-mode principal component model and the three-mode Parafac model. Warped factor analysis (WFA), presented here, further generalizes SFA so that it fits factor variation due to not only the position shifts and the systematic weighting but also more flexible shape changes of sequential factors. A quasi alternating least squares (ALS) procedure is developed for WFA, using a simplified warping of segmented sequential factors. The most unique property of WFA is that it allows sequential factors to change shape distinctively, which is unlike what is implicitly assumed in mapping of data profiles, namely that sequential factors change shape together. Unlike the two-mode factor/component models, two-mode SFA was empirically shown to yield an essentially unique solution, using the additional sources of information provided by fitting systematic position shifts. Likewise, factors recovered by two-mode WFA are likely to be essentially unique due to the systematic shape variance fit by the model, provided that the data contain sufficient sources of shape variation. Copyright © 2009 John Wiley & Sons, Ltd.