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.

Combining tensor decomposition and time warping models for multi-neuronal spike train analysis

“…Then, two new models of neural activity are introduced—time-shifted tensor decomposition (section 1.4) and multi-shift modeling (section 1.5)—which combine different elements of tensor decomposition and time warping. These two models are closely related to previous works from the field of chemometrics Harshman et al 2003; S. Hong and Harshman 2003; S. Hong 2009; Q. Wu et al 2014 and neuroimaging Mørup et al 2008.…”

“…Previous works have explored similar ideas, though with different motivating applications or proposed algorithms (Harshman et al 2003; S. Hong and Harshman 2003; Mørup et al 2008; S. Hong 2009; Q.…”

“…Wu et al 2014). In chemometrics, for example, absorption and emission spectra are often shifted due to measurement variability across samples, and this has been incorporated into tensor decomposition models of these datasets (Harshman et al 2003; S. Hong 2009).…”

Methods to temporally align gait cycle data

“…The first and last POI in each trajectory should be the first and last time points of the GC to ensure that the cycles' endpoints are aligned. We recommend that within-GC POI be determined based on trajectory shape of the data (using an approach described in Hong, 2009) or be kinematically relevant points in the GC (e.g., heel-strike, toe-off, etc. ).…”

Multivariate Analysis Applications in X-ray Diffraction

“…FA has been applied as an alternative to PCA in reducing the number of parameters and various structural descriptors for different molecules in chromatographic datasets [31]. Moreover, developments of the original concepts have been achieved by introducing a certain degree of complexity such as the shifting of factors [32] (factors can have a certain degree of misalignment in the time direction, such as in the time profile for X-ray powder diffraction data) or the shifting and warping (i.e., a time stretching) [33].…”