Parallel Factor Analysis of gait waveform data: A multimode extension of Principal Component Analysis

Abstract

Gait data are typically collected in multivariate form, so some multivariate analysis is often used to understand interrelationships between observed data. Principal Component Analysis (PCA), a data reduction technique for correlated multivariate data, has been widely applied by gait analysts to investigate patterns of association in gait waveform data (e.g., interrelationships between joint angle waveforms from different subjects and/or joints). Despite its widespread use in gait analysis, PCA is for two-mode data, whereas gait data are often collected in higher-mode form. In this paper, we present the benefits of analyzing gait data via Parallel Factor Analysis (Parafac), which is a component analysis model designed for three-or higher-mode data. Using three-mode joint angle waveform data (subjects × time × joints), we demonstrate Parafac's ability to (a) determine interpretable components revealing the primary interrelationships between lowerlimb joints in healthy gait and (b) identify interpretable components revealing the fundamental differences between normal and perturbed subjects' gait patterns across multiple joints. Our results offer evidence of the complex interconnections that exist between lower-limb joints and limb segments in both normal and abnormal gaits, confirming the need for the simultaneous analysis of multi-joint gait waveform data (especially when studying perturbed gait patterns).ii To my loving and supportive family.iii

Analyzing individual and group differences in multijoint multiwaveform gait data using the Parafac2 model

“…Helwig et al . have demonstrated Parafac's ability to provide unique solutions describing fundamental patterns of variation in three‐mode gait waveform data ( subjects × time × joints ) from both healthy and perturbed subjects. However, the methods presented in are only appropriate for the analysis of multimode data that are comparable across all levels within each mode.…”

“…However, the methods presented in are only appropriate for the analysis of multimode data that are comparable across all levels within each mode. Thus, if a researcher desires to simultaneously analyze many different types of waveforms (measured in incomparable units) from multiple body locations of several subjects, an alternative to the methodology in is needed.…”

“…One possible extension of the approach in is to fit a different three‐mode Parafac model to each waveform's three‐mode data array ( subjects × time × joints ). Although seemingly simplistic, this option is undesirable because it produces a different Parafac solution for each type of waveform, and there is no clear way to compare the results across the different waveform types.…”

“…Another possible extension of the method in is to reshape the four‐mode data array into a three‐mode array that can be analyzed by the three‐mode Parafac model. For example, the four‐mode array of subjects × time × joints × waveforms could be reshaped into the three‐mode array of subjects × time * waveforms × joints , where Mode B stacks the different waveforms one after another.…”

“…For example, the four‐mode array of subjects × time × joints × waveforms could be reshaped into the three‐mode array of subjects × time * waveforms × joints , where Mode B stacks the different waveforms one after another. This reshaped array could then be analyzed by the three‐mode Parafac model to produce a solution similar to those discussed in . This method has a few benefits over the previously described approach of fitting separate three‐mode Parafac models for each waveform: (i) analyzing the reshaped array produces only one Parafac solution and (ii) the solution makes it possible to indirectly compare the relative influence of the waveform types via the Mode B weights.…”