Some comments on the reliability of three-index factor analysis models in speech research
Authors: Geng Christian , Hoole Phil 
Published: January 2021

Low- dimensional and speaker-independent linear vocal tract parametrizations can be obtained using the 3-mode PARAFAC factor analysis procedure first introduced by Harshman et al. (1977) and discussed in a series of subsequent papers in the Journal of the Acoustical Society of America (Jackson (1988), Nix et al. (1996), Hoole (1999), Zheng et al. (2003)). Nevertheless, some questions of importance have been left unanswered, e.g. none of the papers using this method has provided a consistent interpretation of the terms usually referred to as "speaker weights". This study attempts an exploration of what influences their reliability as a first step towards their consistent interpretation. With this in mind, we undertook a systematic comparison of the classical PARAFAC1 algorithm with a relaxed version, of it, PARAFAC2. This comparison was carried out on two different corpora acquired by the articulograph, which varied in vowel qualities, consonantal contexts, and the paralinguistic features accent and speech rate. The difference between these statistical approaches can grossly be described as follows: In PARAFAC1, observation units pertain to the same set of variables and the observation units are comparable. In PARAFAC2, observations pertain to the same set of variables, but observation units are not comparable. Such a situation can be easily conceived in a situation such as we are describing: The operationalization we took relies on the comparability of fleshpoint data acquired from different speakers, which need not be a good assumption due to influences like sensor placement and morphological conditions.

In particular, the comparison between the two different approaches is carried out by means of so-called "leverages" on different component matrices originating in regression analysis, calculated as v = diag(A(A A)−1A ) and delivering information on how "influential" a particular loading matrix is for the model. This analysis could potentially be carried out component by component, but we confined ourselves to effects on the global factor structure. For vowels, the most influential loadings are those for the tense cognates of non-palatal vowels. For speakers, the most prominent result is the relative absence of effects of the paralinguistic variables. Results generally indicate that there is quite little influence of the model specification (i.e. PARAFAC1 or PARAFAC2) on vowel and subject components. The patterns for the articulators indicate that there are strong differences between speakers with respect to the most influential measurement as revealed by PARAFAC2: In particular, the most influential y-contribution is the tongue-back for some talkers and the tongue-dorsum for other speakers. With respect to the speaker weights, again, the leverage patterns are very similar for both PARAFAC-versions. These patterns converge with the results of the loading plots, where the articulator profiles seem to be most altered by the use of PARAFAC2. These findings, in general, are interpreted as evidence for the reliability of the PARAFAC1 speaker weights.