Margaret E. Lundy

PARAFAC: parallel factor analysis

We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (xij = ΣRr = 1airbjr) to a trilinear model (xijk = ΣRr = 1airbjrckr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the methods robustness against nonstationary factor structures and qualitative (nonproportional) factor change.

What Accounts for the Appeal of Complementary/Alternative Medicine, and What Makes Complementary/Alternative Medicine “Alternative”?

The goal of this study was to elucidate the basis for the appeal of complementary/alternative medicine (CAM) and the basis upon which people distinguish between CAM and conventional medicine. Undergraduates (N = 173) rated 19 approaches to the treatment of chronic back pain on 16 rating scales. Data were analyzed via 3-mode factor analysis, which extracted conceptual dimensions common to both the scales and the treatments. A 5-factor solution was judged to give the best description of the raters’ perceptions. One of these 5 factors clearly reflected the distinction between conventional versus CAM approaches, and a 2nd factor clearly referred to treatment appeal. The other 3 factors were invasiveness, health care professional versus patient effort, and “druglikeness.” To the extent that treatment was seen as a CAM treatment (as opposed to a conventional treatment), it was seen to be more appealing, less invasive, and less druglike. Simple and partial correlations of the dimension weights indicated that both the appealof CAM and the distinction between CAM and conventional medicine were largely driven by the view that CAM is less invasive than conventional medicine.

“Intelligent” (statistically‐guided) algorithms for vowel normalization

New normalization methods are presented which alter a speakers acoustic space in only “natural” ways, corresponding to identified patterns of naturally occurring speaker variation. These “intelligent” algorithms (i.e., which incorporate statistical‐linguistic knowledge) may stretch or shift some regions of a speakers vowel space more than others, and may do so in directions not parallel to any formant axis. Quantitative descriptions of patterns of speaker variation are obtained by an improved application of PARAFAC three‐way factor analysis. Then the investigator selects those factors of variation he wishes to remove and inputs them into the program, along with the data set to be normalized. The program uses regression methods to remove the identifiable aspects of unwanted variation from each speakers data. Three‐factor normalization reduces within‐cluster variance in the Peterson‐Barney English data by more than 70%, without causing any shifts of vowel cluster centroids. Thus the method seems to be bo...

Reducing speaker variation in Germanic: Improved results with PARAFAC

Previous applications of the PARAFAC factor analytic technique to the problem of vowel normalization [Disner, J. Acoust. Soc. Am. 67, 253 (1980)] have shown it to be well suited to cross‐language comparisons, even though the actual reduction of the variance is moderate. In this paper we present the results of a more powerful PARAFAC‐based technique, which still maintains the correct linguistic relationships in the data. Applied to data from six Germanic languages, extended PARAFAC normalization removes from 65% to more than 80% of the speaker‐related variance, depending on the language; this compares favorably with the 21%–35% removed previously. Many of the factors extracted by PARAFAC are open to linguistic interpretation as dimensions of vowel quality. Factors corresponding to phonetic height, backness, and rounding have been identified across languages; certain other factors appear to reflect rhotacization and other, less readily interpretable dimensions. A comparison has been made of the ranges of th...