Title: Introducing new 'ways' into data analysis:
Making linear models multilinear gives them important new properties

Speaker: Richard A. Harshman, Psychology Dept., University of Western Ontario
London, Canada,  harshman@uwo.ca  http://publish.uwo.ca/~harshman

In 1970, I developed PARAFAC (PARAllel FACtor analysis), a generalization of
factor/component analysis from matrices to three-way arrays of data (e.g.,
to measurements of n cases on m variables on each of p occasions, or to
correlations of n variables with the same n variables in each of p different
circumstances).  The motivation was to enhance validity:  by parallel
factoring of multiple non-identical mixtures of the same patterns, the
three-way model could often overcome the rotational ambiguity of standard
factor/component analysis and uniquely recover the source patterns that
originally generated the mixtures. In the last 10 years there has been a
rapid growth of important PARAFAC applications in diverse fields, ranging
from chemistry and physics (e.g., E-E flourescence and XES x-ray
spectroscopy), to signal engineering (e.g., cell-phone signals, noisy
radar), to neuroscience (EEG and fMRI brain signals), etc.  A Google search
now returns over 50,000 hits. Quite recently, I have been developing similar
generalizations of other common methods of data analysis, which hope will
also have wide application and value.

In this talk I will explain how, by extending standard statistical models
from linear to multilinear, we can substantially increase their power and
give them important new properties. The idea can be briefly explained as
follows: while traditional methods find an optimal linear combination across
one index of a two-way data array (combining columns of data), the
generalized methods find jointly-optimal linear combinations across two (or
more) indices of a three- (or higher)-way array. The figure below shows how
a standard canonical correlation for the General Linear Model (GLM) is
modified for a "level 1" multilinear generalization. The canonical weight
vectors (columns of W on both sides) are chosen so that the correlation
between the left and right canonical variates (columns of C) is maximal.
Note that the data sources on the two sides do not need to have the same
number of 'ways', so either side can be a matrix or a four-way array, etc.

By introducing multilinear generalizations into the General Linear Model,
this approach implicitly also generalizes its many special cases, such as
Discriminant Analysis, (M)ANOVA /(M)ANCOVA, etc.  In many of these
applications, one side of the canonical relation would be a 'design matrix'
or 'design array'. Statistical tests could be based on distribution free
compute-intensive methods such as randomization tests or bootstrapping.

A further kind of generalization will also be described, called "level 2
multilinearity". Here, the patterns themselves are multilinear, and take the
form of matrices or arrays with low-rank outer-product structure. For
example, in the level 2 GLM, the canonical variates become tensors of order
2 or higher. Patterns with such added structure can convey "deeper" or
"higher order" information about the data generating processes, including
how specific latent properties in one 'way' of the array 'interact' or act
jointly with specific latent properties in another.