Page 433 - Jolliffe I. Principal Component Analysis

14. Generalizations and Adaptations of Principal Component Analysis
398
individuals and observations to give a mode with np categories) before
finding cross-products. Details will not be given here (see Tucker (1966) or
Kroonenberg (1983a), where examples may also be found).
The substantial literature on the subject that existed at the time of
Kroonenberg’s (1983a) book has continued to grow. A key reference, col-
lecting together material from many of those working in the area in the late
1980s, is Coppi and Bolasco (1989). Research is still being done on various
extensions, special cases and properties of the three-mode model (see, for
example, Timmerman and Kiers (2000)). One particular extension is to the
case where more than three modes are present. Such data are usually called
‘multiway’ rather than ‘multimode’ data.
Although multiway analysis has its roots in the psychometric literature,
it has more recently been adopted enthusiastically by the chemometrics
community. Volume 14, Issue 3 of the Journal of Chemometrics, published
in 2000, is a special issue on multiway analysis. The issue contains relatively
little on multiway PCA itself, but there is no shortage of articles on it in
the chemometrics literature and in the overlapping field of process control
(see, for example, Dahl et al. (1999)). In process control the three most
commonly encountered modes are different control variables, different time
intervals and different batch runs (Nomikos and MacGregor, 1995).
Another context in which three-mode data arise is in atmospheric science,
where one mode is spatial location, a second is time and a third is a set of
different meteorological variables. It was noted in Section 12.2.1 that the
analysis of such data, which amalgamates the p locations and n different
meteorological variables into a combined set of np variables, is sometimes
known as extended EOF analysis.
An alternative strategy for analysing data of this type is to consider pairs
of two modes, fixing the third, and then perform some form of PCA on each
chosen pair of modes. There are six possible pairs, leading to six possible
analyses. These are known as O-, P-, Q-, R-, S- and T-mode analyses (Rich-
man, 1986), a terminology that has its roots in psychology (Cattell, 1978,
Chapter 12). In atmospheric science the most frequently used mode is S-
mode (locations = variables; times = observations; meteorological variable
fixed), but T-mode (times = variables; locations = observations; mete-
orological variable fixed) is not uncommon (see, for example, Salles et al.
(2001)). Richman (1986) discusses the other four possibilities. Weare (1990)
describes a tensor-based variation of PCA for data having four ‘dimen-
sions,’ three in space together with time. He notes a similarity between his
technique and three-mode factor analysis.
Some types of multiway data convert naturally into other forms. In some
cases one of the modes corresponds to different groups of individuals mea-
sured on the same variables, so that the analyses of Section 13.5 may be
relevant. In other circumstances, different modes may correspond to dif-
ferent groups of variables. For two such groups, Section 9.3 describes a
number of techniques with some connection to PCA, and many of these