Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

398 14. Generalizations and Adaptations of Principal Component Analysisindividuals and observations to give a mode with np categories) beforefinding cross-products. Details will not be given here (see Tucker (1966) orKroonenberg (1983a), where examples may also be found).The substantial literature on the subject that existed at the time ofKroonenberg’s (1983a) book has continued to grow. A key reference, collectingtogether material from many of those working in the area in the late1980s, is Coppi and Bolasco (1989). Research is still being done on variousextensions, special cases and properties of the three-mode model (see, forexample, Timmerman and Kiers (2000)). One particular extension is to thecase 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 chemometricscommunity. Volume 14, Issue 3 of the Journal of Chemometrics, publishedin 2000, is a special issue on multiway analysis. The issue contains relativelylittle on multiway PCA itself, but there is no shortage of articles on it inthe chemometrics literature and in the overlapping field of process control(see, for example, Dahl et al. (1999)). In process control the three mostcommonly encountered modes are different control variables, different timeintervals 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 ofdifferent meteorological variables. It was noted in Section 12.2.1 that theanalysis of such data, which amalgamates the p locations and n differentmeteorological variables into a combined set of np variables, is sometimesknown as extended EOF analysis.An alternative strategy for analysing data of this type is to consider pairsof two modes, fixing the third, and then perform some form of PCA on eachchosen pair of modes. There are six possible pairs, leading to six possibleanalyses. These are known as O-, P-, Q-, R-, S- and T-mode analyses (Richman,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 variablefixed), but T-mode (times = variables; locations = observations; meteorologicalvariable 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 ‘dimensions,’three in space together with time. He notes a similarity between histechnique and three-mode factor analysis.Some types of multiway data convert naturally into other forms. In somecases one of the modes corresponds to different groups of individuals measuredon the same variables, so that the analyses of Section 13.5 may berelevant. In other circumstances, different modes may correspond to differentgroups of variables. For two such groups, Section 9.3 describes anumber of techniques with some connection to PCA, and many of these