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

316 12. PCA for Time Series and Other Non-Independent Data12.2.6 Examples and ComparisonsRelatively few examples have been given in this section, due in part to theircomplexity and their rather specialized nature. Several of the techniquesdescribed claim to be able to detect stationary and propagating wavesin the standard type of spatio-temporal meteorological data. Each of themethods has proponents who give nice examples, both real and artificial, inwhich their techniques appear to work well. However, as with many otheraspects of PCA and related procedures, caution is needed, lest enthusiasmleads to ‘over-interpretation.’ With this caveat firmly in mind, examples ofSSA can be found in Elsner and Tsonis (1996), Vautard (1995); MSSA inPlaut and Vautard (1994), Mote et al. (2000); POPs in Kooperberg andO’Sullivan (1996) with cyclo-stationary POPs in addition in von Storchand Zwiers (1999, Chapter 15); Hilbert EOFs in Horel (1984), Cai andBaines (2001); MTM-SVD in Mann and Park (1999); cyclo-stationary andperiodically extended EOFs in Kim and Wu (1999). This last paper compareseight techniques (PCA, PCA plus rotation, extended EOFs (MSSA),Hilbert EOFs, cyclo-stationary EOFs, periodically extended EOFs, POPsand cyclo-stationary POPs) on artificial data sets with stationary patterns,with patterns that are stationary in space but which have amplitudeschanging in time, and with patterns that have periodic oscillations in space.The results largely confirm what might be expected. The procedures designedto find oscillatory patterns do not perform well for stationary data,the converse holds when oscillations are present, and those techniquesdevised for cyclo-stationary data do best on such data.12.3 Functional PCAThere are many circumstances when the data are curves. A field in whichsuch data are common is that of chemical spectroscopy (see, for example,Krzanowski et al. (1995), Mertens (1998)). Other examples includethe trace on a continuously recording meteorological instrument such as abarograph, or the trajectory of a diving seal. Other data, although measuredat discrete intervals, have an underlying continuous functional form.Examples include the height of a child at various ages, the annual cycle oftemperature recorded as monthly means, or the speed of an athlete duringa race.The basic ideas of PCA carry over to this continuous (functional) case,but the details are different. In Section 12.3.1 we describe the generalset-up in functional PCA (FPCA), and discuss methods for estimating functionalPCs in Section 12.3.2. Section 12.3.3 presents an example. Finally,Section 12.3.4 briefly covers some additional topics including curve registration,bivariate FPCA, smoothing, principal differential analysis, prediction,discrimination, rotation, density estimation and robust FPCA.