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

12.4. PCA and Non-Independent Data—Some Additional Topics 329series, rather than being restricted to real series. It turns out (Brillinger,1981, p. 344) thatB ′ u = 1 ∫ 2π˜B(λ)e iuλ dλ2πC u = 12π0∫ 2π0˜C(λ)e iuλ dλ,where ˜C(λ) isa(p × q) matrix whose columns are the first q eigenvectorsof the matrix F(λ) given in (12.1.4), and ˜B(λ) is the conjugate transposeof ˜C(λ).The q series that form the elements of z t are called the first q PC series ofx t . Brillinger (1981, Sections 9.3, 9.4) discusses various properties and estimatesof these PC series, and gives an example in Section 9.6 on monthlytemperature measurements at 14 meteorological stations. Principal componentanalysis in the frequency domain has also been used on economictime series, for example on Dutch provincial unemployment data (Bartels,1977, Section 7.7).There is a connection between frequency domain PCs and PCs definedin the time domain (Brillinger, 1981, Section 9.5). The connection involvesHilbert transforms and hence, as noted in Section 12.2.3, frequency domainPCA has links to HEOF analysis. Define the vector of variablesyt H (λ) = (x ′ t(λ), x ′Ht (λ)) ′ , where x t (λ) is the contribution to x t at frequencyλ (Brillinger, 1981, Section 4.6), and x H t (λ) is its Hilbert transform.Then the covariance matrix of yt H (λ) is proportional to[ ]Re(F(λ)) Im(F(λ)),− Im(F(λ)) Re(F(λ))where the functions Re(.), Im(.) denote the real and imaginary parts, respectively,of their argument. A PCA of ytH gives eigenvalues that are theeigenvalues of F(λ) with a corresponding pair of eigenvectors[ ] [ ]Re( ˜Cj (λ)) − Im( ˜Cj (λ))Im( ˜C ,j (λ)) Re( ˜C ,j (λ))where ˜C j (λ) isthejth column of ˜C(λ).Horel (1984) interprets HEOF analysis as frequency domain PCA averagedover all frequency bands. When a single frequency of oscillationdominates the variation in a time series, the two techniques become thesame. The averaging over frequencies of HEOF analysis is presumably thereason behind Plaut and Vautard’s (1994) claim that it is less good thanMSSA at distinguishing propagating patterns with different frequencies.Preisendorfer and Mobley (1988) describe a number of ways in whichPCA is combined with a frequency domain approach. Their Section 4ediscusses the use of PCA after a vector field has been transformed intothe frequency domain using Fourier analysis, and for scalar-valued fields