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

72 4. Interpreting Principal Components: Examplessection is taken from Maryon (1979) and is concerned with sea level atmosphericpressure fields, averaged over half-month periods, for most of theNorthern Hemisphere. There were 1440 half-months, corresponding to 60years between 1900 and 1974, excluding the years 1916–21, 1940–48 whendata were inadequate. The pressure fields are summarized by estimatingaverage pressure at p = 221 grid points covering the Northern Hemisphereso that the data set consists of 1440 observations on 221 variables. Datasets of this size, or larger, are commonplace in atmospheric science, anda standard procedure is to replace the variables by a few large-variancePCs. The eigenvectors that define the PCs are often known as empiricalorthogonal functions (EOFs) in the meteorological or climatological literature,and the values of the PCs (the PC scores) are sometimes referredto as amplitude time series (Rasmusson et al., 1981) or, confusingly, ascoefficients (Maryon, 1979) or EOF coefficients (von Storch and Zwiers,1999, Chapter 13). Richman (1986) distinguishes between EOF analysisand PCA, with the former having unit-length eigenvectors and the latterhaving eigenvectors renormalized, as in (2.3.2), to have lengths proportionalto their respective eigenvalues. Other authors, such as von Storchand Zwiers (1999) treat PCA and EOF analysis as synonymous.For each PC, there is a coefficient (in the usual sense of the word), orloading, for each variable, and because variables are gridpoints (geographicallocations) it is possible to plot each loading (coefficient) on a map atits corresponding gridpoint, and then draw contours through geographicallocations having the same coefficient values. The map representation cangreatly aid interpretation, as is illustrated in Figure 4.1.This figure, which comes from Maryon (1979), gives the map of coefficients,arbitrarily renormalized to give ‘round numbers’ on the contours,for the second PC from the pressure data set described above, and is mucheasier to interpret than would be the corresponding table of 221 coefficients.Half-months having large positive scores for this PC will tend to have highvalues of the variables, that is high pressure values, where coefficients on themap are positive, and low values of the variables (low pressure values) atgridpoints where coefficients are negative. In Figure 4.1 this corresponds tolow pressure in the polar regions and high pressure in the subtropics, leadingto situations where there is a strong westerly flow in high latitudes atmost longitudes. This is known as strong zonal flow, a reasonably frequentmeteorological phenomenon, and the second PC therefore contrasts halfmonthswith strong zonal flow with those of opposite character. Similarly,the first PC (not shown) has one of its extremes identified as correspondingto an intense high pressure area over Asia and such situations are again afairly frequent occurrence, although only in winter.Several other PCs in Maryon’s (1979) study can also be interpreted ascorresponding to recognizable meteorological situations, especially whencoefficients are plotted in map form. The use of PCs to summarize pressurefields and other meteorological or climatological fields has been found