A principal component analysis of sulfur concentrations in the Western United States∗

H~r;r~ 5 R %lYbli Conc:ntration Huctu~tions in d smoke plume. ._(:mo,pheu, Enc ~rt.wmwnt18, iiti I -I 106. Sstrsnllls D. D. J 11979at Concenrratlon fUCtUdtlOnS in plumes Ph.D thesis, &tech Eng Dept. L’niversity ot .Aibcrta. Candda. Nettenille D D J. (1979bl Concmrrdtion Fluctuations WI Piurw~ Environ. Research Monograph 19794, Syncrude Cdndds L:d.. 10030-10 Street. Edmonton. Alberta. Canadd TSJ 3ES. Saulbrd B. L. C1983) The efect of Gausstan particle-pair distribution functions in the statistical theory of concentration Huctuations in homogeneous turbulence. Q. JI R. rm:. SO<,.190. 339-354.

AUTHOR’S

REPLY

Dr. Netterville’s interesting comments address the problem of simulating fluctuations in the scalar concentration Iietd by means of various combinations of lateral and vertical terms. He and Csanady have developed one analytical approach and I have suggested another in my paper. In a more recent paper (Hanna, 1984) I recommend yet another simple procedure in which the lateral and vertical components are multiplied rather than added. Nettervtlle objects to my assumptions because of a “lack of theoretical justifications.” Most of the research mentioned in his note has been in wind tunnels, where turbulence tends to be more isotropic than in the atmosphere. My concentration fluctuation models were based more on observations in the atmosphere, where lateral meandering continues to be important long after a plume becomes well-mixed vertically. Also. plumes released near the ground are usually subjected to lateral and vertical turbulence with greatly different integral time scales. Because of this anisotropy in the atmosphere, I feel that it is important to account for the influence on concentration fluctuations of both the large scale lateral meanders and the small scale vertical turbulence. The lateral meanders cause concentration fluctuations by bodily moving a plume back and forth over a monitor, while the small scale vertical turbulence causes concentration fluctuations within the relatively welt-mixed vertical cross-section of the plume. The “one-dimensional model” that Netterville objects to can be tested in the atmosphere if the cross-wind integrated (either vertical or lateral) concentration can be calculated from the observations. For example, the lateral integral, Cy = jCdy. removes the influence of lateral meanders and can be used to study the concentration fluctuations caused by vertical motions. At this point in the study of concentration fluctuations in the atmosphere the data are insufficient to H-arrant choosing one method of combining the components over another. In the future ue will, perhaps, have the luxury of closely examining the bases of the various methods that have been suggested. Encironmental Research & Technology. inc. 696 Virgitlia Rd. Concord, MA 01742, L.S.A

STEVEN R. HANNA

REFERESCE

.I\ PRISCIPAL C’OMPONEST ANALYSIS OF SCLFUR CONCENTRATIONS IS THE WESTERN UNITED STATES* .Ashbaugh c’i dii. (138&l apply pnnctpal component lndlySlS to a data St of u) dir umplsrs to obwtn spatial patterns of sulfur concentrations. There are several points they posit concerning the apphcation and interpretation of the eigenanalysis procedure which cause concern. Spectfically. there are three problem areas: Ii) problems with the terminology of the various eigrntechniques .+nd the definitions oforthogonal vs oblique solutions. [II) the Preisendorfer and Barnett eigenvalue truncation test (Rule N) may be misppli& to the rotated solution and misinterpreted to both unrotated and rotated solutions and (iii) the authors state that “In general, however, an orthogonal rotation is desned” which seems to indicate that they are sanctioning orthogonal rotation for most research. Problems with the terminology ‘ppear throughout the text although Section 2 correctly dttferentiata the factor model from the component model. Furthermore. empirical orthogonal functions or eigenvectors are also confused with components in several instances. most notably in Figs 2-l I where what actually are PC loadings are referred to as eigenvectors. Since these three models can lead to different solutions, a distinction should be made so that other researchers can reproduce the results presented. Part of the terminology problem may arise from the widespread use of statistical packages for principal component analysis. Many of the packages freely intermix the terms for components and factors, thoroughly confusing users. The precise definitions of these terms can be found in standard texts on factor analysis by Harman (1976). Mulaik (1972), Rummel (1970) and in the report by Lorenz (1956) for empirical orthogonal functions which all clearly delineate the differences. (i) Empirical Orthogonal Functions IEOFS) are simply the unit length eigenvectors and corresponding eigenvalues given by the decomposition of a correlation matrix R as RV = VD where b. is the orthogonal matrix of eigenvecrors and D is the diagonal matrix of eigencahes. In the meteorological literature, the term EOF was introduced by Lorenz (1956. p. 20) where he denotes V as empirical orthogonal iunctions of space (due to their spatial orthogonality !. The corresponding time series, X. is given by 2 = .%’I” where 2 are the standardized input data (assumed standardized for this discussion but not a necessary condition) and X are the empirical orthogonal functions of time (also called ‘principal axes’ in the statistical literature and ‘time-dependent amplitudes’ in the meteorological literature). It should be noted that the time/space definitions provided by Lorenz are tailored to his specific application where a series oi weather stations are related through time for one parameter (sea-level pressure). This defines a S-mode analysis and is only one of si.T basic ways of interrelating data for an eigenanalysis (Richman_ l983a p. 66). This type of decomposition (S-mode) is the same as Ashbaugh et d. who relate one parameter (sulfur concentration) over a series of stations (38 sites) through time (months from September 1979 to September 1980) to yield a correlation matrix relatmg the sites in an analysis. Under another type of decomposition where the correlation matrix related the observations (T-mode). the 1. would represent EOFs of time while .Y would represent EOFs of space. (ii) Princtpal components (PCs) are slightly more elegant mathematically compared to EOFs as the spatial --

Hanna S. R. (1984) The exponential probability density function and concentration fluctuations in smoke plumes. Boundary-Layer Met. 29. 361-375.

lAshbaugh Atmospheric

L. L.. Myrup L. 0. and Flocchini Enrironment 18, 753-791.

R.

G. f 1984a)

1973

Discusstons patterns are correlations or covariances between the original data and the PCs. In order to facilitate this, the spatial matrix of PC Ioodinys A = k’D’ ’ where each eigenvector is postmultiplied by the square root of the corresponding eigenvalue. The time series F is known as principal componenls or principal component scores and is found by the equation 2 = FA’. (iii) Factor analysis (FA) is different from EOFs or PCs in that the variance of the original standardized variables, Z, is partitioned into two categories: common variance and unique variance. This results in a more complex model where Z = F A’ i CIV’ where F are the common factors or common jhcror scores, A are the commonfactor loadings. V are the unique foclor scows (which are uncorrelated to F) and V is a diagonal matrix of uniquefbclor loadings (which is uncorrelated to A). Only the common portion of the factor model is typically analyzed as it contains the variance which can be predicted by the remaining variables. The unique variance portion of the equatton accounts for smallscale effects and measurement error, which is treated as residual noise and is discarded. This is in contrast to the principal component model which combines the signal and noise variance into one category which is analyzed. Another point concerning terminology of oblique vs orthogonal rotation (p. 784) is the incorrect distinguishing of the two based on the correlations between eigenvectors [PC loadings]. However, the ditTerence between orthogonal and oblique rotation has to do with the correlations between the PC scores and not the loadings. Walsh et al. (1982, p. 276, their Table 2) illustrate that the spatial EOFs or PC loadings are indeed correlated under an orthogonal rotation. Therefore. both orthogonal and oblique rotations typically yield correlated loadings. However, under orthogonal rotation the PC scores are uncorrelated while they are allowed to become intercorrelated under oblique rotation as discussed in Walsh et al. (1982, p. 274). A second source of confusion arises from the discussion of truncating a series of PCs. The authors state (p. 784) that, in general, truncation ofa series of principal components is used to filter the data to separate the components which account for the signal from those which account for noise. This is not totally true, since the noise (e.g. measurement error) is analyzed along with signal under the PC model (hence all PCs contain some noise). The point in truncating a series of PCs is to attempt to separate those PCs accounting for signal plus noise from those simply accoun ting for noise. This topic again arises in the results section (p. 787) where the Preisendorfer and Bamett (1977) test (referred to as Overland and Preisendorfer, 1982) is applied. The test, known as Rule N, was designed to be applied to unrotated solutions of EOFs or PCs as a series of randomly generated eigenvalues are compared todataeigenvalues. Confidence intervals are added to the random eigenvalue series and the point where the data eigenvalue series crosses the upper confidence interval is where the PC series is truncated at r PCs. In a rotated solution the same r PCS determined from testing the unrotated solution are retained and then rotated to simple structure. The authors applied this test after the solution was rotated. This is likely incorrect for two reasons: (i) the rotated solution has no eigenvalues but only variance explained (hence the test is not applicable as it is designed to test eigenvalues) and (ii) the rotated solution results in a more even distribution of variance explained across PCs. since the r PCs retained are all considered useful signal a priori. Interestingly, Preisendorfer (1984, p. 285) specifically mentions this misapplication of Rule N to the Varimax solution. This possible misapplication led the authors to retain nine rotated PCs while application of the Preisendorfer and Barnett test indicates only four PCs should be retained and rotated. By retaining five extra PCs. which is known as overfactoring. the potential advantages gained by rotation

might be negated. Richman ( I98 I) examined overfactoring on known PC solutions and concluded that, in the case of the rotation used (Varimaxk overfactoring led to distorted spatial PC patterns in his confirmatory study. However, Ashbaugh er al. (1984b. Pers. comm.) indicate that they feel the solution based on nine PCs exhibits more scientifc significance. This may well be true, particularly in light of Preisendorfer’s (1984. p. 189) remark that Rule N sometimes underfactors [retains too few EOFs or PCs]. However, justitication of retaining nine PCs based on Rule N is not warranted. It is unfortunate that, due to the exploratory nature of this work. no quantitative method exists to conclusively test the validity of the PC loadings generated from varying the number of PCs retained. As a final comment, a philosophical disagreement on the comprehensive utility of the widely-used (and sometimes abused) Varimax rotation arises as Ashbaugh et al. (1984a. p, 784) state “lf the incesrigaror expects rhe underlying physical mechanisms

within the system are correlated. then an oblique be desirable. In general, howeurr, an orthogonal is desired.‘* The present author disagrees with this

solution may rolarion

statement for three reasons: (a) Research over the past 25 years from early landmark studies by Coan (I 959) and Cattell and Dickman (1962) through more recent research by Cohen (1983) and Richman (1983a. b; 1985) indicate that oblique rotation is usually more accurate than orthogonal rotation in capturing modes of variation. (b) How is the investigator to surmise if he should “expect that the underlying physical mechanisms within the system are correlated”? In a typical exploratory analysis, such as presented by Ashbaugh et al. (1984a). the researcher has no way of assessing, a priori, whether or not the system is orthogonal. The best way to tind this out is to apply an oblique rotation for the initial analysis, rather than force an orthogonality constraint on a solution, since an oblique rotation will result in an orthogonal solution if the underlying mechanisms are indeed uncorrelated. The intercomponent correlations can be examined to see if they are sufhciently low (e.g. 0.30) for an orthogonal solution to be safely substituted as performed in Lamb and Rtchman (1983) and Richman and Lamb (1985). [Ashbaugh er al. (1984b. pers. comm.) haveapparently done this successfully so point b doesn’t apply to their specific study.] (c) Certain misconceptions exist in the scientific literature that the variance of each oblique PC cannot be calculated. Harman (1976, pp. 268-270) lists a simple procedure to calculate the direct variance contribution of each of the oblique PCs or factors to the total solution. This variance information can then be used in an identical fashion to orthogonal variance statistics. Another misconception is that obliquely determined PC loading patterns are of little value for stattstical procedures requiring orthogonality (e.g. multiple regression). This is only partly true as an orthogonal least-squares target solution, such as the IMSL program OFSCHN. can be employed to make minor adjustments to an oblique PC loading ‘target’ to arrive at a close orthogonal approximation. Therefore, the researcher can typically achieve the advantages of oblique rotation in the best fitting orthogonal rspresentation as an alternative to indiscriminately using Varimax. Illinois

Slate

Wuer

Surcej

MICHAELB. RICHMAN

Climate and Mereorology Section 2204 Grijith Drire Champaign, IL 61820-9050. U.S.A.

AUTHORS’

REPLY

We wish to thank Dr. Richman for hisenlighteningdiscussion of our paper. Dr. Richman has evidently put forth quite an effort in critiquing our work. We agree that there is indeed a problem in terminology, not only in our paper, but throughout the literature on principal component, factor and EOF analysis. This is all the more evident in that we recognized this