Rotation of principal components

Wiley - Tập 6 Số 3 - Trang 293-335 - 1986
Michael B. Richman1
1Climate and Meteorology Section, Illinois State Water Survey, Champaign, Illinois 61820‐7407, U.S.A.

Tóm tắt

AbstractRecent research has pointed to a number of inherent disadvantages of unrotated principal components and empirical orthogonal functions when these techniques are used to depict individual modes of variation of data matrices in exploratory analyses. The various pitfalls are outlined and illustrated with an alternative method introduced to minimize these problems via available linear transformations known as simple structure rotations. The rationale and theory behind simple structure rotation and Procrustes target rotation is examined in the context of meteorological/climatological applications. This includes a discussion of the six unique ways to decompose a rotated data set in order to maximize the physical interpretability of the rotated results.The various analytic simple structure rotations available are compared by a Monte Carlo simulation, which is a modification of a similar technique developed by Tucker (1983), revealing that the DAPPFR and Promax k = 2 rotations are the most accurate in recovering the input structure of the modes of variation over a wide range of conditions. Additionally, these results allow the investigator the opportunity to check the accuracy of the unrotated or rotated solution for specific types of data. This is important because, in the past, the decision of whether or not to apply a specific rotation has been a ‘blind decision’. In response to this, a methodology is presented herein by which the researcher can assess the degree of simple structure embedded within any meteorological data set and then apply known information about the data to the Monte Carlo results to optimize the likelihood of achieving physically meaningful results from a principal component analysis.

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Tài liệu tham khảo

10.1016/0004-6981(84)90262-2

Ashbaugh L. L., 1985, Authors' reply to discussion on A principal component analysis of sulfer concentrations in the western United States, Atmos. Environ., 19

10.1007/BF02423313

Barnston A. G.andLivezey R. E.1985. “High resolution rotated empirical orthogonal function analysis of Northern Hemisphere 700 mb heights for predictive purposes” Preprints Ninth Conf. on Prob. and Stat. in Atmos. Sci. Virginia Beach VA Amer. Meteor. Soc. 290.

10.1111/j.2153-3490.1980.tb00956.x

BMDP., 1981, Biomedical Computer Programs, 94720

10.1175/1520-0450(1976)015<1307:EAFPOT>2.0.CO;2

10.1175/1520-0493(1981)109<1305:SLPPOE>2.0.CO;2

Buell C. E.1975. “The topography of empirical orthogonal functions” Preprints Fourth Conf. on Prob. and Stats in Atmos. Sci. Tallahassee FL Amer. Meteor. Soc.188.

Buell C. E.1979. “On the physical interpretation of empirical orthogonal functions” Preprints Sixth Conf. on Prob. and Stats. in Atmos. Sci. Banff Alta. Amer. Meteor. Soc. 112.

Cahalan R. F.1983. “EOF spectral estimation in climate analysis” Preprints Second International Conf. on Stat. Climat. Lisbon Portugal National Institute of Meteor and Geophysics 4.5.1.

10.1007/BF02289025

10.1126/science.126.3283.1114

Cattell R. B., 1952, Factor Analysis

10.1207/s15327906mbr0102_10

Cattell R. B., 1966, Handbook of Multivariate Experimental Psychology

10.1002/bs.3830080211

Cattell R. B., 1960, The “maxplane” program for factor rotation to oblique simple structure, Educ. and Psych. Measur., 20, 269

10.1175/1520-0493(1966)094<0697:AIOTPO>2.3.CO;2

10.1037/h0025178

10.1175/1520-0450(1983)022<1975:COMHAU>2.0.CO;2

10.2307/2987388

10.1002/qj.49709540510

Crawford C. B., 1967, A general method of rotation for factor analysis

Daultrey S., 1976, Principal Components Analysis

Davies W. K. D., 1971, Varimax and the destruction of generality. A methodological note, Area, 3, 112

Davies W. K. D., 1971, Varimax and generality: A reply, Area, 3, 254

Davies W. K. D., 1972, Varimax and generality: A second reply, Area, 4, 207

10.1175/1520-0485(1976)006<0249:POSSTA>2.0.CO;2

10.1175/1520-0493(1981)109<1285:EAOSTP>2.0.CO;2

10.1175/1520-0493(1981)109<1267:EAOSTP>2.0.CO;2

10.1002/qj.49710143020

10.1002/qj.49710544408

10.1007/BF02288367

10.1175/1520-0450(1985)024<0350:ASAOPF>2.0.CO;2

Fukuoka A., 1951, A study of 10‐day forecast (a synthetic report), Geophys. Mag., 22, 177

10.1175/1520-0450(1978)017<0600:IOASIT>2.0.CO;2

Gilman D. L.1957.Empirical Orthogonal Functions Applied to Thirty‐Day Forecasting. M.I.T. Dept. of Meteorology Science Report No. 1. Contract AF19(604)‐1283.

10.2307/2987387

10.1002/joc.3370010306

10.1002/j.1477-8696.1975.tb05315.x

10.1111/j.2044-8317.1974.tb00531.x

10.1007/BF02291667

10.1207/s15327906mbr1702_3

Harman H. H., 1976, Modern Factor Analysis

10.1007/BF02289601

10.1111/j.2153-3490.1979.tb00879.x

Hayden B. P.1983. “Prediction of seasonal temperatures based on persistences in cyclone frequency EOFs” Preprints Second International Conf. on Stat. Climat. Lisbon Portugal National Institute of Meteor and Geophysics 13.3.1.

10.1111/j.2044-8317.1964.tb00244.x

10.1175/1520-0450(1969)008<0701:AAMFEC>2.0.CO;2

10.1175/1520-0493(1981)109<2080:ARPCAO>2.0.CO;2

10.1175/1520-0450(1984)023<1660:CPCATA>2.0.CO;2

10.1007/BF02289447

10.1037/h0071325

Hsuing J., 1983, The principal nonseasonal modes of variation of global sea surface temperature, J. Phys. Oceanogr., 13

10.1207/s15327906mbr1002_5

10.1002/bs.3830070216

IMSL., 1979, International Mathematical and Statistical Laboratory

10.1007/BF02289465

10.1007/BF02263243

10.1007/BF02289233

10.1177/001316445901900314

10.1207/s15327906mbr0904_9

10.1207/s15327906mbr1304_2

Kaiser H. F., 1959, Analytic determination of common factors, Amer. Psych., 14, 425

10.1007/BF02294056

10.1002/joc.3370020402

10.1175/1520-0450(1982)021<1183:PEITAO>2.0.CO;2

10.1007/BF02291576

Kendall M. G., 1980, Multivariate Analysis

10.1007/BF02266917

10.1007/BF02291763

10.1175/1520-0450(1967)006<0791:EEOSLP>2.0.CO;2

10.1175/1520-0493(1970)098<0708:LSFOMM>2.3.CO;2

Lamb P. J.andRichman M. B.1983a. “Regionalization of central United States for short‐period summer rainfall” Proc. Seventh Climate Diagnostics Workshop Boulder CO 180.

Lamb P. J.andRichman M. B.1983b. “An analysis of the space and time variation of growing season rainfall in the central United States” Preprints Eighth Conf. on Prob. and Stats in Atmos. Sci. Hot Springs AR Amer. Meteor. Soc. 49.

Lamb P. J.andRichman M. B.1986. “On the modes of variation of growing season rainfall in the central United States” to be submitted to Mon. Wea. Rev.

10.1175/1520-0493(1980)108<1992:EAOTVV>2.0.CO;2

10.1007/BF02324658

10.1175/1520-0450(1985)024<0463:SVITUS>2.0.CO;2

Lorenz E. N.1956. “Empirical orthogonal functions and statistical weather prediction” Sci. Rep. 1 Statistical Forecasting Project Dept. of Meteor. Mass. Institute of Technology [NTIS AD 110268].

Mather P. M., 1971, Varimax and generality: A comment, Area, 3, 252

Mather P. M., 1972, Varimax and generality: A second comment, Area, 4, 27

10.1007/BF02423312

10.1175/1520-0450(1983)022<1738:PONIAD>2.0.CO;2

10.1029/WR015i006p01841

Mulaik S. A., 1972, The Foundations of Factor Analysis

10.1111/j.2044-8317.1954.tb00147.x

10.1175/1520-0469(1984)041<0879:EOFANM>2.0.CO;2

10.1175/1520-0493(1982)110<0699:SEITEO>2.0.CO;2

10.1175/1520-0493(1982)110<0001:ASTFPC>2.0.CO;2

Preisendorfer R. W., 1977, Preprints Fifth Conf. on Prob. and Stats. in Atmos. Sci., 169

Preisendorfer R. W., 1981, Foundations of Principal Component Selection Rules

10.1175/1520-0450(1975)014<1223:RSVIMA>2.0.CO;2

10.1175/1520-0450(1981)020<1145:ORPCAI>2.0.CO;2

Richman M. B., 1983, Preprints Second International Conf, on Stat. Climate., 5

Richman M. B.1983b. “Rotation of principal components in climatological research. Part 2: how the various analytic simple structure rotations on the major statistical packages compare on different types of data” Preprints Eighth Conf. on Prob. and Stat. in Atmos. Sci. Hot Springs AR Amer. Meteor. Soc. 115.

Richman M. B.1985. “The eigentechniques pseudo‐random distributions and the interpretation of spurious signal” Preprints Ninth Conf. on Prob. and Stats. in Atmos. Sci. Virginia Beach VA Amer. Meteor. Soc. 262.

10.1175/1520-0450(1985)024<1325:CPAOTA>2.0.CO;2

Richman M. B.andWalsh J. E.1985. “Hemispheric pattern analysis of weekly 500mb data” Preprints Ninth Conf. on Prob. and Stats. in Atmos. Sci. Virginia Beach VA Amer. Meteor. Soc. 276.

10.1175/1520-0493(1976)104<0985:SSTAIT>2.0.CO;2

Ronberg B.andWang W.‐C.1985. “Climate regions derived from Chinese long‐term precipitation records” Preprints Ninth Conf. on Prob. and Stats. in Atmos. Sci. Virginia Beach VA Amer. Meteor. Soc. 301.

Rummel R. J., 1970, Applied Factor Analysis

10.1175/1520-0493(1980)108<1892:NZCIPP>2.0.CO;2

SAS., 1982, SAS Institute Inc. User's Guide: Statistics

10.1007/BF02289800

Saunders D. R., 1962, Trans‐Varimax: Some properties of the Ratiomax and Equamax criteria for blind orthogonal rotation, Amer. Psych., 17, 395

10.1007/BF02289451

Sellers W. D.1957. “A statistical‐dynamic approach to numerical weather prediction” M.I.T. Department of Meteorology Scientific Report No. 2 Statistical Forecasting Project Contract AF19(604)‐1566.

SPSS., 1979, Statistical Package for the Social Sciences

10.1175/1520-0450(1985)024<0716:SAOEPV>2.0.CO;2

10.1002/joc.3370010307

10.1175/1520-0450(1985)024<1245:AMAOMI>2.0.CO;2

Thurstone L. L., 1947, Multiple Factor Analysis

Tucker L. R., 1940, A rotational method based upon the mean principal axis of a subgroup of tests, Psych. Bull., 37, 578

10.1007/BF02288713

10.1007/BF02289018

Tucker L. R.1983. Personal communication.

Tucker L. R.andFinkbeiner C. T.1982. “Transformation of factors by artificial personal probability functions” ETS research report 81–58 test and measurement no. TM 820429 [Available from ERIC clearinghouse Educational Testing Service Princeton NJ 08541 U.S.A.].

10.1007/BF02290601

10.1175/1520-0450(1985)024<0016:ACSFST>2.0.CO;2

Vargas W. M., 1983, Preprints Second International Conf. on Stat. Climat., 5

10.1002/qj.49709138808

10.1175/1520-0493(1980)108<0615:AQAOMA>2.0.CO;2

10.1175/1520-0493(1981)109<0767:SITABS>2.0.CO;2

10.1175/1520-0493(1982)110<0272:SCOMPI>2.0.CO;2

10.1175/1520-0493(1983)111<1838:ICAOLR>2.0.CO;2

10.1002/joc.3370040102

10.1007/BF02263459

10.1007/BF02243232

10.1207/s15327906mbr1702_5