Abstract
It is mathematically possible to extract both R-mode and Q-mode factors simultaneously (RQ-mode factor analysis)by invoking the Eckhart-Young theorem. The resulting factors will be expressed in measures determined by the form of the scalings that have been applied to the original data matrix. Unless the measures for both solutions are meaningful for the problem at hand, the factor results may be misleading or uninterpretable. Correspondence analysis uses a symmetrical scaling of both rows and columns to achieve measures of proportional similarity between objects and variables. In the literature, the resulting similarity is a χ 2 distance appropriate for analysis of enumerated data, the original application of correspondence analysis. Justification for the use of this measure with interval or ratio data is unconvincing, but a minor modification of the scaling procedure yields the profile similarity, which is an appropriate measure. Symmetrical scaling of rows and columns is unnecessary for RQ-mode factor analysis. If the data are scaled so the minor product W'Wis the correlation matrix, the major product WW'is expressed in the Euclidean distances between objects. Therefore, RQ-mode factor analysis can be performed so that the Rmode is a principal components solution and the Qmode is a principal coordinates solution. For applications where the magnitudes of differences are important, this approach will yield more interpretable results than will correspondence analysis.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benzecri, Jean-Paul, 1969, Statistical analysis as a tool to make patterns emerge from data,in S. Watanabe (Ed.) Methodologies of pattern recognition: Academic Press, New York, p. 35–74.
Benzecri, Jean-Paul, F. Benzécri, A. Birou, S. Blumenthal, A. De Bœck, J-P. Bordet, G. Cancelier, P. Cazes, F. da Costa Nicolau, M. Danech-Pajouh, R. Delprat, M. Demonet, B. Escoffier, A. Forcade, Fr. Friant, Y. Grelet, D. Kalogéropoulos, L. Lebart, M.-O. Lebeaux, P. Leroy, J.-F. Marcotorchino, T. Moussa, F. Mutombo, Ch. Nora, A. Prost, A. Rezvani, J. Robert, Ch. Rosenzveig, M. Roux, P. Solety, S. Stépan, N. Tabard, N. Tabet, G. Thauront, M. de Virville, and Y. Vuillaume, 1980, L'Analyse des données, Vol. 2, L'Analyse des Correspondances: Dunod, Paris.
David, M., Dagbert, M., and Beauchemin, Y., 1977, Statistical analysis in geology: Correspondence analysis method: Quart. Colorado Sch. Min., v. 72, no. 1, 57 p.
Eckart, C. and Young, B., 1936, The approximation of one matrix by another of lower rank: Psychometrika, v. 1, no. 3, p. 211–218.
Gabriel, K. R., 1971, The biplot graphic display of matrices with application to principal component analysis: Biometrika, v. 58, no. 3, p. 453–467.
Gower, J. C., 1966, Some distance properties of latent root and vector methods used in multivariate analysis: Biometrika, v. 53, no. 3, 4, p. 325–338.
Gower, J. C., 1967, Multivariate analysis and multidimensional geometry: The Statistician, v. 17, no. 1, p. 13–18.
Hill, M. O., 1974, Correspondence analysis: A neglected multivariate method: Jour. Roy. Stat. Soc., Ser. C: Appl. Stat., v. 23, no. 3, p. 340–354.
Hotelling, H., 1933, Analysis of a complex of statistical variables into principal components: Jour. Educ. Psych., v. 24, p. 417–441, 498–520.
Howarth, R. J., 1973, Preliminary assessment of a nonlinear mapping algorithm in a geological context: Math. Geol., v. 5, no. 1, p. 39–57.
Imbrie, J., 1963, Factor and vector analysis program for analyzing geologic data: Technical Report No. 6, Office of Naval Research, Geography Branch, Northwestern University, 83 p.
Jambu, M., 1980, Cluster analysis for data analysis, 1. Methods: Unpublished manuscript, 328 p.
Johnson, R. M., 1963, On a theorem stated by Eckart and Young: Psychometrika, v. 1, no. 3, p. 259–263.
Jöreskog, K. G., Klovan, J. E., and Reyment, R. A., 1976, Geological factor analysis, methods in geomathematics, 1: Elsevier Scientific Publishing Company, Amsterdam, 178 p.
Klovan, J. E., and Imbrie, J., 1971, An algorithm and FORTRAN IV program for large-scale Q-mode factor analysis and calculation of factor scores:Math. Geol., v. 3, no. 1, p. 61–77.
Krumbein, W. C., 1962, Open and closed number systems in stratigraphic mapping: Bull. Amer. Assoc. Pet. Geol., v. 46, p. 2229–2245.
Krumbein, W. C. and Imbrie, J., 1963, Stratigraphic factor maps: Bull. Amer. Assoc. Pet. Geol., v. 47, p. 698–701.
Lee, P. J., 1969, FORTRAN IV programs for canonical correlation and canonical trend surface analysis: Kansas Geol. Surv. Comput. Contrib., v. 32, 46 p.
Manson, V. and Imbrie, J., 1964, FORTRAN program for factor and vector analysis of geologic data using an IBM 7090 or 7094/1401 computer system: Kansas Geol. Surv. Spec. Distrib. Publ. 13, 46 p.
Miesch, A. T., 1980, Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data: Jour. Math. Geol., v. 12, no. 6, p. 523–538.
Sherman, K. N., Bunker, C. M., and Bush, C. A., 1971, Correlation of uranium, thorium, and potassium with aeroradioactivity in the Berea area, Virginia: Econ. Geol., v. 66, p. 302–308.
Teil, H., 1975, Correspondence factor analysis: An outline of its method: Math. Geol., v. 7, no. 1, p. 3–12.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zhou, D., Chang, T. & Davis, J.C. Dual extraction ofR-mode andQ-mode factor solutions. Mathematical Geology 15, 581–606 (1983). https://doi.org/10.1007/BF01093413
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01093413