Principal components analysis of three-way tables

  • Michael Edward Hohn
Article

Abstract

A three-mode principal components method allows visualization of the structural or taxonomic relationships within three-way data tables. The fundamental model includes three sets of eigenvectors and a “core matrix” relating the principal components of each mode. Formal relationships between the method and the usual principal components formulation allow calculation of “loadings” and “scores” for each mode; taken with the core matrix, these provide a number of points of view in graphical analysis of three-mode data. The model compares favorably with alternative formulations in terms of simplicity of computation, generality, and symmetry of operation among the modes. An organic geochemical example illustrates the method.

Key words

principal components analysis factor analysis organic geochemistry paleoecology petrology 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bartussek, D., 1973, Zur interpretation der Kernmatrix in der dreimodalen Factorenanalyse von R. L. Tucker: Psychol. Beiträge, v. 15, no. 2, p. 169–184.Google Scholar
  2. Benzécri, J. P., 1970, Distance distributionelle et métrique du chideux en analyse factorielle des correspondances: Laboratoire de Statistique Mathématique, Université de Paris, 6. 3e édition.Google Scholar
  3. Bouroche, J.-M. and Dussaix, A. M., 1975, Several alternatives for three-way data analysis: Metra, v. 14, no. 2, p. 299–319.Google Scholar
  4. Bradu, D. and Gabriel, K. R., 1978, The biplot as a diagnostic tool for models of two-way tables: Technometrics, v. 20, no. 1, p. 47–68.Google Scholar
  5. Carroll, J. D. and Chang, J. J., 1970, Analysis of individual differences in multidimensional scaling via aN-way generalization of “Eckart—Young” decomposition: Psychometrika, v. 35, no. 3, p. 238–319.Google Scholar
  6. Dixon, W. J. and Massey, F. J., Jr., 1969, Introduction to statistical analysis: McGraw-Hill, New York, 638 p.Google Scholar
  7. Gleason, T. C. and Staelin, R., 1975, A proposal for handling missing data: Psychometrika, v. 40, no. 2, p. 229–252.Google Scholar
  8. Gower, J. C., 1975, Generalized Procrustes analysis: Psychometrika, v. 40, no. 1, p. 35–51.Google Scholar
  9. Halket, J. Mck. and Reed, R. I., 1975, The analysis of mixtures: Org. Mass Spectrometry, v. 10, no. 9, p. 808–812.Google Scholar
  10. Jöreskog, K. G., Klovan, J. E., and Reyment, R. A., 1976, Geological factor analysis: Elsevier, Amsterdam, 178 p.Google Scholar
  11. Kaiser, H. F., 1958, The varimax criterion for analytic rotation in factor analysis: Psychometrika, v. 23, no. 3, p. 187–200.Google Scholar
  12. Levin, K., 1965, Three-mode factor analysis: Psychol. Bull., v. 64, no. 6, p. 442–452.Google Scholar
  13. Malinowski, E. R. and McCue, M., 1977, Qualitative and quantitative determination of suspected components in mixtures by target transformation factor analysis of their mass spectra: Anal. Chem., v. 49, no. 2, p. 282–287.Google Scholar
  14. Mandel, J., 1972, Principal components, analysis of variance and data structure: Statistica Neerlandica, v. 26, no. 3, p. 119–129.Google Scholar
  15. Neter, J. and Wasserman, W., 1974, Applied linear statistical models: Irwin, Homewood, Illinois, 842 p.Google Scholar
  16. Oudin, J. L., 1979, Analyse géochimique de la matière organique extraite des roches sédimentaires. I. Composés extractibles au chloroforme: Inst. Français Petròle Rev., v. 25, no. 1, p. 3–15.Google Scholar
  17. Pease, M. C., III., 1965, Methods of matrix algebra: Academic Press, New York, 406 p.Google Scholar
  18. Philp, R. P., Brown, S., and Calvin, M., 1978, Isoprenoid hydrocarbons produced by thermal alteration ofNostroc moscorum andRhodopseudomonas spheroides: Geochim. Cosmochim. Acta, v. 42, no. 1, p. 63–68.Google Scholar
  19. Saxena, S. K., 1969, A statistical approach to the study of phase equilibria in multicomponent systems: Lithos, v. 3, no. 1, p. 25–36.Google Scholar
  20. Seifert, W. G. and Moldowan, J. M., 1978, Application of steranes, terpanes and monoaromatics to the maturation, migration and source of crude oils: Geochim. Cosmochim. Acta, v. 42, no. 1, p. 77–95.Google Scholar
  21. Shoenfeld, P. S. and De Voe, J. R., 1976, Statistical and mathematical methods in analytical chemistry: Anal. Chem., v. 48, no. 5, p. 403R-411R.Google Scholar
  22. Temple, J. T., 1978, The use of factor analysis in geology: Math. Geol., v. 10, no. 4, p. 379–387.Google Scholar
  23. Tissot, B., Califet-Debyser, Y., Deroo, G., and Oudin, J. L., 1971, Origin and evolution of hydrocarbons in Early Toarcian shales, Paris Basin, France: Am. Assoc. Pet. Geol. Bull., v. 55, no. 12, p. 2177–2193.Google Scholar
  24. Tucker, L. R., 1964, The extension of factor analysis to three-dimensional matrices,in Contributions to mathematical psychology, Eds.: Frederikson, N. and Gulliksen, H: Holt, Rinehart and Winston, New York, p. 109–127.Google Scholar
  25. Tucker, L. R., 1966, Some mathematical notes on three-mode factor analysis: Psychometrika, v. 31, no. 3, p. 279–311.Google Scholar
  26. Wainer, H., Gruvaeus, G., and Blair, M., 1974, TREBIG: A 360/75 FORTRAN program for three-mode factor analysis designed for big data sets: Behav. Res. Methods Instr., v. 6, no. 1, p. 53–54.Google Scholar
  27. Wakatsuki, T., Farukawa, H., and Kyuma, K., 1977, Geochemical study of the redistribution of elements in soil. I. Evaluation of major elements among the particle size fractions and soil extract: Geochim. Cosmochim. Acta, v. 41, no. 7, p. 891–902.Google Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • Michael Edward Hohn
    • 1
  1. 1.Organic Geochemistry Unit, School of ChemistryThe University of BristolBristolEngland

Personalised recommendations