Mathematical Geology

, Volume 22, Issue 8, pp 933–956 | Cite as

Descriptive statistics forN-dimensional closed arrays: A spherical coordinate approach

  • Clifford R. Stanley
Articles

Abstract

Recent publications have demonstrated that the composition of a system can be described by the angles which define the vector resulting from the projection of the composition onto the unit hypersphere. Although published trigonometric relationships allow determination of the angles in any number of dimensions, no general hyperdimensional recursive formulae exist for the calculation of these quantities and their variances and covariances. A general methodology for calculating the angles which describe the compositions of systems in any number of dimensions is presented. These angles can be used to calculate statistics (central moments and resultant vectors) describing the central tendency and dispersion among rock compositions, as well as to quantify the angular differences between compositions. Equations that relate the variances and covariances of these angular variables to the variances and covariances of the actual component proportion variables are presented, allowing projection of these measurements from one reference frame to another (cartesian to spherical and vice versa).

Key words

closed array spherical coordinates propogation of variance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aitchison, J., 1981, A New Approach to Null Correlations of Proportions. Math. Geol. v. 13, p. 175–189.Google Scholar
  2. Aitchison, J., 1982, The Statistical Analysis of Compositional Data. Jour. Roy. Stat. Soc. B, v. 44, p. 139–177.Google Scholar
  3. Aitchison, J., 1984a, The Statistical Analysis of Geochemical Compositions. Math. Geol. v. 16, p. 532–564.Google Scholar
  4. Aitchison, J., 1984b, Reducing the Dimensionality of Compositional Data Sets. Math. Geol., v. 16, p. 617–635.Google Scholar
  5. Aitchison, J., 1986, The Statistical Analysis of Compositional Data. Chapman and Hall, New York, 416 p.Google Scholar
  6. Aitchison, J., 1989, Reply to “Interpreting and Testing Compositional Data”, by Woronow, A., Love, K. M. and Butler, J. C. Math. Geol., 21, p. 65–71.Google Scholar
  7. Butler, J. C., 1979, Trends on Ternary Petrologic Diagrams—Fact or Fantasy? Am. Mineralogist. v. 64, p 1115–1121.Google Scholar
  8. Carmichael, I. S. E, Hampel, J., and Jack, R. N., 1968, Analytical Data on the U.S.G.S. Standard Rocks. Chem. Geol., v. 3, p 59–64.Google Scholar
  9. Chayes, F., 1949, On Ratio Correlation in Petrology. Jour. Geol., v. 57, p. 239–254.Google Scholar
  10. Chayes, F., 1960, On Correlation Between Variables of Constant Sum. Jour. Geophys. Res., v. 65, p. 4185–4193.Google Scholar
  11. Chayes, F., 1962, Numerical Correlation and Petrographic Variation. Jour. Geol., v. 70, p 440–452.Google Scholar
  12. Chayes, F., and Kruskal, W., 1966, An Approximate Statistical Test for Correlations Between Proportions. Jour. Geol., v. 74, p. 692–702.Google Scholar
  13. Cheeney, R. F., 1983, Statistical Methods in Geology. George Allen and Unwin, Boston, 169 p.Google Scholar
  14. Davis, J. C., 1988, Statistics and Data Analysis in Geology, 2nd Ed: John Wiley & Sons, New York, 646 p.Google Scholar
  15. Fisher, N. I., Lewis, T., and Embleton, B. J. J., 1987, Statistical Analysis of Spherical Data. Cambridge University Press, New York, 329 p.Google Scholar
  16. Fisher, R., 1952, Dispersion on a Sphere. Roy. Soc. London Proc. A, v. 217, p. 295–305.Google Scholar
  17. Imbrie, J., 1963, Factor and Vector Analysis Programs for Analyzing Geologic Data. Office of Naval Research, Geography Branch, Technical Report No. 6 (ONR Task No. 389–135), 83 p.Google Scholar
  18. Imbrie, J., and Purdy, E. G., 1962, Classification of Modern Bahamian Carbonate Sediments,in “Classification of Carbonate Rocks—A Symposium,” American Association of Petroleum Geologists, Memoir 1, p. 253–272.Google Scholar
  19. Imbrie, J., and Van Andel, T. H., 1964, Vector Analysis of Heavy Mineral Data. Geol. Soc. Am. Bull., v. 75, p. 1131–1155.Google Scholar
  20. Ingamells, C. O., and Pitard, F. F., 1986, Applied Geochemical Analysis: John Wiley & Sons, New York, 733 p.Google Scholar
  21. Kendall, M. G., and Stuart, A., 1958, The Advanced Theory of Statistics, Vol. 1, Distribution Theory: Charles Griffin & Co., London, 433 p.Google Scholar
  22. 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, pp. 61–77.Google Scholar
  23. Klovan, J. E., and Miesch, A. T., 1975, Extended CABFAC and QMODEL Computer Programs for Q-Mode Factor Analysis of Compositional Data; Comput. Geosci. v. 1, p. 161–178.Google Scholar
  24. Kretz, R., 1985, Calculation and Illustration of Uncertainty in Geochemical Analyses: Jour. Geol. Ed., v. 33, p. 40–44.Google Scholar
  25. Le Maitre, R. W., 1982, Numerical Petrology: Statistical Interpretation of Geochemical Data: Elsevier Scientific, Amsterdam, 281 p.Google Scholar
  26. Miesch, A. T., 1969, The Constant Sum Problem in Geochemistry,in Merriam, D. F. (Ed.), Computer Applications in Earth Sciences: Plenum Press, New York, p. 161–167.Google Scholar
  27. Miesch, A. T., 1975, Q-Mode Factor Analysis of Compositional Data: Comput. Geosci., v. 1, p. 147–160.Google Scholar
  28. Miesch, A. T., 1976, Q-Mode Factor Analysis of Geochemical and Petrologic Data Matrices with Constant Row-Sums: U.S. Geologic Survey Professional Paper No. 574-G, 47 p.Google Scholar
  29. Meyer, S. L., 1975, Data Analysis for Scientists and Engineers: John Wiley & Sons, New York, 513 p.Google Scholar
  30. Murata, K. J., and Richter, D. H., 1961, Magmatic Differentiation in the Uwekahuna Laccolith, Kilauea Caldera, Hawaii: Journ. Petrology, v. 2, p. 424–437.Google Scholar
  31. Nicholls, J., 1988, The Statistics of Pearce Element Diagrams and the Chayes Closure Problem: Contribut. Mineral. Petrol., v. 99, pp. 36–43.Google Scholar
  32. Pearce, T. H., 1968, A Contribution to the Theory of Variation Diagrams: Contribut. Mineral. Petrol, v. 19, p. 142–157.Google Scholar
  33. Pearson, K., 1897, Mathematical Contribution to the Theory of Evolution: On a Form of Spurious Correlation Which May Arise When Indices are Used in the Measurement of Organs: Roy. Soc. London Proc. B, v. 60, p. 489–498.Google Scholar
  34. Philip, G. M., and Watson, D. F., 1988, Angles Measure Compositional Differences: Geology, v. 16, p. 976–979.Google Scholar
  35. Russell, J. K., and Nicholls, J., 1988, Analysis of Petrologic Hypotheses with Pearce Element Ratios: Contribut. Mineral. Petrol., v. 99, p. 25–35.Google Scholar
  36. Russell, J. K., Nicholls, J., Stanley, C. R., and Pearce, T. H., 1990, Pearce Element Ratios: A Paradigm for the Testing of Petrologic Hypotheses: EOS, v. 71, p. 234–247.Google Scholar
  37. Russell, J. K., and Stanley, C. R., in press, Theoretical Considerations for the Development and Use of Chemical Variation Diagrams: Geochimica et Cosmochimica Acta (submitted).Google Scholar
  38. Russell, J. K., and Stanley, C. R., 1990a, Material Transfer Equations and Chemical Variation Diagrams, GAC Short Course Notes, “Theory and Application of Pearce Element Ratios to Geochemical Data Analysis”: Geological Association of Canada Annual Meeting, May, Vancouver, p. 55–85.Google Scholar
  39. Russell, J. K., and Stanley, C. R., 1990b, Origins of the 1954–1960 Lavas, Kilauea Volcano, Hawaii: Major Element Constraints on Shallow Reservoir Magmatic Processes: Jour. Geophys. Res., v. 95, p. 5021–5047.Google Scholar
  40. Stanley, C. R., 1990, Error Propagation and Regression on Pearce Element Ratio Diagrams, GAC Short Course Notes, “Theory and Application of Pearce Element Ratios to Geochemical Data Analysis”: Geological Association of Canada Annual Meeting, May, Vancouver, p. 179–215.Google Scholar
  41. Stanley, C. R., and Russell, J. K., 1989a, Petrologic Hypothesis Testing with Pearce Element Ratio Diagrams: Derivation of Diagram Axes: Contribut. Mineral. Petrol., v. 103, p. 78–89.Google Scholar
  42. Stanley, C. R., and Russell, J. K., 1989b, PEARCE.PLOT: A Turbo-Pascal Program for the Analysis of Rock Compositions with Pearce Element Ratio Diagrams: Comput. Geosci., v. 15, p. 905–926.Google Scholar
  43. Stanley, C. R., and Russell, J. K., 1989c, PEARCE.PLOT: Interactive Graphics-Supported Software for Testing Petrologic Hypotheses with Pearce Element Ratios: Am. Mineral., v. 74, p. 273–276.Google Scholar
  44. Stanley, C. R., and Russell, J. K., 1990, Matrix Methods for the Development of Pearce Element Ratio Diagrams, GAC Short Course Notes, “Theory and Application of Pearce Element Ratios to Geochemical Data Analysis:” Geological Association of Canada Annual Meeting, May, Vancouver, p. 131–156.Google Scholar
  45. Thompson, J. B., 1982a, Composition Space: An Algebraic and Geometric Approach,in Ferry, J. M. (ed.), Characterization of Metamorphism Through Mineral Equilibria, Reviews in Mineralogy, v. 10; Mineralogical Society of America, p. 1–31.Google Scholar
  46. Thompson, J. B., 1982b, Reaction Space: An Algebaic and Geometric Approach,in Ferry, J. M. (ed.), Characterization of Metamorphism Through Mineral Equilibria, Reviews in Mineralogy, Vol. 10: Mineralogical Society of America, p. 32–52.Google Scholar
  47. Vistelius, A. B., and Sarmonov, O. V., 1961, On the Correlation Between Percentage Values: Major Component Correlation in Ferro-magnesium Micas: Jour. Geol., v. 69, p. 145–153.Google Scholar
  48. Watson, D. F., and Philip, G. M., 1989, Measures of Variability for Geologic Data: Math. Geol., v. 21, p. 233–254.Google Scholar
  49. Watson, G. S., 1983, Statistics on Spheres: John Wiley & Sons, New York, 238 p.Google Scholar
  50. Woronow, A., and Butler, J. C., 1986, Complete subcompositional Independence Testing of Closed Arrays: Comput. Geosci., v. 12, pp. 267–280.Google Scholar
  51. Woronow, A., Love, K. M., and Butler, J. C., 1989, Interpreting and Testing Compositional Data: Math. Geol., v. 21, p. 61–64.Google Scholar

Copyright information

© International Association for Mathematical Geology 1990

Authors and Affiliations

  • Clifford R. Stanley
    • 1
  1. 1.Department of Geology and GeophysicsThe University of CalgaryCalgaryCanada
  2. 2.Department of Geological SciencesQueen's UniversityKingstonCanada

Personalised recommendations