On the Harker Variation Diagrams; A Comment on “The Statistical Analysis of Compositional Data. Where Are We and Where Should We Be Heading?” by Aitchison and Egozcue (2005)

The current theoretical development of the analysis of compositional data in the article by Aitchison and Egozcue neglects the use of Harker’s variation diagrams and other similar plots as “meaningless” or “useless” on compositional data. In this work, it is shown that variation diagrams essentially are not a correlation tool but a graphical representation of the mass actions and mass balances principles in the context of a given geological system, and, when they are used correctly, they provide vital information for the igneous petrologist. The qualitative validity of the “spurious trends” in these diagrams is also shown, when they are interpreted in their proper geological framework. The example previously used by Rollinson to test the usefulness of the log-ratio transformation in the Aitchison and Egozcue article is revisited here in order to fully illustrate the proper use of this tool.

[1]  J. Aitchison,et al.  Logratio Analysis and Compositional Distance , 2000 .

[2]  H. Rollinson.  Another look at the constant sum problem in geochemistry , 1992, Mineralogical Magazine.

[3]  Paul D. Asimow,et al.  Algorithmic modifications extending MELTS to calculate subsolidus phase relations , 1998 .

[4]  The interpretation of igneous rocks , 1979 .

[5]  H. Kuno.  Differentiation of basalt magmas , 1968 .

[6]  T. Irvine,et al.  A Guide to the Chemical Classification of the Common Volcanic Rocks , 1971 .

[7]  V. Pawlowsky-Glahn,et al.  Compositional data and their analysis: an introduction , 2006, Geological Society, London, Special Publications.

[8]  J. Aitchison.  Logratios and Natural Laws in Compositional Data Analysis , 1999 .

[9]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[10]  J. L. Palma,et al.  Deciphering magma mixing: The application of cluster analysis to the mineral chemistry of crystal populations , 2007 .

[11]  M. Baxter,et al.  LOG-RATIO COMPOSITIONAL DATA ANALYSIS IN ARCHAEOMETRY* , 2006 .

[12]  F. Spera.  Energy-constrained open-system magmatic processes I : General model and energy-constrained assimilation and fractional crystallization (EC-AFC) formulation , 2001 .

[13]  F. Chayes.  On correlation between variables of constant sum , 1960 .

[14]  J. D. Winter.  An Introduction to Igneous and Metamorphic Petrology , 2001 .

[15]  John Aitchison,et al.  The Statistical Analysis of Compositional Data , 1986 .

[16]  Mark S. Ghiorso,et al.  Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures , 1995 .

[17]  J. Cortés,et al.  The evolution of the magmatic system of Stromboli volcano during the Vancori period (26–13.8 ky) , 2005 .

[18]  F. Spera,et al.  Open-System Magma Chamber Evolution: an Energy-constrained Geochemical Model Incorporating the Effects of Concurrent Eruption, Recharge, Variable Assimilation and Fractional Crystallization (EC-E′RAχFC) , 2003 .

[19]  R. Shurtz,et al.  Compositional Geometry and Mass Conservation , 2003 .

[20]  H. H. Hess,et al.  Basalts : the Poldervaart treatise on rocks of basaltic composition , 1967 .

[21]  C. Fenner.  The Residual Liquids of Crystallizing Magmas , 1931 .

[22]  William Miller,et al.  Revisiting the Geometry of a Ternary Diagram with the Half-Taxi Metric , 2002 .

[23]  F. Chayes,et al.  Numerical Correlation and Petrographic Variation , 1962, The Journal of Geology.

[24]  K. Pearson.  Mathematical contributions to the theory of evolution.—On a form of spurious correlation which may arise when indices are used in the measurement of organs , 1897, Proceedings of the Royal Society of London.

[25]  A. N. Sadovski.  Algorithm AS 74: L1-norm Fit of a Straight Line , 1974 .

[26]  Alfred Harker,et al.  The Natural History of Igneous Rocks , 2009 .

[27]  J. Aitchison,et al.  Compositional Data Analysis: Where Are We and Where Should We Be Heading? , 2003 .

[28]  C. M. Jackson,et al.  Compositional Data Analysis of Some Alkaline Glasses , 2005 .

[29]  M. G. Best.  Igneous and Metamorphic Petrology , 1982 .

[30]  G. Mateu-Figueras,et al.  Compositional Data Analysis in the Geosciences: From Theory to Practice , 2006 .