Compositional Data Analysis: Where Are We and Where Should We Be Heading?

We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations.

[1]  J. Atchison,et al.  Logistic-normal distributions:Some properties and uses , 1980 .

[2]  J. A. Martín-Fernández,et al.  MARKOV CHAIN MONTE CARLO METHOD APPLIED TO ROUNDING ZEROS OF COMPOSITIONAL DATA: FIRST APPROACH , 2003 .

[3]  G. Mateu-Figueras,et al.  Isometric Logratio Transformations for Compositional Data Analysis , 2003 .

[4]  S. Shen,et al.  The statistical analysis of compositional data , 1983 .

[5]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

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

[7]  C. Raghavendra Rao,et al.  Multivariate Analysis and Its Applications , 1984 .

[8]  J. Aitchison On criteria for measures of compositional difference , 1992 .

[9]  R. Renner The resolution of a compositional data set into mixtures of fixed source compositions , 1993 .

[10]  John Aitchison,et al.  Principles of compositional data analysis , 1994 .

[11]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[12]  J. Aitchison,et al.  Biplots of Compositional Data , 2002 .

[13]  John Aitchison,et al.  Compositional hypotheses of subcompositional stability and specific perturbation change and their testing , 2003 .

[14]  John C. Butler,et al.  Trends in ternary petrologic variation diagrams; fact or fantasy? , 1979 .

[15]  J. Aitchison,et al.  Possible solution of some essential zero problems in compositional data analysis , 2003 .

[16]  J Aitchison,et al.  The one-hour course in compositional data analysis or compositional data analysis is simple , 1997 .

[17]  V. Pawlowsky-Glahn,et al.  Geometric approach to statistical analysis on the simplex , 2001 .

[18]  John Aitchison,et al.  Delusions of uniqueness and ineluctability , 1991 .

[19]  Vera Pawlowsky-Glahn,et al.  Understanding Perturbation on the Simplex: A Simple Method to Better Visualize and Interpret Compositional Data in Ternary Diagrams , 2002 .

[20]  John Aitchison,et al.  Convex linear combinations of compositions , 1999 .

[21]  Glòria Mateu Figueras,et al.  Krigeado de variables positivas. Un modelo alternativo , 2003 .

[22]  Vera Pawlowsky-Glahn,et al.  Modelling Compositional Change: The Example of Chemical Weathering of Granitoid Rocks , 2003 .

[23]  D. F. Watson Reply to comment on “measures of variability for geological data” by D. F. Watson and G. M. Philip , 1990 .

[24]  Felix Chayes,et al.  Detecting nonrandom associations between proportions by tests of remaining-space variables , 1983 .

[25]  Sönke Rehder,et al.  Letter to the Editor: Comment on “Logratio Analysis and Compositional Distance” by J. Aitchison, C. Barceló-Vidal, J. A. Martín-Fernández, and V. Pawlowsky-Glahn , 2001 .

[26]  J. Aitchison,et al.  Kernel Density Estimation for Compositional Data , 1985 .

[27]  Glòria Mateu Figueras Models de distribució sobre el símplex , 2003 .

[28]  V. Pawlowsky-Glahn,et al.  Latent Compositional Factors in The Llobregat River Basin (Spain) Hydrogeochemistry , 2005 .

[29]  A. Rossier Letter to the Editor , 1986, Paraplegia.

[30]  John C. Butler,et al.  The effects of closure on the moments of a distribution , 1979 .

[31]  Pawlowsky-Glahn Statistical modelling on coordinates , .

[32]  John Aitchison,et al.  A plea for precision inMathematical Geology , 1991 .

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

[34]  V. Pawlowsky-Glahn,et al.  BLU Estimators and Compositional Data , 2002 .

[35]  V. Pawlowsky-Glahn,et al.  Groups of Parts and Their Balances in Compositional Data Analysis , 2005 .

[36]  J. Bacon-Shone Ranking methods for compositional data , 1992 .

[37]  D. F. Watson Reply to “delusions of uniqueness and ineluctability” by J. Aitchison , 1991 .

[38]  K. Gabriel,et al.  The biplot graphic display of matrices with application to principal component analysis , 1971 .

[39]  V. Pawlowsky-Glahn,et al.  Zero Replacement in Compositional Data Sets , 2000 .

[40]  John Aitchison,et al.  Measures of location of compositional data sets , 1989 .

[41]  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.

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

[43]  John Aitchison,et al.  Relative variation diagrams for describing patterns of compositional variability , 1990 .

[44]  J. Aitchison,et al.  Some comments on compositional data analysis in archaeometry, in particular the fallacies in Tangri and Wright’s dismissal of logratio analysis , 2002 .

[45]  Antonella Buccianti,et al.  New insights on river water chemistry by using non-centred simplicial principal component analysis: a case study , 2003 .

[46]  E M McGirr,et al.  Doctors as Decision-makers: A Computer-assisted Study of Diagnosis as a Cognitive Skill , 1971, British medical journal.

[47]  J. Mosimann On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions , 1962 .

[48]  Glòria Mateu Figueras,et al.  Una alternativa a la distribución lognormal , 2003 .

[49]  John Aitchison,et al.  The statistical analysis of geochemical compositions , 1984 .

[50]  J. Aitchison A new approach to null correlations of proportions , 1981 .

[51]  J. Aitchison Principal component analysis of compositional data , 1983 .

[52]  John Aitchison,et al.  Comment on “measures of variability for geological data” by D. F. Watson and G. M. Philip , 1990 .

[53]  G. Weltje End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem , 1997 .

[54]  Francis Galton,et al.  XII. The geometric mean, in vital and social statistics , 1879, Proceedings of the Royal Society of London.

[55]  Jane M. Fry,et al.  Compositional data analysis and zeros in micro data , 2000 .

[56]  Carles Barceló-Vidal Mathematical Foundations of Compositional Data Analysis , 2001 .

[57]  P. Guttorp,et al.  Statistical Interpretation of Species Composition , 2001 .

[58]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[59]  Geoffrey S. Watson,et al.  The art of statistical science : a tribute to G.S. Watson , 1992 .

[60]  J. Aitchison,et al.  Principles, practice and performance in decision-making in clinical medicine , 1975 .

[61]  Kai W. Ng,et al.  Characterization of Distributional Forms for Compositional Data and Associated Distributional Tests , 2003 .

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

[63]  K Ruben Gabriel,et al.  Biplot Display of Multivariate Matrices for Inspection of Data and Diagnosis. , 1980 .

[64]  J. Aitchison A General Class of Distributions on the Simplex , 1985 .

[65]  D. Mcalister,et al.  XIII. The law of the geometric mean , 1879, Proceedings of the Royal Society of London.

[66]  John Bacon-Shone Modelling Structural Zeros in Compositional data , 2003 .

[67]  P. Guttorp,et al.  Statistical Analysis and Interpretation of Discrete Compositional Data , 1998 .

[68]  Vera Pawlowsky-Glahn,et al.  Visualization and modeling of sub-populations of compositional data: statistical methods illustrated by means of geochemical data from fumarolic fluids , 2002 .

[69]  John Aitchison,et al.  Exploration of geological variability and possible processes through the use of compositional data analysis: an example using scottish metamorphosed , 2003 .

[70]  Eric Grunsky,et al.  Some aspects of transformations of compositional data and the identification of outliers , 1996 .

[71]  D. F. Watson,et al.  Measures of variability for geological data , 1989 .