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Improved Classification for Compositional Data Using the α-transformation

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Abstract

In compositional data analysis, an observation is a vector containing nonnegative values, only the relative sizes of which are considered to be of interest. Without loss of generality, a compositional vector can be taken to be a vector of proportions that sum to one. Data of this type arise in many areas including geology, archaeology, biology, economics and political science. In this paper we investigate methods for classification of compositional data. Our approach centers on the idea of using the α-transformation to transform the data and then to classify the transformed data via regularized discriminant analysis and the k-nearest neighbors algorithm. Using the α-transformation generalizes two rival approaches in compositional data analysis, one (when α=1) that treats the data as though they were Euclidean, ignoring the compositional constraint, and another (when α = 0) that employs Aitchison’s centered log-ratio transformation. A numerical study with several real datasets shows that whether using α = 1 or α = 0 gives better classification performance depends on the dataset, and moreover that using an intermediate value of α can sometimes give better performance than using either 1 or 0.

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References

  • AITCHISON, J. (1982), “The Statistical Analysis of Compositional Data”, Journal of the Royal Statistical Society: Series B, 44, 139–177.

    MathSciNet  MATH  Google Scholar 

  • AITCHISON, J. (1983), “Principal Component Analysis of Compositional Data”, Biometrika, 70, 57–65.

    Article  MathSciNet  MATH  Google Scholar 

  • AITCHISON, J. (1992), “On Criteria for Measures of Compositional Difference”, Mathematical Geology, 24, 365–379.

    Article  MathSciNet  MATH  Google Scholar 

  • AITCHISON, J. (2003), The Statistical Analysis of Compositional Data (reprinted with additional material by The Blackburn Press), London, UK: Chapman & Hall.

    Google Scholar 

  • AITCHISON, J., BARCELO-VIDAL, C., MARTIN-FERNANDEZ, J.A., and PAWLOWSKY-GLAHN, V. (2000), “Logratio Analysis and Compositional Distance”, Mathematical Geology, 32, 271–275.

    Article  MATH  Google Scholar 

  • BAXTER,M.J. (2001), “Statistical Modelling of Artefact Compositional Data”, Archaeometry, 43, 131–147.

    Article  Google Scholar 

  • BAXTER,M.J., BEARDAH, C. C., COOL, H. E.M., and JACKSON, C. M. (2005), “Compositional Data Analysis of Some Alkaline Glasses”, Mathematical Geology, 37, 183–196.

    Article  Google Scholar 

  • BAXTER, M.J., and FREESTONE, I. C. (2006), “Log-Ratio Compositional Data Analysis in Archaeometry”, Archaeometry, 48, 511–531.

    Article  Google Scholar 

  • BUTLER, A., and GLASBEY, C. (2008), “A Latent Gaussian Model for Compositional Data With Zeros”, Journal of the Royal Statistical Society: Series C, 57, 505–520.

    Article  MathSciNet  Google Scholar 

  • DRDYEN, I.L., and MARDIA, K. V. (1998), Statistical Shape Analysis, New York: Wiley.

    Google Scholar 

  • EGOZQUE, J.J., PAWLOWSKY-GLAHN, V., MATEU-FIGUERAS, G., and BARCELOVIDAL, C. (2003), “Isometric Logratio Transformations for Compositional Data Analysis”, Mathematical Geology, 35, 279–300.

    Article  MathSciNet  MATH  Google Scholar 

  • ENDRES, D.M. and SCHINDELIN, J. E. (2003), “A New Metric for Probability Distributions”, IEEE Transactions on Information Theory, 49, 1858–1860.

    Article  MathSciNet  MATH  Google Scholar 

  • FRY, J.M., FRY, T. R. L., and McLAREN, K. R. (2000), “Compositional Data Analysis and Zeros in Micro Data”, Applied Economics, 32, 953–959.

    Article  Google Scholar 

  • GREENACRE, M. (2009), “Power Transformations in Correspondence Analysis”, Computational Statistics & Data Analysis, 53, 3107–3116.

    Article  MathSciNet  MATH  Google Scholar 

  • GREENACRE, M. (2011), “Measuring Subcompositional Incoherence”, Mathematical Geosciences, 43, 681–693.

    Article  Google Scholar 

  • GUEORGUIEVA, R., ROSENHECK, R., and ZELTERMAN, D. (2008), “Dirichlet Component Regression and Its Applications to Psychiatric Data”, Computational Statistics & Data Analysis, 52, 5344–5355.

    Article  MathSciNet  MATH  Google Scholar 

  • HASTIE, T., TIBSHIRANI, R., and FRIEDMAN, J. (2001), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Berlin: Springer.

    Book  MATH  Google Scholar 

  • IVERSON, S.J., FIELD, C., BOWEN, W.D., and BLANCHARD, W. (2004), “Quantitative Fatty Acid Signature Analysis: A New Method of Estimating Predator Diets”, Ecological Monographs, 74, 211–235.

    Article  Google Scholar 

  • LANCASTER,H.O. (1965), “The Helmert Matrices”, American Mathematical Monthly, 72, 4–12.

    Article  MathSciNet  MATH  Google Scholar 

  • LARROSA, J.M. (2003), “A Compositional Statistical Analysis of Capital Stock”, in Proceedings of the 1st Compositional Data Analysis Workshop, Girona, Spain.

    Google Scholar 

  • NEOCLEOUS, T., AITKEN, C., and ZADORA, G. (2011), “Transformations for Compositional Data With Zeros With an Application to Forensic Evidence Evaluation”, Chemometrics and Intelligent Laboratory Systems, 109, 77–85.

    Article  Google Scholar 

  • OSTERREICHER, F., and VAJDA, I. (2003), “A New Class of Metric Divergences on Probability Spaces and Its Applicability in Statistics”, Annals of the Institute of Statistical Mathematics, 55, 639–653.

    Article  MathSciNet  MATH  Google Scholar 

  • OTERO, N., TOLOSANA-DELGADO, R., SOLER, A., PAWLOWSKY-GLAHN, V., and CANALS, A. (2005), “Relative vs. Absolute Statistical Analysis of Compositions: A Comparative Study of SurfaceWaters of aMediterranean River”, Water Research, 39, 1404–1414.

    Article  Google Scholar 

  • PALAREA-ALBALADEJO, J., MARTIN-FERNANDEZ, J.A., and SOTO, J.A. (2012), “Dealing with Distances and Transformations for Fuzzy C-means Clustering of Compositional Data”, Journal of Classification, 29, 144–169.

    Article  MathSciNet  MATH  Google Scholar 

  • RODRIGUES, P.C., and LIMA, A.T. (2009), “Analysis of an European Union Election Using Principal Component Analysis”, Statistical Papers, 50, 895–904.

    Article  MathSciNet  MATH  Google Scholar 

  • SCEALY, J.L., and WELSH, A.H. (2011), “Regression for Compositional Data by Using Distributions Defined on the Hypersphere”, Journal of the Royal Statistical Society: Series B, 73, 351–375.

    Article  MathSciNet  Google Scholar 

  • SCEALY, J.L., and WELSH, A.H. (2014), “Colours and Cocktails: Compositional Data Analysis: 2013 Lancaster Lecture”, Australian & New Zealand Journal of Statistics, 56, 145–169.

    Article  MathSciNet  MATH  Google Scholar 

  • STEPHENS, M.A. (1982), ”Use of the Von Mises Distribution to Analyse Continuous Proportions”, Biometrika, 69, 197–203.

    Article  MathSciNet  Google Scholar 

  • STEWART, C., and FIELD, C. (2011), “Managing the Essential Zeros in Quantitative Fatty Acid Signature Analysis”, Journal of Agricultural, Biological, and Environmental Statistics, 16, 45–69.

    Article  MathSciNet  MATH  Google Scholar 

  • TSAGRIS, M.T., PRESTON, S., and WOOD, A. T. A. (2011), “A Data-Based Power Transformation for Compositional Data”, in Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain.

    Google Scholar 

  • TSAGRIS, M., and ATHINEOU, G. (2016), Compositional: A Collection of R Functions for Compositional Data Analysis (R package version 1.2.5), https://cran.r-project.org/web/packages/Compositional/

  • UC IRVINE MACHINE LEARNING REPOSITORY (2014), “Forensic Glass Dataset”, http://archive.ics.uci.edu/ml/datasets/Glass+Identification.

  • WORONOW, A. (1997), “The Elusive Benefits of Logratios”, in Proceedings of the 3rd Annual Conference of the International Association for Mathematical Geology, Barcelona, Spain.

    Google Scholar 

  • ZADORA, G., NEOCLEOUS, T., and AITKEN, C. (2010), “A Two-Level Model for Evidence Evaluation in the Presence of Zeros”, Journal of Forensic Sciences, 55, 371–384.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Crete, Heraklion, Greece

    Michail Tsagris

  2. University of Nottingham, Nottingham, UK

    Simon Preston & Andrew T. A. Wood

Authors
  1. Michail Tsagris
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  2. Simon Preston
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  3. Andrew T. A. Wood
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Correspondence to Michail Tsagris.

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Tsagris, M., Preston, S. & Wood, A.T.A. Improved Classification for Compositional Data Using the α-transformation. J Classif 33, 243–261 (2016). https://doi.org/10.1007/s00357-016-9207-5

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