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Profile transformations for reciprocal averaging and singular value decomposition

Author

Listed:
  • Ting-Wu Wang

    (University of Newcastle)

  • Eric J. Beh

    (University of Wollongong
    Stellenbosch University)

  • Rosaria Lombardo

    (University of Campania)

  • Ian W. Renner

    (University of Newcastle)

Abstract

Power transformations of count data, including cell frequencies of a contingency table, have been well understood for nearly 100 years, with much of the attention focused on the square root transformation. Over the past 15 years, this topic has been the focus of some new insights into areas of correspondence analysis where two forms of power transformation have been discussed. One type considers the impact of raising the joint proportions of the cell frequencies of a table to a known power while the other examines the power transformation of the relative distribution of the cell frequencies. While the foundations of the graphical features of correspondence analysis rest with the numerical algorithms like reciprocal averaging, and other analogous techniques, discussions of the role of power transformations in reciprocal averaging have not been described. Therefore, this paper examines this link where a power transformation is applied to the cell frequencies of a two-way contingency table. In doing so, we show that reciprocal averaging can be performed under such a transformation to obtain row and column scores that provide the maximum association between the variables and the greatest discrimination between the categories. Finally, we discuss the connection between performing reciprocal averaging and singular value decomposition under this type of power transformation. The R function, powerRA.exe is included in the Appendix and performs reciprocal averaging of a power transformation of the cell frequencies of a two-way contingency table.

Suggested Citation

  • Ting-Wu Wang & Eric J. Beh & Rosaria Lombardo & Ian W. Renner, 2025. "Profile transformations for reciprocal averaging and singular value decomposition," Computational Statistics, Springer, vol. 40(3), pages 1217-1251, March.
    • Handle: RePEc:spr:compst:v:40:y:2025:i:3:d:10.1007_s00180-024-01517-x
    DOI: 10.1007/s00180-024-01517-x
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    References listed on IDEAS

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    1. Eric J. Beh & Rosaria Lombardo, 2024. "Correspondence Analysis Using the Cressie–Read Family of Divergence Statistics," International Statistical Review, International Statistical Institute, vol. 92(1), pages 17-42, April.
    2. M. O. Hill, 1974. "Correspondence Analysis: A Neglected Multivariate Method," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 23(3), pages 340-354, November.
    3. Yu, Guan, 2009. "Variance stabilizing transformations of Poisson, binomial and negative binomial distributions," Statistics & Probability Letters, Elsevier, vol. 79(14), pages 1621-1629, July.
    4. Greenacre, Michael, 2009. "Power transformations in correspondence analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3107-3116, June.
    5. Beh, Eric J. & Lombardo, Rosaria & Alberti, Gianmarco, 2018. "Correspondence analysis and the Freeman–Tukey statistic: A study of archaeological data," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 73-86.
    6. Ryan E. Truchelut & Philip J. Klotzbach & Erica M. Staehling & Kimberly M. Wood & Daniel J. Halperin & Carl J. Schreck & Eric S. Blake, 2022. "Earlier onset of North Atlantic hurricane season with warming oceans," Nature Communications, Nature, vol. 13(1), pages 1-8, December.
    7. Tuncer, Yalcin & Tanik, Murat M. & Allison, David B., 2008. "An overview of statistical decomposition techniques applied to complex systems," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2292-2310, January.
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