IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i18p3957-d1242128.html
   My bibliography  Save this article

Fitting an Equation to Data Impartially

Author

Listed:
  • Chris Tofallis

    (Statistical Services and Consultancy Unit, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK)

Abstract

We consider the problem of fitting a relationship (e.g., a potential scientific law) to data involving multiple variables. Ordinary (least squares) regression is not suitable for this because the estimated relationship will differ according to which variable is chosen as being dependent, and the dependent variable is unrealistically assumed to be the only variable which has any measurement error (noise). We present a very general method for estimating a linear functional relationship between multiple noisy variables, which are treated impartially, i.e., no distinction between dependent and independent variables. The data are not assumed to follow any distribution, but all variables are treated as being equally reliable. Our approach extends the geometric mean functional relationship to multiple dimensions. This is especially useful with variables measured in different units, as it is naturally scale invariant, whereas orthogonal regression is not. This is because our approach is not based on minimizing distances, but on the symmetric concept of correlation. The estimated coefficients are easily obtained from the covariances or correlations, and correspond to geometric means of associated least squares coefficients. The ease of calculation will hopefully allow widespread application of impartial fitting to estimate relationships in a neutral way.

Suggested Citation

  • Chris Tofallis, 2023. "Fitting an Equation to Data Impartially," Mathematics, MDPI, vol. 11(18), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3957-:d:1242128
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/18/3957/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/18/3957/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Draper, Norman R. & Yang, Yonghong (Fred), 1997. "Generalization of the geometric mean functional relationship," Computational Statistics & Data Analysis, Elsevier, vol. 23(3), pages 355-372, January.
    2. Chris Tofallis, 2003. "Multiple Neutral Data Fitting," Annals of Operations Research, Springer, vol. 124(1), pages 69-79, November.
    3. Peng, Jie & Wang, Pei & Zhou, Nengfeng & Zhu, Ji, 2009. "Partial Correlation Estimation by Joint Sparse Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 735-746.
    4. Healy, John D., 1980. "Maximum likelihood estimation of a multivariate linear functional relationship," Journal of Multivariate Analysis, Elsevier, vol. 10(2), pages 243-251, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Chris Tofallis, 2024. "Fitting an Equation to Data Impartially," Papers 2409.02573, arXiv.org.
    2. Mark H Holmes & Michael Caiola, 2018. "Invariance properties for the error function used for multilinear regression," PLOS ONE, Public Library of Science, vol. 13(12), pages 1-25, December.
    3. Seunghwan Lee & Sang Cheol Kim & Donghyeon Yu, 2023. "An efficient GPU-parallel coordinate descent algorithm for sparse precision matrix estimation via scaled lasso," Computational Statistics, Springer, vol. 38(1), pages 217-242, March.
    4. Colignatus, Thomas, 2017. "Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign," MPRA Paper 80833, University Library of Munich, Germany, revised 17 Aug 2017.
    5. Shaoji Xu, 2014. "A Property of Geometric Mean Regression," The American Statistician, Taylor & Francis Journals, vol. 68(4), pages 277-281, November.
    6. Luo, Shan & Chen, Zehua, 2014. "Edge detection in sparse Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 138-152.
    7. Natalia Bailey & Sean Holly & M. Hashem Pesaran, 2016. "A Two‐Stage Approach to Spatio‐Temporal Analysis with Strong and Weak Cross‐Sectional Dependence," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 31(1), pages 249-280, January.
    8. Claudia Angelini & Daniela De Canditiis & Anna Plaksienko, 2021. "Jewel : A Novel Method for Joint Estimation of Gaussian Graphical Models," Mathematics, MDPI, vol. 9(17), pages 1-24, August.
    9. Matteo Barigozzi & Marc Hallin, 2017. "A network analysis of the volatility of high dimensional financial series," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(3), pages 581-605, April.
    10. Colignatus, Thomas, 2017. "Comparing votes and seats with a diagonal (dis-) proportionality measure, using the slope-diagonal deviation (SDD) with cosine, sine and sign," MPRA Paper 80965, University Library of Munich, Germany, revised 24 Aug 2017.
    11. Guanghui Cheng & Zhengjun Zhang & Baoxue Zhang, 2017. "Test for bandedness of high-dimensional precision matrices," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(4), pages 884-902, October.
    12. Brownlees, Christian & Mesters, Geert, 2021. "Detecting granular time series in large panels," Journal of Econometrics, Elsevier, vol. 220(2), pages 544-561.
    13. Chris Tofallis, 2011. "Investment Volatility: A Critique of Standard Beta Estimation and a Simple Way Forward," Papers 1109.4422, arXiv.org.
    14. Matteo Barigozzi & Christian Brownlees, 2019. "NETS: Network estimation for time series," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 34(3), pages 347-364, April.
    15. Tofallis, Chris, 2008. "Investment volatility: A critique of standard beta estimation and a simple way forward," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1358-1367, June.
    16. Mingyang Ren & Sanguo Zhang & Qingzhao Zhang & Shuangge Ma, 2022. "Gaussian graphical model‐based heterogeneity analysis via penalized fusion," Biometrics, The International Biometric Society, vol. 78(2), pages 524-535, June.
    17. Chi, Eric C. & Lange, Kenneth, 2014. "Stable estimation of a covariance matrix guided by nuclear norm penalties," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 117-128.
    18. Huihang Liu & Xinyu Zhang, 2023. "Frequentist model averaging for undirected Gaussian graphical models," Biometrics, The International Biometric Society, vol. 79(3), pages 2050-2062, September.
    19. Tan, Kean Ming & Witten, Daniela & Shojaie, Ali, 2015. "The cluster graphical lasso for improved estimation of Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 85(C), pages 23-36.
    20. Jie Cheng & Elizaveta Levina & Pei Wang & Ji Zhu, 2014. "A sparse ising model with covariates," Biometrics, The International Biometric Society, vol. 70(4), pages 943-953, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3957-:d:1242128. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.