Fitting an Equation to Data Impartially

Tofallis, C. Fitting an Equation to Data Impartially. (2023). Mathematics, 11(18), 3957;

14 Pages Posted: 21 Sep 2023 Last revised: 28 Nov 2023

See all articles by Chris Tofallis

Chris Tofallis

University of Hertfordshire Business School

Date Written: August 30, 2023


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 minimising distances, but on the symmetric concept of correlation.

The estimated coefficients or parameters 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.

Keywords: functional relationship, data fitting, errors in variables, linear regression, multivariate analysis

JEL Classification: C13, C20

Suggested Citation

Tofallis, Chris, Fitting an Equation to Data Impartially (August 30, 2023). Tofallis, C. Fitting an Equation to Data Impartially. (2023). Mathematics, 11(18), 3957; , Available at SSRN: or

Chris Tofallis (Contact Author)

University of Hertfordshire Business School ( email )

College Lane
Hatfield, Hertfordshire AL10 9AB
United Kingdom


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