Approximate Linear Models

B. Buck and V. A. Macaulay
Theoretical Physics
University of Oxford
1 Keble Road
Oxford OX1 3NP
United Kingdom and


The fitting of straight lines to sets of points determined by pairs of observed quantities is a frequently occurring problem. The usual approach is to assume that the deviations from a linear law arise only from experimental noise. The case where the noise is in the ordinate only is very well known and some progress has been made on the case with errors in both variables. However this treatment is rarely adequate to describe a real situation in physics. Nearly always, approximations have been made in the physical model that led to the linear relation, so that in addition to error in the data due to experimental noise, there is bound to be error in the model. When experimental error can be neglected leaving only model error, straight line fitting is called regression.

We show how to treat regression when experimental error is present in both variables and has been estimated from repeated measurements. Our formulae contain the well-known results as special cases. The idea is developed for the general case in which the two variables of interest are linearly related because each of them depends, with model error, on a third unmeasured variable, which can be thought of as a label. We show how to estimate the parameters in the model, including the probable size of the model error. The method is illustrated on data sets from nuclear physics in which strikingly good straight lines have been discovered, but which no theory of the nucleus could demonstrate to be exact.

This approach can be generalized in many ways. We can treat models other than straight lines; the results are particularly simple in an approximation where no datum departs too much from the model. In addition, we can treat cases with linear relations between more than two variables.

MaxEnt 94 Abstracts /