# Approximate Linear Models

## B. Buck and V. A. Macaulay

Theoretical Physics

University of Oxford

1 Keble Road

Oxford OX1 3NP

United Kingdom

`buck@thphys.ox.ac.uk` and
`vincent@thphys.ox.ac.uk`

### Abstract

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 / mas@mrao.cam.ac.uk