A Least-Squares Solution for a Linear Relation between Two Observed Quantities

Abstract

A LINEAR relation has sometimes to be established between two observed quantities, the precision of each observed value for both quantities being limited and variable from value to value in some known way. Examples in astronomy may be found in the treatment of proper motions in moving clusters, in the evaluation of solar motion by proper motions, and in the evaluation of the linear terms of galactic rotation in the solar neighbourhood. In such cases the line through the intersection of the two regression lines, with a slope equal to the geometric mean of the slopes of the two regression lines, may often be a good estimate of the best linear relation, but it does not seem to be a least-squares solution in a precise sense. Seares¹ discusses at length the best estimates of the regression lines that can be set up, taking into account varying weight for each value and the effect on the regression lines of the distribution of the observed values about the diagram. His best linear relation is evaluated as the weighted geometric mean of the two regression lines and he shows² that in the case of uniform precision in each quantity his solution reduces to that obtained by Hertzsprung³ to cover this particular case. It would seem, however, that generalization of Hertzsprung's result to cases of varying precision should be possible without bringing in the regression lines at all. Jeffreys⁴ apparently gives such a solution (his equation 5, § 4.42), but in fact, as was pointed out to me by Dr. A. J. Wesselink, his solution reduces to one of the usual regression lines in the case of uniform precision and not to Hertzsprung's solution. Trumpler and Weaver⁵ mention several times the importance of ‘two-error’ least squares solutions in astronomy but nowhere make a solution.