Probabilistic and least-squares inference of the parameters of a straight-line model

Abstract

Two methods are presented by which a straight line is to be fitted to a cloud of points in Cartesian coordinates. It is assumed that data are available in the form of a series of measurements in each coordinate, together with an assessment of their covariance matrices. In the first (probabilistic) method, the joint probability density function (PDF) for the two parameters of the straight line is considered. An explicit expression for this PDF is derived; it allows one to compute numerically the expectations, the variances and the covariance between the two parameters of the straight line. The second method is that of least-squares; it renders a non-linear system of equations for the point estimates of the parameters, as well as an approximation to their covariance matrix. In contrast to least-squares, the probabilistic method allows for the exact calculation of the probability that the true values of the parameters lie within specified intervals.