The measured data consist of a set of vectors . Each coordinate of each data point has an associated experimental uncertainty . (The may or may not be known a priori. We assume that at least their relative magnitudes are known.) The experimental errors will cause the data points to lie scattered off the hyperplane of Equation (2). Therefore one seeks a ``best fit'' to the data.
The method of principal component analysis, referred to above, is equivalent to seeking the hyperplane that minimizes the sum of squares of perpendicular distances from the measured points to the hyperplane. As already mentioned, this method is unable to take proper account of the experimental uncertainties of the data, and is not invariant under a change of scale of one or more axes.
The usual method that one finds in the literature for obtaining a best fit of this kind is based on minimizing a sum of squares of the residuals . This is called a regression of against through . The sum of squares of these residuals is either unweighted or weighted by (Bevington, 1991). This approach ignores the uncertainties in coordinates through . It also gives different results depending on which coordinate is chosen as the ``dependent'' coordinate .
The correct treatment that properly takes account of the experimental
uncertainties was formulated by Deming (1943). Given the data
with associated uncertainties , a set of
corresponding ``adjusted'' values are sought which lie
exactly on the hyperplane (2) and minimize the