Definition of problem

In accordance with the forgoing discussion, we assume that the measured data points should ideally lie on a hyperplane of $m-1$ dimensions in a space of $m$ dimensions. If we let $\vec{y}= [y_1, y_2, \ldots, y_m]$ denote a point on this hyperplane, the equation which the data point ideally should satisfy can be written

$$f(\vec{y}) = \sum_{k=1}^{m-1} a_k y_k + a_m - y_m = 0$$ (2)

The measured data consist of a set of $n$ vectors $\vec{Y_i},\ i=1\ldots, n$. Each coordinate $Y_{ik}$ of each data point has an associated experimental uncertainty $\sigma_{ik}$. (The $\sigma_{ik}$ 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 $\vec{Y_i}$ 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 $f(\vec{Y_i})$. This is called a regression of $y_m$ against $y_1$ through $y_{m-1}$. The sum of squares of these residuals is either unweighted or weighted by $1/\sigma_{im}^2$ (Bevington, 1991). This approach ignores the uncertainties in coordinates $y_1$ through $y_{m-1}$. It also gives different results depending on which coordinate is chosen as the ``dependent'' coordinate $y_m$.

The correct treatment that properly takes account of the experimental uncertainties was formulated by Deming (1943). Given the data $Y_{ik}$ with associated uncertainties $\sigma_{ik}$, a set of corresponding ``adjusted'' values $y_{ik}$ are sought which lie exactly on the hyperplane (2) and minimize the variance

$$S = \sum_i \sum_{k=1}^m {1 \over \sigma_{ik}^2} (Y_{ik} - y_{ik})^2$$ (3)

The solution of this formulation of the problem is not straightforward. York (1966) first devised an approach, later improved by Williamson (1968), for the straight-line case. Here we extend Williamson's solution to arbitrary $m \ge 1$.