Deming least-squares fit to multiple hyperplanes

A method is derived to fit a set of multidimensional experimental data points having a priori uncertainties and possibly also covariances in all coordinates to a straight line, plane, or hyperplane of any dimensionality less than the number of coordinates. The least-squares formulation used is that of Deming, which treats all coordinates on an equal basis. Experimentalists needing to fit a linear model to data of this kind have usually performed multiple independent fits in subspaces of the full data space such that each fit has only one dependent coordinate. That procedure does not guarantee mutual consistency of the fits. The present method can be thought of as providing multiple such hyperplane fits in a single simultaneous and therefore consistent solution. As examples, the method is applied to a straight-line fit in three dimensions to synthetic data and to an analysis of xenon isotopes in a lunar rock.