Interpretation of Inaccurate, Insufficient and Inconsistent Data

Many problems in physical science involve the estimation of a number of unknown parameters which bear a linear or quasi-linear relationship to a set of experimental data. The data may be contaminated by random errors, insufficient to determine the unknowns, redundant, or all of the above. This paper presents a method of optimizing the conclusions from such a data set. The problem is formulated as an ill-posed matrix equation, and general criteria are established for constructing an ‘inverse’ matrix. The ‘solution’ to the problem is defined in terms of a set of generalized eigenvectors of the matrix, and may be chosen to optimize the resolution provided by the data, the expected error in the solution, the fit to the data, the proximity of the solution to an arbitrary function, or any combination of the above. The classical ‘least-squares’ solution is discussed as a special case.