LEAST SQUARES ADJUSTMENT IN THE PRESENCE OF DISCREPANT DATA

Least squares methods provide a flexible and efficient approach for analyzing metrology data. Given a model, measurement data values and their associated uncertainty matrix, it is possible to define a least squares analysis method that gives the measurement data values the appropriate ‘degree of belief’ as specified by the uncertainty matrix. Least squares methods also provide, through χ² values and related concepts, a measure of the conformity of the model, data and input uncertainty matrix with each other. If there is conformity, we have confidence in the parameter estimates and their associated uncertainty matrix. If there is nonconformity, we seek methods of modifying the input information so that conformity can be achieved. For example, a linear response may be replaced by a quadratic response, data that has been incorrectly recorded can be replaced by improved values, or the input uncertainty matrix can be adjusted. In this paper, we look at a number of approaches to achieving conformity in which the main element to be adjusted is the input uncertainty matrix. These approaches include the well-known Birge procedure. In particular, we consider the natural extensions of least squares methods to maximum likelihood methods and show how these more general approaches can provide a flexible route to achieving conformity. This work was undertaken as part of the Software Support for Metrology and Quantum Metrology programmes, funded by the United Kingdom Department of Trade and Industry.