An Efficient Leave One Block Out approach to identify outliers

Abstract

In Least Squares (LS), the linearized functional model betweenM observables and N unknown parameters is given. LS provides estimates of parameters, observables, residuals and a posteriori variance. To identify outliers and to estimate accuracies and reliabilities, tests on the model and on the individual residuals can be performed at different levels of significance and power. However, LS is not robust: one outlier could be spread into all the residuals and its identification is difficult.

A possible solution to this problem is given by a Leave One Block Out approach. Let’s suppose that the observation vector can be decomposed into m sub-vectors (blocks) that are reciprocally uncorrelated: in the case of completely uncorrelated observations, m = M. A suspected block is excluded from the adjustment, whose results are used to check it. Clearly, the check is more robust, because one outlier in the excluded block does not affect the adjustment results. The process can be repeated on all the blocks, but can be very slow, because m adjustments must be computed.

To efficiently apply Leave One Block Out, an algorithm has been studied. The usual LS adjustment is performed on all the observations to obtain the ’batch’ results. The contribution of each block is subtracted from the batch results by algebraic decompositions, with a minimal computational effort: this holds for parameters, a posteriori residuals and variance. Therefore all the blocks can be checked. In the paper, the algorithm is discussed. Two examples of ELOBO application are presented: the first testifies ELOBO reliability against classical LS tests. In the second, ELOBO numerical efficiency is analyzed.