Edge Preserving Smoothing by Local Mixture Modelling

Smooth models became more and more popular over the last couple of years. Standard smoothing methods however can not cope with discontinuities in a function or its first derivative. In particular, this implies that structural changes in data may be hidden in smooth estimates. Recently, Chu, Glad, Godtliebsen & Marron (1998) suggest local M estimation as edge preserving smoother. The basic idea behind local M estimation is that observations beyond a jump are considered as outliers and down-weighted or neglected in the estimation. We pursue a different, but related idea here and treat observations beyond a jump as tracing from a different population which differs from the current one by a shift in the mean. This means we impose locally a mixture model where mixing takes place due to different mean values. For fitting we apply a local version of the EM algorithm. The advantage of our approach shows in its general formulation. In particular, it easily extends to non Gaussian data. The procedure is applied in two examples, the first concerning the analysis of structural changes in the duration of unemployment, the second focusing on disease mapping