Robustness and Complex Data Structures by Claudia Becker Roland Fried & Sonja Kuhnt

Author:Claudia Becker, Roland Fried & Sonja Kuhnt
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

12.4 RM-Based Filters with Data-Adaptive Width Selection

Gather and Fried (2004) propose a window width adaption approach for the delayed RM filter. Their idea is adopted by Schettlinger et al. (2010) who develop an RM-based online filter with data-adaptive width selection, the adaptive online RM (aoRM) filter.

12.4.1 The aoRM

The aoRM selects the window width according to the current data situation at each point in time. The window width at time t is denoted by n t in the following. The aoRM uses a goodness-of-fit test to decide whether the window width should be adapted. In short, the aoRM algorithm works as follows: Given a time point t and a window width n t , an RM regression is performed in the time window {t−n t +1,…,n t }. Then the aoRM uses a test to decide whether the RM fit is adequate or not. If the RM fit is not assessed to be adequate, the window width n t is decreased.

The test applied by the aoRM uses the fact that an RM regression results in an equal number of positive and negative residuals. The basic idea of the test is that an RM fit is adequate only if the balance of the residual signs is given for any subset of design points. Given an RM fit in a time window {t−n t +1,…,t}, the aoRM tests the null hypothesis that the median of the distribution of the m≤⌊n t /2⌋ rightmost RM residual signs is zero, against the alternative that the median is not zero. Let denote the residuals of the RM fit in a time window of width n (for simplicity, we omit the time index t here). The aoRM test statistic at time t is then the absolute sum of the signs of the m≤⌊n/2⌋ rightmost RM residuals:

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