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Abstract:
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In this talk we discuss theoretical properties of general multiscale scanning methods which can be used for the detection of anomalies from a known ground truth in a given data set. In our setting, anomalies are active components of a quantity f with respect to a dictionary U from observations Y following a general regression model. Our results rely on a Gaussian approximation of the underlying multiscale statistic with a novel scale penalty and on tail expansions for the distribution of maxima of Gaussian random fields. The scale penalty balances the contribution of the different scales and ensures weak convergence of the statistic towards a Gumbel limit under reasonable assumptions. Based on those results, we obtain a multiple test that allows to identify the active components of the dictionary U at controlled, family-wise error rate, where the general approach allows us to tackle a variety of non-standard situations, in particular, our results can be used to construct multiscale tests for inverse problems.
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