Multiscale scanning with nuisance parameters

TL;DR

This paper introduces a multiscale scanning method for anomaly detection in high-dimensional data with unknown nuisance parameters, ensuring accurate critical values and controlled error rates.

Contribution

It proposes a novel approach to estimate nuisance parameters on the largest scale and adjust multiscale statistics accordingly, with theoretical guarantees and practical validation.

Findings

Uniform invariance principle for adjusted multiscale statistic (AMS)

Method effectively controls family-wise error rate in real data

Simulation confirms theoretical properties

Abstract

We develop a multiscale scanning method to find anomalies in a $d$-dimensional random field in the presence of nuisance parameters. This covers the common situation that either the baseline-level or additional parameters such as the variance are unknown and have to be estimated from the data. We argue that state of the art approaches to determine asymptotically correct critical values for multiscale scanning statistics will in general fail when such parameters are naively replaced by plug-in estimators. Instead, we suggest to estimate the nuisance parameters on the largest scale and to use (only) smaller scales for multiscale scanning. We prove a uniform invariance principle for the resulting adjusted multiscale statistic (AMS), which is widely applicable and provides a computationally feasible way to simulate asymptotically correct critical values. We illustrate the implications of our theoretical results in a simulation study and in a real data example from super-resolution STED microscopy. This allows us to identify interesting regions inside a specimen in a pre-scan with controlled family-wise error rate.