A comparison of fusion policies for local detection statistics in federated monitoring

Federated monitoring of large-scale data streams offers a computationally efficient framework by aggregating local detection statistics into a global monitoring rule. While fusion policies for local detection statistics on the log-likelihood-ratio (LLR) scale, e.g., notably the Cumulative Sum (CUSUM) statistics, are well documented, the properties of fusion on the likelihood-ratio (LR) scale, such as Shiryaev-Roberts-Pollak (SRP) statistics, remain underexplored. In this paper, we conduct a comprehensive numerical study evaluating thirteen fusion policies across both LLR and LR scales. We demonstrate that these two scales yield fundamentally distinct detection patterns: while LLR-scale fusion exhibits a characteristic imbalance in detection delays between sparse and dense changes, the simple sum of local LR-scale detection statistics inherently possesses a “self-shrinkage” property. Crucially, we observe that applying additional explicit shrinkage to local LR-scale detection statistics provides negligible benefit, suggesting that the LR-scale fusion is naturally robust to high-dimensional sparsity. Leveraging these insights, we propose a Rank-Weighted-Sum (RWS) rule for LR-scale fusion, as well as a discussion on general α-scale fusion for future research.