Luckiness Normalized Maximum Likelihood-based Change Detection for High-dimensional Graphical Models with Missing Data

This study focuses on detecting dependency changes in multivariate time series. This is a practically important issue because, for example, changes in the relationships between several stocks may lead to early warning of critical economic events. The corresponding change detection methods should be feasible for addressing high-dimensional cases online. We use the Gaussian graphical model (GGM) to represent the conditional dependencies in multivariate and reduce our issue to change detection in GGM and consider the cases with the size of the graphical model significantly larger than the sample size as high-dimensional. Although there are several studies on GGM change detection, adequately controlling the sparsity of the model to handle high-dimensional data remains a significant challenge. To address this problem, we introduce new change statistics based on the luckiness normalized maximum likelihood (LNML) code length and minimum description length (MDL) adapted for sparse modeling. We demonstrate how to efficiently calculate the LNML-based change statistics and provide a theoretical guarantee in terms of Type I and Type II error probabilities. We expand the methodology to include cases in which missing values exist. We compare the effectiveness of our methodology with that of conventional methods through experiments using synthetic and real-world datasets.