Sequential change-point detection in high-dimensional Gaussian graphical models

High dimensional piecewise stationary graphical models represent a versatile class for modelling time varying networks arising in diverse application areas, including biology, economics, and social sciences. There has been recent work in offline detection and estimation of regime changes in the topology of sparse graphical models. However, the online setting remains largely unexplored, despite its high relevance to applications in sensor networks and other engineering monitoring systems, as well as financial markets. To that end, this work introduces a novel scalable online algorithm for detecting an unknown number of abrupt changes in the inverse covariance matrix of sparse Gaussian graphical models with small delay. The proposed algorithm is based upon monitoring the conditional log-likelihood of all nodes in the network and can be extended to a large class of continuous and discrete graphical models. We also investigate asymptotic properties of our procedure under certain mild regularity conditions on the graph size, sparsity level, number of samples, and pre- and post-changes in the topology of the network. Numerical works on both synthetic and real data illustrate the good performance of the proposed methodology both in terms of computational and statistical efficiency across numerous experimental settings.Comment: 47 pages, 9 figure

Learning the piece-wise constant graph structure of a varying Ising model

This work focuses on the estimation of multiple change-points in a time-varying Ising model that evolves piece-wise constantly. The aim is to identify both the moments at which significant changes occur in the Ising model, as well as the underlying graph structures. For this purpose, we propose to estimate the neighborhood of each node by maximizing a penalized version of its conditional log-likelihood. The objective of the penalization is twofold: it imposes sparsity in the learned graphs and, thanks to a fused-type penalty, it also enforces them to evolve piece-wise constantly. Using few assumptions, we provide two change-points consistency theorems. Those are the first in the context of unknown number of change-points detection in time-varying Ising model. Finally, experimental results on several synthetic datasets and a real-world dataset demonstrate the performance of our method.Comment: 18 pages (9 pages for Appendix), 4 figures, 2 table

A Note on Online Change Point Detection

We investigate sequential change point estimation and detection in univariate nonparametric settings, where a stream of independent observations from sub-Gaussian distributions with a common variance factor and piecewise-constant but otherwise unknown means are collected. We develop a simple CUSUM-based methodology that provably control the probability of false alarms or the average run length while minimizing, in a minimax sense, the detection delay. We allow for all the model parameters to vary in order to capture a broad range of levels of statistical hardness for the problem at hand. We further show how our methodology is applicable to the case in which multiple change points are to be estimated sequentially

Inference on the Change Point for High Dimensional Dynamic Graphical Models

We develop an estimator for the change point parameter for a dynamically evolving graphical model, and also obtain its asymptotic distribution under high dimensional scaling. To procure the latter result, we establish that the proposed estimator exhibits an $O_p(\psi^{-2})$ rate of convergence, wherein $\psi$ represents the jump size between the graphical model parameters before and after the change point. Further, it retains sufficient adaptivity against plug-in estimates of the graphical model parameters. We characterize the forms of the asymptotic distribution under the both a vanishing and a non-vanishing regime of the magnitude of the jump size. Specifically, in the former case it corresponds to the argmax of a negative drift asymmetric two sided Brownian motion, while in the latter case to the argmax of a negative drift asymmetric two sided random walk, whose increments depend on the distribution of the graphical model. Easy to implement algorithms are provided for estimating the change point and their performance assessed on synthetic data. The proposed methodology is further illustrated on RNA-sequenced microbiome data and their changes between young and older individuals.Comment: Software available upon request (built in R

A review on minimax rates in change point detection and localisation

This paper reviews recent developments in fundamental limits and optimal algorithms for change point analysis. We focus on minimax optimal rates in change point detection and localisation, in both parametric and nonparametric models. We start with the univariate mean change point analysis problem and review the state-of-the-art results in the literature. We then move on to more complex data types and investigate general principles behind the optimal procedures that lead to minimax rate-optimal results