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    Online Joint Topology Identification and Signal Estimation with Inexact Proximal Online Gradient Descent

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    Identifying the topology that underlies a set of time series is useful for tasks such as prediction, denoising, and data completion. Vector autoregressive (VAR) model based topologies capture dependencies among time series, and are often inferred from observed spatio-temporal data. When the data are affected by noise and/or missing samples, the tasks of topology identification and signal recovery (reconstruction) have to be performed jointly. Additional challenges arise when i) the underlying topology is time-varying, ii) data become available sequentially, and iii) no delay is tolerated. To overcome these challenges, this paper proposes two online algorithms to estimate the VAR model-based topologies. The proposed algorithms have constant complexity per iteration, which makes them interesting for big data scenarios. They also enjoy complementary merits in terms of complexity and performance. A performance guarantee is derived for one of the algorithms in the form of a dynamic regret bound. Numerical tests are also presented, showcasing the ability of the proposed algorithms to track the time-varying topologies with missing data in an online fashion.Comment: 14 pages including supplementary material, 2 figures, submitted to IEEE Transactions on Signal Processin

    Online Machine Learning for Inference from Multivariate Time-series

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    Inference and data analysis over networks have become significant areas of research due to the increasing prevalence of interconnected systems and the growing volume of data they produce. Many of these systems generate data in the form of multivariate time series, which are collections of time series data that are observed simultaneously across multiple variables. For example, EEG measurements of the brain produce multivariate time series data that record the electrical activity of different brain regions over time. Cyber-physical systems generate multivariate time series that capture the behaviour of physical systems in response to cybernetic inputs. Similarly, financial time series reflect the dynamics of multiple financial instruments or market indices over time. Through the analysis of these time series, one can uncover important details about the behavior of the system, detect patterns, and make predictions. Therefore, designing effective methods for data analysis and inference over networks of multivariate time series is a crucial area of research with numerous applications across various fields. In this Ph.D. Thesis, our focus is on identifying the directed relationships between time series and leveraging this information to design algorithms for data prediction as well as missing data imputation. This Ph.D. thesis is organized as a compendium of papers, which consists of seven chapters and appendices. The first chapter is dedicated to motivation and literature survey, whereas in the second chapter, we present the fundamental concepts that readers should understand to grasp the material presented in the dissertation with ease. In the third chapter, we present three online nonlinear topology identification algorithms, namely NL-TISO, RFNL-TISO, and RFNL-TIRSO. In this chapter, we assume the data is generated from a sparse nonlinear vector autoregressive model (VAR), and propose online data-driven solutions for identifying nonlinear VAR topology. We also provide convergence guarantees in terms of dynamic regret for the proposed algorithm RFNL-TIRSO. Chapters four and five of the dissertation delve into the issue of missing data and explore how the learned topology can be leveraged to address this challenge. Chapter five is distinct from other chapters in its exclusive focus on edge flow data and introduces an online imputation strategy based on a simplicial complex framework that leverages the known network structure in addition to the learned topology. Chapter six of the dissertation takes a different approach, assuming that the data is generated from nonlinear structural equation models. In this chapter, we propose an online topology identification algorithm using a time-structured approach, incorporating information from both the data and the model evolution. The algorithm is shown to have convergence guarantees achieved by bounding the dynamic regret. Finally, chapter seven of the dissertation provides concluding remarks and outlines potential future research directions.publishedVersio
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