This paper introduces a periodic multivariate Poisson autoregression with potentially infinite memory, with a special focus on the network setting. Using contraction techniques, we study the stability of such a process and provide upper bounds on how fast it reaches the periodically stationary regime. We then propose a computationally efficient Markov approximation using the properties of the exponential function and a density result. Furthermore, we prove the strong consistency of the maximum likelihood estimator for the Markov approximation and empirically test its robustness in the case of misspecification. Our model is applied to the prediction of weekly Rotavirus cases in Berlin, demonstrating superior performance compared to the existing PNAR model.

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Network point processes often exhibit latent structure that govern the behaviour of the sub-processes. It is not always reasonable to assume that this latent structure is static, and detecting when and how this driving structure changes is often of interest. In this paper, we introduce a novel online methodology for detecting changes within the latent structure of a network point process. We focus on block-homogeneous Poisson processes, where latent node memberships determine the rates of the edge processes. We propose a scalable variational procedure which can be applied on large networks in an online fashion via a Bayesian forgetting factor applied to sequential variational approximations to the posterior distribution. The proposed framework is tested on simulated and real-world data, and it rapidly and accurately detects changes to the latent edge process rates, and to the latent node group memberships, both in an online manner. In particular, in an application on the Santander Cycles bike-sharing network in central London, we detect changes within the network related to holiday periods and lockdown restrictions between 2019 and 2020.

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Data collected over networks can be modelled as noisy observations of an unknown function over the nodes of a graph or network structure, fully described by its nodes and their connections, the edges. In this context, function estimation has been proposed in the literature and typically makes use of the network topology such as relative node arrangement, often using given or artificially constructed node Euclidean coordinates. However, networks that arise in fields such as hydrology (for example, river networks) present features that challenge these established modelling setups since the target function may naturally live on edges (e.g., river flow) and/or the node-oriented modelling uses noisy edge data as weights. This work tackles these challenges and develops a novel lifting scheme along with its associated (second) generation wavelets that permit data decomposition across the network edges. The transform, which we refer to under the acronym LG-LOCAAT, makes use of a line graph construction that first maps the data in the line graph domain. We thoroughly investigate the proposed algorithm's properties and illustrate its performance versus existing methodologies. We conclude with an application pertaining to hydrology that involves the denoising of a water quality index over the England river network, backed up by a simulation study for a river flow dataset.

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When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of different embeddings—which can be on entirely different scales—by how easy it is to distinguish communities, in an information-theoretical sense. For common types of weighted graphs, such as count networks or p-value networks, we find that transformations such as tempering or thresholding can be highly beneficial, both in theory and in practice.

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In this work, we propose an adaptive wavelet-based approach for extracting primary dynamics in multivariate nonstationary time series.

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Update to GNAR to version 1.1.4

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