Publications

A multiscale method for data collected from network edges via the line graph Cover Image

A multiscale method for data collected from network edges via the line graph

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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Spectral Embedding of Weighted Graphs Cover Image

Spectral Embedding of Weighted Graphs

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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Adaptive Wavelet Domain Principal Component Analysis for Nonstationary Time Series Cover Image

Adaptive Wavelet Domain Principal Component Analysis for Nonstationary Time Series

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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New Methods for Network Count Time Series Cover Image

New Methods for Network Count Time Series

The original generalized network autoregressive models are poor for modelling count data as they are based on the additive and constant noise assumptions, which is usually inappropriate for count data. We introduce two new models (GNARI and NGNAR) for count network time series by adapting and extending existing count-valued time series models. We present results on the statistical and asymptotic properties of our new models and their estimates obtained by conditional least squares and maximum likelihood. We conduct two simulation studies that verify successful parameter estimation for both models and conduct a further study that shows, for negative network parameters, that our NGNAR model outperforms existing models and our other GNARI model in terms of predictive performance. We model a network time series constructed from COVID-positive counts for counties in New York State during 2020--22 and show that our new models perform considerably better than existing methods for this problem.

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New tools for network time series Cover Image

New tools for network time series

Network time series are becoming increasingly important across many areas in science and medicine and are often characterised by a known or inferred underlying network structure, which can be exploited to make sense of dynamic phenomena that are often high-dimensional. For example, the Generalised Network Autoregressive (GNAR) models exploit such structure parsimoniously. We use the GNAR framework to introduce two association measures: the network and partial network autocorrelation functions, and introduce Corbit (correlation-orbit) plots for visualisation. As with regular autocorrelation plots, Corbit plots permit interpretation of underlying correlation structures and, crucially, aid model selection more rapidly than using other tools such as AIC or BIC. We additionally interpret GNAR processes as generalised graphical models, which constrain the processes' autoregressive structure and exhibit interesting theoretical connections to graphical models via utilization of higher-order interactions. We demonstrate how incorporation of prior information is related to performing variable selection and shrinkage in the GNAR context. We illustrate the usefulness of the GNAR formulation, network autocorrelations and Corbit plots by modelling a COVID-19 network time series of the number of admissions to mechanical ventilation beds at 140 NHS Trusts in England & Wales. We introduce the Wagner plot that can analyse correlations over different time periods or with respect to external covariates. In addition, we introduce plots that quantify the relevance and influence of individual nodes. Our modelling provides insight on the underlying dynamics of the COVID-19 series, highlights two groups of geographically co-located `influential' NHS Trusts and demonstrates superior prediction abilities when compared to existing techniques.

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