Visibility graph-based covariance functions for scalable spatial analysis in nonconvex domains

We present a new method for constructing valid covariance functions of Gaussian processes over irregular nonconvex spatial domains such as water bodies, where the geodesic distance agrees with the Euclidean distance only for some pairs of points. Standard covariance functions based on geodesic distances are not positive definite on such domains. Using a visibility graph on the domain, we use the graphical method of "covariance selection" to propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected through the domain. The proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on nonconvex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We evaluate the performance of competing state-of-the-art methods using simulations studies on a contrived nonconvex domain. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains

Detailed study of the Bootes field using 300-500 MHz uGMRT observations: Source Properties and radio--infrared correlations

The dominant source of radio continuum emissions at low frequencies is synchrotron radiation, which originates from star-forming regions in disk galaxies and from powerful jets produced by active galactic nuclei (AGN). We studied the Bootes field using the upgraded Giant Meterwave Radio Telescope (uGMRT) at 400 MHz, achieving a central minimum off-source RMS noise of 35$\mu$Jy beam$^{-1}$ and a catalogue of 3782 sources in $\sim6$ sq. degrees of the sky. The resulting catalogue was compared to other radio frequency catalogues, and the corrected normalised differential source counts were derived. We use standard multi-wavelength techniques to classify the sources in star-forming galaxies (SFGs), radio-loud (RL) AGN, and radio-quiet (RQ) AGN that confirm a boost in the SFGs and RQ\,AGN AGN populations at lower flux levels. For the first time, we investigated the properties of the radio--IR relations at 400\,MHz in this field. The $L_{\rm 400 MHz}$--$L_{\rm TIR}$ relations for SFGs were found to show a strong correlation with non-linear slope values of $1.10\pm0.01$, and variation of $q_{\rm TIR}$ with $z$ is given as, $q_{\rm TIR} = (2.19 \pm 0.07)\ (1+z)^{-0.15 \pm 0.08}$. This indicates that the non-linearity of the radio--IR relations can be attributed to the mild variation of $q_{\rm TIR}$ values with $z$. The derived relationships exhibit similar behaviour when applied to LOFAR at 150 MHz and also at 1.4 GHz. This emphasises the fact that other parameters like magnetic field evolution with $z$ or the number densities of cosmic ray electrons can play a vital role in the mild evolution of $q$ values.Comment: 18 pages, 18 Figures; Accepted for publication in MNRA

Neural networks for geospatial data

Analysis of geospatial data has traditionally been model-based, with a mean model, customarily specified as a linear regression on the covariates, and a covariance model, encoding the spatial dependence. We relax the strong assumption of linearity and propose embedding neural networks directly within the traditional geostatistical models to accommodate non-linear mean functions while retaining all other advantages including use of Gaussian Processes to explicitly model the spatial covariance, enabling inference on the covariate effect through the mean and on the spatial dependence through the covariance, and offering predictions at new locations via kriging. We propose NN-GLS, a new neural network estimation algorithm for the non-linear mean in GP models that explicitly accounts for the spatial covariance through generalized least squares (GLS), the same loss used in the linear case. We show that NN-GLS admits a representation as a special type of graph neural network (GNN). This connection facilitates use of standard neural network computational techniques for irregular geospatial data, enabling novel and scalable mini-batching, backpropagation, and kriging schemes. Theoretically, we show that NN-GLS will be consistent for irregularly observed spatially correlated data processes. To our knowledge this is the first asymptotic consistency result for any neural network algorithm for spatial data. We demonstrate the methodology through simulated and real datasets