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We study the application of a neural network architecture for identifying charged particle trajectories via unsupervised learning of delays and synaptic weights using a spike-time-dependent plasticity rule. In the considered model, the neurons receive time-encoded information on the position of particle hits in a tracking detector for a particle collider, modeled according to the geometry of the Compact Muon Solenoid Phase II detector. We show how a spiking neural network is capable of successfully identifying in a completely unsupervised way the signal left by charged particles in the presence of conspicuous noise from accidental or combinatorial hits. These results open the way to applications of neuromorphic computing to particle tracking, motivating further studies into its potential for real-time, low-power particle tracking in future high-energy physics experiments., Comment: 24 pages, 21 figures, submitted to MDPI Particles

Despite the increasing importance of stochastic processes on linear networks and graphs, current literature on multivariate (vector-valued) Gaussian random fields on metric graphs is elusive. This paper challenges several aspects related to the construction of proper matrix-valued kernels structures. We start by considering matrix-valued metrics that can be composed with scalar- or matrix-valued functions to implement valid kernels associated with vector-valued Gaussian fields. We then provide conditions for certain classes of matrix-valued functions to be composed with the univariate resistance metric and ensure positive semidefiniteness. Special attention is then devoted to Euclidean trees, where a substantial effort is required given the absence of literature related to multivariate kernels depending on the $\ell_1$ metric. Hence, we provide a foundational contribution to certain classes of matrix-valued positive semidefinite functions depending on the $\ell_1$ metric. This fact is then used to characterise kernels on Euclidean trees with a finite number of leaves. Amongst those, we provide classes of matrix-valued covariance functions that are compactly supported.

There exists a plethora of parametric models for positive definite kernels, and their use is ubiquitous in disciplines as diverse as statistics, machine learning, numerical analysis, and approximation theory. Usually, the kernel parameters index certain features of an associated process. Amongst those features, smoothness (in the sense of Sobolev spaces, mean square differentiability, and fractal dimensions), compact or global supports, and negative dependencies (hole effects) are of interest to several theoretical and applied disciplines. This paper unifies a wealth of well-known kernels into a single parametric class that encompasses them as special cases, attained either by exact parameterization or through parametric asymptotics. We furthermore characterize the Sobolev space that is norm equivalent to the RKHS associated with the new kernel. As a by-product, we infer the Sobolev spaces that are associated with existing classes of kernels. We illustrate the main properties of the new class, show how this class can switch from compact to global supports, and provide special cases for which the kernel attains negative values over nontrivial intervals. Hence, the proposed class of kernel is the reproducing kernel of a very rich Hilbert space that contains many special cases, including the celebrated Mat\'ern and Wendland kernels, as well as their aliases with hole effects.

A huge literature in statistics and machine learning is devoted to parametric families of correlation functions, where the correlation parameters are used to understand the properties of an associated spatial random process in terms of smoothness and global or compact support. However, most of current parametric correlation functions attain only non-negative values. This work provides two new families that parameterize negative dependencies (aka hole effects), along with smoothness, and global or compact support. They generalize the celebrated Mat\'ern and Generalized Wendland models, respectively, which are attained as special cases. A link between the two new families is also established, showing that a specific reparameterization of the latter includes the former as a special limit case. Their performance in terms of estimation accuracy and goodness of best linear unbiased prediction is illustrated through synthetic and real data.

The commonly accepted definition of paths starts from a random field but ignores the problem of setting joint distributions of infinitely many random variables for defining paths properly afterwards. This paper provides a turnaround that starts with a given covariance function, then defines paths and finally a random field. We show how this approach retains essentially the same properties for Gaussian fields while allowing to construct random fields whose finite dimensional distributions are not Gaussian. Specifically, we start with a kernel $C$ and the associated Reproducing Kernel Hilbert Space ${\cal H}(C)$, and then assign standardized random values to a deterministic orthonormal expansion in ${\cal H}(C)$. This yields paths as random functions with an explicit representation formula. Using Lo\'eve isometry, we prove that pointwise regularity notions like continuity or differentiability hold on functions of ${\cal H}(C)$, paths, and the random field $R_C$ in precisely the same way. Yet, norms of paths as functions behave differently, as we prove that paths are a.s. not in ${\cal H}(C)$, but in certain larger spaces that can partially be characterized. In case of Matern kernels generating Sobolev space $W_2^m(R^d)$, paths lie almost surely in all $W_2^{p}(R^d)$ for $p

This work addresses the problem of simulating Gaussian random fields that are continuously indexed over a class of metric graphs, termed graphs with Euclidean edges, being more general and flexible than linear networks. We introduce three general algorithms that allow to reconstruct a wide spectrum of random fields having a covariance function that depends on a specific metric, called resistance metric, and proposed in recent literature. The algorithms are applied to a synthetic case study consisting of a street network. They prove to be fast and accurate in that they reproduce the target covariance function and provide random fields whose finite-dimensional distributions are approximately Gaussian.

This paper considers generalised network, intended as networks where (a) the edges connecting the nodes are nonlinear, and (b) stochastic processes are continuously indexed over both vertices and edges. Such topological structures are normally represented through special classes of graphs, termed graphs with Euclidean edges. We build generalised networks in which topology changes over time instants. That is, vertices and edges can disappear at subsequent time instants and edges may change in shape and length. We consider both cases of linear or circular time. For the second case, the generalised network exhibits a periodic structure. Our findings allow to illustrate pros and cons of each setting. Generalised networks become semi-metric spaces whenever equipped with a proper semi-metric. Our approach allows to build proper semi-metrics for the temporally-evolving topological structures of the networks. Our final effort is then devoted to guiding the reader through appropriate choice of classes of functions that allow to build proper reproducing kernels when composed with the temporally-evolving semi-metrics topological structures., Comment: 48 pages, 13 figures

We deal with the comparison of space-time covariance kernels having, either, full, spatially dynamical, or space-time compact support. Such a comparison is based on compatibility of these covariance models under fixed domain asymptotics, having a theoretical background that is substantially coming from equivalence or orthogonality of Gaussian measures. In turn, such a theory is intimately related to the tails of the spectral densities associated with the three models. Models with space-time compact support are still elusive. We taper the temporal part of a model with dynamical support, obtaining a space-time compact support. The spectrum related to such a construction is obtained through temporal convolution of the spatially dynamical spectrum with the spectrum associated with the temporal taper. The solution of such a challenge opens the door to the compatibility-based comparison. Our findings show that indeed these three models can be compatible under some suitable parametric restrictions. As a corollary, we deduce implications in terms of maximum likelihood estimation and misspecified kriging prediction under fixed domain asymptotics., Comment: 28 pages

This work marks the third in a series of experiments that were in a semi-circular, gas-fluidized bed with side jets. In this work, the particles are 1 mm ceramic beads. The bed is operated just at and slightly above and below the minimum fluidization velocity and additional fluidization is provided by two high-speed gas located on the sides of the bed near the flat, front face of the unit. Two primary measurements are taken: high-speed video recording of the front of the bed and bed pressure drop from a tap in the back of the bed. PIV is used to determine particle motion, characterized as a mean Froude number, from the high-speed video. A CFD-DEM model of the bed is presented using the recently released MFIX-Exa code. Four model subvariants are considered using two methods of representing the jets and two drag models, both of which are calibrated to exactly match the experimentally measured minimum fluidization velocity. Although it is more difficult to determine the jet penetration depths in a straightforward manner as in the previous works using Froude number contours, the CFD-DEM results compare quite well to the PIV measurements. Unfortunately, the good agreement of the solids-phase is overshadowed by significant disagreement in the gas-phase data. Specifically, the predicted time-averaged standard deviation of the pressure drop is found to be over an order of magnitude larger than measured. Due to the low value of the measurements, just 1% of the mean bed pressure drop, it seems possible that the data is in error. On the other hand, the model may not be accurately capturing pressure attenuation through an under-fluidized region in the back of the bed. Without the possibility additional experiments to test the validity of the data, this work is simply being reported as is without being able to indicate which, either the simulation or the experiment, is more correct.

The Mat\'ern model has been a cornerstone of spatial statistics for more than half a century. More recently, the Mat\'ern model has been central to disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Mat\'ern-based journey across these disciplines. First, we reflect on the importance of the Mat\'ern model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Mat\'ern model has been influential. Then, we position the Mat\'ern model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Mat\'ern model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.

Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Having covariance functions that allow to specify both local and global properties is certainly on demand. This paper provides a method to find new classes of covariance functions having such properties. We term these models hybrid as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combination of parametric families of measures. In order to illustrate our methodology, we provide new families of covariance functions that are proved to be richer with respect to other well known families that have been proposed by earlier literature. More precisely, we derive a hybrid Cauchy-Mat\'ern model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid Hole-Effect-Mat\'ern model, which is capable of attaining negative values (hole effect), while preserving the local attributes of the traditional Mat\'ern model. Our findings are illustrated through numerical studies with both simulated and real data., Comment: 19 pages, 6 figures

This work provides theoretical foundations for kernel methods in the hyperspherical context. Specifically, we characterise the native spaces (reproducing kernel Hilbert spaces) and the Sobolev spaces associated with kernels defined over hyperspheres. Our results have direct consequences for kernel cubature, determining the rate of convergence of the worst case error, and expanding the applicability of cubature algorithms based on Stein's method. We first introduce a suitable characterisation on Sobolev spaces on the $d$-dimensional hypersphere embedded in $(d+1)$-dimensional Euclidean spaces. Our characterisation is based on the Fourier--Schoenberg sequences associated with a given kernel. Such sequences are hard (if not impossible) to compute analytically on $d$-dimensional spheres, but often feasible over Hilbert spheres. We circumvent this problem by finding a projection operator that allows to Fourier mapping from Hilbert into finite dimensional hyperspheres. We illustrate our findings through some parametric families of kernels.

The advent of data science has provided an increasing number of challenges with high data complexity. This paper addresses the challenge of space-time data where the spatial domain is not a planar surface, a sphere, or a linear network, but a generalized network (termed a graph with Euclidean edges). Additionally, data are repeatedly measured over different temporal instants. We provide new classes of nonseparable space-time stationary covariance functions where {\em space} can be a generalized network, a Euclidean tree, or a linear network, and where time can be linear or circular (seasonal). Because the construction principles are technical, we focus on illustrations that guide the reader through the construction of statistically interpretable examples. A simulation study demonstrates that we can recover the correct model when compared to misspecified models. In addition, our simulation studies show that we effectively recover simulation parameters. In our data analysis, we consider a traffic accident dataset that shows improved model performance based on covariance specifications and network-based metrics.

Let $d,k$ be positive integers. We call generalized spaces the cartesian product of the $d$-dimensional sphere, $\mathbb{S}^d$, with the $k$-dimensional Euclidean space, $\mathbb{R}^k$. We consider the class ${\mathcal P}(\mathbb{S}^d \times \mathbb{R}^k)$ of continuous functions $\varphi: [-1,1] \times [0,\infty) \to \mathbb{R}$ such that the mapping $C: \left ( \mathbb{S}^d \times\mathbb{R}^k \right )^2 \to \mathbb{R}$, defined as $C \Big ( (x,y),(x^{\prime},y^{\prime})\Big ) = \varphi \Big ( \cos \theta(x,x^{\prime}), \|y-y^{\prime}\| \Big )$, $(x,y), \; (x^{\prime},y^{\prime}) \in \mathbb{S}^d \times \mathbb{R}^k$, is positive definite. We propose linear operators that allow for walks through dimension within generalized spaces while preserving positive definiteness.

Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into $d$-dimensional Euclidean spaces, $\R^{d}$, have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggest that the problem is quite challenging. This paper provides extension theorem for multiradial characteristic functions that are defined in balls embedded in $\R^d$ cross, either $\R^{\dd}$ or the unit sphere $\S^{\dd}$ embedded in $\R^{\dd+1}$, for any two positive integers $d$ and $\dd$. We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.

The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.