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In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) L\'{e}vy process. Such processes are intractable for simulation except in certain special edge cases, since the L\'{e}vy density associated with the GIG process is expressed as an integral involving certain Bessel Functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct calculation of the integrals involved. The resulting augmented bivariate process is still intractable and so we propose a novel thinning method based on upper bounds on the intractable integrand. Moreover our approach leads to lower and upper bounds on the Jaeger integral itself, which may be compared with other approximation methods. The shot noise method involves a truncated infinite series of decreasing random variables, and as such is approximate, although the series are found to be rapidly convergent in most cases. We note that the GIG process is the required Brownian motion subordinator for the generalised hyperbolic (GH) L\'{e}vy process and so our simulation approach will straightforwardly extend also to the simulation of these intractable proceses. Our new methods will find application in forward simulation of processes of GIG and GH type, in financial and engineering data, for example, as well as inference for states and parameters of stochastic processes driven by GIG and GH L\'{e}vy processes.

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving numerically these equations is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require to solve both direct and inverse problems in matrix spaces. We present two algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms.

We develop a new method to find the number of volatility regimes in a nonstationary financial time series by applying unsupervised learning to its volatility structure. We use change point detection to partition a time series into locally stationary segments and then compute a distance matrix between segment distributions. The segments are clustered into a learned number of discrete volatility regimes via an optimization routine. Using this framework, we determine a volatility clustering structure for financial indices, large-cap equities, exchange-traded funds and currency pairs. Our method overcomes the rigid assumptions necessary to implement many parametric regime-switching models, while effectively distilling a time series into several characteristic behaviours. Our results provide significant simplification of these time series and a strong descriptive analysis of prior behaviours of volatility. Finally, we create and validate a dynamic trading strategy that learns the optimal match between the current distribution of a time series and its past regimes, thereby making online risk-avoidance decisions in the present.

We consider the problem of uncertainty quantification for an unknown low-rank matrix $\mathbf{X}$, given a partial and noisy observation of its entries. This quantification of uncertainty is essential for many real-world problems, including image processing, satellite imaging, and seismology, providing a principled framework for validating scientific conclusions and guiding decision-making. However, existing literature has mainly focused on the completion (i.e., point estimation) of the matrix $\mathbf{X}$, with little work on investigating its uncertainty. To this end, we propose in this work a new Bayesian modeling framework, called BayeSMG, which parametrizes the unknown $\mathbf{X}$ via its underlying row and column subspaces. This Bayesian subspace parametrization enables efficient posterior inference on matrix subspaces, which represents interpretable phenomena in many applications. This can then be leveraged for improved matrix recovery. We demonstrate the effectiveness of BayeSMG over existing Bayesian matrix recovery methods in numerical experiments, image inpainting, and a seismic sensor network application.

Based on the analysis of the proportion of utility in the supporting transactions used in the field of data mining, high utility-occupancy pattern mining (HUOPM) has recently attracted widespread attention. Unlike high-utility pattern mining (HUPM), which involves the enumeration of high-utility (e.g., profitable) patterns, HUOPM aims to find patterns representing a collection of existing transactions. In practical applications, however, not all patterns are used or valuable. For example, a pattern might contain too many items, that is, the pattern might be too specific and therefore lack value for users in real life. To achieve qualified patterns with a flexible length, we constrain the minimum and maximum lengths during the mining process and introduce a novel algorithm for the mining of flexible high utility-occupancy patterns. Our algorithm is referred to as HUOPM+. To ensure the flexibility of the patterns and tighten the upper bound of the utility-occupancy, a strategy called the length upper-bound (LUB) is presented to prune the search space. In addition, a utility-occupancy nested list (UO-nlist) and a frequency-utility-occupancy table (FUO-table) are employed to avoid multiple scans of the database. Evaluation results of the subsequent experiments confirm that the proposed algorithm can effectively control the length of the derived patterns, for both real-world and synthetic datasets. Moreover, it can decrease the execution time and memory consumption.

Joint modeling of a large number of variables often requires dimension reduction strategies that lead to structural assumptions of the underlying correlation matrix, such as equal pair-wise correlations within subsets of variables. The underlying correlation matrix is thus of interest for both model specification and model validation. In this paper, we develop tests of the hypothesis that the entries of the Kendall rank correlation matrix are linear combinations of a smaller number of parameters. The asymptotic behavior of the proposed test statistics is investigated both when the dimension is fixed and when it grows with the sample size. We pay special attention to the restricted hypothesis of partial exchangeability, which contains full exchangeability as a special case. We show that under partial exchangeability, the test statistics and their large-sample distributions simplify, which leads to computational advantages and better performance of the tests. We propose various scalable numerical strategies for implementation of the proposed procedures, investigate their behavior through simulations and power calculations under local alternatives, and demonstrate their use on a real dataset of mean sea levels at various geographical locations.

A novel combination of two widely-used clustering algorithms is proposed here for the detection and reduction of high data density regions. The Density Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is used for the detection of high data density regions and the k-means algorithm for reduction. The proposed algorithm iterates while successively decrementing the DBSCAN search radius, allowing for an adaptive reduction factor based on the effective data density. The algorithm is demonstrated for a physics simulation application, where a surrogate model for fusion reactor plasma turbulence is generated with neural networks. A training dataset for the surrogate model is created with a quasilinear gyrokinetics code for turbulent transport calculations in fusion plasmas. The training set consists of model inputs derived from a repository of experimental measurements, meaning there is a potential risk of over-representing specific regions of this input parameter space. By applying the proposed reduction algorithm to this dataset, this study demonstrates that the training dataset can be reduced by a factor ~20 using the proposed algorithm, without a noticeable loss in the surrogate model accuracy. This reduction provides a novel way of analyzing existing high-dimensional datasets for biases and consequently reducing them, which lowers the cost of re-populating that parameter space with higher quality data.

Graph neural networks (GNNs) are typically applied to static graphs that are assumed to be known upfront. This static input structure is often informed purely by insight of the machine learning practitioner, and might not be optimal for the actual task the GNN is solving. In absence of reliable domain expertise, one might resort to inferring the latent graph structure, which is often difficult due to the vast search space of possible graphs. Here we introduce Pointer Graph Networks (PGNs) which augment sets or graphs with additional inferred edges for improved model generalisation ability. PGNs allow each node to dynamically point to another node, followed by message passing over these pointers. The sparsity of this adaptable graph structure makes learning tractable while still being sufficiently expressive to simulate complex algorithms. Critically, the pointing mechanism is directly supervised to model long-term sequences of operations on classical data structures, incorporating useful structural inductive biases from theoretical computer science. Qualitatively, we demonstrate that PGNs can learn parallelisable variants of pointer-based data structures, namely disjoint set unions and link/cut trees. PGNs generalise out-of-distribution to 5x larger test inputs on dynamic graph connectivity tasks, outperforming unrestricted GNNs and Deep Sets.

Review-based recommender systems have gained noticeable ground in recent years. In addition to the rating scores, those systems are enriched with textual evaluations of items by the users. Neural language processing models, on the other hand, have already found application in recommender systems, mainly as a means of encoding user preference data, with the actual textual description of items serving only as side information. In this paper, a novel approach to incorporating the aforementioned models into the recommendation process is presented. Initially, a neural language processing model and more specifically the paragraph vector model is used to encode textual user reviews of variable length into feature vectors of fixed length. Subsequently this information is fused along with the rating scores in a probabilistic matrix factorization algorithm, based on maximum a-posteriori estimation. The resulting system, ParVecMF, is compared to a ratings' matrix factorization approach on a reference dataset. The obtained preliminary results on a set of two metrics are encouraging and may stimulate further research in this area.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.