Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gröbner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations. Keywords: Gaussian processes boundary conditions Gröbner bases partial differential equations.

Risk-based active learning is an approach to developing statistical classifiers for online decision-support. In this approach, data-label querying is guided according to the expected value of perfect information for incipient data points. For SHM applications, the value of information is evaluated with respect to a maintenance decision process, and the data-label querying corresponds to the inspection of a structure to determine its health state. Sampling bias is a known issue within active-learning paradigms; this occurs when an active learning process over- or undersamples specific regions of a feature-space, thereby resulting in a training set that is not representative of the underlying distribution. This bias ultimately degrades decision-making performance, and as a consequence, results in unnecessary costs incurred. The current paper outlines a risk-based approach to active learning that utilises a semi-supervised Gaussian mixture model. The semi-supervised approach counteracts sampling bias by incorporating pseudo-labels for unlabelled data via an EM algorithm. The approach is demonstrated on a numerical example representative of the decision processes found in SHM.

Causality is a fundamental part of the scientific endeavour to understand the world. Unfortunately, causality is still taboo in much of psychology and social science. Motivated by a growing number of recommendations for the importance of adopting causal approaches to research, we reformulate the typical approach to research in psychology to harmonize inevitably causal theories with the rest of the research pipeline. We present a new process which begins with the incorporation of techniques from the confluence of causal discovery and machine learning for the development, validation, and transparent formal specification of theories. We then present methods for reducing the complexity of the fully specified theoretical model into the fundamental submodel relevant to a given target hypothesis. From here, we establish whether or not the quantity of interest is estimable from the data, and if so, propose the use of semi-parametric machine learning methods for the estimation of causal effects. The overall goal is the presentation of a new research pipeline which can (a) facilitate scientific inquiry compatible with the desire to test causal theories (b) encourage transparent representation of our theories as unambiguous mathematical objects, (c) to tie our statistical models to specific attributes of the theory, thus reducing under-specification problems frequently resulting from the theory-to-model gap, and (d) to yield results and estimates which are causally meaningful and reproducible. The process is demonstrated through didactic examples with real-world data, and we conclude with a summary and discussion of limitations.

The intersection of learning to rank and choice modeling is an active area of research with applications in e-commerce, information retrieval and the social sciences. In some applications such as recommendation systems, the statistician is primarily interested in recovering the set of the top ranked items from a large pool of items as efficiently as possible using passively collected discrete choice data, i.e., the user picks one item from a set of multiple items. Motivated by this practical consideration, we propose the choice-based Borda count algorithm as a fast and accurate ranking algorithm for top $K$-recovery i.e., correctly identifying all of the top $K$ items. We show that the choice-based Borda count algorithm has optimal sample complexity for top-$K$ recovery under a broad class of random utility models. We prove that in the limit, the choice-based Borda count algorithm produces the same top-$K$ estimate as the commonly used Maximum Likelihood Estimate method but the former's speed and simplicity brings considerable advantages in practice. Experiments on both synthetic and real datasets show that the counting algorithm is competitive with commonly used ranking algorithms in terms of accuracy while being several orders of magnitude faster.

Gaining the ability to make informed decisions on operation and maintenance of structures provides motivation for the implementation of structural health monitoring (SHM) systems. However, descriptive labels for measured data corresponding to health-states of the monitored system are often unavailable. This issue limits the applicability of fully-supervised machine learning paradigms for the development of statistical classifiers to be used in decision-support in SHM systems. One approach to dealing with this problem is risk-based active learning. In such an approach, data-label querying is guided according to the expected value of perfect information for incipient data points. For risk-based active learning in SHM, the value of information is evaluated with respect to a maintenance decision process, and the data-label querying corresponds to the inspection of a structure to determine its health state. In the context of SHM, risk-based active learning has only been considered for generative classifiers. The current paper demonstrates several advantages of using an alternative type of classifier -- discriminative models. Using the Z24 Bridge dataset as a case study, it is shown that discriminative classifiers have benefits, in the context of SHM decision-support, including improved robustness to sampling bias, and reduced expenditure on structural inspections.

Increasingly frequent wildfires significantly affect solar energy production as the atmospheric aerosols generated by wildfires diminish the incoming solar radiation to the earth. Atmospheric aerosols are measured by Aerosol Optical Depth (AOD), and AOD data streams can be retrieved and monitored by geostationary satellites. However, multi-source remote-sensing data streams often present heterogeneous characteristics, including different data missing rates, measurement errors, systematic biases, and so on. To accurately estimate and predict the underlying AOD propagation process, there exist practical needs and theoretical interests to propose a physics-informed statistical approach for modeling wildfire AOD propagation by simultaneously utilizing, or fusing, multi-source heterogeneous satellite remote-sensing data streams. Leveraging a spectral approach, the proposed approach integrates multi-source satellite data streams with a fundamental advection-diffusion equation that governs the AOD propagation process. A bias correction process is included in the statistical model to account for the bias of the physics model and the truncation error of the Fourier series. The proposed approach is applied to California wildfires AOD data streams obtained from the National Oceanic and Atmospheric Administration. Comprehensive numerical examples are provided to demonstrate the predictive capabilities and model interpretability of the proposed approach. Computer code has been made available on GitHub.

Shap explanations are a popular feature-attribution mechanism for explainable AI. They use game-theoretic notions to measure the influence of individual features on the prediction of a machine learning model. Despite a lot of recent interest from both academia and industry, it is not known whether Shap explanations of common machine learning models can be computed efficiently. In this paper, we establish the complexity of computing the Shap explanation in three important settings. First, we consider fully-factorized data distributions, and show that the complexity of computing the Shap explanation is the same as the complexity of computing the expected value of the model. This fully-factorized setting is often used to simplify the Shap computation, yet our results show that the computation can be intractable for commonly used models such as logistic regression. Going beyond fully-factorized distributions, we show that computing Shap explanations is already intractable for a very simple setting: computing Shap explanations of trivial classifiers over naive Bayes distributions. Finally, we show that even computing Shap over the empirical distribution is #P-hard.

We consider the Bayesian calibration of models describing the phenomenon of block copolymer (BCP) self-assembly using image data produced by microscopy or X-ray scattering techniques. To account for the random long-range disorder in BCP equilibrium structures, we introduce auxiliary variables to represent this aleatory uncertainty. These variables, however, result in an integrated likelihood for high-dimensional image data that is generally intractable to evaluate. We tackle this challenging Bayesian inference problem using a likelihood-free approach based on measure transport together with the construction of summary statistics for the image data. We also show that expected information gains (EIGs) from the observed data about the model parameters can be computed with no significant additional cost. Lastly, we present a numerical case study based on the Ohta--Kawasaki model for diblock copolymer thin film self-assembly and top-down microscopy characterization. For calibration, we introduce several domain-specific energy- and Fourier-based summary statistics, and quantify their informativeness using EIG. We demonstrate the power of the proposed approach to study the effect of data corruptions and experimental designs on the calibration results.

One of the significant challenges that smart grid networks face is cyber-security. Several studies have been conducted to highlight those security challenges. However, the majority of these surveys classify attacks based on the security requirements, confidentiality, integrity, and availability, without taking into consideration the accountability requirement. In addition, some of these surveys focused on the Transmission Control Protocol/Internet Protocol (TCP/IP) model, which does not differentiate between the application, session, and presentation and the data link and physical layers of the Open System Interconnection (OSI) model. In this survey paper, we provide a classification of attacks based on the OSI model and discuss in more detail the cyber-attacks that can target the different layers of smart grid networks communication. We also propose new classifications for the detection and countermeasure techniques and describe existing techniques under each category. Finally, we discuss challenges and future research directions.

We investigate the cold posterior effect through the lens of PAC-Bayes generalization bounds. We argue that in the non-asymptotic setting, when the number of training samples is (relatively) small, discussions of the cold posterior effect should take into account that approximate Bayesian inference does not readily provide guarantees of performance on out-of-sample data. Instead, out-of-sample error is better described through a generalization bound. In this context, we explore the connections between the ELBO objective from variational inference and the PAC-Bayes objectives. We note that, while the ELBO and PAC-Bayes objectives are similar, the latter objectives naturally contain a temperature parameter $\lambda$ which is not restricted to be $\lambda=1$. For both regression and classification tasks, in the case of isotropic Laplace approximations to the posterior, we show how this PAC-Bayesian interpretation of the temperature parameter captures the cold posterior effect.