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This paper introduces kDGLM, an R package designed for Bayesian analysis of Generalized Dynamic Linear Models (GDLM), with a primary focus on both uni- and multivariate exponential families. Emphasizing sequential inference for time series data, the kDGLM package provides comprehensive support for fitting, smoothing, monitoring, and feed-forward interventions. The methodology employed by kDGLM, as proposed in Alves et al. (2024), seamlessly integrates with well-established techniques from the literature, particularly those used in (Gaussian) Dynamic Models. These include discount strategies, autoregressive components, transfer functions, and more. Leveraging key properties of the Kalman filter and smoothing, kDGLM exhibits remarkable computational efficiency, enabling virtually instantaneous fitting times that scale linearly with the length of the time series. This characteristic makes it an exceptionally powerful tool for the analysis of extended time series. For example, when modeling monthly hospital admissions in Brazil due to gastroenteritis from 2010 to 2022, the fitting process took a mere 0.11s. Even in a spatial-time variant of the model (27 outcomes, 110 latent states, and 156 months, yielding 17,160 parameters), the fitting time was only 4.24s. Currently, the kDGLM package supports a range of distributions, including univariate Normal (unknown mean and observational variance), bivariate Normal (unknown means, observational variances, and correlation), Poisson, Gamma (known shape and unknown mean), and Multinomial (known number of trials and unknown event probabilities). Additionally, kDGLM allows the joint modeling of multiple time series, provided each series follows one of the supported distributions. Ongoing efforts aim to continuously expand the supported distributions.

In this paper, we propose a generic approach to perform global sensitivity analysis (GSA) for compartmental models based on continuous-time Markov chains (CTMC). This approach enables a complete GSA for epidemic models, in which not only the effects of uncertain parameters such as epidemic parameters (transmission rate, mean sojourn duration in compartments) are quantified, but also those of intrinsic randomness and interactions between the two. The main step in our approach is to build a deterministic representation of the underlying continuous-time Markov chain by controlling the latent variables modeling intrinsic randomness. Then, model output can be written as a deterministic function of both uncertain parameters and controlled latent variables, so that it becomespossible to compute standard variance-based sensitivity indices, e.g. the so-called Sobol' indices. However, different simulation algorithms lead to different representations. We exhibit in this work three different representations for CTMC stochastic compartmental models and discuss the results obtained by implementing and comparing GSAs based on each of these representations on a SARS-CoV-2 epidemic model.

In this paper, we study parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency (as well as the speed of convergence) under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes.

In this paper, we extend our work to the Bayesian inverse problems for inferring unknown forcing and initial condition of the forward Navier-Stokes equation coupled with tracer equation with noisy Lagrangian observation on the positions of the tracers. We consider the Navier-Stokes equations in the two dimensional periodic torus with a tracer equation which is a simple ordinary differential equation. We developed rigorously the theory for the case of the uniform prior where the forcing and the initial condition depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. Numerical experiment using the MLMCMC method produces approximations for posterior expectation of quantities of interest which are in agreement with the theoretical optimal convergence rate established.

In this work, we extend the data-driven It\^{o} stochastic differential equation (SDE) framework for the pathwise assessment of short-term forecast errors to account for the time-dependent upper bound that naturally constrains the observable historical data and forecast. We propose a new nonlinear and time-inhomogeneous SDE model with a Jacobi-type diffusion term for the phenomenon of interest, simultaneously driven by the forecast and the constraining upper bound. We rigorously demonstrate the existence and uniqueness of a strong solution to the SDE model by imposing a condition for the time-varying mean-reversion parameter appearing in the drift term. The normalized forecast function is thresholded to keep such mean-reversion parameters bounded. The SDE model parameter calibration also covers the thresholding parameter of the normalized forecast by applying a novel iterative two-stage optimization procedure to user-selected approximations of the likelihood function. Another novel contribution is estimating the transition density of the forecast error process, not known analytically in a closed form, through a tailored kernel smoothing technique with the control variate method. We fit the model to the 2019 photovoltaic (PV) solar power daily production and forecast data in Uruguay, computing the daily maximum solar PV production estimation. Two statistical versions of the constrained SDE model are fit, with the beta and truncated normal distributions as proxies for the transition density. Empirical results include simulations of the normalized solar PV power production and pathwise confidence bands generated through an indirect inference method. An objective comparison of optimal parametric points associated with the two selected statistical approximations is provided by applying the innovative kernel density estimation technique of the transition function of the forecast error process.

We present a physics-informed neural network (PINN) approach for the discovery of slow invariant manifolds (SIMs), for the most general class of fast/slow dynamical systems of ODEs. In contrast to other machine learning (ML) approaches that construct reduced order black box surrogate models using simple regression, and/or require a priori knowledge of the fast and slow variables, our approach, simultaneously decomposes the vector field into fast and slow components and provides a functional of the underlying SIM in a closed form. The decomposition is achieved by finding a transformation of the state variables to the fast and slow ones, which enables the derivation of an explicit, in terms of fast variables, SIM functional. The latter is obtained by solving a PDE corresponding to the invariance equation within the Geometric Singular Perturbation Theory (GSPT) using a single-layer feedforward neural network with symbolic differentiation. The performance of the proposed physics-informed ML framework is assessed via three benchmark problems: the Michaelis-Menten, the target mediated drug disposition (TMDD) reaction model and a fully competitive substrate-inhibitor(fCSI) mechanism. We also provide a comparison with other GPST methods, namely the quasi steady state approximation (QSSA), the partial equilibrium approximation (PEA) and CSP with one and two iterations. We show that the proposed PINN scheme provides SIM approximations, of equivalent or even higher accuracy, than those provided by QSSA, PEA and CSP, especially close to the boundaries of the underlying SIMs.

The use of Artificial Intelligence (AI) based on data-driven algorithms has become ubiquitous in today's society. Yet, in many cases and especially when stakes are high, humans still make final decisions. The critical question, therefore, is whether AI helps humans make better decisions as compared to a human alone or AI an alone. We introduce a new methodological framework that can be used to answer experimentally this question with no additional assumptions. We measure a decision maker's ability to make correct decisions using standard classification metrics based on the baseline potential outcome. We consider a single-blinded experimental design, in which the provision of AI-generated recommendations is randomized across cases with a human making final decisions. Under this experimental design, we show how to compare the performance of three alternative decision-making systems--human-alone, human-with-AI, and AI-alone. We apply the proposed methodology to the data from our own randomized controlled trial of a pretrial risk assessment instrument. We find that AI recommendations do not improve the classification accuracy of a judge's decision to impose cash bail. Our analysis also shows that AI-alone decisions generally perform worse than human decisions with or without AI assistance. Finally, AI recommendations tend to impose cash bail on non-white arrestees more often than necessary when compared to white arrestees.

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces, establishing that MIONet can approximate mappings with multiple inputs in metric spaces. Subsequently, we construct a set consisting of some appropriate regions and provide a metric on this set thus make it a metric space, which satisfies the approximation condition of MIONet. Building upon the theoretical foundation, we are able to learn the solution mapping of a PDE with all the parameters varying, including the parameters of the differential operator, the right-hand side term, the boundary condition, as well as the domain. Without loss of generality, we for example perform the experiments for 2-d Poisson equations, where the domains and the right-hand side terms are varying. The results provide insights into the performance of this method across convex polygons, polar regions with smooth boundary, and predictions for different levels of discretization on one task. We also show the additional result of the fully-parameterized case in the appendix for interested readers. Reasonably, we point out that this is a meshless method, hence can be flexibly used as a general solver for a type of PDE.

Compared to other techniques, particle swarm optimization is more frequently utilized because of its ease of use and low variability. However, it is complicated to find the best possible solution in the search space in large-scale optimization problems. Moreover, changing algorithm variables does not influence algorithm convergence much. The PSO algorithm can be combined with other algorithms. It can use their advantages and operators to solve this problem. Therefore, this paper proposes the onlooker multi-parent crossover discrete particle swarm optimization (OMPCDPSO). To improve the efficiency of the DPSO algorithm, we utilized multi-parent crossover on the best solutions. We performed an independent and intensive neighborhood search using the onlooker bees of the bee algorithm. The algorithm uses onlooker bees and crossover. They do local search (exploitation) and global search (exploration). Each of these searches is among the best solutions (employed bees). The proposed algorithm was tested on the allocation problem, which is an NP-hard optimization problem. Also, we used two types of simulated data. They were used to test the scalability and complexity of the better algorithm. Also, fourteen 2D test functions and thirteen 30D test functions were used. They also used twenty IEEE CEC2005 benchmark functions to test the efficiency of OMPCDPSO. Also, to test OMPCDPSO's performance, we compared it to four new binary optimization algorithms and three classic ones. The results show that the OMPCDPSO version had high capability. It performed better than other algorithms. The developed algorithm in this research (OMCDPSO) in 36 test functions out of 47 (76.60%) is better than other algorithms. The Onlooker bees and multi-parent operators significantly impact the algorithm's performance.

In recent years, power analysis has become widely used in applied sciences, with the increasing importance of the replicability issue. When distribution-free methods, such as Partial Least Squares (PLS)-based approaches, are considered, formulating power analysis turns out to be challenging. In this study, we introduce the methodological framework of a new procedure for performing power analysis when PLS-based methods are used. Data are simulated by the Monte Carlo method, assuming the null hypothesis of no effect is false and exploiting the latent structure estimated by PLS in the pilot data. In this way, the complex correlation data structure is explicitly considered in power analysis and sample size estimation. The paper offers insights into selecting statistical tests for the power analysis procedure, comparing accuracy-based tests and those based on continuous parameters estimated by PLS. Simulated and real datasets are investigated to show how the method works in practice.