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This paper presents $\Psi$-GNN, a novel Graph Neural Network (GNN) approach for solving the ubiquitous Poisson PDE problems with mixed boundary conditions. By leveraging the Implicit Layer Theory, $\Psi$-GNN models an ''infinitely'' deep network, thus avoiding the empirical tuning of the number of required Message Passing layers to attain the solution. Its original architecture explicitly takes into account the boundary conditions, a critical prerequisite for physical applications, and is able to adapt to any initially provided solution. $\Psi$-GNN is trained using a ''physics-informed'' loss, and the training process is stable by design, and insensitive to its initialization. Furthermore, the consistency of the approach is theoretically proven, and its flexibility and generalization efficiency are experimentally demonstrated: the same learned model can accurately handle unstructured meshes of various sizes, as well as different boundary conditions. To the best of our knowledge, $\Psi$-GNN is the first physics-informed GNN-based method that can handle various unstructured domains, boundary conditions and initial solutions while also providing convergence guarantees.

This paper develops likelihood-based methods for estimation, inference, model selection, and forecasting of continuous-time integer-valued trawl processes. The full likelihood of integer-valued trawl processes is, in general, highly intractable, motivating the use of composite likelihood methods, where we consider the pairwise likelihood in lieu of the full likelihood. Maximizing the pairwise likelihood of the data yields an estimator of the parameter vector of the model, and we prove consistency and, in the short memory case, asymptotic normality of this estimator. When the underlying trawl process has long memory, the asymptotic behaviour of the estimator is more involved; we present some partial results for this case. The pairwise approach further allows us to develop probabilistic forecasting methods, which can be used to construct the predictive distribution of integer-valued time series. In a simulation study, we document the good finite sample performance of the likelihood-based estimator and the associated model selection procedure. Lastly, the methods are illustrated in an application to modelling and forecasting financial bid-ask spread data, where we find that it is beneficial to carefully model both the marginal distribution and the autocorrelation structure of the data.

Current methods for pattern analysis in time series mainly rely on statistical features or probabilistic learning and inference methods to identify patterns and trends in the data. Such methods do not generalize well when applied to multivariate, multi-source, state-varying, and noisy time-series data. To address these issues, we propose a highly generalizable method that uses information theory-based features to identify and learn from patterns in multivariate time-series data. To demonstrate the proposed approach, we analyze pattern changes in human activity data. For applications with stochastic state transitions, features are developed based on Shannon's entropy of Markov chains, entropy rates of Markov chains, entropy production of Markov chains, and von Neumann entropy of Markov chains. For applications where state modeling is not applicable, we utilize five entropy variants, including approximate entropy, increment entropy, dispersion entropy, phase entropy, and slope entropy. The results show the proposed information theory-based features improve the recall rate, F1 score, and accuracy on average by up to 23.01\% compared with the baseline models and a simpler model structure, with an average reduction of 18.75 times in the number of model parameters.

Stability selection represents an attractive approach to identify sparse sets of features jointly associated with an outcome in high-dimensional contexts. We introduce an automated calibration procedure via maximisation of an in-house stability score and accommodating a priori-known block structure (e.g. multi-OMIC) data. It applies to (LASSO) penalised regression and graphical models. Simulations show our approach outperforms non-stability-based and stability selection approaches using the original calibration. Application of multi-block graphical LASSO on real (epigenetic and transcriptomic) data from the Norwegian Women and Cancer study reveals a central/credible and novel cross-OMIC role of LRRN3 in the biological response to smoking. Proposed approaches were implemented in the R package sharp.

This work is motivated by the need to accurately model a vector of responses related to pediatric functional status using administrative health data from inpatient rehabilitation visits. The components of the responses have known and structured interrelationships. To make use of these relationships in modeling, we develop a two-pronged regularization approach to borrow information across the responses. The first component of our approach encourages joint selection of the effects of each variable across possibly overlapping groups related responses and the second component encourages shrinkage of effects towards each other for related responses. As the responses in our motivating study are not normally-distributed, our approach does not rely on an assumption of multivariate normality of the responses. We show that with an adaptive version of our penalty, our approach results in the same asymptotic distribution of estimates as if we had known in advance which variables were non-zero and which variables have the same effects across some outcomes. We demonstrate the performance of our method in extensive numerical studies and in an application in the prediction of functional status of pediatric patients using administrative health data in a population of children with neurological injury or illness at a large children's hospital.

Randomized controlled trials (RCTs) are considered the gold standard for estimating the average treatment effect (ATE) of interventions. One use of RCTs is to study the causes of global poverty -- a subject explicitly cited in the 2019 Nobel Memorial Prize awarded to Duflo, Banerjee, and Kremer "for their experimental approach to alleviating global poverty." Because the ATE is a population summary, anti-poverty experiments often seek to unpack the effect variation around the ATE by conditioning (CATE) on tabular variables such as age and ethnicity that were measured during the RCT data collection. Although such variables are key to unpacking CATE, using only such variables may fail to capture historical, geographical, or neighborhood-specific contributors to effect variation, as tabular RCT data are often only observed near the time of the experiment. In global poverty research, when the location of the experiment units is approximately known, satellite imagery can provide a window into such factors important for understanding heterogeneity. However, there is no method that specifically enables applied researchers to analyze CATE from images. In this paper, using a deep probabilistic modeling framework, we develop such a method that estimates latent clusters of images by identifying images with similar treatment effects distributions. Our interpretable image CATE model also includes a sensitivity factor that quantifies the importance of image segments contributing to the effect cluster prediction. We compare the proposed methods against alternatives in simulation; also, we show how the model works in an actual RCT, estimating the effects of an anti-poverty intervention in northern Uganda and obtaining a posterior predictive distribution over effects for the rest of the country where no experimental data was collected. We make all models available in open-source software.

Survival analysis studies time-modeling techniques for an event of interest occurring for a population. Survival analysis found widespread applications in healthcare, engineering, and social sciences. However, the data needed to train survival models are often distributed, incomplete, censored, and confidential. In this context, federated learning can be exploited to tremendously improve the quality of the models trained on distributed data while preserving user privacy. However, federated survival analysis is still in its early development, and there is no common benchmarking dataset to test federated survival models. This work provides a novel technique for constructing realistic heterogeneous datasets by starting from existing non-federated datasets in a reproducible way. Specifically, we propose two dataset-splitting algorithms based on the Dirichlet distribution to assign each data sample to a carefully chosen client: quantity-skewed splitting and label-skewed splitting. Furthermore, these algorithms allow for obtaining different levels of heterogeneity by changing a single hyperparameter. Finally, numerical experiments provide a quantitative evaluation of the heterogeneity level using log-rank tests and a qualitative analysis of the generated splits. The implementation of the proposed methods is publicly available in favor of reproducibility and to encourage common practices to simulate federated environments for survival analysis.

Analyzing observational data from multiple sources can be useful for increasing statistical power to detect a treatment effect; however, practical constraints such as privacy considerations may restrict individual-level information sharing across data sets. This paper develops federated methods that only utilize summary-level information from heterogeneous data sets. Our federated methods provide doubly-robust point estimates of treatment effects as well as variance estimates. We derive the asymptotic distributions of our federated estimators, which are shown to be asymptotically equivalent to the corresponding estimators from the combined, individual-level data. We show that to achieve these properties, federated methods should be adjusted based on conditions such as whether models are correctly specified and stable across heterogeneous data sets.

Federated Learning (FL) is a decentralized machine-learning paradigm, in which a global server iteratively averages the model parameters of local users without accessing their data. User heterogeneity has imposed significant challenges to FL, which can incur drifted global models that are slow to converge. Knowledge Distillation has recently emerged to tackle this issue, by refining the server model using aggregated knowledge from heterogeneous users, other than directly averaging their model parameters. This approach, however, depends on a proxy dataset, making it impractical unless such a prerequisite is satisfied. Moreover, the ensemble knowledge is not fully utilized to guide local model learning, which may in turn affect the quality of the aggregated model. Inspired by the prior art, we propose a data-free knowledge distillation} approach to address heterogeneous FL, where the server learns a lightweight generator to ensemble user information in a data-free manner, which is then broadcasted to users, regulating local training using the learned knowledge as an inductive bias. Empirical studies powered by theoretical implications show that, our approach facilitates FL with better generalization performance using fewer communication rounds, compared with the state-of-the-art.