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Statistical models typically capture uncertainties in our knowledge of the corresponding real-world processes, however, it is less common for this uncertainty specification to capture uncertainty surrounding the values of the inputs to the model, which are often assumed known. We develop general modelling methodology with uncertain inputs in the context of the Bayes linear paradigm, which involves adjustment of second-order belief specifications over all quantities of interest only, without the requirement for probabilistic specifications. In particular, we propose an extension of commonly-employed second-order modelling assumptions to the case of uncertain inputs, with explicit implementation in the context of regression analysis, stochastic process modelling, and statistical emulation. We apply the methodology to a regression model for extracting aluminium by electrolysis, and emulation of the motivating epidemiological simulator chain to model the impact of an airborne infectious disease.

Discovering causal relations from observational data is important. The existence of unobserved variables (e.g. latent confounding or mediation) can mislead the causal identification. To overcome this problem, proximal causal discovery methods attempted to adjust for the bias via the proxy of the unobserved variable. Particularly, hypothesis test-based methods proposed to identify the causal edge by testing the induced violation of linearity. However, these methods only apply to discrete data with strict level constraints, which limits their practice in the real world. In this paper, we fix this problem by extending the proximal hypothesis test to cases where the system consists of continuous variables. Our strategy is to present regularity conditions on the conditional distributions of the observed variables given the hidden factor, such that if we discretize its observed proxy with sufficiently fine, finite bins, the involved discretization error can be effectively controlled. Based on this, we can convert the problem of testing continuous causal relations to that of testing discrete causal relations in each bin, which can be effectively solved with existing methods. These non-parametric regularities we present are mild and can be satisfied by a wide range of structural causal models. Using both simulated and real-world data, we show the effectiveness of our method in recovering causal relations when unobserved variables exist.

Feature screening is an important tool in analyzing ultrahigh-dimensional data, particularly in the field of Omics and oncology studies. However, most attention has been focused on identifying features that have a linear or monotonic impact on the response variable. Detecting a sparse set of variables that have a nonlinear or non-monotonic relationship with the response variable is still a challenging task. To fill the gap, this paper proposed a robust model-free screening approach for right-censored survival data by providing a new perspective of quantifying the covariate effect on the restricted mean survival time, rather than the routinely used hazard function. The proposed measure, based on the difference between the restricted mean survival time of covariate-stratified and overall data, is able to identify comprehensive types of associations including linear, nonlinear, non-monotone, and even local dependencies like change points. This approach is highly interpretable and flexible without any distribution assumption. The sure screening property is established and an iterative screening procedure is developed to address multicollinearity between high-dimensional covariates. Simulation studies are carried out to demonstrate the superiority of the proposed method in selecting important features with a complex association with the response variable. The potential of applying the proposed method to handle interval-censored failure time data has also been explored in simulations, and the results have been promising. The method is applied to a breast cancer dataset to identify potential prognostic factors, which reveals potential associations between breast cancer and lymphoma.

Accurate time series forecasting is a fundamental challenge in data science. It is often affected by external covariates such as weather or human intervention, which in many applications, may be predicted with reasonable accuracy. We refer to them as predicted future covariates. However, existing methods that attempt to predict time series in an iterative manner with autoregressive models end up with exponential error accumulations. Other strategies hat consider the past and future in the encoder and decoder respectively limit themselves by dealing with the historical and future data separately. To address these limitations, a novel feature representation strategy -- shifting -- is proposed to fuse the past data and future covariates such that their interactions can be considered. To extract complex dynamics in time series, we develop a parallel deep learning framework composed of RNN and CNN, both of which are used hierarchically. We also utilize the skip connection technique to improve the model's performance. Extensive experiments on three datasets reveal the effectiveness of our method. Finally, we demonstrate the model interpretability using the Grad-CAM algorithm.

Model independent techniques for constructing background data templates using generative models have shown great promise for use in searches for new physics processes at the LHC. We introduce a major improvement to the CURTAINs method by training the conditional normalizing flow between two side-band regions using maximum likelihood estimation instead of an optimal transport loss. The new training objective improves the robustness and fidelity of the transformed data and is much faster and easier to train. We compare the performance against the previous approach and the current state of the art using the LHC Olympics anomaly detection dataset, where we see a significant improvement in sensitivity over the original CURTAINs method. Furthermore, CURTAINsF4F requires substantially less computational resources to cover a large number of signal regions than other fully data driven approaches. When using an efficient configuration, an order of magnitude more models can be trained in the same time required for ten signal regions, without a significant drop in performance.

In this paper, we focus our attention on the high-dimensional double sparse linear regression, that is, a combination of element-wise and group-wise sparsity.To address this problem, we propose an IHT-style (iterative hard thresholding) procedure that dynamically updates the threshold at each step. We establish the matching upper and lower bounds for parameter estimation, showing the optimality of our proposal in the minimax sense. Coupled with a novel sparse group information criterion, we develop a fully adaptive procedure to handle unknown group sparsity and noise levels.We show that our adaptive procedure achieves optimal statistical accuracy with fast convergence. Finally, we demonstrate the superiority of our method by comparing it with several state-of-the-art algorithms on both synthetic and real-world datasets.

We propose a new auto-regressive model for the statistical analysis of multivariate distributional time series. The data of interest consist of a collection of multiple series of probability measures supported over a bounded interval of the real line, and that are indexed by distinct time instants. The probability measures are modelled as random objects in the Wasserstein space. We establish the auto-regressive model in the tangent space at the Lebesgue measure by first centering all the raw measures so that their Fr\'echet means turn to be the Lebesgue measure. Using the theory of iterated random function systems, results on the existence, uniqueness and stationarity of the solution of such a model are provided. We also propose a consistent estimator for the model coefficient. In addition to the analysis of simulated data, the proposed model is illustrated with two real data sets made of observations from age distribution in different countries and bike sharing network in Paris. Finally, due to the positive and boundedness constraints that we impose on the model coefficients, the proposed estimator that is learned under these constraints, naturally has a sparse structure. The sparsity allows furthermore the application of the proposed model in learning a graph of temporal dependency from the multivariate distributional time series.

Most existing studies on linear bandits focus on the one-dimensional characterization of the overall system. While being representative, this formulation may fail to model applications with high-dimensional but favorable structures, such as the low-rank tensor representation for recommender systems. To address this limitation, this work studies a general tensor bandits model, where actions and system parameters are represented by tensors as opposed to vectors, and we particularly focus on the case that the unknown system tensor is low-rank. A novel bandit algorithm, coined TOFU (Tensor Optimism in the Face of Uncertainty), is developed. TOFU first leverages flexible tensor regression techniques to estimate low-dimensional subspaces associated with the system tensor. These estimates are then utilized to convert the original problem to a new one with norm constraints on its system parameters. Lastly, a norm-constrained bandit subroutine is adopted by TOFU, which utilizes these constraints to avoid exploring the entire high-dimensional parameter space. Theoretical analyses show that TOFU improves the best-known regret upper bound by a multiplicative factor that grows exponentially in the system order. A novel performance lower bound is also established, which further corroborates the efficiency of TOFU.

Carbon futures has recently emerged as a novel financial asset in the trading markets such as the European Union and China. Monitoring the trend of the carbon price has become critical for both national policy-making as well as industrial manufacturing planning. However, various geopolitical, social, and economic factors can impose substantial influence on the carbon price. Due to its volatility and non-linearity, predicting accurate carbon prices is generally a difficult task. In this study, we propose to improve carbon price forecasting with several novel practices. First, we collect various influencing factors, including commodity prices, export volumes such as oil and natural gas, and prosperity indices. Then we select the most significant factors and disclose their optimal grouping for explainability. Finally, we use the Sparse Quantile Group Lasso and Adaptive Sparse Quantile Group Lasso for robust price predictions. We demonstrate through extensive experimental studies that our proposed methods outperform existing ones. Also, our quantile predictions provide a complete profile of future prices at different levels, which better describes the distributions of the carbon market.

Grade of Membership (GoM) models are popular individual-level mixture models for multivariate categorical data. GoM allows each subject to have mixed memberships in multiple extreme latent profiles. Therefore GoM models have a richer modeling capacity than latent class models that restrict each subject to belong to a single profile. The flexibility of GoM comes at the cost of more challenging identifiability and estimation problems. In this work, we propose a singular value decomposition (SVD) based spectral approach to GoM analysis with multivariate binary responses. Our approach hinges on the observation that the expectation of the data matrix has a low-rank decomposition under a GoM model. For identifiability, we develop sufficient and almost necessary conditions for a notion of expectation identifiability. For estimation, we extract only a few leading singular vectors of the observed data matrix, and exploit the simplex geometry of these vectors to estimate the mixed membership scores and other parameters. Our spectral method has a huge computational advantage over Bayesian or likelihood-based methods and is scalable to large-scale and high-dimensional data. Extensive simulation studies demonstrate the superior efficiency and accuracy of our method. We also illustrate our method by applying it to a personality test dataset.