This paper makes 3 contributions. First, it generalizes the Lindeberg\textendash Feller and Lyapunov Central Limit Theorems to Hilbert Spaces by way of $L^2$. Second, it generalizes these results to spaces in which sample failure and missingness can occur. Finally, it shows that satisfaction of the Lindeberg\textendash Feller Condition in such spaces guarantees the consistency of all inferences from the partial functional data with respect to the completely observed data. These latter two results are especially important given the increasing attention to statistical inference with partially observed functional data. This paper goes beyond previous research by providing simple boundedness conditions which guarantee that \textit{all} inferences, as opposed to some proper subset of them, will be consistently estimated. This is shown primarily by aggregating conditional expectations with respect to the space of missingness patterns. This paper appears to be the first to apply this technique.
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With the ever-increasing availability of data, there has been an explosion of interest in applying modern machine learning methods to fields such as modeling and control. However, despite the flexibility and surprising accuracy of such black-box models, it remains difficult to trust them. Recent efforts to combine the two approaches aim to develop flexible models that nonetheless generalize well; a paradigm we call Hybrid Analysis and modeling (HAM). In this work we investigate the Corrective Source Term Approach (CoSTA), which uses a data-driven model to correct a misspecified physics-based model. This enables us to develop models that make accurate predictions even when the underlying physics of the problem is not well understood. We apply CoSTA to model the Hall-H\'eroult process in an aluminum electrolysis cell. We demonstrate that the method improves both accuracy and predictive stability, yielding an overall more trustworthy model.
We study a class of McKean--Vlasov Stochastic Differential Equations (MV-SDEs) with drifts and diffusions having super-linear growth in measure and space -- the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift's super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient require novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC), and under further assumptions on the model parameters we show an exponential ergodicity property alongside the existence of an invariant distribution. No differentiability or non-degeneracy conditions are required. Further, we present a particle system based Euler-type split-step scheme (SSM) for the simulation of this type of MV-SDEs. The scheme attains, in stepsize, the strong error rate $1/2$ in the non-path-space root-mean-square error metric and we demonstrate the property of mean-square contraction. Our results are illustrated by numerical examples including: estimation of PoC rates across dimensions, preservation of periodic phase-space, and the observation that taming appears to be not a suitable method unless strong dissipativity is present.
Out-of-sample prediction is the acid test of predictive models, yet an independent test dataset is often not available for assessment of the prediction error. For this reason, out-of-sample performance is commonly estimated using data splitting algorithms such as cross-validation or the bootstrap. For quantitative outcomes, the ratio of variance explained to total variance can be summarized by the coefficient of determination or in-sample $R^2$, which is easy to interpret and to compare across different outcome variables. As opposed to the in-sample $R^2$, the out-of-sample $R^2$ has not been well defined and the variability on the out-of-sample $\hat{R}^2$ has been largely ignored. Usually only its point estimate is reported, hampering formal comparison of predictability of different outcome variables. Here we explicitly define the out-of-sample $R^2$ as a comparison of two predictive models, provide an unbiased estimator and exploit recent theoretical advances on uncertainty of data splitting estimates to provide a standard error for the $\hat{R}^2$. The performance of the estimators for the $R^2$ and its standard error are investigated in a simulation study. We demonstrate our new method by constructing confidence intervals and comparing models for prediction of quantitative $\text{Brassica napus}$ and $\text{Zea mays}$ phenotypes based on gene expression data.
Motivated by a real-world application, we model and solve a complex staff scheduling problem. Tasks are to be assigned to workers for supervision. Multiple tasks can be covered in parallel by a single worker, with worker shifts being flexible within availabilities. Each worker has a different skill set, enabling them to cover different tasks. Tasks require assignment according to priority and skill requirements. The objective is to maximize the number of assigned tasks weighted by their priorities, while minimizing assignment penalties. We develop an adaptive large neighborhood search (ALNS) algorithm, relying on tailored destroy and repair operators. It is tested on benchmark instances derived from real-world data and compared to optimal results obtained by means of a commercial MIP-solver. Furthermore, we analyze the impact of considering three additional alternative objective functions. When applied to large-scale company data, the developed ALNS outperforms the previously applied solution approach.
Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For each sampling scheme, the inverse problem is formulated as a regularized convex optimization problem that minimizes the distance between forward model outputs and observations. The optimization problem is solved by the Alternating Direction Method of Multipliers algorithm, which also handles the situation when a linear inequality constraint (e.g., non-negativity) is imposed on the model output. Numerical examples are presented, code is made available on GitHub, and discussions are provided to generate some useful insights of the proposed inverse modeling approaches.
Learning diverse skills is one of the main challenges in robotics. To this end, imitation learning approaches have achieved impressive results. These methods require explicitly labeled datasets or assume consistent skill execution to enable learning and active control of individual behaviors, which limits their applicability. In this work, we propose a cooperative adversarial method for obtaining single versatile policies with controllable skill sets from unlabeled datasets containing diverse state transition patterns by maximizing their discriminability. Moreover, we show that by utilizing unsupervised skill discovery in the generative adversarial imitation learning framework, novel and useful skills emerge with successful task fulfillment. Finally, the obtained versatile policies are tested on an agile quadruped robot called Solo 8 and present faithful replications of diverse skills encoded in the demonstrations.
Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations, and the convergence estimates that are available are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly, and also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods.
Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference, playing a crucial role in optimal treatment allocation, generalizability, subgroup effects, and more. Many flexible methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a nonparametric model where distributional components are Holder-smooth, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. More specifically, our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods; it is shown to be minimax optimal under a condition on how accurately the covariate distribution is estimated. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid. We conclude with some discussion of a few remaining open problems.
Convergence (virtual) bidding is an important part of two-settlement electric power markets as it can effectively reduce discrepancies between the day-ahead and real-time markets. Consequently, there is extensive research into the bidding strategies of virtual participants aiming to obtain optimal bids to submit to the day-ahead market. In this paper, we introduce a price-based general stochastic optimization framework to obtain optimal convergence bid curves. Within this framework, we develop a computationally tractable linear programming-based optimization model, which produces bid prices and volumes simultaneously. We also show that different approximations and simplifications in the general model lead naturally to state-of-the-art convergence bidding approaches, such as self-scheduling and opportunistic approaches. Our general framework also provides a straightforward way to compare the performance of these models, which is demonstrated by numerical experiments on the California (CAISO) market.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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