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The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes. [ABSTRACT FROM AUTHOR]

This paper presents a novel moving horizon estimation approach based on distributionally robust optimisation to tackle the state estimation problem of non-linear systems with missing noise distribution information. The proposed method adopts a fuzzy set to mitigate the impact of uncertainties on state estimation. Specifically, the method derives an empirical distribution within the prediction window using a priori data and constructs a fuzzy sphere set using the Wasserstein metric with the empirical distribution as the sphere centre. This enables the estimation of the state sequence under the worst probability distribution of the fuzzy set. To demonstrate the effectiveness of the proposed method, a simple simulation example is conducted to compare its performance with that of traditional moving horizon estimation. The results provide evidence of the feasibility and superiority of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the general mean-field backward doubly stochastic differential equations (mean-field BDSDEs) whose generator f can be discontinuous in y. We prove the existence theorem of solutions under stochastic linear growth conditions and also obtain the related comparison theorem. Naturally, we present those results under the linear growth condition, which is a special case of the stochastic condition. Finally, a financial claim sale problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance. [ABSTRACT FROM AUTHOR]

This paper introduces link information value obtained by the traffic sensors and presents a traffic sensor location and flow estimation joint optimization model in an urban road network. In contrast to most previous studies, this paper adds new traffic sensors into the existing sensor network and proposes a data-driven path flow measurement method based on Wasserstein metric, which is utilized to measure the distance between the estimated traffic flow distribution and the actual distribution. Furthermore, this paper develops a customized greedy algorithm by combining a search strategy for the link information value to obtain the optimal sensor location scheme and perform traffic flow estimation under different budget conditions. Numerical experiments are conducted on Sioux-Falls test network and Eastern Massachusetts interstate highway subnetwork to verify the accuracy and effectiveness of the proposed model based on Wasserstein metric and the developed solution method. Computational results show that the sensor location scheme generated by the model based on Wasserstein metric can reduce the estimation error of the traffic flow compared with KL divergence model under the same deployment cost. Additionally, the customized greedy algorithm can achieve the better performance than the Brute force algorithm in terms of computing time and solution quality. • A traffic sensor location and path flow estimation joint optimization model is formulated. • A data-driven path flow measurement method based on Wasserstein metric is proposed. • A customized greedy algorithm by combining a search strategy is developed. • Numerical experiments are conducted based on the real-word transportation network. [ABSTRACT FROM AUTHOR]

An optimal control problem in the space of probability measures and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a smooth Fourier--Wasserstein metric. A comparison result between the Lipschitz viscosity sub- and supersolutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. [ABSTRACT FROM AUTHOR]

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of ℝ. An order‐1 autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one‐step conditional Fréchet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalizes or extends certain existing models of transportation‐based regression and autoregression, and in doing so also provides some additional insights on existing models. [ABSTRACT FROM AUTHOR]

We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing δ , AMN is shown to compute the desired zero set up to a factor of δ in the Wasserstein error metric, with failure probability O (δ 4 log 2 (1 / δ)) . We also provide numerical tests and comparison with other algorithms. [ABSTRACT FROM AUTHOR]

By further exploring and deeply analyzing the concrete algebraic structures and essential computational properties about the Wasserstein-1 metric matrices of one- and two-dimensions, we show that they can be essentially expressed by the Neumann series of nilpotent matrices and, therefore, the products of these matrices with a prescribed vector can be accomplished accurately and stably in the optimal computational complexities through solving unit bidiagonal triangular systems of linear equations. We also give appropriate generalizations of these one- and two-dimensional Wasserstein-1 metric matrices, as well as their corresponding extensions to higher dimensions, and demonstrate the algebraic structures and computational properties of these generalized and extended Wasserstein-1 metric matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider a data-driven distributionally robust two-stage stochastic linear optimization problem over 1-Wasserstein ball centered at a discrete empirical distribution. Differently from the traditional two-stage stochastic programming which involves the expected recourse function as the preference criterion and hence is risk-neutral, we take the conditional value-at-risk (CVaR) as the risk measure in order to model its effects on decision making problems. We mainly explore tractable reformulations for the proposed robust two-stage stochastic programming with mean-CVaR criterion by analyzing the first case where uncertainties are only in the objective function and then the second case where uncertainties are only in the constraints. We demonstrate that the first model can be exactly reformulated as a deterministic convex programming. Furthermore, it is shown that under several different support sets, the resulting convex optimization problems can be converted into computationally tractable conic programmings. Besides, the second model is generally NP-hard since checking constraint feasibility can be reduced to a norm maximization problem over a polytope. However, even with the case of uncertainty in constraints, tractable conic reformulations can be established when the extreme points of the polytope are known. Finally, we present numerical results to discuss how to control the risk for the best decisions and illustrate the computational effectiveness and superiority of the proposed models. [ABSTRACT FROM AUTHOR]

In many practical situations, we need to change the spatial distribution of some goods. In such situations, it is desirable to minimize the overall transportation costs. In the 1-D case, the smallest transportation cost of such a change is proportional to what is known as the Wasserstein metric. The same metric can be used to describe the distance between two probability distributions. In the last decades, it turned out that this metric can be successfully used in many economic applications beyond transportation. These successes are somewhat of a mystery: in principle, there are many different metrics on the set of all probability distributions, and it is not immediately clear why namely Wasserstein metric is so successful. In this paper, we show that the Wasserstein metric naturally appears in decision making situations. This fact explains the usefulness of the Wassterstein's metric in economic applications. [ABSTRACT FROM AUTHOR]

The study of events distributed over time which can be quantified as point processes has attracted much interest over the years due to its wide range of applications. It has recently gained new relevance due to the COVID-19 case and death processes associated with SARS-CoV-2 that characterize the COVID-19 pandemic and are observed across different countries. It is of interest to study the behavior of these point processes and how they may be related to covariates such as mobility restrictions, gross domestic product per capita, and fraction of population of older age. As infections and deaths in a region are intrinsically events that arrive at random times, a point process approach is natural for this setting. We adopt techniques for conditional functional point processes that target point processes as responses with vector covariates as predictors, to study the interaction and optimal transport between case and death processes and doubling times conditional on covariates. [ABSTRACT FROM AUTHOR]

In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings ( DRCCPs ) with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. This approximation is essentially exact under certain conditions. We derive a convex relaxation of this approximation model by constructing new decision variables which allows us to eliminate biconvex terms. Specifically, when the constraint function is affine in both the decision variable and the uncertainty, the resulting approximation model is equivalent to a tractable mixed-integer convex reformulation for joint binary DRCCP. Numerical results illustrate the computational effectiveness and superiority of the proposed formulations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic optimization problem with multivariate second-order stochastic dominance (MSSD) constraints where the distribution of uncertain parameters is unknown. Instead, only some historical data are available. Using the Wasserstein metric, we construct an ambiguity set and develop a data-driven distributionally robust optimization model with multivariate second-order stochastic dominance constraints (DROMSSD). By utilizing the linear scalarization function, we transform MSSD constraints into univariate constraints. We present a stability analysis focusing on the impact of the variation of the ambiguity set on the optimal value and optimal solutions. Moreover, we carry out quantitative stability analysis for the DROMSSD problems as the sample size increases. Specially, in the context of the portfolio, we propose a convex lower reformulation of the corresponding DROMSSD models under some mild conditions. Finally, some preliminary numerical test results are reported. We compare the DROMSSD model with the sample average approximation model through out-of-sample performance, certificate and reliability. We also use real stock data to verify the effectiveness of the DROSSM model. [ABSTRACT FROM AUTHOR]

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis κ -Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the κ -objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich–Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich–Rubinstein framework. We call our proposal the κ -Graph-Space Optimal Transport FWI (κ -GSOT-FWI). The results suggest that the κ -GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis κ -statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when κ = 0.6 . [ABSTRACT FROM AUTHOR]

In this paper, a modified conjugate gradient method under the forward-backward splitting framework is proposed to further improve the numerical efficiency for solving the distributionally robust Logistic regression model over the Wasserstein ball, which comprises two phases: in the first phase, a conjugate gradient descent step is performed, and in the second phase, an instantaneous optimization problem is formulated and solved with a trade-off minimization of the regularization term, while simultaneously staying in close proximity to the interim point obtained in the first phase. The modified conjugate gradient method is proven to attain the optimal solution of the Wasserstein distributionally robust Logistic regression model with nonsummable steplength at a convergence rate of 1 / T . Finally, several numerical experiments to validate the effectiveness of theoretical analysis are conducted, which demonstrate that this method outperforms the off-the-shelf solver and the existing first-order algorithmic frameworks. [ABSTRACT FROM AUTHOR]

In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems' application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs. [ABSTRACT FROM AUTHOR]

• A new distributionally robust mean-variance model with Wasserstein metric was developed. • The novel model was transformed into a tractable convex problem by using duality theory. • A nonparametric bootstrap method was introduced to estimate the radius of the Wasserstein ball. • The effects of the parameters and the advantages of the model were demonstrated by some experiments. The mean-variance model formulated by Markowitz for a single period serves as a fundamental method of modern portfolio selection. In this study, we consider a multi-period case with uncertainty that better matches the reality of the financial market. Using the Wasserstein metric to characterize the uncertainty of returns in each period, a new distributionally robust mean-variance model is proposed to solve multi-period portfolio selection problem. We further transform the developed model into a tractable convex problem using duality theory. We also apply a nonparametric bootstrap method and provide a specific algorithm to estimate the radius of the Wasserstein ball. The effects of the parameters on the corresponding strategy and evaluation criteria of portfolios are analyzed using in-sample data. The analysis indicate that the return and risk of our portfolio selections are relatively immune to parameter values. Finally, a series of out-of-sample experiments demonstrate that the proposed model is superior to some other models in terms of final wealth, standard deviation, and Sharpe ratio. [ABSTRACT FROM AUTHOR]

Risk forms are real functionals of two arguments: a bounded measurable function on a Polish space and a probability measure on that space. They are convenient mathematical structures adapting the coherent risk measures to the situation of a variable reference probability measure. We introduce a new class of risk forms called mini-batch forms. We construct them by using a random empirical probability measure as the second argument and by post-composition with the expected value operator. We prove that coherent and law invariant risk forms generate mini-batch risk forms which are well defined on the space of integrable random variables, and we derive their dual representation. We demonstrate how unbiased stochastic subgradients of such risk forms can be constructed. Then, we consider pre-compositions of mini-batch risk forms with nonsmooth and nonconvex functions, which are differentiable in a generalized way, and we derive generalized subgradients and unbiased stochastic subgradients of such compositions. Finally, we study the dependence of risk forms and mini-batch risk forms on perturbation of the probability measure and establish quantitative stability in terms of optimal transport metrics. We obtain finite-sample expected error estimates for mini-batch risk forms involving functions on a finite-dimensional space. [ABSTRACT FROM AUTHOR]

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data are not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron--Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

We develop a class of hierarchically low-rank, scalable optimal transport dissimilarity measures for structured data, bringing the current state-of-the-art optimal transport solvers to a higher level of performance. Given two n-dimensional discrete probability measures supported on two structured grids in R d , we present a fast method for computing an entropically regularized optimal transport distance, referred to as the debiased Sinkhorn distance. The method combines Sinkhorn's matrix scaling iteration with a low-rank hierarchical representation of the scaling matrices to achieve a near-linear complexity O (n ln 4 n) . This provides a fast, scalable, and easy-to-implement algorithm for computing a class of optimal transport dissimilarity measures, enabling their applicability to large-scale optimization problems, where the computation of the classical Wasserstein metric is not feasible. We carry out a rigorous error-complexity analysis for the proposed algorithm and present several numerical examples to verify the accuracy and efficiency of the algorithm and to demonstrate its applicability in tackling real-world problems. [ABSTRACT FROM AUTHOR]

We present a framework for performing regression when both covariate and response are probability distributions on a compact interval. Our regression model is based on the theory of optimal transportation, and links the conditional Fréchet mean of the response to the covariate via an optimal transport map. We define a Fréchet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observations of the regression pairs. Computation of the estimator is shown to reduce to a standard convex optimization problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data. [ABSTRACT FROM AUTHOR]

Magnetic resonance imaging (MRI) and computed tomography (CT) are the prevalent imaging techniques used in treatment planning in radiation therapy. Since MR‐only radiation therapy planning (RTP) is needed in the future for new technologies like MR‐LINAC (medical linear accelerator), MR to CT synthesis model benefits in CT synthesis from MR images generated via MR‐LINAC. A Wasserstein generative adversarial network (WGAN) architecture with a residual UNet based generator and a patch‐based discriminator is proposed in this paper. The WGAN uses the Wasserstein metric with gradient penalty along with mean absolute error (MAE) as the loss function. The WGAN is trained and tested on an NVIDIA Tesla V100 GPU server using mutually aligned MR‐CT brain images of 26 patients. The proposed model generates synthetic CTs with an average PSNR of 31.09 dB, SSIM of 0.9265, MAE of 48.39 HU, and RMSE of 172.75 HU. The treatment planning was performed on real and synthetic CTs. The dose distributions of the synthetic CTs obtained were similar to that of the corresponding real CTs. The synthetic CTs exhibit good visual quality as well. The experimental outcomes and image quality of the model surpass many of the state‐of‐the‐art methods. The proposed CT synthesizer supports MR‐only radiation therapy while minimizing treatment costs and radiation exposure. A significant increase in the training set can further improve the CT prediction accuracy of the proposed model. [ABSTRACT FROM AUTHOR]

There are several approaches for evaluating conflict within belief functions. In this paper, we develop one of them based on axioms and show its connections to the decomposition approach. We describe a class of conflict measures satisfying this system of axioms and show that measuring conflict can be realized through the clusterization of a body of evidence. We also show that well-known conflict measures like the auto-conflict measure and the measure of dissonance do not satisfy the proposed system of axioms. We also tackle the problem of simplifying a body of evidence based on clusterization and show the application of developed theoretical constructions for data processing. [ABSTRACT FROM AUTHOR]

Electrocardiograms (ECG) analysis is one of the most important ways to diagnose heart disease. This paper proposes an efficient ECG classification method based on Wasserstein scalar curvature to comprehend the connection between heart disease and the mathematical characteristics of ECG. The newly proposed method converts an ECG into a point cloud on the family of Gaussian distribution, where the pathological characteristics of ECG will be extracted by the Wasserstein geometric structure of the statistical manifold. Technically, this paper defines the histogram dispersion of Wasserstein scalar curvature, which can accurately describe the divergence between different heart diseases. By combining medical experience with mathematical ideas from geometry and data science, this paper provides a feasible algorithm for the new method, and the theoretical analysis of the algorithm is carried out. Digital experiments on the classical database with large samples show the new algorithm's accuracy and efficiency when dealing with the classification of heart disease. [ABSTRACT FROM AUTHOR]

We propose a distributionally robust optimization formulation with a Wasserstein-based uncertainty set for selecting grouped variables under perturbations on the data for both linear regression and classification problems. The resulting model offers robustness explanations for grouped least absolute shrinkage and selection operator algorithms and highlights the connection between robustness and regularization. We prove probabilistic bounds on the out-of-sample loss and the estimation bias, and establish the grouping effect of our estimator, showing that coefficients in the same group converge to the same value as the sample correlation between covariates approaches 1. Based on this result, we propose to use the spectral clustering algorithm with the Gaussian similarity function to perform grouping on the predictors, which makes our approach applicable without knowing the grouping structure a priori. We compare our approach to an array of alternatives and provide extensive numerical results on both synthetic data and a real large dataset of surgery-related medical records, showing that our formulation produces an interpretable and parsimonious model that encourages sparsity at a group level and is able to achieve better prediction and estimation performance in the presence of outliers. [ABSTRACT FROM AUTHOR]

Hydro-wind-solar integrated operation is a promising way to balance the growing amount of variable renewable energy (RE) and enhance energy utilization efficiency. This study focuses on the short-term reliable-economic equilibrium operation of the hydro-wind-solar energy systems. A Distributionally Robust Optimization based on Wasserstein metric, which considers multiple power supply risks (output shortage, power curtailment and spilled water) and the economy (the water consumption and the stored energy) of hydropower operation simultaneously is proposed to address the above issue. The model is transformed into a mixed-integer linear programming framework using a reformation approach based on ϵ -constraint method, strong duality theory, and linearization technology. Case studies are conducted for a hydro-wind-solar energy system in Southwest China's Beipan River basin. Results indicate that: (1) with more available data, more information about RE uncertainties is involved in the decision-making process to hedge against RE variability's interference. (2) Decisions obtained from the proposed model overcome the issues of stochastic optimization's over-optimism and robust optimization's over-conservativeness. (3) Reliable-economic trade-off mechanisms are derived by imposing a decision-maker's risk attitude, the risks in dry and wet season typical cases decline from 10483.05 MW to 1606.39 MW and from 8394.15 MW to 118.08 MW by adjusting the economy. [ABSTRACT FROM AUTHOR]

Transient electromagnetic method (TEM), as one of the essential time-domain electromagnetic prospecting approaches, has the advantage of expedition, efficiency and convenience. In this paper, we study the transient electromagnetic inversion problem of different geological anomalies. First, Maxwell's differential equations are discretized by the staggered finite-difference (FD) method; then we propose to solve the TEM inversion problem by minimizing the Wasserstein metric, which is related to the optimal transport (OT). Experimental tests based on the layered model and a 3D model are performed to demonstrate the feasibility of our proposed method. [ABSTRACT FROM AUTHOR]

This paper addresses facility location, inventory pre-positioning and allocation of emergency supplies in disaster relief logistics by taking into account both primary and secondary disasters. To characterize the uncertainty associated with post-disaster demand and resource allocation cost, this paper constructs the statistical-distance-based ambiguity sets of possible probability distributions with the Wasserstein metric, which is utilized to measure their distances from the empirical distribution. Armed with the Wasserstein ambiguity set, this paper develops a hybrid risk-averse three-stage distributionally robust chance-constrained (TS-DRCC) model for the considered problem, which measures the risk from both quantitative and qualitative aspects. When the Wasserstein metric uses the l 1 -norm, this paper reformulates the proposed TS-DRCC model as a mixed-integer linear program (MILP) based on the strong duality theory, which can be efficiently solved via CPLEX, thereby enabling decision-makers to use it. Theoretically, this paper also proves that the proposed TS-DRCC model converges to stochastic programming (SP) model as the size of historical data approaches infinity. Finally, this paper conducts a computational study of hurricane threat in the US to indicate the superiority of our proposed TS-DRCC model in terms of demand satisfaction and out-of-sample performance compared to the model considering only primary disasters and the conventional SP model, respectively. Some key managerial insights are summarized as rules of thumb to effectively guide the integrated pre- and post-disaster relief actions in the disaster relief logistics planning practice. • A location-inventory-allocation problem with secondary disaster considerations in DRL is introduced. • A hybrid risk-averse TS-DRCC model for the LIAP under demand and cost ambiguities is formulated. • The Wasserstein ambiguity sets are constructed to address the distributional ambiguities. • Extensive numerical experiments are carried out on a realistic case study. [ABSTRACT FROM AUTHOR]

Modeling conditional distributions for functional data extends the concept of a mean response in functional regression settings, where vector predictors are paired with functional responses. This extension is challenging because of the nonexistence of well‐defined densities, cumulative distributions, or quantile functions in the Hilbert space where the response functions are located. To address this challenge, we simplify the problem by assuming that the response functions are Gaussian processes, which means that the conditional distribution of the responses is determined by conditional mean and conditional covariance. We demonstrate that these quantities can be obtained by applying global and local Fréchet regression, where the local version is more flexible and applicable when the covariate dimension is low and covariates are continuous, while the global version is not subject to these restrictions but is based on the assumption of a more restrictive regression relation. Convergence rates for the proposed estimates are obtained under the framework of M‐estimation. The corresponding estimation of conditional distributions is illustrated with simulations and an application to bike‐sharing data, where predictors include weather characteristics and responses are bike rental profiles. We also show that our methods are applicable to the challenging problem to study functional fragments. Such data are observed in accelerated longitudinal studies and correspond to functional data observed over short domain segments. We demonstrate the utility of conditional distributions in this context by using the time (age) at which a subject enters the domain of a fragment in addition to other covariates as predictor and the function observed over the domain of the fragment as response. [ABSTRACT FROM AUTHOR]

We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on R d , uniform in the Planck constant ℏ ∈ [ 0 , 1 ]. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in ℏ and all parameters involved. [ABSTRACT FROM AUTHOR]

We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover's distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest and provide an overview of various settings leading to tractable formulations. Some of the arising side results, such as the mathematical programming expressions for robustified risk measures in a discrete space, are also of independent interest. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications, and present a numerical study for a novel risk-averse scheduling problem with controllable processing times. Summary of Contribution: In this study, we introduce a new class of optimization problems that simultaneously address distributional and decision-dependent uncertainty. We present a unified modeling framework along with a discussion on possible ways to specify the key model components, and discuss the main computational challenges in solving the complex problems of interest. Special care has been devoted to identifying the settings and problem classes where these challenges can be mitigated. In particular, we provide model reformulation results, including mathematical programming expressions for robustified risk measures, and describe how these results can be utilized to obtain tractable formulations for specific applied problems from the fields of humanitarian logistics and machine scheduling. Toward demonstrating the value of the modeling approach and investigating the performance of the proposed mixed-integer linear programming formulations, we conduct a computational study on a novel risk-averse machine scheduling problem with controllable processing times. We derive insights regarding the decision-making impact of our modeling approach and key parameter choices. [ABSTRACT FROM AUTHOR]

Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions. [ABSTRACT FROM AUTHOR]

We study the distributionally robust linearized stable tail adjusted return ratio (DRLSTARR) portfolio optimization problem, in which the objective is to maximize the worst-case linearized stable tail adjusted return ratio (LSTARR) performance measure under data-driven Wasserstein ambiguity. We consider two types of imperfectly known uncertainties, named uncertain probabilities and continuum of realizations, associated with the losses of assets. We account for two typical combinatorial trading constraints, called buy-in threshold and diversification constraints, to reflect stock market restrictions. Leveraging conic duality theory to tackle the distributionally robust worst-case expectation, the proposed problems are reformulated into mixed-integer linear programming problems. We carry out a series of empirical tests to illustrate the scalability and effectiveness of the proposed solution framework, and to evaluate the performance of the DRLSTARR-constructed portfolios. The cross-validation results obtained using a rolling-horizon procedure show the superior out-of-sample performance of the DRLSTARR portfolios under an uncertain continuum of realizations. [ABSTRACT FROM AUTHOR]

Data consisting of time‐indexed distributions of cross‐sectional or intraday returns have been extensively studied in finance, and provide one example in which the data atoms consist of serially dependent probability distributions. Motivated by such data, we propose an autoregressive model for density time series by exploiting the tangent space structure on the space of distributions that is induced by the Wasserstein metric. The densities themselves are not assumed to have any specific parametric form, leading to flexible forecasting of future unobserved densities. The main estimation targets in the order‐p Wasserstein autoregressive model are Wasserstein autocorrelations and the vector‐valued autoregressive parameter. We propose suitable estimators and establish their asymptotic normality, which is verified in a simulation study. The new order‐p Wasserstein autoregressive model leads to a prediction algorithm, which includes a data driven order selection procedure. Its performance is compared to existing prediction procedures via application to four financial return data sets, where a variety of metrics are used to quantify forecasting accuracy. For most metrics, the proposed model outperforms existing methods in two of the data sets, while the best empirical performance in the other two data sets is attained by existing methods based on functional transformations of the densities. [ABSTRACT FROM AUTHOR]

We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both f-divergences and integral probability metrics (IPMs), such as the 1-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as (f,Γ)-divergences, provide a notion of 'distance' between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The (f,Γ)-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from f-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of f. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation. [ABSTRACT FROM AUTHOR]

We present a novel class of projected methods, to perform statistical analysis on a data set of probability distributions on the real line, with the 2-Wasserstein metric. We focus in particular on Principal Component Analysis (PCA) and regression. To define these models, we exploit a representation of the Wasserstein space closely related to its weak Riemannian structure, by mapping the data to a suitable linear space and using a metric projection operator to constrain the results in the Wasserstein space. By carefully choosing the tangent point, we are able to derive fast empirical methods, exploiting a constrained B-spline approximation. As a byproduct of our approach, we are also able to derive faster routines for previous work on PCA for distributions. By means of simulation studies, we compare our approaches to previously proposed methods, showing that our projected PCA has similar performance for a fraction of the computational cost and that the projected regression is extremely flexible even under misspecification. Several theoretical properties of the models are investigated and asymptotic consistency is proven. Two real world applications to Covid-19 mortality in the US and wind speed forecasting are discussed. [ABSTRACT FROM AUTHOR]

A rise in traffic accidents has led to both traffic congestion and subsequent secondary accidents. Effectively addressing this issue requires rapid accident investigation and management. In this paper, we aim to improve the efficiency of traffic accident assessment and investigation with the aid of drone technologies. Our approach involves strategically pre-positioning drones, enabling traffic supervisory agencies to dispatch drones immediately upon receiving an accident report. Methodology-wise, we present a data-driven robust stochastic optimization (RSO) model, which encapsulates the uncertainty of traffic accidents within a scenario-wise Wasserstein ambiguity set. To the best of our knowledge, this is the first study that incorporates covariates, i.e., weather conditions, into the Wasserstein ambiguity set with the CVaR metric. We demonstrate that the proposed RSO model can be reformulated into a mixed-integer programming model, allowing an efficient solution approach. Via a real-world dataset of London traffic accidents, we validate the practical applicability of the RSO model. Across various parameter settings, our RSO model exhibits superior out-of-sample performance compared with various benchmark models. The numerical results yield valuable insights for traffic supervisory agencies. • We propose a drone pre-positioning problem for traffic accident rapid assessment. • We propose an RSO model that incorporates the uncertain traffic accident. • We demonstrate that the RSO model admits a tractable MILP reformulation. • A real world instance is used to verify the practicality of our model. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence of invariant measures in the Wasserstein sense for fractional stochastic reaction–diffusion equations defined on unbounded domains as the noise intensity approaches zero. Based on uniform estimates of solutions, we prove the family of invariant measures of the stochastic equations converges to the invariant measure of the corresponding deterministic equations in terms of the Wasserstein metric. We also provide the rate of such convergence. [ABSTRACT FROM AUTHOR]

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first-order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide an explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least-squares regression. We consider robustness of call option pricing and deduce a new Black-Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples. [ABSTRACT FROM AUTHOR]

Scenario reduction techniques are widely applied for solving sophisticated dynamic and stochastic programs, especially in energy and power systems, but are also used in probabilistic forecasting, clustering and estimating generative adversarial networks. We propose a new method for ensemble and scenario reduction based on the energy distance which is a special case of the maximum mean discrepancy. We discuss the choice of energy distance in detail, especially in comparison to the popular Wasserstein distance which is dominating the scenario reduction literature. The energy distance is a metric between probability measures that allows for powerful tests for equality of arbitrary multivariate distributions or independence. Thanks to the latter, it is a suitable candidate for ensemble and scenario reduction problems. The theoretical properties and considered examples indicate clearly that the reduced scenario sets tend to exhibit better statistical properties for the energy distance than a corresponding reduction with respect to the Wasserstein distance. We show applications to a Bernoulli random walk and two real data-based examples for electricity demand profiles and day-ahead electricity prices. [ABSTRACT FROM AUTHOR]

We investigate a class of fractional distributionally robust optimization problems with uncertain probabilities. They consist in the maximization of ambiguous fractional functions representing reward-risk ratios and have a semi-infinite programming epigraphic formulation. We derive a new fully parameterized closed-form to compute a new bound on the size of the Wasserstein ambiguity ball. We design a data-driven reformulation and solution framework. The reformulation phase involves the derivation of the support function of the ambiguity set and the concave conjugate of the ratio function. We design modular bisection algorithms which enjoy the finite convergence property. This class of problems has wide applicability in finance, and we specify new ambiguous portfolio optimization models for the Sharpe and Omega ratios. The computational study shows the applicability and scalability of the framework to solve quickly large, industry-relevant-size problems, which cannot be solved in one day with state-of-the-art mixed-integer nonlinear programming (MINLP) solvers. [ABSTRACT FROM AUTHOR]

We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire. [ABSTRACT FROM AUTHOR]

Langevin Monte Carlo (LMC) is a popular Bayesian sampling method. For the logconcave distribution function, the method converges exponentially fast, up to a controllable discretization error. However, the method requires the evaluation of a full gradient in each iteration, and for a problem on Rd, this amounts to d times partial derivative evaluations per iteration. The cost is high when d = 1. In this paper, we investigate how to enhance computational efficiency through the application of RCD (random coordinate descent) on LMC. There are two sides of the theory: By blindly applying RCD to LMC, one surrogates the full gradient by a randomly selected directional derivative per iteration. Although the cost is reduced per iteration, the total number of iteration is increased to achieve a preset error tolerance. Ultimately there is no computational gain; We then incorporate variance reduction techniques, such as SAGA (stochastic average gradient) and SVRG (stochastic variance reduced gradient), into RCD-LMC. It will be proved that the cost is reduced compared with the classical LMC, and in the underdamped case, convergence is achieved with the same number of iterations, while each iteration requires merely one directional derivative. This means we obtain the best possible computational cost in the underdamped-LMC framework. [ABSTRACT FROM AUTHOR]

The interconnection of multiple regional integrated energy systems (RIES) can effectively enhance the low-carbon and flexible operation capabilities of RIES, but the uncertainty and multi-energy interaction of the system pose challenges to the stable operation of RIES. Therefore, a stochastic hierarchical optimization and revenue allocation approach is proposed to optimize the operational strategy of multi-RIES with multi-operator participation, multi-energy interactions, and multiple uncertainties. Firstly, agent-based modeling, Latin hypercube sampling and simultaneous backward reduction based on the Wasserstein metric are proposed to capture the power-side and load-side uncertainties. Secondly, based on the Nash bargaining game and the alternating direction method of multipliers algorithm, multi-RIES peer-to-peer transactions of electricity, thermal, and natural gas are optimized through energy cooperation and scheduling strategies. Subsequently, a Nash-Harsanyi bargaining game revenue allocation method that considers both fairness and renewable energy accommodation is proposed to ensure the stable operation of the alliance. Finally, simulations are conducted on a multi-RIES in northern China, demonstrating that the proposed model and approach can achieve inter-grid flexible resource complementarity, improve RIES economics, increase local renewable energy accommodation rate, and reduce carbon emissions. • Multi-RIES multi-energy internal scheduling and P2P transaction co-optimization. • Wasserstein metric is introduced into the SBR model to capture stochastic scenarios. • A multi-RIES stochastic hierarchical optimization model based on cooperative games. • ADMM is used to solve distributed stochastic hierarchical optimization model. • A revenue allocation method considering equity & renewable energy development is proposed. [ABSTRACT FROM AUTHOR]

Anomalies in dynamical systems mostly occur as deviations between measurement and prediction. Current anomaly detection methods in multivariate time series often require prior clustering, training data, or cannot distinguish local and global anomalies. Furthermore, no generalized metric exists to evaluate and compare different prediction functions regarding their amount of anomalous behavior. We propose a novel methodology to detect local and global anomalies in time series data of dynamical systems. For this purpose, a theoretical density distribution is derived assuming that only noise conceals the time series. If the theoretical and the empirical density distribution yield significantly different entropies, an anomaly is assumed. For a local anomaly detection, the Mahalanobis distance using the theoretical noise distribution’s covariance is applied to evaluate sequences of predictions and measurements. In addition, the Wasserstein metric enables a comparison of predictions using the distance between the noise and empirical distribution as a measure for selecting the best prediction function. The proposed method performs well on nonlinear time series such as logistic growth and enables a useful selection of a prediction model for satellite orbits. Thus, the proposed method improves anomaly detection in time series and model selection for nonlinear systems. [ABSTRACT FROM AUTHOR]

There are numerous industrial settings in which a decision maker must decide whether to enter into long-term contracts to guarantee price (and hence cash flow) stability or to participate in more volatile spot markets. In this paper, we investigate a data-driven distributionally robust optimization (DRO) approach aimed at balancing this tradeoff. Unlike traditional risk-neutral stochastic optimization models that assume the underlying probability distribution generating the data is known, DRO models assume the distribution belongs to a family of possible distributions, thus providing a degree of immunization against unseen and potential worst-case outcomes. We compare and contrast the performance of a risk-neutral model, conditional value-at-risk formulation, and a Wasserstein distributionally robust model to demonstrate the potential benefits of a DRO approach for an "elasticity-aware" price-taking decision maker. • Investigate tradeoffs between long-term supply contracts vs. spot market sales. • Data-driven DRO approach exploiting Wasserstein ambiguity sets. • Compare risk-neutral, conditional value-at-risk, and data-driven DRO approaches. • Case studies in power portfolio optimization for PJM electricity market. [ABSTRACT FROM AUTHOR]