Random Function Research Articles

The model and characteristics of polarized light transmission applicable to polydispersity particle underwater environment

The information about the polarization of underwater objects can be affected by the absorption and scattering of water particles, making it challenging to detect underwater targets accurately. Therefore, it is essential to analyze the characteristics of polarization transmission in water. Traditional Monte Carlo simulations have yet to effectively treat particle size in water, which is unsuitable for the underwater environment of polydispersity particles with significant particle size differences. In this paper, a new simulation model using a Monte Carlo method with random sampling fitting phase functions is proposed, which makes the transmission characteristics of water particles more in line with the actual probability model of complex water bodies. The experimental results demonstrate that the proposed Monte Carlo method produces polarization closer to experimental results. The larger the particle size of water suspensions, the greater change of the polarization degree of polarized light. The polarization-maintaining characteristics of circularly polarized light are better than those of linearly polarized light, providing data support for the transmission and detection of underwater light.

A Model-Free Approach for Solving Choice-Based Competitive Facility Location Problems Using Simulation and Submodularity

This paper considers facility location problems in which a firm entering a market seeks to open facilities on a subset of candidate locations so as to maximize its expected market share, assuming that customers choose the available alternative that maximizes a random utility function. We introduce a deterministic equivalent reformulation of this stochastic problem as a maximum covering location problem with an exponential number of demand points, each of which is covered by a different set of candidate locations. Estimating the prevalence of these preference profiles through simulation generalizes a sample average approximation method from the literature and results in a maximum covering location problem of manageable size. To solve it, we develop a partial Benders reformulation in which the contribution to the objective of the least influential preference profiles is aggregated and bounded by submodular cuts. This set of profiles is selected by a knee detection method that seeks to identify the best tradeoff between the fraction of the demand that is retained in the master problem and the size of the model. We develop a theoretical analysis of our approach and show that the solution quality it provides for the original stochastic problem, its computational performance, and the automatic profile-retention strategy it exploits are directly connected to the entropy of the preference profiles in the population. Computational experiments on existing and new benchmark sets indicate that our approach dominates the classical sample average approximation method on large instances of the competitive facility location problem, can outperform the best heuristic method from the literature under the multinomial logit model, and achieves state-of-the-art results under the mixed multinomial logit model. We characterize a broader class of problems, which includes assortment optimization, to which the solving methodology and the analyses developed in this paper can be extended. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This research was supported by Fonds de Recherche du Québec-Nature et Technologies and Institut de Valorisation des Données through scholarships to R. Legault. E. Frejinger was partially supported by the Canada Research Chair program [Grant 950-232244]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/ijoc.2023.0280 .

Ultra-Efficient MCMC for Bayesian Longitudinal Functional Data Analysis

Functional mixed models are widely useful for regression analysis with dependent functional data, including longitudinal functional data with scalar predictors. However, existing algorithms for Bayesian inference with these models only provide either scalable computing or accurate approximations to the posterior distribution, but not both. We introduce a new MCMC sampling strategy for highly efficient and fully Bayesian regression with longitudinal functional data. Using a novel blocking structure paired with an orthogonalized basis reparameterization, our algorithm jointly samples the fixed effects regression functions together with all subject- and replicate-specific random effects functions. Crucially, the joint sampler optimizes sampling efficiency for these key parameters while preserving computational scalability. Perhaps surprisingly, our new MCMC sampling algorithm even surpasses state-of-the-art algorithms for frequentist estimation and variational Bayes approximations for functional mixed models—while also providing accurate posterior uncertainty quantification—and is orders of magnitude faster than existing Gibbs samplers. Simulation studies show improved point estimation and interval coverage in nearly all simulation settings over competing approaches. We apply our method to a large physical activity dataset to study how various demographic and health factors associate with intraday activity. Supplementary materials for this article are available online.

Exact and Approximate Moment Derivation for Probabilistic Loops With Non-Polynomial Assignments

Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art.

Testing independence for sparse longitudinal data

Abstract With the advance of science and technology, more and more data are collected in the form of functions. A fundamental question for a pair of random functions is to test whether they are independent. This problem becomes quite challenging when the random trajectories are sampled irregularly and sparsely for each subject. In other words, each random function is only sampled at a few time-points, and these time-points vary with subjects. Furthermore, the observed data may contain noise. To the best of our knowledge, there exists no consistent test in the literature to test the independence of sparsely observed functional data. We show in this work that testing pointwise independence simultaneously is feasible. The test statistics are constructed by integrating pointwise distance covariances (Székely et al., 2007) and are shown to converge, at a certain rate, to their corresponding population counterparts, which characterize the simultaneous pointwise independence of two random functions. The performance of the proposed methods is further verified by Monte Carlo simulations and analysis of real data.

Diverse randomized value functions: A provably pessimistic approach for offline reinforcement learning

Offline Reinforcement Learning (RL) faces challenges such as distributional shift and unreliable value estimation, especially for out-of-distribution (OOD) actions. To address these issues, existing uncertainty-based methods penalize the value function with uncertainty quantification and require numerous ensemble networks, leading to computational challenges and suboptimal outcomes. In this paper, we introduce a novel strategy that employs diverse randomized value functions to estimate the posterior distribution of Q-values. This approach provides robust uncertainty quantification and estimates the lower confidence bounds (LCB) of Q-values. By applying moderate value penalties for OOD actions, our method fosters a provably pessimistic approach. We also emphasize diversity within randomized value functions and enhance efficiency by introducing a diversity regularization method, thereby reducing the requisite number of networks. These modules result in reliable value estimation and efficient policy learning from offline data. Theoretical analysis shows that our method recovers the provably efficient LCB-penalty under linear MDP assumptions. Extensive empirical results demonstrate that our proposed method significantly outperforms baseline methods in terms of performance and parametric efficiency.

Accuracy Analysis of Complex Transmission System with Distributed Tooth Profile Errors

Tooth profile errors are the internal excitations that cause gear meshing errors, which are critical error factors affecting gear transmission accuracy. In existing studies, it is usually regarded as a constant or random distribution function. However, the actual machined tooth profile error is not a constant, so this estimation is inconsistent with the actual situation, resulting in an inaccurate evaluation of transmission accuracy. This paper proposes a method for representing tooth profile errors using distribution errors (including systematic and random errors), and a mathematical model of distributed tooth profile errors is presented. The contact stresses of the complex transmission system were compared with those obtained by formulas, proving that tooth profile errors increase contact stress. A method for calculating gear meshing error is proposed to evaluate the actual output accuracy of the complex transmission system. Compared with the test, the output accuracy is reduced by 13.8% under the temperature environment and distributed tooth profile errors. The proposed methods can accurately predict the transmission accuracy of precision transmission systems at the design stage and provide theoretical support for reducing systematic and random errors at the gear machining stage.

Group inference of high-dimensional single-index models

For the supervised and semi-supervised settings, a group inference method is proposed for regression parameters in high-dimensional semi-parametric single-index models with an unknown random link function. The inference procedure is based on least squares, which can be extended to other general convex loss functions. The proposed test statistics are weighted quadratic forms of the regression parameter estimates, in which the weight could be a non-random matrix or the sample covariance matrix of the covariates. The proposed method could detect dense but weak signals and deal with high correlation of covariates inside the group. A ‘contaminated test statistic’ is established in the semi-supervised regime to decrease the variance. The asymptotic properties of the resulting estimators are established. The finite-sample behaviour of the proposed method is evaluated through extensive simulation studies. Applications to two genomic datasets are provided.

Permeability of large‐scale fractures with ununiform proppant distributions in coalbed methane development

AbstractThe coalbed methane (CBM) productivity is directly determined by the fracture permeability during hydraulic fracturing, which is regulated by the distribution of proppants. The proppant may be unevenly distributed in the fracture because of variables like the architecture of the fracture and the characteristics of the sand‐carrying fluid. This study used two types of random functions to produce different ununiform distributions of proppant clusters in large‐scale fractures, with the aim of investigating the effect of these distributions on the overall permeability of the fracture. A model of fluid‐structure coupling is proposed. The closure of large‐scale fractures under in‐situ stress is analyzed using solid mechanics and the penalty function; the CBM flowing in proppant clusters and the high‐speed channel between them is simulated using Darcy's law and the Navier–Stokes equation, respectively; and the overall permeability of fractures is computed using the fluid pressure drop throughout the fracture and the fluid flowing velocity in the fracture's outlet. Since most CBM flows along high‐speed channels between the proppant clusters, the simulated findings show that the overall permeability of fractures with an uneven distribution of proppant clusters is significantly higher than that of the proppant cluster itself. As CBM becomes more discretely distributed, the proportion of CBM flowing within the proppant cluster continuously drops. As the permeability of the proppant cluster increases, the volume ratio of proppant clusters decreases, and the distribution of proppant clusters becomes more discrete, the overall permeability of the fracture continuously increases.

Dynamics of large oscillator populations with random interactions.

We explore large populations of phase oscillators interacting via random coupling functions. Two types of coupling terms, the Kuramoto-Daido coupling and the Winfree coupling, are considered. Under the assumption of statistical independence of the phases and the couplings, we derive reduced averaged equations with effective non-random coupling terms. As a particular example, we study interactions defined via the coupling functions that have the same shape but possess random coupling strengths and random phase shifts. While randomness in coupling strengths just renormalizes the interaction, a distribution of the phase shifts in coupling reshapes the coupling function.

Time-domain dimension-reduction representation for stochastic ground motion utilizing filtered white noise

A method is proposed for characterizing and simulating both stationary and fully non-stationary stochastic ground motions. This method is based on discrete filtered white noise models, including single and double filtered, with the latter is introduced to suppress low-frequency components. Specifically, the proposed method expresses seismic ground motion as a linear combination of products involving orthogonal random variables and deterministic functions. Further, by defining high-dimensional orthogonal random variables as orthogonal functions of extremely low dimensional elementary random variables, efficient dimension-reduction (DR) of primitive ground motion process can be achieved. To illustrate this concept, three distinct categories of random orthogonal functions involving only one or two elementary random variables are examined, employing filtered white noise models to simulate ground motion acceleration processes, thereby demonstrating the accuracy and efficiency of the proposed method. Simultaneously, recommendations for employing the proposed method in simulations are provided based on an analysis of the impacts of various parameters on random ground motion processes. Case studies demonstrate the accuracy and robustness of the proposed method compared to Monte Carlo (MC) methods. Furthermore, case studies on fully non-stationary ground motion highlight the practical applicability of the proposed method in engineering.

STATISTICAL EVALUATION OF PERFORMANCE INDICATORS OF PHOTOVOLTAIC PLANTS AS A SOURCE OF ENERGY FOR WATER DESALINATION IN THE AZOV-BLACK SEA REGION OF UKRAINE

The application of statistical methods for the analysis of random processes to obtain quantitative estimates for the performance indicators of photovoltaic plants as a stochastic energy source is considered. The power generation process is represented by a set of daily random functions with their respective trends and stochastic components. One hour was taken as the minimum stationari-ty interval of trend’s statistical characteristics. A time period of 16 years was studied to obtain statistically stable estimates. The meteorological database SARAH2 was used as a source of hourly data on the density of solar irradiance, ambient temperature and wind speed in the middle part of the Azov-Black Sea region of Ukraine. For mathematical modelling of power generation and electricity production processes, specialized software PVGIS version 5.2 of the European Commission was used. Hourly quantitative estimates of expected daily power generation trends for each month of the year, as well as their correlation functions, were computed. Algorithms were developed and levels of probabilistic assurance for hourly generated power were evaluated. Statistical estimates of the expected daily, monthly, and annual volumes of electricity production by photovoltaic power plants were studied, taking into account the meteorological conditions in the said region.

Obfuscating Verifiable Random Functions for Proof-of-Stake Blockchains

Obfuscating Verifiable Random Functions for Proof-of-Stake Blockchains

Taking into account the random function of parameter deviation in predicting the reliability of equipment

In article the question on forecasting of reliability of the equipment is considered. The characteristics infl uencing change of technical condition and reliability of the car (system) are analysed. Infl uence of stochastic function on a deviation of the actual values of parameter of technical condition is considered and the solution on her account is proposed.

An analysis of the effect of agriculture subsidies on technical efficiency: Evidence from rapeseed production in China

This study was conducted to examine the effects of agricultural subsidies on the technical efficiency of agricultural production technology and on factor input. It utilized a random frontier production function, instrumental variable method, and threshold regression model. The data used for this analysis consisted of 609 field yield measurements from the National Rapeseed Industry Technology System in 2020. The findings indicate that agricultural subsidies have a substantial impacts and it increases the technical efficiency of production process. Specifically, these subsidies encourage the use of land resources while inhibiting the use of chemical fertilizers. However, this does not have a significant effect on the utilization of labor and capital resources. Furthermore, the impact of agricultural subsidies on production technology efficiency varies depending on the scale of the farming operation. The subsidies significantly enhance the production technology efficiency of farmers with a business scale of less than 0.67 ha, but do not significantly improve the production technology efficiency of farmers with a business scale exceeding 0.67 ha. To optimize the effectiveness of agricultural subsidy policy, three methods and recommendations are proposed: increasing the overall amount of subsidies, expanding and diversifying the types of subsidies, and refining the process of disbursing subsidies.

Characterization of supercritical CO2 heat transfer in microporous media under (T/p)-mutability and integrated random function

The thermophysical properties of supercritical carbon dioxide (sCO2) undergo drastic changes under reservoir conditions, which significantly affects transfer characteristics and the storage efficiency. The present work seeks to identity of the heat transfer response of a mutable sCO2 in porous micromodels with integrated Boolean stochastic models. An upgraded variant of Lattice Boltzmann model is used after a thorough validation with literature experiments. The injection conditions of sCO2 are ranged from 295 to 333 K and 8–10 MPa, with a mass flow rate of 0.05–0.20 kg/min. The thermodynamic instabilities are discussed through the joint bonds between temperature and density, local and overall heat transfer, and pressure drop coefficient, under a series of pore-throats dimension from 7.5 to 401.3 μm. The results showed: (i) the remarkable (T/p)-nature of heat transfer capacity of sCO2 to lead the transfer coefficients with a subordinate impact of thermal conductivity near the Widom line, as long as the convective mode primes with increased temperature and pressure; (ii) The heterogeneity of the microporous media promoted the decrease of the transfer capacity while adjusting the spatiotemporal density and the heat transfer orientations inside the pore bodies; (iii) the growth of the pressure drop under increasing temperature has revealed an equilibrium stage where, neither the mutability of sCO2 properties nor the injection intensity can significantly affect the heat transfer of the integrated microporous.

Change-Point Detection in Functional First-Order Auto-Regressive Models

A sample of continuous random functions with auto-regressive structures and possible change-point of the means are considered. We present test statistics for the change-point based on a functional of partial sums. To study their asymptotic behavior, we prove functional limit theorems for polygonal line processes in the space of continuous functions. For some situations, we use a block bootstrap procedure to construct the critical region and provide applications. We also study the finite sample behavior via simulations. Eventually, we apply the statistics to a telecommunications data sample.

Modeling patchy saturation of fluids in nanoporous media probed by ultrasound and optics.

Nanoporous materials provide high surface area per unit mass and are capable of fluids adsorption. While the measurements of the overall amount of fluid adsorbed by a nanoporous sample are straightforward, probing the spatial distribution of fluids is nontrivial. We consider literature data on adsorption and desorption of fluids in nanoporous glasses reported along with the measurements of ultrasonic wave propagation. We analyze these using the so-called dynamic equivalent medium approach, which is based on Biot's theory of dynamic poroelasticity with coefficients that are continuous random functions of position. When further constrained by optical scattering data for similar systems, the calculations show that on adsorption the characteristic patch size is of the order of 100 pore diameters, while on desorption the patch size is comparable to the sample size. Our analysis suggests that one can employ ultrasound to probe the uniformity of the spatial distribution of fluids in nanoporous materials.

A Framework for Developing Systematic Testbeds for Multi-Fidelity Optimization Techniques

Abstract Multi-fidelity (MF) models abound in simulation-based engineering fields. Many MF strategies have been proposed to improve the efficiency in engineering processes, especially in design optimization. When it comes to assessing the performance of MF optimization techniques, existing practice usually relies on test cases involving contrived MF models of seemingly random math functions, due to limited access to real-world MF models. While it is acceptable to use contrived MF models, these models are often manually written up rather than created in a systematic manner. This gives rise to the potential pitfall that the test MF models may be not representative of general scenarios. We propose a framework to generate test MF models systematically and characterize tested MF optimization methods' performances comprehensively. In our framework, the MF models are generated based on given high-fidelity (HF) model and come with two parameters to control their fidelity levels and allow model randomization. In our testing process, MF case problems are systematically formulated using our model creation method. Running the given MF optimization technique on these problems produces what we call “savings curve” that characterizes the method's performance similarly to how ROC curves characterize machine learning classifiers. Our test results also allow plotting “optimality curves” that serve similar functionality to savings curves in certain types of problems. The flexibility of our MF model creation facilitates the development of testing processes for general MF problem scenarios, and our framework can be easily extended to other MF applications than optimization.

Density estimation for time-dependent PDE with random input by a Legendre-based multi-element probabilistic collocation method

This paper proposed a Legendre-based multi-element probabilistic collocation method for time-dependent stochastic differential equations, used for density estimation of unknown functions. This method involves discretizing the stochastic space, and on each element, constructing Lagrange interpolation basis functions based on Legendre–Gauss–Lobatto collocation/quadrature nodes. The proposed method is applied to approximate one-dimensional/two-dimensional smooth/non-smooth functions and is tested for accuracy in approximating random function values, density estimations, and mathematical expectations. This method is applied to stochastic nonlinear Schrödinger equations and coupled stochastic nonlinear Schrödinger equations, and all numerical results are compared with Monte Carlo simulation.