Unknown Probability Research Articles

Design of robust Gaussian approximate filter and smoother with latency probability identification

The Kalman filter is a linear optimal estimator when it can accurately receive measurements and the noise obeys a Gaussian distribution. However, the measurement noise may have heavy-tailed characteristics because of outlier interference. Actually, real-world measurement information may have one-step randomly delayed measurements (ORDMs) with unknown latency probability (LP) due to exogenous disturbances. Moreover, systems are usually not linear. To overcome the influence of the state estimator of nonlinear systems from heavy-tailed measurement noise (HMN) and the unknown LP, this paper focuses on the robust Gaussian approximate (GA) filter and smoother for the above issues. First, the one-step predicted probability density function (PDF) is modeled as a Gaussian distribution, and the likelihood PDF is modeled as a Normal–Gamma–Beta (NGBM) distribution. Furthermore, novel robust Gaussian approximate filter and smoother are proposed based on the variational Bayesian technique. They provide general estimation frameworks for estimating state of nonlinear systems with HMN and ORDMs, and different nonlinear estimators can be implemented using different numerical techniques. In addition, the computational complexity of the proposed algorithms is calculated. Finally, taking a univariate nonstationary growth model, a passive ranging problem, and target tracking as examples, the effectiveness and superiority of the proposed robust algorithms contrasted with the existing algorithms are shown. The outcomes show that the proposed methods have higher estimation precision; however, the computational burden is marginally increased.

A data-driven multi-stage stochastic robust optimization model for dynamic optimal power flow problem

The uncertainty caused by the distributed generations(DG) with inconspicuous patterns has been an essential subject in the optimization scheduling for the distribution network. We propose a novel data-driven approach to deal with the dynamic optimal power flow(DOPF) problem which contains uncertain variables with their unknown probability distribution. The data-driven model is made to learn the joint probability distribution of the uncertain variables and use robust optimization(RO) to solve the multi-stage stochastic linear DOPF by averaging the worst case from each uncertainty set. In contrast to the motivation for traditional RO to find solutions that perform well on the worst-case realization, our proposed approach adds robustness to the historical data as a tool to avoid overfitting as the number of data points tends to infinity. The application verification for the AC OPF problem is presented for the IEEE-33 system. The simulation verifies the feasibility and robustness of the proposed approach and its results are compared with those of other data-driven stochastic optimization methods. We prove that the proposed approach can effectively solve the overvoltage problem caused by the high permeability of photovoltaic generation and achieve a better out-of-sample performance guarantee, and also has obvious economic advantages over other data-driven methods.

A threshold‐based sorting algorithm for dense wireless sensor systems and communication networks

A threshold‐based sorting algorithm for dense wireless sensor systems and communication networks

Integrated risk averse and possibilistic humanitarian logistic model with social costs and material convergence

A stepwise development is proposed for the construction of a model that comprehensively solves the disaster planning and response problem. In addition to traditional logistics costs, the suggested approach incorporates response time sensitivity, social costs related to post-disaster deprivation, the impact of material convergence, and measurable and epistemic uncertainty. Starting from a stochastic model based on previously published scenarios, the global costs of the scenarios are minimized using the expected value of the parameters, in a risk-averse bi-objective function that also optimizes the robustness of the model by minimizing the variability of the parameters whose probabilities are known. Finally, the complexity arising from imprecise parameter values with unknown probabilities is addressed by using chance-constrained fuzzy programming to optimize the risk associated with inter-scenario cost variability and feasibility of the constraints. This formulation is applied to real case study data, demonstrating that both the overall costs and the social cost component of the costs decrease progressively from the base formulation. The model allows the decision-maker to test the sensitivity of the results and thus set the uncertainty parameters and the incorporation of multiple objectives accordingly.

Gambling Under Unknown Probabilities as Proxy for Real World Decisions Under Uncertainty

The subject of decisions under uncertainty about future events, if lacking sufficient theory or data to make confident probability assessments, poses a challenge for any quantitative analysis. This article suggests one way to first look at this subject. We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler’s perceived advantage based on their perceived probability, and their accuracy in estimating true probabilities is measured by mean-squared-error. Within this framework one can study the cost of estimation errors, and seek to formalize the “obvious” notion that in competitive interactions between agents whose actions depend on their perceived probabilities, those who are more accurate at estimating probabilities will generally be more successful than those who are less accurate. This article is an extended version of the Brouwer Medal talk at the 2021 Nederlands Mathematisch Congres.

An Improved Multiple-Target Tracking Scheme Based on IGGM–PMBM for Mobile Aquaculture Sensor Networks

The Poisson multi-Bernoulli Mixture (PMBM) filter, as well as its variants, is a popular and practical multitarget tracking algorithm. There are some pending problems for the standard PMBM filter, such as unknown detection probability, random target newborn distribution, and high energy consumption for limited computational and processing capacity in sensor networks. For the sake of accommodating these existing problems, an improved multitarget tracking method based on a PMBM filter with adaptive detection probability and adaptive newborn distribution is proposed, accompanied by an associated distributed fusion strategy to reduce the computational complexities. Firstly, gamma (GAM) distribution is introduced to present the augmented state of unknown and changing target detection probability. Secondly, the intensity of newborn targets is adaptively derived from the inverse gamma (IG) distribution based on this augmented state. Then, the measurement likelihood is presented as a gamma distribution for the augmented state. On these bases, the detailed recursion and closed-form solutions to the proposed filter are derived by means of approximating the intensity of target birth and potential targets to an inverse gamma Gaussian mixture (IGGM) form and the density of existing Bernoulli components to a single IGGM form. Moreover, the associated distributed fusion strategy generalized covariance intersection (GCI), whose target states are measured by multiple sensors according to their respective fusion weights, is applied to a large-scale aquaculture tracking network. Comprehensive experiments are presented to verify the effectiveness of this IGGM–PMBM method, and comparisons with other multitarget tracking filters also demonstrate that tracking behaviors are largely improved; in particular, tracking energy consumption is reduced sharply, and tracking accuracy is relatively enhanced.

Deep reinforcement learning-based adaptive modulation for OFDM underwater acoustic communication system

Due to the time-varying and space-varying characteristics of the underwater acoustic channel, the communication process may be seriously disturbed. Thus, the underwater acoustic communication system is facing the challenges of alleviating interference and improving communication quality and communication efficiency through adaptive modulation. In order to select the optimal modulation mode adaptively and maximize the system throughput ensuring that the bit error rate (BER) meets the transmission requirements, this paper introduces deep reinforcement learning (DRL) into orthogonal frequency division multiplexing acoustic communication system. The adaptive modulation is mapped into a Markov decision process with unknown state transition probability. Thereby, the underwater communication channel environment is regarded as the state of DRL, and the modulation mode is regarded as action. The system returns channel state information (CSI) and signal–noise ratio in every time slot through the feedback link. Because the Deep Q-Network optimizes in the changing state space of each time slot, it is suitable for a variety of different CSI. Finally, simulations in different underwater environments (SWellEx-96) show that the proposed adaptive modulation scheme can obtain lower BER and improve the system throughput effectively.

ℋ∞ filtering for discrete‐time hidden singular Markov jump systems subject to partially known probability information under DoS attacks

SummaryThis paper concentrates on the problem of asynchronous filtering for singular Markov jump systems under DoS attacks, in which the unknown probability information cases are considered. To better describe cyber security problem, DoS attacks are considered as encountered randomly and modeled by a set of stochastic variables obeying Bernoulli distribution. Besides, hidden Markov model with partially known probability information is proposed to handle the cases that the Markov state information of the system is restrictedly accessed, and the transition probability information and observation probability information can not be directly obtained. A set of criteria are derived to analyze the regularity, causality, and stochastic stability for the filtering error system with an performance index. Furthermore, an asynchronous filter design method is proposed to against DoS attacks. Finally, a numerical example and a practical example modeled by a DC motor are given to demonstrate the effectiveness of the proposed method.

A Deep Learning Approach to Anomaly Sequence Detection for High-Resolution Monitoring of Power Systems

A deep learning approach is proposed to detect data and system anomalies using high-resolution continuous point-on-wave (CPOW) or phasor measurements. Both the anomaly and anomaly-free measurement models are assumed to have unknown temporal dependencies and probability distributions. Historical training samples are assumed for the anomaly-free model, while no training samples are available for the anomaly measurements. By transforming the anomaly-free observations into uniform independent and identically distributed sequences via a generative adversarial network, the proposed approach deploys a uniformity test for anomaly detection at the sensor level. A distributed detection scheme that combines sensor level detections at the control center is also proposed which combines local detections to form more reliable detections. Numerical results demonstrate significant improvement over the state-of-the-art solutions for various bad-data cases using real and synthetic CPOW and PMU data sets.

Convex regularization in statistical inverse learning problems

We consider a statistical inverse learning problem, where the task is to estimate a function $ f $ based on noisy point evaluations of $ Af $, where $ A $ is a linear operator. The function $ Af $ is evaluated at i.i.d. random design points $ u_n $, $ n = 1, ..., N $ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $ p $-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.

Convergence of Policy Gradient Methods for Nash Equilibria in General-sum Stochastic Games

We study Nash equilibria learning of a general-sum stochastic game with an unknown transition probability density function. Agents take actions at the current environment state and their joint action influences the transition of the environment state and their immediate rewards. Each agent only observes the environment state and its own immediate reward and is unknown about the actions or immediate rewards of others. We introduce the concept of weighted asymptotic Nash equilibrium with probability 1 and design a two-loop algorithm by the equivalence of Nash equilibrium and variational inequality problems. In the outer loop, we sequentially update a constructed strongly monotone variational inequality by updating a proximal parameter while employing a single-call extra-gradient algorithm in the inner loop for solving the constructed variational inequality. We show that if the associated Minty variational inequality has a solution, then the designed algorithm converges to the k1/2 -weighted asymptotic Nash equilibrium.

Distributions-free Martingales Test Distributions-shift

A standard assumption of the theory of machine learning is the data are generated from a fixed but unknown probability distribution. Although this assumption is based on the foundations of the theory of probability, however, for most learning problems we usually technically random shuffle the original datasets, such as random split into training and test datasets before the training model, to satisfy the assumption, and then we use the shuffled training dataset to train a machine learning model. However, honestly, for real-life learning applications, the data pairs are observed batch by batch under their own original order and it is not necessary to randomly shuffle the original order in advance. From a mathematical point of view, we test if the random shuffling will play a non-negligible influence on the generalization of learning machines. We reduce the problem of random shuffling into the problem of distribution-shift detection.This paper is devoted to testing the null hypothesis that random shuffling does not affect the generalization of learning machines and introduces a distributions-free martingales method against the null hypothesis. We report the five real-life benchmarks of experimental performance with the help of Support Vector Machines and a multi-layer perceptron model. The results show a bonafide fact that the distribution shift in itself of the data is an inescapable reality when we build machine learning algorithms as the original order.

Robust Optimization for Perishable Product Distribution under Uncertainty of Multimodal Transportation Services

Unusual logistics service uncertainties resulting from recent mega-disruptions, such as COVID-19 outbreak or the Russia-Ukraine conflict, are placing increasing pressure on practitioners and researchers to develop robust solutions to mitigate these uncertainties. Perishable products, such as fresh produce, medicine, and vaccines, which deteriorate quickly and have short shelf-lives, are particularly vulnerable to the lead-time uncertainty from the global logistics services, which causes huge waste or even fatal outcomes. Moreover, modeling such time uncertainties using probability distributions is challenging due to the lack of historical data on unprecedented and complex disruptions. The literature on distribution and transportation service planning for perishable products under unknown probability distributions is also scarce. This paper aims to provide shippers with robust solutions for distribution planning of perishable products considering available transportation services within a multimodal transportation network under uncertainty of transit time including waiting time at sites and transportation time between sites. A robust model is proposed to minimize the total distribution costs composed of transportation, transshipment, and deterioration costs. Our model applies the budget of uncertainty to adjust the degree of conservatism in the solutions. A variable neighborhood search (VNS) algorithm is devised to solve the robust model. Finally, computational experiments are carried out to evaluate the performance and applicability of the proposed model and algorithm.

Stochastic decision tree acceptability analysis with uncertain state probability

In a fast-changing environment, state in the future is difficult to predict. Traditional approaches are unable to support decision-makers to find out optimal alternative effectively when the probability of future’s environmental state is unknown or uncertain. In this study, we propose a stochastic decision tree acceptability analysis (SDTAA), which aims to manage this decision-making problem effectively. In SDTAA, state probability space with random distribution is utilized to capture unknown or uncertain state probabilities and stochastic values or ordinal values are used to model uncertain attributes values. Then, by computing rank acceptability, holistic expected value and value variance of each alternative, SDTAA can help decision makers find the optimal alternative effectively when state probability is uncertain, unknown or missing. In addition, Monte Carlo simulation based algorithms are proposed to calculate the rank acceptability, holistic expected value and value variance. A numerical example is presented to illustrate the SDTAA method.

Finite-time control for discrete-time nonlinear Markov switching LPV systems with DoS attacks

This work studies the finite-time control problem for discrete-time nonlinear Markov switching linear parameter varying (LPV) systems with denial-of-service (DoS) attacks. The aperiodic DoS attacks are introduced into the systems with partially unknown transition probabilities. First, considering the characteristics of aperiodic DoS attacks, an appropriate feedback controller is designed according to that whether it is attacked or not. Furthermore, the finite-time stability criteria of closed-loop system are derived by means of iterative technique and multiple Lyapunov functions. Finally, a turbofan engine model is taken as an example to verify the effectiveness and feasibility of the proposed finite-time control method.

Pinning synchronization for markovian jump neural networks with uncertain impulsive effects

This work concentrates on synchronization of neural networks (NNs) with Markovian parameters, where the Markov chain has partially unknown transition probabilities (PUTP). Due to the existence of interference and noise in practice, we combine the uncertain variable with the complex coupling term as the impulsive disturbance of NNs. A corresponding mode-dependent pinning controller is designed to reduce the control costs, and synchronization error system is also derived, whose impulsive update state is listed separately. A sufficient condition of synchronization for NNs is completed by constructing a Lyapunov functional candidate and a series of iterations. Because the disturbance should avoid being too frequent to guarantee synchronization of NNs, the allowed minimum interval h of the impulsive disturbance is derived. Finally, the correctness and the superiority of the developed result are illustrated by a numerical example.

Manage Inventories with Learning on Demands and Buy-up Substitution Probability

Problem Definition: This paper considers a setting in which an airline company sells seats periodically, and each period consists of two selling phases, an early-bird discount phase and a regular-price phase. In each period, when the early-bird discount seat is stocked out, an early-bird customer who comes for the discounted seat either purchases the regular-price seat as a substitute (called buy-up substitution) or simply leaves. Methodology/Results: The optimal inventory level of the discounted seats reserved for the early-bird sale is a critical decision for the airline company to maximize its revenue. The airline company learns about the demands for both discounted and regular-price seats and the buy-up substitution probability from historical sales data, which, in turn, are affected by past inventory allocation decisions. In this paper, we investigate two information scenarios based on whether lost sales are observable, and we provide the corresponding Bayesian updating mechanism for learning about demand parameters and substitution probability. We then construct a dynamic programming model to derive the Bayesian optimal inventory level decisions in a multiperiod setting. The literature finds that the unobservability of lost sales drives the inventory manager to stock more (i.e., the Bayesian optimal inventory level should be kept higher than the myopic inventory level) to observe and learn more about demand distributions. Here, we show that when the buy-up substitution probability is known, one may stock less, because one can infer some information about the primary demand for the discounted seat from the customer substitution behavior. We also find that to learn about the unknown buy-up substitution probability drives the inventory manager to stock less so as to induce more substitution trials. Finally, we develop a SoftMax algorithm to solve our dynamic programming problem. We show that the obtained stock more (less) result can be utilized to speed up the convergence of the algorithm to the optimal solution. Managerial Implications: Our results shed light on the airline seat protection level decision with learning about demand parameters and buy-up substitution probability. Compared with myopic optimization, Bayesian inventory decisions that consider the exploration-exploitation tradeoff can avoid getting stuck in local optima and improve the revenue. We also identify new driving forces behind the stock more (less) result that complement the Bayesian inventory management literature. Funding: Z. Luo acknowledges the financial support by the Internal Start-up Fund of The Hong Kong Polytechnic University [Grant P0039035]. P. Guo acknowledges the financial support from the Research Grants Council of Hong Kong [Grant 15508518]. Y. Wang’s work was supported by the Research Grants Council of Hong Kong [Grant 15505318] and the National Natural Science Foundation of China [Grant 71971184]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.1169 .

Non-parametric generalised newsvendor model

In the present paper we generalise the classical newsvendor problem for critical perishable commodities having more severe costs than its linear alternative. Piece wise polynomial cost functions are introduced to accommodate the excess severity. Stochastic demand is assumed to follow a completely unknown probability distribution. Non parametric estimator of the optimal order quantity has been developed from an estimating equation using a random sample. Strong consistency of the estimator is proved for unique optimal order quantity and the result is extended for multiple solutions. Simulation results indicate that non parametric estimator is efficient in terms of mean square error. Real life application of the proposed non-parametric estimator has been demonstrated with Avocado demand in the United States of America and Covid-19 test kit demand during second wave of SARS-COV2 pandemic across 86 countries.

Dynamic Event-Triggered State Estimation for Markov Jump Neural Networks With Partially Unknown Probabilities.

This article focuses on the investigation of finite-time dissipative state estimation for Markov jump neural networks. First, in view of the subsistent phenomenon that the state estimator cannot capture the system modes synchronously, the hidden Markov model with partly unknown probabilities is introduced in this article to describe such asynchronization constraint. For the upper limit of network bandwidth and computing resources, a novel dynamic event-triggered transmission mechanism, whose threshold parameter is constructed as an adjustable diagonal matrix, is set between the estimator and the original system to avoid data collision and save energy. Then, with the assistance of Lyapunov techniques, an event-based asynchronous state estimator is designed to ensure that the resulting system is finite-time bounded with a prescribed dissipation performance index. Ultimately, the effectiveness of the proposed estimator design approach combining with a dynamic event-triggered transmission mechanism is demonstrated by a numerical example.

A three-layer joint distributionally robust chance-constrained framework for optimal day-ahead scheduling of e-mobility ecosystem

A high number of electric vehicles (EVs) in the transportation sector necessitates an advanced scheduling framework for e-mobility ecosystem operation to overcome range anxiety and create a viable business model for charging stations (CSs). The framework must account for the stochastic nature of all stakeholders’ operations, including EV drivers, CSs, and retailers and their mutual interactions. In this paper, a three-layer joint distributionally robust chance-constrained (DRCC) model is proposed to plan day-ahead grid-to-vehicle (G2V) and vehicle-to-grid (V2G) operations for e-mobility ecosystems. The proposed three-layer joint DRCC framework formulates the interactions of the stochastic behaviour of the stakeholders in an uncertain environment with unknown probability distributions. The proposed stochastic model does not rely on a specific probability distribution for stochastic parameters. An iterative process is proposed to solve the problem using joint DRCC formulation. To achieve computational tractability, the second-order cone programming reformulation is implemented for double-sided and single-sided chance constraints (CCs). Furthermore, the impact of the temporal correlation of uncertain PV generation on CSs operation is considered in the formulation. A simulation study is carried out for an ecosystem of three retailers, nine CSs, and 600 EVs based on real data from San Francisco, USA. The simulation results show the necessity and applicability of such a scheduling framework for the e-mobility ecosystem in an uncertain environment, e.g., by reducing the number of unique EVs that failed to reach their destination from 272 to 61. In addition, the choice of confidence level significantly affects the cost and revenue of the stakeholders as well as the accuracy of the schedules in real-time operation, e.g., for a low-risk case study, the total net cost of EVs increased by 247.3% compared to a high-risk case study. Also, the total net revenue of CSs and retailers decreased by 26.6% and 10.6%, respectively.