Chebyshev Inequality Research Articles

Optimizing peak shaving operation in hydro-dominated hybrid power systems with limited distributional information on renewable energy uncertainty

The increasing integration of renewable energy sources (RES) in power systems poses challenges for peak shaving operations due to RES uncertainty. However, it is difficult to obtain complete distributional information for uncertainty modeling. This study focuses on optimizing peak shaving in hydro-dominated hybrid power systems under such uncertainty. We utilize limited distributional information of RES forecast errors, specifically the first two moments, to build a moment ambiguity set. Employing distributionally robust chance-constrained programming (DRCCP), we develop a peak shaving model that quantifies the flexibility reserve of hydropower by risk level and the forecast errors. To enhance computational tractability, we apply the Chebyshev inequality to reformulate the moment-based DRCCP model into a mixed-integer linear programming model. Numerical simulations conducted on a provincial power grid in China validate the model's effectiveness. Key findings indicate that: (1) The model effectively leverages hydropower to provide ramping flexibility for peak shaving and quantifies the flexibility reserve needed for RES forecast errors. (2) This uncertainty modeling approach is more practical than probability distribution function-based methods, ensuring reliable peak shaving scheduling and reducing conservatism. (3) Decision-makers can adjust risk level to modify hydropower flexibility reserve, balancing robustness and conservatism of peak shaving scheduling.

Predictive framework for remaining useful life of roller bearings: Utilizing fractional generalized Pareto degradation model in performance evaluation

Research on predictive frameworks for the Remaining Useful Life (RUL) of mechanical systems is limited. This study proposes a comprehensive RUL prediction framework for roller bearings, integrating early failure assessment, adaptive failure threshold determination, and an adaptive drift fractional-order Pareto degradation model. To tackle data insufficiency, multi-sensor fusion combines time-domain, frequency-domain, and time–frequency features. The framework normalizes the health indicator (HI) using Mahalanobis distance and a sigmoid function, and employs the MD-CUSUM technique for early fault detection. Dynamic threshold updating is achieved through BOX-COX transformation and Chebyshev inequality, establishing confidence intervals. The model demonstrates significant results: lowest RMSE of 5.4267, MAE of 3.7857, highest SOR of 0.90936, HD of 0.93543, PICP of 57.143%, and the narrowest MPIW of 9.45, showing enhanced accuracy and reduced uncertainty. Validation with the XJTU-SY bearing dataset confirms improved performance.

Collapse Failure Assessment of Geomaterials behind Steel Structure in Tunnels Using the Chebyshev Inequalities

Collapse Failure Assessment of Geomaterials behind Steel Structure in Tunnels Using the Chebyshev Inequalities

On soft measurement modeling for predicting photovoltaic power with uncertainty based on the Takagi–Sugeno model

Abstract Accurate solar power forecast is becoming more essential for safe and reliable power grid operation with the increasing number of grid-connected photovoltaic (PV) power production units. However, PV power exhibits significant output fluctuation due to both external inputs and intrinsic stochasticity in system dynamics. Therefore, an efficient and reliable soft measurement model of PV power with uncertainty is demanded in practice applications. The technique described in this paper captures the impacts produced by the fundamental uncertainty observed in the data, instead of relying on unrealistic assumptions about uncertainty. A soft measurement model using the Takagi-Sugeno (TS) Fuzzy Logic System (FLS) based on several input-output time series of PV plants is presented in this study. Chebyshev's inequality from probability theory and statistics is adopted to create the confidence interval-based response envelopes for these time series at each moment. An envelope-based measure of output uncertainty and a center-valued response forecasting model can be obtained by the proposed identification technique. PV datasets are employed to demonstrate the concept, which indicates the proposed soft measurement may outperform existing methods in terms of prediction accuracy. The average values of Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and the Correlation Coefficient (R) are only 0.0787, 0.1113, and 0.9979. The average values of the prediction interval coverage probability (PICP) and the prediction interval normalized average width (PINAW) are 0.9806 and 0.1051.

Tracking control of T-S fuzzy systems under probabilistic constraints and fading channels

This work investigates the probabilistic-constrained tracking control problem for T-S fuzzy systems under fading channels described by the memoryless multiplicative noise model. The main control objective is to confine the tracking error into its pre-given bounds no lower than a certain probability. To this end, a tracking controller is firstly synthesised by introducing the fading parameter as an amplification factor towards ‘retrieving’ the original information to some extent. An auxiliary function and the multi-dimensional Chebyshev inequality make it possible to derive sufficient conditions such that the tracking error falls into its specified domain with a frequency larger than a pre-given value. By employing an inequality involving unmatched membership functions of the controller and fuzzy system, the favourite property of matched membership functions can be employed. A numerical example illustrates the effectiveness of the proposed method.

New integral inequalities for synchronous functions via Atangana–Baleanu fractional integral operators

Chebyshev’s inequality is a well-known inequality in the field of inequality theory and yields remarkable results for functions that exhibit synchronous behavior. The study, designed to obtain Chebyshev type inequalities with the help of the Atangana–Baleanu fractional integral operator, which has become the most popular concept of fractional analysis in recent years, aims to obtain general forms for synchronous functions. In the process of obtaining results for synchronized functions within the scope of the study, functional tools such as Atangana–Baleanu fractional integral operator, various mathematical analysis operations and induction were used. The most striking aspect of the findings is that for some special values of the parameter of the fractional integral operator, some of the results can be reduced to the Chebyshev inequality. Compared to the studies in the literature, this study has a unique aspect in that the findings obtained with the help of the Atangana–Baleanu fractional integral operator contain general forms, as well as bringing together a popular concept of synchronous functions and fractional analysis.

A comprehensive power quality confidence evaluation method for microgrid based on Chebyshev inequality

A comprehensive power quality confidence evaluation method for microgrid based on Chebyshev inequality

AN APPROACH OF MEAN-SETS THEORY FOR NEGATIVELY CURVED CONVEX COMBINATION POLISH METRIC SPACES

The choice of a convenient approach to be used is one important issue when attempting to develop methods or obtain results in the setting of Probability on general metric spaces. In this paper, we extend the mean-sets probabilistic approach formally introduced by Mosina on locally finite graphs, and hence (via Cayley graphs) on finitely generated groups, to the field of Negatively Curved Convex Combination Polish (NCCCP) metric spaces. We construct an appropriated Vertex-Weighted Metric (VWM) graph in the framework of this class of geometrical structures. We define a function called convexification function on the direct product of $ n $ copies of the vertex-set of this graph (for a given fixed integer $ n\geq 2$), using the natural convexification operator of the metric space concerned. This function is then used to construct a weighted mean-set that generalizes the notion of convex combination (CC) mean in the sense of Ter{\'a}n and Molchanov, the mean-set concept according to Mosina and the ordinary notion of $ k $-means ($k\geq 2$) of independent identically distributed (i.i.d.) random elements of the metric space. Two numerical examples are given for the cases when the metric space $ X= [0,1]$ and $ X=\mathbf{R}^{2} $. Moreover, an analogue of the Strong Law of Large Numbers (SLLN), the consistency problem and the Chebyshev's inequality for NCCCP spaces are established.

Calibration of Decision-Based Crowd-Behaviour Model

Various methods of calibration are used depending on the model type, application, and individual preferences. While there is no universally applicable method, statistical techniques became popular in recent decades. Introduced calibration concept consists of separate calibration episodes to avoid choosing only a few metrics to describe the whole system and a high computational time increasing exponentially with the number of parameters. These episodes are designed to be separated from each other and to cover one type of pedestrian behaviour captured by some model parameters. The design of the calibration quantities; estimate of the needed simulation time to get stationary results; and the number of iterations by Chebyshev's inequality influencing the quality of the results are discussed. Furthermore, hypothesis testing (James' test) is used to compare the model and experimental data. This calibration process can be applied for any pedestrian model; this paper deals with its application on the crowd-behaviour phase in the author's decision based model.

Chebyshev–Jensen-Type Inequalities Involving χ-Products and Their Applications in Probability Theory

By means of the functional analysis theory, reorder method, mathematical induction and the dimension reduction method, the Chebyshev-Jensen-type inequalities involving the χ-products ⟨·⟩χ and [·]χ are established, and we proved that our main results are the generalizations of the classical Chebyshev inequalities. As applications in probability theory, the discrete with continuous probability inequalities are obtained.

Remaining useful life prediction of rolling bearings based on performance evaluation and multifractional generalized Cauchy model with adaptive drift

The proposed Remaining Useful Life (RUL) prediction framework utilizes several advanced techniques to accurately estimate the remaining service life of rolling bearings. The framework includes early failure assessment, adaptive failure threshold (FT) determination, and a multifractional generalized Cauchy model (MfGC). The early failure assessment is enabled by establishing early FTs and health indicator (HI) curves generated by the Mahalanobis distance cumulative sum (MD-CUSUM) technique. The proposed dynamic fault threshold update method uses the BOX-COX transformation and Chebyshev inequality to determine confidence intervals for evaluating the fault threshold time. The multifractional nature of the MfGC process is characterized by independent, time-varying Hurst indices and fractional dimensions, and the long-range dependence (LRD) characteristics and stochasticity of the process are explained by the diffusion terms generated from the MfGC differential time series. The MfGC model with adaptive drift is constructed for various degenerate trajectories, and a method for estimating the model’s parameters is proposed. The effectiveness of the proposed RUL prediction method is demonstrated using the XJTU-SY bearing dataset.

Distribution-free Bayesian regularized learning framework for semi-supervised learning

In machine learning it is often necessary to assume or know the distribution of the data, however it is difficult to do so in practical applications. Aiming to this problem, this work, we propose a novel distribution-free Bayesian regularized learning framework for semi-supervised learning, which is called Hessian regularized twin minimax probability extreme learning machine (HRTMPELM). In this framework, we attempt to construct two non-parallel hyperplanes by introducing the high separation probability assumption, such that each hyperplane separates samples from one class with maximum probability while moving away from samples from the other class. Subsidiently, the framework can be utilized to construct reasonable semi-supervised classifiers by using the information of the inherent geometric distribution of the samples through the Hessian regularization term. Additionally, the proposed framework controls the misclassification error of samples by minimizing the upper limit of the worst-case misclassification probability, and improves the generalization performance of the model by introducing the idea of regularization to avoid the occurrence of ill-posedness and overfitting problems. More importantly, the framework has no hyperparameters, making the learning process very simplified and efficient. Finally, a simple and reliable algorithm with globally optimal solutions via multivariate Chebyshev inequalities is designed for solving the proposed learning framework. Experiments on multiple datasets demonstrate the reliability and effectiveness of the proposed learning framework compared to other methods. Especially, we applied the framework to Ningxia wolfberry quality detection, which greatly enriches and facilitates the application of machine learning algorithms in the agricultural field.

Statistical Machine Learning Model for Distributed Energy Planning in Industrial Park

With the advancement of industrial modernization, industrial parks have become the main body of new energy production and consumption. However, due to the large demand for energy in industrial agglomeration, the way of energy utilization is changing to sustainable. The direct connection of distributed energy resources in industrial parks, including photovoltaic power generation systems, has an important impact on its planning and operation. Furthermore, weather scenarios can have an impact on distributed photovoltaic generation, and the uncertainty in photovoltaic power output will, in turn, affect the uncertainty in industrial park planning. Therefore, this paper aims to address the issues of inaccurate prediction of distributed electricity generation during the planning period and the non-uniform distribution of energy resources such as electricity, heating, and cooling. This is achieved through the application of statistical machine learning. The paper intends to incorporate the ideas of statistical machine learning into the model for industrial park distributed energy random opportunity-constrained planning, aiming to resolve the problems of non-uniform distribution of distributed energy sources within the park, along with uncertainty in their outputs and high overall investment costs. The model takes the planning, construction and operating costs of the industrial park as the objective function, uses the lost load cost to ensure the safety of the industrial park, and uses the Chebyshev inequality probability to limit the output characteristics of distributed energy equipment. In terms of operation, the planning period is subdivided into heating period, cooling period and transition period, and the balance of electricity, heat and cold is considered. Finally, an actual example of an industrial park is used to verify the effectiveness of this method. Experimental validation shows that this approach can ensure safety requirements in industrial parks during the heating season, cooling season, and transitional periods by flexibly adjusting the confidence threshold. Simultaneously, it delivers significant economic benefits.

Probability and variance

Background: The highly conservative nature of Chebyshev's inequality, with its advantages and disadvantages, calls for an additional approach to the event probability of an individual event across various distributions. Materials and Methods: This investigation commences with the theoretical foundations of event probability, presenting a mathematical framework for the precise quantification of event probability independent of the distribution of the random variable. Simultaneously, we explore the relationship between the event probability of an individual event and the variance of that event. Results: Chebyshev's inequality can function more or less as a rough upper limit for event probability, depending on the number of standard deviations from the mean. The results of this study suggest that the variance and the expected value of an event allow for a very precise determination of the event probability of an individual event, eliminating the need for estimation. Conclusion: In summary, this study sheds light on the dynamic relationship between variance and event probability, emphasizing the limitations of Chebyshev's inequality. This insight can prove particularly valuable in scenarios with non-normally distributed datasets.

Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process

<abstract><p>This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein–Uhlenbeck process. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results.</p></abstract>

Morphological Method for Detecting Abnormal Server States

The paper proposes a computationally simple algorithm for detecting outliers and anomalies based on morphological analysis of the internal structure of multidimensional data. An important advantage of the method is the possibility of simultaneous work with qualitative and quantitative signs. It is also distinguished from its analogues by the simplicity of presentation and interpretation of the results. The values’ confidence range of the studied objects is approximated by combining the values’ confidence ranges of qualitatively homogeneous objects (clusters). The belonging of objects to one cluster is determined by the causal relationships between the features characteristic of the subject area. The method is based on the construction of a finite probability space and each element of binary vector is uniquely assigned to the objects of the sample. Based on the Chebyshev inequality, low-power clusters are taken as emissions. Objects that do not belong to the aggregate confidence area are taken as anomalies. Comparison mechanisms based on the Hamming distance have developed: 1) cluster and cluster; 2) cluster and object; 3) object and object. To demonstrate the effectiveness of the method a software module for detecting abnormal server states based on the Linux operating system has been developed. It can also be used as an auxiliary in professional intrusion detection systems.

Fujiwara’s inequality for synchronous functions and its consequences

Summary The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values of products of random variables. Special cases of this inequality include important classical inequalities such as the Cauchy–Schwarz and Chebyshev inequalities. The purpose of this article is to prove Fujiwara’s inequality in abstract probability spaces with more general assumptions for random variables than those associated with monotonicity as found in the literature. This newly introduced property of random variables will be called synchronicity. As a consequence, we obtain inequalities originated by Chebyshev, Hardy–Littlewood–Pólya and Jensen–Mercer under new conditions. The results are illustrated by some examples constructed for specific probability measures and specific random variables. In biometrics, the obtained inequalities can be used wherever there is a need to compare the characteristics of experimental data based on means or moments, and the data can be modeled using functions that are synchronous or convex.

Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability

This work considers McKean–Vlasov equations driven by Lévy noise in which the coefficients rely not only on the state of the unknown process but also on its probability distribution, and researches the well-posed, attractiveness and stability of solutions with resolvent operator theory. Under global Lipschitz condition, the well-posed of mild solutions is first studied for McKean–Vlasov integro-differential equations by using contraction mapping principle. Then according to Cauchy–Schwartz inequality and Itô isometry formula, one gains global attracting set and quasi-invariant set of solutions of McKean–Vlasov integro-differential equations and also obtained sufficient conditions of stability with continuous dependence on the coefficient of solutions. Moreover, one attains mean-square stability of solutions by Gronwall inequality and almost sure stability by applying Borel–Cantelli lemma and Chebyshev inequality for the aforementioned equation. Finally two examples are presented to verify our results.

Differentially Private Average Consensus With Logarithmic Dynamic Encoding-Decoding Scheme.

This article is concerned with the differentially private average consensus (DPAC) problem for a class of multiagent systems with quantized communication. By constructing a pair of auxiliary dynamic equations, a logarithmic dynamic encoding-decoding (LDED) scheme is developed and then utilized during the process of data transmission, thereby eliminating the effect of quantization errors on the consensus accuracy. The primary purpose of this article is to establish a unified framework that integrates the convergence analysis, the accuracy evaluation, and the privacy level for the developed DPAC algorithm under the LDED communication scheme. By means of the matrix eigenvalue analysis method, the Jury stability criterion, and the probability theory, a sufficient condition (with respect to the quantization accuracy, the coupling strength, and the communication topology) is first derived to ensure the almost sure convergence of the proposed DPAC algorithm, and the convergence accuracy and privacy level are thoroughly investigated by resorting to the Chebyshev inequality and ϵ -differential privacy index. Finally, simulation results are provided to illustrate the correctness and validity of the developed algorithm.

Chebyshev inequalities for unimodal distributions

We provide precise upper bounds for the survival function of bounded unimodal random variables.