Chebyshev Inequality Research Articles

A simple method to determine the sampling numbers in decision-making units with unknown variations of soil cadmium.

Sampling number is one critical issue to achieve credible results when surveying soil contamination and making remediation decisions. Traditional methods based on a normal distribution for determining numbers of samples are not always optimal because most distributions of soil heavy metal concentrations followed a log-normal distribution. Moreover, the variation of soil heavy metal concentrations is a prerequisite for previous methods to determine sampling numbers. Unfortunately, the variation is often unknown before soil sampling. Therefore, a simple method under the log-normal distribution without relying on variation to determine quickly the sampling number (QSN) was developed for soil cadmium and compared with other methods based on classical statistics and Chebyshev inequality. Results showed that an equation as a function of sampling areas could be used to determine QSN (QSN = 18.44 × A0.54 + 8.69, A is sampling areas, km2), with acceptable errors ranging from 13 to 33% at the sampling areas of 0.03-10 km2. The developed simple method for QSN was easy to use and cost-effective without prerequisite on the estimation of variation. Moreover, when the sampling cost was enough and the improved accuracy was requested, the increased sampling numbers were recommended as 1.53 times as the number calculated by the simple method. Therefore, the proposed method is believed as a simple and cost-effective method to determine the sampling numbers of soil Cd in decision-making units with unknown variations.

A note on Chebyshev inequality via k-generalized fractional integrals

A note on Chebyshev inequality via k-generalized fractional integrals

Chebyshev inequality–based inflated convex hull for uncertainty quantification and optimization with scarce samples

Chebyshev inequality–based inflated convex hull for uncertainty quantification and optimization with scarce samples

Uncertainty analysis of reactivity feedback coefficient for LFR core power control system during reactivity insertion accident

To study the influence of the reactivity feedback coefficient uncertainty on reactor core power control system, taking a lead-cooled fast reactor as the object, the core power control system with a BP neural network PID controller is designed. Drawing the input samples with Latin hypercube sampling method, the uncertainty quantitative analysis consists of the Chebyshev inequality and relative standard deviation calculation. Based on the method in this paper, the dynamic simulation of reactor core is performed. The results show that the uncertainties of the system output control performance parameters are less than 1% when introducing 50pcm and 100 pcm step reactivity. The uncertainty of the system output parameters changes and returns to zero with the action of the controller.

A research into the development of models of random variables as part of the structural reliability analysis performed in the absence of some statistical information

Introduction. The scientific review article addresses the approaches to the modeling of random variables performed as part of the structural reliability analysis of elements provided that some statistical information missing (limited). The objectives of the research include the statement of the problem of the probabilistic structural reliability analysis subject to incomplete statistical data, the study of the development of approaches to the generation of models of random variables within the framework of this problem, as well as the assessment of the current state of affairs in this field and some development prospects for the coming years.
 Materials and methods. The principal model of a random variable, considered in the article, represents a p-box (pro­bability box) model. A p-box is an area of possible functions of distributed probabilities of a random variable generated by the two boundary functions of the probability distribution. The article addresses p-boxes generated using the fuzzy set theory, the probability theory, Kolmogorov–Smirnov boundaries, etc.
 Results. The approaches, considered in the article, are illustrated by the numerical examples of p-boxes that use the same statistical data. P-boxes, based on the probability theory, allow to accurately simulate a random variable; however, a priori information about the type of the distribution function is needed. P-boxes, based on the possibility theory, can be used even if an extremely small amount of statistical data is available, and it is also necessary to carefully address the issue of assigning the cutoff (risk) level. P-boxes based on the Chebyshev inequality and the Kolmogorov–Smirnov statistics allow to effectively simulate random variables regardless of the type of the probability distribution. However, these approaches may generate an assessment that is too uninformative for decisions to be made in a number of tasks.
 Conclusions. The choice of a probabilistic model of a random variable for the further reliability analysis of structural elements will depend on the amount and type of statistical data obtained about the random variable. In particular cases, if the statistical information represents a subset of intervals, special approaches based on the Dempster–Shafer theory can be used. A promising and relevant method that underlies both the development of probabilistic models of random variables and the analysis of structural reliability in case of missing statistical information encompasses the employment of numerical modeling methods that employ surrogate models (kriging, Bayesian networks, interval predictors, etc.) and neural network algorithms.

An explainable ensemble feedforward method with Gaussian convolutional filter

The emerging deep learning technologies are leading to a new wave of artificial intelligence, but in some critical applications such as medical image processing, deep learning is inapplicable due to the lack of interpretation, which is essential for a critical application. This work develops an explainable feedforward model with Gaussian kernels, in which the Gaussian mixture model is leveraged to extract representative features. To make the error within the allowable range, we calculate the lower bound of the number of samples through the Chebyshev inequality. In the training processing, we discuss both the deterministic and stochastic feature representations, and investigate the performance of them and the ensemble model. Additionally, we use Shapely additive explanations to analyze the experiment results. The proposed method is interpretable, so it can replace the deep neural network by working with shallow machine learning technologies, such as the Support Vector Machine and Random Forest. We compare our method with baseline methods on Brain Tumor and Mitosis dataset. The experimental results show our method outperforms the RAM (Recurrent Attention Model), VGG19 (Visual Geometry Group 19), LeNET-5, and Explainable Prediction Framework while having strong interpretability.

Type–reduction of Interval Type–2 fuzzy numbers via the Chebyshev inequality

The centroid is one of the most important methods for type–reduction of Type–2 fuzzy sets/numbers and it is also a popular expectation measure, but available methods to compute it are iterative/algorithmic which is an important issue for real–world implementations. This paper presents some theoretical results about the Chebyshev integral inequality for a class of Interval Type–2 fuzzy numbers which leads to obtain non–iterative closed forms of the centroid and its bounds for Type–1 and interval Type–2 fuzzy numbers (by extension). An analysis of the obtained results and a comparison to four well known type–reduction methods: the Karnik–Mendel, Yager Index, Mitchell and Nie–Tan methods are provided where the experimental evidence shows that the proposed method is equivalent to the K–M algorithms.

New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators

In this study, new and general variants have been obtained on Chebyshev’s inequality, which is quite old in inequality theory but also a useful and effective type of inequality. The main findings obtained by using integrable functions and generalized fractional integral operators have generalized many existing results as well as iterating the Chebyshev inequality in special cases.

Data Reconstruction in Heterogeneous Environmental Wireless Sensor Networks Using Robust Tensor Principal Component Analysis

The major challenges during the data acquisition process in an environmental wireless sensor network (EWSN) architecture are the presence of outliers and missing data. Outliers and missing data are ubiquitous in EWSN due to sensor failures, external noise, power dwindling, communication failures etc. Robust tensor principal component analysis (RTPCA) decomposes a noisy data tensor into a low-rank tensor and a sparse tensor, which can be exploited for the data recovery in EWSNs, where the low-rank component represents the intrinsic data tensor and the sparse component represents the gross outlier tensor. In this paper, a novel probabilistic outlier modelling scheme using multivariate Chebyshev's inequality hypothesis is proposed, which maps the sample population and the associated magnitudes of outliers with the spatio-temporal correlations in the acquired data. The inherent spatio-temporal and multi-attribute correlations in the EWSN data are established using singular value methods. A tensor nuclear norm (TNN) which extracts more temporal and multi-attribute correlations in the sensory data through block circulant matricization is used as the minimization function in RTPCA. In the forest surveillance scenario, usage of RTPCA results in a reconstruction accuracy of approximately 90% for a dataset with a high missing ratio of 0.95 and the outliers having largest possible magnitudes considered. For oceanographic data, in which the variance is lower, the simulations show similar performance for different outlier contaminations, although RPTCA performs better than other matrix and tensor data completion methods in terms of reconstruction accuracy.

Non-standard methods for solving inequalities

This article discusses some interesting methods for solving inequalities. And there are some inequalities that will help with this. For example: Cauchy inequality, Cauchy-Schwarz inequality, Titu's Lemma, Holder's inequality, Chebyshev's inequality. Examples of inequalities are also consideredfor better understanding.

Data-based polyhedron model for optimization of engineering structures involving uncertainties

Abstract This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.

Exact converses to a reverse AM-GM inequality, with applications to sums of independent random variables and (super)martingales

For every given real value of the ratio $\mu:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $\mu$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) \equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.

On new Chebyshev inequalities via fractional operators

On new Chebyshev inequalities via fractional operators

Design and Simulation of information processing system based on digital PCR

In order to meet the demand of real-time data acquisition and analysis in the process of digital PCR detection, this paper proposes a high-speed data acquisition system and design scheme based on FPGA. The system uses FPGA as the signal acquisition platform, USB interface as the data transmission module, develops a set of real-time acquisition and processing software, and proposes a peak extraction algorithm based on Chebyshev inequality. The algorithm improves the peak detection accuracy, shortens the peak extraction time, and reduces the requirements for software and hardware. The system can monitor and analyze the data signal. It provides a reliable basis for the later data processing, and has strong engineering practice.

New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel

<abstract><p>The main goal of this article is first to introduce a new generalization of the fractional integral operators with a certain modified Mittag-Leffler kernel and then investigate the Chebyshev inequality via this general family of fractional integral operators. We improve our results and we investigate the Chebyshev inequality for more than two functions. We also derive some inequalities of this type for functions whose derivatives are bounded above and bounded below. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. Finally, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view.</p></abstract>

On outlier detection with the Chebyshev type inequalities

This work considers algorithms of outlier detection based on the Chebyshev inequality. It compares these algorithms with such classical methods as Tukey’s boxplot, the N-sigma rule and its robust modifications based on MAD and FQ scale estimates. To adjust the parameters of the algorithms, a selection procedure is proposed based on the complete knowledge of the data distribution model. Areas of suboptimal parameters are also determined in case of incomplete knowledge of the distribution model. It is concluded that the direct use of the Chebyshev inequality implies the classical N-sigma rule. With the non-classical Chebyshev inequality, a robust outlier detection method is obtained, which slightly outperforms other considered algorithms.

Adaptive threshold modeling algorithm for monitoring indicators of power network server based on Chebyshev inequality

The business system server under the IT automation operation and maintenance platform generates massive data samples, based on which the threshold can be set to realize the allocation and management of hardware resources. The traditional threshold selection method is to determine an appropriate threshold based on human experience. If the threshold is too high, it will not play its due role. But if the threshold is too low, it will frequently produce false positives. To solve this problem, an adaptive threshold method based on Chebyshev inequality theory combined with kernel density estimation is proposed to determine monitoring indexes, and a new dynamic implicit threshold model is established to analyse the data generated by the business system server for real-time monitoring and alarm processing. Through the experimental study on the CPU utilization data of the power grid server, lower missing and false positive rate are obtained, which verifies the feasibility and effectiveness of the proposed method.

Inequalities for D−Synchronous Functions and Related Functionals

We introduce in this paper the concept of quadruple D−synchronous functions which generalizes the concept of a pair of synchronous functions, we establish an inequality similar to Chebyshev inequality and we also provide some Cauchy-Bunyakovsky-Schwarz type inequalities for a functional associated with this quadruple. Some applications for univariate functions of real variable are given. Discrete inequalities are also stated.

Probing nonclassicality with matrices of phase-space distributions

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyonds-parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

The #-Filter Anti-Aliasing Based on Sub-Pixel Continuous Edges

Anti-aliasing plays a decisive role in enhancing immersion experience in games and 3D visualization fields. In general, hardware anti-aliasing is not directly compatible with deferred shading. It is challenging to detect geometry edges accurately from sub-pixel to alleviate artifacts. In this paper, we propose an anti-aliasing algorithm of the #-filter anti-aliasing based on sub-pixel continuous edges. It can solve the geometry edges aliasing and the flicker problem in deferred shading. First, the geometry scene with multi-sampling anti-aliasing (MSAA) is rendered to a G-Buffer designed elaborately. Second, the geometry edges are detected on the sub-pixel-level. We mainly take advantage of the Chebyshev inequality to adaptively determine the edges from the probability statistic and the view frustum location. Third, the continuous geometry edges are reconstructed by a #-filter method. Finally, the edge pixels are shaded adaptively. The implementation demonstrates that our algorithm is efficient and scalable for generating high-quality anti-aliasing geometry and reducing shading calculation overhead.