Search Results (47)

We adapt the classical mixed Poisson process models for investigation of consumer behaviour in a situation where after a random time we can no longer identify a customer despite the customer remaining in the panel and continuing to perform buying actions. We derive explicit expressions for the distribution of the number of purchases by a random customer observed at a random subinterval for a given interval. For the estimation of parameters in the gamma–Poisson scheme, we use the estimator minimizing the Hellinger distance between the sampling and model distributions, and demonstrate that this method is almost as efficient as the maximum likelihood being much simpler. The results can be used for modelling internet user behaviour where cookies and other user identifiers naturally expire after a random time. Full article

Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated. Full article

In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for χ-divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end. Full article

This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes. The general framework and its applications are shown to yield and to relate to the following: f-divergence, Hellinger divergence, Renyi divergence, and Kullback–Leibler divergence (also known as relative entropy); the Lorenz curve and socioeconomic inequality indices; the Gini index and its generalizations; the divergence of renewal processes from the Poisson process; and the divergence of anomalous relaxation from regular relaxation. Presenting a ‘fresh’ perspective on statistical divergence, this paper offers its readers a simple and transparent construction of statistical-divergence gauges, as well as novel paths that lead from statistical divergence to the aforementioned topics. Full article

Numerous robust estimators exist as alternatives to the maximum likelihood estimator (MLE) when a completely observed ground-up loss severity sample dataset is available. However, the options for robust alternatives to a MLE become significantly limited when dealing with grouped loss severity data, with only a handful of methods, like least squares, minimum Hellinger distance, and optimal bounded influence function, available. This paper introduces a novel robust estimation technique, the Method of Truncated Moments (MTuM), pecifically designed to estimate the tail index of a Pareto distribution from grouped data. Inferential justification of the MTuM is established by employing the central limit theorem and validating it through a comprehensive simulation study. Full article

The main issue in this work is to study the limit functions necessary for the reliability assessment of structural steel with the use of the relative entropy apparatus. This will be done using a few different mathematical theories relevant to this relative entropy, namely those proposed by Bhattacharyya, Kullback–Leibler, Jeffreys, and Hellinger. Probabilistic analysis in the presence of uncertainty in material characteristics will be delivered using three different numerical strategies—Monte Carlo simulation, the stochastic perturbation method, as well as the semi-analytical approach. All of these methods are based on the weighted least squares method approximations of the structural response functions versus the given uncertainty source, and they allow efficient determination of the first two probabilistic moments of the structural responses including stresses, displacements, and strains. The entire computational implementation will be delivered using the finite element method system ABAQUS and computer algebra program MAPLE, where relative entropies, as well as polynomial response functions, will be determined. This study demonstrates that the relative entropies may be efficiently used in reliability assessment close to the widely engaged first-order reliability method (FORM). The relative entropy concept enables us to study the probabilistic distance of any two distributions, so that structural resistance and extreme effort in elastoplastic behavior need not be restricted to Gaussian distributions. Full article

In order to solve low-quality problems such as data anomalies and missing data in the condition monitoring data of hydropower units, this paper proposes a monitoring data quality enhancement method based on HDBSCAN-WSGAIN-GP, which improves the quality and usability of the condition monitoring data of hydropower units by combining the advantages of density clustering and a generative adversarial network. First, the monitoring data are grouped according to the density level by the HDBSCAN clustering method in combination with the working conditions, and the anomalies in this dataset are detected, recognized adaptively and cleaned. Further combining the superiority of the WSGAIN-GP model in data filling, the missing values in the cleaned data are automatically generated by the unsupervised learning of the features and the distribution of real monitoring data. The validation analysis is carried out by the online monitoring dataset of the actual operating units, and the comparison experiments show that the clustering contour coefficient (SCI) of the HDBSCAN-based anomaly detection model reaches 0.4935, which is higher than that of the other comparative models, indicating that the proposed model has superiority in distinguishing between the valid samples and anomalous samples. The probability density distribution of the data filling model based on WSGAIN-GP is similar to that of the measured data, and the KL dispersion, JS dispersion and Hellinger’s distance of the distribution between the filled data and the original data are close to 0. Compared with the filling methods such as SGAIN, GAIN, KNN, etc., the effect of data filling with different missing rates is verified, and the RMSE error of data filling with WSGAIN-GP is lower than that of other comparative models. The WSGAIN-GP method has the lowest RMSE error under different missing rates, which proves that the proposed filling model has good accuracy and generalization, and the research results in this paper provide a high-quality data basis for the subsequent trend prediction and state warning. Full article

This paper investigates the distribution characteristics of Fourier, discrete cosine, and discrete sine transform coefficients in T1 MRI images. This paper reveals their adherence to Benford’s law, characterized by a logarithmic distribution of first digits. The impact of Rician noise on the first digit distribution is examined, which causes deviations from the ideal distribution. A novel methodology is proposed for noise level estimation, employing metrics such as the Bhattacharyya distance, Kullback–Leibler divergence, total variation distance, Hellinger distance, and Jensen–Shannon divergence. Supervised learning techniques utilize these metrics as regressors. Evaluations on MRI scans from several datasets coming from a wide range of different acquisition devices of 1.5 T and 3 T, comprising hundreds of patients, validate the adherence of noiseless T1 MRI frequency domain coefficients to Benford’s law. Through rigorous experimentation, our methodology has demonstrated competitiveness with established noise estimation techniques, even surpassing them in numerous conducted experiments. This research empirically supports the application of Benford’s law in transforms, offering a reliable approach for noise estimation in denoising algorithms and advancing image quality assessment. Full article

The challenging nature of nearshore tunnel construction environments introduces a multitude of potential hazards, consequently escalating the likelihood of incidents such as water influx. Existing construction safety risk management methodologies often depend on subjective experiences, leading to inconsistent reliability in assessment outcomes. The multifaceted nature of construction safety risk factors, their sources, and structures complicate the validation of these assessments, thus compromising their precision. Moreover, risk assessments generally occur pre-construction, leaving on-site personnel incapable of recommending pragmatic mitigation strategies based on real-time safety issues. To address these concerns, this paper introduces a construction safety risk assessment approach for nearshore tunnels based on multi-data fusion. In addressing the issue of temporal effectiveness when the conflict factor K in traditional Dempster–Shafer (DS) evidence theory nears infinity, the confidence Hellinger distance is incorporated for improvement. This is designed to accurately demonstrate the degree of conflict between two evidence chains. Subsequently, an integrated evaluation of construction safety risks for a specific nearshore tunnel in Ningbo is conducted through the calculation of similarity, support degree, and weight factors. Simultaneously, the Revit secondary development technology is utilized to visualize risk monitoring point warnings. The evaluation concludes that monitoring point K7+860 exhibits a level II risk, whereas other monitoring points maintain a normal status. Full article

Federated learning (FL) allows the collaborative training of a collective model by a vast number of decentralized clients while ensuring that these clients’ data remain private and are not shared. In practical situations, the training data utilized in FL often exhibit non-IID characteristics, hence diminishing the efficacy of FL. Our study presents a novel privacy-preserving FL algorithm, HW-DPFL, which leverages data label distribution similarity as a basis for its design. Our proposed approach achieves this objective without incurring any additional overhead communication. In this study, we provide evidence to support the assertion that our approach improves the privacy guarantee and convergence of FL both theoretically and empirically. Full article

With the increasing demand for efficient and accurate numerical simulations of spray combustion in jet engines, the necessity for robust models to enhance the capabilities of spray models has become imperative. Existing approaches often rely on ad hoc determinations or simplifications, resulting in information loss and potentially inaccurate predictions for critical spray characteristics, such as droplet diameters, velocities, and positions, especially under extreme operating conditions or temporal fluctuations. In this study, we introduce a novel approach to modeling multivariate spray characteristics using Gaussian mixture models (GMM). By applying this approach to spray data obtained from numerical simulations of the primary atomization in air-blast atomizers, we demonstrate that GMMs effectively capture the spray characteristics across a wide range of operating conditions. Importantly, our investigation reveals that GMMs can handle complex non-linear dependencies by increasing the number of components, thereby enabling the modeling of more complex spray statistics. This adaptability makes GMMs a versatile tool for accurately representing spray characteristics even under extreme operating conditions. The presented approach holds promise for enhancing the accuracy of spray combustion modeling, offering an improved injection model that accurately captures the underlying droplet distribution. Additionally, GMMs can serve as a foundation for constructing meta models, striking a balance between the efficiency of low-order approaches and the accuracy of high-fidelity simulations. Full article

With the steady improvement of advanced manufacturing processes and big data technologies, modern industrial systems have become large-scale. To enhance the sensitivity of fault detection (FD) and overcome the drawbacks of the centralized FD framework in dynamic systems, a new data-driven FD method based on Hellinger distance and subspace techniques is proposed for dynamic systems. Specifically, the proposed approach uses only system input/output data collected via sensor networks, and the distributed residual signals can be generated directly through the stable kernel representation of the process. Based on this, each sensor node can obtain the identical residual signal and test statistic through the average consensus algorithms. In addition, this paper integrates the Hellinger distance into the residual signal analysis for improving the FD performance. Finally, the effectiveness and accuracy of the proposed method have been verified in a real multiphase flow facility. Full article

We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension. Full article

In IoT environments, voluminous amounts of data are produced every single second. Due to multiple factors, these data are prone to various imperfections, they could be uncertain, conflicting, or even incorrect leading to wrong decisions. Multisensor data fusion has proved to be powerful for managing data coming from heterogeneous sources and moving towards effective decision-making. Dempster–Shafer (D–S) theory is a robust and flexible mathematical tool for modeling and merging uncertain, imprecise, and incomplete data, and is widely used in multisensor data fusion applications such as decision-making, fault diagnosis, pattern recognition, etc. However, the combination of contradictory data has always been challenging in D–S theory, unreasonable results may arise when dealing with highly conflicting sources. In this paper, an improved evidence combination approach is proposed to represent and manage both conflict and uncertainty in IoT environments in order to improve decision-making accuracy. It mainly relies on an improved evidence distance based on Hellinger distance and Deng entropy. To demonstrate the effectiveness of the proposed method, a benchmark example for target recognition and two real application cases in fault diagnosis and IoT decision-making have been provided. Fusion results were compared with several similar methods, and simulation analyses have shown the superiority of the proposed method in terms of conflict management, convergence speed, fusion results reliability, and decision accuracy. In fact, our approach achieved remarkable accuracy rates of 99.32% in target recognition example, 96.14% in fault diagnosis problem, and 99.54% in IoT decision-making application. Full article

We present an empirical estimator for the squared Hellinger distance between two continuous distributions, which almost surely converges. We show that the divergence estimation problem can be solved directly using the empirical CDF and does not need the intermediate step of estimating the densities. We illustrate the proposed estimator on several one-dimensional probability distributions. Finally, we extend the estimator to a family of estimators for the family of α-divergences, which almost surely converge as well, and discuss the uniqueness of this result. We demonstrate applications of the proposed Hellinger affinity estimators to approximately bounding the Neyman–Pearson regions. Full article