Empirical Probability Research Articles

Local resolution characteristics of atmospheric turbulence.

D. L. Fried's concept of a "Lucky Image" for turbulent image streams can be seen as creating different degrees of localized resolution in images. These localized regions of resolution can be derived from Fried's equation for the probability of obtaining a Lucky Image. The existence of local resolution variations when imaging through turbulence also implies local variations in the point-spread-functions (PSFs) caused by turbulence. We characterize these local variations by using simple measures on PSFs collected in the presence of atmospheric turbulence. We also compile these variations into an empirical probability density function (PDF) that describes the different resolutions in local regions of turbulent imagery and can be used to characterize specific conditions of turbulence, e.g., the coherence diameter.

Probabilistic Evaluation of Hydraulic Fracture Performance Using Ensemble Machine Learning

Evaluation of hydraulic fracture (HF) performances is critical to develop unconventional resources such as tight oil and gas. We present a probabilistic evaluation approach that integrates ensemble machine learning with Monte Carlo simulation. In the method, we employ the ensemble learning to develop a predictive model between well productivity and its influential factors including both geological properties and HF treatment parameters. Next, coupling the built prediction model with Monte Carlo simulation, an empirical cumulative probability distribution of the well productivity is generated. The well HF performance is assessed by estimating its cumulative probability value. The proposed method is applied to evaluate the HF performances in a developed block of the eastern Sulige region. The study shows that 19% of the wells were fractured with good quality and 55% of the wells were fractured with average quality, while the rest were stimulated with poor quality. The evaluations provide a guideline for optimization of HF designs of wells that have not been hydraulically stimulated in the region.

A proposed cigarette emissions topography protocol reflecting smokers' natural environment use behavior.

The FTC, in 2008, rescinded its 1966 guidance regarding use of the Cambridge Filter Method, noting the yields from the method are relatively poor indicators of tar, nicotine, and carbon monoxide exposure. This article proposes a set of puffing conditions for cigarette emissions testing, with the goal of developing product-specific emissions characterizations which can subsequently be used to realistically model the yield of particulate matter and constituents to the mouth of a smoker, while accounting for the actual puffing behavior of the smoker. Synthesis of data was conducted on data collected from a prior one-week observation of 26 adult cigarette smokers, using their usual brand cigarette in each smokers' natural environment including the puff flow rate, duration, volume and time of day of each puff taken were recorded with a cigarette topography monitor. Data was analyzed to determine the empirical joint probability function and cumulative distribution function of mean puff flow rate and puff duration. The joint CDF was used to define an emissions topography protocol using concepts common to computational grid generation. Analysis of 8,250 cigarette puffs indicated the middle 95% of mean puff flow rates varied between 15 and 121 [mL/s] while the middle 95% of puff duration varied from 0.55 to 3.42 [s]. Thirteen conditions of varying mean puff flow rate and puff duration are proposed for a comprehensive cigarette emissions topography protocol. The proposed protocol addresses inadequacies associated with common machine-puffing profiles used for generating cigarette emissions.

Coupled modeling of the seasonal transmission loss in the Beaufort Sea

An eight-element drifting vertical line array was deployed overwinter 2019–20 in the Beaufort Sea to record transmissions from two moored 35 Hz sources deployed as part of the Coordinated Arctic Acoustic Thermometry Experiment (CAATEX), and seven 925 Hz sources deployed by the Arctic Mobile Observing Systems (AMOS) experiment. Transmissions were received every day on the array at ranges of 10 to 850 km. A probabilistic rough-ice canopy was integrated to a Parabolic Equation code to predict the propagation loss. Sound speed profiles and percent ice cover along the propagation paths were obtained at daily resolution from data-assimilative Global Ice Ocean Prediction System (GIOPS, 20 km spatial resolution). Sea ice roughness and keel properties are empirically determined from the GIOPS ice thickness. The ice roughness is generated using a random pulse-train model with a characteristic wavenumber spectrum slope determined by ice age. Empirical probability distributions of ice keel depth, slope, and spatial densities are used to randomly add keels. Spatial averaging is used to account for signal bandwidth and uncertainty in receiver positions. The environment files and modeling are performed in piece-wise steps to manage memory footprints over long ranges. Modeled propagation losses compare well with the data and reproduce the seasonal variability observed in the measured results.

Sample Size Determination for Interval Estimation of the Prevalence of a Sensitive Attribute Under Randomized Response Models

Studies with sensitive questions should include a sufficient number of respondents to adequately address the research interest. While studies with an inadequate number of respondents may not yield significant conclusions, studies with an excess of respondents become wasteful of investigators’ budget. Therefore, it is an important step in survey sampling to determine the required number of participants. In this article, we derive sample size formulas based on confidence interval estimation of prevalence for four randomized response models, namely, the Warner’s randomized response model, unrelated question model, item count technique model and cheater detection model. Specifically, our sample size formulas control, with a given assurance probability, the width of a confidence interval within the planned range. Simulation results demonstrate that all formulas are accurate in terms of empirical coverage probabilities and empirical assurance probabilities. All formulas are illustrated using a real-life application about the use of unethical tactics in negotiation.

Financial Asset Risk Measurement Based on Smart Sensor Big Data Security Analysis and Bayesian Posterior Probability Model

Following the speeding up of a process of financial globalization, the risks faced by financial markets have become more complex and diversified. Correlated patterns among financial assets exhibit characteristics of nonlinearity, asymmetry, and tail correlation. The original linear correlation analysis method is no longer suitable, but relevant information describing financial risks. In order to confirm whether an asset is safe, the key is to study and master its volatility, and this is based on our mastery of volatility measurement skills. This article is based on smart sensor big data security analysis and Bayesian analysis. The risk measurement of financial assets based on the empirical probability model is studied. The GARCH-t(1,1) model is selected according to the Akaike information criterion (AIC) after the generalized autoregressive conditional heteroskedasticity (GARCH) model is established by the EViews software. According to the results of probability integral transformation, a series of correlation coefficients and degrees of freedom oft-copula are obtained by the maximum likelihood estimation method. This paper uses the risk-adjusted return on capital (RAROC) method to evaluate the risk performance of financial assets. Financial institutions can only retain and absorb the financial market risks that cannot be avoided and transferred. The edge user node sends the service request to the edge server node. The edge server uses the model proposed in this paper to evaluate the user’s trust and selects the corresponding service level according to the trust level corresponding to the calculated credibility results. The data show that the edge calculation takes 0.2581 seconds, while the linear search takes about 64 seconds. The results show that intelligent edge computing improves the accuracy and efficiency of financial asset risk measurement.

Heterogeneous network flow and Petri nets characterize multilayer complex networks

Interacting subsystems are commonly described by networks, where multimodal behaviour found in most natural or engineered systems found recent extension in form of multilayer networks. Since multimodal interaction is often not dictated by network topology alone and may manifest in form of cross-layer information exchange, multilayer network flow becomes of relevant further interest. Rationale can be found in most interacting subsystems, where a form of multimodal flow across layers can be observed in e.g., chemical processes, energy networks, logistics, finance, or any other form of conversion process relying on the laws of conservation. To this end, the formal notion of heterogeneous network flow is proposed, as a multilayer flow function aligned with the theory of network flow. Furthermore, dynamic equivalence is established with the framework of Petri nets, as the baseline model of concurrent event systems. Application of the resulting multilayer Laplacian flow and flow centrality is presented, along with graph learning based inference of multilayer relationships over multimodal data. On synthetic data the proposed framework demonstrates benefits of multimodal flow derivation in critical component identification. It also displays applicability in relationship inference (learning based function approximation) on multimodal time series. On real-world data the proposed framework provides, among others, multimodal flow interpretation of U.S. economic activity, uncovering underlying empirical steady state probability distribution, as well as inherent network (economic) robustness.

Spatial and Temporal Global Patterns of Drought Propagation

Drought is the most expensive natural hazard and one of the deadliest. While drought propagation through standardised indices has been extensively studied at the regional scale, global scale drought propagation, and particularly quantifying the space and time variability, is still a challenging task. Quantifying the space time variability is crucial to understand how droughts have changed globally in order to cope with their impacts. In particular, better understanding of the propagation of drought through the climate, vegetation and hydrological subsystems can improve decision making and preparedness. This study maps spatial temporal drought propagation through different subsystems at the global scale over the last decades. The standardised precipitation index (SPI) based on the gamma distribution, the standardised precipitation evapotranspiration index (SPEI) based on the log-logistic distribution, the standardised vegetation index (SVI) based on z-scores, and the standardised runoff index (SRI) based on empirical runoff probabilities were quantified. Additionally, drought characteristics, including duration, severity and intensity were estimated. Propagation combined the delay in response in the subsystems using drought characteristics, and trends in time were analysed. All these were calculated at 0.05 to 0.25 arc degree pixels. In general, drought propagates rapidly to the response in runoff and streamflow, and a with longer delay in the vegetation. However, this response varies spatially across the globe and depending on the observation scale, and amplifies progressively in duration and severity across large regions from the meteorological to the agricultural/ecological and hydrologic subsystems, while attenuating in intensity. Significant differences exist between major Köppen climate groups in drought characteristics and propagation. Patterns show intensification of drought severity and propagation affecting vegetation and hydrology in regions of southern South America, Australia, and South West Africa.

A Robust Strategy for Leveraging Soft Open Points to Mitigate Load Altering Attacks

Power system cybersecurity is emerging as a critical and urgent problem to the energy sector due to the ongoing power grid modernization initiative. Load altering attack (LAA) is an important category of cyberattacks on the modern power systems, in which the attackers may damage the grid by viciously altering the remotely controllable loads (RCL) that are not properly protected. In order to mitigate the impacts of LAAs on the distribution systems, the promising soft open point (SOP) technology is deployed in this study. A two-stage optimization framework is proposed for the optimal installation and operation of SOPs for defending the distribution systems against LAAs. A chance-constrained optimization model is developed to guarantee the confidence level of the proposed two-stage model of SOPs in mitigating the impacts of LAAs. Further, a Wasserstein metric based distributionally robust chance-constrained (DRCC) optimization method is developed to ensure the robustness of the proposed model against the ambiguity of the empirical probability distribution in practice. Case studies were performed on a 69-bus test system to validate the proposed method. The results of case studies show that the proposed framework is able to mitigate the impacts of LAAs on distribution systems with the installation of SOPs. By applying the DRCC optimization method, the proposed model manages to keep satisfactory confidence levels under the ambiguous probability distributions in the case studies.

Indirect estimation of willingness to pay for energy technology adoption

Adopting energy-efficient and clean technologies is key to climate change mitigation and meeting long-term sustainability goals because they significantly reduce energy consumption and related carbon emissions. Understanding existing barriers and drivers for the adoption of these energy-efficient and clean technologies will be crucial to meeting ambitious national energy and emissions targets, and the customers’ willingness to pay (WTP) is a key factor in understanding the potential for scaling-up adoption. However, in practice, commonly-used WTP estimation methods such as survey or purchase experiments are not always practical or feasible due to budget, time, labor or data constraints. This study proposes a new constrained optimization-based indirect estimation of WTP for energy technology adoption using customers’ implicit life-cycle cost-benefit analysis and market data. The empirical probability distribution of WTP is estimated using the Monte Carlo methods. This new indirect estimation method provides a deeper understanding of the barriers and customers’ willingness to adopt high efficiency and clean energy technologies, and informs the development of supporting policies and programs needed to accelerate market adoption.

Thermophysical modeling of raw glaze liquidus temperature and viscosity

AbstractRaw glazes, which do not contain pre‐melted glass frits, are widely used due to their low cost. The natural variability of the mineral compositions used in raw glazes affects their viscosity during firing. Molten glaze viscosity plays an important role in determining the final surface quality; therefore, it is important to determine how raw glaze composition and firing conditions affect viscosity. Thermophysical modeling provides a way to evaluate the liquidus temperature and viscosity of molten glazes as a function of composition and oxygen activity. Equilibrium phase volume fraction and composition of 242 raw glaze recipes were evaluated under oxidizing and reducing conditions. Undissolved or precipitation of solid phases explains the secondary flux transition from anti‐fluxing to fluxing near the liquidus temperature. The liquidus temperature and viscosity of iron glazes decrease as a function of increasing iron content under a reducing atmosphere. The empirical probability distribution of molten glaze viscosity follows a lognormal distribution, with lead glazes having significantly lower viscosity compared with glazes without lead. The peak firing temperature viscosity of lead‐free glazes is near the working point of glass. Multiple linear regression analysis shows that the peak firing temperature viscosity and liquidus temperature significantly predict the viscosity factor.

Ultrasound Lesion Detectability as a Distance Between Probability Measures.

Lesion detectability (LD) quantifies how easily a lesion or target can be distinguished from the background. LD is commonly used to assess the performance of new ultrasound imaging methods. The contrast-to-noise ratio (CNR) is the most popular measure of LD; however, recent work has exposed its vulnerability to manipulations of dynamic range. The generalized CNR (gCNR) has been proposed as a robust histogram-based alternative that is invariant to such manipulations. Here, we identify key shortcomings of CNR and strengths of gCNR as LD metrics for modern beamformers. Using the measure theory, we pose LD as a distance between empirical probability measures (i.e., histograms) and prove that: 1) gCNR is equal to the total variation distance between probability measures and 2) gCNR is one minus the error rate of the ideal observer. We then explore several consequences of measure-theoretic LD in simulation studies. We find that histogram distances depend on bin selection that LD must be considered in the context of spatial resolution and that many histogram distances are invariant under measure-preserving isomorphisms of the sample space (e.g., dynamic range transformations). Finally, we provide a mathematical interpretation for why quantitative values such as contrast ratio (CR), CNR, and signal-to-noise ratio should not be compared between images with different dynamic ranges or underlying units and demonstrate how histogram matching can be used to reenable such quantitative comparisons.

Modelling the downside risk potential of mutual fund returns

Investors are becoming more sensitive about returns and losses, especially when the investments are exposed to downside risk potential in the financial markets. Despite the computational intensity of the downside risk measures, they are very widely applied to construct a portfolio and evaluate performance in terms of the investors’ loss aversion. Value-at-risk (VaR) has emerged as an industry standard to analyze the market downside risk potential. The approaches used to measure VaR vary from the standard approaches to more recently introduced highly sophisticated volatility models. In this paper, the standard approaches (student-t-distribution, log normal, historical simulation) and sophisticated volatility models (EWMA, GARCH (1,1)) both have been used to estimate the VaR of mutual funds in the Saudi Stock Exchange between June 2017 and June 2020. The VaR approaches have been subjected to conditional coverage backtest to identify the model that is the best at predicting VaR. The empirical coverage probability of the models reveals that EWMA was able to capture VaR better than the other models at a higher significance level followed by GARCH (1,1).

Bayesian parameter estimation in chiral effective field theory using the Hamiltonian Monte Carlo method

The number of low-energy constants (LECs) in chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT}$) grows rapidly with increasing chiral order, necessitating the use of Markov chain Monte Carlo techniques for sampling their posterior probability density function. For this we introduce a Hamiltonian Monte Carlo (HMC) algorithm and sample the LEC posterior up to next-to-next-to-leading order (NNLO) in the two-nucleon sector of $\ensuremath{\chi}\mathrm{EFT}$. We find that the sampling efficiency of HMC is three to six times higher compared to an affine-invariant sampling algorithm. We analyze the empirical coverage probability and validate that the NNLO model yields predictions for two-nucleon scattering data with largely reliable credible intervals, provided that one ignores the leading-order EFT expansion parameter when inferring the variance of the truncation error. We also find that the NNLO truncation error dominates the error budget.

Confidence Intervals for Assessing Non-Inferiority with Assay Sensitivity in a Three-Arm Trial with Normally Distributed Endpoints

Various approaches including hypothesis test and confidence interval (CI) construction have been proposed to assess non-inferiority and assay sensitivity via a known fraction or pre-specified margin in three-arm trials with continuous or discrete endpoints. However, there is little work done on the construction of the non-inferiority margin from historical data and simultaneous generalized CIs (SGCIs) in a three-arm trial with the normally distributed endpoints. Based on the generalized fiducial method and the square-and-add method, we propose two simultaneous CIs for assessing non-inferiority and assay sensitivity in a three-arm trial. For comparison, we also consider the Wald-type Bonferroni simultaneous CI and parametric bootstrap simultaneous CI. An algorithm for evaluating the optimal sample size for attaining the pre-specified power is given. Simulation studies are conducted to investigate the performance of the proposed CIs in terms of their empirical coverage probabilities. An example taken from the mildly asthmatic study is illustrated using the proposed simultaneous CIs. Empirical results show that the proposed generalized fiducial method and the square-and-add method behave better than other two compared CIs.

Prediction intervals of the COVID-19 cases by HAR models with growth rates and vaccination rates in top eight affected countries: Bootstrap improvement

This paper is devoted to modeling and predicting COVID-19 confirmed cases through a multiple linear regression. Especially, prediction intervals of the COVID-19 cases are extensively studied. Due to long-memory feature of the COVID-19 data, a heterogeneous autoregression (HAR) is adopted with Growth rates and Vaccination rates; it is called HAR-G-V model. Top eight affected countries are taken with their daily confirmed cases and vaccination rates. Model criteria results such as root mean square error (RMSE), mean absolute error (MAE), R2, AIC and BIC are reported in the HAR models with/without the two rates. The HAR-G-V model performs better than other HAR models. Out-of-sample forecasting by the HAR-G-V model is conducted. Forecast accuracy measures such as RMSE, MAE, mean absolute percentage error and root relative square error are computed. Furthermore, three types of prediction intervals are constructed by approximating residuals to normal and Laplace distributions, as well as by employing bootstrap procedure. Empirical coverage probability, average length and mean interval score are evaluated for the three prediction intervals. This work contributes three folds: a novel trial to combine both growth rates and vaccination rates in modeling COVID-19; construction and comparison of three types of prediction intervals; and an attempt to improve coverage probability and mean interval score of prediction intervals via bootstrap technique.

New methodology for discrimination of topography diversity of engineering surfaces – Case of grinding

The paper presents the methodology of assessing the degree of differentiation of machined technical surfaces based on the empirical probability distribution of the roughness parameters. The methodology allowed to determine the differentiation level of the analysed surfaces morphology based on the influence of the machining process features on the variability of roughness parameters. The minimum sample size necessary to obtain a statistically significant differentiation level in roughness parameters of the compared surfaces was also determined. The conducted analyses made it possible to determine the information content (indicating the ability to differentiate the features of the analysed surfaces) of the roughness parameters. The robustness of the methodology has been demonstrated. The described methodology was used to evaluate the information content on the example of the grinding process of 100Cr6 bearing steel and Ti-6Al-4V titanium alloy.

A Modified Discretized Chaotic Map and Its Generated Pseudo Binary Random Number

In this manuscript, a modified discretized chaotic map is developed which generates a pseudo-random binary number. This Modified Discretized Chaotic Map (MDCM) not only enlarges the range of system parameter of conventional chaotic map but also improves its performance in terms of key space. The numbers generated from MDCM are random in nature and their randomness is investigated with the help of empirical Probability Density Function (PDF) and Lyapunov Exponent. To investigate the randomness of the numbers, these numbers are quantized into two levels and encoded into 1-bit pseudo random binary numbers and then their randomness behavior is investigated by NIST's (National Institute of Standards and Technology) randomness testing methods. The proposed MDCM is compared with the conventional chaotic map in terms of their Lyapunov Exponent. The numbers generated from MDCM can be used not only in various applications of image processing such as image cryptography, image encryption and others, but also applicable where pseudo numbers are necessarily required.

Molecular Computational Anatomy: Unifying the Particle to Tissue Continuum via Measure Representations of the Brain.

The objective of this research is to unify the molecular representations of spatial transcriptomics and cellular scale histology with the tissue scales of computational anatomy for brain mapping. We present a unified representation theory for brain mapping based on geometric varifold measures of the microscale deterministic structure and function with the statistical ensembles of the spatially aggregated tissue scales. Mapping across coordinate systems in computational anatomy allows us to understand structural and functional properties of the brain at the millimeter scale. New measurement technologies in digital pathology and spatial transcriptomics allow us to measure the brain molecule by molecule and cell by cell based on protein and transcriptomic functional identity. We currently have no mathematical representations for integrating consistently the tissue limits with the molecular particle descriptions. The formalism derived here demonstrates the methodology for transitioning consistently from the molecular scale of quantized particles-using mathematical structures as first introduced by Dirac as the class of generalized functions-to the tissue scales with methods originally introduced by Euler for fluids. We introduce two mathematical methods based on notions of generalized functions and statistical mechanics. We use geometric varifolds, a product measure on space and function, to represent functional states at the micro-scales-electrophysiology, molecular histology-integrated with a Boltzmann-like program to pass from deterministic particle descriptions to empirical probabilities on the functional states at the tissue scales. Our space-function varifold representation provides a recipe for traversing from molecular to tissue scales in terms of a cascade of linear space scaling composed with nonlinear functional feature mapping. Following the cascade implies every scale is a geometric measure so that a universal family of measure norms can be introduced which quantifies the geodesic connection between brains in the orbit independent of the probing technology, whether it be RNA identities, Tau or amyloid histology, spike trains, or dense MR imagery. We demonstrate a unified brain mapping theory for molecular and tissue scales based on geometric measure representations. We call the consistent aggregation of tissue scales from particle and cellular scales, molecular computational anatomy.

Distributionally Robust MPC for Nonlinear Systems

Classical stochastic model predictive control (SMPC) methods assume that the true probability distribution of uncertainties in controlled systems is provided in advance. However, in real-world systems, only partial distribution information can be acquired for SMPC. The discrepancy between the true distribution and the distribution assumed can result in sub-optimality or even infeasibility of the system. To address this, we present a novel distributionally robust data-driven MPC scheme to control stochastic nonlinear systems. We use distributionally robust constraints to bound the violation of the expected state-constraints under process disturbance. Sequential linearization is performed at each sampling time to guarantee that the system's states comply with constraints with respect to the worst-case distribution within the Wasserstein ball centered at the discrete empirical probability distribution. Under this distributionally robust MPC scheme, control laws can be efficiently derived by solving a conic program. The competence of this scheme for disturbed nonlinear systems is demonstrated through two case studies.