Mathematical Estimation Research Articles

Kahramanmaras earthquake on February 6, 2023: mathematical models, estimates and methods of seismic protection

The recent earthquake in Turkey and Syria occurred on February 6, 2023. In the Kahramanmaras region the earthquake had an estimated magnitude of Mw7.8, and reached intensity XI by the modified Mercalli scale. The earthquake had catastrophic consequences, leading to the deaths of more than 52,800 people, as well as the multiple damages of the structures and buildings. The paper relates to the analyses of the seismograms and the detection of an unusually strong delta impulse, which can be associated with the arrival of a horizontally polarized S-wave.

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

Africa Health Research Institute (AHRI) Household Contact study: a study protocol, statistical analysis and modelling plan

Household case-contact studies are an established method to estimate the relative infectiousness of people with pulmonary tuberculosis. Testing of child household contacts with a QuantiFERON TB Gold plus (QFT) assay provides a proxy measure of M. tuberculosis infection in exposed household members. This study aims to generate data about the effect of antiretroviral treatment (ART) and of tuberculosis symptoms on risk of intra-household M. tuberculosis transmission to children and investigate the conditions under which symptom-agnostic screening should be considered, which could provide evidence to support a paradigm shift in tuberculosis screening strategy. Index people diagnosed with bacteriologically-confirmed pulmonary tuberculosis who have been identified by routine health services and starting tuberculosis treatment at one of the 12 health facilities serving the population of the Africa Health Research Institute health and demographic surveillance area will be eligible for recruitment. For each consenting index participant, we will enroll all household members aged 2-14 years who will undergo a symptom screen and provide a blood sample for a QFT and HIV test. We will compare the proportion of child contacts who are QFT-positive in households of (i) index adults who are living with HIV and taking ART vs not taking ART and (ii) index adults engaged in ART care who report no symptoms vs those who report symptoms. We will use a novel mathematical model, incorporating new insights into TB natural history and transmission, to determine the circumstances under which screening for TB, regardless of reported symptoms, would have most impact on tuberculosis incidence and mortality, and explore comparative cost-effectiveness of interventions along the diagnostic pathway at different levels of the health system. Estimates of relative risk of intra-household M. tuberculosis transmission, combined with healthcare utilisation and cost data, will allow us to generate mathematical model estimates of the potential epidemiological impact and relative cost-effectiveness of different approaches to tuberculosis case-finding including symptom-agnostic tuberculosis screening at different levels of the health system.

Quantifying the Temperature Dependence of the Multi-Species, Multi-Reaction Model: Part II. Estimation of Entropy Coefficient for Meso-Carbon Micro-Bead Graphite

In Part 1 of this paper, the temperature impact on the Multi-Species, Multi Reaction (MSMR) model was studied. This was accomplished by acquiring data from slow rate lithiation and delithiation of a meso-carbon micro-bead (MCMB) graphite. Through this analysis, the temperature impact on the total fraction of available host sites in a particular MSMR gallery (Xj), the impact on the reference potential ( Ujo ), and the impact on the parameter detailing the deviation from Nernstian behavior (ω j) was determined. Here, the intercalation material is discussed, compared to traditional methods of acquiring the entropy coefficient, and comparison is made to previous mathematical estimates. Some of the challenges in using temperature dependent constant rate charge and discharge data as compared to the potentiodynamic entropy coefficient calculation method are also discussed, and a recommendation for future applications is proposed.

An innovative trigonometric technique for producing a new range of distributions: Implementations in the medical fields

Introducing new families of statistical distributions for data modeling in the medical sector is a fascinating area of research. Numerous methods have been developed and implemented for such purposes. Many of these methods rely on one or more additional parameters. However, the inclusion of new parameters can lead to certain consequences, such as re-parametrization problems. This paper also presents a novel family of distributions for generating statistical models. The proposed family is based on a trigonometric function and can be referred to as the weighted tangent family of distributions. A notable feature of the weighted tangent family is that it does not require any extra parameters. The paper also derives some mathematical properties and point estimators of the weighted tangent family. Furthermore, the paper focuses on a specific case within the weighted tangent family, namely the weighted tangent Weibull distribution. Finally, the paper considers three applications from the medical sector to demonstrate the effectiveness of the weighted tangent Weibull distribution. The practical illustrations underscore that the weighted tangent approach significantly improves the fitting capacity of the traditional models.

A new extension of XLindley distribution with mathematical properties, estimation, and application on the rainfall data

Data analysis is indeed quite popular and crucial in various fields such as meteorology, hydrology, epidemiology, economics, and biology. So, the purpose of this study is to introduce a new probability distribution to analyze rainfall data. A new extension of XLindley distribution is introduced using the alpha power transformation technique. The new distribution is named “alpha power transformed XLindley – (APTXL) distribution”. Mathematical properties of APTXL distribution are derived such as ordinary moments, moment generating function, quantile function, mean residual life function, and order statistics. The model parameters are estimated using five different estimation methods such as maximum likelihood, Anderson Darling, Cramer von Misses, Ordinary least squares, and weighted least squares. A comprehensive simulation study is utilized to check the behavior and efficiency of the derived estimators. It is found that the weighted least square approach efficiently estimates the model parameter than the other methods. This derivation of a new model not only contributes to theoretical advances in statistical methodology but also provides practical methods for rainfall data modeling. The APTXL distribution is used to evaluate rainfall datasets from Saudi Arabia. It is identified that the APTXL distribution provides efficient results as compared to considered competitive distributions.

Estimating community-wide indirect effects of influenza vaccination: triangulation using mathematical models and bias analysis.

Understanding whether influenza vaccine promotion strategies produce community-wide indirect effects is important for establishing vaccine coverage targets and optimizing vaccine delivery. Empirical epidemiologic studies and mathematical models have been used to estimate indirect effects of vaccines but rarely for the same estimand in the same dataset. Using these approaches together could be a powerful tool for triangulation in infectious disease epidemiology because each approach is subject to distinct sources of bias. We triangulated evidence about indirect effects from a school-located influenza vaccination program using two approaches: a difference-in-difference (DID) analysis, and an age-structured, deterministic, compartmental model. The estimated indirect effect was substantially lower in the mathematical model than in the DID analysis (2.1% (95% Bayesian credible intervals 0.4 - 4.4%) vs. 22.3% (95% CI 7.6% - 37.1%)). To explore reasons for differing estimates, we used sensitivity analyses and probabilistic bias analyses. When we constrained model parameters such that projections matched the DID analysis, results only aligned with the DID analysis with substantially lower pre-existing immunity among school-age children and older adults. Conversely, DID estimates corrected for potential bias only aligned with mathematical model estimates under differential outcome misclassification. We discuss how triangulation using empirical and mathematical modelling approaches could strengthen future studies.

A new two-parameter over-dispersed discrete distribution with mathematical properties, estimation, regression model and applications

This paper focuses on the derivation of a new two-parameter discrete probability distribution. The new model is derived by mixing Poisson and Loai distributions and is named “Poisson Loai Distribution”. The paper explores various mathematical properties of the new model, introducing a count-regression model based on this distribution. The parameters of the model are estimated using the maximum likelihood estimation method. A comprehensive simulation study is utilized to assess the behavior of derived estimators. The importance of the proposed distribution is confirmed through the analysis of three real datasets. It is found that the suggested distribution has the greatest match when compared to all rival distributions, and it may be a viable alternative for assessing dispersed count data.

Wearable Multi-Sensor Positioning Prototype for Rowing Technique Evaluation.

The goal of this study is to determine the feasibility of a wearable multi-sensor positioning prototype to be used as a training tool to evaluate rowing technique and to determine the positioning accuracy using multiple mathematical models and estimation methods. The wearable device consists of an inertial measurement unit (IMU), an ultra-wideband (UWB) transceiver, and a global navigation satellite system (GNSS) receiver. An experiment on a rowing shell was conducted to evaluate the performance of the system on a rower's wrist, against a centimeter-level GNSS reference trajectory. This experiment analyzed the rowing motion in multiple navigation frames and with various positioning methods. The results show that the wearable device prototype is a viable option for rowing technique analysis; the system was able to provide the position, velocity, and attitude of a rower's wrist, with a positioning accuracy ranging between ±0.185 m and ±1.656 m depending on the estimation method.

Studying Continuous, Time-varying, and/or Complex Exposures Using Longitudinal Modified Treatment Policies.

This tutorial discusses a methodology for causal inference using longitudinal modified treatment policies. This method facilitates the mathematical formalization, identification, and estimation of many novel parameters and mathematically generalizes many commonly used parameters, such as the average treatment effect. Longitudinal modified treatment policies apply to a wide variety of exposures, including binary, multivariate, and continuous, and can accommodate time-varying treatments and confounders, competing risks, loss to follow-up, as well as survival, binary, or continuous outcomes. Longitudinal modified treatment policies can be seen as an extension of static and dynamic interventions to involve the natural value of treatment and, like dynamic interventions, can be used to define alternative estimands with a positivity assumption that is more likely to be satisfied than estimands corresponding to static interventions. This tutorial aims to illustrate several practical uses of the longitudinal modified treatment policy methodology, including describing different estimation strategies and their corresponding advantages and disadvantages. We provide numerous examples of types of research questions that can be answered using longitudinal modified treatment policies. We go into more depth with one of these examples, specifically, estimating the effect of delaying intubation on critically ill COVID-19 patients' mortality. We demonstrate the use of the open-source R package lmtp to estimate the effects, and we provide code on https://github.com/kathoffman/lmtp-tutorial.

Examining the Application of Neural Networks in Predicting Temperature and Solar Cell Power

Cost-effective and efficient engineering of solar cells, in addition to creative design, necessitates a detailed examination of the physics involved in solar energy absorption. Solar energy does not operate at night without storage means such as batteries, and cloudy weather can render this technology unreliable during the day. Since solar cells are used to convert light into electricity, they must be composed of materials efficient in light absorption. Like any other power generation source, solar cells face challenges, including reduced output power with increasing cell surface temperature. With each degree rise in temperature, efficiency can decrease by up to 54.0%, highlighting the importance of addressing and implementing solutions to this issue. This study explores different specifications of solar cells after a general overview and modeling. It investigates methods for estimating solar cell temperature using ambient temperature and solar radiation, comparing them with a proposed neural network approach. If the neural network can achieve lower error rates than the desired thresholds, it would offer advantages over mathematical estimators mentioned in the literature. Additionally, this neural network demonstrates the capability to predict future solar cell temperatures, unlike instantaneous mathematical estimators for temperature and radiation. DOI: https://doi.org/10.52783/pst.558

An Extensive Review of MR Sensors with Design and Characteristic Evaluation of Three-layered TMR Sensor

Abstract: The reliability and efficacy of sensor-based automated systems have improved due to the proliferation of electric vehicles, renewable sources, and integrated systems in power industries extensively. This has been accomplished by increasing the power density and decreasing the volume of the system. Background: Mathematical estimation and comparative analysis of the physical factors result in massive usage of operational matrices measured using sensors. Magnetic field sensors, used in industries and biomedical applications, have a high level of precision in the evaluation of measurements. In order to extract the measured parameters such as sensitivity, accuracy, operating cost, the linear range of operation, and power utilisation, these sensors adhere to the physical constraints during their nominal working conditions. The characteristics of the aforementioned sensors are enumerated in detail in this article. Objective: This objective is highly focused on providing a comprehensive overview of classification and the properties of Hall-Effect, anisotropic magnetoresistive (AMR), giant magnetoresistive (GMR), and tunnelling magnetoresistive (TMR) sensors. The dissertation on its properties concludes that TMR is more reliable and sensitive in variable operating conditions. Methods: The methods for selecting the sensors for an application are confined to voltage fluctuations and sensitivity. A three-layered TMR sensor with two magnetic layers and an insulator in between is proposed as a significant advancement compared to the literature. The micromagnetic simulation is carried out at room temperature for a three-layered TMR made up of neodymium alloy, magnesium oxide, and cobalt platinum alloy. Conclusion: Based on the studies executed, it is determined that TMR is more sensitive than both conventional and MR sensors. The proposed schematic claims that the higher free layer thickness offers maximum sensitivity with 77% negative magnetoresistance. The reduced coercivity of 1.9Oe is achieved in this combination at a specified temperature range.

An improved one-parameter weighted distribution with mathematical properties, estimation, and applications

In this study, an improved extension of the Haq distribution was introduced. The new model is named “size-biased Haq distribution (SBHD)”. Its two special cases length-biased and area-biased Haq distribution are discussed. The important mathematical properties such as moments, moment generating function, variance, skewness, kurtosis, Rényi entropy, survival function, hazard function, and mean residual life are derived. The model parameters are estimated using five different estimation methods. A comprehensive simulation study was utilized to illustrate the behavior of derived estimators. Two datasets from different fields are utilized to illustrate the flexibility of the new distributions. It is concluded that the new model is more flexible and efficiently analyzed when the datasets are compared to the considered competitive distributions.

Finding the limits of deep learning clinical sensitivity with fractional anisotropy (FA) microstructure maps.

Quantitative maps obtained with diffusion weighted (DW) imaging, such as fractional anisotropy (FA) -calculated by fitting the diffusion tensor (DT) model to the data,-are very useful to study neurological diseases. To fit this map accurately, acquisition times of the order of several minutes are needed because many noncollinear DW volumes must be acquired to reduce directional biases. Deep learning (DL) can be used to reduce acquisition times by reducing the number of DW volumes. We already developed a DL network named "one-minute FA," which uses 10 DW volumes to obtain FA maps, maintaining the same characteristics and clinical sensitivity of the FA maps calculated with the standard method using more volumes. Recent publications have indicated that it is possible to train DL networks and obtain FA maps even with 4 DW input volumes, far less than the minimum number of directions for the mathematical estimation of the DT. Here we investigated the impact of reducing the number of DW input volumes to 4 or 7, and evaluated the performance and clinical sensitivity of the corresponding DL networks trained to calculate FA, while comparing results also with those using our one-minute FA. Each network training was performed on the human connectome project open-access dataset that has a high resolution and many DW volumes, used to fit a ground truth FA. To evaluate the generalizability of each network, they were tested on two external clinical datasets, not seen during training, and acquired on different scanners with different protocols, as previously done. Using 4 or 7 DW volumes, it was possible to train DL networks to obtain FA maps with the same range of values as ground truth - map, only when using HCP test data; pathological sensitivity was lost when tested using the external clinical datasets: indeed in both cases, no consistent differences were found between patient groups. On the contrary, our "one-minute FA" did not suffer from the same problem. When developing DL networks for reduced acquisition times, the ability to generalize and to generate quantitative biomarkers that provide clinical sensitivity must be addressed.

Determining the optimal parameters of hydrostabilization process of pyro-condensate in the presence of a nickel-chrome catalyst with the method of mathematical statistics

The aim of the work was to obtain mathematical static estimates of the influence of various factors on the degree of hydrostabilization of pyrocondensate obtained during the pyrolysis of straight-run gasoline, and an attempt to determine the most optimal process mode. The experimental data obtained earlier made it possible to determine the temperature range, duration, volume of the catalyst, and the ratio of hydrogen to feedstock necessary for the effective hydrostabilization of the pyrocondensate, which made it possible to narrow the range of variation of the process parameters. At the same time, the task was set to find the optimal conditions that ensure the maximum degree of hydrogenation of the condensate. The planning of the experiment was carried out according to the scheme of a full factorial experiment 24. According to the results of an active experiment carried out using mathematical planning methods, the major role of variable factors was determined, a mathematical model was obtained, and the optimal mode for conducting the pyro-condensate hydrostabilization process with the presence of a nickel-chromium catalyst was determined: temperature - 80° C, the ratio of hydrogen volume to raw material, equal to 0.3, catalyst volume - 5 sm3, process duration - 120 min. The temperature has the greatest influence on the degree of pyro-condensate hydrogenation.

Understanding Stokes Drift Mechanism via Crest and Trough Phase Estimates

Abstract By providing mathematical estimates, this paper answers a fundamental question—“what leads to Stokes drift”? Although overwhelmingly understood for water waves, Stokes drift is a generic mechanism that stems from kinematics and occurs in any nontransverse wave in fluids. To showcase its generality, we undertake a comparative study of the pathline equation of sound (1D) and intermediate-depth water (2D) waves. Although we obtain a closed-form solution x(t) for the specific case of linear sound waves, a more generic and meaningful approach involves the application of asymptotic methods and expressing variables in terms of the Lagrangian phase θ. We show that the latter reduces the 2D pathline equation of water waves to 1D. Using asymptotic methods, we solve the respective pathline equation for sound and water waves, and for each case, we obtain a parametric representation of particle position x(θ) and elapsed time t(θ). Such a parametric description has allowed us to obtain second-order accurate expressions for the time duration, horizontal displacement, and average horizontal velocity of a particle in the crest and trough phases. All these quantities are of higher magnitude in the crest phase in comparison to the trough phase, leading to a forward drift, i.e., Stokes drift. We also explore particle trajectory due to second-order Stokes waves and compare it with linear waves. While finite amplitude waves modify the estimates obtained from linear waves, the understanding acquired from linear waves is generally found to be valid.

Predicting Ballast Fouling Conditions with the Gaussian Mixture Model

Railroad ballast is typically comprised of only large granular particles. However, the degradation of fresh ballast and the arrival of foreign fines result in ballast fouling. Compared with fresh ballast, fouled ballast exhibits reduced resilience and compromised drainage capabilities. To optimize track performance, maintenance activities for the ballast are frequently scheduled based on the fouling severity. An accurate assessment of ballast fouling conditions can enhance maintenance efficiency and reduce costs. Over the years, while many ballast fouling evaluation methods have been developed, their widespread adoption has been hindered by system costs and implementation challenges. This study aims to address this by developing an affordable and easily implemented approach to estimating ballast fouling conditions using the Gaussian Mixture Model (GMM). Initially, images of fouled ballast are characterized by fitting the distributions of each RGB (Red, Green, Blue) channel. Subsequently, two mathematical methods, expectation-maximization and point estimation, are employed to solve the GMM parameters. These derived GMM parameters are then used to backcalculate the sample parameters, facilitating the estimation of ballast fouling conditions. The results of this study reveal a close alignment between the ballast fouling conditions backcalculated with the GMM and those quantified through laboratory sieving analysis. This study thus presents a promising path forward, using images captured from cost-effective cameras to estimate ballast fouling conditions with minimal computational expense.

Optimum Surfactant Concentration for Preparation of Amiodarone Loaded Solid Lipid Nanoparticles: Theoretical Estimation Versus Experimental Results by Box-Behnken Design

Background: Solid lipid nanoparticles (SLNs) are colloidal carriers made up of lipids that are stabilized by surfactant molecules. The lipid matrix remains solid at body temperature. A significant challenge in the preparation of SLNs is determining the optimal concentration of surfactants due to their potential toxicity. Objectives: During the preparation of SLNs, micelle structures tend to form at high concentrations of surfactants. Since micelles exhibit different properties from SLNs, using the optimum concentration of surfactants leads to the preparation of a consistent formulation of SLNs. This was theoretically predicted in this study and then compared with experimental results. Methods: In this study, amiodarone-loaded SLNs were produced via a hot homogenization process. The design of experiments was utilized to explore effective process parameters, as several factors influence the formulation characteristics. The concentration of surfactants was optimized using a Box-Behnken design. The results were compared with a theoretical equation developed in this study, which estimates the concentration of surfactants needed to cover the surface of the particles. Assessments included particle size and morphology, size distribution, drug loading percentage (DL%), and encapsulation efficiency (%). Results: The particle size of the optimum formulation was 74 ± 1.5 nm, with DL% and EE% being 14.81 ± 0.8% and 97.58 ± 2.5%, respectively. The formulation contained 2.3% Witepsol, 0.25% glyceryl monostearate (GMS), 0.5% amiodarone (AMI), 0.02% sodium lauryl sulfate (SLS), 0.05% poloxamer, and 0.17% lecithin. The total surface area of the particles was estimated according to the equation 6× (volume of the lipid phase)/ (diameter of particles), which can be applied to determine the concentration of surfactants required for preparing SLNs. Conclusions: The results indicated that the theoretical equation was suitable for estimating the optimum concentration of surfactant in the aqueous phase to form SLNs and adequately cover the lipid surface. Mathematical estimations were comparable to the experimental results from the Box-Behnken design. Consequently, the formulation consisted of SLNs without any micellar structure, and the applied concentrations of surfactants effectively covered the surface of the particles.

Comparison of Empirically Derived and Model-Based Estimates of Key Population HIV Incidence and the Distribution of New Infections by Population Group in Sub-Saharan Africa.

The distribution of new HIV infections among key populations, including female sex workers (FSWs), gay men and other men who have sex with men (MSM), and people who inject drugs (PWID) are essential information to guide an HIV response, but data are limited in sub-Saharan Africa (SSA). We analyzed empirically derived and mathematical model-based estimates of HIV incidence among key populations and compared with the Joint United Nations Programme on HIV/AIDS (UNAIDS) estimates. We estimated HIV incidence among FSW and MSM in SSA by combining meta-analyses of empirical key population HIV incidence relative to the total population incidence with key population size estimates (KPSE) and HIV prevalence. Dynamic HIV transmission model estimates of HIV incidence and percentage of new infections among key populations were extracted from 94 country applications of 9 mathematical models. We compared these with UNAIDS-reported distribution of new infections, implied key population HIV incidence and incidence-to-prevalence ratios. Across SSA, empirical FSW HIV incidence was 8.6-fold (95% confidence interval: 5.7 to 12.9) higher than total population female 15-39 year incidence, and MSM HIV incidence was 41.8-fold (95% confidence interval: 21.9 to 79.6) male 15-29 year incidence. Combined with KPSE, these implied 12% of new HIV infections in 2021 were among FSW and MSM (5% and 7% respectively). In sensitivity analysis varying KPSE proportions within 95% uncertainty range, the proportion of new infections among FSW and MSM was between 9% and 19%. Insufficient data were available to estimate PWID incidence rate ratios. Across 94 models, median proportion of new infections among FSW, MSM, and PWID was 6.4% (interquartile range 3.2%-11.7%), both much lower than the 25% reported by UNAIDS. Empirically derived and model-based estimates of HIV incidence confirm dramatically higher HIV risk among key populations in SSA. Estimated proportions of new infections among key populations in 2021 were sensitive to population size assumptions and were substantially lower than estimates reported by UNAIDS.

Multi server queuing model with dynamic power shifting and performance efficiency factor

The need for high performance models in the queuing systems; availability in the fields of computing and communication systems and logistics management poses extensive challenges in design and development of the appropriate modeling. In this work, we propose a robust probabilistic model to mitigate these problems by appropriately choosing a multi-server queuing system with dynamic service facility. The arrival substances, such as packets or jobs or customers or consumers, follow a Poisson arrival process and these arrivals enter a queuing system according to FIFO discipline. The system designed in the system is armed with a fixed number of service stations, in which some servers are capable of allocating additional service capacity by adjusting dynamically. When a customer arrives at the system, if a free server is available, it is immediately served by one of the free processors; if no server is free, that is all are busy, the arrived job is accommodated in the queue and waits for service in the system. In this paper, we proposed a stochastic model to handle peak loads efficiently, boosting service capacity during burst arrivals. Numerical results presented in this paper are generated by the spectral expansion method to demonstrate the model’s performance, offering insights into its efficiency and accuracy. Furthermore, we derived some important special cases of speedup factor, which provide the mathematical estimation of system performance in terms of computation and communication times.