Approximate Calculation Method Research Articles

The truncation effect of soil slope acceleration responses

A dynamic centrifugal model test was conducted for a homogeneous sandy slope under an impulsive ground motion to investigate the relationship between seismic response and slope deformation. Firstly, the deformation characteristics and dynamic responses of the slope were analyzed. Then, the main features associated with the response truncation effect were expounded. Subsequently, the Sliding block theory was introduced to explain the mechanism of the acceleration response truncation effect. At last, an approximate calculation method for the sliding mass dynamic response was developed, considering the influence of the truncation effect. The results show the variation pattern of response differs inside the slope and near the slope surface, and the dynamic amplification coefficients calculated by different methods also exhibit significant variations. The truncation of horizontal response acceleration results from the relative sliding in slope, and the truncated response fluctuates around a limited value related to the yield acceleration.

Harnessing 12-lead ECG and MRI data to personalise repolarisation profiles in cardiac digital twin models for enhanced virtual drug testing

Cardiac digital twins are computational tools capturing key functional and anatomical characteristics of patient hearts for investigating disease phenotypes and predicting responses to therapy. When paired with large-scale computational resources and large clinical datasets, digital twin technology can enable virtual clinical trials on virtual cohorts to fast-track therapy development. Here, we present an open-source automated pipeline for personalising ventricular electrophysiological function based on routinely acquired magnetic resonance imaging (MRI) data and the standard 12-lead electrocardiogram (ECG).Using MRI-based anatomical models, a sequential Monte-Carlo approximate Bayesian computational inference method is extended to infer electrical activation and repolarisation characteristics from the ECG. Fast simulations are conducted with a reaction-Eikonal model, including the Purkinje network and biophysically-detailed subcellular ionic current dynamics for repolarisation. For each patient, parameter uncertainty is represented by inferring an envelope of plausible ventricular models rather than a single one, which means that parameter uncertainty can be propagated to therapy evaluation. Furthermore, we have developed techniques for translating from reaction-Eikonal to monodomain simulations, which allows more realistic simulations of cardiac electrophysiology. The pipeline is demonstrated in three healthy subjects, where our inferred pseudo-diffusion reaction-Eikonal models reproduced the patient's ECG with a median Pearson's correlation coefficient of 0.9, and then translated to monodomain simulations with a median correlation coefficient of 0.84 across all subjects. We then demonstrate our digital twins for virtual evaluation of Dofetilide with uncertainty quantification. These evaluations using our cardiac digital twins reproduced dose-dependent QTc and T peak to T end prolongations that are in keeping with large population drug response data.The methodologies for cardiac digital twinning presented here are a step towards personalised virtual therapy testing and can be scaled to generate virtual populations for clinical trials to fast-track therapy evaluation. The tools developed for this paper are open-source, documented, and made publicly available.

Layered target design method for global spectrum optimization of radioisotope production

Targets are irradiated in high-flux reactors to produce transplutonium isotopes. Neutron environment of the target is crucial for the production efficiency of transplutonium isotopes. To improve the production efficiency of transplutonium isotopes, it is necessary to research the optimization design of target. Taking the production of Californium-252 as an example, this study analyzed the impact of self-shielding effect in targets on the yield of transplutonium isotope based on the High Flux Isotope Reactor (HFIR) and High-Flux Fast Reactor (HFFR). The self-shielding effect leads to the hardening of the neutron spectrum inside the target and significantly reduces the conversion rate of nuclides. After conducting a refined energy spectrum analysis, we proposed a layered target design method based on the Genetic Algorithm (GA). To reduce computational costs, we propose a fixed source-burnup coupling approximate calculation method, which can avoid tedious burnup calculation and provide optimization direction. Using this method, we designed an optimal layered target scheme. Compared with non-layered target, the production efficiency of Cf-252 was increased by approximately 4.1 times. This study provides technical support for energy spectrum analysis and target design in producing transplutonium isotopes.

Cross media knowledge information retrieval model based on D-S evidence theory

Cross media knowledge information retrieval provides strong support for information processing and utilization in the information society, but there are problems such as heterogeneity in cross media knowledge information. Therefore, a cross media knowledge information retrieval model using D-S evidence theory is proposed, which involves using approximate calculation methods to improve this theory for information fusion, reducing computational complexity, and using deep networks for fine-grained information retrieval to improve retrieval accuracy. The results showed that the improved theory enhanced computational efficiency by about 27.23 %. The memory usage was <60 %, and the average accuracy of information fusion reached 93.14 %. It also exhibited high recall and low false alarm rates. The cross media knowledge information retrieval model proposed in the study achieved accuracy of 92.64 %, 96.49 %, and 97.46 % on the three datasets used in the experiment, respectively. The study provides an effective, computationally efficient, and highly accurate model for cross media knowledge information retrieval, which is expected to promote research and application in this field. The combination of improved D-S evidence theory and deep networks provides a powerful approach to solving the problem of cross media heterogeneous information retrieval, which has a positive promoting effect on the processing and utilization of information in the information society.

Fairy: fast approximate coverage for multi-sample metagenomic binning

BackgroundMetagenomic binning, the clustering of assembled contigs that belong to the same genome, is a crucial step for recovering metagenome-assembled genomes (MAGs). Contigs are linked by exploiting consistent signatures along a genome, such as read coverage patterns. Using coverage from multiple samples leads to higher-quality MAGs; however, standard pipelines require all-to-all read alignments for multiple samples to compute coverage, becoming a key computational bottleneck.ResultsWe present fairy (https://github.com/bluenote-1577/fairy), an approximate coverage calculation method for metagenomic binning. Fairy is a fast k-mer-based alignment-free method. For multi-sample binning, fairy can be >250×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$> 250 \ imes$$\\end{document} faster than read alignment and accurate enough for binning. Fairy is compatible with several existing binners on host and non-host-associated datasets. Using MetaBAT2, fairy recovers 98.5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$98.5\\%$$\\end{document} of MAGs with >50%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$> 50\\%$$\\end{document} completeness and <5%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$< 5\\%$$\\end{document} contamination relative to alignment with BWA. Notably, multi-sample binning with fairy is always better than single-sample binning using BWA (>1.5×\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$> 1.5\ imes$$\\end{document} more >50%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$>50\\%$$\\end{document} complete MAGs on average) while still being faster. For a public sediment metagenome project, we demonstrate that multi-sample binning recovers higher quality Asgard archaea MAGs than single-sample binning and that fairy’s results are indistinguishable from read alignment.ConclusionsFairy is a new tool for approximately and quickly calculating multi-sample coverage for binning, resolving a computational bottleneck for metagenomics.FUg-6r-bD5B9L5eToLSA3EVideo

Exact and Approximate Moment Derivation for Probabilistic Loops With Non-Polynomial Assignments

Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art.

On a Method for Calculating Bending and Shear Vibrations of a Porous Piezoelement in the Low-Frequency Region

Introduction. Devices for collecting and storing energy from the external environment are low-power sources of electric energy that are actively used. The autonomous devices for monitoring the damaged condition of various structures include them as well. The working element of these devices is a piezoelectric generator (PEG) — a converter of mechanical energy into electrical energy. The design of PEG is associated with the preliminary construction of their mathematical and computer models, with the help of which the calculation and optimization of structures is carried out. One of the ways to model and calculate PEG is to develop approximate calculation methods based on applied theories. The applied theories for calculating bending vibrations of multilayer piezoactive plates are known and previously developed in the literature. However, in the scientific literature there is not enough information about bending and shear vibrations as a tool for improving the efficiency of engineering calculations of the described structures. The objective of this work was to develop an applied method for calculating bending and shear vibrations of piezoceramic plates, including porous ones.Materials and Methods. Piezoceramics PZT-4, including porous ones, were used as the piezoactive material of the plate. When using porous ceramics, the rigidity of the structure decreased to a greater extent than the piezoelectric modules, which made it possible to obtain a more effective PEG under mechanical action. The mathematical formulation was carried out within the framework of the linear theory of electroelasticity with plate polarization in thickness. The sides of the plate were electrodated, the right side was fixed, and a smooth contact in the vertical wall was set on the left side. Steady-state vibrations of the plate were caused by pressure on the front surfaces of the plate or the difference in electrical potentials at the electrodes. To calculate the characteristics of PEG, the authors proposed an applied theory based on hypotheses about the distribution of characteristics of the stress-strain state and the electric field.Results. Transverse vibrations of a piezoceramic plate in the low-frequency region (below the first bending-shear resonance) were studied. Due to the fact that the mathematical formulation was considered within the framework of the linear theory of elasticity, the problem was divided into the sum of two. The first one took into account the mechanical effect: a distributed load and a transverse force at the left end acted on the front surfaces of the plate, and the potentials at the electrodes were zero. In the second task, there were no mechanical loads, but the potential difference was set at the electrodes. Based on hypotheses about the distribution of deformations, mechanical stresses and electric potential, both problems were reduced to a system of ordinary differential equations and boundary conditions. Comparison with the results of calculations by the finite element method in the ACELAN package showed the adequacy of the proposed applied theory in the low-frequency region.Discussion and Conclusion. Since the formulation of the problem was considered in the linear theory of electroelasticity, and the low-frequency region was studied, the work succeeded in dividing the problem of bending-shear vibrations of a porous piezoceramic plate into two: bending — with mechanical action at zero potentials, and shear — when setting the potential difference and zero mechanical action. The corresponding hypotheses about bending and shear were used. Two systems of ordinary differential equations and boundary conditions, which were solved analytically without the use of “heavy” finite element packages, were constructed. To compare the results and confirm the adequacy of the proposed method, the finite element modeling of such tasks was carried out in a specialized ACELAN package. The comparison showed that the error in determining displacements and electric potential when using this approach, in the case of setting mechanical loads and potential differences, did not exceed 6%. The method developed in the paper can be applied in the design of piezoelectric generators for energy storage in the low-frequency region.

Breakdown Voltage Estimation in Transformer Oils with Low-Cost Humidity Sensor

The relative humidity and temperature of the oil used for insulation purposes in transformers directly affect the breakdown voltage of the oil and accordingly the life of the transformer. Continuous monitoring of the moisture content and breakdown voltage of the transformer oil provides the basis for predictive maintenance practices. This study aims to develop a sensor to continuously monitor the moisture content of the insulating oil and to calculate the breakdown voltage approximately. In this context, firstly, measurements were taken with an industrial oil monitoring sensor which calculates the breakdown voltage in transformer oils with high accuracy and was verified with laboratory measurements. At the same time, EE364 by E+E™ and SHT10 by Sensirion™ also used for temperature and humidity measurements. By comparing the measured data with each other, a relationship was formulated between the humidity and temperature values and the breakdown voltage. As a result, an approximate breakdown voltage calculation method that can be used considering the characteristic parameters of transformer oil has been introduced. Thus, a cost-effective function has been developed that can be widely used for transformer monitoring systems.

New sexually transmitted HIV infections from 2016 to 2050 in Guangdong Province, China: a study based on a dynamic compartmental model

BackgroundIn Guangdong Province, China, there is lack of information on the HIV epidemic among high-risk groups and the general population, particularly in relation to sexual transmission, which is a predominant route. The new HIV infections each year is also uncertain owing to HIV transmission from men who have sex with men (MSM) to women, as a substantial proportion of MSM also have female sexual partnerships to comply with social demands in China.MethodsA deterministic compartmental model was developed to predict new HIV infections in four risk groups, including heterosexual men and women and low- and high-risk MSM, in Guangdong Province from 2016 to 2050, considering HIV transmission from MSM to women. The new HIV infections and its 95% credible interval (CrI) were predicted. An adaptive sequential Monte Carlo method for approximate Bayesian computation (ABC-SMC) was used to estimate the unknown parameter, a mixing index. We calibrated our results based on new HIV diagnoses and proportions of late diagnoses. The Morris and Sobol methods were applied in the sensitivity analysis.ResultsNew HIV infections increased during and 2 years after the COVID-19 pandemic, then declined until 2050. New infections rose from 8,828 [95% credible interval (CrI): 6,435–10,451] in 2016 to 9,652 (95% CrI: 7,027–11,434) in 2019, peaking at 11,152 (95% CrI: 8,337–13,062) in 2024 before declining to 7,084 (95% CrI: 5,165–8,385) in 2035 and 4,849 (95% CrI: 3,524–5,747) in 2050. Women accounted for approximately 25.0% of new HIV infections, MSM accounted for 40.0% (approximately 55.0% of men), and high-risk MSM accounted for approximately 25.0% of the total. The ABC-SMC mixing index was 0.504 (95% CrI: 0.239–0.894).ConclusionsGiven that new HIV infections and the proportion of women were relatively high in our calibrated model, to some extent, the HIV epidemic in Guangdong Province remains serious, and services for HIV prevention and control are urgently needed to return to the levels before the COVID-19 epidemic, especially in promoting condom-based safe sex and increasing awareness of HIV prevention to general population.

Numerical analysis of the local damage of RC beams under close‐in explosions based on the calibrated Karagozian &amp; Case (K&amp;C) concrete model

AbstractLimited research has been carried out to investigate the local damage features of reinforced concrete (RC) beams under close‐in explosions. To better capture the local damage features of rectangular cross‐sectional RC beams, a numerical study on the RC beams subjected to close‐in explosions based on the calibrated Karagozian &amp; Case (K&amp;C) concrete model was conducted in LSDYNA. First, a systematic calibration of the K&amp;C model for concrete was performed in the single‐element numerical analysis. Calibration of the K&amp;C model consists of parameter determination for the new strength surfaces, modification of the damage function to better feature the tensile behavior of concrete, elimination of the element size effect by introducing an element size factor and determination of the scaled damage factors. The new strength surfaces are verified and determined through both the approximation calculation method and the parameter fitting method. And the damage function was modified by reducing the termination of λ to better characterize the brittleness of concrete and feature the tensile failure. A comparison between the numerical results and the corresponding experimental results demonstrates that the calibrated K&amp;C model and established FEM model of RC beams subjected to close‐in explosions can well characterize the local damage features of RC beams. On this basis, a further parameter analysis on the factors influencing the local damage response of RC beams was conducted by considering reinforcement details (stirrup spacing and longitudinal reinforcement diameter) and charge details (charge mass and aspect ratio). Results show that the close‐in explosions destroyed the concrete corners in the blast surface with shear stress first and the side shear failure of concrete corners weakens the concrete core strength and contributes to the greater loss of concrete core. Rear spalling damage was initially caused by the bond failure between the tensile longitudinal reinforcement and the surrounding concrete. Furthermore, increasing stirrup spacing, tensile longitudinal reinforcement diameter, and charge mass contribute to the increase in local damage sizes.

Bayesian estimation of large-scale simulation models with Gaussian process regression surrogates

Large scale, computationally expensive simulation models pose a particular challenge when it comes to estimating their parameters from empirical data. Most simulation models do not possess closed-form expressions for their likelihood function, requiring the use of simulation-based inference, such as simulated method of moments, indirect inference, likelihood-free inference or approximate Bayesian computation. However, given the high computational requirements of large-scale models, it is often difficult to run these estimation methods, as they require more simulated runs that can feasibly be carried out. The aim is to address the problem by providing a full Bayesian estimation framework where the true but intractable likelihood function of the simulation model is replaced by one generated by a surrogate model trained on the limited simulated data. This is provided by a Linear Model of Coregionalization, where each latent variable is a sparse variational Gaussian process, chosen for its desirable convergence and consistency properties. The effectiveness of the approach is tested using both a simulated Bayesian computing analysis on a known data generating process, and an empirical application in which the free parameters of a computationally demanding agent-based model are estimated on US macroeconomic data.

FORCE ANALYSIS OF A PLANETARY ROLLER DRIVE TAKING INTO ACCOUNT THE CONTACT COMPLIANCE OF THE MATING THREAD TURNS OF ITS PARTS

The objects of study are planetary roller drives (PRD), which have high performance characteristics and are the most promising mechanical converters of rotational motion into linear motion. They are widely used as part of high-tech machines. Their design requires reliable force analysis. To do this, we need to determine the forces acting on the roller, which is an intermediate part between the nut and the screw. Several works provide approximate methods for the force calculation of the PRD, but in them, as an assumption based on similar calculations of machine parts, the distribution of the load between the roller thread turns interacting with the screw turns and with the nut turns is accepted. In this work, based on the fundamental principles of mechanics, using the necessary assumptions, a numerical method for determining forces is developed. These forces act in the PRD between the mating thread turns of the roller and the screw and between the mating thread turns of the roller and the nut, taking into account only one influencing factor – the momentum that overturns the roller. To implement this method, due to the very large amount of calculations, a computer program was developed. Usually, for force calculations of roller drives, the coefficient of uneven loading (Кн) of roller thread turns is introduced. For the control calculation of a PRD of standard size 20×8 with a four-thread screw and a length of the threaded part of the nut of 56 mm, Кн = 1.509, which is a very large value and shows the large influence of the overturning momentum on the load distribution between the roller threads. To determine the real unevenness of load distribution in the PRD, it is necessary to conduct further research that takes into account other factors influencing the uneven distribution of load in the PRD. In this case, the next most important factor is the uneven distribution of the total axial force between the rollers. This force is applied to the PRD from the actuator.

Метод приближенного вычисления интеграла Кирхгофа

The article proposes a method of approximate calculation of the wave field gradient by its distribution on the only one closed surface. This allows calculating the Kirchhoff's integral without additional calculations or measurements of the electromagnetic field and opens up the opportunity to use it in practice along with the field equivalence principle. The comparison of the Kirchhoff's integral calculation accuracy with its vector forms showed that the proposed approximation for the wave field derivative leads to additional methodological errors, which are equivalent to the stray signal at the level less then −40 dB. At the same time, the use of Kirchhoff's integral instead of a widespread field equivalence principle in the electromagnetic field transformation algorithms reduces their computing complexity to 20 times.

Intrusion Detection Model for Wireless Sensor Networks Based on FedAvg and XGBoost Algorithm

For the characteristics of channel instability in wireless sensor networks, this paper proposes an intrusion detection algorithm based on FedAvg (federated averaging) and XGBoost (extreme gradient boosting) wireless sensor networks using fog computing architecture. First, the network edge is extended by introducing fog computing nodes to reduce the communication delay. It reduces the transmission bandwidth and privacy leakage risk while improving the accuracy of jointly learned global and local models. Then, the histogram-based approximation calculation method is improved to adapt to the unbalanced data characteristics of wireless sensor networks. Finally, by introducing TOP-K gradient selection, the number of model parameter uploads is minimized, and the efficiency of model parameter interaction is improved. The experimental results show that this algorithm has superior detection performance and low energy consumption. It is also compared with other algorithms to demonstrate the high detection rate and low computational complexity of this algorithm.

Разработка способа расчета стоимости монтажа инженерных заграждений при изучении дисциплин инженерно-технической защиты

Specialists in technical information security deal with the physical security of various industrial objects. This article introduces a new method for approximating costs of installing a concertina razor wire obstacle. The method can be implemented as part of information security disciplines. The research involved a review of academic sources, scientific publications, and regulatory documents, which showed that courses of information security and engineering hardly ever cover the issue of costs calculation. The author developed a method for approximate calculation of the costs of installing reinforced barbed wire in its single-level concertina razor variant, including a simplified training sample for a hypothetical protected object. The results obtained can be applied as part of courses of engineering and information security, e.g., as laboratory tasks, tutorials, or research projects. The results may also be of interest to technical specialists involved in the engineering and technical data protection at various facilities that require physical security.

A Quick Calculation Method for Radiation Pattern of Submillimeter Telescope with Deformation and Displacement

Radiation pattern captures the electromagnetic performance of reflector antennas, which is significantly affected by the deformation of the primary reflector due to gravity and the displacement of the secondary reflector. During the design process of large reflector antennas, a substantial amount of time is often dedicated to iteratively adjusting structural parameters and validating electromagnetic performance. To improve the efficiency of the design process, we first propose an approximate calculation method of optical path difference (OPD) for the deformation of the primary reflector under gravity and the displacement of the secondary reflector. Then an OPD fitting function based on the modified Zernike polynomials is proposed to capture the phase difference of radiation over the aperture plane, based on which the radiation pattern will be obtained quickly by the aperture field integration method. Numerical experiments demonstrate the effectiveness of the proposed quick calculation method for analyzing the radiation pattern of a 10.4 m submillimeter telescope antenna at its highest operating frequency of 856 GHz. In comparison with the numerical simulation method based on GRASP (which is an antenna electromagnetic analysis tool combining physical optics (PO) and physical theory of diffraction (PTD)), the quick calculation method reduces the time for radiation pattern analysis from more than one hour to less than two minutes. Furthermore, the quick calculation method exhibits excellent accuracy for the figure of merit (FOM) of the radiation pattern. Therefore, the proposed quick calculation method can obtain the radiation pattern with high speed and accuracy. Compared to the time-consuming numerical simulation method (PO and PTD), it can be employed for quick analysis of the radiation pattern for the lateral displacement of the secondary reflector and the deformation of the primary reflector under gravity in the design process of a reflector antenna.

Approximation method for fast calculation of transmission in multi-mode waveguides.

Freeform dielectric waveguides connect optical chips made of different materials in fully integrated photonic devices. With a spatial extent in the order of 100µm, they constitute a computational challenge and make Maxwell full-wave solvers unhandy for the accelerated design. Therefore, it is of utmost importance to have tools that permit the fast prediction of waveguide loss to enable the rapid optimization of waveguide trajectories. Previously developed methods relied on the assumption that only a single mode propagates in the waveguide. However, the propagation of higher-order modes is not just unavoidable due to the geometry of the waveguides, but also, sometimes, beneficial as it increases the number of channels to transmit information. In this contribution, we present an approximate method for the fast calculation of transmission that accommodates the presence of higher-order waveguide modes, and assess its liability by describing light propagation through selected devices.

Research on the output characteristics of photovoltaic arrays under partial shading conditions based on peak point approximate calculation method

Research on the output characteristics of photovoltaic arrays under partial shading conditions based on peak point approximate calculation method

Optimal cubature formulas for approximate integrals of functions defined on a sphere in three-dimensional space

The development of effective methods of approximate calculation of integrals using optimal cubature formulas and optimal quadrature formulas with trigonometric weights for defined functions on a sphere, the creation of new algorithms for approximate calculation of trigonometric weighted integrals in different classes of functions, as well as the assessment of their errors, is the goal of this work.

Rare Event Approximation Between Subdistribution Hazard Ratio and Cause-specific Hazard Ratio in Survival Analysis With Competing Risks.

Despite the fact that competing risks are inevitable in epidemiological and clinical studies, distinctions between the hazard ratio estimated by handling competing risks as censoring and the subditribution hazard ratio are often overlooked. We derived quantitative relationships between subdistribution hazard ratio and cause-specific hazard ratio and derive an approximate calculation method to transform the two into each other. Numerical examinations of hypothetical six scenarios and published information of a randomized clinical trial of cholesterol-lowering therapy and a registry of acute myeloid leukemia were provided. General and approximate relationships under rare event assumptions between the two types of hazard ratio were given. The approximation formula is based on a survival ratio and has two possible applications. First, one can calculate a subdistribution hazard ratio from published information. Second, this formula allows sample size estimation that takes the presence of competing risks into account. The distinction between the two types of hazard ratio can be addressed by focusing on two quantities. One is how the event of interest and competing risk is rare, and the other is the survival ratio.