Simple Fitting Research Articles

A machine learning approach to Bayesian parameter estimation

Bayesian estimation is a powerful theoretical paradigm for the operation of the approach to parameter estimation. However, the Bayesian method for statistical inference generally suffers from demanding calibration requirements that have so far restricted its use to systems that can be explicitly modeled. In this theoretical study, we formulate parameter estimation as a classification task and use artificial neural networks to efficiently perform Bayesian estimation. We show that the network’s posterior distribution is centered at the true (unknown) value of the parameter within an uncertainty given by the inverse Fisher information, representing the ultimate sensitivity limit for the given apparatus. When only a limited number of calibration measurements are available, our machine-learning-based procedure outperforms standard calibration methods. Our machine-learning-based procedure is model independent, and is thus well suited to “black-box sensors”, which lack simple explicit fitting models. Thus, our work paves the way for Bayesian quantum sensors that can take advantage of complex nonclassical quantum states and/or adaptive protocols. These capabilities can significantly enhance the sensitivity of future devices.

Cosmological boost factor for dark matter annihilation at redshifts of z=10−100 using the power spectrum approach

We compute the cosmological boost factor at high redshifts of $z=10$-$100$ by integrating the non-linear matter power spectrum measured from high-resolution cosmological $N$-body simulations. An accurate boost factor is required to estimate the energy injection from dark matter annihilation, which may affect the cosmological re-ionization process. We combined various box-size simulations (side lengths of $1 \, {\rm kpc}$-$10 \, {\rm Mpc}$) to cover a wide range of scales, i.e. $k=1$-$10^7 \, {\rm Mpc}^{-1}$. The boost factor is consistent with the linear theory prediction at $z \gtrsim 50$ but strongly enhanced at $z \lesssim 40$ as a result of non-linear matter clustering. Although dark matter free-streaming damping was imposed at $k_{\rm fs}=10^6 \, {\rm Mpc}^{-1}$ in the initial power spectrum, the damping disappears at later times of $z\lesssim40$ as a result of the power transfer from large to small scales. Because the simulations do not explore very small-scale clustering at $k>10^7 \, {\rm Mpc}^{-1}$, our result is a lower bound on the boost factor at $z \lesssim 40$. A simple fitting function of the boost factor is also presented.

Numerical simulations of flow through a variable permeability circular cylinder

This paper investigates flow through variable permeability, two-dimensional circular cylinders using a pseudospectral numerical model. Two types of permeability (K) distributions are considered: constant with a lower permeability blockage, and constant with a higher permeability duct. Boundary conditions set by external flow with high Reynolds number lead to streamwise flow asymmetry and more short length scale variability within the cylinder when compared to conditions set by potential flow. High permeability belts are observed to guide flow around regions of lower permeability, while low permeability belts are observed to impede flow from reaching areas surrounded by the low permeability region. Inward surface flux is used to quantify changes in flow through variable permeability cylinders relative to the constant permeability cylinder. For blocking cases, the relationship between ΔK/K0 and the largest possible change in relative surface flux is nearly linear. In ducting simulations, where ΔK/K0∼1 to ∼10, this relationship is no longer linear. Simple polynomial fits are derived for both situations, allowing for the calculation of the change in permeability required to achieve a given increase or reduction in inward flux. Finally, the numerical results are contrasted with theoretical perturbation results for the case of azimuthal variations in permeability, which lead to a fundamentally different pressure distribution.

Tomographic reconstruction of a spatially-extended source from the perimeter of a restricted-access zone using a SCoTSS compton gamma imager

It is a standard procedure in many countries that response to a nuclear or radiological accident or incident would involve mobile aerial- or ground-based survey with highly sensitive gamma-ray detectors to map the distribution of radioactivity. There may however arise situations in which ground- or air-based detectors are not able to access an area to survey for radioactive materials, therefore technologies and techniques that can estimate the position and activity of radioactive materials from a distance are under development. Tomographic reconstruction methods, well-known in medical physics, permit the reconstruction of an N-dimensional map or image, from a number of N-1-dimensional cross-sectional images, or back-projections. We are investigating a tomographic reconstruction method to reconstruct the radioactivity distribution within a restricted-access zone using measurements from a Compton gamma imager placed at several locations around the perimeter of the zone. In this work an extended source of La-140 with an activity of 35 GBq was deposited within a 500 m by 500 m zone that was surveyed from the perimeter at six locations using a Silicon photomultiplier-based Compton Telescope for Safety and Security (SCoTSS) gamma imager. The reconstructed Compton images from multiple viewpoints were then projected back into the zone to reconstruct the distribution of La-140 within it. This tomographic method reconstructed high intensity along the known location of the La-140 source, suggesting that the method is able to localize the radioactive material. A simple fit to measured counts using a point-source approximation of the source distribution yielded a strength estimate of (7 ± 2) GBq at time of deposition, a reasonable result given the presence of soil and snow attenuation. Our method provides an expedient estimate of the distribution of radioactivity using tomographic techniques. It may be used to inform decisions made on the scene in urgent situations where the distribution of radioactivity must be reconstructed from a distance.

Physically-based, lumped-parameter models for the prediction of oxygen concentration during Czochralski growth of silicon crystals

Lumped-parameter models are derived from boundary layer and other physical arguments to describe oxygen concentration levels during the Czochralski (CZ) growth of silicon. These models are assessed against predictions from a detailed, high-fidelity 2D-3D numerical simulation of the entire CZ puller, whose solutions are realistic but require intense computational effort. Comparisons of predictions show that the lumped-parameter model captures the correct trends of melt oxygen levels influenced by melt height, crucible rotation, and crystal rotation. A simple fitting of coefficients provides reasonably good quantitative predictions by the lumped-parameter model, and its near-instantaneous computations make it an interesting candidate for real-time growth optimization and control. Possible model improvements and extensions are discussed.

Effect of lattice excitations on transient near-edge x-ray absorption spectroscopy

Time-dependent and constituent-specific spectral changes in soft near-edge x-ray absorption spectroscopy (XAS) of an ${[\mathrm{Fe}/\mathrm{MgO}]}_{8}$ metal/insulator heterostructure after laser excitation are analyzed at the O K-edge with picosecond time resolution. The oxygen absorption edge of the insulator features a uniform intensity decrease of the fine structure at elevated phononic temperatures, which can be quantified by a simple simulation and fitting procedure presented here. Combining XAS with ultrafast electron diffraction measurements and ab initio calculations demonstrates that the transient intensity changes in XAS can be assigned to a transient lattice temperature. Thus, the sensitivity of transient near-edge XAS to phonons is demonstrated.

BayesicFitting, a PYTHON toolbox for Bayesian fitting and evidence calculation.

BayesicFitting is a comprehensive, general-purpose toolbox for simple and standardized model fitting. Its fitting options range from simple least-squares methods, via maximum likelihood to fully Bayesian inference, working on a multitude of available models. BayesicFitting is open source and has been in development and use since the 1990s. It has been applied to a variety of science applications, chiefly in astronomy. BayesicFitting consists of a collection of PYTHON classes that can be combined to solve quite complicated inference problems. Amongst the classes are models, fitters, priors, error distributions, engines, samples, and of course NestedSampler, our general-purpose implementation of the nested sampling algorithm. Nested sampling is a novel way to perform Bayesian calculations. It can be applied to inference problems, that consist of a parameterized model to fit measured data to. NestedSampler calculates the Bayesian evidence as the numeric integral over the posterior probability of (hyper)parameters of the problem. The solution in terms of the parameters is obtained as a set of weighted samples drawn from the posterior. In this paper, we emphasize nested sampling and all classes that are directly connected to it. Additionally, we present the fitters, which fit the data by the least-squares method or the maximum likelihood method. They can also calculate the Bayesian evidence as a Gaussian approximation. We will discuss the architecture of the toolbox. Which classes are present, what is their function, how they are related and implementational details where it gets complicated. • Python toolbox for generic model fitting. • Easy usage and open source. • Ready-made combinable models. • Least squares, maximum likelihood and Nested Sampling. • Evidence calculation.

Thresholds for Particle Clumping by the Streaming Instability

Abstract The streaming instability (SI) is a mechanism to aerodynamically concentrate solids in protoplanetary disks and trigger the formation of planetesimals. The SI produces strong particle clumping if the ratio of solid to gas surface density—an effective metallicity—exceeds a critical value. This critical value depends on particle sizes and disk conditions such as radial drift-inducing pressure gradients and levels of turbulence. To quantify these thresholds, we perform a suite of vertically stratified SI simulations over a range of dust sizes and metallicities. We find a critical metallicity as low as 0.4% for the optimum particle sizes and standard radial pressure gradients (normalized value of Π = 0.05). This subsolar metallicity is lower than previous results, due to improved numerical methods and computational effort. We discover a sharp increase in the critical metallicity for small solids, when the dimensionless stopping time (Stokes number) is ≤0.01. We provide simple fits to the size-dependent SI clumping threshold, including generalizations to different disk models and levels of turbulence. We also find that linear, unstratified SI growth rates are a surprisingly poor predictor of particle clumping in nonlinear, stratified simulations, especially when the finite resolution of simulations is considered. Our results widen the parameter space for the SI to trigger planetesimal formation.

Analysis of diffuse reflectance spectroscopy by means of Bayesian inference and separation of the parameters for scattering strength and spectral dependence of the scattering.

For many applications in tissue optics, knowledge of the scattering coefficient or at least the reduced scattering coefficient is essential. In addition, its spectral dependence is also an important feature, as this provides information about scatterer sizes. While the characterization of the spectral dependence should be a simple fit, experimental results show strong fluctuations even for the same tissue type. For in-vivo measurements, this problem is even greater. Therefore, it is the aim of this study to analyze the instabilities of the scattering characterizations and find a solution for it by means of Bayesian inference. In this study, this behavior is investigated using the example of diffuse reflectance spectroscopy. It can be shown that the currently used fitting functions are unstable for the fitting as both parameters for characterizing the reduced scattering coefficient describe the spectral dependence as well as the scattering strength. However, by a simple coordinate transform, a stable mathematical description of the scattering is derived. By the fact that a posteriori probability of the reduced scattering coefficient narrows down significantly with the Bayesian inference, the new fitting function is verified.

Development of a Compartmental Pharmacokinetic Model for Molecular Radiotherapy with 131I-CLR1404.

Pharmacokinetic modeling of the radiopharmaceuticals used in molecular radiotherapy is an important step towards accurate radiation dosimetry of such therapies. In this paper, we present a pharmacokinetic model for CLR1404, a phospholipid ether analog that, labeled with 124I/131I, has emerged as a promising theranostic agent. We follow a systematic approach for the model construction based on a decoupling process applied to previously published experimental data, and using the goodness-of-fit, Sobol’s sensitivity analysis, and the Akaike Information Criterion to construct the optimal form of the model, investigate potential simplifications, and study factor prioritization. This methodology was applied to previously published experimental human time-activity curves for 9 organs. The resulting model consists of 17 compartments involved in the CLR1404 metabolism. Activity dynamics in most tissues are well described by a blood contribution plus a two-compartment system, describing fast and slow uptakes. The model can fit both clinical and pre-clinical kinetic data of 124I/131I. In addition, we have investigated how simple fits (exponential and biexponential) differ from the complete model. Such fits, despite providing a less accurate description of time-activity curves, may be a viable alternative when limited data is available in a practical case.

Computational micromechanics of porous brittle solids

Porous brittle solids evidence complex mechanical behavior, where localized failure patterns originate from mechanical processes on the microstructural level. In order to investigate the failure mechanics of porous brittle solids, we outline a general stochastic and numerical microstructure-based approach. To this end, we generate random porous microstructures by level-cutting Gaussian random fields, and conduct numerical simulations using the material point method. This allows investigating both small and large deformation characteristics of irregular porous media where a segmentation into grains and bonds is ambiguous. We demonstrate the versatility of our approach by examining elasticity and failure as a function of a wide range of porosities, from 20% to 80%. Observing that onset of failure can be well described through the second order work, we show that the stress at failure follows a power law similar to that of the elastic modulus. Moreover, we propose that the failure envelope can be approximated by a simple quadratic fitting curve, and that plastic deformation appears to be governed by an associative plastic flow rule. Finally, large deformation simulations reveal a transition in the mode of localization of the deformation, from compaction bands for highly porous samples to shear bands for denser ones.

Binding energy shifts from heavy-ion experiments in a nuclear statistical equilibrium model

Chemical constants extracted from \(^{124}\)Xe+ \(^{124}\)Sn collisions at 32 AMeV are compared to the predictions of an extended nuclear statistical equilibrium model including mean-field interactions and in-medium binding energy shifts for the light (\(Z\le 2\)) clusters. The ion species and density dependence of the in-medium modification is directly extracted from the experimental data. We show that the shift increases with the mass of the cluster and the density of the medium, and we provide a simple linear fit for future use in astrophysical simulations in the framework of the CompOSE data base. The resulting mass fractions are computed in representative thermodynamic conditions relevant for supernova and neutron star mergers. A comparison to the results of a similar analysis of the same data performed in the framework of a relativistic mean-field model shows a good agreement at low density, but significant discrepancies close to the Mott dissolution of clusters in the dense medium.

Design of a novel midstream fluid waste sample collection device for patients

Reducing the contamination rates in mid-stream urine samples is a challenge. A novel design for a mid-stream urine sample collection device, which uses a novel mechanism to separate the mid-stream flow from the fore-stream and end-stream flows based on flow speed, is presented. The device, supplied in two parts, is designed to be clipped to a toilet and used in a seated position, which improves ease of use for disabled patients. As there is no mixing of the flow types during use, it is hypothesised that contamination rates should be lower than many competing designs. The all-polymer design is easy to use by both men and women, due to the addition of a form-fitting funnel and is designed to collect samples directly into containers of the same standard currently issued, allowing easy and hygienic collection. Due to its complex shape, the main body of the device is produced by a non-conventional injection moulding process. The holder can be produced using standard injection moulding technology. The two parts are designed to be assembled together by the user by means of a simple interference fit, which removes the cost of assembly altogether.

Sediment Load Characteristics in South Korean Rivers and Assessment of Sediment Transport Formulas

In this study, using the modified Einstein procedure, data on sediment load were systematically collected and analyzed by year, period, watershed, and main stream and tributaries in South Korean rivers. It was found that the sediment load and concentration differ significantly depending on the region, even in the same area and at the same time, especially in areas with a large minimum-maximum discharge ratio. Therefore, the application of the relationship between discharge and sediment load across all time and space needs to be precise, and data have to be selected separately according to the relational application conditions. In addition, existing sediment transport formulas were compared by applying various goodness-of-fit methods that were well suited to the measured dataset. The results showed that the optimal sediment transport formula varied according to the goodness-of-fit method. Furthermore, while the Ackers and White approach was found to be generally suitable; however, the correlation figures alone were not conclusive. Therefore, a quantitative determination must be made as to how the application of simple fitness methods affects the selection of optimal sediment transport formulas.

Spectral Reconstruction and Bayesian Model Framework for Characterization of Long Period Fiber Gratings

The use of Long-Period Fiber Gratings (LPFGs) as sensors has been thoroughly researched, given the multitude of parameters these structures can monitor by themselves (such as temperature, strain, curvature) and the potential for combination with other materials that allow for monitoring of parameters such as humidity, pH and chemical concentration, at a low price and with easy fabrication processes available. This interest has increased the need for the development of interrogation systems for these sensors, particularly in the C-band spectral region. Given the cost and physical limitations (such as size and weight) of traditional solutions like Optical Spectrum Analyzers (OSA), the development of low-cost approaches for LPFG spectral analysis became an important topic that needed further development. The development of a simple curve fitting routine for LPFG spectra is reported in this article, along with a framework for automatic detection of certain physical phenomena such as corrosion and the presence of chemical species, among others.

The Associations Between Physical Fitness, Complex Vs Simple Movement, And Academic Achievement In Elementary School Students

PURPOSE: The purpose of this study was to analyze the correlation between elementary school students' body composition, physical activity, physical fitness, movement ability, and academic achievement. Movements ranged from simple actions to complex movements requiring executive functioning. METHODS: 110 fourth graders (60 boys, 50 girls) participated in this experiment. Body composition (BMI, % bodyfat), physical activity (1 week by pedometer), physical fitness (muscular strength, endurance, power, and flexibility, as well as VO2max), and complex movement abilities (Illinois Agility test, hand soda pop test, foot soda pop test) were measured. RESULTS: Regression modeling in females of body composition and fitness/activity variables was able to account for 30.5% of variation in total academic scores, but only 4.3% in males. In males, physical activity was correlated with Korean Literature (r = 0.295, p = 0.022), and the Soda Pop hand test times were correlated with math scores (r = -0.260, p = 0.045). In females, sit-and-reach scores correlated with English (r = 0.295, p = 0.038), while the soda pop hand test times correlated with math (r = -0.297, p = 0.036), science (r = -0.287, p-0.043), English (r = -0.411, p = 0.003), and total grade (r = -0.413, p = 0.003). Soda pop foot times correlated with social studies (r = -0.404, p = 0.004), science (r = -0.320, p = 0.023), English (r = -0.341, p = 0.015), and total grade (r = -0.409, p = 0.03). Illinois agility times correlated with Korean literature (r = -0.287, p = 0.043), social studies (r = -0.352, p = 0.012), science (r = -0.0373, p = 0.008), and total grade (r = -0.313, p = 0.027). CONCLUSIONS: Body composition and physical activity level did not correlate with academic achievement, and simple physical fitness showed a low correlation with academic achievement in both boys and girls. On the other hand, complex, cognitive movements such as the Illinois Agility, hand soda pop, and foot soda pop tests had consistent correlations with academic achievement in girls, but not in boys.

Improvement of Regular Exercise on Diabetes Condition of Type II Diabetes Mellitus Elderly Patients

To investigate the effects of 12-week old aerobics and brisk walking exercise on type II diabetes in elderly patients with illness condition, at the same time, after intervention 12 months later, the subsequent movement of patients in the intervention group were followed up, in order to provide a simple and effective, scientific and reasonable exercise prescription for elderly patients. Subjects were selected according to the standard, the sex ratio of 1:1 from 28 over the age of 60 elderly patients with type II diabetes (male and female 14 each), and divided them into intervention group and control group, the control group lived a normal life (without regularity, fitness behavior during the experimental period), the intervention group did aerobics plus walking exercise intervention, two groups of patients were assessed indexes of diabetes before and after the intervention. After 12- week exercise intervention, blood glucose and cholesterol of intervention group decreased significantly, the illness condition of them had improved effectively, while the control group had no obvious change before and after the experiment, after intervention 12 months later, over 85% elderly patients of the intervention group had been in good behavior habits of fitness. Aerobics and walking exercise can improve the illness condition of elderly patients with type II diabetes to a certain extent, think that the old aerobics and rope skipping is a kind of effective and simple fitness behavior, should be targeted according to the condition of patients and health promotion.

Solar cycle variation of ion escape from Mars

Using Mars Express data from 2007 until 2020 we show how ion outflow from Mars varied over more than a solar cycle, from one solar minimum to another. The data was divided into intervals with a length of one Martian year, starting from 30 April 2007 and ending 13 July 2020. The net escape rate was about 5×1024s−1 in the first covered minimum, and 2−3×1024s−1 in the most recent minimum. Ion escape peaked at 1×1025s−1 during the intervening solar maximum. The outflow is a clear function of the solar cycle, in agreement with previous studies which found a clear relationship between solar EUV flux and ion escape at Mars. The outflow during solar maximum is 2.5 to 3 times higher than in the surrounding solar minima. The average solar wind dynamic pressure over a Martian year was investigated, but does not vary much with the solar cycle. The escape rate at solar maximum is in good agreement with some recent MAVEN studies, and dominated by low energy ions at most sampled locations. A simple linear fit to the data gives a prediction of the escape rate for the much stronger solar maximum during the Phobos mission in 1989 that is consistent with observations. The fit also implies a non-linear response of ion escape for low solar EUV, with a lower initial escape response for lower solar EUV levels than those of the studied data set.

An analytical solution for McIntyre’s model of avalanche triggering probability for SPAD compact modeling and performance exploration

Single photon avalanche diodes (SPADs) are widely used to monitor fast and weak optical signals. The modeling of two main figures of merit of SPAD, namely the dark count rate (DCR) and the photon detection probability, requires one to calculate the avalanche triggering probability (ATP), usually obtained by numerically solving two transcendental equations (the McIntyre model) as post processing of technology computer-aided design simulations. This paper proposes an analytical alternative to this approach, exploiting an approximation of the impact ionization rates, in principle valid only under high field conditions, but extended to all fields by a simple fitting procedure. The proposed approximated/analytical ATP calculation can be efficient and relevant for SPAD compact modeling that is compatible with a spice-like simulator. As an illustration, a full analytical calculation for DCR based on both ATP and generation terms for a P+N abrupt diode junction is presented.

Improved gravitational radiation time-scales II: Spin–orbit contributions and environmental perturbations

ABSTRACT Peters’ formula is an analytical estimate of the time-scale of gravitational wave (GW)-induced coalescence of binary systems. It is used in countless applications, where the convenience of a simple formula outweighs the need for precision. However, many promising sources of the Laser Interferometer Space Antenna (LISA), such as supermassive black hole binaries and extreme mass-ratio inspirals (EMRIs), are expected to enter the LISA band with highly eccentric (e ≳ 0.9) and highly relativistic orbits. These are exactly the two limits in which Peters’ estimate performs the worst. In this work, we expand upon previous results and give simple analytical fits to quantify how the inspiral time-scale is affected by the relative 1.5 post-Newtonian (PN) hereditary fluxes and spin–orbit couplings. We discuss several cases that demand a more accurate GW time-scale. We show how this can have a major influence on quantities that are relevant for LISA event-rate estimates, such as the EMRI critical semimajor axis. We further discuss two types of environmental perturbations that can play a role in the inspiral phase: the gravitational interaction with a third massive body and the energy loss due to dynamical friction and torques from a surrounding gas medium ubiquitous in galactic nuclei. With the aid of PN corrections to the time-scale in vacuum, we find simple analytical expressions for the regions of phase space in which environmental perturbations are of comparable strength to the effects of any particular PN order, being able to qualitatively reproduce the results of much more sophisticated analyses.