Least-squares Fitting Method Research Articles

Atomic structure and Stark width calculation for Cl III ion

In this work, we calculated Stark broadening widths of several doubly-ionized chlorine Cl III spectral lines using the modified semi-empirical approach. The theoretical values of Stark widths are calculated using term energies and oscillator strengths of Cl III from the AUTOSTRUCTURE atomic structure code. We compared the calculated Stark widths with available data from the literature. If we need to interpolate these Stark widths at different temperatures for the studied transitions of Cl III, we use a simple and accurate fitting formula based on a least-squares fitting method.

Classical guitar intonation and compensation: The well-tempered guitar.

An intuitive model of classical guitar intonation is presented that includes the effects of the resonant length of the fretted string, linear mass density, tension, and bending stiffness. An expression is derived for the vibration frequencies of a stiff string using asymmetric boundary conditions at the saddle and the fret. Based on logarithmic frequency differences ("cents") that decouple these physical effects, Taylor series expansions are introduced that exhibit clearly the origins of frequency deviations of fretted notes from the corresponding 12-tone equal temperament (12-TET) values. A simple in situ technique is demonstrated for measurement of the changes in frequency of open strings arising from small adjustments in length, and a simple procedure is proposed that any interested guitarist can use to estimate the corresponding frequency shifts due to tension and bending stiffness for their own guitars and string sets. Based on these results, a least-squares fit method is employed to select values of saddle and nut setbacks that map fretted frequencies-for a particular string set and guitar-almost perfectly onto their 12-TET targets. A general approach to tempering an "off-the-shelf" guitar is shown to further reduce the tonal errors inherent in any fretted musical instrument.

Robust inverse parameter fitting of thermal properties from the laser-based Ångstrom method in the presence of measurement noise using physics-informed neural networks (PINNs)

The two-dimensional laser-based Ångstrom method measures the in-plane thermal properties for anisotropic film-like materials. It involves periodic laser heating at the center of a suspended film sample and records its transient thermal response by infrared imaging. These spatiotemporal temperature data must be analyzed to extract the unknown thermal conductivity values in the orthotropic directions, an inverse parameter fitting problem. Previous demonstration of the metrology technique used a least-squares fitting method that relies on numerical differentiation to evaluate the second-order partial derivatives in the differential equation describing transient conduction in the physical system. This fitting approach is susceptible to measurement noise, introducing high uncertainty in the extracted properties when working with noisy data. For example, when noise of a signal-to-noise ratio of 10 is added to simulated amplitude and phase data, the error in the extracted thermal conductivity can exceed 80%. In this work, we introduce a new alternative inverse parameter fitting approach using physics-informed neural networks (PINNs) to increase the robustness of the measurement technique for noisy temperature data. We demonstrate the effectiveness of this approach even for scenarios with extreme levels of noise in the data. Specifically, the PINN-approach accurately extracts the properties to within 5% of the true values even for high noise levels (a signal-to-noise ratio of 1). This offers a promising avenue for improving the robustness and accuracy of advanced thermal metrology tools that rely on inverse parameter fitting of temperature data to extract thermal properties.

An iterative weighted least-square fitting method for crustal anisotropy using receiver functions

SUMMARY The harmonic variation of the P-to-S converted phases (i.e. Pms) observed from receiver functions (RFs) includes information on crustal azimuthal anisotropy. However, this harmonic analysis is easily influenced by low-quality RF traces, and the measurements may be misleading. Here, we propose an improved method, named the iterative weighted least-square method (IWLS), to extract the splitting parameters of the crust and simultaneously retrieve the two- and four-lobed components of backazimuthal variation. The quality and weights of different RF traces are estimated properly in the IWLS method. The weight function is related to the sharpness of the Pms phase and the smearing of other signals. We conduct many synthetic tests, and the IWLS method provides stable measurements for poor backazimuthal coverage, strong noise, weak P-wave azimuthal anisotropy and multiple anisotropic layers. We apply the IWLS method to observational data from two temporary stations on the southeastern Tibetan Plateau and North China Craton, respectively. The measurements are comparable to previous results and provide insight into crustal deformation.

Detection and interpretation of the time-varying seasonal signals in China with multi-geodetic measurements

The time-varying periodic variations in Global Navigation Satellite System (GNSS) stations affect the reliable time series analysis and appropriate geophysical interpretation. In this study, we apply the singular spectrum analysis (SSA) method to characterize and interpret the periodic patterns of GNSS deformations in China using multiple geodetic datasets. These include 23-year observations from the Crustal Movement Observation Network of China (CMONOC), displacements inferred from the Gravity Recovery and Climate Experiment (GRACE), and loadings derived from Geophysical models (GM). The results reveal that all CMONOC time series exhibit seasonal signals characterized by amplitude and phase modulations, and the SSA method outperforms the traditional least squares fitting (LSF) method in extracting and interpreting the time-varying seasonal signals from the original time series. The decrease in the root mean square (RMS) correlates well with the annual cycle variance estimated by the SSA method, and the average reduction in noise amplitudes is nearly twice as much for SSA filtered results compared with those from the LSF method. With SSA analysis, the time-varying seasonal signals for all the selected stations can be identified in the reconstructed components corresponding to the first ten eigenvalues. Moreover, both RMS reduction and correlation analysis imply the advantages of GRACE solutions in explaining the GNSS periodic variations, and the geophysical effects can account for 71% of the GNSS annual amplitudes, and the average RMS reduction is 15%. The SSA method has proved to be useful for investigating the GNSS time-varying seasonal signals. It could be applicable as an auxiliary tool in the improvement of non-linear variations investigations.

Asynchronous progressive iterative approximation method for least squares fitting

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014), in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective.

Numerical analyses of M31 dark matter profiles

In this paper, we reproduce the rotation curve of the Andromeda galaxy (M31) by taking into account its bulge, disk and halo components, considering the last one to contain the major part of dark matter mass. Hence, our prescription is to split the galactic bulge into two components, namely, the inner and main bulges, respectively. Both bulges are thus modeled by exponential density profiles since we underline that the widely accepted de Vaucouleurs law fails to reproduce the whole galactic bulge rotation curve. In addition, we adopt various well-known phenomenological dark matter profiles to estimate the dark matter mass in the halo region. Moreover, we apply the least-squares fitting method to determine from the rotation curve the model free parameters, namely, the characteristic (central) density, scale radius and consequently the total mass. To do so, we perform Markov chain Monte Carlo statistical analyses based on the Metropolis algorithm, maximizing our likelihoods adopting velocity and radii data points of the rotation curves. We do not fit separately the components for bulges, disk and halo, but we perform an overall fit including all the components and employing all the data points. Thus, we critically analyze our corresponding findings and, in particular, we employ the Bayesian information criterion to assess the most accredited model to describe M31 dark matter dynamics.

Wide-band fluctuations of solar active regions probed with SHARP magnetograms

Context. The power spectra of the fluctuation noise of the solar active region (AR) areas and magnetic fluxes sequentially observed in time contain information about their geometrical features and the related fundamental physical processes. These spectra are analysed for five different ARs with various magnetic field structures. Aims. The goal of this work is to detect the characteristic properties of the Fourier and wavelet spectra evaluated for the time series of the fluctuating areas and radial magnetic fluxes of the active regions. Accordingly, this work gathers information on the properties of noise in the different cases considered. Methods. The AR area and radial magnetic flux time series were built using SHARP magnetogram datasets that cover nearly the entire time of the ARs’ transits over the solar disk. Then we applied Fourier and wavelet analyses to these time series using apodization and detrendization methods for the cross-comparison of the results. These methods allow for the detection and removal of the artefact data edge effects. Finally, we used a linear least-squares fitting method for the obtained spectra on a logarithmic scale to evaluate the power-law slopes of the fluctuation spectral power versus frequency (if any). Results. According to our results, the fluctuation spectra of the areas and radial magnetic fluxes of the considered ARs differ from each other to a certain extent, both in terms of the values of the spectral power-law exponents and their frequency bands. Conclusions. The characteristic properties of the fluctuation spectra for the compact, dispersed, and mixed-type ARs exhibit noticeable discrepancies amongst each other. It is plausible to conclude that this difference might be related to distinct physical mechanisms responsible for the vibrations of the AR areas and/or radial magnetic fluxes.

An extended Kalman-Bucy filter for state of charge estimation of 2-RC network modelled Li-ion battery

Li-ion batteries have increased dramatically in the automobile industry over the past few decades. Lithium-ion batteries are viewed more favourably due to their high energy density, specific energy, etc. Optimising the performance of lithium-ion batteries requires accurate modelling and estimation of the state of charge (SOC). This article introduces an efficient second-order resistor-capacitor (2RC) network battery model and an optimised extended Kalman-Bucy filter (EKBF) as the most effective way to model a battery and estimate its state of charge (SOC). The operating parameters of a lithium-ion battery are measured with the least-squares curve fitting method. At a temperature of 20 °C, the proposed method for estimating SOC has a mean absolute error (MAE) of 0.72 % and a root mean square error (RMSR) of 0.71 %. In addition, the operational characteristics of the proposed EKBF were validated by simulating its operation at temperatures ranging from 0 °C to 40 °C. Under all conditions, the proposed method maintained an average error rate of 0.40%. The output of the proposed topology was validated using OPAL-RT (OP5700) hardware-in-the-loop real-time simulator. Consequently, the proposed method is preferable for accurate and dependable SOC estimation in real-time battery management system (BMS) applications.

SIMPLEX: Multiple phase-cycled bSSFP quantitative magnetization transfer imaging with physic-guided simulation learning of neural network

Most quantitative magnetization transfer (qMT) imaging methods require acquiring additional quantitative maps (such as T1) for data fitting. A method based on multiple phase-cycled bSSFP was recently proposed to enable high-resolution 3D qMT imaging based on least square fitting without any extra acquisition, and thus has high potential for simplifying the qMT procedure. However, the quantification of qMT parameters with this method was suboptimal, limiting its potential for clinical application despite its simpler protocol and higher spatial resolution. To improve the fitting of qMT data obtained with multiple phase-cycled bSSFP, we propose SIMulation-based Physics-guided Learning of neural network for qMT parameters EXtraction, or SIMPLEX. In contrast to previous deep learning supervised approaches for quantitative MR that require the acquisition of input data and corresponding ground truth for training, we leveraged the MR signal model to generate training samples without expensive data curation. The network was trained exclusively with simulation data by predicting the simulation parameters. The same network was applied directly to in-vivo data without additional training. The approach was verified with both simulation and in-vivo data. SIMPLEX showed a decrease in fitting mean squared error for all simulation data compared to the existing least-square fitting method. The in-vivo experiment revealed that the network performed well with the real in vivo data unseen during training. For all experiments, we observed that SIMPLEX consistently improved the quantification quality of the qMT parameters whilst being more robust to noise compared to the prior technique. The proposed SIMPLEX will expedite the routine clinical application of qMT by providing qMT parameters (exchange rate, pool fraction) as well as T1, T2, and ΔB0 maps simultaneously with high spatial resolution, better reliability, and reduced processing time.

Wind Turbine Blade Early Fault Detection With Faulty Label Unknown and Labeling Bias

In practical industrial applications, the need for a large number of accurately labeled training samples is a significant challenge for fault detection tasks. However, labeling all training samples is expensive and prone to labeling errors, especially for early fault detection of wind turbine blades. This paper proposes a labeling bias (LB) hypothesis. Assuming the labeler only needs to label parts of normal samples that are easy to judge, we design a probability ratio least-squares importance fitting (PRL-SIF) method based on variable homogeneity. Unlike other state-of-the-art positive unlabeled (PU) learning methods, PRL-SIF does not require knowledge of the class priors to achieve training. Furthermore, to better handle the multi-dimensional time-series data of wind turbines, we provide a data preprocessing method based on functional analysis to achieve time series feature extraction and dimensionality reduction. The effectiveness and robustness of the proposed method are verified on 23 real-world wind turbine datasets. Experimental results show that the proposed method can achieve nearly 90% accuracy while only labeling 20% of normal samples.

Organic geochemical and petrographic characteristics of the Cambrian-Ordovician organic-rich marine shales in Scandinavia

This work used organic geochemistry and organic petrology to study the depositional environment, organic matter characteristics, and thermal maturity of the Cambrian-Ordovician organic-rich marine shales in the Baltic Basin. The main macerals in Cambrian samples include alginite, bituminite and solid bitumen, while zooclastic macerals become the major proportion of organic matter in the Ordovician samples. As the maturity increase, solid bitumen becomes abundant and dispersed. Semifusinite-like maceral was observed only in Furongian of well DBH15/73, which probably indicates the local intrusion of Permo-Carboniferous dolerite dikes. The programmed pyrolysis results show that immature, early mature, and over-mature samples are developed. However, the data of high-uranium Furongian samples were greatly altered by igneous intrusives from local dikes. HIo calculation model is simulated based on pyrolysis data and fitted according to the least-square fitting method. The simulated fitting HIo: 400 mg HC/g TOC (369.5 mg HC/g TOC, 430.5 mg HC/g TOC as 95% confidence bounds) is within the worldwide marine shale HIo and indicates a marine anoxia and transgressive sea with shallow water column (organofacies B).

Calibration and validation of a space electrostatic accelerometer onboard Tianzhou-1 cargo spacecraft using GNSS and attitude data

The first Chinese cargo spacecraft, named Tianzhou-1, provides a very quiet and stable environment to test space accelerometers. In this study, the high-precision electrostatic accelerometer with a resolution of 3×10−10m/s2/Hz1/2 onboard Tianzhou-1 is calibrated and validated by in-orbit measurements. In our analysis, we employ 14.5 hours of accelerometer, Global Navigations Satellite Systems (GNSS), and attitude measurements. Non-gravitational linear accelerations of the spacecraft are estimated from GNSS observables by reduced dynamic orbit determination, and centrifugal accelerations are estimated from the spacecraft attitude data and the center of mass offset. The spacecraft microgravity reaches a level of 1×10−6m/s2/Hz1/2 within the bandwidth of 1×10−4 to 1×10−2 Hz revealed from accelerometer measurements, which agrees well with our estimation of the non-gravitational and centrifugal accelerations. The results show that the centrifugal forces are the dominant signal at the frequencies above 3 cycle per revolution (cpr, 1 cpr ≈1.8×10−4 Hz) while non-gravitational forces occupy the lower frequency band. Furthermore, the bias and scale factor of the accelerometer are obtained by calibration with the estimated non-gravitational and centrifugal accelerations, using the method of least-squares linear fit. Based on the data covering the whole test period (00:33:20-15:02:43, 3 June, 2017, Beijing time), the estimated biases are −(5.31±0.04)×10−5, −(1.75±0.02)×10−6 and (2.65±0.01)×10−6m/s2 for the x, y and z axes, respectively, and the estimated scale factors are 1.001±0.008, 0.999±0.003 and 1.018±0.003 for the x, y and z axes, respectively.

Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter α. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shell have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development.

Super-Resolution Mass Spectrometry Enables Rapid, Accurate, and Highly Multiplexed Proteomics at the MS2 Level.

In tandem mass spectrometry (MS2)-based multiplexed quantitative proteomics, the complement reporter ion approaches (TMTc and TMTproC) were developed to eliminate the ratio-compression problem of conventional MS2-level approaches. Resolving all high m/z complement reporter ions (∼6.32 mDa-spaced) requires mass resolution and scan speeds above the performance levels of OrbitrapTM instruments. Therefore, complement reporter ion quantification with TMT/TMTpro reagents is currently limited to 5 out of 11 (TMT) or 9 out of 18 (TMTpro) channels (∼1 Da spaced). We first demonstrate that a FusionTM LumosTM Orbitrap can resolve 6.32 mDa-spaced complement reporter ions with standard acquisition modes extended with 3 s transients. We then implemented a super-resolution mass spectrometry approach using the least-squares fitting (LSF) method for processing Orbitrap transients to achieve shotgun proteomics-compatible scan rates. The LSF performance resolves the 6.32 mDa doublets for all TMTproC channels in the standard mass range with transients as short as ∼108 ms (Orbitrap resolution setting of 50,000 at m/z 200). However, we observe a slight decrease in measurement precision compared to 1 Da spacing with the 108 ms transients. With 256 ms transients (resolution of 120,000 at m/z 200), coefficients of variation are essentially indistinguishable from 1 Da samples. We thus demonstrate the feasibility of highly multiplexed, accurate, and precise shotgun proteomics at the MS2 level.

Dynamic Differential Image Circle Diameter Measurement Precision Assessment: Application to Burning Droplets.

Dynamic measurement precision assessment has been achieved for a differential circle measurement application. Differential circle diameter measurement, in image analysis, typically requires fitting a circle model that optimizes for image distortions, defects or occlusions. The differential task occurs when precise measurements of diameter change are required given object size variation with time. An automated system was designed to provide diameter measurements and associated measurement precision of images of a fuel droplet undergoing combustion in zero gravity for the FLEX-2 dataset. An image gradient-based, least-squares boundary point fitting method to a circle or ellipse model is used for diameter measurement. The presence of soot aggregates poses significant challenges for diameter measurements when it occludes part of the droplet boundary. The precision of the diameter measurements depends upon the image quality. Using synthetic image simulations that model the soot behavior, we developed a model based on image quality measures that assesses the measurement precision for each individual diameter measurement. Thus, diameter measurements with precision assessments were made available for follow-up scientific analysis. The algorithm's success rate for measurable runs was 98%. In cases of limited occlusion, a measurement precision of ±0.2 pixels for the FLEX-2 dataset was achieved.

Quantum mechanics-based seismic energy absorption analysis for hydrocarbon detection

SUMMARYSeismic attenuation has a considerable impact on resolution reduction and the increase in the dominant frequency period of seismic data. The absorption coefficient estimates, which measure inelastic attenuation, provide a deep understanding of the medium property changes in different geological settings. Conventional absorption coefficient estimation technologies always use time–frequency methods for seismic energy absorption analysis. However, despite continuing efforts to improve the absorption coefficient estimation, the limitation of the time–frequency methods still causes insufficient accuracy of the attenuation estimates, imposing major challenges in oil and gas hydrate exploration. In this study, a quantum mechanics-based seismic absorption coefficient estimation method was proposed for hydrocarbon detection. The seismic data were first projected on a specific basis constructed using the resolution of the Schrödinger equation. Seismic energy absorption analysis was then conducted in the potential-wave function domain. Finally, the quantum absorption coefficient estimates are given by the procedure after using a logarithmic operation and the least-squares fitting method. We examined the merits of these methods using model and field data. The gas reservoir was accurately targeted, which demonstrates that the proposed method has great potential for hydrocarbon detection.

Tree position estimation from TLS data using hough transform and robust least-squares circle fitting

Forest management and planning require information regarding the current state of the forest. Remote sensing techniques allow to obtain geospatial data, also for the forestry sector. As one of the remote-sensed technologies datasets, Terrestrial Laser Scanning data is widely used to derive detailed information about tree and forest stand parameters. This article presents the combination of circular Hough transform, denoising procedure, and robust least-square circle fitting method to extract stem positions from Terrestrial Laser Scanning data. In the proposed approach, initial tree stems position was detected with circular Hough transform. Then, obtained results were denoised to exclude most non-tree trunk points and analyze three-dimensional data from laser scanning to find exact circular tree stems with a robust least-square circle fitting method. The developed algorithm is effective in obtaining the trees’ geodetic positions from laser scanning data. The results generated in this study can be used as basics for further automatic determination of tree characteristics, such as tree species, height, or crown range. In this study, 94.8% tree stems delineation was generated with a mean accuracy of 87.2%, 1.64 cm of root mean square error for stem position, and 1.15 cm for tree radius measured at ground level. The process conducted in this research can be used to detect other circle-shaped objects, such as lamps or power towers, for which obtaining dense Terrestrial Laser Scanning data is available. The detected positions of these objects can power the geographic information systems or thematic industry systems, where it is necessary to determine the geodetic object position results from legal regulations.

Fast time optical variability in Be/X-ray binaries

Context. Classical Be stars, regardless of spectral subtype, display multi-periodic light modulations in the frequency range 0.1–12 c d−1 when observed with a high cadence and a long duration. This behavior is attributed to non-radial pulsations and/or the rotation of the Be star. A similar study on the optical counterparts to Be/X-ray binaries is yet to be carried out. Aims. The main goal of this work is to investigate the fast photometric variability of the optical counterparts to Be/X-ray binaries and compare the general patterns of such variability with the Galactic population of classical Be stars. Methods. The main core of our analysis is based on space-based observations performed by TESS. We analyzed 21 sources with TESS. High-cadence photometry with two ground-based telescopes was also performed for 15 sources. The TESS light curves were created from the full-frame images using the Lightkurve package. The ground-based light curves were obtained through differential photometry between the target and a number of non-variable stars in the same field of view. Standard Fourier analysis and least-squares fitting methods were employed in the frequency analysis. Results. All sources exhibit intra-night light variations with intensity variations of 0.01–0.06 mag in the ground-based observations and up to 5% in flux in TESS observations. This variability manifests itself as multi-periodic signals in the frequency range 0.2–12 c d−1. We find that the patterns of variability of the Be stars in Be/X-ray binaries agree with that of classical early-type Be stars in terms of the general shape of the periodograms. Based on the general shape and number of peaks in the periodograms, Be/X-ray binaries can be classified into different types. The most common case is the presence of groups of closely spaced frequencies (67%), followed by sources that exhibit isolated signals (18%). The remaining source type displays frequency spectra characterized by a mixed pattern of stochastic variability and high-frequency peaks. Conclusions. This study reveals that short-term optical photometric variability is a very common, if not ubiquitous, feature intrinsic to the Be optical companions in Be/X-ray binaries. This variability is mainly attributed to pulsations that originate in the stellar interior.

Bayesian neural networks enabled snapshot polarized light scattering spectroscopy with uncertainty quantification

Polarized light-scattering spectroscopy (PLSS) is a promising noninvasive early cancer detection technique via inversing the nuclear size of epithelial cells from collected single scattered light. Conventionally, the nuclear size is inferred by using model-driven least square fitting (LSF) methods, which is time-consuming and challenging for real-time cancer diagnosis. Although data-driven deep-learning-based techniques can be employed to speed up the inverse process, the labeled data are usually limited and the reliability quantification of prediction is lacking. Herein, by synthesizing a large quantity of single scattered spectra with physical priori, a Bayesian-deep-learning (BDL)-based PLSS framework is presented for the first time as far as we know, and is expected to provide model uncertainty of network prediction for the reliability quantification. Further, to explore the applicability of the BDL uncertain estimation, the neural networks for early cancer diagnosis are designed as a convolutional neural network (CNN) classification problem and a fully connected neural network (FNN) regression problem, respectively. They are verified using the single scattered spectra of polystyrene microsphere tissue models and human colorectal lesion tissue obtained by our proposed snapshot PLSS system. The results show that both BDL networks obtain similar nuclear sizes to the ground truth and their inferring speeds are far faster than LSF methods. Further, it is interesting to note that only the CNN classification network can provide effective uncertainty for reliability quantification. The work presents a new paradigm for the real-time diagnosis of early cancer and provides a reference for the design of snapshot PLSS endoscopy.