Least-squares Fitting Technique Research Articles

Abundances of trace constituents in Jupiter's atmosphere inferred from Herschel/PACS observations

On October 31, 2009, the Photodetector Array Camera and Spectrometer (PACS) on board the Herschel Space Observatory observed far-infrared spectra of Jupiter in the wavelength range between 50 and 220$\ as part of the program "Water and Related Chemistry in the Solar System". The spectra have an effective spectral resolution between 900 and 3500, depending on the wavelength and grating order. We investigate the disk-averaged chemical composition of Jupiter's atmosphere as a function of height using these observations. We used the Planetary Spectrum Generator (PSG) and the least-squares fitting technique to infer the abundances of trace constituents. The PACS data include numerous spectral lines attributable to ammonia (NH$_3$), methane (CH$_4$), phosphine (PH$_3$), water (H$_2$O), and deuterated hydrogen (HD) in the Jovian atmosphere and probe the chemical composition from $p 275$\,mbar to $p 900$\,mbar. From the observations, we infer an ammonia abundance profile that decreases from a mole fraction of $(1.7 $ at $p 900$\,mbar to $(1.7 $ at $p 275$\,mbar, following a fractional scale height of about $0.114$. For phosphine, we find a mole fraction of $(7.2 $ at pressures higher than $(550 100)$\,mbar and a decrease of its abundance at lower pressures following a fractional scale height of $(0.09 0.02)$. Our analysis delivers a methane mole fraction of $(1.49 $. Analyzing the HD $R(0)$ line at $112.1\ yields a new measurement of Jupiter's D/H ratio, $ D/H $. Finally, the PACS data allow us to put the most stringent $3 upper limits yet on the mole fractions of hydrogen halides in the Jovian troposphere. These new upper limits are $<1.1 $ for hydrogen fluoride (HF), $<6.0 $ for hydrogen chloride (HCl), $<2.3 $ for hydrogen bromide (HBr) and $<1.2 $ for hydrogen iodide (HI) and support the proposed condensation of hydrogen halides into ammonium halide salts in the Jovian troposphere.

Fast Zernike fitting of freeform surfaces using the Gauss-Legendre quadrature

Zernike polynomial orthogonality, an established mathematical principle, is leveraged with the Gauss-Legendre quadrature rule in a rapid novel approach to fitting data over a circular domain. This approach provides significantly faster fitting speeds, in the order of thousands of times, while maintaining comparable error rates achieved with conventional least-square fitting techniques. We demonstrate the technique for fitting mid-spatial-frequencies (MSF) prevalent in small-tool-manufacturing typical of aspheric and freeform optics that are poised to soon permeate a wide range of optical technologies.

Contact wire position measurement on trolley-based system using 3D Laser-Scanning with geometric constraint method

The position of overhead contact wire (POCW) is a critical technical parameter for electrified rail transportation systems, and should be measured using contactless approaches. Although various contactless POCW measuring systems have been developed, their limited accuracy and insufficient coverage remain challenges. This paper presents a novel trolley-based system for efficient and accurate POCW measurement. This system integrates the POCW measurement system with an inertial track geometric measurement system, which can be used to measure the absolute POCW. The proposed system uses 3D laser-scanning with a line profiler (3D-LLP) and a robust POCW estimation method to overcome ambient light, speckles, and reflections. The estimation method employs a least-squares circle fitting technique utilizing prior OCW information and geometric photographic constraints. Laboratory and railway experiments demonstrate the system's robustness, accuracy, and precision. Some prospects were proposed for further improving the measurement performance.

Distortion-product otoacoustic emissions measured using synchronized swept-sines.

Swept-sines provide a tool for fast and high-resolution measurement of evoked otoacoustic emissions. During the measurement, a response to swept-sine(s) is recorded by a probe placed in the ear canal. Otoacoustic emissions can then be extracted by various techniques, e.g., Fourier analysis, the heterodyne method, and the least-square-fitting (LSF) technique. This paper employs a technique originally proposed with exponential swept-sines, which allows for direct emission extraction from the measured intermodulation impulse response. It is shown here that the technique can be used to extract distortion-product otoacoustic emissions (DPOAEs) evoked with two simultaneous swept-sines. For proper extraction of the DPOAE phase, the technique employs previously proposed adjusted formulas for exponential swept-sines generating so-called synchronized swept-sines (SSSs). Here, the SSS technique is verified using responses derived from a numerical solution of a cochlear model and responses measured in human subjects. Although computationally much less demanding, the technique yields comparable results to those obtained by the LSF technique, which has been shown in the literature to be the most noise-robust among the emission extraction methods.

Comparative Analysis of Equal and Unequal Grounding Grid Configurations by Compression Ratio and Least Square Curve Fitting Techniques

The primary aim of the power system grounding is to safeguard the person and satisfying the performance of the power system to maintain reliable operation. With equal conductor spacing grounding grid design, the distribution of the current in the grid is not uniform. Hence, unequal grid conductor span in which grid conductors are concentrated more at the periphery is safer to practice than equal spacing. This paper presents the comparative analysis of two novel techniques that create unequal spacing among the grid conductors: the least-square curve fitting technique and the compression ratio technique with equal grid configuration for both square and rectangular grids. Particle Swarm Optimization (PSO) is adopted for finding out one optimal feasible solution among many feasible solutions of equal grid configuration for both square and rectangular grids. Comparative analysis is also carried out between square and rectangular grids using the least square curve fitting technique as it results in only one unequal grid configuration. Simulation results are obtained by the MATLAB software developed. Percentage of improvement in ground potential rise, step voltage, touch voltage, and grid resistance with variation in compression ratios are plotted.

A Stochastic Mathematical Model for Understanding the COVID-19 Infection Using Real Data

Natural symmetry exists in several phenomena in physics, chemistry, and biology. Incorporating these symmetries in the differential equations used to characterize these processes is thus a valid modeling assumption. The present study investigates COVID-19 infection through the stochastic model. We consider the real infection data of COVID-19 in Saudi Arabia and present its detailed mathematical results. We first present the existence and uniqueness of the deterministic model and later study the dynamical properties of the deterministic model and determine the global asymptotic stability of the system for R0≤1. We then study the dynamic properties of the stochastic model and present its global unique solution for the model. We further study the extinction of the stochastic model. Further, we use the nonlinear least-square fitting technique to fit the data to the model for the deterministic and stochastic case and the estimated basic reproduction number is R0≈1.1367. We show that the stochastic model provides a good fitting to the real data. We use the numerical approach to solve the stochastic system by presenting the results graphically. The sensitive parameters that significantly impact the model dynamics and reduce the number of infected cases in the future are shown graphically.

NEWTON’S LAW OF COOLING WITH CAPUTO DERIVATIVE: CONSISTENT DIMENSIONALITY TO COMPARE WITH EXPERIMENTS

The dimensional homogeneity of different non-integer order versions of Newton’s law of cooling was studied. The Caputo derivative was considered for the analysis, as well as an auxiliary parameter [Formula: see text] with a time dimension. Usually, in the open literature this auxiliary parameter is arbitrarily defined by restricting its value or the range of values that it can take. In this investigation, many experimental data from different sources were collected and the most probable values of [Formula: see text] parameter and the parameter representing the non-integer order of derivatives were determined using the standard least-squares fitting technique. The results for the typical system in which an amount of hot water in a container is cooled by natural convection show that the order of the derivative takes values smaller but close to one, implying that the fractional behavior of the cooling curves deviates slightly from that perceived in the integer order solutions. It was also confirmed that the auxiliary parameter [Formula: see text] can be represented as the product between a real number n and the inverse of the cooling constant k of the classic model. Thus, the parameter [Formula: see text] can be interpreted as a kind of time constant, which can be considered reduced if [Formula: see text] and enlarged if [Formula: see text].

An extraction method for areal forward current/voltage characteristics of circular GaN p+n diodes

Anode-size-independent forward current–density/voltage (V) characteristics of circular GaN p+n diodes were extracted (in the V range of 2.7−3.0 V) from measured ones by an estimation technique of the effective anode radius, followed by the least-squares fitting technique of peripheral-to-area ratio analysis. It was found that the extracted diffusion current of diodes with a higher p-type doping level is lower than that of diodes with a lower one and that both currents have an ideality factor of unity. Since these findings agree with the diffusion theory, the demonstrated method should be useful for theoretical analysis of GaN p+n diodes.

Anterior Lens Curvature Matters in the Course of Primary Angle Closure: An Analysis Based on Ultrasound Biomicroscopic Imaging.

Purpose To evaluate the effect of anterior lens curvature in primary angle closure (PAC) and find additional anatomical features of crystalline lens that may predispose primary angle closure to the acute course. Methods 435 eyes (263 subjects) were enrolled in this study. Four groups of eyes were included based on angle configurations and clinical features: (i) acute primary angle closure (APAC, 140 eyes); (ii) chronic primary angle closure (CPAC, 116 eyes); (iii) primary angle closure suspect (PACS, 84 eyes); and (iv) normal controls (95 eyes). All patients underwent thorough ophthalmic exams including applanation tonometry, gonioscopy, low-coherence interferometry, and ultrasound biomicroscopic imaging. Based on the panoramic anterior segment images from ultrasound biomicroscopic imaging measurements, the radius of anterior lens curvature (ALR) was calculated using the least-squares curve fitting technique. ALR, in addition to axial length (AL), anterior chamber depth (ACD), and lens thickness (LT), was compared among different groups using univariate and multivariate analysis with mixed effects linear model. Results APAC, CPAC, and PACS groups all had steeper ALR, shorter AL, shallower ACD, and thicker LT than normal control group. ACD and LT further differ between APAC and CPAC or PACS eyes. Moreover, a steeper ALR was also found in the APAC group as compared to CPAC, PACS, and normal control groups. Conclusions A steeper ALR may predispose the acute attack of PAC. In addition to the relative lens position and size, lens curvature is another variable that contributes to the pathophysiological mechanisms of primary angle closure.

Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey

In the present paper, interactions between COVID-19 and diabetes are investigated using real data from Turkey. Firstly, a fractional order pandemic model is developed both to examine the spread of COVID-19 and its relationship with diabetes. In the model, diabetes with and without complications are adopted by considering their relationship with the quarantine strategy. Then, the existence and uniqueness of solution are examined by using the fixed point theory. The dynamic behaviors of the equilibria and their stability analysis are studied. What is more, with the help of least-squares curve fitting technique (LSCFT), the fitting of the parameters is implemented to predict the direction of COVID-19 by using more accurately generated parameters. By trying to minimize the mean absolute relative error between the plotted curve for the infected class solution and the actual data of COVID-19, the optimal values of the parameters used in numerical simulations are acquired successfully. In addition, the numerical solution of the mentioned model is achieved through the Adams-Bashforth-Moulton predictor-corrector method. Meanwhile, the sensitivity analysis of the parameters according to the reproduction number is given. Moreover, numerical simulations of the model are obtained and the biological interpretations explaining the effects of model parameters are performed. Finally, in order to point out the advantages of the fractional order modeling, the memory trace and hereditary traits are taken into consideration. By doing so, the effect of the different fractional order derivatives on the COVID-19 pandemic and diabetes are investigated.

Caputo SIR model for COVID-19 under optimized fractional order

Everyone is talking about coronavirus from the last couple of months due to its exponential spread throughout the globe. Lives have become paralyzed, and as many as 180 countries have been so far affected with 928,287 (14 September 2020) deaths within a couple of months. Ironically, 29,185,779 are still active cases. Having seen such a drastic situation, a relatively simple epidemiological SIR model with Caputo derivative is suggested unlike more sophisticated models being proposed nowadays in the current literature. The major aim of the present research study is to look for possibilities and extents to which the SIR model fits the real data for the cases chosen from 1 April to 15 March 2020, Pakistan. To further analyze qualitative behavior of the Caputo SIR model, uniqueness conditions under the Banach contraction principle are discussed and stability analysis with basic reproduction number is investigated using Ulam–Hyers and its generalized version. The best parameters have been obtained via the nonlinear least-squares curve fitting technique. The infectious compartment of the Caputo SIR model fits the real data better than the classical version of the SIR model (Brauer et al. in Mathematical Models in Epidemiology 2019). Average absolute relative error under the Caputo operator is about 48% smaller than the one obtained in the classical case (nu =1). Time series and 3D contour plots offer social distancing to be the most effective measure to control the epidemic.

Uncertainty quantification of the DTM2020 thermosphere model

Aims: The semi-empirical Drag Temperature Models (DTM) calculate the Earth’s upper atmosphere’s temperature, density, and composition. They were applied mainly for spacecraft orbit computation. We developed an uncertainty tool that we implemented in the DTM2020 thermosphere model. The model is assessed and compared with the recently HASDM neutral density released publicly in 2020.Methods: The total neutral density dataset covers all high-resolution CHAMP, GRACE, GOCE, and SWARM data spanning almost two solar cycles. We constructed the uncertainty model using statistical binning analysis and least-square fitting techniques, allowing the development of a global sigma error model to function the main variabilities driving the thermosphere state. The model is represented mathematically by a nonlinear manifold approximation in a 6-D space parameter.Results: The results reveal that the altitude parameter presents the most notable error range during quiet and moderate magnetic activity (Kp ≤ 5). However, the most considerable uncertainty appears during severe or extreme geomagnetic activities. The comparison with density data provided by the SET HASDM database highlights some coherent features on the mechanisms occurring in the thermosphere. Moreover, it confirms the tool’s relevance to provide a qualitative database of neutral densities uncertainties values deduced from the DTM2020 model.

Reproducibility of liver iron concentration estimates in MRI through R2* measurement determined by least-squares curve fitting.

Measuring transverse relaxation rate (R2* = 1/T2*) via MRI allows for noninvasive evaluation of multiple clinical parameters, including liver iron concentration (LIC) and fat fraction. Both fat and iron contribute to diffuse liver disease when stored in excess in the liver. This liver damage leads to fibrosis and cirrhosis with an increased risk of developing hepatocellular carcinoma. Liver iron concentration is linearly related to R2* measurements using MRI. A phantom was constructed to assess R2* quantification variability on 1.5 and 3 T MRI systems. Quantification was executed using least‐squares curve fitting techniques. The phantom was created using readily available, low‐cost materials. It contains four vials with R2* values that cover a clinically relevant range (100 to 420 Hz at 1.5 T). Iron content was achieved using ferric chloride solutions contained in glass vials, each affixed in a three‐dimensional (3D)‐printed polylactide (PLA) structure, surrounded by distilled water, all housed in a sealed acrylic cylinder. Multiple phantom stands were also 3D‐printed using PLA for precise orientation of the phantom with respect to the direction of the static magnetic field. Acquisitions at different phantom angles, across multiple MRI systems, and with different pulse sequence parameters were evaluated. The variability between any two R2* measurements, taken in the same vial under these various acquisition conditions, on a 1.5 T MRI system, was <7% for each of the four vials. For 3 T MRI systems, variability was less than 14% in all cases. Variability was <6% for both 1.5 and 3 T acquisitions when unchanged pulse sequence parameters were used. The phantom can be used to mimic a range of clinically relevant levels of R2* relaxation rates, as measured using MRI. These measurements were found to be reproducible relative to the gold‐standard method, liver biopsy, across several different image acquisition conditions.

Fractional modeling for the spread of Hookworm infection under Caputo operator

Abstract It is estimated that, about one billion people mostly from Asia, Sub-Saharan Africa and Latin America are infected with the Hookworm infection. In this paper, we developed and analyzed a model for the transmission dynamics of Hookworm infection in a human population using Caputo fractional order differential operator. Under Caputo operator, existence and uniqueness for the solutions of the new Hookworm infection model have been analyzed using fixed point theorems. Parameters of the model are estimated with the help of real statistics available for the Hookworm infection from a city of Ghana and the best fit is obtained under the nonlinear least-squares curve fitting technique. Further analysis of the proposed model shows that the disease free (infection-absent) equilibrium is locally asymptotically stable whenever a certain reproduction number R 0 1 and the endemic (infection-present) equilibrium point is globally asymptotically stable whenever R 0 1 and unstable if R 0 > 1 . Using forward normalized sensitivity index, the most sensitive parameters are identified that are essential for control of the infection and we obtained different types of simulations for the proposed Hookworm transmission system with the best fitted fractional order parameter (χ). The modelling results show that the chemotherapy treatment, awareness and improvement of personal hygiene are the best measures to be taken for control of the Hookworm infection among vulnerable community.

Parameter Estimation and Sensitivity Analysis of Malaria Model

In this work, we present a model dynamic of malaria disease transmission in Papua province, Indonesia. The parameters of the model are estimated based on the monthly cumulative data of malaria cases in the Papua province in 2018. We use the least-square fitting technique to estimate the parameters. The basic reproduction number (R 0) of the model is estimated as R 0 ≈ 1.2929. Furthermore, the sensitivity analysis is done to determine the significance of the biological parameters in the disease spread. The numerical simulation indicates that the disease can be minimized by reduces the contact between human and mosquito.

L1-NORM FITTING OF ELLIPTIC PARABOLOIDS WITH PRIOR INFORMATION FOR ENHANCED CONIFEROUS TREE LOCALIZATION IN ALS POINT CLOUDS

Abstract. Airborne laser scanning (ALS) is an established tool for deriving various tree characteristics in forests. In some applications, an accurate pointwise estimate of the tree position is required. For dense data acquired by TLS or UAV-mounted scanners, this can be achieved by locating the stem, whose center coordinates are then used for deriving the planimetric tree position. However, in case of standard ALS data this is often not an option due to the low probability of obtaining stem hits in operational scenarios of forest mapping campaigns. This paper presents an alternative, indirect approach where the tree position is approximated as the center of a quadric surface which best represents the tree crown shape. The study targets coniferous trees due to their distinct crown shape which may be approximated by an elliptic paraboloid. It is assumed that individual tree point clusters are given and the task is to find the tree center for each cluster. We first consider the general problem of fitting an elliptic paraboloid with a known axis and an L1 residual norm error criterion, which is more robust to outliers compared to least-squares fitting. We formulate this problem as a quadratically constrained quadratic program (QCQP), and show how prior knowledge on the crown shape and center position can be incorporated. Next, a computationally simpler problem is considered where the paraboloid semiaxis lengths are constrained to be equal, and a corresponding linear program is constructed. Experiments on ALS datasets of forest plots from Bavaria, Germany and Oregon, USA reveal that a reduction in median tree position error of up to 20% can be attained compared to both least-squares fitting and other baseline techniques, resulting in an absolute error of ca. 22 cm on both datasets.

EMISSIVITY MODEL OF STEEL 316L AT 800–1100 K DURING OXIDE LAYER GROWTH ON THE SPECIMEN SURFACE

This work strived to study the variation in the spectral emissivity of steel 316L specimens with wavelength at a certain temperature. The spectral emissivity and radiances from the specimen surface were measured by multispectral radiation thermometry at wavelengths of 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, and 2.1 μm and at temperatures from 800 to 1100 K in increments of 20 K. The specimen was heated and kept at a certain temperature in air for 6 h. The temperature of the specimen surface was measured by two thermocouples, which were symmetrically welded onto the front surface of the specimens near the measuring area for accurate monitoring of the surface temperature. The average of their readings was regarded as the real temperature. Variation in the spectral emissivity with wavelength was studied at different temperatures and different heating times. Effect of surface oxidation on the fitting accuracy of the emissivity models was evaluated. Two favorite models were determined. With the radiant values measured by thermometry and in combination with the least-squares fitting technique, two favorite models were examined. On the whole, both the five-parameter LLWE and LWE models could yield the best overall temperature prediction and could farthest relieve the effect of surface oxidation on the accuracy of temperature prediction over the present wavelength and temperature ranges.

Pressure broadening coefficients in the ν6 band of CH3F in collision with He

In this study we report the first measurements of the pressure line broadening coefficients of 219 lines of CH3F in the PP, RP, RQ, PQ, and RR sub-branches of the ν6 band (8.5 µm-spectral region) at room temperature using a high-resolution Fourier transform spectrometer. The collisional line widths are obtained from the non-linear least-square multi-pressure fitting techniques. A Voigt line shape model was used in the present analysis. Measured values are compared with the results of theoretical calculation based on semi-classical formalism of Robert-Bonamy (RB) model with exact isotropic trajectories. Even with a Smith-Giraud-Cooper intermolecular potential model the calculated values show a quite good agreement with experimental results.

Determination of gravity wave parameters in the airglow combining photometer and imager data

Abstract. Mesospheric airglow measurements of two or three layers were used to characterize both vertical and horizontal parameters of gravity waves. The data set was acquired coincidentally from a multi-channel filter (Multi-3) photometer and an all-sky imager located at São João do Cariri (7.4∘ S, 36.5∘ W) in the equatorial region from 2001 to 2007. Using a least-square fitting and wavelet analysis technique, the phase and amplitude of each observed wave were determined, as well as the amplitude growth. Using the dispersion relation of gravity waves, the vertical and horizontal wavelengths were estimated and compared to the horizontal wavelength obtained from the keogram analysis of the images observed by an all-sky imager. The results show that both horizontal and vertical wavelengths, obtained from the dispersion relation and keogram analysis, agree very well for the waves observed on the nights of 14 October and 18 December 2006. The determined parameters showed that the observed wave on the night of 18 December 2006 had a period of ∼43.8±2.19 min, with the horizontal wavelength of 235.66 ± 11.78 km having a downward phase propagation, whereas that of 14 October 2006 propagated with a period of ∼36.00±1.80 min with a horizontal wavelength of ∼195±9.80 km, and with an upward phase propagation. The observation of a wave taken by a photometer and an all-sky imager allowed us to conclude that the same wave could be observed by both instruments, permitting the investigation of the two-dimensional wave parameter. Keywords. Atmospheric composition and structure (airglow and aurora) – electromagnetics (wave propagation) – history of geophysics (atmospheric sciences)

Detection of Impurities in Organic Crystals by High-Accuracy Terahertz Absorption Spectroscopy

Quantitative detection of impurities in organic crystals was demonstrated by accurately measuring absorption frequencies using a continuous wave gallium phosphide terahertz spectrometer. THz spectra of l-asparagine monohydrate doped with l-aspartic acid at 0.05 to 12.5 wt % were obtained at 10 K. The three lowest frequency absorption peaks were baseline-resolved, allowing them to be examined independently. Using a least-squares curve fitting technique, impurities were detected at levels as low as 500 ppm. The sensitivity and detection limits of the technique depended strongly on the nature of both the host and the impurities. The projected limit of detection using the current system, given optimal materials, was estimated to be 51.7 ppm. In addition to quantitative assessments, impurities may also be identified by comparing frequency shifts of multiple absorptions.