Least-squares Fitting Method Research Articles

Permutationally Invariant Potential Energy Surfaces.

Over the past decade, about 50 potential energy surfaces (PESs) for polyatomics with 4-11 atoms and for clusters have been calculated using the permutationally invariant polynomial method. This is a general, mainly linear least-squares method for precise mathematical fitting of tens of thousands of electronic energies for reactive and nonreactive systems. A brief tutorial of the methodology is given, including several recent improvements. Recent applications to the formic acid dimer (the current record holder in size for a reactive system), the H2-H2O complex, and four protonated water clusters [H+(H2O)n=2,3,4,6] are given. The last application also illustrates extension to large clusters using the many-body representation.

The contribution of gender differences in motor, behavioral and cognitive features to functional capacity, independence and quality of life in patients with Huntington's disease

IntroductionHuntington's disease (HD) is a neurodegenerative autosomal dominant disorder affecting patients' motor, behavioral and cognitive domains leading to total dependency for activities of daily life. This study compares whether gender differences in motor, cognitive and behavioral symptoms affect function and how functional impairment affects quality of life (QoL). MethodsWe recruited 2191 subjects from the REGISTRY data base that provides personal data, HD age of onset, visit date, CAG mutation size, UHDRS and TFC scores from at least one visit. For 1166 participants SF-36 was also available. We calculated Spearman coefficients for correlations between particular symptomatic domains and functional scales, Fisher z-transform was used to test whether differences in correlations between genders were statistically significant. Simultaneous linear regression with least-square fit method was used to determine for how much variability in functional scales the particular symptomatic domains are responsible. ANOVA was used to look for QoL differences between TFC-stage based groups. Baseline statistics showed no significant differences between genders. ResultsMotor, cognitive and behavioral domains contributed significantly to function and independence. The motor domain contributed most followed by the cognitive and to a lesser degree by the behavioral domain. Motor symptoms correlated more with functional ability and influenced function variability more in women than in men. The decline in functional abilities correlated significantly with QoL decline. ConclusionMotor symptoms have highest impact on function in HD, moreover these symptoms affect female function and independence more than males. Results indicate that symptomatic treatment targeting motor symptoms is needed to improve HD function and QoL.

Study on hydrothermal synthesis dynamics of nanoscale xonotlite fibers

The xonotlite crystals were synthesized via the hydrothermal synthesis manner from CaO and SiO2 as the raw materials with their Si/Ca molar ratio of 1.0. Hydrothermal synthesis dynamics of nanoscale xonotlite fibers was explored by masterly measuring the electrical conductivities and the calcium concentrations of product slurries synthesized at various reaction temperature in this paper. The results indicated that the calculated values of the products’ quality at various reaction temperatures were consistent with the measured values. Based on chemical reaction kinetic, using fourth-order Runge-Kutta method, spline interpolation and least-squares fitting method, the dynamic relationship of xonotlite fibers synthesized via hydrothermal synthesis process is of .

Study on hydrothermal synthesis dynamics of xonotlite spherical particles

The xonotlite crystals were synthesized via the hydrothermal synthesis manner from CaO and SiO2 as the raw materials with their Si/Ca molar ratio of 1.0. The products were characterized by XRD and SEM techniques to investigate their crystalline phases and morphologies. Hydrothermal synthesis dynamics of xonotlite spherical particles was explored by masterly measuring the pH and the calcium concentrations of product slurries synthesized at various reaction temperature in this paper. The results indicated that the calculated values of the products’ quality at various reaction temperatures were consistent with the measured values. Based on chemical reaction kinetic, using fourth-order Runge-Kutta method, spline interpolation and least-squares fitting method, the dynamic relationship of xonotlite spherical particles synthesized via hydrothermal synthesis process is of −dcA / dt = kcA0.9.

Determination of the free lunar libration modes from ephemeris DE430

The Moon’s physical librations have been extensively studied, and elaborate researches have been developed for the purpose of deriving accurate modes of free librations. Our motivation comes from the Planetary and Lunar Ephemeris DE430 by JPL/NASA, which was created in April 2013, and is reported to be the most accurate lunar ephemeris today using the data from Gravity Recovery and Interior Laboratory (GRAIL). Therefore, the residuals after fitting the model have reduced owing to improvement in the libration models, and the free librations embedded in the Euler angles have also improved. We use Fourier analysis to extract the approximate frequencies from DE430 and then a quadratic interpolation method is used to determine higher accuracy frequencies. With the frequencies, the linear least-squares fitting method is employed to fit the lunar physical librations to DE430. From this analysis we identified the three modes of free physical librations, and estimated the amplitudes as in longitude, in latitude and for the wobble, with the respective periods of 1056.16, 8806.9 and 27262.99 d. Since the free librations damp with time, they require recent excitation or a continuous stimulating mechanism in order to sustain.

GPS-based PWV for precipitation forecasting and its application to a typhoon event

The temporal variability of precipitable water vapour (PWV) derived from Global Navigation Satellite System (GNSS) observations can be used to forecast precipitation events. A number of case studies of precipitation events have been analysed in Zhejiang Province, and a forecasting method for precipitation events was proposed. The PWV time series retrieved from the Global Positioning System (GPS) observations was processed by using a least-squares fitting method, so as to obtain the line tendency of ascents and descents over PWV. The increment of PWV for a short time (two to six hours) and PWV slope for a longer time (a few hours to more than ten hours) during the PWV ascending period are considered as predictive factors with which to forecast the precipitation event. The numerical results show that about 80%–90% of precipitation events and more than 90% of heavy rain events can be forecasted two to six hours in advance of the precipitation event based on the proposed method. 5-minute PWV data derived from GPS observations based on real-time precise point positioning (RT-PPP) were used for the typhoon event that passed over Zhejiang Province between 10 and 12 July, 2015. A good result was acquired using the proposed method and about 74% of precipitation events were predicted at some ten to thirty minutes earlier than their onset with a false alarm rate of 18%. This study shows that the GPS-based PWV was promising for short-term and now-casting precipitation forecasting.

Solar differential rotation in the period 1964–2016 determined by the Kanzelhöhe data set

The main aim of this work is to determine the solar differential rotation by tracing sunspot groups during the period 1964-2016, using the Kanzelh\"ohe Observatory for Solar and Environmental Research (KSO) sunspot drawings and white light images. Two procedures for the determination of the heliographic positions were applied: an interactive procedure on the KSO sunspot drawings (1964 - 2008, solar cycles nos. 20 - 23) and an automatic procedure on the KSO white light images (2009 - 2016, solar cycle no. 24). For the determination of the synodic angular rotation velocities two different methods have been used: a daily shift (DS) method and a robust linear least-squares fit (rLSQ) method. Afterwards, the rotation velocities had to be converted from synodic to sidereal, which were then used in the least-squares fitting for the solar differential rotation law. For the test data from 2014, we found the rLSQ method for calculating rotational velocities to be more reliable than the DS method. The best fit solar differential rotation profile for the whole time period is $\omega(b)$ = (14.47 $\pm$ 0.01) - (2.66 $\pm$ 0.10) $\sin^2b$ (deg/day) for the DS method and $\omega(b)$ = (14.50 $\pm$ 0.01) - (2.87 $\pm$ 0.12) $\sin^2b$ (deg/day) for the rLSQ method. A barely noticeable north - south asymmetry is observed for the whole time period 1964 - 2016 in the present paper. Rotation profiles, using different data sets (e.g. Debrecen Photoheliographic Data, Greenwich Photoheliographic Results), presented by other authors for the same time periods and the same tracer types, are in good agreement with our results. Therefore, the KSO data set is suitable for the investigation of the long-term variabilities in the solar rotation profile.

Real-time reference-based dynamic phase retrieval algorithm for optical measurement.

To study dynamic behaviors of a phenomenon, measuring the evolving field of a specimen/material/structure is required. Optical interferometry, as a full-field, non-contact, and highly sensitive optical measurement technique, has been applied, where the evolving field is represented as dynamic phase distribution. A dynamic phase retrieval algorithm, called least-squares with 3 unknowns (LS3U), which estimates the phase change between each two consecutive patterns by a least-squares fitting method and denoises the phase change by a windowed Fourier filtering (WFF) algorithm, has been shown to be a simple yet effective algorithm. However, LS3U is computationally expensive, restricting its potential application in real-time dynamic phase retrieval systems. In this paper, a real-time LS3U algorithm powered by GPU parallel computing is proposed, with which frame rates of up to 64.5 frames per second (fps) and 131.8fps are achieved on NVIDIA's GTX 680 and GTX 1080 graphics cards, respectively.

Three dimensional least-squares fitting of ellipsoids and hyperboloids

Spatial continuity can be described as a variogram model that has an ellipsoid anisotropy. In previous research, two-dimensional least-square ellipse fitting method by Fitzgibbon, Pilu and Fisher has been applied to the analysis of spatial continuity for coal deposits. However, it is not easy to generalize their method to three-dimensional least-square ellipsoid fitting. In this research, we obtain a three-dimensional least-square fitting for ellipsoids and hyperboloids by generalizing two-dimensional least-square ellipse fitting method introduced by Gander, Golub and Strebel.

A New Algorithm for Surface Currents Inversion With High-Frequency Over-the-Horizon Radar

The conventional method of extracting ocean surface currents by high-frequency over-the-horizon radar is based on the fixed first-order Bragg frequency formula and ignores the effects caused by the environment, especially in near-shore areas. In this letter, a current inversion model based on 2-D Fourier series expansion was developed. The first-order Bragg frequency and the Doppler offset induced by radial current are dealt with the bivariate functions of group distance and azimuth angle in the proposed method. By solving an overdetermined matrix equation with the least-square fitting method, the current at each detection grid can be estimated. As the Bragg frequency obtained by this new algorithm is adaptive to the environment, the accuracy of current measurement will be improved. The feasibility and effectiveness of the new method are verified with simulations and experimental results. The currents estimated by the traditional method and the new algorithm are compared with two in situ buoys. Results indicate that the new algorithm possesses comparable accuracy for far-shore areas and better accuracy for near-shore areas when compared with the conventional method.

Daily solar irradiance in Daily useful energy of Domestic solar water heater

Daily useful energy was studied on different daily solar irradiance, which is important to the coefficient of thermal performance(CTP) of domestic solar water heater. The numerical simulation of 8 hours thermal performance experimental was carried out by Fluent, and the thermal energy equation was obtained by least-squares fitting method, the influence between daily solar irradiance and daily useful energy was analyzed quantitatively, The feasibility of the equation was verified. The accuracy of the test results could be increased by the application of the equation.

Transport properties of dilute ammonia-noble gas mixtures from new intermolecular potential energy functions

Abstract Previously we have determined the dilute mixture transport properties of slightly polar fluorocarbons using the inverted intermolecular potential energies ( Ind. Eng. Chem. Res. 45 (2006) 9211–9223). In the present paper, the corresponding states correlations for reduced viscosity collision integrals were employed to obtain effective unlike interaction potential models for dilute binary mixtures of highly polar molecule ammonia with noble gases. The inverted potentials were fitted to the Morse–Spline–van der Waals (MSV), model potential. The method of least-squares fitting was then applied to identify best consistence force parameters for each ammonia-noble gas mixture, taking advantage of experimental viscosities, diffusion coefficients and thermal conductivities. The proposed potential models were compared with those obtained from other sources, in order to assess the extent of their validity. The potentials were later employed to calculate transport properties of the studied mixtures. Then, results were compared with those reported in the literature, which led to the acceptable agreement.

A sphere fitting approach to determine the hip joint centre of the horse

Accurate identification of the hip joint centre (HJC) is crucial for the correct estimation of knee and hip joint loads and kinematics, which is particularly relevant in orthopaedic surgery and musculoskeletal modelling. Several methods have been described for calculation of the HJC in humans, however, no studies have used these methods in the horse despite a similar need for improved evaluation of hip joint biomechanics in rehabilitation and musculoskeletal modelling. This preliminary study uses the commonly used functional method (least-squares sphere fit) to determine the HJC in three equid cadavers. Bone pins with reflective markers attached were drilled into the tuber coxae (TC), tuber ischium (TI), tuber sacrale (TS), greater trochanter (GT), third trochanter (TT) and lateral femoral condyle (FC) of the uppermost limb of the cadavers positioned in lateral recumbency. Three repetitions of passive movements consisting of pro-and retraction, ab- and adduction and circumduction were performed. The HJC was calculated using a least-squares sphere fitting method and presented as a distance from the TC based on a percentage of the TC to TI vector magnitude. Mean (± standard deviation) of the HJC is located 52.4% (± 3.9) caudally, 0.2% (± 6.5) dorsally, and 19.8% (± 4.2) medially from the TC. This study is the first to quantify the HJC in horses ex vivo using a functional method. Further work (in vivo and imaging) is required to validate the findings of the present study.

Application of NSGA-Ⅱ Algorithm in the Design of Car Body Lateral Crashworthiness

Car side impact is a highly non-linear process, involving numerous parts, in the optimization of crashworthiness, it is necessary to take into account the amount of invasion and intrusion speed, but also the impact on lightweight design. Thicknesses of main components of car side are set as design variables, invasion of front and back door and intrusion speed of B-pillar are set as design targets. By use of orthogonal experimental design and method of least squares fitting, a high precision response surface model is built. A platform is built using ISIGHT software for the multi-objective optimization based on NSGA-i algorithm. The result shows that, compared to the original model, the optimized amount of invasion and intrusion speed can both achieve over 10% improvement and total mass reduces by nearly 2kg. So it proves that the optimization is effective and feasible.

Unsteady Aerodynamic Force Sensing from Strain Data

A simple approach for computing unsteady aerodynamic forces from simulated measured strain data is proposed in this study. First, the deflection and slope of the structure are computed from the unsteady strain. Velocities and accelerations of the structure are computed using the autoregressive moving average model, online parameter estimator, low-pass filter, and a least-squares curve fitting method, together with analytical derivatives with respect to time. Finally, aerodynamic forces over the wing are computed using modal aerodynamic influence coefficient matrices, a rational function approximation, and a time-marching algorithm. A cantilevered rectangular wing is used to validate the simple approach. Unsteady aerodynamic forces as well as wing deflections, velocities, accelerations, and strains are computed using the CFL3D computational fluid dynamics code and the MSC/NASTRAN finite element analysis code; and these CFL3D/NASTRAN-based results are assumed as measured quantities. Computed deflections, velocities, accelerations, and unsteady aerodynamic forces are compared with the CFL3D/NASTRAN-based results. Computed aerodynamic forces based on lifting-surface theory at subsonic speeds are in good agreement with the target aerodynamic forces generated using CFL3D code with the Euler equation. This research demonstrates the feasibility of obtaining induced drag and lift forces through the use of distributed sensor technology with measured strain data.

Spins of superdeformed rotational bands in Tl isotopes

The two-parameter model defined for even-even nuclei viz. soft-rotor formula is used to assign the band-head spin of the 17 rotational bands in Tl isotopes. The least-squares fitting method is employed to obtain the spins of these bands in the \(A\sim 190\) mass region. The calculated transition energies are found to depend sensitively on the proposed spin. Whenever a correct spin assignment is made, the calculated and experimental transition energies coincide very well. The dynamic moment of inertia is also calculated and its variation with rotational frequency is explored.

Improved FRFT-based method for estimating the physical parameters from Newton’s rings

Newton’s rings are often encountered in interferometry, and in analyzing them, we can estimate the physical parameters, such as curvature radius and the rings’ center. The fractional Fourier transform (FRFT) is capable of estimating these physical parameters from the rings despite noise and obstacles, but there is still a small deviation between the estimated coordinates of the rings' center and the actual values. The least-squares fitting method is popularly used for its accuracy but it is easily affected by the initial values. Nevertheless, with the estimated results from the FRFT, it is easy to meet the requirements of initial values. In this paper, the proposed method combines the advantages of the fractional Fourier transform (FRFT) with the least-squares fitting method in analyzing Newton’s rings fringe patterns. Its performance is assessed by analyzing simulated and actual Newton’s rings images. The experimental results show that the proposed method is capable of estimating the parameters in the presence of noise and obstacles. Under the same conditions, the estimation results are better than those obtained with the original FRFT-based method, especially for the rings’ center. Some applications are shown to illustrate that the improved FRFT-based method is an important technique for interferometric measurements.

Microwave spectrum, structure and dipole moment of 4-fluorophenylacetylene (4FPA)

Using a chirped-pulse Fourier-transform microwave (CP-FTMW) spectrometer, a 6–18 GHz spectrum of 4-fluorophenylacetylene (4FPA) was measured and only a-type R-branch transitions were observed up to J = 9. Rotational constants and quartic centrifugal distortion constants for the normal isotopic species were determined based on Watson-S reduction: A = 5652.812(22) MHz, B = 966.92885(11) MHz, C = 825.67680(11) MHz, DJ = 0.01377(60) kHz, and DJK = 0.2468(61) kHz, with other three distortion constants fixed as DK = 0.6629 kHz, d1 = 2.386 Hz, and d2 = 0.989 Hz from ab initio results. For six kinds of carbon-13 isotopic species, 10–15 transitions were detected by a resonant cavity FTMW spectrometer in natural abundance, and rotational constants of each species were also determined by fitting transition frequencies. Gaseous molecular structures of 4FPA were derived via the least-squares fitting (r0) and substitution (rs) methods, and ab initio optimization (re). They were compared to the structures of benzene derivatives having fluorine and the acetylenic group as substituents. In addition, dipole moment component of 4FPA was also determined to be μa = μtotal = 0.8935(9) D from Stark effect measurements.

Band head spin assignment of superdeformed bands in 86Zr

Two parameter expressions for rotational spectra viz. variable moment of inertia (VMI), ab formula and three parameter Harris ω2 expansion are used to assign the band head spins (I0) of four rotational superdeformed bands in 86Zr. The least-squares fitting method is employed to obtain the band head spins of these four bands in the A ∼ 80 mass region. Model parameters are extracted by fitting of intraband γ-ray energies, so as to obtain a minimum root-mean-square (rms) deviation between the calculated and the observed transition energies. The calculated transition energies are found to depend sensitively on the assigned spins. Whenever an accurate band head spin is assigned, the calculated transition energies are in agreement with the experimental transition energies. The dynamic moment of inertia is also extracted and its variation with rotational frequency is investigated. Since a better agreement of band head spin with experimental results is found using the VMI model, it is a more powerful tool than the ab formula and Harris ω2 expansion.

Theoretical simulation of 87Rb absorption spectrum in a thermal cell**Project supported by the National Basic Research Program of China (Grant No. 2013CB922003) and the National Natural Science Foundation of China (Grant Nos. 91421305, 91121005, and 11174329).

In this paper, we present a theoretical simulation of 87Rb absorption spectrum in a thermal cm-cell which is adaptive to the experimental observation. In experiment, the coupling and probe beams are configured to copropagate but perpendicular polarized, making up to five velocity selective optical pumping (VSOP) absorption dips able to be identified. A Λ-type electromagnetically induced transparency (EIT) is also observed for each group of velocity-selected atoms. The spectrum by only sweeping the probe beam can be decomposed into a combination of Doppler-broadened background and three VSOP dips for each group of velocity-selected atoms, accompanied by an EIT peak. This proposed theoretical model can be used to simulate the spectrum adaptive to the experimental observation by the non-linear least-square fit method. The fit for the high quality of experimental observation can determine valuable transition parameters such as decaying rates and coupling beam power accurately.