Linear Least-squares Research Articles

Adjustment of JEFF-3.3 data for U-235 and Pu-239 in the fast, non-resonant energy range

A stochastic technique based upon the Serpent code is used along with the JEFF-3.3 nuclear data library consistently including their covariance, to adjust fast, non-resonant ENDF-6 format data for U-235 and Pu-239, and this in conjunction with the Asymptotic Generalized Linear Least-Squares (AGLLS) assimilation methodology. The current investigations are focused on simple, well documented critical systems such as Godiva and Jezebel exhibiting much stronger sensitivities of the assimilated parameters to this nuclear data. Correspondingly, the assimilation database consists of effective multiplication factors along with ratios of fission reaction rates per atom for Pu-239 and U-235 which are spectral indices measured at core center.By not identifying the fission cross-section of U-235 with a standard, we find that this data set should be decreased between 2 keV and 2.2 MeV; in relative terms by maximum 5.9 % occurring in the energy range between 41 keV and 67 keV which is less than two standard deviations (σ) of 6.6 %, obtained from the covariance files of the JEFF-3.3 library. Whereas ν¯ needs to be increased by up to ∼ 0.3 %, corresponding to half a σ. Also to reduce is the elastic scattering P1 cross-section between 111 keV and 10 MeV with a stronger decrease amounting to 12.7 % in the range 2.2 MeV-3.7 MeV.For the capture cross-section, we find a decrease of maximum 3 % between 111 keV and 3.7 MeV, in which case the one-σ limit is about 10 %.For Pu-239, the results of the adjustment study also indicate a reduction of the elastic scattering P1 cross-section in the same energy range as for U-235, however by only 5.7 %. The fission and inelastic scattering cross-section changes are much smaller than 1 %; as opposed to U-235, ν¯ is to decrease, and this by less than 0.5 % corresponding to one σ. The additionally proposed fission spectrum change not exceeding the one-σ limit somewhat shifts the median emission energy to a lower value. Differently from U-235, the adjustment of the capture cross-section is an increase of maximum 1.8 % in the range 2.2 MeV-3.7 MeV, which is well below the one-σ limit of the JEFF-3.3 library.

Angular Velocity Estimation and Error-Covariance Analysis from Accelerometers Using Least Squares

In this work angular velocity solutions are presented from accelerometers. Two main formulations are presented. The first is based on using multiple vector sets of accelerometers, and the second is based on using multiple single-axis accelerometers. Linear least-squares approaches are derived for both, along with error-covariance expressions for the angular velocity estimates. Nonlinear least-squares approaches are also derived for both, which also provide the Cramér–Rao lower bound as a byproduct. All approaches have a sign ambiguity in the angular velocity solutions, which is resolved using a simple linear Kalman filter. Simulation results are shown to compare the formulations, along with the linear and nonlinear least-squares solutions.

Correlation dispersion as a measure to better estimate uncertainty in remotely sensed glacier displacements

Abstract. In recent years a vast amount of glacier surface velocity data from satellite imagery has emerged based on correlation between repeat images. Thereby, much emphasis has been put on the fast processing of large data volumes and products with complete spatial coverage. The metadata of such measurements are often highly simplified when the measurement precision is lumped into a single number for the whole dataset, although the error budget of image matching is in reality neither isotropic nor constant over the whole velocity field. The spread of the correlation peak of individual image offset measurements is dependent on the image structure and the non-uniform flow of the ice and is used here to extract a proxy for measurement uncertainty. A quantification of estimation error or dispersion for each individual velocity measurement can be important for the inversion of, for instance, rheology, ice thickness and/or bedrock friction. Errors in the velocity data can propagate into derived results in a complex and exaggerating way, making the outcomes very sensitive to velocity noise and outliers. Here, we present a computationally fast method to estimate the matching precision of individual displacement measurements from repeat imaging data, focusing on satellite data. The approach is based upon Gaussian fitting directly on the correlation peak and is formulated as a linear least-squares estimation, making its implementation into current pipelines straightforward. The methodology is demonstrated for Sermeq Kujalleq (Jakobshavn Isbræ), Greenland, a glacier with regions of strong shear flow and with clearly oriented crevasses, and Malaspina Glacier, Alaska. Directionality within an image seems to be the dominant factor influencing the correlation dispersion. In our cases these are crevasses and moraine bands, while a relation to differential flow, such as shear, is less pronounced on the correlation spread.

Spatially Variant Ultrasound Attenuation Mapping Using a Regularized Linear Least-Squares Approach

Quantitative ultrasound methods aim to estimate the acoustic properties of the underlying medium, such as the attenuation and backscatter coefficients, and have applications in various areas including tissue characterization. In practice, tissue heterogeneity makes the coefficient estimation challenging. In this work, we propose a computationally efficient algorithm to map spatial variations of the attenuation coefficient. Our proposed approach adopts a fast, linear least-squares strategy to fit the signal model to data from pulse-echo measurements. As opposed to existing approaches, we directly estimate the attenuation map, that is, the local attenuation coefficient at each axial location by solving a joint estimation problem. In particular, we impose a physical model that couples all these local estimates and combine it with a smooth regularization to obtain a smooth map. Compared to the conventional spectral log difference method and the more recent ALGEBRA approach, we demonstrate that the attenuation estimates obtained by our method are more accurate and better correlate with the ground-truth attenuation profiles over a wide range of spatial and contrast resolutions.

Theoretical and experimental investigation of a lateral broad-crested contraction as a flow measurement device

The article looked at the possibility of making use of a lateral contraction as a flow measurement device in a rectangular open channel. The contraction under consideration consisted of two removable prismatic elements placed on either side of the walls of the channel. This configuration resulted in the creation of a throat of length L and opening width b less than the channel width B so that the contraction rate has been defined as β = b/B. This ratio actually reflects the degree of the lateral contraction while there is no vertical contraction because the device is devoid of crest height. Its longitudinal axis thus merges with that of the channel, which gives the device a self-cleaning property. Due to the changeover of the subcritical flow upstream of the device to a supercritical flow inside the throat, firmly confirmed by laboratory observations, a control section has been created in a given section at the entrance of the throat. This is the prerequisite for the correct functioning of a flow measuring device in open channels.The essential steps undertaken during the study were firstly the research for the theoretical relationship governing the flow rate Q for forthwith deriving the resulting discharge coefficient Cd equation. This aim has been successfully achieved through the simultaneous application of the momentum and the energy equations taking care to include the effect of the approach flow velocity. The obtained theoretical flow rate QTh was identified in the form of explicit functions depending both on the geometric characteristics of the device and the upstream depthh1 which is fully consistent with semi-modular devices. Likewise, the resulting discharge coefficient Cd,Th was formally identified as being exclusively dependent on the contraction rate β and this finding was predicted by the dimensional analysis.The statistical analysis of the 157 collected experimental values allowed observing an excellent agreement between the theoretical and experimental discharge coefficients. The maximum deviation calculated over the tested range of β was only 1.22%. The use of linear least-squares fitting method involving experimental and theoretical data gave the following trend line relationship Cd,Exp=0.9911Cd,Th obtained with a coefficient of determination R2=0.9906. Thereby, using the previous equation, the accuracy on the discharge coefficient Cd,Th computation has been highly improved since the maximum deviation reduced to only 0.66%.

Comparison on BER Performance Using Linear Filtering Least Square Method and Least Square Method for MIMO-OFDM Based Communication Systems

In Multiple Input Multiple Output Orthogonal Frequency Division Multiplexing (MIMO-OFDM) systems, the Bit Error Rate is poor by using Linear Filtering Least Square method (LFLS) and Least Square (LS) method for massive MIMO-OFDM systems to extend the strength of the signal and to scale back the noise. Therefore the objective of the project is to maintain better Bit Error Rate performance. Materials and methods: Using Linear Filtering Least Square method, different values are used to enhance the Bit Error Rate (BER). The proposed method is compared with the least square method, the results are obtained by using Matlab simulation software. Results: The performance of Bit Error Rate is calculated by using the Linear Filtering Least Square method and enhanced Least Square method parameters. At 20 dB SNR, the linear filtering least-square provides the BER value is 10-3, whereas the Novel enhanced least-square scheme provides the BER value around 10-5. Conclusion: Based on experimental results and statistical analysis using the independent sample T-test, the Least Square method significantly performs better than the Linear Filtering Least Square method.

Dielectric Response Model for Transformer Insulation Using Frequency Domain Spectroscopy and Vector Fitting

This paper proposes a rational approximation-based approach to find positive real parameters for the extended Debye model (EDM), aimed at condition assessment of insulation systems of power transformers. The EDM can model the slow and fast polarization phenomenon, including relaxation mechanisms with different relaxation times within a composite dielectric material. In the proposed approach, the complex permittivity of the transformer’s composite insulation is approximated via rational functions, as given by the vector fitting (VF) software tool, and the EDM parameters are identified from the obtained poles/residues. To guarantee positive real parameters, i.e., a physically realizable circuit, VF is internally modified to calculate the final residues of the rational approximation via a constrained linear least-squares problem without resorting to further post-processing algorithms, as in existing methods, hence without affecting fitting accuracy. The effectiveness of the parametrized EDM is demonstrated in two ways: (a) by reconstructing frequency domain spectroscopy (FDS) curves provided via measurements in new oil-immersed power transformers and (b) by the comparison of the calculated polarization current given by EDM versus real measurements in time domain. The achieved fitting accuracy in most of the cases is above 99 percent for the reconstructed FDS curves, while the polarization current waveform is reproduced with good agreement.

Augmented Nonlinear Least Squares Estimation With Applications to Localization

Least squares (LS) estimation is simple yet effective for parameter estimation. Most real-world problems are nonlinear. In practice, nonlinear LS problems are linearized first and then solved by a linear LS estimator. However, since these methods solve a nonlinear LS problem by a linear way, there is room for improvement only if estimators being nonlinear in the original measurement. In this article, an LS approach is proposed for nonlinear LS problems, along with theoretical analysis and justification. First, for a general nonlinear measurement model with any form of noise, a unified linearization (UL) approach is proposed under certain assumptions and a linearized least squares (LLS) estimator is defined based on the UL model. Second, an augmented nonlinear least squares (ANLS) estimator is proposed based on LLS estimation. In the ANLS estimator, the original measurement is augmented by its nonlinear conversion intended to further utilize the nonlinear measurement. Moreover, we prove that the ANLS estimator outperforms LLS estimation-based estimators. Finally, the UL and the ANLS estimator are applied to source localization with time difference of arrival and/or frequency difference of arrival measurement, and a two-step ANLS localization algorithm is developed. The performance of the proposed algorithm is evaluated via simulation of some typical localization scenarios.

The Kinematic Calibration of an Industrial Robot With an Improved Beetle Swarm Optimization Algorithm

The article proposes a kinematic calibration method based on an improved beetle swarm optimization algorithm for an industrial robot to finish drilling and riveting. Specifically, a preference random substitution method to improve the update strategy was presented, which comprises a new boundary-processing strategy that avoids stagnation states and local optimization traps by adopting a dynamic parameter adjustment strategy to increase the speed and stability of searches. The effectiveness of the proposed method was verified by simulations and calibration experiments involving a robot drilling and riveting system with an industrial robot KUKA KR500L340-2. In addition, comparisons were conducted with linear least-squares, particle swarm optimization, and beetle swarm optimization algorithms. According to the results of calibration simulations and experiments, the fitness function value of the proposed algorithm can decrease rapidly in the first 20 iterations, while the mean value of the end-effector position error is reduced from 2.95mm to 0.20mm by using the proposed algorithm. Compared with other three algorithms, the positioning accuracy is improved by more than 60%.

Quantitative evaluation of SPH in TIG spot welding

While the application of the Smoothed Particle Hydrodynamics (SPH) method for the modeling of welding processes has become increasingly popular in recent years, little is yet known about the quantitative predictive capability of this method. We propose a novel SPH model for the simulation of the tungsten inert gas (TIG) spot welding process and conduct a thorough comparison between our SPH implementation and two finite element method (FEM)-based models. In order to be able to quantitatively compare the results of our SPH simulation method with grid-based methods, we additionally propose an improved particle to grid interpolation method based on linear least-squares with an optional hole-filling pass which accounts for missing particles. We show that SPH is able to yield excellent results, especially given the observed deviations between the investigated FEM methods and as such, we validate the accuracy of the method for an industrially relevant engineering application.

Precision dosimetry in yttrium-90 radioembolization through CT imaging of radiopaque microspheres in a rabbit liver model

PurposeTo perform precision dosimetry in yttrium-90 radioembolization through CT imaging of radiopaque microspheres in a rabbit liver model and to compare extracted dose metrics to those produced from conventional PET-based dosimetry.Materials and methodsA CT calibration phantom was designed containing posts with nominal microsphere concentrations of 0.5 mg/mL, 5.0 mg/mL, and 25.0 mg/mL. The mean Hounsfield unit was extracted from the post volumes to generate a calibration curve to relate Hounsfield units to microsphere concentration. A nominal bolus of 40 mg of microspheres was administered to the livers of eight rabbits, followed by PET/CT imaging. A CT-based activity distribution was calculated through the application of the calibration curve to the CT liver volume. Post-treatment dosimetry was performed through the convolution of yttrium-90 dose-voxel kernels and the PET- and CT-based cumulated activity distributions. The mean dose to the liver in PET- and CT-based dose distributions was compared through linear regression, ANOVA, and Bland–Altman analysis.ResultsA linear least-squares fit to the average Hounsfield unit and microsphere concentration data from the calibration phantom confirmed a strong correlation (r2 > 0.999) with a slope of 14.13 HU/mg/mL. A poor correlation was found between the mean dose derived from CT and PET (r2 = 0.374), while the ANOVA analysis revealed statistically significant differences (p < 10−12) between the MIRD-derived mean dose and the PET- and CT-derived mean dose. Bland–Altman analysis predicted an offset of 15.0 Gy between the mean dose in CT and PET. The dose within the liver was shown to be more heterogeneous in CT than in PET with an average coefficient of variation equal to 1.99 and 1.02, respectively.ConclusionThe benefits of a CT-based approach to post-treatment dosimetry in yttrium-90 radioembolization include improved visualization of the dose distribution, reduced partial volume effects, a better representation of dose heterogeneity, and the mitigation of respiratory motion effects. Post-treatment CT imaging of radiopaque microspheres in yttrium-90 radioembolization provides the means to perform precision dosimetry and extract accurate dose metrics used to refine the understanding of the dose–response relationship, which could ultimately improve future patient outcomes.

RECYCLING OF WET-STRENGTH TISSUE PAPER. PART 1. KINETICS OF PAPER DISINTEGRATION AT REPULP-ING PROCESS

The present investigation focuses on disintegration kinetics of wet-strength tissue paper in various termo-alkaline and chemical conditions of batch low-consistency repulping process. As chlorine-free repulping aids, we applied sodium persulfate in its original state and in combination with peroxy activator (chemically activated form of sodium persulfate). Based on a first-order kinetic model, we estimated the influence of repulping conditions on the intensity of wet-strength paper defiberization process by the disintegration rate constant kd. The values of kd were found by linear least-squares curve fitting of the experimental data in the t, ln (F'/100) coordinate system, where F' is non-disintegrated or non-defibrated portion of paper, t – repulping time. The results have shown that the termo-alkaline repulping of wet-strength tissue paper (T=60 °C, pH=10.5) compared with mild conditions of repulping (T=40 °C, pH=8.5) leads to an increase of the rate constant of disintegration kd from 5.6·10-3 min-1 to 9.7·10-3 min-1. Addition of 2% persulfate at termo-alkaline repulping increases the rate constant of disintegration kd up to 14,0·10-3 min-1. Application of chemically activated persulfate under the same conditions allows to further increase the kd up to 23.4·10-3 min-1. Raising the temperature and pH of the pulp during persulfate repulping increases the disintegration rate constant kd nonlinearly. Finally, addition of persulfate along with thermo-alkaline pulp treatment lead to decline repulping time and energy by 31% and 29%. It should be noted, that more significant savings of time and energy (57% and 59% respectively) we found at the application of persulfate in a chemically activated form.

Creating effective visuals for destination marketing videos: Scenery vs people

By drawing from imagery and consumer choice theories, we examine how visuals of faces and social groups can foster behavioural intention and emotional effects in the case of nature tourism video marketing. We implemented an experimental study by creating three different videos with altering levels of human presence. Data were analysed using linear regression and partial least-squares path analysis methods. Unlike prior research had suggested, video content with fewer direct visuals of human beings was found to be more entertaining. Thus, scenery performed better than people as visuals in our destination marketing case. The visuals of faces and social groups had no effect on consumers’ behavioural intention or emotions, even if they did arouse feelings of social presence. Behavioural intention was mainly associated with entertainment value, while social presence had only a minor effect. Gender, nationality and outdoor activity also affected the media effects of videos, showing them to be complex and context-dependent phenomena.

Summer Nighttime Anomalies of Ionospheric Electron Content at Midlatitudes: Comparing Years of Low and High Solar Activities Using Observations and Tidal/Planetary Wave Features

In this study, midlatitude summer nighttime anomalies (MSNAs) are analyzed via observations and tidal/planetary wave features using measurements from the Formosat-3/Constellation Observing System for Meteorology, Ionosphere, and Climate (F3C) for 2007, a year with low solar activity, and 2014, a year with high solar activity. The total ionospheric electron content, ECion, an integrated quantity derived from F3C measurements, was used to compare the observational data. The ECion values were derived from accurate radio-occultation-retrieved electron density profiles without assuming spherical symmetry and from a model that separated the ground total electron content into the plasmaspheric and the ionospheric electron content contributions. An analysis of the ECion data set confirmed that MSNAs were present in three different regions of the world for the months surrounding the local summer solstice during both 2007 and 2014. In the southern hemisphere, the so-called Weddell Sea Anomaly showed a maximum increase in ECion, measured as the difference between nighttime and midday values, that was more than three times that in the northern MSNAs. For each individual MSNA, the corresponding maximum increases in electron content were similar between the two years analyzed, so they were not significantly affected by solar activity. Then, linear least-square fit to the frequency–wave number basis functions was used to derive the tidal and planetary wave components contributing to MSNAs. The main component that appears to produce the Weddell Sea Anomaly is D0, followed by SPW1, DW2, and DE1, in this order, which make secondary but still relevant contributions. The presence of MSNAs in the northern hemisphere was clearly supported by the migrating tide SW2 in combination with DE1. SW2 also supported an early morning MSNA being observed in the northern hemisphere. The main tidal and planetary wave signatures producing the MSNAs did not significantly differ between 2007 and 2014.

Computer Vision-Based Localization With Visible Light Communications

Visible light positioning and computer vision-based localization have the potential to be cost-effective technologies for accurate indoor localization. However, the feasibility of existing methods in this domain is limited. In this paper, a novel visible light communication (VLC)-assisted perspective-four-line algorithm (V-P4L) is proposed for practical indoor localization. The basic idea of V-P4L is to jointly use VLC and computer vision techniques to achieve high localization accuracy regardless of LED height differences. In particular, the space-domain information is first exploited to estimate the orientation and coordinate information of a single rectangular LED luminaire in the camera coordinate system based on plane geometry theory and solid geometry theory. Then, by using time-domain information transmitted by VLC and the estimated luminaire information, the proposed V-P4L can estimate the position and pose of the camera using single-view geometry theory and the linear least square (LLS) method. To further mitigate the effect of height differences among LEDs on localization accuracy, a correction algorithm based on the LLS method and a simple optimization method is proposed. Due to the combination of time- and space-domain information, V-P4L can achieve accurate localization using a single luminaire without limitation on the correspondences between the features and their projections in conventional perspective-n-line (PnL) algorithms. Simulation results show that the position error caused by the proposed V-P4L algorithm is always less than 15 cm and the orientation error is always less than 4° using popular indoor luminaires. Experimental results with real hardware show that the average position error is less than 3 cm under both similar and different heights for the LEDs.

Quantitative MRI Differentiates Electromyography Severity Grades of Denervated Muscle in Neuropathy of the Brachial Plexus.

Quantitative MRI (qMRI) metrics reflect microstructural skeletal muscle changes secondary to denervation and may correspond to conventional electromyography (EMG) assessments of motor unit recruitment (MUR) and denervation. Differences in quantitative T2 , diffusion-based apparent fiber diameter (AFD), and fat fraction (FF) exist between EMG grades, in patients with clinically suspected neuropathy of the brachial plexus. Prospective. A total of 30 subjects (age=37.5 ± 17.5, 21M/9F) with suspected brachial plexopathy. 3-Tesla; qMRI using fast spin echo (T2 -mapping), multi-b-valued diffusion-weighted echo planar imaging (for AFD), and dual-echo Dixon gradient echo (FF-mapping) sequences. qMRI values were compared against EMG grades (MUR and denervation). qMRI values (T2 , AFD, and FF) were obtained for five regional shoulder muscles. A 4-point scale was used for MUR/denervation severity. Linear mixed models and least-squares pairwise comparisons were used to evaluate qMRI differences between EMG grades. Predictive accuracy of EMG grades from qMRI was quantified by 10-fold cross-validated logistic models. A P value < 0.05 was considered statistically significant. Mean (95% confidence interval) qMRI for "full" MUR were T2 =39.40 msec (35.72-43.08 msec), AFD=78.35 μm (72.52-84.19 μm), and FF=4.54% (2.11-6.97%). Significant T2 increases (+8.36 to +14.67 msec) and significant AFD decreases (-11.04 to -21.58 μm) were observed with all abnormal MUR grades as compared to "full" MUR. Significant changes in both T2 and AFD were observed with increased denervation (+9.59 to +15.04 msec, -16.25 to -18.66 μm). There were significant differences in FF between some MUR grades (-1.45 to +2.96%), but no significant changes were observed with denervation (P=0.089-0.662). qMRI prediction of abnormal MUR or denervation was strong (mean accuracy=0.841 and 0.810, respectively) but moderate at predicting individual grades (accuracy=0.492 and 0.508, respectively). Quantitative T2 and AFD differences were observed between EMG grades in assessing muscle denervation. 2 TECHNICAL EFFICACY: Stage 1.

Simultaneous parametrization of torsional and third-neighbor interaction terms in force-field development: The LLS-SC algorithm.

The calibration of torsional interaction terms by fitting relative gas-phase conformational energies against their quantum-mechanical values is a common procedure in force-field development. However, much less attention has been paid to the optimization of third-neighbor nonbonded interaction parameters, despite their strong coupling with the torsions. This article introduces an algorithm termed LLS-SC, aimed at simultaneously parametrizing torsional and third-neighbor interaction terms based on relative conformational energies. It relies on a self-consistent (SC) procedure where each iteration involves a linear least-squares (LLS) regression followed by a geometry optimization of the reference structures. As a proof-of-principle, this method is applied to obtain torsional and third-neighbor interaction parameters for aliphatic chains in the context of the GROMOS 53A6 united-atom force field. The optimized parameter set is compared to the original one, which has been fitted manually against thermodynamic properties for small linear alkanes. The LLS-SC implementation is freely available under http://github.com/mssm-labmmol/profiler.

Design of Digital Constrained Linear Least-Squares Multiple-Resonator-Based Harmonic Filtering

Although voiced speech signals are physical signals which are approximately harmonic and electric power signals are true harmonic, the algorithms used for harmonic analysis in electric power systems can be successfully used in speech processing, including in speech enhancement, noise reduction, speaker recognition, and hearing aids. The discrete Fourier transform (DFT), which has been widely used as a phasor estimator due to its simplicity, has led to the development of new DFT-based algorithms because of its poor performance under dynamic conditions. The multiple-resonator (MR) filter structure proposed in previous papers has proven to be a suitable approach to dynamic harmonic analysis. In this article, optimized postprocessing compensation filters are applied to obtain frequency responses of the transfer functions convenient for fast measurements in dynamic conditions. An optimization design method based on the constrained linear least-squares (CLLS) is applied. This way, both the flatness in the passband and the equiripple attenuation in the stopband are satisfied simultaneously, and the latency is reduced.

Machine learned interatomic potentials using random features

We present a method to model interatomic interactions such as energy and forces in a computationally efficient way. The proposed model approximates the energy/forces using a linear combination of random features, thereby enabling fast parameter estimation by solving a linear least-squares problem. We discuss how random features based on stationary and non-stationary kernels can be used for energy approximation and provide results for three classes of materials, namely two-dimensional materials, metals and semiconductors. Force and energy predictions made using the proposed method are in close agreement with density functional theory calculations, with training time that is 96% lower than standard kernel models. Molecular Dynamics calculations using random features based interatomic potentials are shown to agree well with experimental and density functional theory values. Phonon frequencies as computed by random features based interatomic potentials are within 0.1% of the density functional theory results. Furthermore, the proposed random features-based potential addresses scalability issues encountered in this class of machine learning problems.

Methods for Fitting the Limit State Function of the Residual Strength of Damaged Ships

The limit state function is important for the assessment of the longitudinal strength of damaged ships under combined bending moments in severe waves. As the limit state function cannot be obtained directly, the common approach is to calculate the results for the residual strength and approximate the limit state function by fitting, for which various methods have been proposed. In this study, four commonly used fitting methods are investigated: namely, the least-squares method, the moving least-squares method, the radial basis function neural network method, and the weighted piecewise fitting method. These fitting methods are adopted to fit the limit state functions of four typically sample distribution models as well as a damaged tanker and damaged bulk carrier. The residual strength of a damaged ship is obtained by an improved Smith method that accounts for the rotation of the neutral axis. Analysis of the results shows the accuracy of the linear least-squares method and nonlinear least-squares method, which are most commonly used by researchers, is relatively poor, while the weighted piecewise fitting method is the better choice for all investigated combined-bending conditions.