Nonlinear Least Research Articles

A Field Calibration Method for Low-Cost MEMS Accelerometer Based on the Generalized Nonlinear Least Square Method

This paper proposes a field calibration method for an accelerometer without the need of having any external devices for calibration. In the proposed calibration method, generalized nonlinear least-square (GNLS) is used to estimate deterministic errors. A novel sensor’s data collection procedure is developed to collect data of an accelerometer along all three axes and all possible orientations where the expectation of influence of all possible errors is very high. The proposed calibration method is verified by applying it to two different accelerometers. The proposed calibration method achieved an accurate estimation of calibration parameters. The results of the proposed GNLS based calibration method are compared with two other commonly used algorithms, such as Levenberg–Marquardt (LM) and Gauss–Newton (GN). Simulation and experimental results show that the proposed GNLS-based calibration method is slightly more accurate than the LM and GN. The GNLS convergence rate for estimating the calibration parameters is also faster than the LM and GN.

A CALIBRATION METHOD OF TWO MOBILE LASER SCANNING SYSTEM UNITS FOR RAILWAY MEASUREMENT

Abstract. This paper proposes a methodology to calibrate the laser scanner of a Mobile Laser Scanning System (MLS) with the trajectory of the other MLS, both of which are installed directly above the top of both rails. Railway vehicle laser scanners systems of MLS are able to obtain 3D scanning map of the rail environment. In order to adapt the actual site condition of the maintenance works, we propose a calibration method with non-linear Least Mean Square calculation which use point clouds around poles along rails and sleepers of rails as cylindrical and planner constraints. The accuracy of 0.006 m between two laser point clouds can be achieved with this method. With the common planar and cylinder condition Leven-Marquardt method has been applied for this method. This method can execute without a good initial value for the extrinsic parameter and can shorten the processing time compared with the linear type of Least Mean Square method.

Mechanical Properties Analysis of Scaffold Material Using Nonlinear Least Squares Fitting by Hyperelastic Model

Mechanical Properties Analysis of Scaffold Material Using Nonlinear Least Squares Fitting by Hyperelastic Model

Logistic growth modelling of COVID-19 proliferation in China and its international implications

ObjectiveAs the coronavirus disease 2019 (COVID-19) pandemic continues to proliferate globally, this paper shares the findings of modelling the outbreak in China at both provincial and national levels. This paper examines the applicability of the logistic growth model, with implications for the study of the COVID-19 pandemic and other infectious diseases. MethodsAn NLS (Non-Linear Least Squares) method was employed to estimate the parameters of a differentiated logistic growth function using new daily COVID-19 cases in multiple regions in China and in other selected countries. The estimation was based upon training data from January 20, 2020 to March 13, 2020. A restriction test was subsequently implemented to examine whether a designated parameter was identical among regions or countries, and the diagnosis of residuals was also conducted. The model's goodness of fit was checked using testing data from March 14, 2020 to April 18, 2020. ResultsThe model presented in this paper fitted time-series data exceedingly well for the whole of China, its eleven selected provinces and municipalities, and two other countries - South Korea and Iran - and provided estimates of key parameters. This study rejected the null hypothesis that the growth rates of outbreaks were the same among ten selected non-Hubei provinces in China, as well as between South Korea and Iran. The study found that the model did not provide reliable estimates for countries that were in the early stages of outbreaks. Furthermore, this study concured that the R2 values might vary and mislead when compared between different portions of the same non-linear curve. In addition, the study identified the existence of heteroskedasticity and positive serial correlation within residuals in some provinces and countries. ConclusionsThe findings suggest that there is potential for this model to contribute to better public health policy in combatting COVID-19. The model does so by providing a simple logistic framework for retrospectively analyzing outbreaks in regions that have already experienced a maximal proliferation in cases. Based upon statistical findings, this study also outlines certain challenges in modelling and their implications for the results.

Fast Nonlinear Least Squares Optimization of Large‐Scale Semi‐Sparse Problems

AbstractMany problems in computer graphics and vision can be formulated as a nonlinear least squares optimization problem, for which numerous off‐the‐shelf solvers are readily available. Depending on the structure of the problem, however, existing solvers may be more or less suitable, and in some cases the solution comes at the cost of lengthy convergence times. One such case is semi‐sparse optimization problems, emerging for example in localized facial performance reconstruction, where the nonlinear least squares problem can be composed of hundreds of thousands of cost functions, each one involving many of the optimization parameters. While such problems can be solved with existing solvers, the computation time can severely hinder the applicability of these methods. We introduce a novel iterative solver for nonlinear least squares optimization of large‐scale semi‐sparse problems. We use the nonlinear Levenberg‐Marquardt method to locally linearize the problem in parallel, based on its first‐order approximation. Then, we decompose the linear problem in small blocks, using the local Schur complement, leading to a more compact linear system without loss of information. The resulting system is dense but its size is small enough to be solved using a parallel direct method in a short amount of time. The main benefit we get by using such an approach is that the overall optimization process is entirely parallel and scalable, making it suitable to be mapped onto graphics hardware (GPU). By using our minimizer, results are obtained up to one order of magnitude faster than other existing solvers, without sacrificing the generality and the accuracy of the model. We provide a detailed analysis of our approach and validate our results with the application of performance‐based facial capture using a recently‐proposed anatomical local face deformation model.

Self-Adaptive Nonlinear Least Squares Optimization for Geometric Calculations

Self-Adaptive Nonlinear Least Squares Optimization for Geometric Calculations

A novel method for Estimating Time of Flight of ultrasonic echoes through short-time Fourier transforms

The accurate estimation of time of flight (TOF) of an ultrasonic signal is essential in ultrasonic non-destructive testing (NDT). A novel method for TOF estimation based on the short-time Fourier transform (STFT) is proposed in this paper. Firstly, the time and frequency components of ultrasonic echoes are represented by a STFT. Then, the STFT spectrum of the echoes is projected onto the time domain. Finally, the TOF is estimated by calculating the location of the maximum modulus of STFT spectrum of the signal. A nonlinear Least Squares (LS) estimation method based on Modified Gauss–Newton (MGN) is employed to improve the estimation accuracy. The TOFs of multi-component echo signals are estimated by successive windowing. Numerical simulations and experiment tests show the performance of the proposed method.

The Sequential Fusion Estimation Algorithms Based on Gauss-Newton Method Over Multi-Agent Networked Systems

In multi-agent networked systems, parameter estimation problems arising in many practical applications are often required to solve Non-Linear Least Squares (NLLS) problems with the usual objective function (i.e., sum of squared residuals). The aim is to estimate a global parameter of interest across the network, such that the discrepancy between the estimation model and the real output of the system is minimized. There are challenges to face when applying the conventional Gauss-Newton method, such as non-cooperation and prosaic learning behavior. In this paper, we propose two Gauss-Newton type fusion estimation algorithms for solving overdetermined NLLS optimization problems arising frequently in multi-agent networked environment. One is the cycle-based Gauss-Newton (CGN) algorithm that is more attractive in performance due to its distributed nature than its peer: the known centralized Gauss-Newton algorithm. On the basis of CGN, we put emphasis on developing a simple but effective learning scheme leveraging an incremental technique, which is distributed on each computing agent over network. Such scheme results in the Incremental Gauss-Newton (IGN) algorithm that achieves a clear increase on convergence rate at the expense of higher computation cost than the CGN algorithm as well as the centralized one by deeper learning over the networking cycle. Both algorithms utilize Gauss-Newton iteration update in a cyclic cooperative manner, which offers the flexibility in exploiting the network topology. We provide the detailed analysis and the sufficient conditions for convergence of proposed IGN algorithm. By applying to target localization in wireless sensor networks, the numerical results confirm our convergence analysis and show that the proposed incremental scheme outperforms the centralized one in term of convergence performance.

Modelling an Indoor Crowd Monitoring System based on RSSI-based Distance

This paper reports a real-time localization algorithm system that has a main function to determine the location of devices accurately. The model can locate the smartphone position passively (which do not need a set on a smartphone) as long as the Wi-Fi is turned on. The algorithm uses Intersection Density, and the Nonlinear Least Square Algorithm (NLS) method that utilizes the Lavenberg-Marquart method. To minimize the localization error, Kalman Filter (KF) is used. The algorithm is computed under Matlab approach. The most obtained model will be implemented in this Wi-Fi tracker system using RSSI-based distance for indoor crowd monitoring. According to the experiment result, KF can improve Hit ratio of 81.15 %. Hit ratio is predicting results of a location that is less than 5 m from the actual area (location). It can be obtained from several RSSI scans, the calculation is as follows: the number of non-error results divided by the number of RSSI scans and multiplied by 100%.

Allometric equations for total aboveground dry biomass and carbon content of Pinus occidentalis trees

Although precise species-specific aboveground tree biomass equations are needed in the Dominican Republic to quantify potential carbon storage in the context of climate change and sustainable forest management, there has been a lack of available information concerning total aboveground biomass and/or carbon content for naturally occurring Pinus occidentalis trees. The objectives of the study were to develop species-specific allometric biomass models for P. occidentalis, as well as to assess variation in carbon concentration among stem plus bark, branches and foliage by means of chemical analyses. Predictor variables included diameter at breast height (DBH) and total tree height (H), alone and in combination. Model fitting methods employing linear and nonlinear regression techniques were used and evaluated based on goodness-of-fit criteria. Two competing biomass models produced similar goodness-of-fit statistics: model 4, fitted by the Weighted Non-linear Least Squares technique, , and model 6, fitted by modeling the variance component and applying the Maximum Likelihood method, . Average carbon concentration (%) was highest in foliage tissue (49.8%), followed by branches (46.37%) and, lastly, stem plus bark (45.95%). On average, each individual tree is storing 175 kg of total aboveground carbon. To our knowledge, this is the first account reporting carbon fraction for this tropical species. Calculated wood carbon fractions from the study can be easily incorporated into forest C accounting, and may reduce errors in forest C valuations, which nowadays are performed using global estimates published elsewhere.

A Low-Complexity Nonlinear Least Mean Squares Filter Based on a Decomposable Volterra Model

Nonlinear signal processing is important in various applications, however it usually requires high computational cost. This paper proposes a low-complexity nonlinear adaptive least mean squares (LMS) filter based on decomposable Volterra kernels. The decomposability condition comes from a well-posed approximation problem, which imposes a rank-one structure on the full Volterra model, resulting in a system equivalent to a product of ordinary linear filters. A mean-square estimation problem is posed over such a decomposable Volterra model (DVM) and a solution via a steepest descent algorithm is introduced, subsequently motivating an adaptive implementation. The resulting DVM-LMS filter is nonlinear in the input signal as well as in its parameters, however leading to an exponential decrease in computational complexity, as compared to the full Volterra kernel. The nonlinearity in the parameters may introduce instabilities, which are tackled via normalizing strategies, introduced together with heuristics for proper initialization. Necessary conditions for stability are derived and serve as guidance to design the step-size, while simulations show that the algorithm is unbiased after convergence. Additionally, an expression for the steady-state excess mean square error is derived. Several simulations show the new filter's performance and competitiveness against celebrated methods in the literature.

Parameter estimation for a ship's roll response model in shallow water using an intelligent machine learning method

In order to accurately identify the ship's roll model parameters in shallow water, and solve the problems of difficult estimating nonlinear damping coefficients by traditional methods, a novel Nonlinear Least Squares - Support Vector Machine (NLS-SVM) is introduced. To illustrate the validity and applicability of the proposed method, simulation and decay tests data are combined and utilized to estimate unknown parameters and predict the roll motions. Firstly, simulation data is applied in the NLS-SVM model to obtain estimated damping parameters, compared with pre-defined parameters to verify the validity of the proposed method. Subsequently, decay tests data are used in identifying unknown parameters by utilizing traditional models and the new NLS-SVM model, the results illustrate that the intelligent method can improve the accuracy of parametric estimation, and overcome the conventional algorithms' weakness of difficult identification of the nonlinear damping parameter in the roll model. Finally, to show the wide applicability of the proposed model in shallow water, experimental data from various speeds and Under Keel Clearances (UKCs) are applied to identify the damping coefficients. Results reveal the potential of using the NLS-SVM for the problem of the roll motion in shallow water, and the effectiveness and accuracy are verified as well.

(Invited) A Critical View on the Application of Distribution of Relaxation Times

Electrochemical Impedance Spectroscopy (EIS) has been an important tool in Solid State Ionics since the sixties of the 20th century. Evolving from graphical analysis to fully automated Complex Nonlinear Least Squares analysis (CNLS), it has contributed to a better understanding of charge and mass transfer and storage processes in solid state devices, e.g. batteries, sensors, fuel cells etc. In many cases the use of an Equivalent Circuit (EqC) gave access to important materials parameters. Due to the increasing complexity of e.g. electrodes, it is not always simple to find an appropriate EqC. The transformation of the impedance spectra to a distribution of relaxation times in the τ-domain has gained increased attention over the last decades, as it is seen as a model independent representation. The main problem, however, is that this transformation is an ill-posed inverse problem. Various methods (e.g. FFT, Tikhonov Regularization, Max Entropy) have been designed to provide the distributions, but they all require adjustment of a ‘smoothing’ parameter. In this presentation the advantages and limitations of the use of a DRT will be presented with selected practical examples.

A comparison between Nonlinear Least Squares and Maximum Likelihood estimation for the prediction of tumor growth on experimental data of human and rat origin

Several mathematical models have been developed to explain the growth of tumors and used to fit experimental or clinical data. Their predictive power – i.e. their ability to forecast the future growth on the basis of present knowledge – however, has been rarely explored. Here, we investigate whether a Hidden Markov Model (HMM) based on the well-established Gompertz tumor growth function with additive Gaussian noise could effectively be used to predict the future growth of experimental tumors. The idea behind this work is that one might achieve more accurate predictions if estimates of the unknown parameters of the HMM are used instead of those obtained by fits of the deterministic Gompertz model to the data. We use the principle of Maximum Likelihood (ML) to estimate unknown parameters related to growth dynamics and noise, and we compare its effectiveness to the classical Nonlinear Least Squares minimization approach. The analyses show that our approach can provide better growth predictions when the data contain adequate information concerning the tumors saturation phase. The forecasts could also be improved by taking into account prior knowledge about the unknown parameters when the information concerning the saturation phase was inadequate. We conclude that by using HMMs in combination with the principle of ML, one can obtain more reliable growth predictions for individual tumors.

A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model

Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.

Comparison and Combination of Techniques for Determining the Parameters of a Magnetic Hysteresis Model

The Jiles-Atherton scalar hysteresis model presents five parameters used to represent the material tested and used to calculate the magnetic losses. This article presents a comparative analysis of the performance of two methods of identifying these parameters. In the first method, the equations of Jiles-Atherton were assembled into a single non-linear ordinary differential equation as a function of the variables of interest. An algebraic system of five equations with five unknowns is obtained by evaluating the non-linear ordinary differential equation in five points belonging to the branch of the experimental hysteresis loop. The parameters are obtained by solving this system of equations using the method of Non-Linear Least Squares (NLLS). In the second method, the inverse model of Jiles-Atherton is used to calculate the magnetic field H from the experimental values of magnetic induction B. Using the method of genetic algorithms (MGA), the objective function given by the sum of the relative error of calculated magnetic field and experimental magnetic field along the hysteresis loop is minimized. To validate methods the experimental curves were compared with calculated ones. When applying the methods, it was verified that NLLS besides providing more accurate results, it is faster when compared to MGA. In the MGA the convergence of the calculated magnitudes to the experimental magnitudes improves when one of the chromosomes of the initial population is the solution obtained applying NLLS.

Characterization and Correction of the Geometric Errors in Using Confocal Microscope for Extended Topography Measurement. Part I: Models, Algorithms Development and Validation

This work presents a method for characterizing and correcting the geometric errors of the movement of the lateral stage of Imaging Confocal Microscope (CM) in extended topography measurement. For an extended topography measurement, a defined number of 2D images are taken and stitched by correlation methods. Inaccuracies due to linear displacement, vertical and horizontal straightness errors, angular errors, and squareness errors based on the assumption of the rigid body kinematics are described. A mathematical model for the scale calibration of the X- and Y- coordinates is derived according to the system kinematics, the axis chain vector of CM, and the geometric error functions and their approximations by Legendre polynomials. The correction coefficients of the kinematic modelling are determined by the measured and certified data of a dot grid target standard artefact. To process the measurement data, algorithms for data partitions, fittings of cylinder centers, and determinations of coefficients are developed and validated. During which methods such as form removal, K-means clustering, linear and non-linear Least Squares are implemented. Results of the correction coefficients are presented in Part II based on the experimental studies. The mean residual reduces 29.6% after the correction of the lateral stage errors.

Sequential Quadratic Programming Method for Nonlinear Least Squares Estimation and Its Application

In this study, we propose a direction-controlled nonlinear least squares estimation model that combines the penalty function and sequential quadratic programming. The least squares model is transformed into a sequential quadratic programming model, allowing for the iteration direction to be controlled. An ill-conditioned matrix is processed by our model; the least squares estimate, the ridge estimate, and the results are compared based on a combination of qualitative and quantitative analyses. For comparison, we use two equality indicators: estimated residual fluctuation of different methods and the deviation between estimated and true values. The root-mean-squared error and standard deviation are used for quantitative analysis. The results demonstrate that our proposed model has a smaller error than other methods. Our proposed model is thereby found to be effective and has high precision. It can obtain more precise results compared with other classical unwrapping algorithms, as shown by unwrapping using both simulated and real data from the Jining area in China.

A nonlinear least squares framework for periodic grating identification with a high–order perturbation of surfaces implementation

The scattering of linear electromagnetic waves by a layered structure is a central model in many problems of engineering interest. In this contribution, we focus on the interaction of visible and near–visible radiation with periodic structures on the micron or nanometer scales which are relevant for many applications in nanoplasmonics. The fabrication of such structures is extremely difficult and costly, and even the most sophisticated laboratories have difficulty measuring the dimensions of gratings they have constructed. We present a new, extremely rapid and robust, identification algorithm for providing precisely this information. It is built upon a Nonlinear Least Squares framework, and implemented with a High–Order Perturbation of Surfaces methodology which is orders of magnitude faster than volumetric solvers, while outperforming surface methods (such as Boundary Integral Methods) for the geometries we consider here. In addition to a full derivation and specification of the algorithm, we also support our claims with a number of illustrative simulations.

Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.