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For distributed estimation arising in the nonlinear least squares (NLLSs) problems over adaptive networks, where every node has the abilities of data processing and learning, only the incomplete local data are exploited by the traditional noncooperative method, thereby resulting in the degradation on estimation performance. In this article, a cooperative diffusion strategy is proposed by using a Gauss-Newton (GN) method in order to fully utilize the diversity of temporal-spatial data on local updates. The proposed algorithm includes two steps, i.e., aggregate then update (ATU), where the aggregating step collects in real time the global information instead of local information due to the diffusion strategy, and the updating step implements the local GN iteration. The resulting ATU diffusion algorithm is a distributed and cooperative system without any increase on communication cost, as compared with the noncooperative version. Based on the detailed convergence analysis for ATU, which is fundamental to the promotion of this algorithm, the sufficient conditions for convergence are derived and the evidences of faster convergence than the noncooperative version are provided. The simulation results confirm the obtained theoretical derivations by applying the ATU algorithm to an NLLS-based target localization problem and show the cooperation gains in many aspects, such as the convergence rate, steady-state accuracy, and robustness to noisy range, step size, node, and link failures.

When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this type. In this paper, we are concerned with the computation of the minimal-norm solution of an underdetermined nonlinear least-squares problem. We present a Gauss–Newton type method, which relies on two relaxation parameters to ensure convergence, and which incorporates a procedure to dynamically estimate the two parameters, as well as the rank of the Jacobian matrix, along the iterations. Numerical results are presented.

Non-Newtonian fluids are commonly used in a wide range of industries; one example are in biogas power plants. Proper measurements and modeling of such fluids can be crucial from the design and operations point of view. Results presented in this study covered seven samples from three plants (a sewage sludge treatment plant, utilization biogas station and a biogas plant in a sugar factory), including mechanically thickened excessive activated sludge (MTEAS), sugar beet pulp (SBP), liquid fruit and vegetable waste (FVW), beet roots (BR) and corn waste (CW); their mixtures were prepared as in a real plant. The total solid content remained below 6.8% for all samples. The apparent viscosity (15 RPM) did not exceed 10 Pas in any sample. A correlation analysis for solvent type influence on the viscosity was carried out. The obtained results were analyzed, and the Herschel–Bulkley rheological model was selected for the fluid description. Then, the Moullinex method was applied to determine the H–B model parameters. The obtained results may contribute to the proper design and operation of various biogas power plants, in which viscosity seems to be one of the crucial flow parameters that influences the device types used, as well as energy consumption.

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and calculates multiple roots simultaneously from a given multiplicity structure and initial root approximations. To fulfill the input requirement of Algorithm I, we develop a numerical GCD-finder containing a successive singular value updating and an iterative GCD refinement as the main engine of Algorithm II that calculates the multiplicity structure and the initial root approximation. While limitations of our algorithm exist in identifying the multiplicity structure in certain situations, the combined method calculates multiple roots with high accuracy and consistency in practice without using multiprecision arithmetic even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the sensitivity of the multiple roots, a structure-preserving condition number is proposed and error bounds are established. According to our computational experiments and error analysis, a polynomial being ill-conditioned in the conventional sense can be well conditioned with the multiplicity structure being preserved, and its multiple roots can be computed with high accuracy.

The conditional, unconditional, or the exact maximum likelihood estimation and the least-squares estimation involve minimizing either the conditional or the unconditional residual sum of squares. The maximum likelihood estimation (MLE) approach and the nonlinear least squares (NLS) procedure involve an iterative search technique for obtaining global rather than local optimal estimates. Several authors have presented brief overviews of algorithms for solving NLS problems. Snezana S. Djordjevic (2019) presented a review of some unconstrained optimization methods based on the line search techniques. Mahaboob et al. (2017) proposed a different approach to estimate nonlinear regression models using numerical methods also based on the line search techniques. Mohammad, Waziri, and Santos (2019) have briefly reviewed methods for solving NLS problems, paying special attention to the structured quasi-Newton methods which are the family of the search line techniques. Ya-Xiang Yuan (2011) reviewed some recent results on numerical methods for nonlinear equations and NLS problems based on online searches and trust regions techniques, particularly on Levenberg-Marquardt type methods, quasi-Newton type methods, and trust regions algorithms. The purpose of this paper is to review some online searches and trust region's more well-known robust numerical optimization algorithms and the most used in practice for the estimation of time series models and other nonlinear regression models. The line searches algorithms considered are: Gradient algorithm, Steepest Descent (SD) algorithm, Newton-Raphson (NR) algorithm, Murray’s algorithm, Quasi-Newton (QN) algorithm, Gauss-Newton (GN) algorithm, Fletcher and Powell algorithm (FP), Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. While the only trust-region algorithm considered is the Levenberg-Marquardt (LM) algorithm. We also give some main advantages and disadvantages of these different algorithms.

Large-scale optimization problems at the core of many graphics, vision, and imaging applications are often implemented by hand in tedious and error-prone processes in order to achieve high performance (in particular on GPUs), despite recent developments in libraries and DSLs. At the same time, these hand-crafted solver implementations reveal that the key for high performance is a problem-specific schedule that enables efficient usage of the underlying hardware. In this work, we incorporate this insight into Thallo, a domain-specific language for large-scale non-linear least squares optimization problems. We observe various code reorganizations performed by implementers of high-performance solvers in the literature, and then define a set of basic operations that span these scheduling choices, thereby defining a large scheduling space. Users can either specify code transformations in a scheduling language or use an autoscheduler. Thallo takes as input a compact, shader-like representation of an energy function and a (potentially auto-generated) schedule, translating the combination into high-performance GPU solvers. Since Thallo can generate solvers from a large scheduling space, it can handle a large set of large-scale non-linear and non-smooth problems with various degrees of non-locality and compute-to-memory ratios, including diverse applications such as bundle adjustment, face blendshape fitting, and spatially-varying Poisson deconvolution, as seen in Figure 1. Abstracting schedules from the optimization, we outperform state-of-the-art GPU-based optimization DSLs by an average of 16× across all applications introduced in this work, and even some published hand-written GPU solvers by 30%+.

We present a sparse Gauss-Newton solver for accelerated sensitivity analysis with applications to a wide range of equilibrium-constrained optimization problems. Dense Gauss-Newton solvers have shown promising convergence rates for inverse problems, but the cost of assembling and factorizing the associated matrices has so far been a major stumbling block. In this work, we show how the dense Gauss-Newton Hessian can be transformed into an equivalent sparse matrix that can be assembled and factorized much more efficiently. This leads to drastically reduced computation times for many inverse problems, which we demonstrate on a diverse set of examples. We furthermore show links between sensitivity analysis and nonlinear programming approaches based on Lagrange multipliers and prove equivalence under specific assumptions that apply for our problem setting., 10 pages, 7 figures

In this paper, we propose a novel visible light positioning (VLP) scheme under multi-path propagation, which is a practical scenario that has not been well studied for VLP. The new scheme exploits the so-called dual-mode light-emitting diode (DM-LED) of different radiation lobe mode numbers at the transmitter and a photodiode at the receiver. Specifically, we devise a method that utilizes a radiation angle measurement approach for DM-LED, derive the Cramer-Rao lower bound (CRLB) of the estimated distance and analyze the characteristics of key parameters. In addition, two localization algorithms based on linear and nonlinear least squares methods are developed. Simulation results show that under the ideal light-of-sight scenario, the CRLB of the proposed VLP scheme can be close to that of the conventional received signal strength (RSS) aided VLP scheme. Furthermore, we implement the first ever prototype DM-LED lamp to validate the new DM-LED aided VLP system model. Both simulation and experiment results demonstrate that the proposed system outperforms its RSS-aided VLP counterpart in terms of positioning accuracy in the more realistic, and thus more challenging multi-path scenario, even with a tilting receiver.

The article presents a velocity estimation algorithm through the wheel encoder-based odometry and wheel circumference identification. The motivation of the paper is that a proper model can improve the motion estimation in poor sensor performance cases. For example, when the GNSS signals are unavailable, or when the vision-based methods are incorrect due to the insufficient number of features, furthermore, when the IMU-based method fails due to the lack of frequent accelerations. In these situations, the wheel encoders can be an appropriate choice for state estimation. However, this type of estimation suffers from parameter uncertainty. In the paper, a wheel circumference identification is proposed to improve the velocity estimation. The algorithm listens to the incoming sensor measurements and estimates the wheel circumferences recursively with a nonlinear least squares method. The experimental results demonstrate that with the application of the identified parameters in the wheel odometry model, accurate velocity estimation can be obtained with high frequency. Thus, the presented algorithm can improve the motion estimation in the driver assistant functions of autonomous vehicles.

Producción Científica, To accurately estimate the partial volume fraction of free water in the white matter from diffusion MRI acquisitions not demanding strong sensitizing gradients and/or large collections of different b-values. Data sets considered comprise~32- 64 gradients near b =1000 s/mm2 plus~6 gradients near b =500 s/mm2.Theory and Methods: The spherical means of each diffusion MRI set with the same b- value are computed. These means are related to the inherent diffusion parameters within the voxel (free- and cellular- water fractions; cellular- water diffusivity), which are solved by constrained nonlinear least squares regression.Results: The proposed method outperforms those based on mixtures of two Gaussians for the kind of data sets considered. W.r.t. the accuracy, the former does not introduce significant biases in the scenarios of interest, while the latter can reach a bias of 5%– 7% if fiber crossings are present. W.r.t. the precision, a variance near 10%, compared to 15%, can be attained for usual configurations.Conclusion: It is possible to compute reliable estimates of the free- water fraction inside the white matter by complementing typical DTI acquisitions with few gradients at a lowb- value. It can be done voxel- by- voxel, without imposing spatial regularity constraints., Ministerio de Ciencia e Innovación, (grant RTI2018- 094569- B- I00)

A new method to estimate the size distribution of non-interacting colloidal particles from small-angle scattering data is presented. The method demonstrates that the distribution can be efficiently retrieved through features of the scattering data when plotted in the Porod representation, thus avoiding the standard fitting procedure of nonlinear least squares. The present approach is elaborated using log-normal and Weibull distributions. The method can differentiate whether the distribution actually follows the functionality of either of these two distributions, unlike the standard fitting procedure which requires a prior assumption of the functionality of the distribution. After validation with various simulated scattering profiles, the formalism is used to estimate the size distribution from experimental small-angle X-ray scattering data from two different dilute dispersions of silica. At present the method is limited to monomodal distributions of dilute spherical particles only.

Modelling tree height-diameter relationships in complex tropical rain forest ecosystems remains a challenge because of characteristics of multi-species, multi-layers, and indeterminate age composition. Effective modelling of such complex systems required innovative techniques to improve prediction of tree heights for use for aboveground biomass estimations. Therefore, in this study, deep learning algorithm (DLA) models based on artificial intelligence were trained for predicting tree heights in a tropical rain forest of Nigeria. The data consisted of 1736 individual trees representing 116 species, and measured from 52 0.25 ha sample plots. A K-means clustering was used to classify the species into three groups based on height-diameter ratios. The DLA models were trained for each species-group in which diameter at beast height, quadratic mean diameter and number of trees per ha were used as input variables. Predictions by the DLA models were compared with those developed by nonlinear least squares (NLS) and nonlinear mixed-effects (NLME) using different evaluation statistics and equivalence test. In addition, the predicted heights by the models were used to estimate aboveground biomass. The results showed that the DLA models with 100 neurons in 6 hidden layers, 100 neurons in 9 hidden layers and 100 neurons in 7 hidden layers for groups 1, 2, and 3, respectively, outperformed the NLS and NLME models. The root mean square error for the DLA models ranged from 1.939 to 3.887 m. The results also showed that using height predicted by the DLA models for aboveground biomass estimation brought about more than 30% reduction in error relative to NLS and NLME. Consequently, minimal errors were created in aboveground biomass estimation compared to those of the classical methods.

In this paper, we consider the nonlinear least squares (NLLSs) problems in adaptive networks, where a collection of nodes with adaptation ability by learning are required to estimate a global vector parameter by minimizing the specified convex cost function. Although the global Gauss–Newton (GN) method is an excellent candidate for solving such problems, many challenges need to be addressed for practical realization due to its natures of centralization and noncooperation. Without specialized design for routing, we motivate and propose new diffusion GN methods with cooperative strategy among local neighborhoods. The good performances of diffusion cooperation schemes have been proved in different literatures, such as distribution, robustness, and easy implementation. The proposed cooperative diffusion strategies are named as aggregation-then-update (ATU) and update-then-aggregation (UTA), which reach fully information diffusion across network and consist of two steps in a reversible way including aggregation of local estimates and local GN update. Although all implementations are local, the cooperation between nodes is network wide. Based on the steady-state equilibria theory in the nonlinear discrete dynamical system, the convergence analysis of proposed algorithms is provided. The results show that the global convergence can be achieved when the sufficient conditions are satisfied. We also provide performance comparisons and analysis together with simulation to confirm the applicability and effectiveness of proposed diffusion GN algorithms.

The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are contaminated by some noise. For ill-posed TLS poblems, regularization techniques are necessary to stabilize the computed solution; otherwise, TLS produces a noise-dominant output. In this paper, we show that the regularized total least squares (RTLS) problem can be reformulated as a nonlinear least squares problem and can be solved by the Gauss–Newton method. Due to the nature of the RTLS problem, we present an appropriate method to choose a good regularization parameter and also a good initial guess. Finally, the efficiency of the proposed method is examined by some test problems.

The radial basis function network-based state-dependent autoregressive (RBF-AR) model has been widely used in modeling and prediction of nonlinear time series. The parameter identification of RBF-AR model can be reformulated as a separable nonlinear least squares problem. The variable projection (VP) algorithm has been proven to be valuable in solving such problems. However, for ill-posed problems, the classical VP algorithm usually yields unstable models. In this paper, we consider a novel regularized separable algorithm that takes advantage of the VP method and the expectation–maximization (EM) method. The proposed algorithm utilizes the VP algorithm to optimize the nonlinear parameters and automatically picks out the regularization parameters during the search process. Numerical results on real-world data and synthetic time series confirm the effectiveness of the proposed algorithm.

Air quality is a vital concern globally, especially in China. To improve fine particulate matter (PM2.5) forecasts, a nonlinear least squares four-dimensional variational (NLS-4DVar) data assimilation system was established and applied into the Weather Research and Forecasting model coupled with the “offline” Community Multiscale Air Quality (WRF-CMAQ) model. By assimilating hourly surface PM2.5 observations, the optimal initial conditions (ICs) of the state variable were solved iteratively with the NLS-4DVar method, which uses a Gauss–Newton iterative scheme to handle nonlinearity without a tangent linear or adjoint model, thereby rendering the aerosol assimilation process fast and simple. Observing system simulation experiments (OSSEs) were designed from 10 to 16 November 2018 to evaluate the effectiveness of the NLS-4DVar data assimilation system for PM2.5 forecasts (NASM) assimilation system. The results derived from the OSSEs indicated that the NASM system could effectively assimilate multi-time PM2.5 observations, reduce uncertainty in surface initial PM2.5 concentrations, and thus improve the accuracy of predictions.

Objective: The absolute image reconstruction problem of electrical impedance tomography (EIT) is ill-posed. Traditional methods usually solve a nonlinear least squares problem with some kind of regularization. These methods suffer from low accuracy, poor anti-noise performance, and long computation time. Besides, the integration of a priori information is not very flexible. This work tries to solve EIT inverse problem using a machine learning algorithm for the application of thorax imaging. Methods: We developed the supervised descent learning EIT (SDL-EIT) inversion algorithm based on the idea of supervised descent method (SDM). The algorithm approximates the mapping from measured data to the conductivity image by a series of descent directions learned from training samples. We designed a training data set in which the thorax contour, and some general structure of lungs, and heart are embedded. The algorithm is implemented in both two-, and three-dimensional cases, and is evaluated using synthetic, and measured thoracic data. Results, and conclusion: For synthetic data, SDL-EIT shows better accuracy, and anti-noise performance compared with traditional Gauss–Newton inversion (GNI) method. For measured data, the result of SDL-EIT is reasonable compared with computed tomography (CT) scan image. Significance: Using SDL-EIT, prior information can be easily integrated through the specifically designed training data set, and the image reconstruction process can be accelerated. The algorithm is effective in inverting measured thoracic data. It is a potential algorithm for human thorax imaging.

Accurate metro ridership prediction can guide passengers in efficiently selecting their departure time and simultaneously help traffic operators develop a passenger organization strategy. However, short-term passenger flow prediction needs to consider many factors, and the results of the existing models for short-term subway passenger flow forecasting are often unsatisfactory. Along this line, we propose a parallel architecture, called the seasonal and nonlinear least squares support vector machine (SN-LSSVM), to extract the periodicity and nonlinearity characteristics of passenger flow. Various forecasting models, including auto-regressive integrated moving average, long short-term memory network, and support vector machine, are employed for evaluating the performance of the proposed architecture. Moreover, we first applied the method to the Tiyu Xilu station which is the most crowded station in the Guangzhou metro. The results indicate that the proposed model can effectively make all-weather and year-round passenger flow predictions, thus contributing to the management of the station.

1. The quantification of thermal performance curves (TPCs) for biological rates has many applications to problems such as predicting species9 responses to climate change. There is currently no widely used open-source pipeline to fit mathematical TPC models to data, which limits the transparency and reproducibility of the curve fitting process underlying applications of TPCs. 2. We present a new pipeline in R that currently allows for reproducible fitting of 24 different TPC models using non-linear least squares (NLLS) regression. The pipeline consists of two packages - rTPC and nls.multstart - that allow multiple start values for NLLS fitting and provides helper functions for setting start parameters. This pipeline overcomes previous problems that have made NLLS fitting and estimation of key parameters difficult or unreliable. 3. We demonstrate how rTPC and nls.multstart can be combined with other packages in R to robustly and reproducibly fit multiple models to multiple TPC datasets at once. In addition, we show how model selection or averaging, weighted model fitting, and bootstrapping can easily be implemented within the pipeline. 4. This new pipeline provides a flexible and reproducible approach that makes the challenging task of fitting multiple TPC models to data accessible to a wide range of users.

Monitoring systems are common in small and large power plants, providing measurements of basic electrical, mechanical and thermal variables. These measurements can be used for the identification of the synchronous generator parameters. In this paper, a simple method for the identification of a subset of synchronous generator parameters, using basic measurements provided by monitoring systems, is proposed. The identification is performed in normal operating conditions using a synchronous machine simplified model suited for operating conditions under small disturbances. Three identification models are derived, each one depending on a subset of parameters. Three optimization subproblems are then solved by minimizing the residues between each model output and a system measurement. The resulting nonlinear least squares problems are solved by the interior-point method taking into account constraints on the parameters range. A method is proposed to estimate the load angle which is required in the identification. Results are presented for synthetic data and real data, from a 25 MVA generator in a hydroelectric power plant.

This paper presents an improved new method with differential evolution and the cubic spline approach is proposed to retrieve sea level height based on GNSS SNR observations from a single geodetic receiver. Considering the B-spline function is unstable at the beginning or end, and the feature that B-spline functions do not pass through nodes may introduce errors. Thus, the cubic spline is applied to the retrieval process and accounts for a continuous and smooth in sea level retrieval time series. Besides, the biases caused by tropospheric delay and dynamic sea level are considered and corrected. Testing data from two stations with different tidal range and the final solution agrees well with measurements from co-located tide gauges, reaching the RMSE of 3.67 cm at Friday Harbor, Washington, and 1.36 cm at Onsala, Sweden. Comparison of the nonlinear least squares, this method leads to a clear increase in precision of the sea level retrievals within 50%. Additionally, referring to the result of Purnell et al. (2020) and the IAG inter-comparison campaign, the results of this paper show more potential.

Intravascular photoacoustic tomography (IVPAT) is a newly developed imaging modality for the diagnosis and intervention of coronary artery diseases. It is an ill-posed nonlinear least squares (NLS)...

While classical measurement error in the dependent variable in a linear regression framework results only in a loss of precision, nonclassical measurement error can lead to estimates, which are biased and inference which lacks power. Here, we consider a particular type of nonclassical measurement error: skewed errors. Unfortunately, skewed measurement error is likely to be a relatively common feature of many outcomes of interest in political science research. This study highlights the bias that can result even from relatively “small” amounts of skewed measurement error, particularly, if the measurement error is heteroskedastic. We also assess potential solutions to this problem, focusing on the stochastic frontier model and Nonlinear Least Squares. Simulations and three replications highlight the importance of thinking carefully about skewed measurement error as well as appropriate solutions.

In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence ratio and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval., 23 pages, 7 figures

The resection in 3D space is a common problem in surveying engineering and photogrammetry based on observed distances, angles, and coordinates. This resection problem is nonlinear and comprises redundant observations which is normally solved using the least-squares method in an iterative approach. In this paper, we introduce a vigorous angular based resection method that converges to the global minimum even with very challenging starting values of the unknowns. The method is based on deriving oblique angles from the measured horizontal and vertical angles by solving spherical triangles. The derived oblique angles tightly connected the rays enclosed between the resection point and the reference points. Both techniques of the nonlinear least square adjustment either using the Gauss-Newton or Levenberg – Marquardt are applied in two 3D resection experiments. In both numerical methods, the results converged steadily to the global minimum using the proposed angular resection even with improper starting values. However, applying the Levenberg – Marquardt method proved to reach the global minimum solution in all the challenging situations and outperformed the Gauss-Newton method.

Background: This study aims to determine the effective atomic number (Zeff) from dual-energy image sets obtained using a conventional computed tomography (CT) simulator. The estimated Zeff can be used for deriving the stopping power and material decomposition of CT images, thereby improving dose calculations in radiation therapy. Materials and Methods: An electron-density phantom was scanned using Philips Brilliance CT Big Bore at 80 and 140 kVp. The estimated Zeff values were compared with those obtained using the calibration phantom by applying the Rutherford, Schneider, and Joshi methods. The fitting parameters were optimized using the nonlinear least squares regression algorithm. The fitting curve and mass attenuation data were obtained from the National Institute of Standards and Technology. The fitting parameters obtained from stopping power and material decomposition of CT images, were validated by estimating the residual errors between the reference and calculated Zeff values. Next, the calculation accuracy of Zeff was evaluated by comparing the calculated values with the reference Zeff values of insert plugs. The exposure levels of patients under additional CT scanning at 80, 120, and 140 kVp were evaluated by measuring the weighted CT dose index (CTDIw). Results and Discussion: The residual errors of the fitting parameters were lower than 2%. The best and worst Zeff values were obtained using the Schneider and Joshi methods, respectively. The maximum differences between the reference and calculated values were 11.3% (for lung during inhalation), 4.7% (for adipose tissue), and 9.8% (for lung during inhalation) when applying the Rutherford, Schneider, and Joshi methods, respectively. Under dual-energy scanning (80 and 140 kVp), the patient exposure level was approximately twice that in general single- energy scanning (120 kVp). Conclusion: Zeff was calculated from two image sets scanned by conventional single-energy CT simulator. The results obtained using three different methods were compared. The Zeff calculation based on single-energy exhibited appropriate feasibility.

We propose structured spectral gradient algorithms for solving nonlinear least squares problems based on a modified structured secant equation. The idea was to integrate more details of the Hessian...

Ultraviolet light-emitting diodes (UV LEDs) play an important role in inactivating novel coronavirus pneumonia, but the lack of rapid lifetime prediction can easily cause untimely failure detection, long product development cycles, and high costs. This study predicts the lifetime of UV LEDs based on the long short-term memory (LSTM) recurrent neural network (RNN). First, the equipment setup was designed to conduct an aging test to obtain a predicted length of life for the UV LED samples using a Weibull distribution. Next, LSTM RNN was employed to predict the lifetime of the UV LEDs based on the radiation power degradation. The results were then compared with those from nonlinear least squares (NLS) regression recommended by the IESNA TM-21 industry standard. Finally, the robustness of the two methods was analyzed by changing the starting times of the predictions. The results showed that the LSTM RNN proposed in this letter reveals not only a 29.7% lower lifetime prediction error compared with the NLS regression, but also a more stable robustness. Thus, the LSTM RNN method is found to be more accurate and more robust in predicting the lifetime of UV LEDs.

[EN] This work presents a hybrid modeling methodology based on dynamic response, filtering and identification techniques, in order to determine a ball mill representative model. In essence, we have provided models for electrical drive, mechanical reduction and load, without the need for physical decoupling. The electrical parameters are determined using state variable filtering, linear regression and recursive least square techniques. The mechanical parameters are identified considering the system acceleration time. A final adjustment stage considering the parameters set, is carried out using the nonlinear least squares method. Based on the ball mill complete model, a load torque estimator is proposed, using high-pass filters, and a load torque estimate. The numerical simulations, under different operating conditions, show suitable approximation with experimental results. Therefore, the proposed hybrid methodology, based on both dynamic modeling and signal analysis, has the potential to assist in the design for supervision and control systems of a ball mill., [ES] Este trabajo presenta una metodología híbrida de modelado basada en técnicas de respuesta dinámica, filtrado e identificación, considerando el dominio del tiempo y la frecuencia, para determinar el modelo representativo de un molino de bolas de acoplamiento fijo. Se proponen modelos para el accionamiento eléctrico, reductor mecánico y carga, sin la necesidad de desacoplamiento físico. Los parámetros eléctricos se determinan utilizando técnicas de filtrado de variable de estado, regresión lineal y mínimos cuadrados recursivos, y los parámetros mecánicos se identifican considerando solo el tiempo de aceleración del sistema. Se realiza un ajuste final del conjunto de parámetros mediante la técnica de mínimos cuadrados no lineales. Basado en el modelo completo del molino, se propone un estimador del par de carga, utilizando filtros de paso alto, y se presenta una estimación de la cantidad de carga del molino. Las simulaciones numéricas del modelo determinado, en diferentes condiciones de operación del molino, muestran una buena aproximación con resultados experimentales. Por lo tanto, la metodología híbrida propuesta, basada tanto en el modelado dinámico como en análisis de señales, presenta potencial para ayudar en el proyecto de procesos de supervisión y control del molino de bolas de acoplamiento fijo.

In this paper, we study the strong consistency and asymptotic normality properties of nonlinear least squares (NLS) estimator of the periodic EXPAR ( 1 ) model. The general statistical literature o...

A new forecasting system—the System of Multigrid Nonlinear Least-squares Four-dimensional Variational (NLS-4DVar) Data Assimilation for Numerical Weather Prediction (SNAP)—was established by building upon the multigrid NLS-4DVar data assimilation scheme, the operational Gridpoint Statistical Interpolation (GSI)-based data-processing and observation operators, and the widely used Weather Research and Forecasting numerical model. Drawing upon lessons learned from the superiority of the operational GSI analysis system, for its various observation operators and the ability to assimilate multiple-source observations, SNAP adopts GSI-based data-processing and observation operator modules to compute the observation innovations. The multigrid NLS-4DVar assimilation framework is used for the analysis, which can adequately correct errors from large to small scales and accelerate iteration solutions. The analysis variables are model state variables, rather than the control variables adopted in the conventional 4DVar system. Currently, we have achieved the assimilation of conventional observations, and we will continue to improve the assimilation of radar and satellite observations in the future. SNAP was evaluated by case evaluation experiments and one-week cycling assimilation experiments. In the case evaluation experiments, two six-hour time windows were established for assimilation experiments and precipitation forecasts were verified against hourly precipitation observations from more than 2400 national observation sites. This showed that SNAP can absorb observations and improve the initial field, thereby improving the precipitation forecast. In the one-week cycling assimilation experiments, six-hourly assimilation cycles were run in one week. SNAP produced slightly lower forecast RMSEs than the GSI 4DEnVar (Four-dimensional Ensemble Variational) as a whole and the threat scores of precipitation forecasts initialized from the analysis of SNAP were higher than those obtained from the analysis of GSI 4DEnVar.

Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

An improved method for decomposition of random and deterministic jitter measurements using tail fitting of the probability density function of total jitter is proposed. The currently employed techniques assume that the tails of the probability density function tend to a Gaussian function. We show that to be inaccurate by deriving analytical expressions for the accurate asymptotic form of the probability density and cumulative distribution functions. In doing so, we prove that the tails approach a Gaussian function multiplied by a term that is inversely proportional to the total jitter. Monte Carlo simulations of jitter consisting of a combination of random and deterministic components are performed and nonlinear least squares fits to the derived asymptotic forms are used to estimate the root mean square of the random component and the bounds of the deterministic component for a few examples.

A measurement system for nondestructive electromagnetic characterization of two-layer absorbers made of a dielectric layer on top of a lossy magnetic layer is presented. The characterization includes simultaneous determination of the complex permittivity of the first layer and the complex permittivity and permeability of the second layer over a broad frequency bandwidth. The measurement system is composed of two flanged open-ended coaxial probes coupled to each other via a two-layer absorber. A full-wave analytical model is worked out to determine the theoretical reflection and transmission coefficients. The material properties of each layer are extracted by solving the inverse problem via nonlinear least squares. Numerical and experimental results for a two-layer absorber are reported and compared with those of standard 7 mm coaxial measurements to validate the proposed approach.

This paper provides an approach to estimation and inference for nonlinear conditional mean panel data models, in the presence of cross‐sectional dependence. We modify Pesaran's (Econometrica, 2006, 74(4), 967–1012) common correlated effects correction to filter out the interactive unobserved multifactor structure. The estimation can be carried out using nonlinear least squares, by augmenting the set of explanatory variables with cross‐sectional averages of both linear and nonlinear terms. We propose pooled and mean group estimators, derive their asymptotic distributions, and show the consistency and asymptotic normality of the coefficients of the model. The features of the proposed estimators are investigated through extensive Monte Carlo experiments. We also present two empirical exercises. The first explores the nonlinear relationship between banks' capital ratios and riskiness. The second estimates the nonlinear effect of national savings on national investment in OECD countries depending on countries' openness.

We extend the notion of a two-part fractional regression model with conditional free disposal hull efficiency responses to accommodate two-stage regression analysis. The two-part regression model includes the binomial model with a nonlinear specification for the expected response in (0,1] and is a more general formulation in the context of fractional regressions. We use nonlinear least squares to assess the effect of covariates in the conditional efficiency response. The approach is applied to Brazilian agricultural county data, as reported in the Brazilian agricultural census of 2006. The efficiency measure is output oriented and assumes variable returns to scale. Output is rural gross income and inputs are land expenses, labor expenses and expenses on other technological inputs. The covariates affecting production are credit, technical assistance, a rural development index, income concentration, measured by the Gini index, and regional dummies. Overall Brazilian rural production performance responds positively to all covariates.

This letter proposes a robust direction-of-arrival (DoA) estimator in the presence of phase noise (PN). For a multi-antenna system with independent oscillators, the received signals will be corrupted by multiple independent PN sources, resulting in severe performance degradation of the DoA estimator, especially for millimeter-wave systems. However, due to the time-varying nature of PN, the number of associated unknowns grows with the size of the snapshots, making the calibration challenging. To tackle this problem, we use a piece-wise linear model to approximate PN so that the number of associated unknowns can be reduced. Then we estimate the DoA and the PN parameters iteratively via a nonlinear least squares estimator. To evaluate the effectiveness of the proposed scheme, the Cramer-Rao lower bound (CRLB) of the DoA estimation in the presence of PN is derived. Numerical results show a significant performance improvement of the proposed DoA estimator compared to the results without calibration.

In this article, we use Newton's method and the nonlinear least squares method to solve a nonlinear problem in electromagnetic compatibility. Both techniques provide a way to identify the temporal profile of a voltage source that would produce a target voltage over a given time interval at a point of the transmission line network presenting nonlinear components. After comparing the results of a test problem, the most efficient method is considered to identify the temporal source in order to bring a software correction to a nonlinear faulty lossy transmission line network. The software correction process may cancel any unintentional perturbations produced by one or more faults regardless of their type and location.

This research aims to predict the end of the COVID-19 pandemic in Indonesia based on parametric growth models. The models are chosen by considering their fitness with the data of Taiwan which is believed to have passed over the peak of the pandemic and have gone through all phases in the growth curves. The models are parameterized using the nonlinear least squares method. The deviation and confidence interval of each parameter is estimated using the k-fold cross-validation and the bootstrap techniques. Using the total cases per million population data from March 2 to June 18, 2020, it was found that two growth models fit the data, i.e. logistic and modified Gompertz, where the latter performs better. Using the information about the deviation of each model parameter, a simulation model is developed to predict the time at which the total cases curve starts to flatten, which is an indication of the end of the pandemic. It was found with 95% confidence level that based on the modified Gompertz model the pandemic will end somewhere between March 9 – September 7, 2021 with total cases per million of 206 - 555. Meanwhile, based on the logistic growth model, the end of the pandemic is between August 28 – September 23, 2020 with total cases per million of 180 - 375. This model can be extended by making comparative scenario with Taiwan based on measures that represent the quality of the pandemic mitigation such as test ratio and the intensity of social restriction.

Radar echoes produced by a target contain the target motion information and physical characteristics. In a stepped frequency (SF) signal, a range-Doppler coupling occurs where the movement of the target affects the target range profile. Using this characteristic, in this paper, an algorithm for estimating the size and rotation vector of the rotating target is proposed. For this, we consider a distributed radar system to utilize target-sensor geometry. By executing signature extraction from the high resolution range profiles (HRRPs) on time-range images and two nonlinear least squares (NLS) estimations, the desired parameters are successfully estimated. Through several simulations, the performance of the proposed method is analyzed and the favorable results are obtained.

A nondestructive evaluation technique for determining complex permittivities and permeabilities of uniaxial anisotropic sheet media is presented. An existing technique, the two-flanged waveguide measurement technique (tFWMT), has demonstrated good results for the nondestructive electromagnetic characterization of isotropic materials. This article extends the tFWMT for uniaxially anisotropic materials and presents the experimental determination of the permittivity and permeability of uniaxially anisotropic media. The measured scattering parameters are compared to theoretical scattering parameters, and the complex permittivity and permeability are extracted using a nonlinear least squares method. To find theoretical scattering parameters, Love’s equivalence principle and the spectral-domain Green’s function are used to form a set of coupled magnetic-field integral equations (MFIEs). This set of coupled MFIEs is solved utilizing the method of moments. To validate the new method, electromagnetic characterization of two honeycomb materials is made by using the two-flanged waveguides measurement technique and the results are compared to those obtained using the established methods.

This paper addresses two fundamental issues in the peak deconvolution method of cellulose XRD data analysis: there is no standard model for amorphous cellulose and common peak functions such as Gauss, Lorentz and Voigt functions do not fit the amorphous profile well. It first examines the effects of ball milling on three types of cellulose and results show that ball milling transforms all samples into a highly amorphous phase exhibiting nearly identical powder X-ray diffraction (XRD) profiles. It is hypothesized that short range order within a glucose unit and between adjacent units survives ball milling and generates the characteristic amorphous XRD profiles. This agrees well with cellulose I d-spacing measurements and oligosaccharide XRD analysis. The amorphous XRD profile is modeled using a Fourier series equation where the coefficients are determined using the nonlinear least squares method. A new peak deconvolution method then is proposed to analyze cellulose XRD data with the amorphous Fourier model function in conjunction with standard Voigt functions representing the crystalline peaks. The impact of background subtraction method has also been assessed. Analysis of several cellulose samples was then performed and compared to the conventional peak deconvolution methods with common peak fitting functions and background subtraction approach. Results suggest that prior peak deconvolution methods overestimate cellulose crystallinity.

Purpose The purpose of this paper is to focus on the performance of three typical nonlinear least-squares estimation algorithms in atmospheric density model calibration. Design/methodology/approach The error of Jacchia-Roberts atmospheric density model is expressed as an objective function about temperature parameters. The estimation of parameter corrections is a typical nonlinear least-squares problem. Three algorithms for nonlinear least-squares problems, Gauss–Newton (G-N), damped Gauss–Newton (damped G-N) and Levenberg–Marquardt (L-M) algorithms, are adopted to estimate temperature parameter corrections of Jacchia-Roberts for model calibration. Findings The results show that G-N algorithm is not convergent at some sampling points. The main reason is the nonlinear relationship between Jacchia-Roberts and its temperature parameters. Damped G-N and L-M algorithms are both convergent at all sampling points. G-N, damped G-N and L-M algorithms reduce the root mean square error of Jacchia-Roberts from 20.4% to 9.3%, 9.4% and 9.4%, respectively. The average iterations of G-N, damped G-N and L-M algorithms are 3.0, 2.8 and 2.9, respectively. Practical implications This study is expected to provide a guidance for the selection of nonlinear least-squares estimation methods in atmospheric density model calibration. Originality/value The study analyses the performance of three typical nonlinear least-squares estimation methods in the calibration of atmospheric density model. The non-convergent phenomenon of G-N algorithm is discovered and explained. Damped G-N and L-M algorithms are more suitable for the nonlinear least-squares problems in model calibration than G-N algorithm and the first two algorithms have slightly fewer iterations.

Many 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 firstorder 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.

We propose and study an iterative method for the solution of a nonlinear least-squares problem with nondifferentiable operator. In this method, instead of the Jacobian, we use the sum of derivative of the differentiable part of the operator and divided difference with specially chosen points of the nondifferentiable part of operator. We prove a theorem substantiating the process of convergence of the proposed method and establish its rate. We also present the results of numerical experiments carried out for the test problems with nondifferentiable operators.

An inverse design approach is proposed to compute wind turbine blade geometries which maximize the aggregate power output from a wind farm. An iterative inverse algorithm is used to solve the optimization problem. The algorithm seeks to minimize the target function, f = − C P , av , where C P , av is the average normalized mechanical power of all the turbines in the wind farm. An upper bound on the blade planform area, representative of the blade weight, is imposed to demonstrate how to incorporate constraints in the design process. The power coefficients ( C P ) of the turbines in the farm are computed by solving the Reynolds Averaged Navier Stokes equations with the turbine rotors modeled as momentum sources using the actuator disk model. The inverse design is carried out using the trust-region-reflective method, which is a nonlinear least squares regression solver. The computation cost is reduced by computing the Jacobian once every few iterations and approximating it using Broyden's method in between. The proposed design approach is first demonstrated to maximize the isolated performance of single- and dual-rotor wind turbines and subsequently used to design the blades for a 3-turbine array and a ten-turbine array in which the downstream turbines operate directly in the wake of the upstream turbines. For a turbine-turbine spacing of four rotor diameters, the farm-optimized blade designs increase the farm power output by over five percent and the optimized blade geometries are found to be considerably different from the blade geometry optimized for isolated turbine operation. As the turbine-turbine spacing is increased to eight rotor diameters, the difference between the blade geometry optimized for farm operation versus that for isolated operation, is reduced.

The classic approach for estimating parameters of a model using historical data is to solve a nonlinear least squares optimization problem using numerical methods. We develop an I-frame methodology to solve the nonlinear least squares problem quickly which can be applied to both offline and online (where data is streamed in real time) parameter estimation. Using the concept of I-frames from imaging and animation, we approximate a solution to the nonlinear least squares problem via a two-step process, an I-frame optimization, and an incremental optimization. The I-frame optimization solves for the parameters using a subset of data points and the incremental optimization adjusts the parameters in between the I-frames. We show that the criterion of generating I-frames can affect the average squared error of the final solution. Our methodology benefits from being scalable as the number of parameters and amount of data increases with an appropriate I-frame generation criterion.