Model Structural Errors Research Articles

A method for temperature sensor and model selection for machine tool thermal error modelling using ANFIS and ANN

AbstractThermal errors account for a significant part of the dimensional errors of components produced by precision machine tools. These errors are commonly compensated using predictions from temperature-based empirical models. The accuracy and robustness of these predictions are affected by the locations of temperature sensors used to obtain the model’s input data. Methods for sensor selection found in literature are often difficult to replicate and automate because they require tuning of several hyperparameters. This work presents a sensor and model selection approach that uses proper orthogonal decomposition (POD) and QR pivoting to select a subset of sensors that have been preinstalled in the machine tool as possible model inputs. These sensors are then sorted according to their correlation with the thermal error being modelled. The final set of inputs and thermal error model structure is chosen using Bayesian Information Criterion (BIC) to limit model overfitting. The approach was tested by modelling the Y-axis thermal error measured from air-cutting experiments performed under different spindle speeds and feed rates. This enabled determination of the structure and the choice inputs for Adaptive Neuro Fuzzy Inference System (ANFIS) and Artificial Neural Network (ANN) thermal error models. The accuracy of these models was compared to that of models trained using inputs selected by conventional approaches: the least absolute shrinkage and selection operator (LASSO), fuzzy c-means clustering (FCM), and principal component regression (PCR). The presented approach had better or comparable results to the conventional approaches while using fewer inputs. The presented approach is also well suited for automation compared to conventional approaches that require expert input.

Modal test and finite element updating of sprayer boom truss

In addressing the finite element model and actual structural error of the sprayer boom truss, this study aims to achieve high-precision dynamic characteristics, enhance simulation credibility, make informed optimization decisions, and reduce testing costs. The research investigates the dynamic behavior of the sprayer boom truss through modal experiments and finite element simulations. Initially, modal parameters of the sprayer boom are obtained through experimental testing, validating their reasonableness and reliability. Subsequently, Ansys Workbench18.0 simulation software was employed to analyze the finite element model of the sprayer boom, revealing a maximum relative error of 11.93% compared to experimental results. To improve accuracy, a kriging-based response surface model was constructed, and multi-objective parameter adjustments using the MOGA algorithm reduce the maximum relative error to 4.6%. Sensitivity analysis further refines the model by optimizing target parameters, resulting in a maximum relative error of 4.96%. These findings demonstrate the effective enhancement of the corrected finite element model’s precision, with the response surface method outperforming sensitivity analysis the maximum relative error between the updated finite element model and experimental results was within the engineering allowable range, confirming the effectiveness of the updated model.

How Important are Commodity Terms of Trade Shocks in Explaining Government Tax Revenue Fluctuations? Evidence from Ethiopia

Using a new database of commodity terms of trade and identifying country-specific commodity export and import price shocks separately, this article mainly aims to examine the importance of commodity export and import price shocks in explaining government tax revenue fluctuations using Ethiopia as an example over the sample period 1980–2019. This article employs a structural vector autoregressive model and forecast error variance decomposition analysis to address this objective. The results show that commodity export and import price shocks account for around 48% of government tax revenue fluctuations. While commodity export price shocks contribute to around 37% of fluctuations in government tax revenue, commodity import price shocks account for only 11% of government tax revenue fluctuations. These findings are discussed in terms of their policy implications. JEL Classification C32, F41, F44

Dynamic force identification considering modeling errors using modal expansion method and relevant vector regression algorithm

Accurate identification and estimation of external forces on launch vehicles, aircraft, and other in-service engineering structures plays a vital role in structural design and health monitoring. Considering structural modeling errors, this paper develops a novel force identification method based on modal expansion and relevant vector regression (RVR). Initially, the incomplete and noisy experimental modal data are used to modify the stiffness and mass matrices of the finite element (FE) model, reducing the impact of modeling errors on force identification accuracy. Subsequently, to account for noise disturbances and potential information about unknown initial conditions in the measured acceleration responses, the external forces and the unknown initial conditions are respectively expanded using trigonometric functions and the mode shapes based on the function fitting technique. On this basis, the force identification equation is established for the updated model. The RVR algorithm is integrated into modal expansion and force identification. Firstly, the mode shapes of the FE model are utilized to expand the incomplete and noisy experimental mode shapes to their full degree of freedom, significantly enhancing the accuracy of the model updating. Secondly, by combining the measured acceleration responses, the base coefficients can be solved sparsely, effectively identifying the dynamic forces even when unknown initial conditions are present.

Developing Parametric and Nonparametric Models for Model-Informed Precision Dosing: A Quality Improvement Effort in Vancomycin for Patients With Obesity.

Both parametric and nonparametric methods have been proposed to support model-informed precision dosing (MIPD). However, which approach leads to better models remains uncertain. Using open-source software, these 2 statistical approaches for model development were compared using the pharmacokinetics of vancomycin in a challenging subpopulation of class 3 obesity. Patients on vancomycin at the University of Vermont Medical Center from November 1, 2021, to February 14, 2023, were entered into the MIPD software. The inclusion criteria were body mass index (BMI) of at least 40 kg/m 2 and 1 or more vancomycin levels. A parametric model was created using nlmixr2/NONMEM, and a nonparametric model was created using Pmetrics. Then, a priori and a posteriori predictions were evaluated using the normalized root mean squared error (nRMSE) for precision and the mean percentage error (MPE) for bias. The parametric model was evaluated in a simulated MIPD context using an external validation dataset. In total, 83 patients were included in the model development, with a median age of 56.6 years (range: 24-89 years), and a median BMI of 46.3 kg/m 2 (range: 40-70.3 kg/m 2 ). Both parametric and nonparametric models were 2-compartmental, with creatinine clearance and fat-free mass as covariates to clearance and volume parameters, respectively. The a priori MPE and nRMSE for the parametric versus nonparametric models were -6.3% versus 2.69% and 27.2% versus 30.7%, respectively. The a posteriori MPE and RMSE were 0.16% and 0.84%, and 13.8% and 13.1%. The parametric model matched or outperformed previously published models on an external validation dataset (n = 576 patients). Minimal differences were found in the model structure and predictive error between the parametric and nonparametric approaches for modeling vancomycin class 3 obesity. However, the parametric model outperformed several other models, suggesting that institution-specific models may improve pharmacokinetics management.

Identification of Continuous-Time Dynamic Systems With Uncertainties Measured by Fuzzy Sets Subject to Model Structure Errors

Identification of Continuous-Time Dynamic Systems With Uncertainties Measured by Fuzzy Sets Subject to Model Structure Errors

Transformer Based Long-Term Prognostics for Dynamic Operating PEM Fuel Cells

Proton Exchange Membrane Fuel Cell (PEMFC) is considered as a clean alternative energy that has been widely used in many fields. However, the large-scale commercialization of PEMFC is still limited by its durability performance. Prognostics and Health Management (PHM) is considered as a helpful solution to improve the durability of PEMFC with degradation prediction and health-based control and maintenance methods. The long-term PEMFC prognostic task, as a primary component in PHM, has received extensive attention from both academia and industry. Recently, the combination of deep neural networks and PEMFC prognostics has expressed a broad research perspective. The deep-learning-based methods, such as Convolutional Neural Network (CNN), Echo State Network (ESN), and Recurrent Neural Network (RNN) family methods, have been well-studied by many researchers and have shown satisfactory performance in degradation prediction. However, the long-term prediction performance is still limited by the model structure and accumulative errors. This paper focuses on the degradation of the PEMFC and a Transformer-based PEMFC prognostic framework is proposed to predict the long-term degradation of the PEMFC system. The Transformer model is applied for the first time to predict the degradation of the PEMFC, and a series-attention mechanism is proposed to replace the self-attention mechanism in the Vanilla Transformer which could improve the health indicator (HI) prediction performance. Finally, the PEMFC dynamic durability test data is utilized to evaluate the performance of the proposed framework in both multi-step-ahead prediction and long-term prognostic conditions, and the experimental results illustrate the feasibility and effectiveness of the proposed method.

Bayesian two-stage structural identification with equivalent formulation and EM algorithm

For structural model identification using a Bayesian two-stage approach, modal properties (e.g., natural frequencies) are first extracted from measured data in Stage I and then used for determining structural properties (e.g., stiffness) in Stage II. With sufficient data that allow the problem to be globally identifiable, the computational problem often reduces to optimizing a ‘measure-of-fit’ function to yield the ‘most probable value’ (MPV) that informs the ‘best estimate’, and determining the Hessian of the function to yield the posterior covariance matrix that informs the remaining uncertainty. Recent developments deal with the presence of model error, and the increasing complexity calls for proper computational strategy. In this spirit, assuming Gaussian model error, this work develops a hypothetical yet mathematically equivalent formulation for the two-stage problem, which facilitates development of effective algorithms for MPV using Expectation-Maximization techniques. By treating hypothetically the MPV of modal properties in Stage I as ‘data’ and model error as latent variables, the Q-function in the M−step can be expressed as a sum of two terms that can be optimized with respect to structural parameters and model error parameters. The optimization of structural parameters reduces to that of a Stage II problem without model error, for which existing algorithms can be applied. Analytical expressions are also derived for computing the posterior covariance matrix. The proposed methodology is investigated with synthetic data for model errors associated with sensor misalignment and model simplification, lab data for the effect of number of modes and measured degrees of freedom (DoFs), and field data for reality tests.

A flexible and efficient knowledge-guided machine learning data assimilation (KGML-DA) framework for agroecosystem prediction in the US Midwest

Process-based models are widely used to predict the agroecosystem dynamics, but such modeled results often contain considerable uncertainty due to the imperfect model structure, biased model parameters, and inaccurate or inaccessible model inputs. Data assimilation (DA) techniques are widely adopted to reduce prediction uncertainty by calibrating model parameters or dynamically updating the model state variables using observations. However, high computational cost, difficulties in mitigating model structural error, and low flexibility in framework development hinder its applications in large-scale agroecosystem predictions. In this study, we addressed these challenges by proposing a novel DA framework that integrates a Knowledge-Guided Machine Learning (KGML)-based surrogate with tensorized ensemble Kalman filter (EnKF) and parallelized particle swarm optimization (PSO) to effectively assimilate historical and in-season multi-source remote sensing data. Specifically, we incorporate knowledge from a process-based model, ecosys, into a Gated Recurrent Unit (GRU)-based hierarchical neural network. The hierarchical architecture of KGML-DA mimics key processes of ecosys and builds a causal relationship between target variables. Using carbon budget quantification in the US Corn-Belt as a context, we evaluated KGML-DA's performance in predicting key processes of the carbon cycle at three agricultural sites (US-Ne1, US-Ne2, US-Ne3), along with county-level (627 counties) and 30-m pixel-level (Champaign County, IL) grain yield. The site experiments show that updating the upstream variable, e.g., gross primary production (GPP), improved the prediction of downstream variables such as ecosystem respiration, net ecosystem exchange, biomass, and leaf area index (LAI), with RMSE reductions ranging from 9.2% to 30.5% for corn and 4.8% to 24.6% for soybean. Uncertainty in downstream variables was automatically constrained after correcting the upstream variables, demonstrating the effectiveness of the causality linkages in the hierarchical surrogate. We found joint use of in-season GPP and evapotranspiration (ET) products along with historical GPP and surveyed yields achieved the best prediction for county-level yields, while assimilating in-season LAI observations benefitted the prediction in extreme years. Uncertainty and error analysis of regional yield estimation demonstrated that KGML-DA could reduce prediction error by 26.5% for corn and 36.2% for soybean. Remarkably, the GPU-based tensor operation design makes this DA framework more than 7000 times faster than the PB model with a High-Performance Computing system, indicating the high potential of the proposed framework for in-season, high-resolution agroecosystem predictions.

Data worth analysis within a model-free data assimilation framework for soil moisture flow

Abstract. Conventional data worth (DW) analysis for soil water problems depends on physical dynamic models. The widespread occurrence of model structural errors and the strong nonlinearity of soil water flow may lead to biased or wrong worth assessment. By introducing the nonparametric data worth analysis (NP-DWA) framework coupled with the ensemble Kalman filter (EnKF), this real-world case study attempts to assess the worth of potential soil moisture observations regarding the reconstruction of fully data-driven soil water flow models prior to data gathering. The DW of real-time soil moisture observations after Gaussian process training and Kalman update was quantified with three representative information metrics, including the trace, Shannon entropy difference and relative entropy. The sequential NP-DWA framework was examined by a number of cases in terms of the variable of interest, spatial location, observation error, and prior data content. Our results indicated that, similarly to the traditional DW analysis based on physical models, the overall increasing trend of the DW from the sequential augmentation of additional observations within the NP-DWA framework was also susceptible to interruptions by localized surges due to never-experienced atmospheric conditions (i.e., rainfall events). The difference is that this biased DW in the former is caused by model structural errors triggered by contrasting scenarios, which is difficult to be compensated for by assimilating more prior data, while this performance degradation in the NP-DWA can be effectively alleviated by enriching training scenarios or the appropriate amplification of observational noise under extreme meteorological conditions. Nevertheless, a substantial expansion of the prior data content may cause an unexpected increase in the DW of future potential observations due to the possible introduction of ensuing observation noises. Hence, high-quality and representative small data may be a better choice than unfiltered big data. Compared with the observations in the surface layer with the strongest time variability, the soil water content in the middle layer robustly exhibited remarkable superiority in the construction of model-free soil moisture models. We also demonstrated that the DW assessment performance was jointly determined by 3C, i.e., the capacity of potential observation realizations to capture actual observations, the correlation of potential observations with the variables of interest and the choice of DW indicators. Direct mapping from regular meteorological data to soil water content within the NP-DWA mitigated the adverse effects of nonlinearity-related interference, which thus facilitated the identification of the soil moisture covariance matrix, especially the cross-covariance.

Evolving Neuro-Fuzzy Systems-Based Design of Experiments in Process Identification

This paper presents a new design of experiment approach based on an evolving neuro-fuzzy model. The input of the process is proposed by a space-filling method that uses a sequence of step functions to maximise the coverage of the antecedent clusters in the input space of the evolving model, which is called the heuristic step sequence (HSS). When the system reaches a steady state, a new rule is added, existing rules are merged based on the similarity of the consequent transfer functions, and a new optimal excitation is calculated. The output error model structure was chosen because it assumes a noise distribution common to real processes and is suitable for identification, while the step function input signal is one of the most commonly used signals in identification and control. A filtered recursive least squares method is used to identify the consequent parameters of the output error models and the optimisation filter is adapted based on the the confidence interval of the local model. Evaluations were performed on a Hammerstein type model comparing the HSS method with a staircase excitation and on a real plate heat-exchanger pilot plant. The experiments show that the proposed HSS method outperforms the staircase excitation and can be used to identify nonlinear dynamical systems of Hammerstein-Wiener type.

Synthesizing forecasts to inform decision‐making and advance ecological theory

Synthesizing forecasts to inform decision‐making and advance ecological theory

From Inertial Measurements to Smart Cities [About This Issue

Autonomous vehicle technology is advancing rapidly. Their control capabilities often rely on high-bandwidth state estimation incorporating inertial measurements. High-performance state estimation incorporates inertial measurement error models through the process of state augmentation to enable on-the-fly instrument calibration. The feature article “Inertial Measurement Unit Error Modeling Tutorial” by Jay A. Farrell, Felipe O. Silva, Farzana Rahman, and J. Wendel provides a tutorial describing the process and issues related to developing a state-space model for the stochastic errors affecting an inertial measurement unit (IMU). The starting point is the instrument error characterization data sheet provided by the manufacturer, which is typically either an Allan standard deviation graph or the Allan variance parameters extracted from that graph. The desired output of the modeling process is a linear, discrete-time, state-space model of the IMU stochastic errors suitable for augmentation to the INS error-state model. Along with this tutorial, supplementary open source software is available. One software component does the following: 1) Given a continuous-time state-space IMU stochastic error model selected by the designer to match the Allan variance, the software computes a discrete-time equivalent state-space model. 2) Given that discrete-time model, it produces a stochastic error sequence suitable for Allan variance computations. 3) Given a sequence of stochastic errors, it computes and plots the Allan variance. Given Allan variance data and a specific continuous-time state-space IMU stochastic error model structure, a second software component implements an optimization-based approach for selecting the model parameters to match the Allan variance data.

Issues in calibrating models with multiple unbalanced constraints: the significance of systematic model and data errors

Abstract Calibrating process‐based models using multiple constraints often improves the identifiability of model parameters, helps to avoid several errors compensating each other and produces model predictions that are more consistent with underlying processes. However, using multiple constraints can lead to predictions for some variables getting worse. This is particularly common when combining data sources with very different sample sizes. Such unbalanced model‐data fusion efforts are becoming increasingly common, for example when combining manual and automated measurements. Here we use a series of simulated virtual data experiments that aim to demonstrate and disentangle the underlying cause of issues that can occur when calibrating models with multiple unbalanced constraints in combination with systematic errors in models and data. We propose a diagnostic tool to help identify whether a calibration is failing due to these factors. We also test the utility of adding terms representing uncertainty in systematic model/data systematic error in calibrations. We show that unbalanced data by itself is not the problem—when fitting simulated data to the ‘true’ model, we can correctly recover model parameters and the true dynamics of latent variables. However, when there are systematic errors in the model or the data, we cannot recover the correct parameters. Consequently, the modelled dynamics of the low data volume variables departs significantly from the true values. We demonstrate the utility of the diagnostic tool and show that it can also be used to identify the extent of the imbalance before the calibration starts to ignore the more sparse data. Finally, we show that representing uncertainty in model structural errors and data biases in the calibration can greatly improve the model fit to low‐volume data, and improve coverage of uncertainty estimates. We conclude that the underlying issue is not one of sample size or information content per se, despite the popularity of ad hoc approaches that focus on ‘weighting’ datasets to achieve balance. Our results emphasize the importance of considering model structural deficiencies and data systematic biases in the calibration of process‐based models.

A robust objective function for calibration of groundwater models in light of deficiencies of model structure and observations

• Need for robust objective functions in model calibration and parameter estimation. • Common squared error_based criteria have issues with outliers and extreme values. • Suggestion: robust objective function based on CRPS as heuristic norm. • Comparison of CRPS with other performance criteria in groundwater model calibration. Groundwater models require parameter optimization based on the minimization of objective functions describing, for example, the residual between observed and simulated groundwater head. At larger scales, constraining these models requires large datasets of groundwater head observations. These observations are typically only available from databases comprised of varying quality data from a variety of sources and will be associated with unknown observational uncertainty. At the same time the model structure, especially the hydrogeological description, will inevitably be a simplification of the complex natural system. As a result, calibration of groundwater models often results in parameter compensation for model structural deficiency, or can be affected by observation errors. This problem can be amplified by the application of common squared error-based performance criteria, which are most sensitive to the largest errors. In our context, we assume that the residuals that remain large during the optimization process likely do so because of either model structural error or observation error. Based on this assumption it is desirable to design an objective function that is less sensitive to these large residuals of low probability, and instead favours the majority of observations that can fit the given model structure and likely are free of large observation errors. We suggest a Continuous Ranked Probability Score (CRPS) based objective function that limits the influence of large residuals in the optimization process as the metric puts more emphasis on the position of the residual along the cumulative distribution function than on the magnitude of the residual. The CRPS-based objective function was applied in the calibration of regional-scale coupled surface-groundwater models and compared to conventional objective functions based on mean of squared, absolute and root errors, using synthetic as well as real observations. The optimization tests illustrated that the novel CRPS-based objective function successfully limited the dominance of large residuals in the optimization process and consistently reduced overall bias.

Estimating rainfall–runoff modeling using the rainfall prognostic model-based artificial framework with a well-ordered selective genetic algorithm

Abstract Rainfall–runoff modeling is one of the most well-known applications of hydrology. The goal of rainfall–runoff modeling is to simulate the peak river flow caused by an actual or hypothetical rainfall force. In existing methods, the rainfall–runoff relationships are quantified to predict the daily streamflow of each catchment from its landscape attributes to measure the daily rainfall. However, the structural model error, infiltration rate, and the steep slopes of the hill affect the prediction process. To tackle these issues, this paper proposed a novel rainfall prognostic model-based artificial framework, which predicts day-to-day rainfall to prevent environmental disasters. The day-to-day predictions minimize the risks to life and property and also manage the agricultural farms in a better way because the possibility of rainfall has been estimated earlier. Furthermore, the posterior fire-breathing network is utilized to estimate model errors in the computational runoff by using time-dependent and random noise to the model's internal storage to solve the uncertainty problem. Since the model errors are estimated, there are limits to the infiltration rate and thus a prophetic multilayer network is utilized which relies on the soil runoff levels. Moreover, the network measures the dynamics of soil moisture to regulate the infiltration rate according to the rural or urban section. Moreover, to measure the surface water from the steep slopes, the system offered a well-ordered selective genetic algorithm to calculate the velocity of runoff in different bend areas to overcome the numerical problem. Thus, the model results showed that the work effectively predicts the rainfall from the investigation of model errors, infiltration rates, and velocity to achieve a better prediction range in the rainfall.

Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case.

Models of many engineering and natural systems are imperfect. The discrepancy between the mathematical representations of a true physical system and its imperfect model is called the model error. These model errors can lead to substantial differences between the numerical solutions of the model and the state of the system, particularly in those involving nonlinear, multi-scale phenomena. Thus, there is increasing interest in reducing model errors, particularly by leveraging the rapidly growing observational data to understand their physics and sources. Here, we introduce a framework named MEDIDA: Model Error Discovery with Interpretability and Data Assimilation. MEDIDA only requires a working numerical solver of the model and a small number of noise-free or noisy sporadic observations of the system. In MEDIDA, first, the model error is estimated from differences between the observed states and model-predicted states (the latter are obtained from a number of one-time-step numerical integrations from the previous observed states). If observations are noisy, a data assimilation technique, such as the ensemble Kalman filter, is employed to provide the analysis state of the system, which is then used to estimate the model error. Finally, an equation-discovery technique, here the relevance vector machine, a sparsity-promoting Bayesian method, is used to identify an interpretable, parsimonious, and closed-form representation of the model error. Using the chaotic Kuramoto-Sivashinsky system as the test case, we demonstrate the excellent performance of MEDIDA in discovering different types of structural/parametric model errors, representing different types of missing physics, using noise-free and noisy observations.

An ineffective antidote for hawkmoths

In recent publications we have drawn attention to the fact that if the dynamics of a model is structurally unstable, then the presence of structural model error places in-principle limits on the model’s ability to generate decision-relevant probability forecasts. Writing with a varying array of co-authors, Eric Winsberg has now produced at least four publications in which he dismisses our points as unfounded; the most recent of these appeared in this journal. In this paper we respond to the arguments of Winsberg and his co-workers, and we point out that their criticisms fail. We take this as an opportunity to restate and explain our arguments, and to point to fruitful directions for future research.

Influence of plant ecophysiology on ozone dry deposition: comparing between multiplicative and photosynthesis-based dry deposition schemes and their responses to rising CO<sub>2</sub> level

Abstract. Dry deposition is a key process for surface ozone (O3) removal. Stomatal uptake is a major component of O3 dry deposition, which is parameterized differently in current land surface models and chemical transport models. We developed and used a standalone terrestrial biosphere model, driven by a unified set of prescribed meteorology, to evaluate two widely used dry deposition modeling frameworks, Wesely (1989) and Zhang et al. (2003), with different configurations of stomatal resistance: (1) the default multiplicative method in the Wesely scheme (W89) and Zhang et al. (2003) scheme (Z03), (2) the traditional photosynthesis-based Farquhar–Ball–Berry (FBB) stomatal algorithm, and (3) the Medlyn stomatal algorithm (MED) based on optimization theory. We found that using the FBB stomatal approach that captures ecophysiological responses to environmental factors, especially to water stress, can generally improve the simulated dry deposition velocities compared with multiplicative schemes. The MED stomatal approach produces higher stomatal conductance than FBB and is likely to overestimate dry deposition velocities for major vegetation types, but its performance is greatly improved when spatially varying slope parameters based on annual mean precipitation are used. Large discrepancies were also found in stomatal responses to rising CO2 levels from 390 to 550 ppm: the multiplicative stomatal method with an empirical CO2 response function produces reduction (−35 %) in global stomatal conductance on average much larger than that with the photosynthesis-based stomatal method (−14 %–19 %). Our results show the potential biases in O3 sink caused by errors in model structure especially in the Wesely dry deposition scheme and the importance of using photosynthesis-based representation of stomatal resistance in dry deposition schemes under a changing climate and rising CO2 concentration.

Varying coefficient single-index regression model with missing responses under rank-based modelling

A varying coefficients single-index regression model with responses missing at random is considered. Rank-based estimators of the index coefficient and the functional coefficients are studied, and their asymptotic properties (consistency and asymptotic normality) are established under mild conditions. To demonstrate the performance of the proposed approach, Monte Carlo simulation experiments are carried out and show that the proposed approach provides robust and more efficient estimators compared to its least-squares counterpart. This is demonstrated under different model error structures, including the standard normal, the t and the contaminated model error distributions. Finally, a real data example is given to illustrate our proposed method.