Errors In Parameters Research Articles

Effect of Increasing Sample Size on Multi-Collinearity in Multilevel Non-Linear Model

A multilevel logistic regression model demonstrating high correlations among predictor variables is susceptible to multi-collinearity. Multi-collinearity significantly impacts the robustness and interpretability of multilevel non-linear models. In multilevel non-linear models, the effects of multi-collinearity can be amplified, leading to distorted parameter estimates and inflated standard errors. This phenomenon contributes to an escalation in the variances of parameter estimates, thereby resulting in inaccurate inferences regarding the relationships between the response and explanatory factors. The primary objective of this study is to investigate the impact of multi-collinearity on multilevel non-linear models. The research aims to assess whether the quantity of independent variables influences the Multilevel Variance Inflation Factor and to explore the effect of altering the correlation degree at one level on multi-collinearity within a multilevel non-linear model. Additionally, the research seeks to determine how multi-collinearity affects the standard errors of parameters in a multilevel non-linear model. In a 2-level logistic regression, a binary variable was the dependent variable, while pre-established standard variables functioned as regressors. The Monte Carlo analysis incorporated three distinct correlation strengths (0.2, 0.5, and 0.9) and sample sizes (500, 100, and 30). The Multilevel Variance Inflation Factor was employed for multi-collinearity diagnosis. The outcomes revealed that, within the logistic multilevel regression model, an increase in sample size correlated with a reduction in multi-collinearity. Notably, the influence of multi-collinearity on standard errors in a multilevel non-linear model was more pronounced. It was observed that increasing the sample size remains an effective strategy to mitigate multi-collinearity errors in a multilevel non-linear model. This approach is particularly crucial due to the reliance on maximum likelihood estimation in logistic regression, as opposed to ordinary least squares (OLS) regression, which contrasts with the methodology of OLS regression.

Accurate superposition modeling method of grinding-shot peening surfaces

Abstract The coexistence of initial surface texture features and shot peening texture features in the final surface topography is a significant feature of shot peening. After the initial surface topography is strengthened by shot peening, the wave peak will produce more substantial deformation relative to the wave valley, which has not been considered in the existing superposition modeling methods. And this will affect the correlation study between surface topography and performance. Taking the surface generated by the grinding-shot peening process as the research object, based on the feature extraction of measured topography, the concept of ‘Shot peening texture depth distribution surface’ is put forward. And the proposed precise superposition method uses the texture depth distribution surface to optimize the characteristic baseline of shot peening based on the existing theory. The measured grinding-shot peening surfaces verify the given method. It is found that the topographical and spatial features of the surfaces superimposed by this method are consistent with those of the measured surfaces, and the error of roughness parameters defined by ISO25178 are all less than 8%. Among them, the percentage errors of the critical parameters Spd, Sdq, and Sdr related to friction and lubrication are significantly reduced to 5%.

Learning a Tracking Controller for Rolling μbots.

Micron-scale robots (μbots) have recently shown great promise for emerging medical applications. Accurate control of μbots, while critical to their successful deployment, is challenging. In this work, we consider the problem of tracking a reference trajectory using a μbot in the presence of disturbances and uncertainty. The disturbances primarily come from Brownian motion and other environmental phenomena, while the uncertainty originates from errors in the model parameters. We model the μbot as an uncertain unicycle that is controlled by a global magnetic field. To compensate for disturbances and uncertainties, we develop a nonlinear mismatch controller. We define the model mismatch error as the difference between our model's predicted velocity and the actual velocity of the μbot. We employ a Gaussian Process to learn the model mismatch error as a function of the applied control input. Then we use a least-squares minimization to select a control action that minimizes the difference between the actual velocity of the μbot and a reference velocity. We demonstrate the online performance of our joint learning and control algorithm in simulation, where our approach accurately learns the model mismatch and improves tracking performance. We also validate our approach in an experiment and show that certain error metrics are reduced by up to 40%.

Adaptive Compensation Tracking Control for Parallel Robots Actuated by Pneumatic Artificial Muscles With Error Constraints

As pneumatic artificial muscles (PAMs) are similar to biological muscles in structure and movement mechanisms, parallel robots actuated by PAMs have development prospects in rehabilitation and industry, with advantages such as compliance, high safety, strong bearing capacity, and satisfactory dynamic performance. However, the parameter uncertainties and model complexity related to inherent characteristics of parallel robots actuated by PAMs (e.g., time-varying, coupling, hysteresis, creep, and high nonlinearity), bring challenges to accurate dynamic modeling and controller design. Therefore, to achieve satisfactory tracking performance, this paper presents an adaptive compensation tracking controller with error constraints for parallel robots actuated by PAMs. The proposed controller deals with parameter uncertainties by estimating system parameters to ensure accurate tracking, which is indicated as an effective solution for a combination of PAMs and parallel robots. Furthermore, using desired trajectory signals in the complicated regression matrix, the online computational burden is significantly reduced. Moreover, to improve operation safety further, an auxiliary term with a theoretical demonstration guarantees that the tracking errors are maintained within allowable ranges. Then, the closed-loop stability is demonstrated by Lyapunov techniques. As far as we know, it is the first time that the challenges of parameter uncertainties, computational burdens, and error constraints of parallel robots actuated by PAMs are simultaneously addressed, which has both theoretical significance and practical value. Finally, the hardware experiments are implemented under different scenarios, and the results indicate that the proposed method achieves satisfactory tracking performance.

Global Stabilization of Antipodal Points on $n$-Sphere With Application to Attitude Tracking

Existing approaches to robust global asymptotic stabilization of a pair of antipodal points on unit $n$-sphere $\mathbb{S}^n$ typically involve the non-centrally synergistic hybrid controllers for attitude tracking on unit quaternion space. However, when switching faults occur due to parameter errors, the non-centrally synergistic property can lead to the unwinding problem or in some cases, destabilize the desired set. In this work, a hybrid controller is first proposed based on a novel centrally synergistic family of potential functions on $\mathbb{S}^n$, which is generated from a basic potential function through angular warping. The synergistic parameter can be explicitly expressed if the warping angle has a positive lower bound at the undesired critical points of the family. Next, the proposed approach induces a new quaternion-based controller for global attitude tracking. It has three advantageous features over existing synergistic designs: 1) it is consistent, i.e., free from the ambiguity of unit quaternion representation; 2) it is switching-fault-tolerant, i.e., the desired closed-loop equilibria remain asymptotically stable even when the switching mechanism does not work; 3) it relaxes the assumption on the parameter of the basic potential function in literature. Comprehensive simulation confirms the high robustness of the proposed centrally synergistic approach compared with existing non-centrally synergistic approaches.

Nearly Optimal Control for Mixed Zero-Sum Game Based on Off-Policy Integral Reinforcement Learning.

In this article, we solve a class of mixed zero-sum game with unknown dynamic information of nonlinear system. A policy iterative algorithm that adopts integral reinforcement learning (IRL), which does not depend on system information, is proposed to obtain the optimal control of competitor and collaborators. An adaptive update law that combines critic-actor structure with experience replay is proposed. The actor function not only approximates optimal control of every player but also estimates auxiliary control, which does not participate in the actual control process and only exists in theory. The parameters of the actor-critic structure are simultaneously updated. Then, it is proven that the parameter errors of the polynomial approximation are uniformly ultimately bounded. Finally, the effectiveness of the proposed algorithm is verified by two given simulations.

Construction and optimization of representative actual driving cycles based on the improved autoencoder

In this study, much work has been performed to accurately and efficiently develop representative actual driving cycles. Electric vehicle road tests were conducted and the associated data were gathered based on the manual driving method, and the Changsha Driving Cycle Construction (CS-DCC) method was proposed to achieve systematical construction of a representative driving cycle from the original data. The results show that the refined data exhibit greater stability and a smoother pattern in contrast to the original data after noise reduction by five-scale wavelet analysis. The Gaussian Kernel Principal Component Analysis (KPCA) algorithm is chosen to reduce the dimensionality of the characteristic matrix, and the number of principal components is selected as 5 with a cumulative contribution rate of 85.99%. The average error of the characteristic parameters between the optimized drive cycle and the total data is further reduced from 13.6 to 6.1%, with a reduction ratio of 55.1%. Meanwhile, the constructed driving cycle has prominent local characteristics compared with four standard driving cycles, demonstrating the necessity of constructing an actual driving cycle that reflects localized driving patterns. The findings present a powerful application of artificial intelligence in advancing engineering technologies.

Research on Textile Dyeing Formulation Based on Young's Double-Slit Interference Experiment Optimization Algorithm

With the development of intelligent algorithms, computer color matching has become an important method to improve product quality and save labor cost in printing and dyeing industry. In this paper, based on the K-M optical theory and the color difference theory of CIELAB uniform color space, and combined with the latest heuristic optimization algorithm "Young's Double-Slit Interference Experiment Optimizer (YDSE)" proposed in 2023, a feasible solution to generate textile dyeing formulas quickly is proposed. The model is tested with 12 textile target samples, and the experiments show that the proposed dyeing formulation model can converge quickly, and the color parameters of the generated scheme are closer to those of the real samples, with the maximum parameter error not exceeding 10-2 orders of magnitude, and the predicted color differences of the optimal scheme are all in the range of 10-6 orders of magnitude or less. Simultaneous testing of the sample formulations using three other common optimization algorithms, WOA, HHO, and IHHO, shows that their predicted color differences are greater than those of YDSE, proving the reliability of the model.

Reliability of assessing skeletal muscle architecture and tissue organization of the gastrocnemius medialis and vastus lateralis muscle using ultrasound and spatial frequency analysis.

The purpose of this study was to investigate inter- and intra-rater reliability as well as the inter-rater interpretation error of ultrasound measurements assessing skeletal muscle architecture and tissue organization of the gastrocnemius medialis (GM) and vastus lateralis (VL) muscle. The GM and VL of 13 healthy adults (22 ± 3 years) were examined thrice with sagittal B-mode ultrasound: intraday test-retest examination by one investigator (intra-rater) and separate examinations by two investigators (inter-rater). Additionally, images from one investigator were analysed by two interpretators (interpretation error). Muscle architecture was assessed by muscle thickness [MT], fascicle length [FL], as well as superior and inferior pennation angle [PA]. Muscle tissue organization was determined by spatial frequency analysis (SFA: peak spatial frequency radius, peak -6 dB width, PSFR/P6, normalized peak value of amplitude spectrum [Amax], power within peak [PWP], peak power percent). Reliability of ultrasound examination and image interpretation are presented as intraclass correlation coefficient (ICC), test-retest variability, standard error of measurement as well as bias and limits of agreement. GM and VL demonstrated excellent ICCs for inter- and intra-rater reliability, along with excellent ICCs for interpretation error of MT (0.91-0.99), showing minimal variability (<5%) and SEM% (<5%). Systematic bias for MT was less than 1 mm. For PA and FL poor to good ICCs for inter- and intra-rater reliability were revealed (0.41-0.90), with moderate variability (<12%), low SEM% (<10%) and systematic bias between 0.1-1.4°. Tissue organization analysis indicated moderate to good ICCs for inter- and intra-rater reliability. Notably, Amax and PWP consistently held the highest ICC values (0.77-0.87) across all analyses but with higher variability (<24%) and SEM% (<18%), compared to lower variability (<9%) and SEM% (<8%) in other tissue organization parameters. Interpretation error of all muscle tissue organization parameters showed excellent ICCs (0.96-0.999) with very low variability (≤1%) and SEM% (<2%), except Amax & PWP (TRV%: <6%; SEM%: <7%). Our findings demonstrated excellent inter- and intra-rater reliability for MT. However, agreement for PA, FL, and SFA parameters was not as strong. Additionally, MT and all SFA parameters exhibited excellent agreement for inter-rater interpretation error. Therefore, the SFA seems to offer the possibility of objectively and reliably evaluating ultrasound images.

MPMC-frame: Multiplatform migration control framework for manipulator control

Existing robot control models suffer from poor generalization performance due to varied tasks between manipulators, configuration differences, and physical limits of motion. To address this problem, a multiplatform migration control framework (MPMC-frame) based on a constrained dynamics model is proposed in this paper. First, a model of the robotic manipulator dynamics with configuration adjustment (CA) is constructed based on a neural network to unify the physical joint data of different manipulators. Second, the model-based controller that can be integrated into the framework is designed, and the constraint on the upper bound on the error of uncertain parameters in the control law to guarantee the control robustness and accuracy of the controller for different manipulator platforms. Final, the generic potential function is designed based on multiplatform task requirements, and a task parameter table is constructed to improve the joint motion control performance of MPMC-frame. The feasibility of the method proposed in this paper are verified through simulations and experiments based on the ABB_irb120 and AUBO_i5 robotic manipulators.

A Fast Self-Calibration Method for Dual-Axis Rotational Inertial Navigation Systems Based on Invariant Errors.

In order to ensure that dual-axis rotational inertial navigation systems (RINSs) maintain a high level of accuracy over the long term, there is a demand for periodic calibration during their service life. Traditional calibration methods for inertial measurement units (IMUs) involve removing the IMU from the equipment, which is a laborious and time-consuming process. Reinstalling the IMU after calibration may introduce new installation errors. This paper focuses on dual-axis rotational inertial navigation systems and presents a system-level self-calibration method based on invariant errors, enabling high-precision automated calibration without the need for equipment disassembly. First, navigation parameter errors in the inertial frame are expressed as invariant errors. This allows the corresponding error models to estimate initial attitude even more rapidly and accurately in cases of extreme misalignment, eliminating the need for coarse alignment. Next, by utilizing the output of a gimbal mechanism, angular velocity constraint equations are established, and the backtracking navigation is introduced to reuse sensor data, thereby reducing the calibration time. Finally, a rotation scheme for the IMU is designed to ensure that all errors are observable. The observability of the system is analyzed based on a piecewise constant system method and singular value decomposition (SVD) observability analysis. The simulation and experimental results demonstrate that this method can effectively estimate IMU errors and installation errors related to the rotation axis within 12 min, and the estimated error is less than 4%. After using this method to compensate for the calibration error, the velocity and position accuracies of a RINS are significantly improved.

Generic characterization method for nano-gratings using deep-neural-network-assisted ellipsometry

Abstract As a non-destructive and rapid technique, optical scatterometry has gained widespread use in the measurement of film thickness and optical constants. The recent advances in deep learning have presented new and powerful approaches to the resolution of inverse scattering problems. However, the application of deep-neural-network-assisted optical scatterometry for nanostructures still faces significant challenges, including poor stability, limited functionalities, and high equipment requirements. In this paper, a novel characterization method is proposed, which employs deep-neural-network-assisted ellipsometry to address these challenges. The method processes ellipsometric angles, which are measured by basic ellipsometers, as functional signals. A comprehensive model is developed to profile nano-gratings fabricated by diverse techniques, by incorporating rounded corners, residual layers, and optical constants into an existing model. The stability of the model is enhanced by implementing several measures, including multiple sets of initial values and azimuth-resolved measurements. A simple compensation algorithm is also introduced to improve accuracy without compromising efficiency. Experimental results demonstrate that the proposed method can rapidly and accurately characterize nano-gratings fabricated by various methods, with relative errors of both geometric and optical parameters well controlled under 5 %. Thus, the method holds great promise to serve as an alternative to conventional characterization techniques for in-situ measurement.

Changes in Corneal Higher-Order Aberrations and Ocular Biometric Measurements after Phacoemulsification Combined with Goniosynechialysis in Primary Angle Closure/Glaucoma Patients.

To compare corneal higher-order aberrations (HOAs), refractive error, and ocular biological parameters before and after phacoemulsification combined with goniosynechialysis (Phaco-GSL) in primary angle closure/glaucoma (PAC/PACG) patients with different axial lengths (ALs). In this prospective study, cataract patients diagnosed with PAC/PACG were categorized into two groups based on their ALs: the short AL group (AL ≤ 22.5 mm) and the normal AL group (22.5 < AL ≤ 24.5 mm). The pre- and postsurgery measurements of intraocular pressure (IOP) and best-corrected visual acuity (BCVA) were conducted at 1 day, 1 week, 1 month, 3 months, 6 months, and 12 months. Additionally, the assessments included corneal HOAs, the number of antiglaucoma medications, visual field parameters, manifest refraction, and other ocular biological parameters before surgery and at the final follow-up. Prior to surgery, the two groups exhibited no significant differences, except for AL, curvature value, and Z (4, 0) of the posterior corneal surface (all P < 0.01). Following surgery, BCVA improved, and IOP decreased significantly in both groups (P < 0.01). Both anterior and total corneal HOAs, along with Z (3, -3), increased in the two groups (all P < 0.05), with the normal AL group exhibiting a significantly greater increase in total cornea Z (3, -3) than the short AL group (P=0.047). The normal AL group also exhibited a slight tendency towards hyperopia (P < 0.01). Significant changes were observed in the visual field index and mean deviation in both groups (P < 0.05). Phaco-GSL resulted in an increased corneal HOAs, particularly trefoil, with variations based on the patient's AL. Patients with normal ALs tended to shift towards hyperopia after surgery.

Comparison of Multiple Machine Learning Methods for Correcting Groundwater Levels Predicted by Physics-Based Models

Accurate groundwater level (GWL) prediction is crucial in groundwater resource management. Currently, it relies mainly on physics-based models for prediction and quantitative analysis. However, physics-based models used for prediction often have errors in structure, parameters, and data, resulting in inaccurate GWL predictions. In this study, machine learning algorithms were used to correct the prediction errors of physics-based models. First, a MODFLOW groundwater flow model was created for the Hutuo River alluvial fan in the North China Plain. Then, using the observed GWLs from 10 monitoring wells located in the upper, middle, and lower parts of the alluvial fan as the test standard, three algorithms—random forest (RF), extreme gradient boosting (XGBoost), and long short-term memory (LSTM)—were compared for their abilities to correct MODFLOW’s predicted GWLs of these 10 wells under two sets of feature variables. The results show that the RF and XGBoost algorithms are not suitable for correcting predicted GWLs that exhibit continuous rising or falling trends, but the LSTM algorithm has the ability to correct them. During the prediction period, the LSTM2 model, which incorporates additional source–sink feature variables based on MODFLOW’s predicted GWLs, can improve the Pearson correlation coefficient (PR) for 80% of wells, with a maximum increase of 1.26 and a minimum increase of 0.02, and can reduce the root mean square error (RMSE) for 100% of the wells with a maximum decrease of 1.59 m and a minimum decrease of 0.17 m. And it also outperforms the MODFLOW model in capturing the long-term trends and short-term seasonal fluctuations of GWLs. However, the correction effect of the LSTM1 model (using only MODFLOW’s predicted GWLs as a feature variable) is inferior to that of the LSTM2 model, indicating that multiple feature variables are superior to a single feature variable. Temporally and spatially, the greater the prediction error of the MODFLOW model, the larger the correction magnitude of the LSTM2 model.

Research and application of a novel selective stacking ensemble model based on error compensation and parameter optimization for AQI prediction

With the ongoing process of industrialization, the issue of declining air quality is increasingly becoming a critical concern. Accurate prediction of the Air Quality Index (AQI), considered as an all-inclusive measure representing the extent of pollutants present in the atmosphere, is of paramount importance. This study introduces a novel methodology that combines stacking ensemble and error correction to improve AQI prediction. Additionally, the reptile search algorithm (RSA) is employed for optimizing model parameters. In this study, four distinct regional AQI data containing a collection of 34864 data samples are collected. Initially, we perform cross-validation on ten commonly used single models to obtain prediction results. Then, based on evaluation indices, five models are selected for ensemble. The results of the study show that the model proposed in this paper achieves an improvement of around 10% in terms of accuracy when compared to the conventional model. Thus, the model introduced in this study offers a more scientifically grounded approach in tackling air pollution.

Stepwise Identification Method of Thermal Load for Box Structure Based on Deep Learning.

Accurate and rapid thermal load identification based on limited measurement points is crucial for spacecraft on-orbit monitoring. This study proposes a stepwise identification method based on deep learning for identifying structural thermal loads that efficiently map the local responses and overall thermal load of a box structure. To determine the location and magnitude of the thermal load accurately, the proposed method segments a structure into several subregions and applies a cascade of deep learning models to gradually reduce the solution domain. The generalization ability of the model is significantly enhanced by the inclusion of boundary conditions in the deep learning models. In this study, a large simulated dataset was generated by varying the load application position and intensity for each sample. The input variables encompass a small set of structural displacements, while the outputs include parameters related to the thermal load, such as the position and magnitude of the load. Ablation experiments are conducted to validate the effectiveness of this approach. The results show that this method reduces the identification error of the thermal load parameters by more than 45% compared with a single deep learning network. The proposed method holds promise for optimizing the design and analysis of spacecraft structures, contributing to improved performance and reliability in future space missions.

Observed Fisher information for radar clutter modeled with sub-Gaussian α-stable distribution

It is well known that the clutter in such applications as synthetic aperture radar (SAR) follows heavy-tailed distributions. In such applications, knowledge of the exact statistical properties and/or errors in estimation of the clutter parameters plays an important role in successful implementation of the related applications like target tracking using SAR. In this study, the observed Fisher information (OFI) matrix will be derived for the received clutter assuming that the statistical behavior of clutter follows a sub-Gaussian α-stable (SGS) distribution along with a density function for an SGS distribution based on a newly proposed expansion. The new expansion is characterized with high speed computation in one hand, and avoids issues related to numerical computation of SGS density function, on the other. The minimum standard error of the clutter parameters, then, will be derived as the OFI matrix is inversely related to the Cramer-Rao lower bound (CRLB) of the clutter's estimated parameters. In order to compute the density function and OFI of the SGS distribution, the performance of the proposed methodology has been tested through numerical simulation where the implemented computer code of the proposed numerical method has been demonstrated to be fast and accurate. The R code is available at https://github.com/mahditeimouri/Sub-Gaussian/blob/main/OFI-sub-Gaussian.r.

Optimization design of load-sharing and light weight of the twin rotors concentric face gear power-split transmission system based on gear tooth matching condition

To improve the dynamic load-sharing performance and achieve lightweight, the twin rotors concentric face gear power-split transmission system (TRFGPSTS) was optimized. The gear tooth matching conditions of the system based on the design characteristics of power flow closed-loop and virtual constraint configuration of the system are established, and the basic parameters of each gear meet the requirements of the system load-sharing layout and synchronous meshing of each branch movement are calculated. Based on the lumped mass parameter method, combined with the calculation results of gear tooth matching, the 45-DOF bending torsional coupling dynamic model of the system was established considering time-varying meshing stiffness, support stiffness, manufacturing and installation errors, and other factors, and the influence of each factor on the dynamic load-sharing performance of the system was analyzed. Combined with the influence law of various factors, a multi-objective dynamic load-sharing optimization design model was established with the best dynamic load-sharing performance and the minimum system quality as the objective function. Through the weighted combination method, the global optimal solution of the system optimization design was determined, and the influence of each parameter on the optimization objective function was analyzed. The correctness of the results was verified by using the genetic algorithm based on Pareto frontier distribution. The results show that the dynamic load-sharing characteristics of the system become worse with the increase of error and torsional stiffness, and are more sensitive to the influence of various error parameters. The optimization process focuses on the geometric parameters such as the Module, the teeth Number, the Pressure angle, and the Tooth width of each gear, which can reflect the degree of influence on the dynamic load-sharing and lightweight of the system. After optimization, the dynamic load sharing coefficient (DLSC) was reduced to 1.017, and the overall geometric volume was reduced to 1.473 × 108 mm3. It provides a theoretical basis for the design and research of a new type of helicopter power transmission system with a compact structure, good maneuverability, and high reliability.

Quantifying and correcting geolocation error in spaceborne LiDAR forest canopy observations using high spatial accuracy data: A Bayesian model approach

AbstractGeolocation error in spaceborne sampling light detection and ranging (LiDAR) measurements of forest structure can compromise forest attribute estimates and degrade integration with georeferenced field measurements or other remotely sensed data. Data integration is especially problematic when geolocation error is not well quantified. We propose a general model that uses airborne laser scanning data to quantify and correct geolocation error in spaceborne sampling LiDAR. To illustrate the model, LiDAR data from NASA Goddard's LiDAR Hyperspectral and Thermal Imager (G‐LiHT) was used with a subset of LiDAR data from NASA's Global Ecosystem Dynamics Investigation (GEDI). The model accommodates multiple canopy height metrics derived from a simulated GEDI footprint kernel using spatially coincident G‐LiHT, and incorporates both additive and multiplicative mapping between the canopy height metrics generated from both datasets. A Bayesian implementation provides probabilistic uncertainty quantification in both parameter and geolocation error estimates. Results show a systematic geolocation error of 9.62 m in the southwest direction. In addition, estimated geolocation errors within GEDI footprints were highly variable, with results showing a 0.45 probability the true footprint center is within 20 m. Estimating and correcting geolocation error via the model outlined here can help inform subsequent efforts to integrate spaceborne LiDAR data, like GEDI, with other georeferenced data.

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.