Optimal Regularization Parameter Research Articles

Improved Bacterial Foraging Optimization Algorithm with Machine Learning-Driven Short-Term Electricity Load Forecasting: A Case Study in Peninsular Malaysia

Accurate electricity demand forecasting is crucial for ensuring the sustainability and reliability of power systems. Least square support vector machines (LSSVM) are well suited to handle complex non-linear power load series. However, the less optimal regularization parameter and the Gaussian kernel function in the LSSVM model have contributed to flawed forecasting accuracy and random generalization ability. Thus, these parameters of LSSVM need to be chosen appropriately using intelligent optimization algorithms. This study proposes a new hybrid model based on the LSSVM optimized by the improved bacterial foraging optimization algorithm (IBFOA) for forecasting the short-term daily electricity load in Peninsular Malaysia. The IBFOA based on the sine cosine equation addresses the limitations of fixed chemotaxis constants in the original bacterial foraging optimization algorithm (BFOA), enhancing its exploration and exploitation capabilities. Finally, the load forecasting model based on LSSVM-IBFOA is constructed using mean absolute percentage error (MAPE) as the objective function. The comparative analysis demonstrates the model, achieving the highest determination coefficient (R2) of 0.9880 and significantly reducing the average MAPE value by 28.36%, 27.72%, and 5.47% compared to the deep neural network (DNN), LSSVM, and LSSVM-BFOA, respectively. Additionally, IBFOA exhibits faster convergence times compared to BFOA, highlighting the practicality of LSSVM-IBFOA for short-term load forecasting.

Improving electrocardiographic imaging solutions: A comprehensive study on regularization parameter selection in L-curve optimization in the Atria

BackgroundIn electrocardiographic imaging (ECGI), selecting an optimal regularization parameter (λ) is crucial for obtaining accurate inverse electrograms. The effects of signal and geometry uncertainties on the inverse problem regularization have not been thoroughly quantified, and there is no established methodology to identify when λ is sub-optimal due to these uncertainties. This study introduces a novel approach to λ selection using Tikhonov regularization and L-curve optimization, specifically addressing the impact of electrical noise in body surface potential map (BSPM) signals and geometrical inaccuracies in the cardiac mesh. MethodsNineteen atrial simulations (5 of regular rhythms and 14 of atrial fibrillation) ensuring variability in substrate complexity and activation patterns were used for computing the ECGI with added white Gaussian noise from 40 dB to -3dB. Cardiac mesh displacements (1–3 cm) were applied to simulate the uncertainty of atrial positioning and study its impact on the L-curve shape. The regularization parameter, the maximum curvature, and the most horizontal angle of the L-curve (β) were quantified. In addition, BSPM signals from real patients were used to validate our findings. ResultsThe maximum curvature of the L-curve was found to be inversely related to signal-to-noise ratio and atrial positioning errors. In contrast, the β angle is directly related to electrical noise and remains unaffected by geometrical errors. Our proposed adjustment of λ, based on the β angle, provides a more reliable ECGI solution than traditional corner-based methods. Our findings have been validated with simulations and real patient data, demonstrating practical applicability. ConclusionAdjusting λ based on the amount of noise in the data (or on the β angle) allows finding optimal ECGI solutions than a λ purely found at the corner of the L-curve. It was observed that the relevant information in ECGI activation maps is preserved even under the presence of uncertainties when the regularization parameter is correctly selected. The proposed criteria for regularization parameter selection have the potential to enhance the accuracy and reliability of ECGI solutions.

Adaptive Wind Field Estimation Using an Empirical Bayesian Approach

In lidar-based gust load alleviation, the wind profile ahead of the aircraft cannot be measured directly but has to be reconstructed (estimated) based on the acquired line-of-sight measurements. Such wind reconstruction algorithms typically include regularization in order to adequately handle the noise within the data. This paper presents an empirical Bayesian approach to choosing optimal regularization parameters for any given set of measurements. Using simulations of flight through turbulence, the Bayesian approach is compared with a former approach (based on engineering guess) and an omniscient optimizer, which yields the best achievable results for a constant set of parameters by using the full knowledge of the wind field. The Bayesian approach is shown to outperform the engineering guess and performs close to the omniscient optimizer while purely relying on the lidar measurement data.

Identification of Time-Wise Thermal Diffusivity, Advection Velocity on the Free-Boundary Inverse Coefficient Problem

This paper is concerned with finding solutions to free-boundary inverse coefficient problems. Mathematically, we handle a one-dimensional non-homogeneous heat equation subject to initial and boundary conditions as well as non-localized integral observations of zeroth and first-order heat momentum. The direct problem is solved for the temperature distribution and the non-localized integral measurements using the Crank–Nicolson finite difference method. The inverse problem is solved by simultaneously finding the temperature distribution, the time-dependent free-boundary function indicating the location of the moving interface, and the time-wise thermal diffusivity or advection velocities. We reformulate the inverse problem as a non-linear optimization problem and use the lsqnonlin non-linear least-square solver from the MATLAB optimization toolbox. Through examples and discussions, we determine the optimal values of the regulation parameters to ensure accurate, convergent, and stable reconstructions. The direct problem is well-posed, and the Crank–Nicolson method provides accurate solutions with relative errors below 0.006% when the discretization elements are M=N=80. The accuracy of the forward solutions helps to obtain sensible solutions for the inverse problem. Although the inverse problem is ill-posed, we determine the optimal regularization parameter values to obtain satisfactory solutions. We also investigate the existence of inverse solutions to the considered problems and verify their uniqueness based on established definitions and theorems.

Rational Polynomial Coefficient Estimation via Adaptive Sparse PCA-Based Method

The Rational Function Model (RFM) is composed of numerous highly correlated Rational Polynomial Coefficients (RPCs), establishing a mathematical relationship between two-dimensional images and three-dimensional spatial coordinates. Due to the existence of ill-posedness and overparameterization, the estimated RPCs are sensitive to any slight perturbations in the observation data, particularly when handling a limited number of Ground Control Points (GCPs). Recently, Principal Component Analysis (PCA) has demonstrated significant performance improvements in the RFM optimization problem. In the PCA-based RFM, each Principal Component (PC) is a linear combination of all variables in the design matrix. However, some original variables are noise related and have very small or almost zero contributions to the construction of PCs, which leads to the overparameterization problem and makes the RPC estimation process ill posed. To address this problem, in this paper, we propose an Adaptive Sparse Principal Component Analysis-based RFM method (ASPCA-RFM) for RPC estimation. In this method, the Elastic Net sparsity constraint is introduced to ensure that each PC contains only a small number of original variables, which automatically eliminates unnecessary variables during PC computation. Since the optimal regularization parameters of the Elastic Net vary significantly in different scenarios, an adaptive regularization parameter approach is proposed to dynamically adjust the regularization parameters according to the explained variance of PCs and degrees of freedom. By adopting the proposed method, the noise and error in the design matrix can be reduced, and the ill-posedness and overparameterization of the RPC estimation can be significantly mitigated. Additionally, we conduct extensive experiments to validate the effectiveness of our method. Compared to existing state-of-the-art methods, the proposed method yields markedly improved or competitive performance.

Learning-Assisted Fast Determination of Regularization Parameter in Constrained Image Reconstruction.

To leverage machine learning (ML) for fast selection of optimal regularization parameter in constrained image reconstruction. Constrained image reconstruction is often formulated as a regularization problem and selecting a good regularization parameter value is an essential step. We solved this problem using an ML-based approach by leveraging the finding that for a specific constrained reconstruction problem defined for a fixed class of image functions, the optimal regularization parameter value is weakly subject-dependent and the dependence can be captured using few experimental data. The proposed method has four key steps: a) solution of a given constrained reconstruction problem for a few (say, 3) pre-selected regularization parameter values, b) extraction of multiple approximated quality metrics from the initial reconstructions, c) predicting the true quality metrics values from the approximated values using pre-trained neural networks, and d) determination of the optimal regularization parameter by fusing the predicted quality metrics. The effectiveness of the proposed method was demonstrated in two constrained reconstruction problems. Compared with L-curve-based method, the proposed method determined the regularization parameters much faster and produced substantially improved reconstructions. Our method also outperformed state-of-the-art learning-based methods when trained with limited experimental data. This paper demonstrates the feasibility and improved reconstruction quality by using machine learning to determine the regularization parameter in constrained reconstruction. The proposed method substantially reduces the computational burden of the traditional methods (e.g., L-curve) or relaxes the requirement of large training data by modern learning-based methods, thus enhancing the practical utility of constrained reconstruction.

The Modified Ambiguity Function Approach with regularization for instantaneous precise GNSS positioning

Abstract The Modified Ambiguity Function Approach (MAFA) implicitly conducts the search procedure of carrier phase GNSS integer ambiguity resolution (IAR) in the coordinate domain using the integer least squares (ILS) principle, i.e. MAFA-ILS. One of the still open scientific problems is an accurate definition of the search region, especially in the context of instantaneous IAR. In doing so, the float solution results, which encompass float position (FP) and its variance-covariance (VC) matrix, must be improved as these are necessary for defining the search region. For this reason, the ambiguity parameters are separately regularized, and then the baseline parameters are conditioned on regularized float ambiguities. The conditional-regularized estimation is thus designed, obtaining the regularized FP (RFP) and its VC-matrix. This solution is promising because its accuracy is enhanced in the sense of mean squared error (MSE) thanks to the improved precision at the cost of regularized bias. The optimal regularization parameter (RP) values obtained for ambiguity parameters balance the contributions of improved precision and bias in the regularized float baseline solution’s MSE. Therefore, the regularized search region is defined accurately in the coordinate domain to contain such approximate coordinates that more frequently give the correct ILS solution. It also contains fewer MAFA-ILS candidates, improving the search procedure’s numerical efficiency. The regularized ILS estimator performs well with the presence of bias, increasing the probability of correct IAR in the coordinate domain.

Iterative reconstruction methods and the resolution principle for fast-ion loss detector measurements

Fast-ion loss detectors (FILDs) are crucial for analyzing fast-ion dynamics in magnetically confined fusion plasmas. A core challenge is to derive an accurate ion velocity distribution, requiring treatment of thousands of remapped camera frames for a full discharge. The ill-posed nature of this task necessitates regularization with a well-chosen regularization parameter and computationally efficient methods. In this work, we introduce the ‘resolution principle,’ a novel criterion for selecting the optimal regularization parameter, providing a distinction between genuine features and artefacts smaller than the diagnostic resolution in the reconstruction, thereby preventing misinterpretations. This principle, coupled with three iterative reconstruction techniques—Kaczmarz’s method, coordinate descent, and Cimmino’s method—demonstrates enhanced reconstruction capabilities compared to conventional methods like Tikhonov regularization. Utilizing these techniques allows rapid processing of measurements from full discharges, removing the computational bottleneck and facilitating between-discharge reconstructions. By reconstructing 6000 camera frames from an ELMy H-mode discharge at ASDEX Upgrade, we capture the temporal evolution of gyroradii and pitch angles, unveiling a direct correlation between pitch-angle behavior and changes in the toroidal magnetic field for a specific subset of lost ions accelerated by edge-localized modes (ELMs) to energies approximately twice that of the injection energy.

Testing the suitability of v-Support Vector Machine for hyperspectral image classification

ABSTRACT This research presents the novel application of v-Support Vector Machine (v-SVM) to hyperspectral image classification. The essence of this work is to test the suitability of v-SVM for hyperspectral image classification. The v-SVM provides an enhancement to the regularisation parameter in the classical Support Vector Machine (SVM). The regularisation parameter controls the trade-off between obtaining a high training error and a low training error which is the ability of the model to generalise the unseen data (or test data). The value of the regularisation parameter in the classical SVM ranges from 0 to + ∞ ; this makes it usually challenging to determine the most appropriate optimal regularisation parameter value. The invention of the v-SVM has made it easier to find an appropriate optimal regularisation parameter value since the regularisation parameter is narrowed down from a wide range of 0 to + ∞ to a narrow range of 0 to 1 . In this study, two hyperspectral images of Indian Pines region in Northwest Indiana, USA and University of Pavia, Italy are used as test beds for the experiment. The result of the experiment shows that the v-SVM performed fairly better than the classical SVM; however, it fell short of the notable conventional classifiers.

A regularization strategy for discontinuity when modelling coupled water and heat flow in freezing unsaturated soil

Freezing tightly couples the water and heat flow. In most porous media, the interface between liquid and frozen water is not sharp and a slushy zone is present. There are two distinct approaches for mathematical modelling of freezing. It is the equilibrium approach which allows an instant freezing under given conditions and non-equilibrium approach where a specific timing in the freezing process is considered. In this contribution, we specifically target the equilibrium approach.The key mathematical model for the equilibrium approach is the Clausius–Clapeyron equation, which allows the derivation of a soil freezing curve relating temperature to pressure head. Implementing freezing soil accurately is not a straight-forward. Using the Clausius–Clapeyron equation creates a discontinuity in the freezing rate and latent heat at the freezing point. Little attention has been paid to the adequate description of the numerical treatment of this phenomenon and to the computational challenges that it poses. Numerical approximation of this discontinuous system is prone to spurious oscillations. In this contribution, we show the application of regularization of the discontinuous term. This treatment successfully stabilizes the computation and can remove oscillations. To avoid over-regularization, we present here a minimax strategy to determine optimal regularization parameters. We further compare an over-regularized setup with the non-equilibrium approach, where we show that a successful regularization is equivalent to incorporating a minimal timing to the freezing process. Finally – for validating our implementation and computational approach – experimental laboratory data of the volumetric water content and temperature profiles from previously published soil freezing experiments were represented here with the optimally regulized equilibrium approach.

Analysis of Logistic Regression Regularization in Wild Elephant Classification with VGG-16 Feature Extraction

The research article explores the intersection of image-based wildlife classification and logistic regression regularization, focusing on the classification of wild elephant species. It begins by highlighting the significance of ecological research in biodiversity monitoring and conservation and introduces Convolutional Neural Networks (CNNs) as potent tools for feature extraction from images. The VGG-16 model is particularly emphasized for its ability to capture hierarchical representations of visual features crucial for classification tasks. The integration of VGG-16 feature extraction with logistic regression regularization is proposed as a compelling approach, offering a balance between sophisticated feature representation and efficient classification algorithms. The literature review delves into image-based wildlife classification, emphasizing the role of CNNs, especially VGG-16, in extracting discriminative features. It discusses the fusion of VGG-16 features with logistic regression and the challenges in this field, such as dataset annotation and environmental variability. The method section outlines the dataset acquisition, feature extraction using the VGG-16 architecture, and model configuration using logistic regression with lasso and ridge regularization. The process of finding the optimal regularization parameter (lambda) and model evaluation through cross-validation is detailed. Results showcase the optimal lambda values for lasso and ridge regularization and compare the performance of logistic lasso and logistic ridge models. Misclassification analysis reveals factors influencing classification accuracy, including feature variability and contextual complexity. The discussion reflects on the implications of the findings, emphasizing the importance of lambda selection and addressing challenges in wildlife classification. It suggests avenues for further research, such as advanced modeling techniques and feature engineering approaches. In conclusion, the study contributes to advancing wildlife classification efforts by leveraging state-of-the-art techniques and sheds light on opportunities to enhance classification accuracy in wildlife conservation.

Single shot interferenceless coded aperture correlation holography via a learnable Wiener deconvolution network

One of the challenges in interferenceless coded aperture correlation holography (I-COACH) is high-quality single-shot reconstruction, as conventional cross-correlation reconstruction algorithm requires recording multiple holograms and complex reconstruction operations, resulting in serious background noise. Here, we present a deep learning based single-shot reconstruction by introducing a learnable Wiener deconvolution network in I-COACH. By feeding the point spread hologram (PSH) of I-COACH system and object hologram (OH) into Wiener deconvolution, a mapping relationship between Wiener deconvolution result and ground truth is generated. A Res-UNet is trained to learn optimal filter and noise regularization parameters, thereby achieving high quality reconstruction. Moreover, the training dataset used in our network are simulated using PSH that is obtained through only one experiment acquisition in advance, greatly simplifying experimental OH dataset acquisition. Compared to the conventional correlation reconstruction methods, the proposed Wiener deconvolution network can conveniently realize high-quality single-shot reconstruction in I-COACH, and the axial sensitivity in 3D reconstruction can be demonstrated by varying scattering degree of coded phase mask. Importantly, this Wiener deconvolution solution will provide an enlightening reference for dynamic imaging in I-COACH.

Conjugate gradient method with regularization in estimating mold surface heat flux during continuous casting

The reconstruction of the boundary heat flux between solidifying steel and water-cooled mold wall using measured temperature data is recognized as an inverse heat conduction problem (IHCP). A Tikhonov spatial regularization approach to enhance the smoothness of the Conjugate Gradient Method (CGMr) was proposed which incorporates three different orders of spatial regularization: zeroth, first, and second order. The optimal regularization parameter was selected using a modified L-curve method. The accuracy of the CGMr is investigated with Case 1: a triangular time-spatial variation heat flux, and Case 2: a step change in the form of rectangular variation heat flux. The effects of measurement noise, grid points and time-step size are investigated. The results show that the minimum relative error (eRMS) of the predicted Case 1 heat flux is 1.98%, 7.64%, and 8.02% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 3.82%, 13.42%, and 14.91%. Subsequently, CGMr algorithm is applied to calculate the heat flux in a mold simulator experiment. By comparing the relationship between heat fluxes reconstructed by CGMr after different iteration numbers, it is observed that the recovered heat flux of 2.14 MW/m2 with zeroth regularization remains highly stable.

Magnetics only real-time equilibrium reconstruction on ASDEX Upgrade

Real-time reconstruction of the magnetic equilibrium provides fundamental control of plasma shape and position in a tokamak. Details of the implementation of the equilibrium reconstruction code developed for the ASDEX Upgrade tokamak (JANET++) are summarized. Cubic Hermite splines are introduced as current density basis functions for solving the Grad–Shafranov equation. The choice of the optimal Tikhonov regularization parameter is discussed. The code is validated by comparing the results of the equilibrium reconstruction with those of further equilibrium reconstructions available on ASDEX Upgrade (CLISTE and IDE). In a high time resolution study of a discharge with edge localized modes (ELM), the poloidal asymmetry of the fits and magnetic probe measurements suggest that the real-time equilibrium reconstruction captures the essential features of the current density redistribution in an ELMing edge plasma. An efficient algorithm to locate multiple X-points and identify the active one in advanced X-divertor and snowflake divertor configurations is presented.

An improved composite ship magnetic field model with ellipsoid and magnetic dipole arrays.

In order to simultaneously maintain the ship magnetic field modeling accuracy, reduce the number of coefficient matrix conditions and the model computational complexity, an improved composite model is designed by introducing the magnetic dipole array model with a single-axis magnetic moment on the basis of the hybrid ellipsoid and magnetic dipole array model. First, the improved composite model of the ship's magnetic field is established based on the magnetic dipole array model with 3-axis magnetic moment, the magnetic dipole array model with only x-axis magnetic moment, and the ellipsoid model. Secondly, the set of equations for calculating the magnetic moments of the composite model is established, and for the problem of solving the pathological set of equations, the least-squares estimation, stepwise regression method, Tikhonov, and truncated singular value decomposition regularization methods are introduced in terms of the magnetic field, and generalized cross-validation is used to solve the optimal regularization parameters. Finally, a ship model test is designed to compare and analyze the effectiveness of the composite and hybrid models in four aspects: the number of coefficient matrix conditions of the model equation set, the relative error of magnetic field fitting, the relative error of magnetic field extrapolation, and the computational time complexity. The modeling results based on the ship model test data show that the composite model can be used for modeling the magnetic field of ships, and compared with the hybrid model, it reduces the number of coefficient matrix conditions and improves the computational efficiency on the basis of retaining a higher modeling accuracy, and it can be effectively applied in related scientific research and engineering.

Using learned priors to regularize the Helmholtz equation least-squares method

The Helmholtz equation least-squares (HELS) method is a valuable tool for estimating equivalent sound sources of a radiating object. It solves an inverse problem by mapping measured pressures to a set of basis functions satisfying the Helmholtz equation in spherical coordinates. However, this problem is often ill-posed, necessitating additional regularization methods, in which often variations of Ridge or Lasso are used. These conventional methods do not explicitly consider the distribution underlying the source radiations (besides sparsity) and are often used in the context of obtaining only a point estimate, even in the presence of ambiguity in the data. In this work, we propose the use of empirical priors through a normalizing flow model to enhance the inversion results obtained with the HELS method. We first validate our approach using numerical data and subsequently demonstrate its superior performance in interpolating a measured violin directivity compared to Lasso and Ridge methods, even when optimal regularization parameters are selected.

Efficient learning methods for large-scale optimal inversion design

In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $ p $ and $ q $ norms for $ {\rm L}^p-{\rm L}^q $ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.

Impedance analysis of alkaline water electrolysis based on distribution of relaxation time

Electrochemical Impedance Spectroscopy (EIS) is an important tool for identifying electrochemical processes and evaluating electrolyzer performance. Fitting EIS data with equivalent circuits often requires prior knowledge and involves inherent uncertainties. This work performed the experimental measurements of visualized zero-gap water electrolyzer equipped with four different flow channels, and then explored an efficient impedance analysis procedure for Alkaline Water Electrolysis (AWE) coupling with the Distribution Relaxation Time (DRT) method. By determining the optimal regularization parameter λ, typical characteristic peaks were identified corresponding to ion transfer, charge transfer, and mass transfer impedance, respectively. The flow channel structure significantly impacted two-phase flow distribution, with the pin-type channel showing a more uniform flow. Based on the optimal pin-type flow channel structure, the influences of operating parameters (current density, flow rate, and temperature) on the electrolysis performance were experimentally measured, and the impact mechanism was disclosed using the DRT method. The results illustrate that the charge transfer impedance Rt is greatly associated with current density and ohmic resistance RO is highly dependent on flow rate and temperature. At high current densities, the bubble coverage plays an indispensable role in the mass transfer impedance process. This work offers a quantitative method to characterize the AWE performance, which can be employed to optimize the structure and operating parameters of electrolyzers.

DETERMINATION OF THE PARAMETERS OF STRESS-CORROSION CRACKS OF THE MAIN GAS PIPELINE BASED ON THE RESULTS OF EDDY CURRENT CONTROL

In order to increase the accuracy of estimating the depth of stress-corrosion cracks in the main gas pipeline from the measured signal of the overhead eddy-current transducer, a mathematical recognition method based on the regularization method is proposed. Analytical expressions are obtained for the smoothing function and the optimal regularization parameter, which makes it possible to obtain stable estimates of the depth of stress-corrosion cracks in the main gas pipeline. In the absence of spatial resolution of a group of stress-corrosion cracks based on the signal of an overhead eddy-current transducer, a mathematical model of a group of cracks with the same depths and a constant interval is proposed. The procedure for determining the values of the parameters used in the regularization parameter of the smoothing functional is described. Based on the analysis of eddy-current transducer signals by the regularization method obtained on artificial models of stress-corrosion cracks in the main gas pipeline, it is shown that the average relative error in estimating the depth of a group of identical cracks does not exceed 8.5 %. It is shown that the estimate of the depth of a continuity defect obtained by recognizing a group of unequal cracks is close to the average value of this parameter established for a group of cracks, while the average relative error in estimating the average depth of a group of unequal cracks does not exceed 8 %.

Classifying dynamic motor imagery with the locals-balanced extreme learning machine

The dynamic motor imagery (dMI) provides additional benefits as compared to the traditional motor imagery (MI) in training and neurorehabilitation field. The objective of this work is to develop a brain-computer interface (BCI) for dMI electroencephalograph (EEG). We propose a novel method by combining synchronization likelihood (SL) based functional brain network (FBN) and modified extreme learning machine (ELM) to interpret EEG. The proposed method 1) uses L1-norm and adaptive threshold instead of L2-norm and empirical threshold in SL method; 2) improves the procedure of FBN calculations (namely, SL matrix binarization) by combining threshold and MST methods; 3) detects two defects in standard ELM in fusion, and proposes the locals-balanced ELM (LBELM); 4) employs the special version of leave-one-out cross validation (LOO) approach improved the computational complexity of optimal threshold for binarization, the most effective fusion of two complementary features, and the optimal regularization parameters for regularized LBELM. As compared with the empirical threshold, the recognition rate of the optimal threshold increases by 1.39–11.11 %. Similarly, as compared with the regularized ELM, the recognition rate of LBELM increases by 2.78–9.73 %.