Comparative Analysis of Radial Basis Functions and Cubic Splines for Data Imputation

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

Nowadays , nonlinear time series and artificial neural networks (ANN) models are used for forecasting in the field of business, agriculture and soon. Recent studies have shown, ANN have been successfully used for forecasting of financial and agriculture data series The classical methods used for time series prediction like Box-Jenkins or ARIMA assumes that there is a linear relationship between inputs and outputs. ANN have more advantages that can approximate to model both linear and nonlinear structures in time series, they are not able to handling both structures equally well. The autoregressive integrated moving average (ARIMA) model and two ANN models namely, Radial basis function neural networks (RBFNN), and Elman recurrent neural networks (ERNN) methods were applied to Hyderabad airport traffic data. The data obtained for 15 years from 2002–2003 to 2016–2017 about domestic and international passenger of International Airport of Hyderabad, India. In this research paper, we compared the performances of ARIMA, RBFNN and ERNN were based on three measures: mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE). The results showed that RBFNN obtained the smallest MAE, MAPE and RMSE in both the modeling and forecasting processes. The performances of the three models ranked in ascending order were: ARIMA, ERNN and the RBFNN model. Keywords: T ime series, forecasting, artificial neural networks, ARIMA models, radial basis function neural networks, and Elman recurrent neural networks Cite this Article R. Ramakrishna, Berhe Aregay, Tewodros Gebregergs. The Comparison in Time Series Forecasting of Air Traffic Data by Autoregressive Integrated Moving Average Model, Radial Basis Function and Elman Recurrent Neural Networks. Research & Reviews: Journal of Statistics . 2018; 7(3): 75–90p.

In response to the problem that Near-Infrared Spectroscopy (NIRS) is difficult to model and easy to fit under small sample conditions, this paper proposes a regression detection model based on the Kolmogorov-Arnold Network (KAN) combined with Bidirectional Gated Recurrent Units (BiGRU), termed BiGRU-KAN. This model innovatively combines the functional approximation capability of the KAN model with the temporal feature extraction capability of BiGRU, and replaces the B-spline function in traditional KAN with Gaussian radial basis functions (RBF). The introduction of RBF reduces the parameter space, achieving a parameter reduction ratio of 21.4025%. This resolves the issues of parameter inflation and boundary instability caused by B-spline functions under high-dimensional inputs. Overfitting risks are further mitigated through parameter tuning and regularization settings. The experimental results show that the model performs well on the verification set, with the decision coefficient (R2) reaching 0.9886, the root mean square error (RMSE) as low as 0.1983, and the relative analysis error (RPD) as high as 9.0764. Compared to traditional models such as Partial Least Squares (PLS), the original KAN, and BiGRU-MLP, its performance shows significant improvement, providing algorithmic support for the development of MEMS-based micro-miniature near-infrared spectrometers.

A study is presented using a mesh-free approach and a radial basis function generated finite difference (RBF-FD) method for numerically modelling 3-D gravity data. The gravity responses, that is, vertical gravity and gravity gradients, are obtained by solving the partial differential equation (PDE), that is, the Poisson’s equation for gravitational potential. The mesh-free approach discretizes PDEs using exclusively a cloud of unconnected nodes, instead of traditional tessellated meshes as used by mesh-based numerical methods such as finite difference, finite element and finite volume. Thus, the potentially computationally expensive and unstable creation and manipulation of 3-D meshes can be entirely avoided. A new type of finite-smoothness radial basis functions (RBFs), namely, the quintic-order polyharmonic spline (PHS) RBF, is proposed here in the RBF-FD frame for solving the gravity problem. Previous geophysical data modelling studies using RBF-FD have employed the infinitely smooth RBFs, such as the popular Gaussian (GA) RBFs. Here, both GA and PHS RBFs were tested with different numbers of nodes per mesh-free subdomain and with various shape parameter values (only GA RBFs have a shape parameter). The test results show that the PHS RBF-FD method is more computationally efficient than the GA RBF-FD counterpart. To achieve more efficiency, unstructured node distributions are proposed in discretizing the density models. For both quasi-uniform and unstructured node distributions, numerical results from the proposed PHS RBF-FD demonstrate that the computed vertical gravity and gravitational potential values agree well with analytical solutions with a reasonable number of degrees of freedom. A comparison study of modelling a complex density model with the PHS RBF-FD scheme and nodal finite-element method shows that the RBF-FD scheme generates sparse, asymmetric, linear systems of equations, supports unstructured nodal discretization and local refinement, and can have nonlinear h-convergence under refinement. Finally, the proposed RBF-FD method was applied to obtain the vertical gravity and gravity gradients over a real-world density model, where the benefits of mesh-free discretization are clearly illustrated.

In this study numerical solutions for the compressible fluid flow around obstacles, based on a meshless method with Radial Basis Functions (RBFs) are presented. The numerical solutions are obtained by a hybrid method that uses the first step of the boundary element method, namely a singular boundary integral equation, which is an equivalent mathematical model for the problem of the compressible fluid around an obstacle, and then a meshless method based on RBFs for solving this singular boundary integral equation. Two types of RBFs are used, namely the Multiquadric RBFs and Gauss-type RBFs. An analytical check is made in order to study the numerical solutions' accuracy. Using computer simulations, we compare our results with an analytical solution available for a specific case. We examine how different parameters used to obtain the numerical solutions affect their accuracy, and we compare these numerical solutions to determine which RBF is most suitable for this scenario. We also note that the shape parameters of RBFs have a great influence on numerical solutions accuracy and, through simulations, optimal values are found. For evaluating the integrals of singular kernels which appear, the truncation method is applied. The influence of the parameter used to evaluate these integrals is also highlighted in the paper.

Prediction of suspended sediment discharge in a catchments area is very important because it can be used to evaluation the erosion hazard, management of its water resources, water quality, hydrology project management (dams, reservoirs, and irrigation) and to determine the extent of the damage that occurred in the catchments. Multiple Linear Regression analysis and artificial neural network can be used to predict the amount of daily suspended sediment discharge. Regression analysis using the least square method, whereas artificial neural networks using Radial Basis Function (RBF) and feedforward multilayer perceptron with three learning algorithms namely Levenberg-Marquardt (LM), Scaled Conjugate Descent (SCD) and Broyden-Fletcher-Goldfarb-Shanno Quasi-Newton (BFGS). The number neuron of hidden layer is three to sixteen, while in output layer only one neuron because only one output target. The mean absolute error (MAE), root mean square error (RMSE), coefficient of determination (R2 ) and coefficient of efficiency (CE) of the multiple linear regression (MLRg) value Model 2 (6 input variable independent) has the lowest the value of MAE and RMSE (0.0000002 and 13.6039) and highest R2 and CE (0.9971 and 0.9971). When compared between LM, SCG and RBF, the BFGS model structure 3-7-1 is the better and more accurate to prediction suspended sediment discharge in Jenderam catchment. The performance value in testing process, MAE and RMSE (13.5769 and 17.9011) is smallest, meanwhile R2 and CE (0.9999 and 0.9998) is the highest if it compared with the another BFGS Quasi-Newton model (6-3-1, 9-10-1 and 12-12-1). Based on the performance statistics value, MLRg, LM, SCG, BFGS and RBF suitable and accurately for prediction by modeling the non-linear complex behavior of suspended sediment responses to rainfall, water depth and discharge. The comparison between artificial neural network (ANN) and MLRg, the MLRg Model 2 accurately for to prediction suspended sediment discharge (kg/day) in Jenderan catchment area.

Dissolved oxygen (DO) as one of the most fundamental parameters of water quality plays a vital role in aquatic life. This study was conducted to predict DO, biological oxygen demand (BOD), and chemical oxygen demand (COD) in an intensive rainbow trout rearing system with different biomass (B). The multilayer perceptron (MLP) and the radial basis function (RBF) neural networks were employed for evaluating the impacts of food parameters (crude protein (CP), consumed feed (CF)), fish parameters (different values of B, and weight gain (WG)), and water quality parameters including temperature (T) and flow rate (Q) on variation of DO, BOD, and COD concentrations. This study's results showed that although both MLP and RBF neural networks are capable to estimate DO, BOD, and COD concentrations, RBF neural network showed better performance compared to MLP neural network. The results of sensitivity analysis indicated that the parameter CF has the highest effect on DO concentration estimation. Independent variables CF, CP, WG, and B showed the highest to the lowest rank of impacts on BOD estimation, respectively. The results also illustrated a decreasing trend of the effects on the estimation error of COD changes simulation by all independent variables, including B, T, WG, CF, CP, and Q, respectively. RBF neural network based on better stability and generalization ability with average root mean square error (RMSE) and mean absolute percentage error (MAPE) values of less than 0.12 and 3% was superior to MLP in DO, BOD, and COD concentration prediction. Moreover, CF was identified as the most effective factor in estima12tion process. Based on the present study results, there are direct relationships between DO, BOD, and COD concentrations and water quality parameters, fish parameters, and food parameters. Food parameters relative to fish and water quality parameters imposed the greatest effects. Improvement in feeding process such as application of intelligence feeding methods and change in fish diet and feeding time can considerably reduce losses in production system. Graphical abstract.

This paper presents a new method to construct response surface function and a new hybrid optimization method. For the response surface function, the radial basis function is used for a zeroth-order approximation, while new bases is proposed for the moving least squares method for a first-order approximation. For the new hybrid optimization method, the gradient-based algorithm and pattern search algorithm are integrated for robust and efficient optimization process. These methods are based on: (1) multi-point approximations of the objective and constraint functions; (2) a multi-quadric radial basis function for the zeroth-order function representation or radial basis function plus polynomial based moving least squares approximation for the first-order function approximation; and (3) a pattern search algorithm to impose a descent condition. Several numerical examples are presented to illustrate the accuracy and computational efficiency of the proposed method for both function approximation and design optimization. The examples for function approximation indicate that the multi-quadric radial basis function and the proposed radial basis function plus polynomial based moving least squares method can yield accurate estimates of arbitrary multivariate functions. Results also show that the hybrid method developed provides efficient and convergent solutions to both mathematical and structural optimization problems.

Knowing the exchangeable sodium percentage (ESP) variations and its values in sodic or saline-sodic soils is essential in order to estimate the amount of soil amendments and better land management. ESP calculated from cation exchange capacity (CEC), and since CEC measurement is difficult and time-consuming, ESP computation is costly and subject to error. Thus, presenting a method to estimate ESP indirectly, by an easily available index is much more efficient and economical. In this study, 296 soil samples collected and analyzed from Sistan plain, southeastern Iran. Soil ESP were predicted by using artificial neural networks, comprising radial basis functions (RBFN) and multilayer perceptron (MLP) and adaptive neuro-fuzzy inference systems (ANFIS), and results compared with stepwise linear regression method. Results indicated that the linear regression models performed poorly in order to estimate ESP (R2 ≤ 0.58 and root mean square error (RMSE) ≥ 4.31). Applying fewer inputs (electrical conductivity (EC) and pH), ANFIS showed better results (R2=0.80, RMSE=2.34), while increasing inputs (clay and organic carbon) decreased the accuracy (R2=0.82, RMSE=4.20). Using more inputs, RBFN resulted in better performance in comparison with other methods (R2=0.83, RMSE=2.85). Sensitivity analysis using the connection weight method demonstrated that EC, pH, clay percentage and bulk density are the most important variables in order to explain ESP variability in the region, respectively. Generally, considering the evaluation criteria and the number of used variables of models, ANFIS (with EC and pH as inputs) is the most appropriate method for estimating ESP in Sistan plain.

Remote sensing technology is the important tool of digital earth, it can facilitate nutrient management in sustainable cropping systems. In the study, two types of radial basis function (RBF) neural network approaches, the standard radial basis function (SRBF) neural networks and the modified type of RBF, generalized regression neural networks (GRNN), were investigated in estimating the nitrogen concentrations of oilseed rape canopy using vegetation indices (VIs) and hyperspectral reflectance. Comparison analyses were performed to the spectral variables and the approaches. The Root Mean Square Error (RMSE) and determination coefficients (R2) were used to assess their predictability of nitrogen concentrations. For all spectral variables (VIs and hyperspectral reflectance), the GRNN method produced more accurate estimates of nitrogen concentrations than did the SRBF method at all ranges of nitrogen concentrations, and the better agreements between the measured and the predicted nitrogen concentration were obtained with the GRNN method. This indicated that the GRNN method is prior to the SRBF method in estimation of nitrogen concentrations. Among the VIs, the Modified Chlorophyll Absorption in Reflectance Index (MCARI), MCARI1510, and Transformed Chlorophyll Absorption in Reflectance Index are better than the others in estimating oilseed rape canopy nitrogen concentrations. Compared to the results from VIs, the hyperspectral reflectance data also gave an acceptable estimation. The study showed that nitrogen concentrations of oilseed rape canopy could be monitored using remotely sensed data and the RBF method, especially the GRNN method, is a useful explorative tool for oilseed rape nitrogen concentration monitoring when applied on hyperspectral data.

In this paper, a fractional order digital differentiator is designed by using Inverse multiquadric radial basis function (RBF). First, the RBF interpolation approach is described. Then, the non-integer delay sample estimation is derived by using RBF approach. Next, the Grunwald-Letnikov derivative and non-integer delay sample delay are applied to obtain the transfer function of the proposed method i.e. fractional order digital differentiator. The design accuracy of the proposed method is better then the conventional methods like examples Time domain least squares method, Fractional sample delay method and Frequency response approximation method.

This paper explores the application of the mesh-free Local Radial Basis Function Collocation Method (LRBFCM) [1] in solution of coupled heat transfer and fluid flow problems associated with solidification of a binary alloy. The involved temperature, velocity, species and pressure fields are represented on overlapping sub-domains through collocation by using multiquadrics Radial Basis Functions (RBF). The involved first and second derivatives of the fields are calculated from the respective derivatives of the RBF’s. The energy and momentum equations are solved through explicit time stepping. The pressure-velocity coupling is calculated iteratively, with pressure correction, predicted from the local continuity equation violation [2,3]. The solution procedure is demonstrated in simulation of solidification of 8 wt % Pb-Sn alloy and 5 wt % Sn-Pb alloy in rectangular cavity at conditions, defined in [4]. The numerical examples include studies with different uniform discretization and its influence on final macrosegregation map. The solution is assessed by comparison with the reference results of the finite volume and finite element method. The advantages of the method are simplicity, accuracy and straightforward applicability in non-uniform node arrangements. The methodology will be use in complementing our heat transfer model [5] of the continuous casting of steel.

This study introduced a new predictive approach for estimating the bearing capacity of driven piles. To this end, the required data based on literature such as hammer strikes, soil properties, geometry of the pile, and friction angle between pile and soil were gathered as a suitable database. Then, three predictive models i.e., gene expression programming (GEP), radial basis function type neural networks (RBFNN) and multivariate nonlinear regression (MVNR) were applied and developed for pile bearing capacity prediction. After proposing new models, their performance indices i.e., root mean square error (RMSE) and coefficient of determination (R2) were calculated and compared to each other in order to select the best one among them. The obtained results indicated that the RBFNN model is able to provide higher performance prediction level in comparison with other predictive techniques. In terms of R2, results of 0.9976, 0.9466 and 0.831 were obtained for RBFNN, GEP and MVNR models respectively, which confirmed that, the developed RBFNN model could be selected as a new model in piling technology. Definitely, other researchers and engineers can utilize the procedure and results of this study in order to get better design of driven piles.

Purpose- The purpose of this study is to compare the success of alternative data imputation techniques with missing data. The study distinguishes itself from the rest of the literature by proposing an appropriate technique for mixed data on financial performance and environmental, social and governance (ESG) metrics of companies. In addition to simple imputation techniques, we also use machine learning techniques that allow working with more complex data. Methodology- We first employ ad-hoc methods such as mean, median, mode, constant, most frequent and regression imputation. In what follows, we handle multivariate imputation techniques such as multiple imputation by chained equations (MICE). Finally, we run imputation methods with machine learning (ML) classification such as K-nearest Neighbor (KNN), Ridge and Random Forest. To consider the assumptions of missing data, we first check the normality of the variables with Kolmogorov-Smirnov test and employ Rubin’s classification technique that defines the relationship among variables with the probability of missing data. The success of imputation techniques applied to missing data changes when the missing data are classified with Rubin’s technique according to randomness. Consequently, we apply listwise deletion at various levels and alternative data imputation techniques. We then compare their performances. The raw data contain parametric as well as categorical variables (binary and others). Among these are time-series (yearly) financial series such as sales and total assets obtained from financial statements, ESG scores as well as float ratios for firms from several countries and industries. Imputation is done randomly on a sample varying from 5% to 30% of the dataset and results are compared to true data based on accuracy or other measures such as root mean square errors (RMSE) or mean absolute percentage error (MAPE). Several robustness checks have been performed to supplement the analysis. Findings- Results show that ML methods such as KNN have a superior performance than others. Moreover, when multidimensional nature of the data is taken into account, the prediction performance improves. Hence, an optimality can be reached based on parameters. Conclusion- Based upon the analysis, we conclude that the selected imputation technique and how it is employed matter to attain a higher accuracy and a better prediction of the missing values on selected mixed panel data in finance. Keywords: Imputation techniques, Panel data, Machine learning, Financial performance, ESG JEL Codes: C55, C81, M14, Q51

ABSTRACTComplex void space structure and flow patterns in karstic aquifers render behaviour prediction of karstic springs difficult. Four support vector regression-based models are proposed to predict flow rates from two adjacent karstic springs in Greece (Mai Vryssi and Pera Vryssi). Having no accurate estimates of the groundwater flow pattern, we used four kernels: linear, polynomial, Gaussian radial basis function and exponential radial basis function (ERBF). The data used for training and testing included daily and mean monthly precipitation, and spring flow rates. The support vector machine (SVM) performance depends on hyper-parameters, which were optimized using a grid search approach. Model performance was evaluated using root mean square error and correlation coefficient. Polynomial kernel performed better for Mai Vryssi and the ERBF for Pera Vryssi. All models except one performed better for Pera Vryssi. Our models performed better than generalized regression neural network, radial basis function neural network and ARIMA models.