Generalized Cross-validation Criterion Research Articles

Mathematical modelling of drilling mud plastic viscosity at downhole conditions using multivariate adaptive regression splines

Viscosity is an essential mud property which enables it to transport cuttings out of a well. Determining the real time value of mud viscosity at downhole conditions is complex and till date lacks a precise model. Literature is filled with models that are only applicable at surface conditions with most of the models being artificial neural network (ANN) based. Neural networks have been criticized for their long training time because they rely on a trial-and-error approach to determine their optimal architecture. The few models related to mud viscosity at downhole conditions are based on non-linear regression which are not flexible in nature. However, Multivariate Adaptive Regression Splines (MARS) has at no time been used notwithstanding some of its known benefits over ANN and other regression methods. The focus of this study is to model mud plastic viscosity at downhole conditions with the MARS method using parameters such as temperature, pressure and initial mud viscosity. MARS is a flexible non-parametric splines-based technique that extends linear regression models by including nonlinearities and interactions between predictors. An experimental database comprising 88 and 149 data points culled from literature for oil-based mud (OBM) and water-based mud (WBM) respectively were utilized for developing the models. For each mud type, the most suitable model was determined by choosing the model with the maximum coefficient of determination (R2) value and the least values of mean square error (MSE), root mean square error (RMSE), and generalized cross-validation criterion (GCV). Using these criteria, the R2, MSE and RMSE of the MARS models according to the mud type were found to be 0.64, 104.35, 10.21 and 0.71, 12.8, 3.57 for OBM and WBM respectively while the GCV values were 129.3 and 14.38 for OBM and WBM respectively. Due to lack of mud rheology data at downhole conditions, it was difficult to compare the performance of the developed models with existing models. Parametric importance analysis indicate that temperature has the highest effect on mud viscosity estimation at downhole conditions. The developed models are valid for 20 °C ≤ Temperature ≤315 °C and 0 psi ≤ Pressure ≤40,000 psi. This pioneer work on using MARS in drilling fluid property prediction points out the implementation protocol, the dataset utilized, the relative importance of input variables and the explicit model for estimating mud viscosity at downhole conditions.

Model selection in multivariate adaptive regressions splines (MARS) using alternative information criteria

Multivariate Adaptive Regression Splines (MARS) is a useful non-parametric regression analysis method that can be used for model selection in high-dimensional data. Since MARS can identify and model complex, non-linear relationships between the dependent variable and independent variables without requiring any assumptions, it has advantage over simple linear regression techniques. Also, for simplifying the model building process and preventing overfitting, MARS can select automatically the variables to be included in the model, which is useful for datasets with many variables. While MARS is a flexible non-parametric regression method, generalized cross validation (GCV) technique is used within the MARS framework to avoid overfitting and to select the best model. GCV criterion is widely used and can be effective in many situations, however it has some criticism. These criticism are the arbitrary value of the smoothing parameter used in the algorithm of the GCV criterion and the models obtained using this criterion are high-dimensional. In this paper, it is aimed to obtain the barest model that best explains the relationship between the dependent variable and independent variables by using alternative information criteria (Akaike information criterion (AIC), Schwarz Bayesian criterion (SBC) and information complexity criterion (ICOMP(IFIM)PEU)) instead of the use of smoothing parameters in order to put an end to the criticism. To achieve this goal, a simulation study was first conducted with a data set composed of variables that do and do not contribute to the dependent variable to test the success of the information criteria. As a consequence of this simulation work, when variables (which do not contribute to the dependent variable) are not included in the regression model, it demonstrates the success of the criteria in model selection. As a real data set, the reasons for loan defaults were investigated between the years 2005–2019 by utilizing data from 18 banks operating in Türkiye. The results obtained reveal the success of ICOMP(IFIM)PEU criterion in model selection.

A fuzzy nonparametric regression model based on an extended center and range method

When dealing with nonparametric regression problems, kernel-based methods are often employed. In this paper, we investigate nonparametric regression problems with fuzzy responses and exact predictors by implementing an extended center and range method within a three-stage procedure. For this purpose, we utilize the Nadaraya–Watson estimator for estimating the unknown fuzzy smooth function. In each stage, the unknown bandwidth is determined with the help of a generalized cross validation criterion. To evaluate and to compare the performance of our proposed fuzzy nonparametric regression model with established regression models, different goodness-of-fit criteria are considered. Finally, we investigate the practical applicability and show the superiority of the proposed regression model by means of a simulation study and real data applications.

A Monte Carlo study on the ridge parameter of the seemingly unrelated ridge regression models

The ridge regression approach can also be applied to deal with multicollinearity in the seemingly unrelated regression equations (SURE) models. However, to get the ridge-type estimators of the coefficients in a SURE model, choosing the ridge parameter has always been an important and challenging task. Herein, to cope with this issue, an optimal choice of the ridge parameter is proposed based on the generalized cross-validation (GCV) criterion. Moreover, some existing estimators of this parameter, in the context of the ordinary ridge regression models, are extended to be used in the SURE models. All these estimators, including the proposed and extended ones, and several state-of-the-art alternatives (altogether 36 different estimators) are compared with each other in terms of the GCV criterion. Lastly, an application of the methodology is given on chronic renal failure effect data.

Revisiting Akaike’s Final Prediction Error and the Generalized Cross Validation Criteria in Regression from the Same Perspective: From Least Squares to Ridge Regression and Smoothing Splines

Revisiting Akaike’s Final Prediction Error and the Generalized Cross Validation Criteria in Regression from the Same Perspective: From Least Squares to Ridge Regression and Smoothing Splines

Geographically Weighted Poisson Regression Model with Adaptive Bisquare Weighting Function (Case study: data on number of leprosy cases in Indonesia 2020)

Abstract Geographically Weighted Poisson Regression (GWPR) is a Poisson regression model which is applied on spatial data. The parameter estimation of GWPR is done in each observation location through spatial weighting. This study aims to determine the GWPR model of the number of leprosy cases in each province of Indonesia 2020 and to find the influencing factors. The research uses secondary data collected from Indonesian Ministry of Health and Central Statistics Agency. The spatial weighting is calculated by using the adaptive bisquare function, while the optimum bandwidth is determined by using Generalized Cross-Validation criteria (GCV). The parameter estimation of GWPR uses Maximum Likelihood Estimation (MLE) method. The result of research show that the closed form of Maximum Likelihood (ML) estimator can not be found analytically and that the approximation of ML estimator is found by using Newton-Raphson iterative method. Based on the parameter significance test of the GWPR model, the factors that influenced the number of leprosy cases locally are the percentage of households that have access to proper sanitation, population density, the percentage of people who experience health complaints and outpatient, the number of health workers, the percentage of poor people, the percentage of districts/cities that carry out healthy living community movement (GERMAS) and the percentage of habitable houses. While the factors that globally affected the number of leprosy cases are the percentage of households that have access to proper sanitation, population density, the percentage of people who experience health complaints and outpatient, the number of health workers, the percentage of poor people, the percentage of districts/cities that carry out GERMAS.

A Joint Inversion Algorithm for the Establishment of High-Accuracy 3-D Marine Gravity Field

The 3-D marine gravity field plays an important role in many practical applications, such as matching navigation, object monitoring, and resource exploitation. However, the accuracy of the gravity field generated by most existing methods is relatively low. To solve this problem, this study innovatively puts forward a new method for establishing a high-accuracy 3-D marine gravity field by using a joint inversion algorithm. The presented method incorporates gravity fields observed at different altitudes, especially those observed near the seafloor, because they contain more prominent short-wavelength signals of the sources. The final joint inversion formulas are developed from the standard form of independent inversion of single gravity datasets, so it is conducive to include different weighting matrices. To achieve the efficiency of the presented method, the nonlinear objective function is minimized iteratively using a fast randomized singular value decomposition (RSVD) technique. The employing of RSVD also facilitates the determination of an optimal regularization parameter based on the generalized cross-validation criterion. The presented method is tested on both synthetic gravity fields simulated by a theoretical density model and real gravity fields collected in the South China Sea. Numerical results demonstrate that the presented method can provide more accurate 3-D gravity fields than the commonly used iterative downward continuation method in the wavenumber domain (DCW) and the conventional equivalent source (CES) method.

A novel strategy for force identification of nonlinear structures

Dynamic force is the key indicator for monitoring the condition of a mechanical product. These mechanical structures always encompass some nonlinear factors. Most previous studies focused on obtaining the dynamic force of linear structures. Consequently, this study focuses on the nonlinear mechanical structure, and a novel identification strategy is proposed to indirectly identify the excitation force. For the identification strategy, based on a nonlinear state-space model, a force identification equation for the nonlinear structure is built, wherein the transfer matrix consists of coefficient matrices of the nonlinear state-space model, and these coefficient matrices are calculated by a nonlinear subspace identification algorithm. Then, under the generalized cross-validation criterion, the truncated total least squares method is introduced to solve the ill-posed equation to eventually obtain the excitation force of the nonlinear structure. The identification results from two numerical simulation cases and one experimental case illustrate that the proposed identification strategy can stably and accurately identify the excitation force of nonlinear structures.

Non-contact detection of single-cell lead-acid battery electrodes’ defects through conductivity reconstruction by magnetic induction tomography

The change of electrodes’ conductivity is a crucial parameter during battery aging process, non-contact detection of battery electrodes’ defects through conductivity reconstruction is an innovative technology. In this paper, the magnetic induction tomography (MIT) was applied to reconstruct the conductivity of electrodes, the simplified battery models with complete and broken electrodes were chosen as target A and target Brespectively, and an eight-channel MIT system was designed to measure the change of mutual induced voltage. A method of computing the forward problem called ‘A·E’ was adopted to compute the coefficient matrix for the forward problem and the frequency 12.665kHz was chosen as the excitation frequency. Based on the conductivity reconstruction images, relative errors of reconstructed conductivity and objective function, ‘A·E’ method exhibited advantages in terms of computation time and accuracy compared to coefficient sensitivity method with the simulation data. In order to test the accuracy of ‘A·E’ method, data measured by MIT system was applied in the inverse problem with four regularization parameter selection methods. L-curve criterion and generalized cross-validation criterion identified defects successfully as they were more sensitive to the change of conductivity.

Vehicle's Instantaneous Velocity Reconstruction by Combining GNSS Doppler and Carrier Phase Measurements Through Tikhonov Regularized Kernel Learning

GNSS has become a widely accessible technique for vehicle instantaneous velocimetry. GNSS time difference carrier phase (TDCP) velocimetry can provide high-accuracy displacement increments, through which the between-epoch average velocity can be derived. However, there are always so-called modelling errors in such velocity, i.e., the deviation between the average velocity and the instantaneous one. GNSS Doppler velocimetry offers exactly instantaneous velocity, but its measurement is much noisier. In this work, we propose to integrate TDCP with Doppler for estimating vehicle's instantaneous velocity. The TDCP-derived displacements and the Doppler-derived instantaneous velocity are treated as two sets of measurements, whereas the vehicle's kinematics is represented by kernel model. Rather than directly solving for vehicle's velocity, we indirectly seek for the kernel weights to establish an analytical kernel model of vehicle's motion state. Tikhonov regularization is introduced to deal with the ill-conditioned problem in kernel weights estimation, and it can significantly smooth/denoise the noisy Doppler measurements. The hyperparameters involved are optimized using generalized cross validation criterion. The constructed kernel model can provide vehicle's velocity at any instants, not necessarily at the sampling epochs. The static and dynamic vehicle field experiments demonstrate that the proposed TDCP/Doppler integrated velocimetry can provide both high accuracy and efficiency.

Stable numerical solution of an inverse coefficient problem for a time fractional reaction-diffusion equation

‎In this paper‎, ‎an inverse problem of determining an unknown reaction coefficient in a one-dimensional time-fractional reaction-diffusion equation is considered‎. ‎This inverse problem is generally ill-posed‎. ‎For this reason‎, ‎the mollification regularization technique with the generalized cross-validation criteria will be employed to find an equivalent stable problem‎. ‎Afterward‎, ‎a finite difference marching scheme is introduced to solve this regularized problem‎. ‎The stability and convergence of the numerical solution are investigated‎. ‎In the end‎, ‎some numerical examples are presented to verify the ability and effectiveness of the proposed algorithm‎.

Current distribution reconstruction in switching arcs by means of regularization based on GSVD

The knowledge of the variation of arc current is helpful in improving the breaking performance of a vacuum circuit breaker. As a novel non-intrusive method, arc magnetic performance testing technology has attracted attention. In view of the widely used regularization method of the standard form in arc current reconstructions cannot fully overcome the ill-posed nature of the Biot-Savart equation, the regularization approach based on generalized singular value decomposition (GSVD) is proposed to improve the situation. Moreover, a discrete 3-D arc model with the help of the finite element technique incorporating the current divergence-free and boundary conditions is introduced to improve the authenticity of the model. The performance of the truncated generalized singular value decomposition (TGSVD) and general-form Tikhonov regularization with derivative operators L 1 and L 2 are studied in numerical tests. In addition, the effectiveness of generalized cross validation criterion, quasi optimality criterion and normalized cumulative periodic graph method in selecting regularization parameters are compared. The methodology can help improve arc control and guide the design of vacuum circuit breakers with improved performance parameters.

The potential of generalized additive modelling for the prediction of radial growth of Norway spruce from Central Germany

During the past decades managed forest ecosystems in Central Europe underwent vast changes, induced by extreme climate conditions and occasionally adverse forest management. Tree ring width patterns mirror these changes and thus have been widely examined as environmental archives and reliable empirical data sources in ‘tree growth modelling’. Dendrochronologists often suppose linear co-variation among the covariates, variable independence and homoscedasticity. Conventionally, these assumptions were achieved by eliminating biological age trends (detrending) and removing the autocorrelation from the time series (pre-whitening). Particularly detrending might be biased according to the scientific problem and sometimes inflexible age models. In this study, we tackle these issues and examine the suitability of a flexible Generalized Additive Model (GAM) on recently developed tree ring width time series of 30 Norway spruce stands (Picea abies [L.] H. Karst) from Central Germany.The model was established to simultaneously cope with the mentioned detrending issue, to unravel nonlinear climate-growth relationships and to predict mean ring width time series for spruce stands in the region. Particularly the latter was of primary interest, since recent forest planning relies on static yield tables that often underestimate the actual growth.The model reliably captured the empirical data, indicated by a small Generalized Cross Validation criterion (GCV = 0.045) and a deviance explained of 88.6 %. The flexible additive smoothers accounted for the social status of individual trees, captured low frequency variations of changing growth conditions adequately and displayed a rather flat biological age trend. The radial increment responded positively to summer season precipitation of the current and previous year. Positive temperature responses were found during the early vegetation period, whereas high summer season temperatures negatively affected the radial growth. The seasonal transition from spring to summer in June induced a shift in the climate response of the linear predictor, leading to a distinct negative effect of temperature and a no-role of precipitation on the linear predictor.Most important, utilizing the calibrated GAM for the purely climate-driven prediction of mean ring width time series from five independent spruce sites revealed proper coherencies. Herein, the mean ring width for sites located within the climatic-optimum for spruce growth were more exactly predicted than for sites with adverse spruce growth conditions. In addition, large mean ring widths were systematically underestimated, whereas small mean ring widths were precisely predicted. Overall, we strongly recommend GAMs as a powerful tool for the investigation of nonlinear climate-growth relationships and for the prediction of radial growth in managed forest ecosystems.

Comparative studies on the criteria for regularization parameter selection based on moving force identification

The studies on inverse problems exist extensively in aerospace, mechanical, identification, detection, scanning imaging and other fields. Its ill-posed characteristics often lead to large oscillations in the solution of the inverse problem. In this study, the truncated generalized singular value decomposition (TGSVD) method is introduced to identify two kinds of moving forces, single and multi-axial forces. The truncating point is the most influential regularization parameter of TGSVD, which is initially selected by two classic regularization parameter selection criteria, namely, the L-curve criterion and the generalized cross-validation (GCV) criterion. Due to numerical non-uniqueness and noise disturbance in moving force identification (MFI), numerical simulation results show that neither of the two criteria can effectively help select the optimal truncating point of TGSVD. Hence, a relative percentage error (RPE) criterion is proposed for selecting the truncating point of TGSVD. Comparative studies show that the RPE criterion can be used to select the optimal truncating point of TGSVD more accurately against the GCV criterion and L-curve criterion. Moreover, the RPE criterion can be used to reflect the connections between certain properties and the ill-posedness problem existing in MFI, which should be adopted priority for the optimal truncating point selection of TGSVD.

Generalized Cross-Validation for Simultaneous Optimization of Tuning Parameters in Ridge Regression

When multicollinearity exists in the context of robust regression, ridge rank regression estimator can be used as an alternative to the rank estimator. Performance of the ridge rank regression estimator is highly dependent on the ridge parameter, here the tuning parameter. On the other hand, suppose we are provided with some non-sample uncertain prior information (UPI) about regression coefficients. Shrinkage estimation is a well-known strategy to improve estimation, under the UPI, where the amount of shrinkage is controlled by a tuning parameter. Hence, optimization of both tuning parameters can be a problem of interest. In this study, theoretical development of the generalized cross-validation (GCV) is considered and some numerical illustrations are given to validate the theoretical findings. Our results demonstrated that using the proposed GCV criterion, the shrinkage ridge rank regression estimator behaves well in the sense of minimum risk function.

Visualization of rainfall data using functional data analysis

In recent years, functional data analysis (FDA) has been used in many applications in order to analyze data that provide detail on curves, surfaces or other components of a continuum. The purpose of this research is mainly to incorporate and adapt visualization tools for the FDA of rainfall in the Taiz Region. Moreover, visualizing the temporal variations of rainfall in order to provide a clear understanding of the rainfall patterns and also to help in predicting the future. The current study has been conducted based on average monthly rainfall over the Taiz Region during the period of 1998–2018. The R software was used to process the data obtained from the Tropical Rainfall Measuring Mission in Taiz City. The functional rainfall data have been smoothed using penalized smoothing according to generalized cross-validation criteria. The results showed that the rainfall profiles in the Taiz Region depend significantly on their temporal locations due to the monsoonal influences, which reflect the distribution of rains in the spring and the summer seasons.

Regularized Cubic B-Spline Collocation Method With Modified L-Curve Criterion for Impact Force Identification

The time history of the impact force, especially for the peak force, is vital to monitor the performance of mechanical products over the life-time. However, considering the limitation of sensing technology and inaccessibility of installing, it is always difficult or even impossible to measure the force directly in engineering practice. Therefore, a regularized cubic B-spline collocation (RCBSC) method combined with the modified L-curve criterion (RCBSC-ML) is presented to identify the impact force by easily measurable dynamic response. Because the profile of cubic B-spline is close to that of impact force, a linear combination of cubic B-spline functions is used to approximate the unknown impact force. Then the force identification equation is reformed, wherein the unknown variable changes from impact force vector to weight coefficient vector. Furthermore, the modified L-curve criterion is developed to select the optimal number of collocation points which is the regularization parameter of RCBSC method. Rich simulation studies on a five degree-of-freedoms structure and experimental studies on a simplified artillery test-bed are performed to validate the performance of RCBSC-ML method. The results illustrate that RCBSC-ML method can be used to accurately identify the unknown impact force, and compared with modified generalized cross validation criterion, the presented modified L-curve criterion is more suitable to determine the optimal regularization parameter of RCBSC method. Furthermore, compared with classical Tikhonov regularization method, RCBSC-ML method is more efficient and accurate in impact force identification.

Bandlimited force identification based on sinc-dictionaries and Tikhonov regularization

A bandlimited force identification approach is proposed in this paper. It consists of two main ingredients. On the one hand, the sinc-dictionaries are established for bandlimited force representation. This motivates from the sampling formula that a bandlimited signal can be well interpolated by sinc functions. It is also noteworthy that using the sinc-dictionaries, the amount of variables is dramatically reduced. On the other hand, the Tikhonov regularization is introduced to circumvent the possible ill-posedness of force identification. In proceeding so, the generalized cross validation criterion is adopted to determine a proper regularization parameter. Numerical examples are conducted to testify the accuracy and efficiency of the proposed approach for bandlimited force identification.

Stabilized Solution for a Time-Fractional Inverse Problem with an Unknown Nonlinear Condition

In this paper, we consider a time-fractional inverse problem in which the nonlinear boundary conditions contain an unknown function. A finite difference scheme will be proposed to solve numerically the inverse problem. This inverse problem is generally ill-posed. For this reason, we will employ the mollification regularization method with the generalized cross-validation criterion to find a stable solution. The stability and convergence of numerical solutions are investigated. Finally, some numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

A Novel Approach for Solving an Inverse Reaction–Diffusion–Convection Problem

In this paper, we consider an inverse reaction–diffusion–convection problem in which one of the boundary conditions is unknown. A sixth-kind Chebyshev collocation method will be proposed to solve numerically this problem and to obtain the unknown boundary function. Since this inverse problem is generally ill-posed, to find an optimal stable solution, we will utilize a regularization method based on the mollification technique with the generalized cross-validation criterion. The error estimate of the numerical solution is investigated. Finally, to authenticate the validity and effectiveness of the proposed algorithm, some numerical test problems are presented.