Multiple Nonlinear Regression Model Research Articles

Utilizing regression models to find functions for determining ripping production based on laboratory tests

The selection of suitable overburden loosening method has a crucial importance in several applications of geotechnical engineering. Factors such as rock properties and environmental constrains play a significant role in the selection of the needed equipment for overburden loosening. This paper presents several new models/equations for prediction of ripping production (Q) using rock material properties. To this end, three sites in Malaysia were selected and a total of 52 direct ripping tests were conducted in Johor state on sandstone and shale rock types. In addition, using the collected block samples, point load test, Brazilian test, slake-durability test, p-wave velocity test and uniaxial compressive strength test were also carried out in the laboratory. Numerous equations have been proposed to predict Q considering simple regression, linear multiple regression (LMR), and non-linear multiple regression (NLMR) models. Simple regression analysis indicated that the relationships between rock material properties and Q were meaningful and acceptable. Furthermore, both LMR and NLMR equations indicated similar performance capacity in predicting Q. Nevertheless; the use of NLMR equations resulted in prediction performance with higher accuracy in estimating Q compared to LMR equations.

Response surface optimization, modeling and uncertainty analysis of mass loss response of co-combustion of sewage sludge and water hyacinth

The present study aims at quantifying mass loss percentage (MLP, %) predictions and their stochastic uncertainty when co-combustion of sewage sludge (SS) and water hyacinth (WH) are applied as alternative biomass materials under different blend ratios (BR), heating rates (HR, °C/min) and temperatures (T, °C). Optimization and validation of experimental data through Box–Behnken design pointed to 630.9°C for T, 60.1% SS for BR, and 29.9°C/min for HR as the optimal co-combustion parameters to achieve the maximum MLP of 92.4%. Monte Carlo (MC) simulations were used to quantify uncertainty in MLP predictions of the best-fit multiple non-linear regression (MNLR) model derived from the entire experimental data as a function of MC-generated T as the only continuous predictor of the MNLR. Mean MLP value of the MNLR predictions was higher by 19% than that of the MC-simulated T whose mean was higher by only 1% than mean measured T. Incorporating the uncertainty estimation based on Monte Carlo simulations with response surface approach for co-combustion of SS and WH was one of the main novel contributors of the present study to related literature.

The prediction of ultimate pure bending moment of concrete-filled steel tubes by adaptive neuro-fuzzy inference system (ANFIS)

In this study, different modelling techniques such as multiple regression and adaptive neuro-fuzzy inference system (ANFIS) are used for predicting the ultimate pure bending of concrete-filled steel tubes (CFTs). The behaviour of CFT under pure bending is complex and highly nonlinear; therefore, forward modelling techniques can have considerable limitations in practical situations where fast and reliable solutions are required. Linear multiple regression (LMR), nonlinear multiple regression (NLMR) and ANFIS models were trained and checked using a large database that was constructed and populated from the literature. The database comprises 72 pure bending tests conducted on fabricated and cold-formed tubes filled with concrete. Out of 72 tests, 48 tests were conducted by the second author. Input variables for the models are the same with those used by existing codes and practices such as the tube thickness, tube outside diameter, steel yield strength, strength of concrete and shear span. A practical application example, showing the translation of constructed ANFIS model into design equations suitable for hand calculations, was provided. A sensitivity analysis was conducted on ANFIS and multiple regression models. It was found that the ANFIS model is more sensitive to change in input variables than LMR and NLMR models. Predictions from ANFIS models were compared with those obtained from LMR, NLMR, existing theory and a number of available codes and standards. The results indicate that the ANFIS model is capable of predicting the ultimate pure bending of CFT with a high degree of accuracy and outperforms other common methods.

Ripping Production Prediction in Different Weathering Zones According to Field Data

In response to the environmental restrictions and the blasting problems, ripping method as a surface excavation method is the most commonly-used in construction of many civil engineering systems. So, it is essential to provide a more applicable rippability model that can effectively predict ripping production (Q) in the field. This paper presents several new models/equations for prediction of Q in diverse weathering zones (grade from II to V) based on field observations and in situ tests. To do this, four sites in Johor state, Malaysia were selected and a total of 123 direct ripping tests were carried out on two types of sedimentary rocks, namely, sandstone and shale. Based on literature’s suggestions and possible conducted field works, point load strength index, sonic velocity, Schmidt hammer rebound number and joint spacing were chosen to estimate Q in different weathering zones. Then, simple and multiple regression analyses, namely linear multiple regression (LMR) and non-linear multiple regression (NLMR) were performed to predict Q. The simple regression analysis generally showed an acceptable and meaningful correlation between the Q and input variables. Additionally, a range of 0.582–0.966 was obtained for coefficient of determination (R2) values of developed LMR models while this range was observed from 0.586 to 0.949 for proposed NLMR models. As a result, both the LMR and NLMR models deliver almost the same predictive performance in estimating the Q for various weathering zones. Nevertheless, in most of the cases, NLMR models can provide higher performance prediction in estimating Q compared to LMR models.

Risk assessment and prediction of safety factor for circular failure slope using rock engineering systems

Circular failure is generally observed in the slope of soil, highly jointed rock mass, mine dump and weak rock. Accurate estimation of the safety factor (SF) of slopes and their performance is not an easy task. In this research, based on rock engineering systems (RES), a new approach for the estimation of the SF is presented. The introduced model involves six effective parameters on SF [unit weight (γ), pore pressure ratio (r u), height (H), angle of internal friction (φ), cohesion (C) and slope angle ( $$\beta$$ )], while retaining simplicity as well. In the case of SF prediction, all the datasets were divided randomly to training and testing datasets for proposing the RES model. For comparison purposes, nonlinear multiple regression models were also employed for estimating SF. The performances of the proposed predictive models were examined according to two performance indices, i.e., coefficient of determination (R 2) and mean square error. The obtained results of this study indicated that the RES is a reliable method to predict SF with a higher degree of accuracy in comparison with nonlinear multiple regression models.

Regression analysis of major parameters affecting the intensity of coal and gas outbursts in laboratory

Estimating the intensity of outbursts of coal and gas is important as the intensity and frequency of outbursts of coal and gas tend to increase in deep mining. Fully understanding the major factors contributing to coal and gas outbursts is significant in the evaluation of the intensity of the outburst. In this paper, we discuss the correlation between these major factors and the intensity of the outburst using Analysis of Variance (ANOVA) and Contingency Table Analysis (CTA). Regression analysis is used to evaluate the impact of these major factors on the intensity of outbursts based on physical experiments. Based on the evaluation, two simple models in terms of multiple linear and nonlinear regression were constructed for the prediction of the intensity of the outburst. The results show that the gas pressure and initial moisture in the coal mass could be the most significant factors compared to the weakest factor - porosity. The P values from Fisher’s exact test in CTA are: moisture (0.019), geostress (0.290), porosity (0.650), and gas pressure (0.031). P values from ANOVA are moisture (0.094), geostress (0.077), porosity (0.420), and gas pressure (0.051). Furthermore, the multiple nonlinear regression model (RMSE: 3.870) is more accurate than the linear regression model (RMSE: 4.091).

Prediction and minimization of blast-induced ground vibration using two robust meta-heuristic algorithms

During blasting operations in open pit mines, most of the blast energy (approximately 40%) is wasted due to ground vibration. This phenomenon causes damages to the surrounding structures and pit slope. Therefore, prediction of ground vibration with an appropriate degree of accuracy is important to identify safety area of blasting. In this paper, in the first step, a predictive equation based on gene expression programming (GEP) was developed to estimate ground vibrations of blasting operations conducted in Gol-E-Gohar iron mine. For this aim, 115 blasting operations were identified and the most effective parameters on peak particle velocity (PPV), i.e., burden, spacing, stemming, hole-depth, hole-diameter, powder factor, maximum charge per delay and distance from the blast face were collected from the mine. Capability of the developed GEP model was compared with a non-linear multiple regression (NLMR) model and five conventional equations. The obtained results revealed that the developed GEP model is more efficient compared to the other models in predicting PPV. In the second step, to optimize the GEP predictive model, cuckoo optimization algorithm (COA) was employed and proposed. To do this, several strategies were defined and then several optimized blasting patterns were determined for each strategy. It was found that by developing the COA model, a significant reduction can be happened in PPV values.

Optimization of hard modified asphalt formula for gussasphalt based on uniform experimental design

Surfacing steel decks is one of the key technical issues facing engineers during construction of long-span steel bridges. The objective of this study was to improve the high-temperature performance of gussasphalt (GA), one of the preferred surfacing materials for steel deck bridges around the world, by optimizing its binder formula. Three types of asphalt binders (i.e., styrene-butadienestyrene [SBS] modified asphalt, Trinidad lake asphalt [TLA], and NES-1 rock asphalt) were selected as basic constituents to achieve an optimized asphalt binder, which is known as hard modified asphalt. Abiding by the principles of uniform experimental design, a series of laboratory tests for penetration, ductility, ring-and-ball (R&B) softening point, and viscosity were conducted to assess and optimize the binder performance. With the laboratory testing results, quadratic polynomial stepwise regression analysis was performed to obtain four nonlinear multiple regression models for the relationships between performance and compositions of the hard modified asphalt. It is shown that NES-1 rock asphalt has more significant influence than Trinidad lake asphalt on improving the high-temperature performance of the hard modified asphalt. In order to ensure mixing workability and low-temperature performance of gussasphalt, the contents of NES-1 rock asphalt and Trinidad lake asphalt should not exceed 5% and 30%, respectively. The optimum formula of hard modified asphalt for gussasphalt is 75% SBS modified asphalt, 20% Trinidad lake asphalt, and 5% NES-1 rock asphalt. The performance of the optimized hard modified asphalt was validated by laboratory tests.

Estimation of biogas and methane yields in an UASB treating potato starch processing wastewater with backpropagation artificial neural network

Three-layered feedforward backpropagation (BP) artificial neural networks (ANN) and multiple nonlinear regression (MnLR) models were developed to estimate biogas and methane yield in an upflow anaerobic sludge blanket (UASB) reactor treating potato starch processing wastewater (PSPW). Anaerobic process parameters were optimized to identify their importance on methanation. pH, total chemical oxygen demand, ammonium, alkalinity, total Kjeldahl nitrogen, total phosphorus, volatile fatty acids and hydraulic retention time selected based on principal component analysis were used as input variables, whiles biogas and methane yield were employed as target variables. Quasi-Newton method and conjugate gradient backpropagation algorithms were best among eleven training algorithms. Coefficient of determination (R2) of the BP-ANN reached 98.72% and 97.93% whiles MnLR model attained 93.9% and 91.08% for biogas and methane yield, respectively. Compared with the MnLR model, BP-ANN model demonstrated significant performance, suggesting possible control of the anaerobic digestion process with the BP-ANN model.

Caracterizando a interação entre aguas subterrâneas e superficiais no aquífero cárstico de Fangshan, Pequim (China)

Correct understanding of groundwater/surface-water (GW–SW) interaction in karst systems is of greatest importance for managing the water resources. A typical karst region, Fangshan in northern China, was selected as a case study. Groundwater levels and hydrochemistry analyses, together with isotope data based on hydrogeological field investigations, were used to assess the GW–SW interaction. Chemistry data reveal that water type and the concentration of cations in the groundwater are consistent with those of the surface water. Stable isotope ratios of all samples are close to the local meteoric water line, and the 3H concentrations of surface water and groundwater samples are close to that of rainfall, so isotopes also confirm that karst groundwater is recharged by rainfall. Cross-correlation analysis reveals that rainfall leads to a rise in groundwater level with a lag time of 2 months and groundwater exploitation leads to a fall within 1 month. Spectral analysis also reveals that groundwater level, groundwater exploitation and rainfall have significantly similar response periods, indicating their possible inter-relationship. Furthermore, a multiple nonlinear regression model indicates that groundwater level can be negatively correlated with groundwater exploitation, and positively correlated with rainfall. The overall results revealed that groundwater level has a close correlation with groundwater exploitation and rainfall, and they are indicative of a close hydraulic connection and interaction between surface water and groundwater in this karst system.

Feasibility of PSO–ANFIS model to estimate rock fragmentation produced by mine blasting

Desired rock fragmentation is the main goal of the blasting operation in surface mines, civil and tunneling works. Therefore, precise prediction of rock fragmentation is very important to achieve an economically successful outcome. The primary objective of this article is to propose a new model for forecasting the rock fragmentation using adaptive neuro-fuzzy inference system (ANFIS) in combination with particle swarm optimization (PSO). The proposed PSO–ANFIS model has been compared with support vector machines (SVM), ANFIS and nonlinear multiple regression (MR) models. To construct the predictive models, 72 blasting events were investigated, and the values of rock fragmentation as well as five effective parameters on rock fragmentation, i.e., specific charge, stemming, spacing, burden and maximum charge used per delay were measured. Based on several statistical functions [e.g., coefficient of correlation (R 2) and root-mean-square error (RMSE)], it was found that the PSO–ANFIS (with R 2 = 0.89 and RMSE = 1.31) performs better than the SVM (with R 2 = 0.83 and RMSE = 1.66), ANFIS (with R 2 = 0.81 and RMSE = 1.78) and nonlinear MR (with R 2 = 0.57 and RMSE = 3.93) models. Finally, the sensitivity analysis shows that the burden and maximum charge used per delay have the least and the most effects on the rock fragmentation, respectively.

Modelling of adsorption in rotating packed bed using artificial neural networks (ANN)

Rotating packed bed (RPB) exhibits good performance for absorption, distillation and extraction. However, the development of RPB was restricted by complicated liquid flow pattern on activated carbon in adsorption. Therefore, a model of artificial neural network (ANN) was adopted to predict the adsorption in RPB. The experimental data were classified into two groups, and they were training ones for establishing the model and testing ones for validating the model. Different types of ANN models, including Cascade-forward back propagation neural network (CFBPNN), Elman-forward back propagation neural network (EFBPNN) and Feed-forward back propagation neural network (FFBPNN), were investigated in this study. High gravity factor, liquid Reynolds number, adsorption time to the maximum adsorption time and packing density to liquid concentration were used as input data. While, the adsorption amount to the maximum adsorption amount was taken as output data for each model. Optimal hidden neurons for FFBPNN, EBPNN, CFBPNN were 9, 12 and 8, respectively. Experimental adsorption amount data were in good agreement with the predicted data indicating that the ANN models had a superior performance. Besides, the ANN models exhibited a more accurate prediction and had more generalization ability than multiple nonlinear regression model (MNR). The proposed FFBPNN model of absorption in RPB will provide an optimization tool to maximize the adsorption amount.

Likelihood ratio tests for a dose-response effect using multiple nonlinear regression models.

We consider the problem of testing for a dose-related effect based on a candidate set of (typically nonlinear) dose-response models using likelihood-ratio tests. For the considered models this reduces to assessing whether the slope parameter in these nonlinear regression models is zero or not. A technical problem is that the null distribution (when the slope is zero) depends on non-identifiable parameters, so that standard asymptotic results on the distribution of the likelihood-ratio test no longer apply. Asymptotic solutions for this problem have been extensively discussed in the literature. The resulting approximations however are not of simple form and require simulation to calculate the asymptotic distribution. In addition, their appropriateness might be doubtful for the case of a small sample size. Direct simulation to approximate the null distribution is numerically unstable due to the non identifiability of some parameters. In this article, we derive a numerical algorithm to approximate the exact distribution of the likelihood-ratio test under multiple models for normally distributed data. The algorithm uses methods from differential geometry and can be used to evaluate the distribution under the null hypothesis, but also allows for power and sample size calculations. We compare the proposed testing approach to the MCP-Mod methodology and alternative methods for testing for a dose-related trend in a dose-finding example data set and simulations.

Estimation of daily PM2.5 concentration and its relationship with meteorological conditions in Beijing

When investigating the impact of air pollution on health, particulate matter less than 2.5 μm in aerodynamic diameter (PM2.5) is considered more harmful than particulates of other sizes. Therefore, studies of PM2.5 have attracted more attention. Beijing, the capital of China, is notorious for its serious air pollution problem, an issue which has been of great concern to the residents, government, and related institutes for decades. However, in China, significantly less time has been devoted to observing PM2.5 than for PM10. Especially before 2013, the density of the PM2.5 ground observation network was relatively low, and the distribution of observation stations was uneven. One solution is to estimate PM2.5 concentrations from the existing data on PM10. In the present study, by analyzing the relationship between the concentrations of PM2.5 and PM10, and the meteorological conditions for each season in Beijing from 2008 to 2014, a U-shaped relationship was found between the daily maximum wind speed and the daily PM concentration, including both PM2.5 and PM10. That is, the relationship between wind speed and PM concentration is not a simple positive or negative correlation in these wind directions; their relationship has a complex effect, with higher PM at low and high wind than for moderate winds. Additionally, in contrast to previous studies, we found that the PM2.5/PM10 ratio is proportional to the mean relative humidity (MRH). According to this relationship, for each season we established a multiple nonlinear regression (MNLR) model to estimate the PM2.5 concentrations of the missing periods.

Function development for appraising brittleness of intact rocks using genetic programming and non-linear multiple regression models

Brittleness of rock is one of the most critical features for design of underground excavation project. Therefore, proper assessing of rock brittleness can be very useful for designers and evaluators of geotechnical applications. In this study, feasibility of genetic programming (GP) model and non-linear multiple regression (NLMR) in predicting brittleness of intact rocks is examined. For this purpose, a dataset developed by conducting various rock tests including uniaxial compressive strength, Brazilian tensile strength, unit weight and brittleness via punch penetration on rock samples gathered from 48 tunnels projects around the world is utilized herein. Considering multiple inputs, several GP models were constructed to estimate brittleness index of the rock and finally, the best GP model was selected. Note that, GP can make an equation for predicting output of the system using model inputs. To show applicability of the developed GP model, non-linear multiple regression (NLMR) was also applied and developed. Considering some model performance indices, performance prediction of the GP and NLMR models were evaluated and it was found that the GP model is superior to NLMR one. Based on coefficient of determination (R2) of testing datasets, by proposing GP model, it can be improved from 0.882 (obtained by NLMR model) to 0.904. It is worth mentioning that the proposed predictive models in this study should be planned and used for the similar types of rock and the established inputs ranges.

Risk assessment and prediction of rock fragmentation produced by blasting operation: a rock engineering system

Poor fragmentation is one of the most side effects induced by blasting operations. Therefore, risk assessment and prediction of rock fragmentation are essential to reduce the mentioned effects. In the present study, an attempt has been made to evaluate the risk associated with rock fragmentation as well as its prediction at Sarcheshmeh copper mine, Iran, proposing the rock engineering system (RES) technique. A total number of 52 blasting events were collected and considered and the values of 10 key effective parameters in rock fragmentation were carefully measured in the mine. These 10 key parameters were only related to blasting design and rock mass properties were not considered in the analysis of this study due to some limitations regarding their measurements in the mine. The RES result showed that the level of overall risk, based on the considered blast events, is in the range of medium–high. Furthermore, it was found that the burden is the most interaction factor in the rock fragmentation. In case of rock fragmentation prediction, all of datasets were divided randomly to training and testing datasets for proposing RES model. For comparison purpose, non-linear multiple regression (NLMR) was also employed for estimating rock fragmentation. The performances of the proposed predictive models were examined according to three performance indices, i.e. coefficient of determination (R 2), root mean square error (RMSE) and variance account for (VAF). The obtained results of this study indicated that the RES is a reliable method to predict rock fragmentation with a higher degree of accuracy in comparison to NLMR model. For instance, RMSE values of 1.95 and 4.002 for testing datasets of RES and NLMR models, respectively, suggest the superiority of the RES model in predicting rock fragmentation compared to other developed model.

Prediction of ground vibration due to quarry blasting based on gene expression programming: a new model for peak particle velocity prediction

Blasting is a widely used technique for rock fragmentation in opencast mines and tunneling projects. Ground vibration is one of the most environmental effects produced by blasting operation. Therefore, the proper prediction of blast-induced ground vibrations is essential to identify safety area of blasting. This paper presents a predictive model based on gene expression programming (GEP) for estimating ground vibration produced by blasting operations conducted in a granite quarry, Malaysia. To achieve this aim, a total number of 102 blasting operations were investigated and relevant blasting parameters were measured. Furthermore, the most influential parameters on ground vibration, i.e., burden-to-spacing ratio, hole depth, stemming, powder factor, maximum charge per delay, and the distance from the blast face were considered and utilized to construct the GEP model. In order to show the capability of GEP model in estimating ground vibration, nonlinear multiple regression (NLMR) technique was also performed using the same datasets. The results demonstrated that the proposed model is able to predict blast-induced ground vibration more accurately than other developed technique. Coefficient of determination values of 0.914 and 0.874 for training and testing datasets of GEP model, respectively show superiority of this model in predicting ground vibration, while these values were obtained as 0.829 and 0.790 for NLMR model.

Development of a new model for predicting flyrock distance in quarry blasting: a genetic programming technique

This research was aimed at developing a new model to predict flyrock distance based on a genetic programming (GP) technique. For this purpose, six granite quarry mines in the Johor area of Malaysia were investigated, for which various controllable blasting parameters were recorded. A total of 262 datasets consisting of six variables (i.e., powder factor, stemming length, burden-to-spacing ratio, blast-hole diameter, maximum charge per delay, and blast-hole depth) were collected applied to developing the flyrock predictive model. To identify the optimum model, several GP models were developed to predict flyrock. In the same way, using non-linear multiple regression (NLMR) analysis, various models were established to predict flyrock. Finally, to compare the performance of the developed models, regression coefficient (R2), root mean square error (RMSE), variance account for (VAF), and simple ranking methods were computed. According to the results obtained from the test dataset, the best flyrock predictive model was found to be the GP based model, with R2 = 0.908, RMSE = 17.638 and VAF = 89.917, while the corresponding values for R2, RMSE and VAF for the NLMR model were 0.816, 26.194, and 81.041, respectively.

Constitutive equations of 1060 pure aluminum based on modified double multiple nonlinear regression model

In order to study the work-ability and establish the optimum hot formation processing parameters for industrial 1060 pure aluminum, the compressive deformation behavior of pure aluminum was investigated at temperatures of 523–823 K and strain rates of 0.005–10 s−1 on a Gleeble–1500 thermo-simulation machine. The influence rule of processing parameters (strain, strain rate and temperature) on flow stress of pure aluminum was investigated. Nine analysis factors consisting of material parameters and according weights were optimized. Then, the constitutive equations of multilevel series rules, multilevel parallel rules and multilevel series & parallel rules were established. The correlation coefficients (R) are 0.992, 0.988 and 0.990, respectively, and the average absolute relative errors (AAREs) are 6.77%, 8.70% and 7.63%, respectively, which proves that the constitutive equations of multilevel series rules can predict the flow stress of pure aluminum with good correlation and precision.

RETRACTED ARTICLE: Function finding for indirect determination of rock brittleness based on genetic programming and non-linear multiple regression models

Brittleness of the rock is one of the most critical features for design of underground excavation project. Therefore, proper estimating of rock brittleness can be very useful for designers and evaluators of geotechnical applications. In this study, feasibility of genetic programming (GP) model in predicting brittleness of the rock samples is examined. In this regard, several laboratory tests including uniaxial compressive strength, Brazilian, density and punch penetration were conducted on 48 rock samples collected from various locations of India. Considering multiple-inputs, several GP models were constructed to estimate brittleness index (BI) of the rock and finally, the best GP model was selected. Note that, GP can make an equation for predicting output of the system using model inputs. To show applicability of the developed GP model, non-linear multiple regression (NLMR) was also applied and developed. Considering some model performance indices, performance prediction of the GP and NLMR models were evaluated and it was found that the GP model is superior to NLMR one. Based on coefficient of determination (R2) of testing datasets, by proposing GP model, it can be improved from 0.882 (which was obtained by NLMR model) to 0.904. It is worth mentioning that the proposed predictive models in this study should be planned and used for the same conditions and the established inputs ranges.