Multivariate adaptive regression splines for analysis of geotechnical engineering systems

With the rapid increases in processing speed and memory of low-cost computers, it is not surprising that various advanced computational learning tools such as neural networks have been increasingly used for analyzing or modeling highly nonlinear multivariate engineering problems. These algorithms are useful for analyzing many geotechnical problems, particularly those that lack a precise analytical theory or understanding of the phenomena involved. In situations where measured or numerical data are available, neural networks have been shown to offer great promise for mapping the nonlinear interactions (dependency) between the system’s inputs and outputs. Unlike most computational tools, in neural networks no predefined mathematical relationship between the dependent and independent variables is required. However, neural networks have been criticized for its long training process since the optimal configuration is not known a priori. This paper explores the use of a fairly simple nonparametric regression algorithm known as multivariate adaptive regression splines (MARS) which has the ability to approximate the relationship between the inputs and outputs, and express the relationship mathematically. The main advantages of MARS are its capacity to produce simple, easy-to-interpret models, its ability to estimate the contributions of the input variables, and its computational efficiency. First the MARS algorithm is described. A number of examples are then presented that explore the generalization capabilities and accuracy of this approach in comparison to the back-propagation neural network algorithm.     

http://www.sciencedirect.com/science/article/pii/S1674987114001364

Piles are long, slender structural elements used to transfer the loads from the superstructure through weak strata onto stiffer soils or rocks. For driven piles, the impact of the piling hammer induces compression and tension stresses in the piles. Hence, an important design consideration is to check that the strength of the pile is sufficient to resist the stresses caused by the impact of the pile hammer. Due to its complexity, pile drivability lacks a precise analytical solution with regard to the phenomena involved.In situations where measured data or numerical hypothetical results are available, neural networks stand out in mapping the nonlinear interactions and relationships between the system's predictors and dependent responses. In addition, unlike most computational tools, no mathematical relationship assumption between the dependent and independent variables has to be made. Nevertheless, neural networks have been criticized for their long trial-and-error training process since the optimal configuration is not known a priori. This paper investigates the use of a fairly simple nonparametric regression algorithm known as multivariate adaptive regression splines(MARS), as an alternative to neural networks, to approximate the relationship between the inputs and dependent response, and to mathematically interpret the relationship between the various parameters. In this paper, the Back propagation neural network(BPNN) and MARS models are developed for assessing pile drivability in relation to the prediction of the Maximum compressive stresses(MCS), Maximum tensile stresses(MTS), and Blow per foot(BPF). A database of more than four thousand piles is utilized for model development and comparative performance between BPNN and MARS predictions.     

Multivariate Adaptive Regression Spline (Mars) for Prediction of Elastic Modulus of Jointed Rock Mass

This article presents multivariate adaptive regression spline (MARS) for determination of elastic modulus (E_j) of jointed rock mass. MARS is a technique to estimate general functions of high-dimensional arguments given sparse data. It is a nonlinear and non-parametric regression methodology. The input variables of model are joint frequency (J_n), joint inclination parameter (n), joint roughness parameter (r), confining pressure (σ₃) and elastic modulus (E_i) of intact rock. The developed MARS gives an equation for determination of E_j of jointed rock mass. The results from the developed MARS model have been compared with those of artificial neural networks (ANNs) using average absolute error. The developed MARS gives a robust model for determination of E_j of jointed rock mass.     

Reliability Analysis of Quick Sand Condition

First order reliability method (FORM) is generally used for reliability analysis in geotechnical engineering. This article adopts generalized regression neural network (GRNN) based FORM, Gaussian process regression (GPR) based FORM and multivariate adaptive regression spline (MARS) based FORM for reliability analysis of quick sand condition. GRNN is related to the radial basis function (RBF) network. GPR is developed based on probabilistic framework. MARS is a nonparametric regression technique. A comparative study has been carried out between the developed models. The performance of GPR based FORM and MARS based FORM match well with the FORM. This article gives the alternative methods for reliability analysis of quick sand condition.     

Determination of ultimate capacity of driven piles in cohesionless soil: A Multivariate Adaptive Regression Spline approach

The determination of ultimate capacity (Q) of driven piles in cohesionless soil is an important task in geotechnical engineering. This article adopts Multivariate Adaptive Regression Spline (MARS) for prediction Q of driven piles in cohesionless soil. MARS uses length (L), angle of shear resistance of the soil around the shaft (?_shaft), angle of shear resistance of the soil at the tip of the pile (?_tip), area (A), and effective vertical stress at the tip of the pile as input variables. Q is the output of MARS. The results of MARS are compared with that of the Generalized Regression Neural Network model. An equation has been also presented based on the developed MARS. The results show the strong potential of MARS to be applied to geotechnical engineering as a regression tool. Copyright © 2011 John Wiley & Sons, Ltd.     

A Methodology for the Definition of Geotechnical Mine Sectors Based on Multivariate Cluster Analysis

This paper offers a new method for the definition of geotechnical sectors in open pit mines based on multivariate cluster analysis. A geological-geotechnical data set of a manganese open pit mine was used to demonstrate the methodology. The data set consists of a survey of geological and geotechnical parameters of the rock mass, measured directly in several points of the mine, structured initially in twenty-eight variables. After the preprocessing of the data set, the clustering technique was applied using the k-Prototype algorithm. The squared Euclidean distance was used to quantify the proximity between numerical variables, and the Jaccard's coefficient of similarity was used to quantify the proximity between the nominal variables. The different cluster results obtained were validated by the multivariate analysis of variance. The identification of cluster structures was achieved by plotting them on the mine map for spatial visualization and definition of geotechnical sectors. These sectors are spatially contiguous and relatively homogeneous regarding their geological–geotechnical properties, indicated by a high density of points of the same group. It was possible to observe a great adherence of the proposed sectors to the mine geology, demonstrating the practical representativeness of the clustering results and the proposed sectors.

     

对岩土工程数值分析的几点思考

首先,介绍了笔者对我国岩土工程数值分析现状的调查结果及分析,然后,分析了采用连续介质力学分析岩土工程问题的关键,并讨论分析了岩土本构理论发展现状,提出对岩土本构理论发展方向的思考,最后对数值分析在岩土工程分析中的地位作了分析。分析表明,岩土工程数值分析结果是岩土工程师在岩土工程分析过程中进行综合判断的重要依据之一;采用连续介质力学模型求解岩土工程问题的关键是如何建立岩土的工程实用本构方程;建立多个工程实用本构方程结合积累大量工程经验才能促使数值方法在岩土工程中由用于定性分析转变到定量分析。     

Ground movements induced by installation of twin large diameter deeply-buried caissons: 3D numerical modeling

The assessment and control of ground movements during the installation of large diameter deeply-buried (LDDB) caissons are critically important to maintain the stability of surrounding infrastructures. However, for twin LDDB caissons which have been installed worldwide, no well-documented guidelines for assessing the induced ground movements are available due to the complexities of caisson–soil interaction. To this end, considering the mechanical boundaries of caissons and mechanized installation process, this paper presents a simple kinematic mechanical model balancing both computational cost and accuracy, which can be easily incorporated in commercial finite-element (FE) programs. Based on a project of twin LDDB caissons alternately installed employing a newly developed installation technology in wet ground with stiff clays in Zhenjiang, China, a three-dimensional (3D) numerical model is developed to capture the ground movements in terms of surface settlements and radial displacements induced by the installation of twin LDDB caissons. Moreover, hardening soil model with small-strain stiffness (HSSmall model) conceptually capable of capturing the nonlinear soil stiffness from very small to large strain levels is used to simulate undrained ground. The validations against field observations, empirical predictions and centrifuge test data are carried out to demonstrate the accuracy and validity of the developed FE model. Subsequently, the comparisons of ground movements numerically obtained in three frequently used installation schemes (i.e., synchronous, asynchronous and alternating installation) are conducted for installation sequence optimization of twin caissons. It is found that synchronous installation is the optimal scheme for limiting ground movements. Parametric studies considering the effects of horizontal spacing between twin caissons, staged penetration depth, inner diameter, controllable soil-plugging height, frictional coefficient between caisson–soil interface, as well as cutting edge gradient are thus performed in synchronous installation scheme. Based on an artificial data set generated through FE calculation, the multivariate adaptive regression splines (MARS) model capable of accurately capturing the nonlinear relationships between a set of input variables and output variables in multi-dimensions is used to analyze the sensitivity of caisson design parameters. Finally, the MARS mathematical equations for predicting the maximum surface settlement and radial displacement used in preliminary caisson design are proposed.

     

Multivariate Modelling of Geometallurgical Variables by Projection Pursuit

The integration of geological and geometallurgical data can significantly improve decision-making and optimize mining production due to a better understanding of the resources and their metallurgical performances. The primary-response rock property framework is an approach to the modelling of geometallurgy in which quantitative and qualitative primary properties are used as proxies of metallurgical responses. Within this framework, primary variables are used to fit regression models to predict metallurgical responses. Whilst primary rock property data are relatively abundant, metallurgical response property data are not, which makes it difficult to establish predictive response relationships. Relationships between primary input variables and geometallurgical responses are, in general, complex, and the response variables are often non-additive which further complicates the prediction process. Consequently, in many cases, the traditional multivariate linear regression models (MLR) of primary-response relationships perform poorly and a better alternative is required for prediction. Projection pursuit is a powerful exploratory statistical modelling technique in which data from a number of variables are projected onto a set of directions that optimize the fit of the model. The purpose of the projection is to reveal underlying relationships. Projection pursuit regression (PPR) fits standard regression models to the projected data vectors. In this paper, PPR is applied to the modelling of geometallurgical response variables. A case study with six geometallurgical variables is used to demonstrate the modelling approach. The results from the proposed PPR models show a significant improvement over those from MLR models. In addition, the models were bootstrapped to generate distributions of feasible scenarios for the response variables. Our results show that PPR is a robust technique for modelling geometallurgical response variables and for assessing the uncertainty associated with these variables.     

Implementing statistical fitting and reliability analysis for geotechnical engineering problems in R

Reliability analysis and multivariate statistical fitting are valuable techniques that enhance the scientific basis of regulatory decisions in geotechnical problems. This study introduces the use of several R packages specifically developed to assist risk assessors in their geotechnical projects. Firstly, the fitting of parameterised models either to the distribution of observed samples or to characterise the dependence structures among variables, or both is presented. Secondly, the most popular reliability analysis methods, such as the first- and second-order reliability methods and the random sampling simulation method, are implemented in R. The efficiency of implementing these classical approximation methods is demonstrated through two example problems.     

Mapping landslide susceptibility with logistic regression, multiple adaptive regression splines, classification and regression trees, and maximum entropy methods: a comparative study

Four statistical techniques for modelling landslide susceptibility were compared: multiple logistic regression (MLR), multivariate adaptive regression splines (MARS), classification and regression trees (CART), and maximum entropy (MAXENT). According to the literature, MARS and MAXENT have never been used in landslide susceptibility modelling, and CART has been used only twice. Twenty independent variables were used as predictors, including lithology as a categorical variable. Two sets of random samples were used, for a total of 90 model replicates (with and without lithology, and with different proportions of positive and negative data). The model performance was evaluated using the area under the receiver operating characteristic curve (AUC) statistic. The main results are (a) the inclusion of lithology improves the model performance; (b) the best AUC values for single models are MLR (0.76), MARS (0.76), CART (0.77), and MAXENT (0.78); (c) a smaller amount of negative data provides better results; (d) the models with the highest prediction capability are obtained with MAXENT and CART; and (e) the combination of different models is a way to evaluate the model reliability. We further discuss some key issues in landslide modelling, including the influence of the various methods that we used, the sample size, and the random replicate procedures.     

A FORM-based approach for probabilistic analysis in geotechnics: Application to a reinforced concrete drainage culvert

As numerical models are increasingly used as a design tool in geotechnical engineering, it is highly desirable if geotechnical reliability analysis can be conducted based on numeral models. Currently, the practical use of geotechnical reliability analysis-based numerical models is still quite limited. In this study, an easy to access method is derived to conduct geotechnical reliability analysis based on numerical models. To facilitate its application, a procedure is outlined to implement the suggested method such that geotechnical reliability analysis can be automated using existing geotechnical numerical packages. The procedure is illustrated in detail with an example, and the source codes provided can be easily adapted to analyze other similar problems. The method described in this paper is used to study the reliability of a deteriorating reinforced concrete drainage culvert in Shanghai, China. The suggested method provides a convenient means for reliability analysis of complex geotechnical problems.     

Optimal Spatial Sampling Design in a Multivariate Framework

The problem of spatial sampling design for estimating a multivariate random field from information obtained by sampling related variables is considered. A formulation assigning different degrees of importance to the variables and locations involved is introduced. Adopting an entropy-based approach, an objective function is defined as a linear combination in terms of the amount of information on the variables and/or the locations of interest contained in the data. In the multivariate Gaussian case, the objective function is obtained as a geometric mean of conditional covariance matrices. The effect of varying the degrees of importance for the variables and/or the locations of interest is illustrated in some numerical examples.     

Reliability assessment of ultimate limit state of twin caverns

Construction of a cavern in close proximity to an existing cavern modifies the state of stresses and movements in a zone around the existing cavern, as some degree of interaction between these two caverns generally takes place. This study investigates the interaction of two parallel caverns and the influence of such interaction on stress-induced global stability in terms of a global factor of safety. A series of finite difference analyses were performed to derive the global factor of safety of a system of two parallel and adjacent caverns. A mathematical response surface model was then built using the multivariate adaptive regression splines (MARS) approach and a series of charts based on this surrogate model were developed to relate the global factor of safety to the critical parameters. The built MARS model is of high accuracy and is simple to interpret and can be used to perform probabilistic assessment of ultimate limit state of twin caverns.     

Multivariate oceanographic sampling

Increasing attention in recent years has been devoted to the application of statistical techniques in the analysis and interpretation of geologic and oceanographic data. Equally important, but less well explored, are methods for efficient experimental design. The theory of linear programming provides plans for optimal sampling of geologic and oceanographic phenomena. Of particular significance are solutions to problems of multivariate sampling. Often, a single field sample may be analyzed for a number of oxides, or a number of minerals, or a number of textural parameters. In general, these variables differ in the degree to which they are diagnostic of changes in the phenomenon of interest, and thus they must be known with different levels of precision if they are to be useful. Similarly, the variables differ in the ease with which they may be measured. If a sampling plan is to be most efficient, it must provide the requisite levels of precision for the minimum expenditure of time and effort. Sampling for a single variable may be optimized directly. Sampling for several variables simultaneously usually introduces special difficulties, but if the objective function can be generalized to hold for all variables, solutions can be determined even in this situation.     

基于PSO-MLSSVR的土体参数反演方法在深基坑工程中的应用

为了确保基坑工程安全,常常会采用数值模拟的方法预测支护结构的位移,其中岩土体力学参数的选取对于结果的影响最大.本文使用了一种粒子群(PSO)算法结合多输出最小二乘支持向量回归机(MLSSVR)的基坑土体参数位移反分析法,以深圳某深基坑的支护桩顶水平位移监测数据为依据,基于正交设计生成具有代表性的土体参数组合,通过有限元...     

Numerical Modelling of a Shaking Table Test for Soil-Foundation-Superstructure Interaction by Means of a Soil Constitutive Model Implemented in a FEM Code

Dynamic soil-structure interaction (DSSI) plays a fundamental role in many geotechnical and/or structural design situations, as clearly shown by the damage which occurred during several recent earthquakes (Kobe 1995; Koaceli 1999; Chi-Chi 1999; L’Aquila 2009). For a long time civil engineering researchers have devoted increasing attention to this subject. Thanks to their efforts, several technical regulations, such as EC8 (2003), have taken DSSI into account. However, many steps are still necessary in order to increase our knowledge regarding this complex phenomenon, as well as to make all the results achieved known to academics and practitioners. This paper presents the results of a shaking table test performed on a scaled physical model consisting of a 3-D steel frame resting on a bed of sand. The experimental results are compared with the numerical ones obtained using a sophisticated elasto-plastic constitutive model recently implemented in the FEM code utilised. The solution of geotechnical problems requires the use of appropriate constitutive models. Many interesting constitutive models have been developed, but only a few of these have been implemented into commercial numerical codes; which is particularly so when dynamic analyses are required. The described experimental results, as well as the comparison between them and the numerical results, allow interesting considerations to be drawn on dynamic soil-structure interaction and on its numerical simulation.     

System reliability analysis of soil slopes with general slip surfaces using multivariate adaptive regression splines

A data driven multivariate adaptive regression splines (MARS) based algorithm for system reliability analysis of earth slopes having random soil properties under the framework of limit equilibrium method of slices is considered. The theoretical formulation is developed based on Spencer method (valid for general slip surfaces) satisfying all conditions of static equilibrium coupled with a nonlinear programming technique of optimization. Simulated noise is used to take account of inevitable modeling inaccuracies and epistemic uncertainties. The proposed MARS based algorithm is capable of achieving high level of computational efficiency in the system reliability analysis without significantly compromising the accuracy of results.     

A generalized surrogate response aided-subset simulation approach for efficient geotechnical reliability-based design

This paper aims to develop an efficient geotechnical reliability-based design (RBD) approach using Monte Carlo simulation (MCS). The proposed approach combines a recently developed MCS-based RBD approach, namely expanded RBD approach, with an advanced MCS method called “Subset Simulation (SS)” to improve the computation efficiency at small probability levels that are often concerned in geotechnical design practice. To facilitate the integration of SS and expanded RBD, a generalized surrogate response f is proposed to define the driving variable, which is a key parameter in SS, for expanded RBD of geotechnical structures (e.g., soil retaining structures and foundations). With the aid of the proposed surrogate response, failure probabilities of all the possible designs in a prescribed design space are calculated from a single run of SS. Equations are derived for integration of the surrogate response-aided SS and expanded RBD, and are illustrated using an embedded sheet pile wall design example and two drilled shaft design examples. Results show that the proposed approach provides reasonable estimates of failure probabilities of different designs using a single run of the surrogate response-aided SS, and significantly improves the computational efficiency at small probabilities levels in comparison with direct MCS-based expanded RBD. The surrogate response-aided SS is able to, simultaneously, approach the failure domains of all the possible designs in the design space by a single run of simulation and to generate more complete design information, which subsequently yields feasible designs with a wide range of combinations of design parameters. This is mainly attributed to the strong correlation between the surrogate response and target response (e.g., factor of safety) of different designs concerned in geotechnical RBD.     

Landslide susceptibility mapping: A comparison of logistic regression and neural networks methods in a medium scale study, Hendek region (Turkey)

Landslide susceptibility mapping is one of the most critical issues in Turkey. At present, geotechnical models appear to be useful only in areas of limited extent, because it is difficult to collect geotechnical data with appropriate resolution over larger regions. In addition, many of the physical variables that are necessary for running these models are not usually available, and their acquisition is often very costly. Conversely, statistical approaches are currently pursued to assess landslide hazard over large regions. However, these approaches cannot effectively model complicated landslide hazard problems, since there is a non-linear relationship between nature-based problems and their triggering factors. Most of the statistical methods are distribution-based and cannot handle multisource data that are commonly collected from nature. In this respect, logistic regression and neural networks provide the potential to overcome drawbacks and to satisfy more rigorous landslide susceptibility mapping requirements. In the Hendek region of Turkey, a segment of natural gas pipeline was damaged due to landslide. Re-routing of the pipeline is planned but it requires preparation of landslide susceptibility map. For this purpose, logistic regression analysis and neural networks are applied to prepare landslide susceptibility map of the problematic segment of the pipeline. At the end, comparative analysis is conducted on the strengths and weaknesses of both techniques. Based on the higher percentages of landslide bodies predicted in very high and high landslide susceptibility zones, and compatibility between field observations and the important factors obtained in the analyses, the result found by neural network is more realistic.