]. .. , Classification of uncertainties

, Space of two random variable (r, s) and their joint density function f RS (r, s); their marginal density functions f R and f S ; the failure domain D

. .. , Reliability methods based on limit state functions [25], p.10

. .. , Evolution from physical model to surrogate model, p.11

, Basic procedures to build a surrogate model

, Basic procedure of engineering structural reliability analysis, p.14

. .. Safe, 20 2.2 Classification of strategies to estimate probability of failure, p.22

, Graphical representation of the linearization of the limit-state function around the design point at the basis of the FORM estimation of P f, p.23

, Some descriptions of failure probability convergence by MCS, p.28

P. Estimated and . .. Ci,

. .. P-f, , p.30

. .. Svm-for-regression,

A. .. General-structure-of,

. .. Process, , p.41

, Example regression tree

, Example classification tree

.. .. Inducing-process-of-adaboost,

. .. , Comparison of RF and ETs in node splitting, vol.46

, Illustration of the general framework for uncertainty quantification

. .. , 2 Classification of uncertainty propagation methods [43], p.51

, Theoretical framework to approximate P C f (X) by statistical learning model 54

, Structural responses induced by different random excitations, p.55

.. .. ,

, Ten sampled outputs according to the uncertainties of the coefficients, p.60

, The first five eigenvalues of the Fredholm integral equation of the second kind 60

, The first five eigenfunctions of the Fredholm integral equation of the 2nd kind, p.61

, The five scaled functions corresponding to Figure 3.23, p.61

, Twenty sampled realizations with

, Illustration of an symmetric elementary failure region, p.68

, Ten storey shear building under earthquake excitations

, Evolution of CPU time according to different values of P f (by standard MCS) 72

, Evolution of CPU time according to different values of P f (by KL-IS), p.73

, Impulse (absolute) response of different DOFs

, Impulse (relative) response of different DOFs

. .. , Evolution of failure probabilities of different DOFs, p.76

M. Tmd and .. .. ,

, 2 The schematic of the proposed method

. .. , A response process containing exceedance events, vol.87

, X N ×n is the input matrix that contains N observations of the input variables, n is the number of input variables, Y N ×1 is a column vector that contains the output values

, An illustrative node splitting process. The symbol ' ?' means that the variable used to carry out the next splitting needs to be determined, p.92

. .. , Illustrative example: prediction by a single tree, p.94

, Model evaluation on different data(×1000) for 1-DOF structure, p.96

, Model evaluation on different data(×1000) for 2-DOF structure, p.97

, Model evaluation on different data(×1000) for 3-DOF structure, p.98

. .. , Simulation result based on different standard deviations, vol.100

. .. , Simulation result based on different standard deviations, vol.101

, F I evaluation of jth feature via Tree i in RF

. .. , F Is of the structural properties (with SDs), p.105

. .. , Simulation results of the object structure via RF, p.108

, Ten-DOF uncertain structure subjected to stochatic excitations

. .. Rf, , p.111

. .. , P f estimations (with error bars) by standard MCS, p.112

. .. , Relative response of different DOFs of the structure, p.116

. General and . .. Stacking,

, 2 Diagram of Stacking method (include model evaluation on test data), p.122

, CVs (K=5) to create 2nd-level data. In practice, K=10 is used, p.124

, Both Stacking1 and Stacking2 take GB as meta-model. Stacking1 takes RF, ETs as base models; Stacking2 takes RF, ETs, GB as base models

, Performances of Stacking models that have two base learners and a metalearner. The form 'A&B-C' denotes that A and B are base learners and C the meta-learner

, Performances of Stacking models that have three base learners and a metalearner. The form 'A&B&C-D' denotes that A, B and C are base learners and D the meta-learner

. .. , A special case: 'two' is better than 'three', p.128

, Stacking models that have two base learners

. .. , 131 5.10 Change of RMSE with respect to the number of trees in GB, p.131

. .. Svr, RMSE with respect to the kernel type of, p.132

, RMSE in terms of the number of sampled features in each splitting in RF, vol.132

, RMSE in terms of the number of sampled features for each splitting in GB 133

, Bias-variance of single models in terms of maxFeatures in each split, p.136

, Bias-variance of Stacking model in terms of K in each split in base learner RF

, Bias-variance values in terms of K in each split in meta-model, p.138

. .. , 144 List of Tables 2.1 characteristics of different types of single trees, 17 CPU time of the models employed in the simulations, p.45

, Sampling scheme according to ISD proposed in eq

, 1 Nominal values of the structure

. .. , Standard deviations of the uncertain properties, p.99

, List of feature importances

.. .. Rmses,

. .. , Structure parameters (i = 1, 2, 3; j = 1, 2), p.107

. .. , Statistical properties of the structural parameters, vol.109

R. .. Parameters-of,

, 111 4.10 Comparisons between standard MCS and RF results, p.113

. .. , Statistical properties of the structural parameters, p.114

. .. , Thresholds of interest to evaluate failure probability [107], p.115

. .. , Reliability estimation results from different thresholds, p.117

. .. , 2 Structure parameters (i = 1, 2, 3; j = 1, 2), p.125

, Hyper-parameters of the base models

, Pseudo-code of bias&variance calculation

, Bias-variance decomposition result

, Time complexities (average) of the base models in Stacking2, p.141

. .. , Compare Stacking and RF in reliability estimations, p.145

, Compare Stacking and RF when multi-thresholds are assumed, p.145

, Compare Stacking and RF when the structural parameters are all lognormal 146

, Compare Stacking and RF when the structural parameters are all Gamma 146

, Compare Stacking and RF when the structural parameters are a mixture of Lognormal and Gamma

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