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
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.30
,
,
, , p.41
, Example regression tree
, Example classification tree
,
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 ,
,
, 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
, , p.111
P f estimations (with error bars) by standard MCS, p.112 ,
Relative response of different DOFs of the structure, p.116 ,
,
, 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 ,
, 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
,
Structure parameters (i = 1, 2, 3; j = 1, 2), p.107 ,
Statistical properties of the structural parameters, vol.109 ,
,
, 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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