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Table of contents
1 Introduction
1.1 Context
1.2 Objectives
1.3 Organization of the manuscript
2 Avalanche models, Bayesian inference and global sensitivity analysis
2.1 A short introduction to 1-D snow avalanche dynamic modeling
2.1.1 Model input parameters
2.1.2 Sensitivity analyses for avalanche models
2.1.3 Few words about land-use planning
2.2 Model calibration using Bayesian inference
2.3 Global sensitivity analysis
2.3.1 Screening approaches
2.3.2 Variance-based measures
2.3.3 Moment-free measures
2.3.4 Other measures
2.3.5 A little bit about metamodels
2.3.6 Multivariate and/or functional model outputs
2.4 Dimension reduction techniques for functional outputs
Part I Model calibration using Bayesian inference
3 Bayesian calibration of an avalanche model
3.1 Introduction
3.2 Avalanche model calibration principle
3.2.1 Sliding block propagation model
3.2.2 Statistical model formulation
3.2.3 Bayesian framework
3.2.4 Metropolis-Hastings algorithm
3.2.5 Model selection using Bayes Factor
3.3 Application data
3.3.1 Avalanche data from the Lautaret test-site
3.3.2 Synthetic data generation
3.4 Application, results and discussion
3.4.1 The avalanche
3.4.2 Synthetic data
3.5 Conclusion and outlook
3.6 Appendix A: Likelihood of an AR(1) process
3.7 Appendix B: Numerical evaluation of the Bayes Factor
3.8 Appendix C: Prior sensitivity analysis
3.9 Appendix D: Images used in the photogrammetric process
Part II Global sensitivity analysis
4 Nonparametric estimation of aggregated Sobol’ indices
4.1 Introduction
4.2 Aggregated Sobol’ indices
4.2.1 Nonparametric estimation procedure
4.2.2 Bias correction and bandwidth selection
4.2.3 Dimension reduction based on principal component analysis
4.3 Test cases
4.3.1 Toy functions
4.3.2 A functional example: the mass-spring model
4.4 Application: the avalanche model
4.4.1 Sobol’ indices
4.4.2 Scalar Sobol’ indices
4.4.3 Aggregated Sobol’ indices
4.5 Conclusions and perspectives
4.6 Appendix
5 Aggregated Shapley eects
5.1 Introduction
5.2 Aggregated Shapley eects
5.2.1 Denition
5.2.2 Properties
5.3 Estimation procedure for scalar and aggregated Shapley eects
5.3.1 Given data estimator of scalar Shapley eects proposed by [Broto et al., 2020]
5.3.2 Estimator of the aggregated Shapley eects
5.3.3 Dimension reduction: functional principal component analysis
5.4 Bootstrap condence intervals with percentile bias correction
5.5 Test cases
5.5.1 Multivariate linear Gaussian model in dimension d = 2
5.5.2 Mass-spring model
5.6 Snow avalanche modeling
5.6.1 Model
5.6.2 Scenario 1
5.6.3 Scenario 2
5.7 Conclusions and perspectives
5.8 Comparison between the uniform allocation Nu = Ntot=(2d 2) , and the one introduced in [Broto et al., 2020] N u = j Ntot d juj 1 (d 1)1 k , ; u ( f1; : : : ; dg
5.9 Comparison between dierent values of NI
5.10 Estimation of Shapley eects using one or two samples
5.11 Functional principal components and Shapley eects for scenario 2
6 Conclusions and perspectives
6.1 Conclusions
6.2 Perspectives
6.2.1 For calibration using Bayesian inference
6.2.2 For the Global sensitivity analysis
Appendices
A Given data estimation methods for Sobol’ indices
A.1 Given data method
A.2 The spectral approach EASI
A.3 Test cases
A.3.1 Ishigami function
A.3.2 Bartley et al. function
B Correlations between Shapley eects and local slope
Bibliography
