Handbook of Probabilistic Models

1. Introduction......Page 9
2. Important steps in reliability evaluation......Page 10
3. Elements of set theory......Page 12
4. Quantification of uncertainties in random variables......Page 13
5. Transformation of uncertainty from parameter to the system level......Page 14
5.1.1 Exact solution......Page 15
5.1.2 Partial Solutions......Page 17
5.1.3 Approximate solutions—general function of multiple RVs......Page 18
5.3 Regression analysis......Page 19
5.3.1 Simple linear regression......Page 20
6. Fundamentals of reliability analysis......Page 21
6.1 Limit state equations or functions......Page 23
6.1.1 Serviceability limit state equation or function......Page 24
6.2 Reliability evaluation methods......Page 25
6.2.1 Advanced FOSM for statistically independent normal variables (Hasofer–Lind method)......Page 26
6.2.2 First-order reliability method with two-parameter distributions......Page 28
6.2.3 Examples......Page 30
6.3 Reliability analysis with correlated variables......Page 32
6.4 Reliability analysis for implicit limit state analysis......Page 36
7. Performance-based seismic design......Page 37
8. Monte Carlo simulation......Page 38
10. Computer programs......Page 40
Acknowledgments......Page 41
References......Page 42
1. Introduction......Page 44
2. Theoretical framework......Page 48
2.1 Artificial neural network......Page 49
2.2 Minimax probability machine regression......Page 52
2.3 Genetic programming......Page 53
3.2 Study region......Page 54
3.3 Development of data-intelligent models......Page 60
3.4 Model performance evaluation criteria......Page 73
4. Results......Page 76
5. Discussion: limitations and opportunity for further research......Page 82
6. Conclusion......Page 85
References......Page 88
3.1 Introduction......Page 95
3.2.1 Markov Chain Monte Carlo–based statistical copula models......Page 97
3.3.1 Rainfall data set and study region......Page 99
3.3.2 Model performance criteria......Page 100
3.4 Results......Page 102
3.5 Discussion: limitations and opportunity for further research......Page 104
References......Page 107
1. Introduction......Page 112
2. Methodology......Page 114
2.1 Deterministic model......Page 115
2.2 Probabilistic model and data—@RISK......Page 116
2.3 Agent-based model......Page 117
2.3.2 Rules governing an agent's interactions for the present case......Page 118
2.4 Generating automatic event trees—general rules......Page 119
3. Case study......Page 120
4. Results and discussion......Page 122
4.1 Property damage due to fire......Page 124
4.2 Varying reliability values of fire safety systems......Page 126
4.3 The effects of different fire growth rate......Page 128
5. Conclusions......Page 131
References......Page 132
Further reading......Page 134
1. Introduction......Page 135
2. Governing equations......Page 137
3. Polynomial neural network......Page 138
4. Results and discussion......Page 139
References......Page 142
1. Introduction......Page 145
2.1.1 Standardized Precipitation-Evapotranspiration Index......Page 148
2.1.2 Copula theory......Page 149
2.1.3 Vine copula......Page 151
2.1.5 Joint return periods......Page 153
2.2 Study area and data......Page 154
2.2.1 Characterization of drought properties......Page 155
2.2.2 Copula-statistical model development......Page 156
2.2.4 Selection of copulae......Page 158
2.2.5 Dependence modeling......Page 162
3.1 Applications on the Standardized Precipitation-Evapotranspiration Index and climate indices......Page 163
3.2 Applications on drought properties and climate mode indices......Page 166
3.3 Applications on drought properties......Page 173
4. Further discussion......Page 176
5. Summary......Page 178
References......Page 179
1. Introduction......Page 183
2.1 Robust design optimization......Page 185
3. Proposed surrogate-assisted robust design optimization framework......Page 186
4.1 Integrating high-dimensional model representation and Polynomial Chaos Expansion (PCE) into Kriging trend......Page 189
4.2.1 Sparse recovery using Bayesian learning......Page 190
4.2.2 Sparse recovery using accelerated Bayesian learning......Page 191
5.1.1 Example 1: Welded beam design......Page 193
5.1.2 Example 2: Multiobjective robust design optimization of a vibrating platform......Page 194
5.2 Problem type 2: finite element model of a building frame......Page 198
6. Summary and conclusions......Page 200
A.2 Example 2: Vibrating platform......Page 202
References......Page 203
1. Introduction......Page 206
2.1 Cumulative rainfall index......Page 208
2.3.1 Copula theorem......Page 211
2.3.3 Semiparametric d-vine quantile regression......Page 213
2.3.4 Linear quantile regression......Page 214
3.1 Climate–rainfall relationships......Page 215
3.2 Rainfall quantile forecast......Page 220
4. Discussion......Page 224
5. Conclusions......Page 225
References......Page 227
2. Difference between geostatistics and classical statistics methods......Page 231
3. Regionalized variables......Page 232
5. Variogram and semivariogram......Page 233
7. Range......Page 234
12. Spherical model......Page 235
14. Gaussian model......Page 236
19. Selection of the theoretical variogram models......Page 237
22. Geometric anisotropy......Page 238
26. Kriging equations......Page 239
28. Ordinary Kriging......Page 240
33. Disjunctive Kriging......Page 241
37. Neighborhood......Page 242
References......Page 243
1. Introduction......Page 245
2.1 Standard H∞ filter......Page 246
2.2 Conservativeness and optimization......Page 247
3.1 Switched H∞ Kalman filter......Page 248
3.2 Optimal-switched H∞ Kalman filter......Page 252
4.1 Orbital relative motion model......Page 254
5. Numerical simulations......Page 257
6. Conclusions......Page 260
7. Appendix......Page 261
References......Page 265
1. Introduction......Page 267
2. R installations, help and advantages......Page 268
3. Operators in R......Page 270
3.1 Data entry in R......Page 271
4.1 Remarks......Page 272
5. Loops and if/else statements in R......Page 273
6. Curve plotting in R......Page 275
7. Maximum likelihood estimation......Page 277
7.1 Maximum likelihood estimate for censored data......Page 279
7.2 The GILD likelihoods and survival estimates......Page 280
8.1 Complete case: maximum flood levels data......Page 281
8.2 Censored case: head and neck cancer data......Page 284
References......Page 288
1. Introduction......Page 290
3. Methodology......Page 291
4. Results and discussion......Page 296
References......Page 304
1. Introduction......Page 306
2. Challenges and trends in risk evaluation......Page 308
3. State-of-the-art in estimating risk of dynamic structural systems......Page 309
4. A novel structural risk estimation procedure for dynamic loadings applied in time domain......Page 311
4.1.1 Selection of center point......Page 312
4.1.2 Factorial design schemes......Page 313
4.2 Reliability estimation using IRS and AFD......Page 314
5.1 Total Number of Deterministic Analyses......Page 315
6. Accuracy in generating an IRS......Page 316
6.1 Moving least squares method......Page 317
6.2 Kriging method......Page 319
7.2 Performance levels......Page 322
7.3 Incorporation of uncertainties......Page 323
8.1 Example 1—verification on AFD—two-Story steel frame......Page 324
8.2 A case study to document the capabilities of the proposed reliability evaluation concept—failure of 13-Story steel frame lo .........Page 328
8.3 Example 3—implementation of the PBSD concept for 3- and 9-Story steel frames......Page 331
9.1 Reliability of electronic packaging—thermomechanical loading......Page 340
10. Further improvements of Kriging method......Page 341
References......Page 343
1. Introduction......Page 348
2.1 Parzen windows......Page 351
2.3 Potential functions......Page 353
3. Structure of probabilistic neural networks......Page 356
4. Improving memory performance......Page 360
4.1 Feature reduction using principal component analysis......Page 361
4.2 Pattern layer size reduction using clustering......Page 362
5. Simple probabilistic neural network example in Python......Page 363
References......Page 367
1. Introduction......Page 369
2.1 Basic formulation......Page 371
2.3 Polynomial chaos expansion model accuracy......Page 372
3.1 Monte Carlo sampling......Page 373
3.4 Importance sampling......Page 374
4.1 Analytical problem: Ishigami function......Page 375
4.2.1 Truss structure......Page 376
4.2.2 Tensile membrane structure......Page 378
5. Summary......Page 380
References......Page 381
1.1 Introduction to stochastic analysis of structural systems......Page 382
1.2 Forward propagation of uncertainty—an introduction to different approaches......Page 385
1.3 Introduction to multiply supported secondary systems......Page 387
2. Details of the generalized polynomial chaos method......Page 388
2.1 Details of orthogonal basis......Page 390
2.2 Convergence of PC expansion......Page 391
2.3 Normal distribution and Hermite polynomial expansion......Page 392
2.4 PC expansion for multidimensional random variable......Page 393
2.5.1 Intrusive methods—stochastic Galerkin......Page 395
2.5.2.1 Discrete projection......Page 396
2.6 Postprocessing of polynomial chaos expansion......Page 397
2.6.3 Sensitivity analysis—Sobol’ indices......Page 398
2.6.4 Reliability analysis......Page 399
3. Deterministic model of base-isolated SDOF and base-isolated MDOF structure with secondary system......Page 400
3.2 Details of fixed-base primary structure with fixed-base secondary system and base-isolated primary structure with fixed-bas .........Page 401
3.3.1 Fixed-base primary structure with secondary system......Page 404
3.3.3 Details of laminated rubber bearing base-isolation system......Page 405
4.1 PC expansion of time-independent input uncertainties......Page 407
4.2 PC expansions of time-dependant QoI......Page 408
4.2.1 Selection of collocation points......Page 409
4.3 Postprocessing of results......Page 410
5.1 Base-isolated SDOF system with random inputs......Page 411
5.1.1 Stochastic time history response of base-isolated SDOF......Page 412
5.1.2 Probability measures of QoI......Page 414
5.2 Deterministic response of fixed-base primary structure with fixed-base secondary system and base-isolated primary structure .........Page 416
5.3 Stochastic response of fixed-base primary structure with fixed-base secondary system and base-isolated primary structure wi .........Page 418
5.3.1 Stochastic response of the primary structure and secondary system......Page 420
5.3.2 Probability measures of QoI......Page 422
5.3.3 Sensitivity analysis......Page 428
5.3.4 Computing probability of failure from PC expansion......Page 429
6. Conclusions......Page 431
References......Page 432
1. Introduction......Page 435
3. Search and optimization......Page 436
4. Markov chain model......Page 439
5. Convergence of the stochastic diffusion search......Page 442
6. Time complexity......Page 443
7. Conclusions......Page 444
References......Page 445
1. Introduction......Page 447
2. Rao-Blackwellized Monte Carlo Data Association......Page 449
3. Resampling techniques......Page 451
3.2 Stratified/systematic resampling method......Page 452
3.3 Residual resampling (remainder resampling)......Page 453
3.4 Residual systematic resampling technique......Page 454
3.5.1 Dynamic threshold......Page 455
3.5.2 Fixed threshold......Page 456
4.1 Tracking unknown number of targets in 3D......Page 457
4.2 Simulation results......Page 459
5. Conclusions......Page 461
References......Page 462
1. Introduction......Page 464
2. Back-propagation neural network methodologies......Page 465
3. Development of the back-propagation neural network model......Page 466
3.3 Back-propagation neural network model architecture......Page 467
5. Modeling results......Page 475
7. Model interpretabilities......Page 476
Appendix A BPNN pile settlement model......Page 478
Appendix B weights and bias values for BPNN pile settlement model......Page 480
References......Page 482
1. Introduction......Page 485
2.1 Criteria and constraints......Page 486
2.2 Methods in solar site selection......Page 488
3.2 Monte Carlo simulation approach......Page 491
4. Conclusion......Page 493
References......Page 497
1. Introduction......Page 501
2. Power spectral density......Page 502
2.2 The autocorrelation function......Page 503
2.4 Ergodicity......Page 505
3. Input–output relationship......Page 506
3.2 Example......Page 507
4. Monte Carlo simulation......Page 510
4.1 Wind sample generation......Page 511
5. Fluid viscous dampers......Page 514
5.1 Statistical linearization technique......Page 516
6. Conclusion......Page 518
References......Page 521
1. Introduction......Page 522
2. Governing equations for composite plates......Page 523
3. Artificial neural network......Page 528
4. Polynomial neural network......Page 529
5. Stochastic approach using neural network model......Page 532
6. Results and discussion......Page 534
References......Page 538
Index......Page 543