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Monte Carlo strategies in scientific computing:

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Bibliographische Detailangaben
1. Verfasser:	Liu, Jun S. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York, NY Springer 2008
Ausgabe:	1. softcover print.
Schriftenreihe:	Springer series in statistics
Schlagworte:	Naturwissenschaft Monte Carlo method Science > Statistical methods Monte-Carlo-Simulation
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVI, 343 S. Ill., graph. Darst.
ISBN:	9780387763699 9780387952307

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adam_text	Contents Preface vii 1 Introduction and Examples 1 1.1 The Need of Monte Carlo Techniques ............. 1 1.2 Scope and Outline of the Book ................ 3 1.3 Computations in Statistical Physics ............. 7 1.4 Molecular Structure Simulation ................ 9 1.5 Bioinformatics: Finding Weak Repetitive Patterns ..... 10 1.6 Nonlinear Dynamic System: Target Tracking ........ 14 1.7 Hypothesis Testing for Astronomical Observations ..... 16 1.8 Bayesian Inference of Multilevel Models ........... 18 1.9 Monte Carlo and Missing Data Problems .......... 19 2 Basic Principles: Rejection, Weighting, and Others 23 2.1 Generating Simple Random Variables ............ 23 2.2 The Rejection Method ..................... 24 2.3 Variance Reduction Methods ................. 26 2.4 Exact Methods for Chain-Structured Models ........ 28 2.4.1 Dynamic programming ................. 29 2.4.2 Exact simulation .................... 30 2.5 Importance Sampling and Weighted Sample ......... 31 2.5.1 An example ....................... 31 2.5.2 The basic idea ..................... 33 2.5.3 The rule of thumb for importance sampling .... 34 xii Contents 2.5.4 Concept of the weighted sample ............ 36 2.5.5 Marginalization in importance sampling ....... 37 2.5.6 Example: Solving a linear system ........... 38 2.5.7 Example: A Bayesian missing data problem ..... 40 2.6 Advanced Importance Sampling Techniques ......... 42 2.6.1 Adaptive importance sampling ............ 42 2.6.2 Rejection and weighting ................ 43 2.6.3 Sequential importance sampling ............ 46 2.6.4 Rejection control in sequential importance sampling 48 2.7 Application of SIS in Population Genetics .......... 49 2.8 Problems ............................ 51 3 Theory of Sequential Monte Carlo 53 3.1 Early Developments: Growing a Polymer ........... 55 3.1.1 A simple model of polymer: Self-avoid walk ..... 55 3.1.2 Growing a polymer on the square lattice ....... 56 3.1.3 Limitations of the growth method .......... 59 3.2 Sequential Imputation for Statistical Missing Data Problems 60 3.2.1 Likelihood computation ................ 60 3.2.2 Bayesian computation ................. 62 3.3 Nonlinear Filtering ....................... 64 3.4 A General Framework ..................... 67 3.4.1 The choice of the sampling distribution ....... 69 3.4.2 Normalizing constant ................. 69 3.4.3 Pruning, enrichment, and resampling ......... 71 3.4.4 More about resampling ................ 72 3.4.5 Partial rejection control ................ 75 3.4.6 Marginalization, look-ahead, and delayed estimate . 76 3.5 Problems ............................ 77 4 Sequential Monte Carlo in Action 79 4.1 Some Biological Problems ................... 79 4.1.1 Molecular Simulation ................. 79 4.1.2 Inference in population genetics ............ 81 4.1.3 Finding motif patterns in DNA sequences ...... 84 4.2 Approximating Permanents .................. 90 4.3 Counting 0-1 Tables with Fixed Margins ........... 92 4.4 Bayesian Missing Data Problems ............... 94 4.4.1 Murray s data ...................... 94 4.4.2 Nonparametric Bayes analysis of binomial data ... 95 4.5 Problems in Signal Processing ................. 98 4.5.1 Target tracking in clutter and mixture Kalman filter 98 4.5.2 Digital signal extraction in fading channels ..... 102 4.6 Problems ............................ 103 Contents xiii Metropolis Algorithm and Beyond 105 5.1 The Metropolis Algorithm ................... 106 5.2 Mathematical Formulation and Hastings s Generalization . Ill 5.3 Why Does the Metropolis Algorithm Work? ......... 112 5.4 Some Special Algorithms .................... 114 5.4.1 Random-walk Metropolis ............... 114 5.4.2 Metropolized independence sampler ......... 115 5.4.3 Configurational bias Monte Carlo ........... 116 5.5 Multipoint Metropolis Methods ................ 117 5.5.1 Multiple independent proposals ............ 118 5.5.2 Correlated multipoint proposals ............ 120 5.6 Reversible Jumping Rule .................... 122 5.7 Dynamic Weighting ...................... 124 5.8 Output Analysis and Algorithm Efficiency .......... 125 5.9 Problems ............................ 127 The Gibbs Sampler 129 6.1 Gibbs Sampling Algorithms .................. 129 6.2 Illustrative Examples ...................... 131 6.3 Some Special Samplers ..................... 133 6.3.1 Slice sampler ...................... 133 6.3.2 Metropolized Gibbs sampler .............. 133 6.3.3 Hit-and-run algorithm ................. 134 6.4 Data Augmentation Algorithm ................ 135 6.4.1 Bayesian missing data problem ............ 135 6.4.2 The original DA algorithm .............. 136 6.4.3 Connection with the Gibbs sampler ......... 137 6.4.4 An example: Hierarchical Bayes model ........ 138 6.5 Finding Repetitive Motifs in Biological Sequences ..... 139 6.5.1 A Gibbs sampler for detecting subtle motifs ..... 140 6.5.2 Alignment and classification .............. 141 6.6 Covariance Structures of the Gibbs Sampler ......... 143 6.6.1 Data Augmentation .................. 143 6.6.2 Autocovariances for the random-scan Gibbs sampler 144 6.6.3 More efficient use of Monte Carlo samples ...... 146 6.7 Collapsing and Grouping in a Gibbs Sampler ........ 146 6.8 Problems ............................ 151 Cluster Algorithms for the Ising Model 153 7.1 Ising and Potts Model Revisit ................. 153 7.2 The Swendsen-Wang Algorithm as Data Augmentation . . . 154 7.3 Convergence Analysis and Generalization .......... 155 7.4 The Modification by Wolff ................... 157 7.5 Further Generalization ..................... 157 7.6 Discussion ............................ 158 xiv Contents 7.7 Problems............................ 159 8 General Conditional Sampling 161 8.1 Partial Resampling ....................... 161 8.2 Case Studies for Partial Resampling ............. 163 8.2.1 Gaussian random field model ............. 163 8.2.2 Texture synthesis .................... 165 8.2.3 Inference with multivariate t-distribution ...... 169 8.3 Transformation Group and Generalized Gibbs ........ 171 8.4 Application: Parameter Expansion for Data Augmentation . 174 8.5 Some Examples in Bayesian Inference ............ 176 8.5.1 Probit regression .................... 176 8.5.2 Monte Carlo bridging for stochastic differential equa¬ tion ........................... 178 8.6 Problems ........................... . 181 9 Molecular Dynamics and Hybrid Monte Carlo 183 9.1 Basics of Newtonian Mechanics ................ 184 9.2 Molecular Dynamics Simulation ................ 185 9.3 Hybrid Monte Carlo ...................... 189 9.4 Algorithms Related to HMC .................. 192 9.4.1 Langevin-Euler moves ................. 192 9.4.2 Generalized hybrid Monte Carlo ........... 193 9.4.3 Surrogate transition method .............. 194 9.5 Multipoint Strategies for Hybrid Monte Carlo ........ 195 9.5.1 Neal s window method ................. 195 9.5.2 Multipoint method ................... 197 9.6 Application of HMC in Statistics ............... 198 9.6.1 Indirect observation model .............. 199 9.6.2 Estimation in the stochastic volatility model .... 201 10 Multilevel Sampling and Optimization Methods 205 10.1 Umbrella Sampling ....................... 206 10.2 Simulated Annealing ...................... 209 10.3 Simulated Tempering ...................... 210 10.4 Parallel Tempering ....................... 212 10.5 Generalized Ensemble Simulation ............... 215 10.5.1 Multicanonical sampling ................ 216 10.5.2 The l/i-ensemble method ............... 217 10.5.3 Comparison of algorithms ............... 218 10.6 Tempering with Dynamic Weighting ............. 219 10.6.1 Ising model simulation at sub-critical temperature . 221 10.6.2 Neural network training ................ 222 Contents xv 11 Population-Based Monte Carlo Methods 225 11. X Adaptive Direction Sampling: Snooker Algorithm ...... 226 11.2 Conjugate Gradient Monte Carlo ............... 227 11.3 Evolutionary Monte Carlo ................... 228 11.3.1 Evolutionary movements in binary-coded space . . . 230 11.3.2 Evolutionary movements in continuous space .... 231 11.4 Some Further Thoughts .................... 233 11.5 Numerical Examples ...................... 235 11.5.1 Simulating from a bimodal distribution ....... 235 11.5.2 Comparing algorithms for a multimodal example . . 236 11.5.3 Variable selection with binary-coded EMC ...... 237 11.5.4 Bayesian neural network training ........... 239 11.6 Problems ............................ 242 12 Markov Chains and Their Convergence 245 12.1 Basic Properties of a Markov Chain ............. 245 12.1.1 Chapman-Kolmogorov equation ............ 247 12.1.2 Convergence to stationarity .............. 248 12.2 Coupling Method for Card Shuffling ............. 250 12.2.1 Random-to-top shuffling ................ 250 12.2.2 Riffle shuffling ..................... 251 12.3 Convergence Theorem for Finite-State Markov Chains . . . 252 12.4 Coupling Method for General Markov Chain ......... 254 12.5 Geometric Inequalities ..................... 256 12.5.1 Basic setup ....................... 257 12.5.2 Poincaré inequality ................... 257 12.5.3 Example: Simple random walk on a graph ...... 259 12.5.4 Cheeger s inequality .................. 261 12.6 Functional Analysis for Markov Chains ............ 263 12.6.1 Forward and backward operators ........... 264 12.6.2 Convergence rate of Markov chains .......... 266 12.6.3 Maximal correlation .................. 267 12.7 Behavior of the Averages ................... 269 13 Selected Theoretical Topics 271 13.1 MCMC Convergence and Convergence Diagnostics ..... 271 13.2 Iterative Conditional Sampling ................ 273 13.2.1 Data augmentation ................... 273 13.2.2 Random-scan Gibbs sampler ............. 275 13.3 Comparison of Metropolis-Type Algorithms ......... 277 13.3.1 Peskun s ordering .................... 277 13.3.2 Comparing schemes using Peskun s ordering ..... 279 13.4 Eigenvalue Analysis for the Independence Sampler ..... 281 13.5 Perfect Simulation ....................... 284 13.6 A Theory for Dynamic Weighting ............... 287 13.6.1 Definitions ....................... 287 xvi Contents 13.6.2 Weight behavior under different scenarios ...... 288 13.6.3 Estimation with weighted samples .......... 291 13.6.4 A simulation study ................... 292 A Basics in Probability and Statistics 295 A.I Basic Probability Theory ................... 295 A. 1.1 Experiments, events, and probability ......... 295 A. 1.2 Univariate random variables and their properties . . 296 A. 1.3 Multivariate random variable ............. 298 A. 1.4 Convergence of random variables ........... 300 A. 2 Statistical Modeling and Inference .............. 301 A.2.1 Parametric statistical modeling ............ 301 A.2.2 FVequentist approach to statistical inference ..... 302 A.2.3 Bayesian methodology ................. 304 A.3 Bayes Procedure and Missing Data Formalism ........ 306 A.3.1 The joint and posterior distributions ......... 306 A.3. 2 The missing data problem ............... 308 A.4 The Expectation-Maximization Algorithm .......... 310 References 313 Author Index 333 Subject Index 338
adam_txt	Contents Preface vii 1 Introduction and Examples 1 1.1 The Need of Monte Carlo Techniques . 1 1.2 Scope and Outline of the Book . 3 1.3 Computations in Statistical Physics . 7 1.4 Molecular Structure Simulation . 9 1.5 Bioinformatics: Finding Weak Repetitive Patterns . 10 1.6 Nonlinear Dynamic System: Target Tracking . 14 1.7 Hypothesis Testing for Astronomical Observations . 16 1.8 Bayesian Inference of Multilevel Models . 18 1.9 Monte Carlo and Missing Data Problems . 19 2 Basic Principles: Rejection, Weighting, and Others 23 2.1 Generating Simple Random Variables . 23 2.2 The Rejection Method . 24 2.3 Variance Reduction Methods . 26 2.4 Exact Methods for Chain-Structured Models . 28 2.4.1 Dynamic programming . 29 2.4.2 Exact simulation . 30 2.5 Importance Sampling and Weighted Sample . 31 2.5.1 An example . 31 2.5.2 The basic idea . 33 2.5.3 The "rule of thumb" for importance sampling . 34 xii Contents 2.5.4 Concept of the weighted sample . 36 2.5.5 Marginalization in importance sampling . 37 2.5.6 Example: Solving a linear system . 38 2.5.7 Example: A Bayesian missing data problem . 40 2.6 Advanced Importance Sampling Techniques . 42 2.6.1 Adaptive importance sampling . 42 2.6.2 Rejection and weighting . 43 2.6.3 Sequential importance sampling . 46 2.6.4 Rejection control in sequential importance sampling 48 2.7 Application of SIS in Population Genetics . 49 2.8 Problems . 51 3 Theory of Sequential Monte Carlo 53 3.1 Early Developments: Growing a Polymer . 55 3.1.1 A simple model of polymer: Self-avoid walk . 55 3.1.2 Growing a polymer on the square lattice . 56 3.1.3 Limitations of the growth method . 59 3.2 Sequential Imputation for Statistical Missing Data Problems 60 3.2.1 Likelihood computation . 60 3.2.2 Bayesian computation . 62 3.3 Nonlinear Filtering . 64 3.4 A General Framework . 67 3.4.1 The choice of the sampling distribution . 69 3.4.2 Normalizing constant . 69 3.4.3 Pruning, enrichment, and resampling . 71 3.4.4 More about resampling . 72 3.4.5 Partial rejection control . 75 3.4.6 Marginalization, look-ahead, and delayed estimate . 76 3.5 Problems . 77 4 Sequential Monte Carlo in Action 79 4.1 Some Biological Problems . 79 4.1.1 Molecular Simulation . 79 4.1.2 Inference in population genetics . 81 4.1.3 Finding motif patterns in DNA sequences . 84 4.2 Approximating Permanents . 90 4.3 Counting 0-1 Tables with Fixed Margins . 92 4.4 Bayesian Missing Data Problems . 94 4.4.1 Murray's data . 94 4.4.2 Nonparametric Bayes analysis of binomial data . 95 4.5 Problems in Signal Processing . 98 4.5.1 Target tracking in clutter and mixture Kalman filter 98 4.5.2 Digital signal extraction in fading channels . 102 4.6 Problems . 103 Contents xiii Metropolis Algorithm and Beyond 105 5.1 The Metropolis Algorithm . 106 5.2 Mathematical Formulation and Hastings's Generalization . Ill 5.3 Why Does the Metropolis Algorithm Work? . 112 5.4 Some Special Algorithms . 114 5.4.1 Random-walk Metropolis . 114 5.4.2 Metropolized independence sampler . 115 5.4.3 Configurational bias Monte Carlo . 116 5.5 Multipoint Metropolis Methods . 117 5.5.1 Multiple independent proposals . 118 5.5.2 Correlated multipoint proposals . 120 5.6 Reversible Jumping Rule . 122 5.7 Dynamic Weighting . 124 5.8 Output Analysis and Algorithm Efficiency . 125 5.9 Problems . 127 The Gibbs Sampler 129 6.1 Gibbs Sampling Algorithms . 129 6.2 Illustrative Examples . 131 6.3 Some Special Samplers . 133 6.3.1 Slice sampler . 133 6.3.2 Metropolized Gibbs sampler . 133 6.3.3 Hit-and-run algorithm . 134 6.4 Data Augmentation Algorithm . 135 6.4.1 Bayesian missing data problem . 135 6.4.2 The original DA algorithm . 136 6.4.3 Connection with the Gibbs sampler . 137 6.4.4 An example: Hierarchical Bayes model . 138 6.5 Finding Repetitive Motifs in Biological Sequences . 139 6.5.1 A Gibbs sampler for detecting subtle motifs . 140 6.5.2 Alignment and classification . 141 6.6 Covariance Structures of the Gibbs Sampler . 143 6.6.1 Data Augmentation . 143 6.6.2 Autocovariances for the random-scan Gibbs sampler 144 6.6.3 More efficient use of Monte Carlo samples . 146 6.7 Collapsing and Grouping in a Gibbs Sampler . 146 6.8 Problems . 151 Cluster Algorithms for the Ising Model 153 7.1 Ising and Potts Model Revisit . 153 7.2 The Swendsen-Wang Algorithm as Data Augmentation . . . 154 7.3 Convergence Analysis and Generalization . 155 7.4 The Modification by Wolff . 157 7.5 Further Generalization . 157 7.6 Discussion . 158 xiv Contents 7.7 Problems. 159 8 General Conditional Sampling 161 8.1 Partial Resampling . 161 8.2 Case Studies for Partial Resampling . 163 8.2.1 Gaussian random field model . 163 8.2.2 Texture synthesis . 165 8.2.3 Inference with multivariate t-distribution . 169 8.3 Transformation Group and Generalized Gibbs . 171 8.4 Application: Parameter Expansion for Data Augmentation . 174 8.5 Some Examples in Bayesian Inference . 176 8.5.1 Probit regression . 176 8.5.2 Monte Carlo bridging for stochastic differential equa¬ tion . 178 8.6 Problems . . 181 9 Molecular Dynamics and Hybrid Monte Carlo 183 9.1 Basics of Newtonian Mechanics . 184 9.2 Molecular Dynamics Simulation . 185 9.3 Hybrid Monte Carlo . 189 9.4 Algorithms Related to HMC . 192 9.4.1 Langevin-Euler moves . 192 9.4.2 Generalized hybrid Monte Carlo . 193 9.4.3 Surrogate transition method . 194 9.5 Multipoint Strategies for Hybrid Monte Carlo . 195 9.5.1 Neal's window method . 195 9.5.2 Multipoint method . 197 9.6 Application of HMC in Statistics . 198 9.6.1 Indirect observation model . 199 9.6.2 Estimation in the stochastic volatility model . 201 10 Multilevel Sampling and Optimization Methods 205 10.1 Umbrella Sampling . 206 10.2 Simulated Annealing . 209 10.3 Simulated Tempering . 210 10.4 Parallel Tempering . 212 10.5 Generalized Ensemble Simulation . 215 10.5.1 Multicanonical sampling . 216 10.5.2 The l/i-ensemble method . 217 10.5.3 Comparison of algorithms . 218 10.6 Tempering with Dynamic Weighting . 219 10.6.1 Ising model simulation at sub-critical temperature . 221 10.6.2 Neural network training . 222 Contents xv 11 Population-Based Monte Carlo Methods 225 11. X Adaptive Direction Sampling: Snooker Algorithm . 226 11.2 Conjugate Gradient Monte Carlo . 227 11.3 Evolutionary Monte Carlo . 228 11.3.1 Evolutionary movements in binary-coded space . . . 230 11.3.2 Evolutionary movements in continuous space . 231 11.4 Some Further Thoughts . 233 11.5 Numerical Examples . 235 11.5.1 Simulating from a bimodal distribution . 235 11.5.2 Comparing algorithms for a multimodal example . . 236 11.5.3 Variable selection with binary-coded EMC . 237 11.5.4 Bayesian neural network training . 239 11.6 Problems . 242 12 Markov Chains and Their Convergence 245 12.1 Basic Properties of a Markov Chain . 245 12.1.1 Chapman-Kolmogorov equation . 247 12.1.2 Convergence to stationarity . 248 12.2 Coupling Method for Card Shuffling . 250 12.2.1 Random-to-top shuffling . 250 12.2.2 Riffle shuffling . 251 12.3 Convergence Theorem for Finite-State Markov Chains . . . 252 12.4 Coupling Method for General Markov Chain . 254 12.5 Geometric Inequalities . 256 12.5.1 Basic setup . 257 12.5.2 Poincaré inequality . 257 12.5.3 Example: Simple random walk on a graph . 259 12.5.4 Cheeger's inequality . 261 12.6 Functional Analysis for Markov Chains . 263 12.6.1 Forward and backward operators . 264 12.6.2 Convergence rate of Markov chains . 266 12.6.3 Maximal correlation . 267 12.7 Behavior of the Averages . 269 13 Selected Theoretical Topics 271 13.1 MCMC Convergence and Convergence Diagnostics . 271 13.2 Iterative Conditional Sampling . 273 13.2.1 Data augmentation . 273 13.2.2 Random-scan Gibbs sampler . 275 13.3 Comparison of Metropolis-Type Algorithms . 277 13.3.1 Peskun's ordering . 277 13.3.2 Comparing schemes using Peskun's ordering . 279 13.4 Eigenvalue Analysis for the Independence Sampler . 281 13.5 Perfect Simulation . 284 13.6 A Theory for Dynamic Weighting . 287 13.6.1 Definitions . 287 xvi Contents 13.6.2 Weight behavior under different scenarios . 288 13.6.3 Estimation with weighted samples . 291 13.6.4 A simulation study . 292 A Basics in Probability and Statistics 295 A.I Basic Probability Theory . 295 A. 1.1 Experiments, events, and probability . 295 A. 1.2 Univariate random variables and their properties . . 296 A. 1.3 Multivariate random variable . 298 A. 1.4 Convergence of random variables . 300 A. 2 Statistical Modeling and Inference . 301 A.2.1 Parametric statistical modeling . 301 A.2.2 FVequentist approach to statistical inference . 302 A.2.3 Bayesian methodology . 304 A.3 Bayes Procedure and Missing Data Formalism . 306 A.3.1 The joint and posterior distributions . 306 A.3. 2 The missing data problem . 308 A.4 The Expectation-Maximization Algorithm . 310 References 313 Author Index 333 Subject Index 338
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id	DE-604.BV023115644
illustrated	Illustrated
index_date	2024-07-02T19:49:54Z
indexdate	2025-02-20T06:44:31Z
institution	BVB
isbn	9780387763699 9780387952307
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016318171
oclc_num	191760006
open_access_boolean
owner	DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-739 DE-91G DE-BY-TUM DE-83 DE-11 DE-862 DE-BY-FWS
owner_facet	DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-739 DE-91G DE-BY-TUM DE-83 DE-11 DE-862 DE-BY-FWS
physical	XVI, 343 S. Ill., graph. Darst.
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Springer
record_format	marc
series2	Springer series in statistics
spellingShingle	Liu, Jun S. Monte Carlo strategies in scientific computing Naturwissenschaft Monte Carlo method Science Statistical methods Monte-Carlo-Simulation (DE-588)4240945-7 gnd
subject_GND	(DE-588)4240945-7
title	Monte Carlo strategies in scientific computing
title_auth	Monte Carlo strategies in scientific computing
title_exact_search	Monte Carlo strategies in scientific computing
title_exact_search_txtP	Monte Carlo strategies in scientific computing
title_full	Monte Carlo strategies in scientific computing Jun S. Liu
title_fullStr	Monte Carlo strategies in scientific computing Jun S. Liu
title_full_unstemmed	Monte Carlo strategies in scientific computing Jun S. Liu
title_short	Monte Carlo strategies in scientific computing
title_sort	monte carlo strategies in scientific computing
topic	Naturwissenschaft Monte Carlo method Science Statistical methods Monte-Carlo-Simulation (DE-588)4240945-7 gnd
topic_facet	Naturwissenschaft Monte Carlo method Science Statistical methods Monte-Carlo-Simulation
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016318171&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT liujuns montecarlostrategiesinscientificcomputing

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