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# Applied Statistics for Civil and Environmental Engineers

De
736 pages
Civil and environmental engineers need an understanding of mathematical statistics and probability theory to deal with the variability that affects engineers' structures, soil pressures, river flows and the like. Students, too, need to get to grips with these rather difficult concepts.

This book, written by engineers for engineers, tackles the subject in a clear, up-to-date manner using a process-orientated approach. It introduces the subjects of mathematical statistics and probability theory, and then addresses model estimation and testing, regression and multivariate methods, analysis of extreme events, simulation techniques, risk and reliability, and economic decision making.

325 examples and case studies from European and American practice are included and each chapter features realistic problems to be solved.

For the second edition new sections have been added on Monte Carlo Markov chain modeling with details of practical Gibbs sampling, sensitivity analysis and aleatory and epistemic uncertainties, and copulas. Throughout, the text has been revised and modernized.
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Contents
Dedication
Preface to the First Edition
Preface to the Second Edition
1
2
Introduction
Preliminary Data Analysis 1.1 Graphical Representation 1.1.1 Line diagram or bar chart 1.1.2 Dot diagram 1.1.3 Histogram 1.1.4 Frequency polygon 1.1.5 Cumulative relative frequency diagram 1.1.6 Duration curves 1.1.7 Summary of Section 1.1 1.2 Numerical Summaries of Data 1.2.1 Measures of central tendency 1.2.2 Measures of dispersion 1.2.3 Measure of asymmetry 1.2.4 Measure of peakedness 1.2.5 Summary of Section 1.2 1.3 Exploratory Methods 1.3.1 Stemandleaf plot 1.3.2 Box plot 1.3.3 Summary of Section 1.3 1.4 Data Observed in Pairs 1.4.1 Correlation and graphical plots 1.4.2 Covariance and the correlation coefﬁcient 1.4.3 QQ plots 1.4.4 Summary of Section 1.4 1.5 Summary for Chapter 1 References Problems
Basic Probability Concepts 2.1 Random Events 2.1.1 Sample space and events 2.1.2 The null event, intersection, and union 2.1.3 Venn diagram and event space 2.1.4 Summary of Section 2.1
xiii
xiv
xvi
1
3 3 4 4 5 8 9 10 11 11 12 15 19 19 19 20 20 22 23 23 23 24 26 27 27 28 29
38 39 39 41 43 49
v
vi
Contents
3
4
2.2 Measures of Probability 2.2.1 Interpretations of probability 2.2.2 Probability axioms 2.2.3 Addition rule 2.2.4 Further properties of probability functions 2.2.5 Conditional probability and multiplication rule 2.2.6 Stochastic independence 2.2.7 Total probability and Bayes theorems 2.2.8 Summary of Section 2.2 2.3 Summary for Chapter 2 References Problems
Random Variables and Their Properties 3.1 Random Variables and Probability Distributions 3.1.1 Random variables 3.1.2 Probability mass function 3.1.3 Cumulative distribution function of a discrete random variable 3.1.4 Probability density function 3.1.5 Cumulative distribution function of a continuous random variable 3.1.6 Summary of Section 3.1 3.2 Descriptors of Random Variables 3.2.1 Expectation and other population measures 3.2.2 Generating functions 3.2.3 Estimation of parameters 3.2.4 Summary of Section 3.2 3.3 Multiple Random Variables 3.3.1 Joint probability distributions of discrete variables 3.3.2 Joint probability distributions of continuous variables 3.3.3 Properties of multiple variables 3.3.4 Summary of Section 3.3 3.4 Associated Random Variables and Probabilities 3.4.1 Functions of a random variable 3.4.2 Functions of two or more variables 3.4.3 Properties of derived variables 3.4.4 Compound variables 3.4.5 Summary of Section 3.4 3.5 Copulas 3.6 Summary for Chapter 3 References Problems
Probability Distributions 4.1 Discrete Distributions 4.1.1 Bernoulli distribution 4.1.2 Binomial distribution 4.1.3 Poisson distribution 4.1.4 Geometric and negative binomial distributions
50 50 52 53 55 56 61 65 72 72 73 74
83 83 83 84
85 86
88 90 90 90 99 103 112 112 113 118 124 132 132 133 135 143 151 154 154 157 157 160
165 165 166 167 171 181
5
4.1.5 Logseries distribution 4.1.6 Multinomial distribution 4.1.7 Hypergeometric distribution 4.1.8 Summary of Section 4.1 4.2 Continuous Distributions 4.2.1 Uniform distribution 4.2.2 Exponential distribution 4.2.3 Erlang and gamma distribution 4.2.4 Beta distribution 4.2.5 Weibull distribution 4.2.6 Normal distribution 4.2.7 Lognormal distribution 4.2.8 Summary of Section 4.2 4.3 Multivariate Distributions 4.3.1 Bivariate normal distribution 4.3.2 Other bivariate distributions 4.4 Summary for Chapter 4 References Problems
Contents
Model Estimation and Testing 5.1 A Review of Terms Related to Random Sampling 5.2 Properties of Estimators 5.2.1 Unbiasedness 5.2.2 Consistency 5.2.3 Minimum variance 5.2.4 Efﬁciency 5.2.5 Sufﬁciency 5.2.6 Summary of Section 5.2 5.3 Estimation of Conﬁdence Intervals 5.3.1 Conﬁdence interval estimation of the mean when the standard deviation is known 5.3.2 Conﬁdence interval estimation of the mean when the standard deviation is unknown 5.3.3 Conﬁdence interval for a proportion 5.3.4 Sampling distribution of differences and sums of statistics 5.3.5 Interval estimation for the variance: chisquared distribution 5.3.6 Summary of Section 5.3 5.4 Hypothesis Testing 5.4.1 Procedure for testing 5.4.2 Probabilities of Type I and Type II errors and the power function 5.4.3 NeymanPearson lemma 5.4.4 Tests of hypotheses involving the variance 5.4.5 TheFdistribution and its use 5.4.6 Summary of Section 5.4 5.5 Nonparametric Methods 5.5.1 Sign test applied to the median 5.5.2 Wilcoxon signedrank test for association of paired observations
vii
185 187 189 192 194 194 196 200 203 205 209 215 217 217 219 222 222 223 224
230 230 231 231 232 232 234 234 235 236
236
239 242 242 243 247 247 248
254 256 257 258 259 260 261
262
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Contents
6
5.5.3 KruskalWallis test for paired observations inksamples 5.5.4 Tests on randomness: runs test 5.5.5 Spearmans rank correlation coefﬁcient 5.5.6 Summary of Section 5.5 5.6 GoodnessofFit Tests 5.6.1 Chisquared goodnessofﬁt test 5.6.2 KolmogorovSmirnov goodnessofﬁt test 5.6.3 KolmogorovSmirnov twosample test 5.6.4 AndersonDarling goodnessofﬁt test 5.6.5 Other methods for testing the goodnessofﬁt to a normal distribution 5.6.6 Summary of Section 5.6 5.7 Analysis of Variance 5.7.1 Oneway analysis of variance 5.7.2 Twoway analysis of variance 5.7.3 Summary of Section 5.7 5.8 Probability Plotting Methods and Visual Aids 5.8.1 Probability plotting for uniform distribution 5.8.2 Probability plotting for normal distribution 5.8.3 Probability plotting for Gumbel or EV1 distribution 5.8.4 Probability plotting of other distributions 5.8.5 Visual ﬁtting methods based on the histogram 5.8.6 Summary of Section 5.8 5.9 Identiﬁcation and Accommodation of Outliers 5.9.1 Hypothesis tests 5.9.2 Test statistics for detection of outliers 5.9.3 Dealing with nonnormal data 5.9.4 Estimation of probabilities of extreme events when outliers are present 5.9.5 Summary of Section 5.9 5.10 Summary of Chapter 5 References Problems
Methods of Regression and Multivariate Analysis 6.1 Simple Linear Regression 6.1.1 Estimates of the parameters 6.1.2 Properties of the estimators and errors 6.1.3 Tests of signiﬁcance and conﬁdence intervals 6.1.4 The bivariate normal model and correlation 6.1.5 Summary of Section 6.1 6.2 Multiple Linear Regression 6.2.1 Formulation of the model 6.2.2 Linear least squares solutions using the matrix method 6.2.3 Properties of least squares estimators and error variance 6.2.4 Model testing 6.2.5 Model adequacy 6.2.6 Residual plots 6.2.7 Inﬂuential observations and outliers in regression 6.2.8 Transformations
264 267 268 269 270 271 273 274 277
281 282 283 284 288 294 295 296 297 300 301 303 305 305 306 307 309
311 312 312 313 316
326 327 328 332 337 339 342 342 343 343 346 350 355 356 358 365
7
8
6.2.9 Conﬁdence intervals on mean response and prediction 6.2.10 Ridge regression 6.2.11 Other methods and discussion of Section 6.2 6.3 Multivariate Analysis 6.3.1 Principal components analysis 6.3.2 Factor analysis 6.3.3 Cluster analysis 6.3.4 Other methods and summary of Section 6.3 6.4 Spatial Correlation 6.4.1 The estimation problem 6.4.2 Spatial correlation and the semivariogram 6.4.3 Some semivariogram models and physical aspects 6.4.4 Spatial interpolations and Kriging 6.4.5 Summary of Section 6.4 6.5 Summary of Chapter 6 References Problems
Frequency Analysis of Extreme Events 7.1 Order Statistics 7.1.1 Deﬁnitions and distributions 7.1.2 Functions of order statistics 7.1.3 Expected value and variance of order statistics 7.1.4 Summary of Section 7.1 7.2 Extreme Value Distributions 7.2.1 Basic concepts of extreme value theory 7.2.2 Gumbel distribution 7.2.3 Fréchet distribution 7.2.4 Weibull distribution as an extreme value model 7.2.5 General extreme value distribution 7.2.6 Contagious extreme value distributions 7.2.7 Use of other distributions as extreme value models 7.2.8 Summary of Section 7.2 7.3 Analysis of Natural Hazards 7.3.1 Floods, storms, and droughts 7.3.2 Earthquakes and volcanic eruptions 7.3.3 Winds 7.3.4 Sea levels and highest sea waves 7.3.5 Summary of Section 7.3 7.4 Summary of Chapter 7 References Problems
Simulation Techniques for Design 8.1 Monte Carlo Simulation 8.1.1 Statistical experiments 8.1.2 Probability integral transform 8.1.3 Sample size and accuracy of Monte Carlo experiments 8.1.4 Summary for Section 8.1 8.2 Generation of Random Numbers
Contents
ix
366 368 370 373 373 379 383 385 386 387 387 389 391 394 394 395 398
405 406 406 409 411 415 415 415 422 429 432 435 439 445 450 453 453 461 465 470 473 474 474 478
487 488 488 493 495 501 501
x
Contents
9
8.2.1 Random outcomes from standard uniform variates 8.2.2 Random outcomes from continuous variates 8.2.3 Random outcomes from discrete variates 8.2.4 Random outcomes from jointly distributed variates 8.2.5 Summary of Section 8.2 8.3 Use of Simulation 8.3.1 Distributions of derived design variates 8.3.2 Sampling statistics 8.3.3 Simulation of time or spacevarying systems 8.3.4 Design alternatives and optimal design 8.3.5 Summary of Section 8.3 8.4 Sensitivity and Uncertainty Analysis 8.5 Summary and Discussion of Chapter 8 References Problems
Risk and Reliability Analysis 9.1 Measures of Reliability 9.1.1 Factors of safety 9.1.2 Safety margin 9.1.3 Reliability index 9.1.4 Performance function and limiting state 9.1.5 Further practical solutions 9.1.6 Summary of Section 9.1 9.2 Multiple Failure Modes 9.2.1 Independent failure modes 9.2.2 Mutually dependent failure modes 9.2.3 Summary of Section 9.2 9.3 Uncertainty in Reliability Assessments 9.3.1 Reliability limits 9.3.2 Bayesian revision of reliability 9.3.3 Summary of Section 9.3 9.4 Temporal Reliability 9.4.1 Failure process and survival time 9.4.2 Hazard function 9.4.3 Reliable life 9.4.4 Summary of Section 9.4 9.5 ReliabilityBased Design 9.6 Summary for Chapter 9 References Problems
10 Bayesian Decision Methods and Parameter Uncertainty 10.1 Basic Decision Theory 10.1.1 Bayes rules 10.1.2 Decision trees 10.1.3 The minimax solution 10.1.4 Summary of Section 10.1 10.2 Posterior Bayesian Decision Analysis 10.2.1 Subjective probabilities
501 506 511 513 514 514 514 517 519 524 530 530 531 531 533
541 542 542 547 550 558 568 577 577 578 584 592 592 592 593 597 597 597 602 605 606 606 612 613 615
623 624 624 627 630 632 632 633
10.2.2 Loss and utility functions 10.2.3 The discrete case 10.2.4 Inference with conditional binomial and prior beta 10.2.5 Poisson hazards and gamma prior 10.2.6 Inferences with normal distribution 10.2.7 Likelihood ratio testing 10.2.8 Summary of Section 10.2 10.3 Markov Chain Monte Carlo Methods 10.4 JamesStein Estimators 10.5 Summary and Discussion of Chapter 10 References Problems
Contents
Appendix A: Further mathematics A.1 Chebyshev Inequality A.2 Convex Function and Jensen Inequality A.3 Derivation of the Poisson distribution A.4 Derivation of the normal distribution A.5 MGF of the normal distribution A.6 Central limit theorem A.7 Pdf of StudentsTdistribution A.8 Pdf of theFdistribution A.9 Wilcoxon signedrank test: mean and variance of the test statistic A.10 Spearmans rank correlation coefﬁcient
Appendix B: Glossary of Symbols
Appendix C: Tables of Selected Distributions
Appendix D: Brief Answers to Selected Problems
Appendix E: Data Lists
Index
xi
634 635 636 638 639 642 643 643 650 653 653 656
659 659 659 659 660 661 662 663 664 664 665
667
673
684
687
707