Data Analysis Statistical and Computational Methods for Scientists and Engineers

✍ Scribed by Brandt, Siegmund

• Publisher: Springer
• Year: 2014
• Tongue: English
• Leaves: 532
• Edition: 4th ed. 2014
• Category: Library

✦ Synopsis

The fourth edition of this successful textbook presents a comprehensive introduction to statistical and numerical methods for the evaluation of empirical and experimental data. Equal weight is given to statistical theory and practical problems. The concise mathematical treatment of the subject matter is illustrated by many examples and for the present edition a library of Java programs has been developed. It comprises methods of numerical data analysis and graphical representation as well as many example programs and solutions to programming problems. The programs (source code, Java classes and documentation) and extensive appendices to the main text are available for free download from the book s page atwww.springer.com.ContentsProbabilities. Random variables.Random numbers and the Monte Carlo Method.Statistical distributions (binomial, Gauss, Poisson). Samples. Statistical tests.Maximum Likelihood. Least Squares. Regression. Minimization.Analysis of Variance. Time series analysis.AudienceThe book is conceived both as an introduction and as a work of reference. In particular it addresses itself to students, scientists and practitioners in science and engineering as a help in the analysis of their datain laboratory courses, in working for bachelor or master degrees, in thesis work, in research and professional work." The book is concise, but gives a sufficiently rigorous mathematical treatment of practical statistical methods for data analysis; it can be of great use to all who are involved with data analysis. "Physicalia""" Serves as a nice reference guide for any scientist interested in the fundamentals of data analysis on the computer. "The American Statistician" This lively and erudite treatise covers the theory of the main statistical tools and their practical applications a first rate university textbook, and good background material for the practicing physicist. "Physics BulletinThe AuthorSiegmund Brandt is Emeritus Professor of Physics at the University of Siegen. With his group he worked on experiments in elementary-particle physics at the research centers DESY in Hamburg and CERN in Geneva in which the analysis of the experimental data plays an important role. He is author or coauthor of textbooks which have appeared in ten languages.

✦ Table of Contents

Preface to the Fourth English Edition......Page 6
Contents......Page 8
List of Examples......Page 16
Frequently Used Symbols and Notation......Page 20
1.1 Typical Problems of Data Analysis......Page 22
1.2 On the Structure of this Book......Page 23
1.3 About the Computer Programs......Page 26
2.1 Experiments, Events, Sample Space......Page 28
2.2 The Concept of Probability......Page 29
2.3 Rules of Probability Calculus: Conditional Probability......Page 31
2.4.1 Probability for n Dots in the Throwing of TwoDice......Page 32
2.4.2 Lottery 6 Out of 49......Page 33
2.4.3 Three-Door Game......Page 34
3.2 Distributions of a Single Random Variable......Page 35
3.3 Functions of a Single Random Variable, Expectation Value, Variance, Moments......Page 37
3.4 Distribution Function and Probability Density of TwoVariables: Conditional Probability......Page 45
3.5 Expectation Values, Variance, Covariance, and Correlation......Page 47
3.6 More than Two Variables: Vector and Matrix Notation......Page 50
3.7 Transformation of Variables......Page 53
3.8 Linear and Orthogonal Transformations: ErrorPropagation......Page 56
4.1 Random Numbers......Page 61
4.2 Representation of Numbers in a Computer......Page 62
4.3 Linear Congruential Generators......Page 64
4.4 Multiplicative Linear Congruential Generators......Page 65
4.5 Quality of an MLCG: Spectral Test......Page 67
4.6 Implementation and Portability of an MLCG......Page 70
4.7 Combination of Several MLCGs......Page 72
4.8.1 Generation by Transformation of the UniformDistribution......Page 75
4.8.2 Generation with the von Neumann Acceptance–Rejection Technique......Page 78
4.9 Generation of Normally Distributed Random Numbers......Page 82
4.10 Generation of Random Numbers Accordingto a Multivariate Normal Distribution......Page 83
4.11 The Monte Carlo Method for Integration......Page 84
4.12 The Monte Carlo Method for Simulation......Page 86
4.13 Java Classes and Example Programs......Page 87
5.1 The Binomial and Multinomial Distributions......Page 89
5.2 Frequency: The Law of Large Numbers......Page 92
5.3 The Hypergeometric Distribution......Page 94
5.4 The Poisson Distribution......Page 98
5.5 The Characteristic Function of a Distribution......Page 101
5.6 The Standard Normal Distribution......Page 104
5.7 The Normal or Gaussian Distribution......Page 106
5.8 Quantitative Properties of the Normal Distribution......Page 108
5.9 The Central Limit Theorem......Page 110
5.10 The Multivariate Normal Distribution......Page 114
5.11.1 Folding Integrals......Page 120
5.11.2 Convolutions with the Normal Distribution......Page 123
5.12 Example Programs......Page 126
6.1 Random Samples. Distributionof a Sample. Estimators......Page 129
6.2 Samples from Continuous Populations: Meanand Variance of a Sample......Page 131
6.3 Graphical Representation of Samples: Histogramsand Scatter Plots......Page 135
6.4 Samples from Partitioned Populations......Page 142
6.5 Samples Without Replacement from Finite DiscretePopulations. Mean Square Deviation. Degrees ofFreedom......Page 147
6.6 Samples from Gaussian Distributions: χ2-Distribution......Page 150
6.7 χ2 and Empirical Variance......Page 155
6.8 Sampling by Counting: Small Samples......Page 156
6.9 Small Samples with Background......Page 162
6.10 Determining a Ratio of Small Numbers of Events......Page 164
6.11 Ratio of Small Numbers of Events with Background......Page 167
6.12 Java Classes and Example Programs......Page 169
7.1 Likelihood Ratio: Likelihood Function......Page 173
7.2 The Method of Maximum Likelihood......Page 175
7.3 Information Inequality. Minimum VarianceEstimators. Sufficient Estimators......Page 177
7.4 Asymptotic Properties of the Likelihood Functionand Maximum-Likelihood Estimators......Page 184
7.5 Simultaneous Estimation of Several Parameters:Confidence Intervals......Page 187
7.6 Example Programs......Page 193
8.1 Introduction......Page 194
8.2 F-Test on Equality of Variances......Page 196
8.3 Student's Test: Comparison of Means......Page 199
8.4 Concepts of the General Theory of Tests......Page 204
8.5 The Neyman–Pearson Lemma and Applications......Page 210
8.6 The Likelihood-Ratio Method......Page 213
8.7.1 χ2-Test with Maximal Number of Degreesof Freedom......Page 218
8.7.3 χ2-Test and Empirical Frequency Distribution......Page 219
8.8 Contingency Tables......Page 222
8.9 2 2 Table Test......Page 223
8.10 Example Programs......Page 224
9.1 Direct Measurements of Equal or Unequal Accuracy......Page 227
9.2 Indirect Measurements: Linear Case......Page 232
9.3 Fitting a Straight Line......Page 236
9.4.1 Fitting a Polynomial......Page 240
9.4.2 Fit of an Arbitrary Linear Function......Page 242
9.5 Indirect Measurements: Nonlinear Case......Page 244
9.6 Algorithms for Fitting Nonlinear Functions......Page 246
9.6.1 Iteration with Step-Size Reduction......Page 247
9.6.2 Marquardt Iteration......Page 252
9.7 Properties of the Least-Squares Solution: χ2-Test......Page 254
9.8 Confidence Regions and Asymmetric Errorsin the Nonlinear Case......Page 258
9.9 Constrained Measurements......Page 261
9.9.1 The Method of Elements......Page 262
9.9.2 The Method of Lagrange Multipliers......Page 265
9.10 The General Case of Least-Squares Fitting......Page 269
9.11 Algorithm for the General Case of Least Squares......Page 273
9.12 Applying the Algorithm for the General Caseto Constrained Measurements......Page 276
9.13 Confidence Region and Asymmetric Errorsin the General Case......Page 278
9.14 Java Classes and Example Programs......Page 279
10.1 Overview: Numerical Accuracy......Page 284
10.2 Parabola Through Three Points......Page 290
10.4 Bracketing the Minimum......Page 292
10.5 Minimum Search with the Golden Section......Page 294
10.7 Minimization Along a Direction in n Dimensions......Page 297
10.8 Simplex Minimization in n Dimensions......Page 298
10.9 Minimization Along the Coordinate Directions......Page 301
10.10 Conjugate Directions......Page 302
10.11 Minimization Along Chosen Directions......Page 304
10.13 Minimization Along Conjugate Gradient Directions......Page 305
10.15 Marquardt Minimization......Page 309
10.16 On Choosing a Minimization Method......Page 312
10.17 Consideration of Errors......Page 313
10.18 Examples......Page 315
10.19 Java Classes and Example Programs......Page 320
11.1 One-Way Analysis of Variance......Page 324
11.2 Two-Way Analysis of Variance......Page 328
11.3 Java Class and Example Programs......Page 336
12.1 Orthogonal Polynomials......Page 337
12.2 Regression Curve: Confidence Interval......Page 341
12.3 Regression with Unknown Errors......Page 342
12.4 Java Class and Example Programs......Page 345
13.1 Time Series: Trend......Page 346
13.2 Moving Averages......Page 347
13.4 Confidence Intervals......Page 351
13.5 Java Class and Example Programs......Page 355
Literature......Page 356
A Matrix Calculations......Page 361
A.1 Definitions: Simple Operations......Page 362
A.2 Vector Space, Subspace, Rank of a Matrix......Page 365
A.3 Orthogonal Transformations......Page 367
A.3.1 Givens Transformation......Page 368
A.3.2 Householder Transformation......Page 370
A.3.4 Permutation Transformation......Page 373
A.4 Determinants......Page 374
A.5 Matrix Equations: Least Squares......Page 376
A.6 Inverse Matrix......Page 379
A.7 Gaussian Elimination......Page 381
A.8 LR-Decomposition......Page 383
A.9 Cholesky Decomposition......Page 386
A.10 Pseudo-inverse Matrix......Page 389
A.11 Eigenvalues and Eigenvectors......Page 390
A.12 Singular Value Decomposition......Page 393
A.13 Singular Value Analysis......Page 394
A.14.1 Strategy......Page 399
A.14.2 Bidiagonalization......Page 400
A.14.3 Diagonalization......Page 402
A.15 Least Squares with Weights......Page 406
A.16 Least Squares with Change of Scale......Page 407
A.17 Modification of Least Squares According to Marquardt......Page 408
A.18 Least Squares with Constraints......Page 410
A.19 Java Classes and Example Programs......Page 413
B Combinatorics......Page 418
C.2 Hypergeometric Distribution......Page 421
C.4 Normal Distribution......Page 422
C.5 χ2-Distribution......Page 424
C.7 t-Distribution......Page 425
C.8 Java Class and Example Program......Page 426
D.1 The Euler Gamma Function......Page 427
D.4 Computing Continued Fractions......Page 430
D.6 Incomplete Beta Function......Page 432
D.7 Java Class and Example Program......Page 434
E.1 Numerical Differentiation......Page 436
E.3 Interactive Input and Output Under Java......Page 438
E.4 Java Classes......Page 439
F.2 Graphical Workstations: Control Routines......Page 441
F.3.1 Coordinate Systems......Page 442
F.3.2 Linear Transformations: Window – Viewport......Page 443
F.4 Transformation Methods......Page 445
F.5 Drawing Methods......Page 446
F.6 Utility Methods......Page 449
F.8 Java Classes and Example Programs......Page 451
G.1 Problems......Page 457
G.2 Hints and Solutions......Page 466
G.3 Programming Problems......Page 480
H Collection of Formulas......Page 497
I Statistical Tables......Page 512
List of Computer Programs......Page 524
Index......Page 526