Data Analysis. Statistical and Computational Methods for Scientists and Engineers

✍ Scribed by Siegmund Brandt

• Publisher: Springer
• Year: 2014
• Tongue: English
• Leaves: 533
• Edition: 4
• Category: Library

✦ Table of Contents

Preface to the Fourth English Edition
Contents
List of Examples
Frequently Used Symbols and Notation
1 Introduction
1.1 Typical Problems of Data Analysis
1.2 On the Structure of this Book
1.3 About the Computer Programs
2 Probabilities
2.1 Experiments, Events, Sample Space
2.2 The Concept of Probability
2.3 Rules of Probability Calculus: Conditional Probability
2.4 Examples
2.4.1 Probability for n Dots in the Throwing of TwoDice
2.4.2 Lottery 6 Out of 49
2.4.3 Three-Door Game
3 Random Variables: Distributions
3.1 Random Variables
3.2 Distributions of a Single Random Variable
3.3 Functions of a Single Random Variable, Expectation Value, Variance, Moments
3.4 Distribution Function and Probability Density of TwoVariables: Conditional Probability
3.5 Expectation Values, Variance, Covariance, and Correlation
3.6 More than Two Variables: Vector and Matrix Notation
3.7 Transformation of Variables
3.8 Linear and Orthogonal Transformations: ErrorPropagation
4 Computer Generated Random Numbers: The MonteCarlo Method
4.1 Random Numbers
4.2 Representation of Numbers in a Computer
4.3 Linear Congruential Generators
4.4 Multiplicative Linear Congruential Generators
4.5 Quality of an MLCG: Spectral Test
4.6 Implementation and Portability of an MLCG
4.7 Combination of Several MLCGs
4.8 Generation of Arbitrarily Distributed Random Numbers
4.8.1 Generation by Transformation of the UniformDistribution
4.8.2 Generation with the von Neumann Acceptance–Rejection Technique
4.9 Generation of Normally Distributed Random Numbers
4.10 Generation of Random Numbers Accordingto a Multivariate Normal Distribution
4.11 The Monte Carlo Method for Integration
4.12 The Monte Carlo Method for Simulation
4.13 Java Classes and Example Programs
5 Some Important Distributions and Theorems
5.1 The Binomial and Multinomial Distributions
5.2 Frequency: The Law of Large Numbers
5.3 The Hypergeometric Distribution
5.4 The Poisson Distribution
5.5 The Characteristic Function of a Distribution
5.6 The Standard Normal Distribution
5.7 The Normal or Gaussian Distribution
5.8 Quantitative Properties of the Normal Distribution
5.9 The Central Limit Theorem
5.10 The Multivariate Normal Distribution
5.11 Convolutions of Distributions
5.11.1 Folding Integrals
5.11.2 Convolutions with the Normal Distribution
5.12 Example Programs
6 Samples
6.1 Random Samples. Distributionof a Sample. Estimators
6.2 Samples from Continuous Populations: Meanand Variance of a Sample
6.3 Graphical Representation of Samples: Histogramsand Scatter Plots
6.4 Samples from Partitioned Populations
6.5 Samples Without Replacement from Finite DiscretePopulations. Mean Square Deviation. Degrees ofFreedom
6.6 Samples from Gaussian Distributions: χ2-Distribution
6.7 χ2 and Empirical Variance
6.8 Sampling by Counting: Small Samples
6.9 Small Samples with Background
6.10 Determining a Ratio of Small Numbers of Events
6.11 Ratio of Small Numbers of Events with Background
6.12 Java Classes and Example Programs
7 The Method of Maximum Likelihood
7.1 Likelihood Ratio: Likelihood Function
7.2 The Method of Maximum Likelihood
7.3 Information Inequality. Minimum VarianceEstimators. Sufficient Estimators
7.4 Asymptotic Properties of the Likelihood Functionand Maximum-Likelihood Estimators
7.5 Simultaneous Estimation of Several Parameters:Confidence Intervals
7.6 Example Programs
8 Testing Statistical Hypotheses
8.1 Introduction
8.2 F-Test on Equality of Variances
8.3 Student's Test: Comparison of Means
8.4 Concepts of the General Theory of Tests
8.5 The Neyman–Pearson Lemma and Applications
8.6 The Likelihood-Ratio Method
8.7 The χ2-Test for Goodness-of-Fit
8.7.1 χ2-Test with Maximal Number of Degreesof Freedom
8.7.2 χ2-Test with Reduced Number of Degreesof Freedom
8.7.3 χ2-Test and Empirical Frequency Distribution
8.8 Contingency Tables
8.9 2 2 Table Test
8.10 Example Programs
9 The Method of Least Squares
9.1 Direct Measurements of Equal or Unequal Accuracy
9.2 Indirect Measurements: Linear Case
9.3 Fitting a Straight Line
9.4 Algorithms for Fitting Linear Functionsof the Unknowns
9.4.1 Fitting a Polynomial
9.4.2 Fit of an Arbitrary Linear Function
9.5 Indirect Measurements: Nonlinear Case
9.6 Algorithms for Fitting Nonlinear Functions
9.6.1 Iteration with Step-Size Reduction
9.6.2 Marquardt Iteration
9.7 Properties of the Least-Squares Solution: χ2-Test
9.8 Confidence Regions and Asymmetric Errorsin the Nonlinear Case
9.9 Constrained Measurements
9.9.1 The Method of Elements
9.9.2 The Method of Lagrange Multipliers
9.10 The General Case of Least-Squares Fitting
9.11 Algorithm for the General Case of Least Squares
9.12 Applying the Algorithm for the General Caseto Constrained Measurements
9.13 Confidence Region and Asymmetric Errorsin the General Case
9.14 Java Classes and Example Programs
10 Function Minimization
10.1 Overview: Numerical Accuracy
10.2 Parabola Through Three Points
10.3 Function of n Variables on a Linein an n-Dimensional Space
10.4 Bracketing the Minimum
10.5 Minimum Search with the Golden Section
10.6 Minimum Search with Quadratic Interpolation
10.7 Minimization Along a Direction in n Dimensions
10.8 Simplex Minimization in n Dimensions
10.9 Minimization Along the Coordinate Directions
10.10 Conjugate Directions
10.11 Minimization Along Chosen Directions
10.12 Minimization in the Direction of Steepest Descent
10.13 Minimization Along Conjugate Gradient Directions
10.14 Minimization with the Quadratic Form
10.15 Marquardt Minimization
10.16 On Choosing a Minimization Method
10.17 Consideration of Errors
10.18 Examples
10.19 Java Classes and Example Programs
11 Analysis of Variance
11.1 One-Way Analysis of Variance
11.2 Two-Way Analysis of Variance
11.3 Java Class and Example Programs
12 Linear and Polynomial Regression
12.1 Orthogonal Polynomials
12.2 Regression Curve: Confidence Interval
12.3 Regression with Unknown Errors
12.4 Java Class and Example Programs
13 Time Series Analysis
13.1 Time Series: Trend
13.2 Moving Averages
13.3 Edge Effects
13.4 Confidence Intervals
13.5 Java Class and Example Programs
Literature
A Matrix Calculations
A.1 Definitions: Simple Operations
A.2 Vector Space, Subspace, Rank of a Matrix
A.3 Orthogonal Transformations
A.3.1 Givens Transformation
A.3.2 Householder Transformation
A.3.3 Sign Inversion
A.3.4 Permutation Transformation
A.4 Determinants
A.5 Matrix Equations: Least Squares
A.6 Inverse Matrix
A.7 Gaussian Elimination
A.8 LR-Decomposition
A.9 Cholesky Decomposition
A.10 Pseudo-inverse Matrix
A.11 Eigenvalues and Eigenvectors
A.12 Singular Value Decomposition
A.13 Singular Value Analysis
A.14 Algorithm for Singular Value Decomposition
A.14.1 Strategy
A.14.2 Bidiagonalization
A.14.3 Diagonalization
A.14.4 Ordering of the Singular Values and Permutation
A.14.5 Singular Value Analysis
A.15 Least Squares with Weights
A.16 Least Squares with Change of Scale
A.17 Modification of Least Squares According to Marquardt
A.18 Least Squares with Constraints
A.19 Java Classes and Example Programs
B Combinatorics
C Formulas and Methods for the Computation of StatisticalFunctions
C.1 Binomial Distribution
C.2 Hypergeometric Distribution
C.3 Poisson Distribution
C.4 Normal Distribution
C.5 χ2-Distribution
C.6 F-Distribution
C.7 t-Distribution
C.8 Java Class and Example Program
D The Gamma Function and Related Functions: Methodsand Programs for Their Computation
D.1 The Euler Gamma Function
D.2 Factorial and Binomial Coefficients
D.3 Beta Function
D.4 Computing Continued Fractions
D.5 Incomplete Gamma Function
D.6 Incomplete Beta Function
D.7 Java Class and Example Program
E Utility Programs
E.1 Numerical Differentiation
E.2 Numerical Determination of Zeros
E.3 Interactive Input and Output Under Java
E.4 Java Classes
F The Graphics Class DatanGraphics
F.1 Introductory Remarks
F.2 Graphical Workstations: Control Routines
F.3 Coordinate Systems, Transformationsand Transformation Methods
F.3.1 Coordinate Systems
F.3.2 Linear Transformations: Window – Viewport
F.4 Transformation Methods
F.5 Drawing Methods
F.6 Utility Methods
F.7 Text Within the Plot
F.8 Java Classes and Example Programs
G Problems, Hints and Solutions, and Programming Problems
G.1 Problems
G.2 Hints and Solutions
G.3 Programming Problems
H Collection of Formulas
I Statistical Tables
List of Computer Programs
Index