This second edition focuses on the application of statistical methods in the field of hydrology and hydroclimatology. Among the latest theories being used in these fields, the book introduces the theory of copulas and its applications in this context. The purpose is to develop an understanding and illustrate the usefulness of the statistical techniques with detailed theory and numerous worked out examples. Apart from this, sample scripts based on MATLAB, Python and R for some examples are also provided to assist the readers to handle real life data. Besides serving as a textbook for graduate courses on stochastic modeling in hydrology and related disciplines, the book offers a valuable resource for researchers and professionals involved in the field of hydrology and climatology.
Author(s): Rajib Maity
Series: Springer Transactions in Civil and Environmental Engineering
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 455
City: Singapore
Preface to the Second Edition
Preface to the First Edition
Acknowledgements
Contents
About the Author
1 Introduction
1.1 Definitions and Scope
1.2 Role of Statistical Methods
1.2.1 Hydrologic and Hydroclimatic Variability
1.2.2 Need of Statistical Methods
1.3 Organization of the Book
2 Basic Concepts of Probability and Statistics
2.1 Concepts of Random Experiments and Random Variables
2.1.1 Random Experiments, Sample Space and Events
2.1.2 Concept of Random Variables and Events
2.2 Basic Concepts of Probability
2.2.1 The Axioms of Probability
2.2.2 Some Elementary Properties on Probability
2.3 Conditional Probability Theorem
2.4 Total Probability Theorem and Bayes Rule
2.5 Univariate and Bivariate Probability Distribution of Random Variables
2.5.1 Discrete Random Variable
2.5.2 Continuous Random Variable
2.6 Marginal and Conditional Probability Distribution
2.6.1 Marginal Probability Distribution
2.6.2 Conditional Distribution Function
2.7 Independence Between Random Variables
2.8 Functions of Random Variables
2.8.1 Univariate Random Variable
2.8.2 Bivariate Random Variables
2.9 Coding Examples with MATLAB, PYTHON and R
Exercise
3 Basic Statistical Properties of Data
3.1 Descriptive Statistics
3.1.1 Measures of Central Tendency
3.1.2 Measure of Dispersion
3.1.3 Measure of Symmetry
3.1.4 Measure of Tailedness
3.2 Concept of Moments and Expectation
3.2.1 Expectation
3.3 Moment Generating Functions
3.4 Characteristic Functions
3.5 Statistical Properties of Jointly Distributed Random Variables
3.5.1 Expectation
3.5.2 Moment about the Origin
3.5.3 Moment about the Mean (Central Moment)
3.5.4 Moment Generating Function
3.5.5 Covariance
3.5.6 Correlation Coefficient
3.5.7 Further Properties of Moments
3.6 Properties of the Estimator
3.6.1 Unbiasedness
3.6.2 Consistency
3.6.3 Efficiency
3.6.4 Sufficiency
3.7 Parameter Estimation
3.7.1 Method of Moments
3.7.2 Maximum Likelihood
3.8 Chebyshev Inequality
3.9 Law of Large Number
3.10 Coding Examples with MATLAB, PYTHON and R
Exercise
4 Probability Distributions and Their Applications
4.1 Discrete Probability Distributions
4.1.1 Binomial Distribution
4.1.2 Negative Binomial Distribution
4.1.3 Multinomial Distribution
4.1.4 Hypergeometric Distribution
4.1.5 Geometric Distribution
4.1.6 Poisson Distribution
4.2 Continuous Probability Distributions
4.2.1 Uniform Distribution
4.2.2 Exponential Distribution
4.2.3 Normal Distribution
4.2.4 Lognormal Distribution
4.2.5 Gamma Distribution
4.2.6 Extreme Value Distribution
4.2.7 Beta Distribution
4.2.8 Pearson and Log-Pearson Type III Distribution
4.3 Mixed Distribution
4.4 Some Important Distributions of Sample Statistics
4.4.1 Chi-Square Distribution
4.4.2 The t-Distribution
4.4.3 The F-Distribution
4.5 Coding Examples with MATLAB, PYTHON and R
Exercise
5 Frequency Analysis, Risk and Uncertainty in Hydroclimatic Analysis
5.1 Concept of Return Period
5.2 Probability Plotting and Plotting Positions Formulae
5.3 Probability Paper
5.3.1 Mathematical Construction of Probability Paper
5.3.2 Graphical Construction of Probability Paper
5.4 Frequency Analysis of Hydroclimatic Extremes
5.4.1 Normal Distribution
5.4.2 Lognormal Distribution
5.4.3 Log-Pearson Type III Distribution
5.4.4 Extreme Value Type I Distribution
5.5 Risk and Reliability in Hydrologic Design
5.6 Concept of Uncertainty
5.6.1 Analysis of Uncertainty
5.6.2 Measures of Uncertainty
5.7 Reliability, Resilience and Vulnerability of Hydrologic Time Series
5.7.1 Reliability
5.7.2 Resilience
5.7.3 Vulnerability
5.8 Coding Examples with MATLAB, PYTHON and R
Exercise
6 Hypothesis Testing and Non-parametric Test
6.1 Populations and Samples
6.2 Random Samples
6.3 Sampling Distribution
6.3.1 Sampling Distribution of the Mean
6.3.2 Sampling Distribution of the Variance
6.4 Statistical Inference
6.4.1 Point Estimation
6.4.2 Interval Estimation
6.4.3 Hypothesis Testing
6.4.4 Goodness-of-Fit Test
6.4.5 Non-parametric Test
6.5 Coding Examples with MATLAB, PYTHON and R
Exercise
7 Regression Analysis and Curve Fitting
7.1 Simple Linear Regression
7.2 Curvilinear Regression
7.2.1 Model Transformable to Linear Regression
7.2.2 Model Not Transformable to Linear Regression
7.3 Multiple Linear Regression
7.4 Evaluation of Regression Model
7.5 Correlation and Regression
7.6 Correlation and Causality
7.7 Confidence Interval
7.8 Coding Examples with MATLAB, PYTHON and R
8 Multivariate Analysis
8.1 Principal Component Analysis
8.1.1 Determination of Principal Components
8.2 Supervised Principal Component Analysis
8.3 Dimensionality Reduction Using PCA and SPCA
8.4 Canonical Correlation Analysis
8.5 Empirical Orthogonal Function
8.6 Data Generation
8.6.1 Univariate Data Generation
8.6.2 Multivariate Data Generation
8.7 Analysis of Variance in Hydrology and Hydroclimatology
8.7.1 One-Way Analysis of Variance
8.7.2 Two-Way Analysis of Variance
8.7.3 Multiple Comparisons
8.8 Coding Examples with MATLAB, PYTHON and R
9 Time Series Analysis
9.1 Data Representation in Hydroclimatology
9.2 Stationary and Non-stationary Time Series
9.3 Ensemble and Realization
9.4 Trend Analysis
9.4.1 Tests for Randomness and Trend
9.4.2 Trend Removal
9.5 Analysis of Periodicity
9.5.1 Harmonic Analysis
9.5.2 Spectral Analysis
9.6 Data Transformation
9.6.1 Test for Normal Distribution
9.7 Time Series Modeling in Hydroclimatology
9.7.1 Measures of Linear Association in Time Series
9.7.2 Statistical Operators on Time Series
9.7.3 Properties of Time Series Models
9.7.4 Auto-Regressive (AR) Model
9.7.5 Moving Average (MA) Model
9.7.6 Auto-Regressive Moving Average (ARMA) Model
9.7.7 Auto-regressive Integrated Moving Average (ARIMA) Model
9.7.8 Auto-regressive Moving Average Model with Exogenous Inputs (ARMAX)
9.7.9 Forecasting with ARMA/ARMAX
9.7.10 Parsimony of Time Series Models
9.7.11 Diagnostic Check for ARMA Models
9.8 Wavelet Analysis
9.8.1 Haar Wavelet
9.8.2 Multi-resolution Analysis
9.9 Coding Examples with MATLAB, PYTHON and R
Exercise
10 Theory of Copula in Hydrology and Hydroclimatology
10.1 Introduction
10.2 Preliminary Concepts
10.2.1 Definition of Copula
10.2.2 Graphical Representation of Copula
10.3 Sklar's Theorem
10.4 Basic Properties of a Copula Function
10.4.1 Basic Terminologies
10.5 Non-parametric Measures of Association
10.6 Copula and Function of Random Variables
10.7 Survival Copula
10.8 Most Commonly Used Copula Function
10.8.1 Elliptical Copula
10.8.2 Archimedean Copula
10.9 Selection of Best-Fit Copula
10.9.1 Test Using Empirical Copula
10.9.2 Test Using Kendall's Transform
10.9.3 Test Using Rosenblatt Probability Integral Transformation
10.10 Use of Copulas
10.10.1 Data Generation
10.10.2 Probabilistic Prediction Using Copulas
10.11 Coding Examples with MATLAB, PYTHON and R
Exercise
References
Appendix A Data Set
Appendix B Statistical Tables
Index
