Table Of ContentSpringer Transactions in Civil
and Environmental Engineering
Rajib Maity
Statistical Methods
in Hydrology and
Hydroclimatology
Springer Transactions in Civil
and Environmental Engineering
More information about this series at http://www.springer.com/series/13593
Rajib Maity
Statistical Methods
in Hydrology
and Hydroclimatology
123
Rajib Maity
Department ofCivil Engineering
Indian Institute of Technology Kharagpur
Kharagpur
India
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ISBN978-981-10-8778-3 ISBN978-981-10-8779-0 (eBook)
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Dedicated to my
Parents
and my wife
Mitali
Preface
In the areas of hydrology and hydroclimatology, usage of different statistical
methods is inevitable due to inherent uncertainty. Hydrology and climatology are
two areas of science that involve studies related to hydrologic and climatic
systems/subsystems, respectively. In connection with the climate change and its
impactsonwaterresourceengineering,hydrologicandhydroclimaticproblemsare
nowbeingaddressedhandinhand.Randomvariabilityofhydrologicvariableshas
alonghistorysinceitsrecognition,andseveralstatisticaltechniquesarecurrentlyin
use. Further, the correspondence between climatic variability and hydrologic
variability has produced a relatively new interdisciplinary field, known as hydro-
climatology. It provides a platform to analyze the relationship between climatic
factorsandhydrologicvariablesoverspaceandtime. Spatio-temporalevolutionof
such relationship is essential in the context of climate change. Several statistical
methodologies arecurrentlybeing developedandintroduced inthissubjectarea to
tackle new emerging challenges.
This book focuses on a wide range of statistical methods ranging from funda-
mentalconceptstoadvancedtheoriesthatarefoundtobepotentialandessentialto
deal with the real-life problems in the fields of hydrology and hydroclimatology.
Besidesotheradvancedtheories,thebookalsointroducesthetheoryofcopulasand
its applications in a chapter with many illustrative examples and MATLAB-based
small codes to deal with the problems and solutions in hydrology and
hydroclimatology.
Part of the book is intended to serve as a textbook for graduate courses on
stochastic methods in hydrology and related disciplines. The book may also be a
valuableresourceforresearchers,professionals,anddoctoratestudentsintheareas
ofhydrology,hydroclimatology,andrelatedfields.Thisbookisbroadlyorganized
as follows: Chapter 1 provides a basic introduction on the subject area and role of
statistical methods in it. Chapters 2 and 3 are introductory in nature and present a
thoroughdiscussiononthebasicconceptsofrandomexperiment,randomvariables,
and some basic exploratory statistical properties. Chapter 4 provides mathematical
and conceptual foundations of commonly used probability distributions in the
domains of hydrology and hydroclimatology. Chapter 5 deals with frequency
vii
viii Preface
analysis, risk, and uncertainty in hydroclimatic analysis. Hypothesis testing and
nonparametric tests are discussed in Chap. 6. Regression analysis and multivariate
analysis including ANOVA and wavelet analysis are covered in Chaps. 7 and 8,
respectively. Chapter 9 presents the concepts of hydroclimatic time series analysis
and forecasting including stationarity, homogeneity, periodicity. Chapter 10 por-
trays the potential of copula theory in hydrology and hydroclimatology. Copulas
help to develop the joint distribution between multiple associated hydroclimatic
variables.Itspotentialinfrequencyanalysis,multivariatemodeling,simulation,and
prediction is discussed for hydroclimatic problems.
Kharagpur, India Rajib Maity
Acknowledgements
There are many individuals who directly or indirectly contributed to this book. It
starts with many professors and academicians in India and abroad with whom I
interacted through various collaborations.
Iwouldalsoliketoacknowledgethesupportofferedbymanyresearchstudents.
Specifically, help from Ph.D. students—Mayank Suman and Subharthi Sarkar—
andMSstudentRiyaDuttaishighlyappreciatedandacknowledged.Technicalhelp
fromMayankisalsoacknowledged.SupportfromSubbaraoPichuka,ANarenand
indirect support from Manali Pal and Subhasmita Dash are also acknowledged.
Finally,itismywifeMitali,afriend,philosopher,andguide,whoisalwayswith
me in all my ups and downs. To state the least, I must mention her dedication,
sacrifice, and love without which everything is meaningless.
ix
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Definitions and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Role of Statistical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Hydrologic and Hydroclimatic Variability . . . . . . . . 2
1.2.2 Need of Statistical Methods. . . . . . . . . . . . . . . . . . . 2
1.3 Organization of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Basic Concepts of Probability and Statistics . . . . . . . . . . . . . . . . . . 7
2.1 Concepts of Random Experiments and Random Variables. . . . 7
2.1.1 Random Experiments, Sample Space, and Events. . . 7
2.1.2 Concept of Random Variables and Events . . . . . . . . 8
2.2 Basic Concepts of Probability . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 The Axioms of Probability . . . . . . . . . . . . . . . . . . . 11
2.2.2 Some Elementary Properties on Probability . . . . . . . 12
2.3 Conditional Probability Theorem . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Total Probability Theorem and Bayes’ Rule . . . . . . . . . . . . . . 18
2.5 Univariate and Bivariate Probability Distribution
of Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Discrete Random Variable. . . . . . . . . . . . . . . . . . . . 23
2.5.2 Continuous Random Variable . . . . . . . . . . . . . . . . . 26
2.6 Marginal and Conditional Probability Distribution. . . . . . . . . . 31
2.6.1 Marginal Probability Distribution. . . . . . . . . . . . . . . 32
2.6.2 Conditional Distribution Function . . . . . . . . . . . . . . 38
2.7 Independence between Random Variables. . . . . . . . . . . . . . . . 43
2.8 Functions of Random Variables . . . . . . . . . . . . . . . . . . . . . . . 44
2.8.1 Univariate Random Variable . . . . . . . . . . . . . . . . . . 44
2.8.2 Bivariate Random Variables . . . . . . . . . . . . . . . . . . 45
2.9 MATLAB Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Exercise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
xi
