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R语言模拟与数值计算.pdf
c6872_c000.pdf
INTRODUCTION TO: Scientific Programming and Simulation Using R
Preface
Course structure options
Chapter outlines
Bibliography/further reading
The R language
Scientific programming/numerical techniques
Stochastic modelling and simulation
Caveat computator
Thanks
Contents
Glossary of R commands
Programs and functions developed in the text
c6872_c001.pdf
Table of Contents
PART I: Programming
CHAPTER 1: Setting up
1.1 Installing R
1.2 Starting R
1.3 Working directory
1.4 Writing scripts
1.5 Help
1.6 Supporting material
1.6.1 Installing and loading the package when you have write privileges
1.6.2 Installing and loading the package with limited write privileges
Glossary of R commands
Programs and functions developed in the text
c6872_c002.pdf
Table of Contents
CHAPTER 2: R as a calculating environment
2.1 Arithmetic
2.2 Variables
2.3 Functions
2.4 Vectors
2.4.1 Example: mean and variance
2.4.2 Example: simple numerical integration
2.4.3 Example: exponential limit
2.5 Missing data
2.6 Expressions and assignments
2.7 Logical expressions
2.7.1 Example: rounding error
2.7.2 Sequential && and ||
2.8 Matrices
2.9 The workspace
2.10 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c003.pdf
Table of Contents
CHAPTER 3: Basic programming
3.1 Introduction
3.1.1 Example: roots of a quadratic 1 quad1.r
3.2 Branching with if
3.2.1 Example: roots of a quadratic 2 quad2.r
3.3 Looping with for
3.3.1 Example: summing a vector
3.3.2 Example: n factorial 1 nfact1.r
3.3.3 Example: pension value pension.r
3.3.4 Example: redimensioning an array
3.4 Looping with while
3.4.1 Example: Fibonacci numbers fibonacci.r
3.4.2 Example: compound interest compound.r
3.5 Vector-based programming
3.6 Program flow
3.6.1 Pseudo-code
3.7 Basic debugging
3.8 Good programming habits
3.9 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c004.pdf
Table of Contents
CHAPTER 4: I/O: Input and Output
4.1 Text
4.2 Input from a file
4.2.1 Example: file input quartiles1.r
4.3 Input from the keyboard
4.3.1 Example: roots of a quadratic 2b quad2b.r
4.4 Output to a file
4.5 Plotting
4.6 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c005.pdf
Table of Contents
CHAPTER 5: Programming with functions
5.1 Functions
5.1.1 Example: roots of a quadratic 3 quad3.r
5.1.2 Example: n choose r n_choose_r.r
5.1.3 Example: Winsorised mean wmean.r
5.1.4 Program flow using functions
5.2 Scope and its consequences
5.3 Optional arguments and default values
5.4 Vector-based programming using functions
5.4.1 Example: density of primes primedensity.r
5.5 Recursive programming
5.5.1 Example: n factorial 2 nfact2.r
5.5.2 Example: Sieve of Eratosthenes primesieve.r
5.6 Debugging functions
5.7 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c006.pdf
Table of Contents
CHAPTER 6: Sophisticated data structures
6.1 Factors
6.2 Dataframes
6.2.1 Attaching
6.3 Lists
6.3.1 Example: Australian rules football
6.4 The apply family
6.4.1 tapply
6.4.2 Applying functions to lists lapply and sapply
6.4.3 Example: tree growth
6.5 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c007.pdf
Table of Contents
CHAPTER 7: Better graphics
7.1 Introduction
7.2 Graphics parameters: par
7.3 Graphical augmentation
7.4 Mathematical typesetting
7.5 Permanence
7.6 Grouped graphs: lattice
7.7 3D-plots
7.8 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c008.pdf
Table of Contents
CHAPTER 8: Pointers to further programming techniques
8.1 Packages
8.1.1 Package management
8.1.2 Package construction
8.2 Frames and environments
8.3 Debugging again
8.4 Object-oriented programming: S3
8.4.1 Generic functions
8.4.2 Example: seed dispersal
8.5 Object-oriented programming: S4
8.6 Compiled code
8.6.1 Writing
8.6.2 Compiling
8.6.3 Attaching
8.6.4 Call
8.7 Further reading
8.8 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c009.pdf
Table of Contents
PART II: Numerical techniques
CHAPTER 9: Numerical accuracy and program efficiency
9.1 Machine representation of numbers
9.1.1 Integers
9.1.2 Real numbers
9.2 Significant digits
9.2.1 Example: sin(x) − x near 0
9.2.2 Example: range reduction
9.3 Time
9.4 Loops versus vectors
9.4.1 Example: column sums of a matrix
9.5 Memory
9.6 Caveat
9.7 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c010.pdf
Table of Contents
CHAPTER 10: Root-finding
10.1 Introduction
10.1.1 Example: loan repayments
10.2 Fixed-point iteration
10.2.1 Example: finding the root of f(x) = log(x) – exp(–x)
10.3 The Newton-Raphson method
10.4 The secant method
10.5 The bisection method
10.6 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c011.pdf
Table of Contents
CHAPTER 11: Numerical integration
11.1 Trapezoidal rule
11.2 Simpson’s rule
11.2.1 Example: Phi(z) Phi.r
11.2.2 Example: convergence of Simpson’s rule simpson_test.r
11.2.3 Achieving a set tolerance
11.3 Adaptive quadrature
11.4 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c012.pdf
Table of Contents
CHAPTER 12: Optimisation
12.1 Newton’s method for optimisation
12.2 The golden-section method
12.3 Multivariate optimisation
12.4 Steepest ascent
12.4.1 Line search
12.4.2 Example: sin(x2/–y2/4) cos(2x–exp(y))
12.5 Newton’s method in higher dimensions
12.5.1 Example: sin(x2/2–y2/4) cos(2x–exp(y))
12.5.2 On differentiation
12.6 Optimisation in R and the wider world
12.7 A curve fitting example
12.8 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c013.pdf
Table of Contents
PART III: Probability and statistics
CHAPTER 13: Probability
13.1 The probability axioms
13.1.1 Counting probability
13.2 Conditional probability
13.2.1 Example: life tables
13.2.2 Example: indigenous deaths in custody
13.3 Independence
13.3.1 Example: disjoint events
13.3.2 Example: the Chevalier de Meré
13.4 The Law of Total Probability
13.5 Bayes’ theorem
13.5.1 Example: prostate cancer screening
13.6 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c014.pdf
Table of Contents
CHAPTER 14: Random variables
14.1 Definition and distribution function
14.2 Discrete and continuous random variables
14.3 Empirical cdf’s and histograms
14.3.1 Example: Cavendish’s experiments
14.4 Expectation and finite approximations
14.4.1 Example: numerical calculation of the mean expex.r
14.4.2 Example: truncated normal
14.4.3 Infinite range
14.4.4 Example: gamma function
14.5 Transformations
14.5.1 Transforming a discrete rv
14.5.2 Example: transforming a continuous rv
14.5.3 Expectation of a transformed random variable
14.5.4 Sums of random variables
14.6 Variance and standard deviation
14.7 The Weak Law of Large Numbers
14.7.1 Sample proportion
14.7.2 Sample variance
14.8 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c015.pdf
Table of Contents
CHAPTER 15: Discrete random variables
15.1 Discrete random variables in R
15.2 Bernoulli distribution
15.3 Binomial distribution
15.3.1 Example: sampling a manufacturing line
15.4 Geometric distribution
15.4.1 Example: lighting a Barbeque
15.4.2 Example: two-up
15.5 Negative binomial distribution
15.5.1 Example: quality control
15.6 Poisson distribution
15.6.1 Example: the dreaded lurgy
15.6.2 Poisson as a binomial limit
15.7 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c016.pdf
Table of Contents
CHAPTER 16: Continuous random variables
16.1 Continuous random variables in R
16.2 Uniform distribution
16.3 Lifetime models: exponential and Weibull
16.3.1 Exponential distribution
16.3.2 Example: radioactive decay
16.3.3 Weibull distribution
16.3.4 Example: time to the next disaster
16.4 The Poisson process and the gamma distribution
16.4.1 A paradox?
16.4.2 Merging and Thinning
16.4.3 Gamma distribution
16.4.4 Example: discrete simulation of a queue
16.5 Sampling distributions: normal, X2, and t
16.5.1 Normal or Gaussian distribution
16.5.2 Example: normal percentage points
16.5.3 The sum of independent normals
16.5.4 X2 distribution
16.5.5 Student’s t distribution
16.6 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c017.pdf
Table of Contents
CHAPTER 17: Parameter Estimation
17.1 Point Estimation
17.1.1 Example: Kew rainfall
17.1.2 Example: truncated normal
17.2 The Central Limit Theorem
17.2.1 Proof of the Central Limit Theorem
17.2.2 Normal approximation to the binomial
17.2.3 Continuity correction
17.2.4 Example: insurance risk
17.2.5 Normal approximation to the Poisson
17.2.6 Normal approximation to the negative binomial and gamma
17.3 Confidence intervals
17.3.1 Confidence interval for a proportion
17.3.2 Example: accuracy of an opinion poll
17.3.3 Small sample confidence intervals
17.4 Monte-Carlo confidence intervals
17.4.1 Example: meta-analysis of opinion polls
17.5 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c018.pdf
Table of Contents
PART IV: Simulation
CHAPTER 18: Simulation
18.1 Simulating iid uniform samples
18.1.1 Congruential generators
18.1.2 Seeding
18.2 Simulating discrete random variables
18.2.1 Example: binomial
18.2.2 Sequences of independent trials
18.3 Inversion method for continuous rv
18.3.1 Example: uniform distribution
18.3.2 Example: exponential distribution
18.4 Rejection method for continuous rv
18.4.1 Example: triangular density
18.4.2 General rejection method
18.4.3 Efficiency
18.4.4 Example: gamma
18.5 Simulating normals
18.5.1 Central Limit Theorem
18.5.2 Rejection with exponential envelope
18.5.3 Box-Muller algorithm
18.6 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c019.pdf
Table of Contents
CHAPTER 19: Monte-Carlo integration
19.1 Hit-and-miss method
19.2 (Improved) Monte-Carlo integration
19.2.1 Lemma
19.2.2 Accuracy in higher dimensions
19.3 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c020.pdf
Table of Contents
CHAPTER 20: Variance reduction
20.1 Antithetic sampling
20.1.1 Example: Bu on’s needle and cross
20.1.2 General antithetic variate technique
20.1.3 Example: improved Monte-Carlo integration
20.1.4 Antithetic pairs through inversion
20.2 Importance sampling
20.2.1 Example: evaluating a simple integral three di erent ways
20.2.2 Example: standard normal tail probability
20.2.3 Example: standard normal central probability
20.3 Control variates
20.3.1 Example: standard normal central probability
20.4 Exercises
Glossary of R commands
Programs and functions developed in the text
c6872_c021.pdf
Table of Contents
CHAPTER 21: Case studies
21.1 Introduction
21.2 Epidemics
21.2.1 SIR model
21.2.2 Branching processes
21.2.3 Forest fire model
21.3 Inventory
21.3.1 Continuous Review Inventory Model
21.3.2 Simulated inventory level
21.3.3 A two-stage inventory system
21.4 Seed dispersal
21.4.1 Simulating the radial distance R
21.4.2 Object-oriented programming implementation
Glossary of R commands
Programs and functions developed in the text
c6872_c022.pdf
Table of Contents
CHAPTER 22: Student projects
22.1 The level of a dam
22.1.1 Height and volume
22.1.2 Tracking height over time
22.2 Roulette
22.2.1 Simulation
22.2.2 Verification
22.2.3 Variation
22.3 Buffon’s needle and cross
22.3.1 Theoretical analysis
22.3.2 Simulation estimates
22.3.3 Buffon’s cross
22.4 Insurance risk
22.4.1 Simulating X
22.4.2 Simulating Z
22.4.3 Profit taking
22.5 Squash
22.5.1 Status of the game
22.5.2 Simulating a game
22.5.3 Probability of winning
22.5.4 Length of a game
22.6 Stock prices
22.6.1 Simulating S
22.6.2 Estimating ES(t)
22.6.3 Down-and-out call option
Glossary of R commands
Programs and functions developed in the text
c6872_glossary.pdf
Table of Contents
Glossary of R commands
Workspace and help
Objects
Packages
Flow and control and function definition
Mathematical and logical operators and functions
Vectors
Matrices
Dataframes, factors and lists
Input and output
Plotting
Random numbers and probability distributions
Programs and functions developed in the text
c6872_Programs_functions.pdf
Table of Contents
Programs and functions developed in the text
Glossary of R commands
I n t rod u c tI on to
Scientific Programming
and Simulation using r
© 2009 by Taylor & Francis Group, LLC
I n t rod u c tI on to
Scientific Programming
and Simulation using r
Owen Jones, Robert Maillardet,
and Andrew Robinson
© 2009 by Taylor & Francis Group, LLC
Chapman & Hall/CRC
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2009 by Taylor & Francis Group, LLC
Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number-13: 978-1-4200-6872-6 (Hardcover)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher can-
not assume responsibility for the validity of all materials or the consequences of their use. The
authors and publishers have attempted to trace the copyright holders of all material reproduced
in this publication and apologize to copyright holders if permission to publish in this form has not
been obtained. If any copyright material has not been acknowledged please write and let us know so
we may rectify in any future reprint.
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transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
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Visit the Taylor & Francis Web site at
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© 2009 by Taylor & Francis Group, LLC
Preface
This book has two principal aims: to teach scientific programming and to
introduce stochastic modelling. Stochastic modelling in particular, and math-
ematical modelling in general, are intimately linked to scientific programming
because the numerical techniques of scientific programming enable the practi-
cal application of mathematical models to real-world problems. In the context
of stochastic modelling, simulation is the numerical technique that enables us
to analyse otherwise intractable models.
Simulation is also the best way we know of developing statistical intuition.
This book assumes that users have completed or are currently undertaking
a first year university level calculus course. The material is suitable for first
and second year science/engineering/commerce students and masters level stu-
dents in applied disciplines. No prior knowledge of programming or probability
is assumed.
It is possible to use the book for a first course on probability, with an empha-
sis on applications facilitated by simulation. Modern applied probability and
statistics are numerically intensive, and we give an approach that integrates
programming and probability right from the start.
We chose the programming language R because of its programming features.
We do not describe statistical techniques as implemented in R (though many
of them are admittedly quite remarkable), but rather show how to turn algo-
rithms into code. Our intended audience is those who want to make tools, not
just use them.
Complementing the book is a package, spuRs, containing most of the code
and data we use. Instructions for installing it are given in the first chapter. In
the back of the book we also provide an index of the programs developed in
the text and a glossary of R commands.
Course structure options
This book has grown out of the notes prepared for a first year course consist-
ing of 36 lectures, 12 one-hour tutorials, and 12 two-hour lab classes. However
it now contains more material than would fit in such a course, which permits
© 2009 by Taylor & Francis Group, LLC
v
vi
PREFACE
its use for a variety of course structures, for example to meet prerequisite
requirements for follow-on subjects. We found the lab classes to be particu-
larly important pedagogically, as students learn programming through their
own experimentation. Instructors may straightforwardly compile lab classes
by drawing on the numerous examples and exercises in the text, and these are
supplemented by the programming projects contained in Chapter 22, which
are based on assignments we gave our students.
Core content The following chapters contain our core material for a course
on scientific programming and simulation.
Part I: Core knowledge of R and programming concepts. Chapters 1–6.
Part II: Thinking about mathematics from a numerical point of view: applying
Part I concepts to root finding and numerical integration. Chapters 9–11.
Part III: Essentials of probability, random variables, and expectation required
to understand simulation. Chapters 13–15 plus the uniform distribution.
Part IV: Stochastic modelling and simulation: random number generation,
Monte-Carlo integration, case studies and projects. Chapters 18.1–18.2, 19,
21.1–21.2 and 22.
Additional stochastic material The core outlined above only uses discrete
random variables, and for estimation only uses the concept of a sample average
converging to a mean. Chapters 16 and 17 add continuous random variables,
the Central Limit Theorem and confidence intervals. Chapters 18.3–18.5 and
20 then look at simulating continuous random variables and variance reduc-
tion. With some familiarity of continuous random variables the remaining case
studies, Chapter 21.3–21.4, become accessible.
Note that some of the projects in Chapter 22 use continuous random variables,
but can be easily modified to use discrete random variables instead.
Additional programming and numerical material For the core material ba-
sic plotting of output is sufficient, but for those wanting to produce more
professional graphics we provide Chapter 7. Chapter 8, on further program-
ming, acts as a bridge to more specialised texts, for those who wish to pursue
programming more deeply.
Chapter 12 deals with univariate and multivariate optimisation. Sections 12.3–
12.7 on multivariate optimisation, are harder than the rest of the book, and
require a basic familiarity with vector calculus. This material is self-contained,
with the exception of Example 17.1.2, which uses the optim function. However,
if you are prepared to use optim as a black box, then this example is also quite
accessible without reading the multivariate optimisation sections.
© 2009 by Taylor & Francis Group, LLC
PREFACE
Chapter outlines
vii
1: Setting up. Here we describe how to obtain and install R, and the package
spuRs which complements the book.
2: R as a calculating environment. This chapter shows you how to use R to
do arithmetic calculations; create and manipulate variables, vectors, and ma-
trices; work with logical expressions; call and get help on built-in R functions;
and to understand the workspace.
3: Basic programming. This chapter introduces a set of basic programming
structures that are the building blocks of many programs. Some structures are
common to numerous programming languages, for example if, for and while
statements. Others, such as vector-based programming, are more specialised,
but are arguably just as important for efficient R coding.
4: Input and output. This chapter describes some of the infrastructure that R
provides for importing data for subsequent analysis, and for displaying and
saving results of that analysis. More details on the construction of graphics
are available in Chapter 7, and we provide more information about importing
data in Chapter 6.
5: Programming with functions. This chapter extends Chapter 3 to include
user-defined functions. We cover the creation of functions, the rules that they
must follow, and how they relate to the environments from which they are
called. We also present some tips on the construction of efficient functions,
with especial reference to how they are treated in R.
6: Sophisticated data structures. In this chapter we study R’s more sophisti-
cated data structures—lists and dataframes—which simplify data representa-
tion, manipulation, and analysis. The dataframe is like a matrix but extended
to allow for different data modes in different columns, and the list is a general
data storage object that can house pretty much any other kind of R object.
We also introduce the factor, which is used to represent categorical objects.
7: Better graphics. This chapter provides a deeper exposition of the graphical
capabilities of R, building on Chapter 4. We explain the individual pieces that
make up the default plot. We discuss the graphics parameters that are used
to fine-tune individual graphs and multiple graphics on a page. We show how
to save graphical objects in various formats. Finally, we demonstrate some
graphical tools for the presentation of multivariate data (lattice graphs), and
3D-graphics.
8: Further programming. This chapter briefly mentions some more advanced
aspects of programming in R. We introduce the management of and interac-
tion with packages. We present details about how R arranges the objects that
we create within the workspace, and within functions that we are running.
We provide further suggestions for debugging your own functions. Finally, we
© 2009 by Taylor & Francis Group, LLC
viii
PREFACE
present some of the infrastructure that R provides for object-oriented program-
ming, and for executing code that has been compiled from another computer
language, for example, C.
9: Numerical accuracy and program efficiency. In this chapter we consider
technical details about how computers operate, and their ramifications for
programming practice, particularly within R. We look at how computers rep-
resent numbers, and the effect that this has on the accuracy of computation
results. We also discuss the time it takes to perform a computation, and pro-
gramming techniques for speeding things up. Finally we consider the effects
of memory limitations on computation efficiency.
10: Root-finding. This chapter presents a suite of different techniques for find-
ing roots. We cover fixed-point iteration, the Newton-Raphson method, the
secant method, and the bisection method.
11: Numerical integration. This chapter introduces numerical integration. The
problem with integration is that often a closed form of the antiderivative is not
available. Under such circumstances we can try to approximate the integral
using computational methods. We cover the trapezoidal rule, Simpson’s rule,
and adaptive quadrature.
12: Optimisation. This chapter covers the problem of finding the maximum
or minimum of a possibly multivariate function. We introduce the Newton
method and the golden-section method in the context of a univariate function,
and steepest ascent/descent and Newton’s method for multivariate functions.
We then provide some further information about the optimisation tools that
are available in R.
13: Probability. In this chapter we introduce mathematical probability, which
allows us to describe and think about uncertainty in a precise fashion. We cover
the probability axioms and conditional probability. We also cover the Law of
Total Probability, which can be used to decompose complicated probabilities
into simpler ones that are easier to compute, and Bayes’ theorem, which is
used to manipulate conditional probabilities in very useful ways.
14: Random variables. In this chapter we introduce the concept of a random
variable. We define discrete and continuous random variables and consider
various ways of describing their distributions, including the distribution func-
tion, probability mass function, and probability density function. We define
expectation, variance, independence, and covariance. We also consider trans-
formations of random variables and derive the Weak Law of Large Numbers.
15: Discrete random variables. In this chapter we study some of the most
important discrete random variables, and summarise the R functions relating
to them. We cover the Bernoulli, binomial, geometric, negative binomial, and
the Poisson distribution.
16: Continuous random variables. This chapter presents the theory, applica-
tions of, and R representations of, a number of continuous random variables.
© 2009 by Taylor & Francis Group, LLC
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