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Advanced R programming-By Hadley Wickham.pdf

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    Introduction
    Who should read this book
    What you will get out of this book
    Meta-techniques
    Recommended reading
    Getting help
    Acknowledgements
    Colophon
    I Foundations
    Data structures
    Quiz
    Vectors
    Atomic vectors
    Lists
    Exercises
    Attributes
    Factors
    Exercises
    Matrices and arrays
    Data frames
    Creation
    Testing and coercion
    Combining data frames
    Special columns
    Subsetting
    Data types
    Atomic vectors
    Lists
    Matrices and arrays
    Data frames
    S3 objects
    S4 objects
    Exercises
    Subsetting operators
    Simplifying vs. preserving subsetting
    $
    Missing/out of bounds indices
    Exercises
    Subsetting and assignment
    Applications
    Lookup tables (character subsetting)
    Matching and merging by hand (integer subsetting)
    Random samples/bootstrap (integer subsetting)
    Ordering (integer subsetting)
    Expanding aggregated counts (integer subsetting)
    Removing columns from data frame (character subsetting)
    Selecting rows based on a condition (logical subsetting)
    Boolean algebra vs sets (logical & integer subsetting)
    Exercises
    Vocabulary
    The basics
    Common data structures
    Statistics
    Working with R
    I/O
    Functions
    Components of a function
    Primitive functions
    Exercises
    Lexical scoping
    Name masking
    Functions vs. variables
    A fresh start
    Dynamic lookup
    Exercises
    Every operation is a function call
    Function arguments
    Calling functions
    Calling a function given a list of arguments
    Default and missing arguments
    Lazy evaluation
    ...
    Exercises
    Special calls
    Infix functions
    Replacement functions
    Exercises
    Return values
    on.exit()
    Exercises
    OO field guide
    Introduction
    Base types
    S3
    Recognising objects, generic functions and methods
    Defining classes and creating objects
    Creating new methods and generics
    Method dispatch
    Exercises
    S4
    Recognising objects, generic functions and methods
    Defining classes and creating objects
    Creating new methods and generics
    Method dispatch
    Exercises
    RC
    Defining classes and creating objects
    Recognising objects and methods
    Method dispatch
    Exercises
    Picking a system
    Environments
    Introduction
    Environment basics
    What is an environment?
    Manipulating and inspecting environments
    Special environments
    Where
    Exercises
    Function environments
    The environment where the function was created
    The environment where the function lives
    The environment created every time a function is run
    The environment where the function was called
    Exercises
    Explicit scoping with local
    Assignment: binding names to values
    Regular binding
    <<-
    Delayed bindings
    Active bindings
    Exercises
    Debugging, condition handling and defensive programming
    Debugging techniques
    Debugging tools
    Determining the sequence of calls
    Browsing on error
    Browsing arbitrary code
    The call stack: traceback(), where and recover().
    Other types of failure
    Condition handling
    Ignore errors with try()
    Handle conditions with tryCatch()
    withCallingHandlers()
    Custom signal classes
    Exercises
    Defensive programming
    Exercises
    II Functional programming
    Functional programming
    Other languages
    Motivation
    Anonymous functions
    Exercises
    Introduction to closures
    Function factories
    Mutable state
    Exercises
    Lists of functions
    Moving lists of functions to the global environment
    Exercises
    Case study: numerical integration
    Exercises
    Functionals
    Introduction
    My first functional: lapply()
    Looping patterns
    Exercises
    For loop functionals: friends of lapply()
    Vector output: sapply and vapply
    Multiple inputs: Map (and mapply)
    Rolling computations
    Parallelisation
    Exercises
    Data structure functionals
    Matrix and array operations
    Group apply
    The plyr package
    Exercises
    Functional programming
    Reduce()
    Predicate functionals
    Exercises
    Mathematical functionals
    Exercises
    Converting loops to functionals, and when it's not possible
    Modifying in place
    Recursive relationships
    While loops
    A family of functions
    Exercises
    Function operators
    In other languages
    Behavioural FOs
    Useful behavioural FOs
    Memoisation
    Capturing function invocations
    Exercises
    Output FOs
    Minor modifications
    Changing what a function does
    Exercises
    Input FOs
    Prefilling function arguments: partial function application
    Changing input types
    Exercises
    Combining FOs
    Function composition
    Logical predicates and boolean algebra
    Exercises
    The common pattern and a subtle bug
    Exercises
    III Computing on the language
    Metaprogramming
    Capturing expressions
    Non-standard evaluation in subset
    Exercises
    Scoping issues
    Exercises
    Calling from another function
    Substitute
    Non-standard evaluation in substitute
    Capturing unevaluated …
    The downsides of non-standard evaluation
    Applications
    plyr::. and ggplot2::aes
    Plyr: summarise, mutate and arrange
    Conclusion
    Expressions
    Structure of expressions
    Constants
    Names
    Calls
    Exercises
    Parsing and deparsing
    Sourcing files from disk
    Exercises
    Capturing the current call
    Other uses of call capturing
    Exercises
    Creating a function
    Unenclose
    Exercises
    Walking the call tree with recursive functions
    Finding F and T
    Finding all variables created by assignment
    Modifying the call tree
    Exercises
    Formulas
    Formula as a quoting function
    xtabs()
    Formulas for modelling
    Special contexts
    Evaluate code in a special context
    with_something
    capture.output
    Evaluating code in a new environment
    Exercises
    Anaphoric functions
    With connection
    Special environments for run-time checking
    Logical abbreviations
    Test that
    Domain specific languages
    HTML
    Goal
    Escaping
    Basic tag functions
    Tag functions
    Processing all tags
    Exercises
    LaTeX
    LaTeX mathematics
    Goal
    to_math
    Known symbols
    Unknown symbols
    Known functions
    Unknown functions
    Exercises
    IV Performance
    Performance
    Micro-benchmarking
    Method dispatch
    Extracting variables out of a data frame
    Vectorised operations on a data frame
    Brainstorming
    Caching
    Memoisation
    Byte code compilation
    Other people's code
    Important vectorised functions
    Rewrite in a lower-level language
    Profiling and benchmarking
    Performance profiling
    Timing
    Performance testing
    Memory
    object.size()
    Exercises
    Total memory use
    Garbarge collection
    Memory profiling with lineprof
    Exercises
    Modification in place
    Loops
    Exercises
    High performance functions with Rcpp
    Getting started
    Getting started with C++
    No inputs, scalar output
    Scalar input, scalar output
    Vector input, scalar output
    Vector input, vector output
    Matrix input, vector output
    Using sourceCpp
    Exercises
    Rcpp classes and methods
    Rcpp sugar
    Arithmetic and logical operators
    Logical summary functions
    Vector views
    Other useful functions
    Missing values
    Scalars
    Strings
    Boolean
    Vectors
    Exercises
    The STL
    Using iterators
    Algorithms
    Data structures
    Vectors
    Sets
    Map
    Exercises
    Case studies
    Gibbs sampler
    R vectorisation vs. C++ vectorisation
    Using Rcpp in a Package
    Adding Rcpp to an existing package (Rcpp <= 0.10.6)
    Adding Rcpp to an existing package (Rcpp >= 0.10.7)
    More details
    Learning more
    More Rcpp
    More C++
    Acknowledgements
    R's C interface
    Introduction
    Differences between R and C
    Calling C functions from R
    Basic data structures
    Character vectors
    Coercion and object creation
    Object creation and garbage collection
    Allocation shortcuts
    Extracting C vectors
    Modifying strings
    Coercing scalars
    Modifying objects
    Pairlists and symbols
    Missing and non-finite values
    Checking types in C
    Finding the C source code for a function
    .External
    Using C code in a package
    V Packages
    Package development philosophy
    Getting started
    Introduction to devtools
    Package basics
    Package essentials
    Optional components
    Getting started
    The R/ directory
    A minimal DESCRIPTION file
    Other DESCRIPTION components
    Source, binary and bundled packages
    Package libraries
    Installing packages
    Exercises
    The package development cycle
    Dev mode
    Key functions
    Development cycles
    Confirmatory programming
    Exploratory programming
    Package level documentation
    README
    plyr
    stringr
    NEWS
    CITATION
    Vignettes
    Demos
    Documenting functions
    Help
    Roxygen process
    Common documentation tags
    Documenting functions (and methods)
    Documenting S3 generic functions and methods
    Documenting S4 generic and methods
    Documenting R5 methods
    Documenting classes
    Documenting a S3 class
    Documenting an S4 class
    Documenting an R5 class
    Documenting datasets
    Documenting packages
    Text formatting
    Lists
    Tables
    Mathematics
    Character formatting
    Dynamic help
    Testing
    Motivation
    testthat test structure
    Expectations
    Tests
    Contexts
    Running tests
    Testing files and directories
    Autotest
    R CMD check
    Style guide
    Notation and naming
    File names
    Identifiers
    Syntax
    Spacing
    Curly braces
    Line length
    Indentation
    Assignment
    Organisation
    Commenting guidelines
    Namespaces
    How do namespaces work
    Exporting
    Importing
    Compiled code
    How do they work
    Git and github
    Git
    The basics
    Github
    Releasing a package
    Checking
    Version numbers
    Publishing on CRAN
    Publicising
    Advanced R programming Author: Hadley Wickham
    Contents 1 Introduction Who should read this book . . . . . . . . . . . . . . . . . . . . . . . . What you will get out of this book . . . . . . . . . . . . . . . . . . . . Meta-techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recommended reading . . . . . . . . . . . . . . . . . . . . . . . . . . . Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Colophon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Foundations 2 Data structures Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vectors Atomic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrices and arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 17 19 19 20 21 21 22 22 24 25 26 26 27 28 30 31 32 34 34 37
    CONTENTS Creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing and coercion . . . . . . . . . . . . . . . . . . . . . . . . . Combining data frames . . . . . . . . . . . . . . . . . . . . . . . Special columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Subsetting Data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atomic vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrices and arrays . . . . . . . . . . . . . . . . . . . . . . . . . Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3 objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subsetting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplifying vs. preserving subsetting . . . . . . . . . . . . . . . . $ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Missing/out of bounds indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises Subsetting and assignment . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lookup tables (character subsetting) . . . . . . . . . . . . . . . . Matching and merging by hand (integer subsetting) . . . . . . . Random samples/bootstrap (integer subsetting) . . . . . . . . . . Ordering (integer subsetting) . . . . . . . . . . . . . . . . . . . . Expanding aggregated counts (integer subsetting) . . . . . . . . . Removing columns from data frame (character subsetting) . . . . Selecting rows based on a condition (logical subsetting) . . . . . Boolean algebra vs sets (logical & integer subsetting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 2 37 37 38 39 41 41 42 44 44 45 46 46 46 47 48 49 50 51 51 52 52 53 54 55 56 56 57 59 60
    CONTENTS 4 Vocabulary The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common data structures . . . . . . . . . . . . . . . . . . . . . . . . . . Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working with R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Functions Components of a function . . . . . . . . . . . . . . . . . . . . . . . . . Primitive functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises Lexical scoping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Name masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions vs. variables . . . . . . . . . . . . . . . . . . . . . . . . A fresh start . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic lookup . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Every operation is a function call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function arguments Calling functions . . . . . . . . . . . . . . . . . . . . . . . . . . . Calling a function given a list of arguments . . . . . . . . . . . . Default and missing arguments . . . . . . . . . . . . . . . . . . . Lazy evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises Special calls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Infix functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Replacement functions . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Return values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . on.exit() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 61 61 63 64 64 65 67 68 68 69 69 70 72 73 73 75 75 77 77 79 79 80 83 84 84 85 86 88 88 91 91
    CONTENTS 4 6 OO field guide Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Base types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recognising objects, generic functions and methods . . . . . . . . Defining classes and creating objects . . . . . . . . . . . . . . . . Creating new methods and generics . . . . . . . . . . . . . . . . . Method dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises 92 92 93 94 94 96 98 99 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Recognising objects, generic functions and methods . . . . . . . . 103 Defining classes and creating objects . . . . . . . . . . . . . . . . 104 Creating new methods and generics . . . . . . . . . . . . . . . . . 106 Method dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 RC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Defining classes and creating objects . . . . . . . . . . . . . . . . 107 Recognising objects and methods . . . . . . . . . . . . . . . . . . 110 Method dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Picking a system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7 Environments 112 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Environment basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 What is an environment? . . . . . . . . . . . . . . . . . . . . . . 112 Manipulating and inspecting environments . . . . . . . . . . . . . 113 Special environments . . . . . . . . . . . . . . . . . . . . . . . . . 116 Where . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Exercises Function environments . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
    CONTENTS 5 The environment where the function was created . . . . . . . . . 120 The environment where the function lives . . . . . . . . . . . . . 121 The environment created every time a function is run . . . . . . 122 The environment where the function was called . . . . . . . . . . 123 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Exercises Explicit scoping with local . . . . . . . . . . . . . . . . . . . . . . . . 126 Assignment: binding names to values . . . . . . . . . . . . . . . . . . . 127 Regular binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 <<- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Delayed bindings . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Active bindings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8 Debugging, condition handling and defensive programming Debugging techniques Debugging tools 134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Determining the sequence of calls . . . . . . . . . . . . . . . . . . 138 Browsing on error . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Browsing arbitrary code . . . . . . . . . . . . . . . . . . . . . . . 141 The call stack: traceback(), where and recover(). . . . . . . . 142 Other types of failure . . . . . . . . . . . . . . . . . . . . . . . . 142 Condition handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Ignore errors with try() . . . . . . . . . . . . . . . . . . . . . . . 144 Handle conditions with tryCatch() . . . . . . . . . . . . . . . . 146 withCallingHandlers() . . . . . . . . . . . . . . . . . . . . . . 148 Custom signal classes . . . . . . . . . . . . . . . . . . . . . . . . . 149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Exercises Defensive programming . . . . . . . . . . . . . . . . . . . . . . . . . . 151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Exercises
    CONTENTS II Functional programming 6 154 9 Functional programming Other languages 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Anonymous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Introduction to closures . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Function factories . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Mutable state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Exercises Lists of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Moving lists of functions to the global environment . . . . . . . . 169 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Case study: numerical integration . . . . . . . . . . . . . . . . . . . . 170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Exercises Exercises 10 Functionals 175 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 My first functional: lapply() . . . . . . . . . . . . . . . . . . . . . . . 176 Looping patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Exercises For loop functionals: friends of lapply() . . . . . . . . . . . . . . . . 181 Vector output: sapply and vapply . . . . . . . . . . . . . . . . . 182 Multiple inputs: Map (and mapply) . . . . . . . . . . . . . . . . . 184 Rolling computations . . . . . . . . . . . . . . . . . . . . . . . . . 186 Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Exercises Data structure functionals . . . . . . . . . . . . . . . . . . . . . . . . . 189 Matrix and array operations . . . . . . . . . . . . . . . . . . . . . 189 Group apply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 The plyr package . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
    CONTENTS 7 Exercises Exercises Mathematical functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Functional programming . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Reduce() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Predicate functionals . . . . . . . . . . . . . . . . . . . . . . . . . 195 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 . . . . . . 199 Modifying in place . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Recursive relationships . . . . . . . . . . . . . . . . . . . . . . . . 200 While loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A family of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Converting loops to functionals, and when it’s not possible Exercises 11 Function operators In other languages 208 . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Behavioural FOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 . . . . . . . . . . . . . . . . . . . . . . . 211 Useful behavioural FOs Memoisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Capturing function invocations . . . . . . . . . . . . . . . . . . . 215 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Output FOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Minor modifications . . . . . . . . . . . . . . . . . . . . . . . . . 218 Changing what a function does . . . . . . . . . . . . . . . . . . . 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Exercises Input FOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Prefilling function arguments: partial function application . . . . 221 Changing input types . . . . . . . . . . . . . . . . . . . . . . . . 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Exercises Combining FOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Function composition . . . . . . . . . . . . . . . . . . . . . . . . 226
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