2. Introductory Examples #

Contents

Introductory Examples

2.1. Overview #

We’re now ready to start learning the Julia language itself.

2.1.1. Level#

Our approach is aimed at those who already have at least some knowledge of programming — perhaps experience with Python, MATLAB, Fortran, C or similar.

In particular, we assume you have some familiarity with fundamental programming concepts such as

variables
arrays or vectors
loops
conditionals (if/else)

2.1.2. (Optional) Resources for Introductory Programming#

If Julia is your first programming language, then you may want to examine other resources before proceeding through these chapters.

First, you will want to ensure that you have installed Conda, Julia in the previous section, and are able to launch Jupyter Lab. Then, even without downloading these lecture notes you can start learning Julia.

Some resources for users with little to no programming experience are the:

Introduction to Julia which is appropriate for those with little programming experience.
- You will need to sign up for this course to access it, but it is free.
- Avoid the initial lecture on the installation of Julia and nteract.
- The Jupyter Notebooks for this class are available online
Julia for Beginners from the julia for talented amateurs channel.
- If you watch these, ignore the environment. It is using a different open-source editor (i.e. Atom, which is being slowly phased out) and the REPL. However the content itself is independent of the environment. Skip any setup instructions and just use Jupyter.
Julia Programming for Nervous Beginners youtube tutorials.
- These get advanced fairly quickly, but have plenty of great content.
- This may be less gentle of an introduction for new users given the tooling choices.

However, if you have even introductory experience with any programming language (e.g. you do not need to review what a loop, array, variable, or conditional “if” statement is), you should be able to start directly with these lectures.

2.1.3. Approach#

In this lecture we will write and then pick apart small Julia programs.

At this stage the objective is to introduce you to basic syntax and data structures.

Deeper concepts—how things work—will be covered in later lectures.

Since we are looking for simplicity the examples are a little contrived

In this lecture, we will often start with a direct MATLAB/FORTRAN approach which often is poor coding style in Julia, but then move towards more elegant code which is tightly connected to the mathematics.

We assume that you’ve worked your way through our getting started lecture already.

The definitive reference is Julia’s own documentation. The manual is thoughtfully written but is also quite dense.

The presentation in this and our remaining lectures is more of a tutorial style based around examples.

2.2. Example: Plotting a White Noise Process #

To begin, let’s suppose that we want to simulate and plot the white noise process $\epsilon_0, \epsilon_1, \ldots, \epsilon_T$, where each draw $\epsilon_t$ is independent standard normal.

2.2.1. Introduction to Packages#

Assuming that you followed our getting started lecture you will have installed all packages required for your notebooks to run.

More generally, there are two ways to install packages and versions (where the first method is discouraged, since Julia provides a state-of-the-art setup for reproducible environments)

add the packages directly into your global installation (e.g. Pkg.add("MyPackage") or ] add MyPackage) without any project activated.
- We installed the IJulia package in this way, since it must be accessible in every project.
- As a reminder, the ] enters the package mode, and with some terminals you can directly copy in the whole ] add MyPackage string.
using a directory with a Project.toml and Manifest.toml files, which provides an isolated set of packages for a particular project.
- If you load a jupyter notebook, it will automatically look up the tree for the project files to activate, but will not automatically install them. This can be done with ] instantiate, as we did in our installation
- The project files can be in parent folders relative to the notebooks and sourcecode.

The project provides the environment for running code is reproducible, so that anyone can replicate the precise set of package and versions used in construction.

After the installation and activation, using provides a way to say that a particular code or notebook will use the package.

using LinearAlgebra, Statistics, Plots, LaTeXStrings

Warning

If these packages were not in your project file or global package environment, you will see an error requiring installation. To reinforce the point above: Ideally you should be using a local Project.toml with these in them, in which case you might simply need to ] instantiate it. Without a project file, you can manually add them as required (e.g. ] add Plots). See here for more on instantiation and activation of projects.

2.2.2. Using Functions from a Package#

Some functions are built into the base Julia, such as randn, which returns a single draw from a normal distribution with mean 0 and variance 1 if given no parameters.

randn()

0.38463987865787813

Other functions require importing all of the names from an external library

n = 100
ep = randn(n)
plot(1:n, ep)

Let’s break this down and see how it works.

The effect of the statement using Plots is to make all the names exported by the Plots module available.

If a project file was activated (i.e., running jupyter with the project files local to the notebook), it will use whatever version of Plots.jl that was specified in the Project.toml and Manifest.toml files.

The other packages LinearAlgebra and Statistics are base Julia libraries, but require an explicit using.

The arguments to plot are the numbers 1,2, ..., n for the x-axis, a vector ep for the y-axis, and (optional) settings.

The function randn(n) returns a column vector n random draws from a normal distribution with mean 0 and variance 1.

Note

As a language intended for mathematical and scientific computing, Julia has strong support for using unicode characters.

Rather than using a symbol such as ep we could have used ϵ and many other symbols can be typed in most Julia editor by providing the LaTeX and <TAB>, i.e. \epsilon<TAB>.

2.2.3. Arrays#

The return type is one of the most fundamental Julia data types: an array

typeof(ep)

Vector{Float64} (alias for Array{Float64, 1})

ep[1:5]

5-element Vector{Float64}:
  1.062940485763283
  1.5459741657413155
  2.0899773851503958
 -0.8163645022016492
 -0.15482053344207383

The information from typeof() tells us that ep is an array of 64 bit floating point values, of dimension 1.

In Julia, one-dimensional arrays are interpreted as column vectors for purposes of linear algebra.

The ep[1:5] returns an array of the first 5 elements of ep.

Notice from the above that

array indices start at 1 (like MATLAB and Fortran, but unlike Python and C)
array elements are referenced using square brackets (unlike MATLAB and Fortran)

To get help and examples in Jupyter or other julia editors, use the ? before a function name or syntax.

?typeof

search: typeof typejoin TypeError

Get the concrete type of x.

Examples

julia> a = 1//2;

julia> typeof(a)
Rational{Int64}

julia> M = [1 2; 3.5 4];

julia> typeof(M)
Array{Float64,2}

2.2.4. For Loops#

Although there’s no need in terms of what we wanted to achieve with our program, for the sake of learning syntax let’s rewrite our program to use a for loop for generating the data.

Note

The rules for variables accessed in for and while loops can be sensitive to how they are used (and variables can sometimes require a global as part of the declaration). We strongly advise you to avoid top level (i.e. outside of functions) for and while loops when working with .jl. This scoping issue does not apply when used within functions, and will not be seen in the REPL or Jupyter notebooks.

Starting with the most direct version, and pretending we are in a world where randn can only return a single value

# poor style
n = 100
ep = zeros(n)
for i in 1:n
    ep[i] = randn()
end

Here we first declared ep to be a vector of n numbers, initialized by the floating point 0.0.

The for loop then populates this array by successive calls to randn().

Like all code blocks in Julia, the end of the for loop code block (which is just one line here) is indicated by the keyword end.

The word in from the for loop can be replaced by either ∈ or =.

The index variable is looped over for all integers from 1:n – but this does not actually create a vector of those indices.

Instead, it creates an iterator that is looped over – in this case the range of integers from 1 to n.

While this example successfully fills in ep with the correct values, it is very indirect as the connection between the index i and the ep vector is unclear.

To fix this, use eachindex

# better style
n = 100
ep = zeros(n)
for i in eachindex(ep)
    ep[i] = randn()
end

Here, eachindex(ep) returns an iterator of indices which can be used to access ep.

While iterators are memory efficient because the elements are generated on the fly rather than stored in memory, the main benefit is (1) it can lead to code which is clearer and less prone to typos; and (2) it allows the compiler flexibility to creatively generate fast code.

In Julia you can also loop directly over arrays themselves, like so

ep_sum = 0.0 # careful to use 0.0 here, instead of 0
m = 5
for ep_val in ep[1:m]
    ep_sum = ep_sum + ep_val
end
ep_mean = ep_sum / m

0.4428692054182136

where ep[1:m] returns the elements of the vector at indices 1 to m.

Of course, in Julia there are built in functions to perform this calculation which we can compare against

ep_mean ≈ mean(ep[1:m])
isapprox(ep_mean, mean(ep[1:m])) # equivalent
ep_mean ≈ sum(ep[1:m]) / m

true

In these examples, note the use of ≈ or isapprox to test equality, rather than ==, which is appropriate for integers and other types.

Approximately equal, typed with \approx<TAB>, is the appropriate way to compare any floating point numbers due to the standard issues of floating point math. Note that x ≈ y is the same as isapprox(x, y).

2.2.5. User-Defined Functions#

For the sake of the exercise, let’s go back to the for loop but restructure our program so that generation of random variables takes place within a user-defined function.

To make things more interesting, instead of directly plotting the draws from the distribution, let’s plot the squares of these draws

# poor style
function generatedata(n)
    ep = zeros(n)
    for i in eachindex(ep)
        ep[i] = (randn())^2 # squaring the result
    end
    return ep
end

data = generatedata(10)
plot(data)

Here

function is a Julia keyword that indicates the start of a function definition
generatedata is an arbitrary name for the function
return is a keyword indicating the return value, as is often unnecessary

Let us make this example slightly better by “remembering” that randn can return a vectors.

# still poor style
function generatedata(n)
    ep = randn(n) # use built in function

    for i in eachindex(ep)
        ep[i] = ep[i]^2 # squaring the result
    end

    return ep
end
data = generatedata(5)

5-element Vector{Float64}:
17598050406449844
14983718640304147
047858013018347335
3563774286259527
2844817802107032

While better, the looping over the i index to square the results is difficult to read.

Instead of looping, we can broadcast the ^2 square function over a vector using a ..

To be clear, unlike Python, R, and MATLAB (to a lesser extent), the reason to drop the for is not for performance reasons, but rather because of code clarity.

Loops of this sort are at least as efficient as vectorized approach in compiled languages like Julia, so use a for loop if you think it makes the code more clear.

# better style
function generatedata(n)
    ep = randn(n) # use built in function
    return ep .^ 2
end
data = generatedata(5)

5-element Vector{Float64}:
47847463731407613
4174253078170076
2794720312587563
0642693382903423
052157568997771735

We can even drop the function if we define it on a single line.

# good style
generatedata(n) = randn(n) .^ 2
data = generatedata(5)

5-element Vector{Float64}:
6542681214118663
21773043138676815
6479317478057668
0043729154016878
006384047362589526

Finally, we can broadcast any function, where squaring is only a special case.

# good style
f(x) = x^2 # simple square function
generatedata(n) = f.(randn(n)) # broadcasts on f
data = generatedata(5)

5-element Vector{Float64}:
3680925176726574
32590578698563594
024354667030861324
606168954505449
23208767458974666

As a final – abstract – approach, we can make the generatedata function able to generically apply to a function.

generatedata(n, gen) = gen.(randn(n)) # broadcasts on gen
f(x) = x^2 # simple square function
data = generatedata(5, f) # applies f

5-element Vector{Float64}:
09522098921534289
026050351835772194
16988569413013166
9110569100416455
11740226501122988

Whether this example is better or worse than the previous version depends on how it is used.

High degrees of abstraction and generality, e.g. passing in a function f in this case, can make code either clearer or more confusing, but Julia enables you to use these techniques with no performance overhead.

For this particular case, the clearest and most general solution is probably the simplest.

# direct solution with broadcasting, and small user-defined function
n = 100
f(x) = x^2

x = randn(n)
plot(f.(x), label = L"x^2")
plot!(x, label = L"x") # layer on the same plot

While broadcasting above superficially looks like vectorizing functions in MATLAB, or Python ufuncs, it is much richer and built on core foundations of the language.

The other additional function plot! adds a graph to the existing plot.

This follows a general convention in Julia, where a function that modifies the arguments or a global state has a ! at the end of its name.

The L in front of the labels is using the LaTeXStrings package, which will try to interpret the text as latex for display.

2.2.5.1. A Slightly More Useful Function#

Let’s make a slightly more useful function.

This function will be passed in a choice of probability distribution and respond by plotting a histogram of observations.

In doing so we’ll make use of the Distributions package, which we assume was instantiated above with the project.

Here’s the code

using Distributions

function plothistogram(distribution, n)
    ep = rand(distribution, n)  # n draws from distribution
    histogram(ep)
end

lp = Laplace()
plothistogram(lp, 500)

Let’s have a casual discussion of how all this works while leaving technical details for later in the lectures.

First, lp = Laplace() creates an instance of a data type defined in the Distributions module that represents the Laplace distribution.

The name lp is bound to this value.

When we make the function call plothistogram(lp, 500) the code in the body of the function plothistogram is run with

the name distribution bound to the same value as lp
the name n bound to the integer 500

2.2.5.2. A Mystery#

Now consider the function call rand(distribution, n).

This looks like something of a mystery.

The function rand() is defined in the base library such that rand(n) returns n uniform random variables on $[0, 1)$.

rand(3)

3-element Vector{Float64}:
42508901007930777
09483617173470305
39909036221500804

On the other hand, distribution points to a data type representing the Laplace distribution that has been defined in a third party package.

So how can it be that rand() is able to take this kind of value as an argument and return the output that we want?

The answer in a nutshell is multiple dispatch, which Julia uses to implement generic programming.

This refers to the idea that functions in Julia can have different behavior depending on the particular arguments that they’re passed.

Hence in Julia we can take an existing function and give it a new behavior by defining how it acts on a new type of value.

The compiler knows which function definition to apply to in a given setting by looking at the types of the values the function is called on.

In Julia these alternative versions of a function are called methods.

2.3. Example: Variations on Fixed Points #

Take a mapping $f : X \to X$ for some set $X$.

If there exists an $x^* \in X$ such that $f(x^*) = x^*$, then $x^*$: is called a “fixed point” of $f$.

For our second example, we will start with a simple example of determining fixed points of a function.

The goal is to start with code in a MATLAB style, and move towards a more Julian style with high mathematical clarity.

2.3.1. Fixed Point Maps#

Consider the simple equation, where the scalars $p,\beta$ are given, and $v$ is the scalar we wish to solve for

\[ v = p + \beta v \]

Of course, in this simple example, with parameter restrictions this can be solved as $v = p/(1 - \beta)$.

Rearrange the equation in terms of a map $f(x) : \mathbb R \to \mathbb R$

(2.1)#\[v = f(v)\]

where

\[ f(v) := p + \beta v \]

Therefore, a fixed point $v^*$ of $f(\cdot)$ is a solution to the above problem.

2.3.2. While Loops#

One approach to finding a fixed point of (2.1) is to start with an initial value, and iterate the map

(2.2)#\[v^{n+1} = f(v^n)\]

For this exact f function, we can see the convergence to $v = p/(1-\beta)$ when $|\beta| < 1$ by iterating backwards and taking $n\to\infty$

\[ v^{n+1} = p + \beta v^n = p + \beta p + \beta^2 v^{n-1} = p \sum_{i=0}^{n-1} \beta^i + \beta^n v_0 \]

To implement the iteration in (2.2), we start by solving this problem with a while loop.

The syntax for the while loop contains no surprises, and looks nearly identical to a MATLAB implementation.

# poor style
p = 1.0 # note 1.0 rather than 1
beta = 0.9
maxiter = 1000
tolerance = 1.0E-7
v_iv = 0.8 # initial condition

# setup the algorithm
v_old = v_iv
normdiff = Inf
iter = 1
while normdiff > tolerance && iter <= maxiter
    v_new = p + beta * v_old # the f(v) map
    normdiff = norm(v_new - v_old)

    # replace and continue
    v_old = v_new
    iter = iter + 1
end
println("Fixed point = $v_old
  |f(x) - x| = $normdiff in $iter iterations")

Fixed point = 9.999999173706609
  |f(x) - x| = 9.181037796679448e-8 in 155 iterations

The while loop, like the for loop should only be used directly in Jupyter or the inside of a function.

Here, we have used the norm function (from the LinearAlgebra base library) to compare the values.

The other new function is the println with the string interpolation, which splices the value of an expression or variable prefixed by $ into a string.

An alternative approach is to use a for loop, and check for convergence in each iteration.

# setup the algorithm
v_old = v_iv
normdiff = Inf
iter = 1
for i in 1:maxiter
    v_new = p + beta * v_old # the f(v) map
    normdiff = norm(v_new - v_old)
    if normdiff < tolerance # check convergence
        iter = i
        break # converged, exit loop
    end
    # replace and continue
    v_old = v_new
end
println("Fixed point = $v_old
  |f(x) - x| = $normdiff in $iter iterations")

Fixed point = 9.999999081896231
  |f(x) - x| = 9.181037796679448e-8 in 154 iterations

The new feature there is break , which leaves a for or while loop.

2.3.3. Using a Function#

The first problem with this setup is that it depends on being sequentially run – which can be easily remedied with a function.

# better, but still poor style
function v_fp(beta, ρ, v_iv, tolerance, maxiter)
    # setup the algorithm
    v_old = v_iv
    normdiff = Inf
    iter = 1
    while normdiff > tolerance && iter <= maxiter
        v_new = p + beta * v_old # the f(v) map
        normdiff = norm(v_new - v_old)

        # replace and continue
        v_old = v_new
        iter = iter + 1
    end
    return (v_old, normdiff, iter) # returns a tuple
end

# some values
p = 1.0 # note 1.0 rather than 1
beta = 0.9
maxiter = 1000
tolerance = 1.0E-7
v_initial = 0.8 # initial condition

v_star, normdiff, iter = v_fp(beta, p, v_initial, tolerance, maxiter)
println("Fixed point = $v_star
  |f(x) - x| = $normdiff in $iter iterations")

Fixed point = 9.999999173706609
  |f(x) - x| = 9.181037796679448e-8 in 155 iterations

While better, there could still be improvements.

2.3.4. Passing a Function#

The chief issue is that the algorithm (finding a fixed point) is reusable and generic, while the function we calculate p + beta * v is specific to our problem.

A key feature of languages like Julia, is the ability to efficiently handle functions passed to other functions.

# better style
function fixedpointmap(f, iv, tolerance, maxiter)
    # setup the algorithm
    x_old = iv
    normdiff = Inf
    iter = 1
    while normdiff > tolerance && iter <= maxiter
        x_new = f(x_old) # use the passed in map
        normdiff = norm(x_new - x_old)
        x_old = x_new
        iter = iter + 1
    end
    return (x_old, normdiff, iter)
end

# define a map and parameters
p = 1.0
beta = 0.9
f(v) = p + beta * v # note that p and beta are used in the function!

maxiter = 1000
tolerance = 1.0E-7
v_initial = 0.8 # initial condition

v_star, normdiff, iter = fixedpointmap(f, v_initial, tolerance, maxiter)
println("Fixed point = $v_star
  |f(x) - x| = $normdiff in $iter iterations")

Fixed point = 9.999999173706609
  |f(x) - x| = 9.181037796679448e-8 in 155 iterations

Much closer, but there are still hidden bugs if the user orders the settings or returns types wrong.

2.3.5. Named Arguments and Return Values#

To enable this, Julia has two features: named function parameters, and named tuples

# good style
function fixedpointmap(f, iv; tolerance = 1E-7, maxiter = 1000)
    # setup the algorithm
    x_old = iv
    normdiff = Inf
    iter = 1
    while normdiff > tolerance && iter <= maxiter
        x_new = f(x_old) # use the passed in map
        normdiff = norm(x_new - x_old)
        x_old = x_new
        iter = iter + 1
    end
    return (; value = x_old, normdiff, iter) # A named tuple
end

# define a map and parameters
p = 1.0
beta = 0.9
f(v) = p + beta * v # note that p and beta are used in the function!

sol = fixedpointmap(f, 0.8; tolerance = 1.0E-8) # don't need to pass
println("Fixed point = $(sol.value)
  |f(x) - x| = $(sol.normdiff) in $(sol.iter) iterations")

Fixed point = 9.999999918629035
  |f(x) - x| = 9.041219328764782e-9 in 177 iterations

In this example, all function parameters after the ; in the list, must be called by name.

Furthermore, a default value may be enabled – so the named parameter iv is required while tolerance and maxiter have default values.

The return type of the function also has named fields, value, normdiff, and iter – all accessed intuitively using ..

Finally, this shows how named tuples and keyword arguments are constructed from local variable names. In particular, (;value = x_old, normdiff, iter) is identical to (value = x_old, normdiff = normdiff, iter = iter) where anything after the ; is assumed to be a keyword with the name identical to the local variable.

The named tuple notation is also used for unpacking values. In particular we could have written the execution of this with

(; value, normdiff, iter) = fixedpointmap(f, 0.8; tolerance = 1.0E-8)
println("Fixed point = $value
  |f(x) - x| = $normdiff in $iter iterations")

Fixed point = 9.999999918629035
  |f(x) - x| = 9.041219328764782e-9 in 177 iterations

That is, (; value, normdiff, iter) = expression is the same as exp = expression(); value = exp.value, normdiff = exp.normdiff, iter = exp.iter.

To show the flexibility of this code, we can use it to find a fixed point of the non-linear logistic equation, $x = f(x)$ where $f(x) := r x (1-x)$.

r = 2.0
f(x) = r * x * (1 - x)

sol = fixedpointmap(f, 0.8) # the ; is optional but generally good style
println("Fixed point = $(sol.value)
  |f(x) - x| = $(sol.normdiff) in $(sol.iter) iterations")

Fixed point = 0.4999999999999968
  |f(x) - x| = 3.979330237546819e-8 in 7 iterations

2.3.6. Using a Package#

But best of all is to avoid writing code altogether.

# best style
using NLsolve

p = 1.0
beta = 0.9
f(v) = p .+ beta * v # broadcast the +
sol = fixedpoint(f, [0.8]; m = 0)
normdiff = norm(f(sol.zero) - sol.zero)
println("Fixed point = $(sol.zero)
  |f(x) - x| = $normdiff in $(sol.iterations) iterations")

Fixed point = [9.999999909587816]
  |f(x) - x| = 9.041219328764782e-9 in 176 iterations

The fixedpoint function from the NLsolve.jl library implements the simple fixed point iteration scheme above.

Since the NLsolve library only accepts vector based inputs, we needed to make the f(v) function broadcast on the + sign, and pass in the initial condition as a vector of length 1 with [0.8].

While a key benefit of using a package is that the code is clearer, and the implementation is tested, by using an orthogonal library we also enable performance improvements.

# best style
p = 1.0
beta = 0.9
iv = [0.8]
sol = fixedpoint(v -> p .+ beta * v, iv)
fnorm = norm(f(sol.zero) - sol.zero)
println("Fixed point = $(sol.zero)
  |f(x) - x| = $fnorm  in $(sol.iterations) iterations
  converged = $(sol.f_converged)")

Fixed point = [9.999999999999972]
  |f(x) - x| = 3.552713678800501e-15  in 3 iterations
  converged = true

Note that this completes in 3 iterations vs 176 for the naive fixed point iteration algorithm.

Since Anderson iteration is doing more calculations in an iteration, whether it is faster or not would depend on the complexity of the f function.

But this demonstrates the value of keeping the math separate from the algorithm, since by decoupling the mathematical definition of the fixed point from the implementation in (2.2), we were able to exploit new algorithms for finding a fixed point.

The only other change in this function is the move from directly defining f(v) and using an anonymous function.

Similar to anonymous functions in MATLAB, and lambda functions in Python, Julia enables the creation of small functions without any names.

The code v -> p .+ beta * v defines a function of a dummy argument, v with the same body as our f(x).

2.3.7. Composing Packages#

A key benefit of using Julia is that you can compose various packages, types, and techniques, without making changes to your underlying source.

As an example, consider if we want to solve the model with a higher-precision, as floating points cannot be distinguished beyond the machine epsilon for that type (recall that computers approximate real numbers to the nearest binary of a given precision; the machine epsilon is the smallest nonzero magnitude).

In Julia, this number can be calculated as

eps()

2.220446049250313e-16

For many cases, this is sufficient precision – but consider that in iterative algorithms applied millions of times, those small differences can add up.

The only change we will need to our model in order to use a different floating point type is to call the function with an arbitrary precision floating point, BigFloat, for the initial value.

# use arbitrary precision floating points
p = 1.0
beta = 0.9
iv = [BigFloat(0.8)] # higher precision

# otherwise identical
sol = fixedpoint(v -> p .+ beta * v, iv)
normdiff = norm(f(sol.zero) - sol.zero)
println("Fixed point = $(sol.zero)
  |f(x) - x| = $normdiff in $(sol.iterations) iterations")

Fixed point = BigFloat[10.00000000000000222044604925031357388532909931412848377287867809936811555686155]
  |f(x) - x| = 0.0 in 3 iterations

Here, the literal BigFloat(0.8) takes the number 0.8 and changes it to an arbitrary precision number.

The result is that the residual is now exactly 0.0 since it is able to use arbitrary precision in the calculations, and the solution has a finite-precision solution with those parameters.

2.3.8. Multivariate Fixed Point Maps#

The above example can be extended to multivariate maps without any modifications to the fixed point iteration code.

Using our own, homegrown iteration and simply passing in a bivariate map:

p = [1.0, 2.0]
beta = 0.9
iv = [0.8, 2.0]
f(v) = p .+ beta * v # note that p and beta are used in the function!

sol = fixedpointmap(f, iv; tolerance = 1.0E-8)
println("Fixed point = $(sol.value)
  |f(x) - x| = $(sol.normdiff) in $(sol.iter) iterations")

Fixed point = [9.999999961080519, 19.999999923853192]
  |f(x) - x| = 9.501826248250528e-9 in 184 iterations

This also works without any modifications with the fixedpoint library function.

using NLsolve

p = [1.0, 2.0, 0.1]
beta = 0.9
iv = [0.8, 2.0, 51.0]
f(v) = p .+ beta * v

sol = fixedpoint(v -> p .+ beta * v, iv)
normdiff = norm(f(sol.zero) - sol.zero)
println("Fixed point = $(sol.zero)
  |f(x) - x| = $normdiff in $(sol.iterations) iterations")

Fixed point = [10.0, 20.000000000000004, 0.9999999999999929]
  |f(x) - x| = 6.661338147750939e-16 in 3 iterations

Finally, to demonstrate the importance of composing different libraries, use a StaticArrays.jl type, which provides an efficient implementation for small arrays and matrices.

using NLsolve, StaticArrays
p = @SVector [1.0, 2.0, 0.1]
beta = 0.9
iv = [0.8, 2.0, 51.0]
f(v) = p .+ beta * v

sol = fixedpoint(v -> p .+ beta * v, iv)
normdiff = norm(f(sol.zero) - sol.zero)
println("Fixed point = $(sol.zero)
  |f(x) - x| = $normdiff in $(sol.iterations) iterations")

Fixed point = [10.0, 20.000000000000004, 0.9999999999999929]
  |f(x) - x| = 6.661338147750939e-16 in 3 iterations

The @SVector in front of the [1.0, 2.0, 0.1] is a macro for turning a vector literal into a static vector.

All macros in Julia are prefixed by @ in the name, and manipulate the code prior to compilation.

We will see a variety of macros, and discuss the “metaprogramming” behind them in a later lecture.

2.4. Exercises #

2.4.1. Exercise 1#

Recall that $n!$ is read as “$n$ factorial” and defined as $n! = n \times (n - 1) \times \cdots \times 2 \times 1$.

In Julia you can compute this value with factorial(n).

Write your own version of this function, called factorial2, using a for loop.

2.4.2. Exercise 2#

The binomial random variable $Y \sim Bin(n, p)$ represents

number of successes in $n$ binary trials
each trial succeeds with probability $p$

Using only rand() from the set of Julia’s built-in random number generators (not the Distributions package), write a function binomial_rv such that binomial_rv(n, p) generates one draw of $Y$.

Hint: If $U$ is uniform on $(0, 1)$ and $p \in (0,1)$, then the expression U < p evaluates to true with probability $p$.

2.4.3. Exercise 3#

Compute an approximation to $\pi$ using Monte Carlo.

For random number generation use only rand().

Your hints are as follows:

If $U$ is a bivariate uniform random variable on the unit square $(0, 1)^2$, then the probability that $U$ lies in a subset $B$ of $(0,1)^2$ is equal to the area of $B$.
If $U_1,\ldots,U_n$ are iid copies of $U$, then, as $n$ gets larger, the fraction that falls in $B$ converges to the probability of landing in $B$.
For a circle, area = π * $radius^2$.

2.4.4. Exercise 4#

Write a program that prints one realization of the following random device:

Flip an unbiased coin 10 times.
If 3 consecutive heads occur one or more times within this sequence, pay one dollar.
If not, pay nothing.

Once again use only rand() as your random number generator.

2.4.5. Exercise 5#

Simulate and plot the correlated time series

\[ x_{t+1} = \alpha \, x_t + \epsilon_{t+1} \quad \text{where} \quad x_0 = 0 \quad \text{and} \quad t = 0,\ldots,n \]

The sequence of shocks $\{\epsilon_t\}$ is assumed to be iid and standard normal.

Set $n = 200$ and $\alpha = 0.9$.

2.4.6. Exercise 6#

Plot three simulated time series, one for each of the cases $\alpha = 0$, $\alpha = 0.8$ and $\alpha = 0.98$.

(The figure will illustrate how time series with the same one-step-ahead conditional volatilities, as these three processes have, can have very different unconditional volatilities)

2.4.7. Exercise 7#

This exercise is more challenging.

Take a random walk, starting from $x_0 = 1$

\[ x_{t+1} = \, \alpha \, x_t + \sigma\, \epsilon_{t+1} \quad \text{where} \quad x_0 = 1 \quad \text{and} \quad t = 0,\ldots,t_{\max} \]

Furthermore, assume that the $x_{t_{\max}} = 0$ (i.e. at $t_{\max}$, the value drops to zero, regardless of its current state).
The sequence of shocks $\{\epsilon_t\}$ is assumed to be iid and standard normal.
For a given path $\{x_t\}$ define a first-passage time as $T_a = \min\{t\, |\, x_t \leq a\}$, where by the assumption of the process $T_a \leq t_{\max}$.

Start with $\sigma = 0.2, \alpha = 1.0$

calculate the first-passage time, $T_0$, for 100 simulated random walks – to a $t_{\max} = 200$ and plot a histogram
plot the sample mean of $T_0$ from the simulation for $\alpha \in \{0.8, 1.0, 1.2\}$

2.4.8. Exercise 8(a)#

This exercise is more challenging.

The root of a univariate function $f(\cdot)$ is an $x$ such that $f(x) = 0$.

One solution method to find local roots of smooth functions is called Newton’s method.

Starting with an $x_0$ guess, a function $f(\cdot)$ and the first-derivative $f'(\cdot)$, the algorithm is to repeat

\[ x^{n+1} = x^n - \frac{f(x^n)}{f'(x^n)} \]

until $| x^{n+1} - x^n|$ is below a tolerance

Use a variation of the fixedpointmap code to implement Newton’s method, where the function would accept arguments f, f_prime, x_0, tolerance, maxiter.
Test it with $f(x) = (x-1)^3$ and another function of your choice where you can analytically find the derivative.

2.4.9. Exercise 8(b)#

For those impatient to use more advanced features of Julia, implement a version of Exercise 8(a) where f_prime is calculated with auto-differentiation.

using ForwardDiff

# operator to get the derivative of this function using AD
D(f) = x -> ForwardDiff.derivative(f, x)

# example usage: create a function and get the derivative
f(x) = x^2
f_prime = D(f)

f(0.1), f_prime(0.1)

(0.010000000000000002, 0.2)

Using the D(f) operator definition above, implement a version of Newton’s method that does not require the user to provide an analytical derivative.
Test the sorts of f functions which can be automatically integrated by ForwardDff.jl.

2.5. Solutions #

2.5.1. Exercise 1#

function factorial2(n)
    k = 1
    for i in 1:n
        k *= i  # or k = k * i
    end
    return k
end

factorial2(4)

factorial2(4) == factorial(4) # built-in function

true

2.5.2. Exercise 2#

function binomial_rv(n, p)
    count = 0
    U = rand(n)
    for i in 1:n
        if U[i] < p
            count += 1 # or count = count + 1
        end
    end
    return count
end

for j in 1:25
    b = binomial_rv(10, 0.5)
    print("$b, ")
end

7, 3, 4, 6, 2, 5, 5, 5, 4, 5, 5, 4, 6, 6, 5, 4, 7, 5, 4, 7, 7, 6, 3, 8, 6,

2.5.3. Exercise 3#

Consider a circle with diameter 1 embedded in a unit square.

Let $A$ be its area and let $r = 1/2$ be its radius.

If we know $\pi$ then we can compute $A$ via $A = \pi r^2$.

But the point here is to compute $\pi$, which we can do by $\pi = A / r^2$.

Summary: If we can estimate the area of the unit circle, then dividing by $r^2 = (1/2)^2 = 1/4$ gives an estimate of $\pi$.

We estimate the area by sampling bivariate uniforms and looking at the fraction that fall into the unit circle.

n = 1000000
count = 0
for i in 1:n
    u, v = rand(2)
    d = sqrt((u - 0.5)^2 + (v - 0.5)^2)  # distance from middle of square
    if d < 0.5
        count += 1
    end
end

area_estimate = count / n

print(area_estimate * 4)  # dividing by radius**2

3.144524

2.5.4. Exercise 4#

payoff = 0
count = 0

print("Count = ")

for i in 1:10
    U = rand()
    if U < 0.5
        count += 1
    else
        count = 0
    end
    print(count)
    if count == 3
        payoff = 1
    end
end
println("\npayoff = $payoff")

Count = 1010101200
payoff = 0

We can simplify this somewhat using the ternary operator. Here are some examples

a = 1 < 2 ? "foo" : "bar"

"foo"

a = 1 > 2 ? "foo" : "bar"

"bar"

Using this construction:

payoff = 0.0
count = 0.0

print("Count = ")

for i in 1:10
    U = rand()
    count = U < 0.5 ? count + 1 : 0
    print(count)
    if count == 3
        payoff = 1
    end
end
println("\npayoff = $payoff")

Count = 0012001200
payoff = 0.0

2.5.5. Exercise 5#

Here’s one solution

using Plots
alpha = 0.9
n = 200
x = zeros(n + 1)

for t in 1:n
    x[t + 1] = alpha * x[t] + randn()
end
plot(x)

2.5.6. Exercise 6#

alphas = [0.0, 0.8, 0.98]
n = 200
p = plot() # naming a plot to add to

for alpha in alphas
    x = zeros(n + 1)
    x[1] = 0.0
    for t in 1:n
        x[t + 1] = alpha * x[t] + randn()
    end
    plot!(p, x, label = "alpha = $alpha") # add to plot p
end
p # display plot

2.5.7. Exercise 7: Hint#

As a hint, notice the following pattern for finding the number of draws of a uniform random number until it is below a given threshold

function drawsuntilthreshold(threshold; maxdraws = 100)
    for i in 1:maxdraws
        val = rand()
        if val < threshold # checks threshold
            return i # leaves function, returning draw number
        end
    end
    return Inf # if here, reached maxdraws
end

draws = drawsuntilthreshold(0.1; maxdraws = 100)

Additionally, it is sometimes convenient to add to just push numbers onto an array without indexing it directly

vals = zeros(0) # empty vector

for i in 1:100
    val = rand()
    if val < 0.5
        push!(vals, val)
    end
end
println("There were $(length(vals)) below 0.5")

There were 56 below 0.5