Distributions.jl

The Distributions package provides an extensive collection of probabilistic distributions and related functions. Each distribution is defined as a custom type, which allows creating instances of distributions.

julia> using Distributions
julia> D = Normal(2, 0.5)Distributions.Normal{Float64}(μ=2.0, σ=0.5)

The example above creates the normal distribution with mean μ = 2 and standard deviation σ = 0.5. The Distributions package provides functions to compute mean, variance, or quantiles.

julia> mean(D)2.0
julia> var(D)0.25
julia> quantile(D, 0.9)2.6407757827723

The package also provides ways to evaluate probability or cumulative density functions.

julia> pdf(D, 2)0.7978845608028654
julia> cdf(D, 2)0.5

In combination with the StatsPlots package, it is possible to plot probability density functions.

using StatsPlots
plot(
    plot(D; title = "pdf"),
    plot(D; func = cdf,  title = "cdf");
    legend = false,
    xlabel = "x",
    ylabel = "f(x)",
    ylims = (0,1),
    linewidth = 2,
    layout = (1,2),
    size = (800, 400)
)

The Distributions package also provides methods to fit a distribution to a given set of samples.

julia> x = rand(Normal(2, 0.5), 10000); # generate 10000 random numbers from Normal(2, 0.5)
julia> D = fit(Normal, x)Distributions.Normal{Float64}(μ=1.9966787650664388, σ=0.4935102108825582)

The fit function chooses a reasonable way to fit the distribution, which, in most cases, is the maximum likelihood estimation. However, this is not supported for all distributions. We can quickly check that the fit by using a histogram.

histogram(x; normalize = :pdf, legend = false, opacity = 0.5)
plot!(D; linewidth = 2, xlabel = "x", ylabel = "pdf(x)")

Solution:

The easiest way to create multiple distributions is to use the broadcasting system.

Ds = Gamma.([2, 9, 7.5, 0.5], [2, 0.5, 1, 1])

Similarly, we use broadcasting to create a vector of labels.

labels = reshape(string.("Gamma", params.(Ds)), 1, :)

We need to reshape the labels to become a row vector. The reason is that we want to plot multiple distributions, and the Plot package expects that labels will be a row vector. Now, we call the plot function to plot all distributions.

plot(Ds;
    xaxis = ("x", (0, 20)),
    yaxis = ("pdf(x)", (0, 0.5)),
    labels = labels,
    linewidth = 2,
    legend = :topright,
)

A plot of the cumulative probability functions cannot be done in the same way. However, StatsPlots provides the func keyword argument that allows specifying which function should be plotted.

plot(Ds;
    func = cdf,
    xaxis = ("x", (0, 20)),
    yaxis = ("cdf(x)", (0, 1.05)),
    labels = labels,
    linewidth = 2,
    legend = :bottomright,
)

Another possibility is to use the Plots package directly. To do so, we need to define a function with one argument, which at a given point returns the value of the cumulative probability function. Such functions for all our distributions can be easily defined as anonymous functions.

cdfs = [x -> cdf(D, x) for D in Ds]

The previous expression returns a vector of functions. Now we can use the plot function to create a curve for each element of the vector of cumulative probability functions. The example below creates these curves for $x$ from $0$ to $20$.

plot(cdfs, 0, 20;
    xaxis = ("x", (0, 20)),
    yaxis = ("cdf(x)", (0, 1.05)),
    labels = labels,
    linewidth = 2,
    legend = :bottomright,
)

"/home/runner/work/Julia-for-Optimization-and-Learning/Julia-for-Optimization-and-Learning/docs/build/lecture_03/Gamma_cdf.svg"

BSON.jl

BSON is a package for working with the Binary JSON serialization format. It can be used as a general store for Julia's data structures. To save the data, BSON provides the bson function. The data can be passed to the function directly via keyword arguments

julia> using BSON
julia> BSON.bson("test2.bson", a = [1+2im, 3+4im], b = "Hello, World!")

or as a dictionary

julia> data = Dict(:a => [1+2im, 3+4im], :b => "Hello, World!")Dict{Symbol, Any} with 2 entries:
  :a => Complex{Int64}[1+2im, 3+4im]
  :b => "Hello, World!"
julia> BSON.bson("test1.bson", data)

To load the data, BSON provides the load function that accepts the path to the data.

julia> BSON.load("test1.bson")Dict{Symbol, Any} with 2 entries:
  :a => Complex{Int64}[1+2im, 3+4im]
  :b => "Hello, World!"
julia> BSON.load("test2.bson")Dict{Symbol, Any} with 2 entries:
  :a => Complex{Int64}[1+2im, 3+4im]
  :b => "Hello, World!"

The package also provides an alternative way to saving and loading data using the @save and @load macro.

julia> using BSON: @save, @load
julia> a = [1+2im, 3+4im];
julia> b = "Hello, World!";
julia> @save "test.bson" a b # Same as above
julia> @load "test.bson" a b # Loads `a` and `b` back into the workspace

ProgressMeter.jl

The ProgressMeter package provides excellent utilities for printing progress bars for long computations. The package provides the @showprogress macro that prints the progress bar for for loops.

julia> using ProgressMeter

julia> @showprogress 1 "Computing..." for i in 1:50
           sleep(0.1)
       end
Computing... 20%|███████▊                               |  ETA: 0:00:04

The same syntax can also be used with the map, pmap or reduce functions. Progress bars can also be created manually, which allows additional formatting of the output. For example, it is possible to print and update information related to the computation by the showvalues keyword.

julia> x, n = 1 , 10;

julia> p = Progress(n);

julia> for iter in 1:10
           x *= 2
           sleep(0.5)
           ProgressMeter.next!(p; showvalues = [(:iter, iter), (:x, x)])
       end
Progress: 100%|█████████████████████████████████████████| Time: 0:00:10
  iter:  10
  x:     1024

BenchmarkTools.jl

The BenchmarkTools package provides a framework for writing and running groups of benchmarks as well as comparing benchmark results. The primary macro provided by BenchmarkTools is the @benchmark macro

julia> using BenchmarkTools
julia> @benchmark sin(x) setup=(x=rand())BenchmarkTools.Trial: 10000 samples with 1000 evaluations.
 Range (min … max):  6.171 ns … 25.197 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     6.192 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   6.629 ns ±  0.901 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

  █▁▅                                                    ▆▃▂ ▁
  ███▅▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▃▁▃▁▆▁▄▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁███ █
  6.17 ns      Histogram: log(frequency) by time     8.14 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

The setup expression is run once per sample and is not included in the timing results. Another handy macro provided by the package is the @btime macro. The output of this macro is similar to the built-in @time macro.

julia> A = rand(3,3);
julia> @btime inv($A);  483.005 ns (4 allocations: 1.86 KiB)

We use $ to interpolate variable A into the benchmark expression. Any expression that is interpolated in such a way is "pre-computed" before the benchmarking begins.