Load and save DataFrames

Load and save DataFrames#

We do not cover all features of the packages. Please refer to their documentation to learn them.

Here we’ll load CSV.jl to read and write CSV files and Arrow.jl, JLSO.jl, and serialization, which allow us to work with a binary format and JSONTables.jl for JSON interaction. Finally we consider a custom JDF.jl format.

using DataFrames
using Arrow
using CSV
using Serialization
using JLSO
using JSONTables
using CodecZlib
using ZipFile
using JDF
using StatsPlots ## for charts
using Mmap ## for compression

Let’s create a simple DataFrame for testing purposes,

x = DataFrame(
    A=[true, false, true], B=[1, 2, missing],
    C=[missing, "b", "c"], D=['a', missing, 'c']
)
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc

and use eltypes to look at the columnwise types.

eltype.(eachcol(x))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, Char}

CSV.jl#

Let’s use CSV to save x to disk; make sure x1.csv does not conflict with some file in your working directory.

CSV.write("x1.csv", x)

"x1.csv"

Now we can see how it was saved by reading x.csv.

print(read("x1.csv", String))

A,B,C,D
true,1,,a
false,2,b,
true,,c,c

We can also load it back as a data frame

y = CSV.read("x1.csv", DataFrame)
3×4 DataFrame
RowABCD
BoolInt64?String1?String1?
1true1missinga
2false2bmissing
3truemissingcc

Note that when loading in a DataFrame from a CSV the column type for columns :C :D have changed to use special strings defined in the InlineStrings.jl package.

eltype.(eachcol(y))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, InlineStrings.String1}
 Union{Missing, InlineStrings.String1}

Serialization by JDF.jl and JLSO.jl#

Now we use serialization to save x.

There are two ways to perform serialization. The first way is to use the Serialization.serialize as below:

Note that in general, this process will not work if the reading and writing are done by different versions of Julia, or an instance of Julia with a different system image.

open("x.bin", "w") do io
    serialize(io, x)
end

Now we load back the saved file to y variable. Again y is identical to x. However, please beware that if you session does not have DataFrames.jl loaded, then it may not recognize the content as DataFrames.jl

y = open(deserialize, "x.bin")
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc
eltype.(eachcol(y))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, Char}

JDF.jl#

JDF.jl is a relatively new package designed to serialize DataFrames. You can save a DataFrame with the savejdf function. For more details about design assumptions and limitations of JDF.jl please check out xiaodaigh/JDF.jl.

JDF.save("x.jdf", x);

To load the saved JDF file, one can use the loadjdf function

x_loaded = JDF.load("x.jdf") |> DataFrame
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc

You can see that they are the same

isequal(x_loaded, x)

true

JDF.jl offers the ability to load only certain columns from disk to help with working with large files. set up a JDFFile which is a on disk representation of x backed by JDF.jl

x_ondisk = jdf"x.jdf"

JDF.JDFFile{String}("x.jdf")

We can see all the names of x without loading it into memory

names(x_ondisk)

4-element Vector{Symbol}:
 :A
 :B
 :C
 :D

The below is an example of how to load only columns :A and :D

xd = JDF.load(x_ondisk; cols=["A", "D"]) |> DataFrame
3×2 DataFrame
RowAD
BoolChar?
1truea
2falsemissing
3truec

JLSO.jl#

Another way to perform serialization is by using the JLSO.jl library:

JLSO.save("x.jlso", :data => x)

Now we can load back the file to y

y = JLSO.load("x.jlso")[:data]
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc
eltype.(eachcol(y))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, Char}

JSONTables.jl#

Often you might need to read and write data stored in JSON format. JSONTables.jl provides a way to process them in row-oriented or column-oriented layout. We present both options below.

open(io -> arraytable(io, x), "x1.json", "w")

106

open(io -> objecttable(io, x), "x2.json", "w")

76

print(read("x1.json", String))

[{"A":true,"B":1,"C":null,"D":"a"},{"A":false,"B":2,"C":"b","D":null},{"A":true,"B":null,"C":"c","D":"c"}]

print(read("x2.json", String))

{"A":[true,false,true],"B":[1,2,null],"C":[null,"b","c"],"D":["a",null,"c"]}

y1 = open(jsontable, "x1.json") |> DataFrame
3×4 DataFrame
RowABCD
BoolInt64?String?String?
1true1missinga
2false2bmissing
3truemissingcc
eltype.(eachcol(y1))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, String}

y2 = open(jsontable, "x2.json") |> DataFrame
3×4 DataFrame
RowABCD
BoolInt64?String?String?
1true1missinga
2false2bmissing
3truemissingcc
eltype.(eachcol(y2))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, String}

Arrow.jl#

Finally we use Apache Arrow format that allows, in particular, for data interchange with R or Python.

Arrow.write("x.arrow", x)

"x.arrow"

y = Arrow.Table("x.arrow") |> DataFrame
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc
eltype.(eachcol(y))

4-element Vector{Type}:
 Bool
 Union{Missing, Int64}
 Union{Missing, String}
 Union{Missing, Char}

Note that columns of y are immutable

try
    y.A[1] = false
catch e
    show(e)
end

ReadOnlyMemoryError()

This is because Arrow.Table uses memory mapping and thus uses a custom vector types:

y.A

3-element Arrow.BoolVector{Bool}:
 1
 0
 1

y.B

3-element Arrow.Primitive{Union{Missing, Int64}, Vector{Int64}}:
 1
 2
  missing

You can get standard Julia Base vectors by copying a data frame

y2 = copy(y)
3×4 DataFrame
RowABCD
BoolInt64?String?Char?
1true1missinga
2false2bmissing
3truemissingcc
y2.A

3-element Vector{Bool}:
 1
 0
 1

y2.B

3-element Vector{Union{Missing, Int64}}:
 1
 2
  missing

Basic benchmarking#

Next, we’ll create some files, so be careful that you don’t already have these files in your working directory! In particular, we’ll time how long it takes us to write a DataFrame with 1000 rows and 100000 columns.

bigdf = DataFrame(rand(Bool, 10^4, 1000), :auto)

bigdf[!, 1] = Int.(bigdf[!, 1])
bigdf[!, 2] = bigdf[!, 2] .+ 0.5
bigdf[!, 3] = string.(bigdf[!, 3], ", as string")

println("First run")

First run

println("CSV.jl")
csvwrite1 = @elapsed @time CSV.write("bigdf1.csv", bigdf)
println("Serialization")
serializewrite1 = @elapsed @time open(io -> serialize(io, bigdf), "bigdf.bin", "w")
println("JDF.jl")
jdfwrite1 = @elapsed @time JDF.save("bigdf.jdf", bigdf)
println("JLSO.jl")
jlsowrite1 = @elapsed @time JLSO.save("bigdf.jlso", :data => bigdf)
println("Arrow.jl")
arrowwrite1 = @elapsed @time Arrow.write("bigdf.arrow", bigdf)
println("JSONTables.jl arraytable")
jsontablesawrite1 = @elapsed @time open(io -> arraytable(io, bigdf), "bigdf1.json", "w")
println("JSONTables.jl objecttable")
jsontablesowrite1 = @elapsed @time open(io -> objecttable(io, bigdf), "bigdf2.json", "w")
println("Second run")
println("CSV.jl")
csvwrite2 = @elapsed @time CSV.write("bigdf1.csv", bigdf)
println("Serialization")
serializewrite2 = @elapsed @time open(io -> serialize(io, bigdf), "bigdf.bin", "w")
println("JDF.jl")
jdfwrite2 = @elapsed @time JDF.save("bigdf.jdf", bigdf)
println("JLSO.jl")
jlsowrite2 = @elapsed @time JLSO.save("bigdf.jlso", :data => bigdf)
println("Arrow.jl")
arrowwrite2 = @elapsed @time Arrow.write("bigdf.arrow", bigdf)
println("JSONTables.jl arraytable")
jsontablesawrite2 = @elapsed @time open(io -> arraytable(io, bigdf), "bigdf1.json", "w")
println("JSONTables.jl objecttable")
jsontablesowrite2 = @elapsed @time open(io -> objecttable(io, bigdf), "bigdf2.json", "w")

CSV.jl
  7.184657 seconds (44.69 M allocations: 1.127 GiB, 2.91% gc time, 70.36% compilation time)
Serialization
  0.250513 seconds (224.29 k allocations: 10.830 MiB, 31.57% compilation time)
JDF.jl
  0.163586 seconds (68.08 k allocations: 147.947 MiB, 22.95% gc time, 35.80% compilation time)
JLSO.jl
  1.219439 seconds (284.76 k allocations: 20.358 MiB, 7.93% compilation time)
Arrow.jl
  5.980401 seconds (5.44 M allocations: 275.143 MiB, 0.50% gc time, 97.45% compilation time)
JSONTables.jl arraytable
 20.148658 seconds (229.62 M allocations: 5.422 GiB, 14.65% gc time, 0.07% compilation time)
JSONTables.jl objecttable
  0.882334 seconds (96.64 k allocations: 309.084 MiB, 51.19% gc time, 15.89% compilation time)
Second run
CSV.jl
  2.400129 seconds (44.40 M allocations: 1.113 GiB, 7.17% gc time)
Serialization
  0.219526 seconds (15.01 k allocations: 400.680 KiB, 7.47% compilation time)
JDF.jl
  0.131295 seconds (35.13 k allocations: 146.224 MiB, 15.43% gc time)
JLSO.jl
  1.285567 seconds (33.34 k allocations: 7.460 MiB)
Arrow.jl
  0.154724 seconds (81.35 k allocations: 5.426 MiB)
JSONTables.jl arraytable
 16.327581 seconds (229.62 M allocations: 5.422 GiB, 13.53% gc time, 0.10% compilation time)
JSONTables.jl objecttable
  0.513288 seconds (20.71 k allocations: 305.221 MiB, 52.74% gc time, 1.88% compilation time)

0.513486184

groupedbar(
    repeat(["CSV.jl", "Serialization", "JDF.jl", "JLSO.jl", "Arrow.jl", "JSONTables.jl\nobjecttable"],
        inner=2),
    [csvwrite1, csvwrite2, serializewrite1, serializewrite1, jdfwrite1, jdfwrite2,
        jlsowrite1, jlsowrite2, arrowwrite1, arrowwrite2, jsontablesowrite2, jsontablesowrite2],
    group=repeat(["1st", "2nd"], outer=6),
    ylab="Second",
    title="Write Performance\nDataFrame: bigdf\nSize: $(size(bigdf))"
)
_images/1359b3d2524b1d4819a73f2d41d48f109df234af1230839bb02d183939568afe.png 
data_files = ["bigdf1.csv", "bigdf.bin", "bigdf.arrow", "bigdf1.json", "bigdf2.json"]
df = DataFrame(file=data_files, size=getfield.(stat.(data_files), :size))
append!(df, DataFrame(file="bigdf.jdf", size=reduce((x, y) -> x + y.size,
    stat.(joinpath.("bigdf.jdf", readdir("bigdf.jdf"))),
    init=0)))
sort!(df, :size)
6×2 DataFrame
Rowfilesize
StringInt64
1bigdf.arrow1742786
2bigdf.bin5199248
3bigdf.jdf5221164
4bigdf1.csv55084818
5bigdf2.json55088819
6bigdf1.json124029926
@df df plot(:file, :size / 1024^2, seriestype=:bar, title="Format File Size (MB)", label="Size", ylab="MB")
_images/a173a4080cab9e56e8b77ebf77054d70202dcb0bf2bed193d53c0aeeb087f88b.png 
println("First run")
println("CSV.jl")
csvread1 = @elapsed @time CSV.read("bigdf1.csv", DataFrame)
println("Serialization")
serializeread1 = @elapsed @time open(deserialize, "bigdf.bin")
println("JDF.jl")
jdfread1 = @elapsed @time JDF.load("bigdf.jdf") |> DataFrame
println("JLSO.jl")
jlsoread1 = @elapsed @time JLSO.load("bigdf.jlso")
println("Arrow.jl")
arrowread1 = @elapsed @time df_tmp = Arrow.Table("bigdf.arrow") |> DataFrame
arrowread1copy = @elapsed @time copy(df_tmp)
println("JSONTables.jl arraytable")
jsontablesaread1 = @elapsed @time open(jsontable, "bigdf1.json")
println("JSONTables.jl objecttable")
jsontablesoread1 = @elapsed @time open(jsontable, "bigdf2.json")
println("Second run")
csvread2 = @elapsed @time CSV.read("bigdf1.csv", DataFrame)
println("Serialization")
serializeread2 = @elapsed @time open(deserialize, "bigdf.bin")
println("JDF.jl")
jdfread2 = @elapsed @time JDF.load("bigdf.jdf") |> DataFrame
println("JLSO.jl")
jlsoread2 = @elapsed @time JLSO.load("bigdf.jlso")
println("Arrow.jl")
arrowread2 = @elapsed @time df_tmp = Arrow.Table("bigdf.arrow") |> DataFrame
arrowread2copy = @elapsed @time copy(df_tmp)
println("JSONTables.jl arraytable")
jsontablesaread2 = @elapsed @time open(jsontable, "bigdf1.json")
println("JSONTables.jl objecttable")
jsontablesoread2 = @elapsed @time open(jsontable, "bigdf2.json");

First run
CSV.jl
  2.710564 seconds (3.51 M allocations: 205.433 MiB, 1.17% gc time, 186 lock conflicts, 136.91% compilation time)
Serialization
  0.399885 seconds (9.50 M allocations: 155.494 MiB, 8.97% gc time, 8.71% compilation time)
JDF.jl
  0.207842 seconds (169.43 k allocations: 158.443 MiB, 9.80% gc time, 73.23% compilation time)
JLSO.jl
  0.366240 seconds (9.52 M allocations: 158.104 MiB, 4.90% gc time, 6.73% compilation time)
Arrow.jl
  0.419136 seconds (545.46 k allocations: 26.029 MiB, 98.38% compilation time)
  0.061942 seconds (14.50 k allocations: 10.258 MiB, 19.60% gc time)
JSONTables.jl arraytable
  6.467178 seconds (271.07 k allocations: 1.838 GiB, 9.44% gc time)
JSONTables.jl objecttable
  0.351149 seconds (7.43 k allocations: 403.797 MiB, 3.13% gc time, 0.02% compilation time)
Second run
  0.604527 seconds (152.55 k allocations: 34.426 MiB)
Serialization
  0.330234 seconds (9.48 M allocations: 154.588 MiB, 5.84% gc time)
JDF.jl
  0.063943 seconds (77.27 k allocations: 153.746 MiB, 9.45% gc time)
JLSO.jl
  0.306189 seconds (9.50 M allocations: 157.235 MiB, 4.20% gc time)
Arrow.jl
  0.005701 seconds (83.57 k allocations: 3.533 MiB)
  0.049116 seconds (14.50 k allocations: 10.258 MiB)
JSONTables.jl arraytable
  6.340624 seconds (271.07 k allocations: 1.838 GiB, 9.15% gc time)
JSONTables.jl objecttable
  0.328206 seconds (7.08 k allocations: 403.781 MiB, 1.92% gc time)

Exclude JSONTables due to much longer timing

groupedbar(
    repeat(["CSV.jl", "Serialization", "JDF.jl", "JLSO.jl", "Arrow.jl", "Arrow.jl\ncopy", ##"JSON\narraytable",
            "JSON\nobjecttable"], inner=2),
    [csvread1, csvread2, serializeread1, serializeread2, jdfread1, jdfread2, jlsoread1, jlsoread2,
        arrowread1, arrowread2, arrowread1 + arrowread1copy, arrowread2 + arrowread2copy,
        # jsontablesaread1, jsontablesaread2,
        jsontablesoread1, jsontablesoread2],
    group=repeat(["1st", "2nd"], outer=7),
    ylab="Second",
    title="Read Performance\nDataFrame: bigdf\nSize: $(size(bigdf))"
)
_images/39f3a44395b7da7d5e4a649cb3418ed381b9e0b47fc1d6f380a2feeb71bc9162.png

Using gzip compression#

A common user requirement is to be able to load and save CSV that are compressed using gzip. Below we show how this can be accomplished using CodecZlib.jl. The same pattern is applicable to JSONTables.jl compression/decompression. Again make sure that you do not have file named df_compress_test.csv.gz in your working directory. We first generate a random data frame.

df = DataFrame(rand(1:10, 10, 1000), :auto)
10×1000 DataFrame
900 columns omitted
Rowx1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29x30x31x32x33x34x35x36x37x38x39x40x41x42x43x44x45x46x47x48x49x50x51x52x53x54x55x56x57x58x59x60x61x62x63x64x65x66x67x68x69x70x71x72x73x74x75x76x77x78x79x80x81x82x83x84x85x86x87x88x89x90x91x92x93x94x95x96x97x98x99x100⋯
Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64⋯
1274101212722710674361281081035281063258105747910222751947546886258355732177983611021651473336736213932228633⋯
2524105796487510996810113536109778236537269631024810651033757243946979374194216619578446109584168877571010410268⋯
33735937948297743399917991104845105824277105726358228104627494108736104484104664410719685249838997994261186109⋯
4210186241091031977296106978593752810581038852987547981526531351049745295777619310184567264951023381023925105156⋯
582881588410458663372213110234619123838493241596171057641056839312881086251062958679669366579857675384110169⋯
6931101048192418168341351031051843255747232236739791034521102998668279510108281017815107785110654966814171252227⋯
7274442610342481729668218531410837624176565865210528795541641237101872198653877510103973728973244297911921010⋯
86632911158445889444412613468136199661497375227844865325663106698109899254733331126774149210628999249892⋯
971499475356339910692122271086111743245116102552892995894104310310107674108105299854861036218184228321015788410102⋯
10351086541017545154932551029961022141339518729198410599519396445957226988863337622886101039964196265108729422⋯

GzipCompressorStream comes from CodecZlib

open("df_compress_test.csv.gz", "w") do io
    stream = GzipCompressorStream(io)
    CSV.write(stream, df)
    close(stream)
end

df2 = CSV.File(transcode(GzipDecompressor, Mmap.mmap("df_compress_test.csv.gz"))) |> DataFrame
10×1000 DataFrame
900 columns omitted
Rowx1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29x30x31x32x33x34x35x36x37x38x39x40x41x42x43x44x45x46x47x48x49x50x51x52x53x54x55x56x57x58x59x60x61x62x63x64x65x66x67x68x69x70x71x72x73x74x75x76x77x78x79x80x81x82x83x84x85x86x87x88x89x90x91x92x93x94x95x96x97x98x99x100⋯
Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64Int64⋯
1274101212722710674361281081035281063258105747910222751947546886258355732177983611021651473336736213932228633⋯
2524105796487510996810113536109778236537269631024810651033757243946979374194216619578446109584168877571010410268⋯
33735937948297743399917991104845105824277105726358228104627494108736104484104664410719685249838997994261186109⋯
4210186241091031977296106978593752810581038852987547981526531351049745295777619310184567264951023381023925105156⋯
582881588410458663372213110234619123838493241596171057641056839312881086251062958679669366579857675384110169⋯
6931101048192418168341351031051843255747232236739791034521102998668279510108281017815107785110654966814171252227⋯
7274442610342481729668218531410837624176565865210528795541641237101872198653877510103973728973244297911921010⋯
86632911158445889444412613468136199661497375227844865325663106698109899254733331126774149210628999249892⋯
971499475356339910692122271086111743245116102552892995894104310310107674108105299854861036218184228321015788410102⋯
10351086541017545154932551029961022141339518729198410599519396445957226988863337622886101039964196265108729422⋯
df == df2

true

Using zip files#

Sometimes you may have files compressed inside a zip file. In such a situation you may use ZipFile.jl in conjunction an an appropriate reader to read the files. Here we first create a ZIP file and then read back its contents into a DataFrame.

df1 = DataFrame(rand(1:10, 3, 4), :auto)
3×4 DataFrame
Rowx1x2x3x4
Int64Int64Int64Int64
151037
291067
385109
df2 = DataFrame(rand(1:10, 3, 4), :auto)
3×4 DataFrame
Rowx1x2x3x4
Int64Int64Int64Int64
138110
26817
39341

And we show yet another way to write a DataFrame into a CSV file: Writing a CSV file into the zip file

w = ZipFile.Writer("x.zip")

f1 = ZipFile.addfile(w, "x1.csv")
write(f1, sprint(show, "text/csv", df1))

# write a second CSV file into zip file
f2 = ZipFile.addfile(w, "x2.csv", method=ZipFile.Deflate)
write(f2, sprint(show, "text/csv", df2))

close(w)

Now we read the compressed CSV file we have written:

z = ZipFile.Reader("x.zip");
# find the index index of file called x1.csv
index_xcsv = findfirst(x -> x.name == "x1.csv", z.files)
# to read the x1.csv file in the zip file
df1_2 = CSV.read(read(z.files[index_xcsv]), DataFrame)
3×4 DataFrame
Rowx1x2x3x4
Int64Int64Int64Int64
151037
291067
385109
df1_2 == df1

true

# find the index index of file called x2.csv
index_xcsv = findfirst(x -> x.name == "x2.csv", z.files)
# to read the x2.csv file in the zip file
df2_2 = CSV.read(read(z.files[index_xcsv]), DataFrame)
3×4 DataFrame
Rowx1x2x3x4
Int64Int64Int64Int64
138110
26817
39341
df2_2 == df2

true

Note that once you read a given file from z object its stream is all used-up (reaching its end). Therefore to read it again you need to close the file object z and open it again. Also do not forget to close the zip file once you are done.

close(z)

Remove generated files

rm("x.arrow")
rm("x.bin")
rm("x.zip")
rm("x.jlso")
rm("x1.csv")
rm("x1.json")
rm("x2.json")
rm("x.jdf", recursive=true)
rm("bigdf.jdf", recursive=true)
rm("df_compress_test.csv.gz")
rm("bigdf1.json")
rm("bigdf1.csv")
rm("bigdf2.json")
rm("bigdf.jlso")
rm("bigdf.bin")
rm("bigdf.arrow")

This notebook was generated using Literate.jl.