text:
### A Pluto.jl notebook ###
# v0.17.0

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ ac443ae2-d0ac-4568-acf8-bdc5fd6a9aea
using Turing

# ╔═╡ def3863a-819e-48a7-8887-4b5682b66bc8
begin
	using OffsetArrays
	A = Float64.(reshape(1:15, 3, 5))
	OA = OffsetArray(A, -1:1, 0:4)
	OA[-1,0]
end

# ╔═╡ 1cfba614-6a6a-40aa-9cae-8295b2de69b7
begin
	using DataFrames
	using CSV

	function download_titanic()
	    url = "https://www.openml.org/data/get_csv/16826755/phpMYEkMl"
	    return DataFrame(CSV.File(download(url); missingstring = "?"))
	end
end

# ╔═╡ 6659ee0e-cdfa-4f85-a9a3-bc5e0a3ae523
using PlutoUI

# ╔═╡ 69ebdee0-a9e7-409f-9bf2-a681809758b5
md"

TL;DR: [Julia](https://julialang.org) attempts to be a fast language for scientific/numerical work with the expressivity of a langauge like Python.

"

# ╔═╡ 399bbb98-332c-11ec-2228-75a09609463a
md"# Introduction to the Julia language 🤹

Originally developed by [Jeff Bezanson](https://github.com/JeffBezanson), [Stefan Karpinski](https://karpinski.org), [Viral B. Shah](https://github.com/ViralBShah), and [Alan Edelman](http://www-math.mit.edu/~edelman/) in 2009 with first release in 2012.

In 2015 founded the company [Julia Computing](https://juliacomputing.com/about-us/) to offer JuliaHub, a Software as a Service (SaaS) platform for Julia.

###### Notable People in Julia community:
- [Dr. Chris Rackauckas](https://www.chrisrackauckas.com): Applied Mathematics Instructor at MIT with a background in computational pharmacology, works on scientific applications of machine learning. Blog: [Stochastic Lifestyle](https://www.stochasticlifestyle.com)
- [Fons van der Plas](https://github.com/fonsp) main developer of Juypter notebook alternative [Pluto.jl](https://github.com/fonsp/Pluto.jl)

###### Noteable Julia Software
- [Flux](https://fluxml.ai): machine learning library reminiscent of Python's [Jax](https://github.com/google/jax).
    - [TextAnalysis.jl](https://github.com/JuliaText/TextAnalysis.jl)
    - [Zygote.jl](https://github.com/FluxML/Zygote.jl): Autodiff engine
    - [Transformers.jl](https://github.com/chengchingwen/Transformers.jl)
- [Pluto.jl](https://github.com/fonsp/Pluto.jl): Reactive Computational Notebook
- [Turing](https://turing.ml/stable/): probablistic programming library akin to [Stan](https://mc-stan.org) or [PyMC3](https://github.com/pymc-devs/pymc)
- [DataFrames.jl](https://github.com/JuliaData/DataFrames.jl): Julia's alternative to [pandas](https://pandas.pydata.org) or [dplyr](https://dplyr.tidyverse.org)
"

# ╔═╡ f8bb328a-2cad-4d6e-9519-a982a87897db
md"""Unicode variable names are allowed!"""

# ╔═╡ fb9c6a2c-5a85-4a11-afb2-e3a9e6ed592f
δ = 0.00001

# ╔═╡ a64b65ef-bae1-4bc0-9b72-7bdd9ccaa5f4
q = 5

# ╔═╡ e2f41584-ee59-4115-ac3f-92c72f58a04e
r = 10

# ╔═╡ 07cc4bbd-5783-4724-8217-699d29b833b6

# ╔═╡ d04137f1-8f36-4994-9808-b541aa12ecf1
function addOne(x)
	x + 1
end

# ╔═╡ cf17c2de-ac75-4d73-beb9-011586c5ecb7
addOne(3)

# ╔═╡ 7392ee62-2610-4544-8081-38ee07153c99
multiplyTwo(y) = y*2

# ╔═╡ c907152b-db94-4c20-b441-5191cfc0a739
multiplyTwo(4)

# ╔═╡ 94555af6-c7b1-44c2-a840-8dc6b8787f0b
multiplyTwoInt(y :: Int32) = y*2

# ╔═╡ aef78259-8b28-4c34-bb19-12a53fbc86ad
multiplyTwoInt(2.0)

# ╔═╡ b81f4d83-49d3-48a5-9b90-f05e375f4b5e
concatStr = (s1, s2) -> s1 * s2

# ╔═╡ da44693b-eb6e-420b-9016-9c231b05504c
concatStr("hi", " world")

# ╔═╡ 5370629c-2ed4-4ec3-b7d2-6968b045ef07
sumTuple((fst, snd)) = fst + snd

# ╔═╡ 3ec812e3-e096-4fb7-99e3-f35fe2d4fd72
begin
	tup = (2,3)
	sumTuple(tup)
end

# ╔═╡ ab791022-8afe-4db5-877f-5387f0b029a6
md"""
# Strings
"""

# ╔═╡ d89f53f6-579a-49aa-8f8d-a85cb01ae41e
c1 = 'a'

# ╔═╡ fc8d28c7-63b4-4800-879c-3a1eeba3d178
typeof(c1)

# ╔═╡ 9c2ea4be-0216-4b93-9e5e-a1068110b667
s1 = "Hello world"

# ╔═╡ b8132a67-17dd-4cb8-bd94-c0a8868fbb10
s2 = "1 + 2 = $(1 + 2)"

# ╔═╡ 867895e7-66b8-4dfd-9452-5b07f2e617d4
md"""
# Arrays

MATLAB-like syntax
"""

# ╔═╡ e996fa3f-dc54-4800-b600-50ddc380b577
a = [ 1 2 3 4]

# ╔═╡ 3ff82b9c-7859-44a5-b7eb-8bb6a3a9ab49
b = [1; 2; 3; 4]

# ╔═╡ 8161496c-d7d4-459a-bb19-9b3ee2bd216d
b[1]

# ╔═╡ 37fe3655-77f4-4ddb-8c17-9374e7056135
ones(Float64, (4,5,2))

# ╔═╡ d7751bb3-f048-4016-831b-2bb930770790
A2 = fill(7, (3,4,5))

# ╔═╡ a1fb7cfe-9612-4731-8452-48bebe52c0f7
size(A2)

# ╔═╡ 0f401fe7-a745-4898-a9e5-26970af4f5c0
setindex!(A2, [1, 3], [1, 2])

# ╔═╡ c8914f66-3c73-4e8f-b742-608b755c3c09
accumulate((x,y) -> x + y, [1,2,3])

# ╔═╡ e523b8af-00b8-4941-8533-c6911e6aa384
+(1, 20, 4)

# ╔═╡ 38247fa2-1cb8-4ea9-afe1-4851a7fac7b8
D = [0 1 0]

# ╔═╡ 9d5ae8c1-9d06-4ca9-95bf-6cbccfcaa7bb
C = [3 4 5; 7 8 9]

# ╔═╡ 56ec1cb6-7f59-4fea-bef7-0eda2c1dec43
vcat(D, C)

# ╔═╡ 66cf41c7-c5ff-471a-9874-dbb70f9478e4

# ╔═╡ 7c27d71d-7e26-4566-8ef7-c65491885823
parse.(Int, ["1", "2"])

# ╔═╡ 9d25812f-e8b0-4303-be86-f2f7742c5ae1
broadcast(+, 1.0, (0, -2.0))

# ╔═╡ 8a1ef977-8e35-4542-8f16-19e127871fff
with_terminal() do
	for x in 1:10
		println(x)
	end
end

# ╔═╡ 9a436fd7-81b2-4fea-9858-23dd1fdfb37f
[x for x in 1:10]

# ╔═╡ 8a81c35c-dad4-4d48-a88d-d006cc1b4576
begin
	x1 = 1.0:3.0
	y1 = similar(x1)
	 @. y1 = x1 + 3 * sin(x1)
end

# ╔═╡ 95452dbb-4fa4-490f-9632-27b49863e059
titanic = download_titanic()

# ╔═╡ 3491db16-8e1f-4cba-8602-196b7bc2ec62
begin
	using StatsPlots, IndexedTables
	@df titanic plot(:age)
end

# ╔═╡ 1ee0ec78-8f7b-438e-9720-b8558ae71ebd
titanic.name

# ╔═╡ c68a4ee8-0f77-4302-8dbe-cb0f3a7508c6
md"""# Metaprogramming"""

# ╔═╡ 1e1de466-ebe7-4b41-b9b0-e387ac4935bc
Meta.parse("1 + 2")

# ╔═╡ 971ecd6d-e7d1-4735-8bd9-96ca06742e04
typeof(:(1 + 2))

# ╔═╡ b5766225-d125-424b-9334-8ba0e33cd6d5
typeof(:x)

# ╔═╡ c558bf3b-bd5c-4a5f-bafc-c26f21c3f6ff

# ╔═╡ 5802f939-1f5b-4e62-92e9-9ccece9372f5
Meta.parse("1 + 2").head

# ╔═╡ 92698b28-0fb3-4a17-93ea-62231c10d56b
Meta.parse("1 + 2").args

# ╔═╡ 3e56b2c8-7f35-4ebc-994f-81af680df79a
with_terminal() do
    dump(Meta.parse("1 + 2"))
end

# ╔═╡ fcfa2823-8194-4375-8bf5-24008976a7fc
macro replaceAdd(exp)
	Expr(:call, :*, exp.args[2], exp.args[3])
end

# ╔═╡ d20ffbc6-2ae4-4482-9f85-71bf28200d7d
@replaceAdd 1 + 3

# ╔═╡ c6d39690-22d7-4420-9240-dfb8fb918682
ex = quote
           x = 1
           y = 2
           x + y
       end

# ╔═╡ 34f1664f-0cf9-4303-8384-fd930d59629e
ex2 = :($a + b)

# ╔═╡ 22ccf99a-2c24-457d-9eee-5fd03e0dc9b8
eval(ex)

# ╔═╡ 5cb21e3d-7508-463a-b0bf-8b15a74fc472
struct Animal
    name  :: String
	age :: Int32
end

# ╔═╡ 144467e8-147e-4a2f-8ff1-e7df17510b34
pluto = Animal("Pluto",4)

# ╔═╡ 2d054021-1f15-4024-ac6a-89d85976d184
@bind val Slider(20:0.1:30, default=25)

# ╔═╡ e002934b-c0b0-4b8b-aac2-f7ec1aa1da1d
md"""
# Pluto.jl UI

The value is $(val)
"""

# ╔═╡ bf59a6b0-f0aa-4bf1-8711-93153d1e40cc
@bind x html"<textinput></textinput>"

# ╔═╡ 38bd2f1f-72d6-4331-942d-70b2027125f0
with_terminal() do
	if q < r
    	println("q is less than r")
	elseif x > y
    	println("q is greater than r")
	else
    	println("q is equal to r")
	end
end

# ╔═╡ b3638992-3433-4c1e-81d2-b58d3d92bf69
with_terminal(() -> println("Hello $(x)"))

# ╔═╡ 023a2628-8fbe-4924-9e45-1aba22fa9b55
md"""
# Resources For Learning Julia
- [Official YouTube channel](https://www.youtube.com/c/TheJuliaLanguage/)
- List of college courses using it: [https://julialang.org/learning/classes/](https://julialang.org/learning/classes/)
    - MIT's [Computational Thinking](https://computationalthinking.mit.edu/Spring21/) course, website is an interactive [Pluto.jl](https://github.com/fonsp/Pluto.jl) notebook
- [Stochastic Lifestyle](https://www.stochasticlifestyle.com) blog for deep dives in scientific computing with Julia
    - ex: [Post about algorithms not optimized by major machine learning libraries](https://www.stochasticlifestyle.com/useful-algorithms-that-are-not-optimized-by-jax-pytorch-or-tensorflow/)
- [Julia Academy](https://juliaacademy.com)
- Dataframes.jl [cheat sheet](https://www.ahsmart.com/assets/pages/data-wrangling-with-data-frames-jl-cheat-sheet/DataFramesCheatSheet_v1.x_rev1.pdf)
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
IndexedTables = "6deec6e2-d858-57c5-ab9b-e6ca5bd20e43"
OffsetArrays = "6fe1bfb0-de20-5000-8ca7-80f57d26f881"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"

[compat]
CSV = "~0.8.5"
DataFrames = "~1.2.2"
IndexedTables = "~1.0.0"
OffsetArrays = "~1.10.7"
PlutoUI = "~0.7.16"
StatsPlots = "~0.14.28"
Turing = "~0.18.0"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

[[AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "485ee0867925449198280d4af84bdb46a2a404d0"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.0.1"

[[AbstractMCMC]]
deps = ["BangBang", "ConsoleProgressMonitor", "Distributed", "Logging", "LoggingExtras", "ProgressLogging", "Random", "StatsBase", "TerminalLoggers", "Transducers"]
git-tree-sha1 = "db0a7ff3fbd987055c43b4e12d2fa30aaae8749c"
uuid = "80f14c24-f653-4e6a-9b94-39d6b0f70001"
version = "3.2.1"

[[AbstractPPL]]
deps = ["AbstractMCMC"]
git-tree-sha1 = "15f34cc635546ac072d03fc2cc10083adb4df680"
uuid = "7a57a42e-76ec-4ea3-a279-07e840d6d9cf"
version = "0.2.0"

[[AbstractTrees]]
git-tree-sha1 = "03e0550477d86222521d254b741d470ba17ea0b5"
uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
version = "0.3.4"

[[Adapt]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.3.1"

[[AdvancedHMC]]
deps = ["AbstractMCMC", "ArgCheck", "DocStringExtensions", "InplaceOps", "LinearAlgebra", "ProgressMeter", "Random", "Requires", "Setfield", "Statistics", "StatsBase", "StatsFuns", "UnPack"]
git-tree-sha1 = "0a655e9a59ee1c8bafc3af18e96d90f980b08600"
uuid = "0bf59076-c3b1-5ca4-86bd-e02cd72cde3d"
version = "0.3.2"

[[AdvancedMH]]
deps = ["AbstractMCMC", "Distributions", "Random", "Requires"]
git-tree-sha1 = "8ad8bfddf8bb627d689ecb91599c349cbf15e971"
uuid = "5b7e9947-ddc0-4b3f-9b55-0d8042f74170"
version = "0.6.6"

[[AdvancedPS]]
deps = ["AbstractMCMC", "Distributions", "Libtask", "Random", "StatsFuns"]
git-tree-sha1 = "06da6c283cf17cf0f97ed2c07c29b6333ee83dc9"
uuid = "576499cb-2369-40b2-a588-c64705576edc"
version = "0.2.4"

[[AdvancedVI]]
deps = ["Bijectors", "Distributions", "DistributionsAD", "DocStringExtensions", "ForwardDiff", "LinearAlgebra", "ProgressMeter", "Random", "Requires", "StatsBase", "StatsFuns", "Tracker"]
git-tree-sha1 = "130d6b17a3a9d420d9a6b37412cae03ffd6a64ff"
uuid = "b5ca4192-6429-45e5-a2d9-87aec30a685c"
version = "0.1.3"

[[ArgCheck]]
git-tree-sha1 = "dedbbb2ddb876f899585c4ec4433265e3017215a"
uuid = "dce04be8-c92d-5529-be00-80e4d2c0e197"
version = "2.1.0"

[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"

[[Arpack]]
deps = ["Arpack_jll", "Libdl", "LinearAlgebra"]
git-tree-sha1 = "2ff92b71ba1747c5fdd541f8fc87736d82f40ec9"
uuid = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
version = "0.4.0"

[[Arpack_jll]]
deps = ["Libdl", "OpenBLAS_jll", "Pkg"]
git-tree-sha1 = "e214a9b9bd1b4e1b4f15b22c0994862b66af7ff7"
uuid = "68821587-b530-5797-8361-c406ea357684"
version = "3.5.0+3"

[[ArrayInterface]]
deps = ["Compat", "IfElse", "LinearAlgebra", "Requires", "SparseArrays", "Static"]
git-tree-sha1 = "68bbd3f4e89924e1ebec2e2650f6b31ee7fa8ba9"
uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
version = "3.1.37"

[[Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[AxisAlgorithms]]
deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"]
git-tree-sha1 = "66771c8d21c8ff5e3a93379480a2307ac36863f7"
uuid = "13072b0f-2c55-5437-9ae7-d433b7a33950"
version = "1.0.1"

[[AxisArrays]]
deps = ["Dates", "IntervalSets", "IterTools", "RangeArrays"]
git-tree-sha1 = "d127d5e4d86c7680b20c35d40b503c74b9a39b5e"
uuid = "39de3d68-74b9-583c-8d2d-e117c070f3a9"
version = "0.4.4"

[[BangBang]]
deps = ["Compat", "ConstructionBase", "Future", "InitialValues", "LinearAlgebra", "Requires", "Setfield", "Tables", "ZygoteRules"]
git-tree-sha1 = "0ad226aa72d8671f20d0316e03028f0ba1624307"
uuid = "198e06fe-97b7-11e9-32a5-e1d131e6ad66"
version = "0.3.32"

[[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[Baselet]]
git-tree-sha1 = "aebf55e6d7795e02ca500a689d326ac979aaf89e"
uuid = "9718e550-a3fa-408a-8086-8db961cd8217"
version = "0.1.1"

[[Bijectors]]
deps = ["ArgCheck", "ChainRulesCore", "Compat", "Distributions", "Functors", "IrrationalConstants", "LinearAlgebra", "LogExpFunctions", "MappedArrays", "Random", "Reexport", "Requires", "Roots", "SparseArrays", "Statistics"]
git-tree-sha1 = "369af32fcb9be65d496dc43ad0bb713705d4e859"
uuid = "76274a88-744f-5084-9051-94815aaf08c4"
version = "0.9.11"

[[Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
version = "1.0.8+0"

[[CSV]]
deps = ["Dates", "Mmap", "Parsers", "PooledArrays", "SentinelArrays", "Tables", "Unicode"]
git-tree-sha1 = "b83aa3f513be680454437a0eee21001607e5d983"
uuid = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
version = "0.8.5"

[[Cairo_jll]]
deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "f2202b55d816427cd385a9a4f3ffb226bee80f99"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+0"

[[ChainRules]]
deps = ["ChainRulesCore", "Compat", "LinearAlgebra", "Random", "RealDot", "Statistics"]
git-tree-sha1 = "035ef8a5382a614b2d8e3091b6fdbb1c2b050e11"
uuid = "082447d4-558c-5d27-93f4-14fc19e9eca2"
version = "1.12.1"

[[ChainRulesCore]]
deps = ["Compat", "LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "3533f5a691e60601fe60c90d8bc47a27aa2907ec"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
version = "1.11.0"

[[Clustering]]
deps = ["Distances", "LinearAlgebra", "NearestNeighbors", "Printf", "SparseArrays", "Statistics", "StatsBase"]
git-tree-sha1 = "75479b7df4167267d75294d14b58244695beb2ac"
uuid = "aaaa29a8-35af-508c-8bc3-b662a17a0fe5"
version = "0.14.2"

[[ColorSchemes]]
deps = ["ColorTypes", "Colors", "FixedPointNumbers", "Random"]
git-tree-sha1 = "a851fec56cb73cfdf43762999ec72eff5b86882a"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
version = "3.15.0"

[[ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.0"

[[Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.8"

[[Combinatorics]]
git-tree-sha1 = "08c8b6831dc00bfea825826be0bc8336fc369860"
uuid = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
version = "1.0.2"

[[CommonSolve]]
git-tree-sha1 = "68a0743f578349ada8bc911a5cbd5a2ef6ed6d1f"
uuid = "38540f10-b2f7-11e9-35d8-d573e4eb0ff2"
version = "0.2.0"

[[CommonSubexpressions]]
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uuid = "4af54fe1-eca0-43a8-85a7-787d91b784e3"

[[LeftChildRightSiblingTrees]]
deps = ["AbstractTrees"]
git-tree-sha1 = "71be1eb5ad19cb4f61fa8c73395c0338fd092ae0"
uuid = "1d6d02ad-be62-4b6b-8a6d-2f90e265016e"
version = "0.1.2"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+0"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
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[[Libtask]]
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git-tree-sha1 = "90c6ed7f9ac449cddacd80d5c1fca59c97d203e7"
uuid = "6f1fad26-d15e-5dc8-ae53-837a1d7b8c9f"
version = "0.5.3"

[[Libtask_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "901fc8752bbc527a6006a951716d661baa9d54e9"
uuid = "3ae2931a-708c-5973-9c38-ccf7496fb450"
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[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
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[[LinearAlgebra]]
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[[LogExpFunctions]]
deps = ["ChainRulesCore", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "6193c3815f13ba1b78a51ce391db8be016ae9214"
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[[Logging]]
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[[LoggingExtras]]
deps = ["Dates", "Logging"]
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[[MCMCChains]]
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git-tree-sha1 = "04c3fd6da28ebd305120ffb05f0a3b8f9baced0a"
uuid = "c7f686f2-ff18-58e9-bc7b-31028e88f75d"
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[[MCMCDiagnosticTools]]
deps = ["AbstractFFTs", "DataAPI", "Distributions", "LinearAlgebra", "MLJModelInterface", "Random", "SpecialFunctions", "Statistics", "StatsBase", "Tables"]
git-tree-sha1 = "f3f0c23f0ebe11db62ff1e81412919cf7739053d"
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[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "5455aef09b40e5020e1520f551fa3135040d4ed0"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
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[[MLJModelInterface]]
deps = ["Random", "ScientificTypesBase", "StatisticalTraits"]
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uuid = "e80e1ace-859a-464e-9ed9-23947d8ae3ea"
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[[MacroTools]]
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[[MappedArrays]]
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[[Markdown]]
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[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
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[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
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[[MicroCollections]]
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[[Missings]]
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[[Mmap]]
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[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[MultivariateStats]]
deps = ["Arpack", "LinearAlgebra", "SparseArrays", "Statistics", "StatsBase"]
git-tree-sha1 = "8d958ff1854b166003238fe191ec34b9d592860a"
uuid = "6f286f6a-111f-5878-ab1e-185364afe411"
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[[NNlib]]
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uuid = "872c559c-99b0-510c-b3b7-b6c96a88d5cd"
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[[NaNMath]]
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uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
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[[NamedArrays]]
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git-tree-sha1 = "2fd5787125d1a93fbe30961bd841707b8a80d75b"
uuid = "86f7a689-2022-50b4-a561-43c23ac3c673"
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[[NaturalSort]]
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[[NearestNeighbors]]
deps = ["Distances", "StaticArrays"]
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[[NetworkOptions]]
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[[Observables]]
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[[OffsetArrays]]
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version = "1.10.7"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
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[[OnlineStatsBase]]
deps = ["AbstractTrees", "Dates", "LinearAlgebra", "OrderedCollections", "Statistics", "StatsBase"]
git-tree-sha1 = "e80d7c35fb08b9e8e96c30daf4daa598d154c5b0"
uuid = "925886fa-5bf2-5e8e-b522-a9147a512338"
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[[OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"

[[OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
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[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
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[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
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[[OrderedCollections]]
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[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[PDMats]]
deps = ["LinearAlgebra", "SparseArrays", "SuiteSparse"]
git-tree-sha1 = "4dd403333bcf0909341cfe57ec115152f937d7d8"
uuid = "90014a1f-27ba-587c-ab20-58faa44d9150"
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[[Parsers]]
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uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "1.1.2"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
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[[Pkg]]
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uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
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[[PlotUtils]]
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git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b"
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[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "ca7d534a27b1c279f05cd094196cb70c35e3d892"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
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[[PlutoUI]]
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git-tree-sha1 = "4c8a7d080daca18545c56f1cac28710c362478f3"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
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[[PooledArrays]]
deps = ["DataAPI", "Future"]
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[[Preferences]]
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[[PrettyTables]]
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[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[ProgressLogging]]
deps = ["Logging", "SHA", "UUIDs"]
git-tree-sha1 = "80d919dee55b9c50e8d9e2da5eeafff3fe58b539"
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[[ProgressMeter]]
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[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
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[[QuadGK]]
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[[REPL]]
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uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
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[[RangeArrays]]
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[[Ratios]]
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[[RealDot]]
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[[RecipesBase]]
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[[RecipesPipeline]]
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[[Reexport]]
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[[Requires]]
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[[Rmath]]
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[[Rmath_jll]]
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[[Roots]]
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[[SHA]]
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[[ScientificTypesBase]]
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[[Scratch]]
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[[SentinelArrays]]
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[[Serialization]]
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[[Setfield]]
deps = ["ConstructionBase", "Future", "MacroTools", "Requires"]
git-tree-sha1 = "def0718ddbabeb5476e51e5a43609bee889f285d"
uuid = "efcf1570-3423-57d1-acb7-fd33fddbac46"
version = "0.8.0"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "f0bccf98e16759818ffc5d97ac3ebf87eb950150"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.8.1"

[[SplittablesBase]]
deps = ["Setfield", "Test"]
git-tree-sha1 = "39c9f91521de844bad65049efd4f9223e7ed43f9"
uuid = "171d559e-b47b-412a-8079-5efa626c420e"
version = "0.1.14"

[[Static]]
deps = ["IfElse"]
git-tree-sha1 = "e7bc80dc93f50857a5d1e3c8121495852f407e6a"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.4.0"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[StatisticalTraits]]
deps = ["ScientificTypesBase"]
git-tree-sha1 = "730732cae4d3135e2f2182bd47f8d8b795ea4439"
uuid = "64bff920-2084-43da-a3e6-9bb72801c0c9"
version = "2.1.0"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "eb35dcc66558b2dda84079b9a1be17557d32091a"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.12"

[[StatsFuns]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "95072ef1a22b057b1e80f73c2a89ad238ae4cfff"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.12"

[[StatsPlots]]
deps = ["Clustering", "DataStructures", "DataValues", "Distributions", "Interpolations", "KernelDensity", "LinearAlgebra", "MultivariateStats", "Observables", "Plots", "RecipesBase", "RecipesPipeline", "Reexport", "StatsBase", "TableOperations", "Tables", "Widgets"]
git-tree-sha1 = "eb007bb78d8a46ab98cd14188e3cec139a4476cf"
uuid = "f3b207a7-027a-5e70-b257-86293d7955fd"
version = "0.14.28"

[[StructArrays]]
deps = ["DataAPI", "Tables"]
git-tree-sha1 = "ad1f5fd155426dcc879ec6ede9f74eb3a2d582df"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.4.2"

[[SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableOperations]]
deps = ["SentinelArrays", "Tables", "Test"]
git-tree-sha1 = "e383c87cf2a1dc41fa30c093b2a19877c83e1bc1"
uuid = "ab02a1b2-a7df-11e8-156e-fb1833f50b87"
version = "1.2.0"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[TableTraitsUtils]]
deps = ["DataValues", "IteratorInterfaceExtensions", "Missings", "TableTraits"]
git-tree-sha1 = "78fecfe140d7abb480b53a44f3f85b6aa373c293"
uuid = "382cd787-c1b6-5bf2-a167-d5b971a19bda"
version = "1.0.2"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[TerminalLoggers]]
deps = ["LeftChildRightSiblingTrees", "Logging", "Markdown", "Printf", "ProgressLogging", "UUIDs"]
git-tree-sha1 = "d620a061cb2a56930b52bdf5cf908a5c4fa8e76a"
uuid = "5d786b92-1e48-4d6f-9151-6b4477ca9bed"
version = "0.1.4"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[Tracker]]
deps = ["Adapt", "DiffRules", "ForwardDiff", "LinearAlgebra", "MacroTools", "NNlib", "NaNMath", "Printf", "Random", "Requires", "SpecialFunctions", "Statistics"]
git-tree-sha1 = "bf4adf36062afc921f251af4db58f06235504eff"
uuid = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
version = "0.2.16"

[[Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "bccb153150744d476a6a8d4facf5299325d5a442"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.67"

[[Turing]]
deps = ["AbstractMCMC", "AdvancedHMC", "AdvancedMH", "AdvancedPS", "AdvancedVI", "BangBang", "Bijectors", "DataStructures", "Distributions", "DistributionsAD", "DocStringExtensions", "DynamicPPL", "EllipticalSliceSampling", "ForwardDiff", "Libtask", "LinearAlgebra", "MCMCChains", "NamedArrays", "Printf", "Random", "Reexport", "Requires", "SpecialFunctions", "Statistics", "StatsBase", "StatsFuns", "Tracker", "ZygoteRules"]
git-tree-sha1 = "e22a11c2029137b35adf00a0e4842707c653938c"
uuid = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
version = "0.18.0"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[WeakRefStrings]]
deps = ["DataAPI", "Random", "Test"]
git-tree-sha1 = "28807f85197eaad3cbd2330386fac1dcb9e7e11d"
uuid = "ea10d353-3f73-51f8-a26c-33c1cb351aa5"
version = "0.6.2"

[[Widgets]]
deps = ["Colors", "Dates", "Observables", "OrderedCollections"]
git-tree-sha1 = "80661f59d28714632132c73779f8becc19a113f2"
uuid = "cc8bc4a8-27d6-5769-a93b-9d913e69aa62"
version = "0.6.4"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─69ebdee0-a9e7-409f-9bf2-a681809758b5
# ╟─399bbb98-332c-11ec-2228-75a09609463a
# ╠═ac443ae2-d0ac-4568-acf8-bdc5fd6a9aea
# ╟─f8bb328a-2cad-4d6e-9519-a982a87897db
# ╠═fb9c6a2c-5a85-4a11-afb2-e3a9e6ed592f
# ╠═a64b65ef-bae1-4bc0-9b72-7bdd9ccaa5f4
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text:
# Reactive message-passing in linear Gaussian dynamical system
# Experiment to pass messages without a schedule
#
# Wouter Kouw, BIASlab
# 12-08-2019

using Random
using Revise
using CPUTime
using ForneyLab
using Plots
pyplot()

# Functions for generating data
include(joinpath(@__DIR__, "../gen_data.jl"))

"""
Experiment parameters
"""

# Signal time horizon
T = 50

# Secondary clock for number of iterations
TT = 10

# Known transition and observation matrices
gain = 0.8
emission = 1.0

# Known noises (variance form)
process_noise = 2.0
measurement_noise = 0.5

# Parameters for state x_0
mean_0 = 0.0
precision_0 = 1.0

# Generate data
Random.seed!(256)
observed, hidden = gendata_LGDS(gain,
                                emission,
                                process_noise,
                                measurement_noise,
                                mean_0,
                                precision_0,
                                time_horizon=T)

"""
Model specification
"""

CPUtic()

g = FactorGraph()

@RV s_0 ~ GaussianMeanPrecision(mean_0, precision_0)

global s = Vector{Variable}(undef, T)
global x = Vector{Variable}(undef, T)
global s_t_min = s_0
for t = 1:T
    @RV s[t] ~ GaussianMeanPrecision(s_t_min, inv(process_noise))
    @RV x[t] ~ GaussianMeanVariance(s[t], inv(measurement_noise))

    global s_t_min = s[t]

    placeholder(x[t], :x, index=t)
end

"""
Mean-field algorithm generation
"""

# Define recognition factorization
RecognitionFactorization()

q_s_0 = RecognitionFactor(s_0)

q_s = Vector{RecognitionFactor}(undef, T)
for t=1:T
    q_s[t] = RecognitionFactor(s[t])
end

# Compile algorithm
algo_s_mf = variationalAlgorithm([q_s_0; q_s], name="SMF")
algo_F_mf = freeEnergyAlgorithm(name="MF");

"""
Run inference algorithm
"""

# Load algorithm
eval(Meta.parse(algo_s_mf))
eval(Meta.parse(algo_F_mf))

# Initialize data
global data = Dict(:x => observed)

# Initial recognition distributions
global marginals_mf = Dict{Symbol, ProbabilityDistribution}()
for t = 0:T
    global marginals_mf[:s_*t] = vague(GaussianMeanPrecision)
end

# Run algorithm
global estimated_states = zeros(T,2,TT)
global F_mf = Vector{Float64}(undef, TT)
for tt = 1:TT
    stepSMF!(data, marginals_mf)

    for t = 1:T
        estimated_states[t,1,tt] = mean(marginals_mf[:s_*t])
        estimated_states[t,2,tt] = sqrt(var(marginals_mf[:s_*t]))
    end

    # Free energy
    F_mf[tt] = freeEnergyMF(data, marginals_mf)
end

CPUtoc()

"""
Visualize experimental results
"""

# Visualize final estimates over time
scatter(1:T, observed, color="black", label="observations")
plot!(hidden[2:end], color="red", label="states")
plot!(estimated_states[:,1,end], color="blue", label="estimates")
plot!(estimated_states[:,1,end],
      ribbon=[estimated_states[:,2,end], estimated_states[:,2,end]],
      linewidth=2,
      color="blue",
      fillalpha=0.2,
      fillcolor="blue",
      label="")
xlabel!("time (t)")
savefig(joinpath(@__DIR__, "viz/baseline_state_estimates.png"))

# Visualize free energy
plot(1:TT, F_mf, color="green", label="")
xlabel!("#iterations")
ylabel!("free energy (F)")
savefig(joinpath(@__DIR__, "viz/baseline_fe-iters.png"))

ext: jl

max_stars_repo_name: wmkouw/schedulefreeMP

max_issues_repo_name: wmkouw/schedulefreeMP

max_forks_repo_name: wmkouw/schedulefreeMP


text:
module VStatistic
import GSL:sf_hyperg_2F1
# could have also used scipy.special.hyp2f1 via @pyimports

function vstat(n::Integer, p::Integer, R²)
    R² = R² <= 0 ? .0001 : R²

    r = ((p - 1) * (1 - R²)) / ((n - p) * R²)
    g = min(r, 1)
    g = .4999 < g < .5001 ? .5001 : g

    z = (g - √(g - g^2)) / (2g - 1)
    α = acos((1 - z) / √(1 - 2z * (1 - z)))

    v = begin
        ((2 * cos(α) * gamma((p + 2) / 2)) /
        (√(π) * gamma((p + 1) / 2))) *
        (sf_hyperg_2F1(.5, (1 - p) / 2, 3/2, cos(α)^2) -
        sin(α)^(p - 1))
    end

    return v
end

function sample_size(p::Integer, R²; v=.8)
    n = p
    while vstat(n, p, R²) < v
        n += 1
    end
    return n, round(n / p, 2)
end

end


text:
"""
Represents a static array with N dimensions whose size is
exactly 2 along each dimension. This makes templating on
the number of dimensions easier than with a regular
SArray.
"""
struct TwosArray{N, T, L} <: StaticArray{NTuple{N, 2}, T, N}
    data::NTuple{L, T}
end

@generated function TwosArray(x::NTuple{L, T}) where {L, T}
    @assert log2(L) == round(log2(L)) "L must equal 2^N for integer N"
    N = Int(log2(L))
    :(TwosArray{$N, T, L}(x))
end

getindex(b::TwosArray, i::Int) = b.data[i]

"""
Highly specialized selectdim implementation for TwosArray (an array of size
2 along every dimension). Returns a TwosArray with N-1 dimensions. See
`test/twosarray.jl` for exhaustive testing of this function. This is about
100 times faster than Julia's base selectdim().
"""
@generated function Base.selectdim(A::TwosArray{N, T}, d::Integer, i::Integer) where {N, T}
    quote
        x = 2^(d - 1)
        j = 1
        k = 1
        TwosArray($(Expr(:tuple, [quote
            z = j
            j += 1
            if k < x
                k += 1
            else
                j += x
                k = 1
            end
            A[z + (i - 1) * x]
        end for n in 1:(2^(N - 1))]...)))
    end
end

similar_type(::Type{TwosArray{N, T, L}}, ::Type{T2}) where {N, T, L, T2} = TwosArray{N, T2, L}


text:
@inline function _for_each_node(f::F, graph::PharmGraph) where {F}
    for vertex_integer in MetaGraphs.vertices(graph.g)
        class = _vertex_integer_to_class(graph, vertex_integer)::PharmClass
        f(class)
    end
    return nothing
end

@inline function MetaGraphs.outneighbors(graph::PharmGraph,
                                         class::PharmClass;
                                         normalization::Bool = true)
    if normalization
        return _outneighbors(graph, normalize(class))
    else
        return _outneighbors(graph, class)
    end
end

@inline function _outneighbors(graph::PharmGraph, class::PharmClass)
    vert_int = _class_to_vertex_integer_nocreate(graph, class)
    outneighbors_vert_ints = MetaGraphs.outneighbors(graph.g, vert_int)
    outneighbors_classes = _vertex_integers_to_class(graph, outneighbors_vert_ints)
    unique!(outneighbors_classes)
    sort!(outneighbors_classes)
    return outneighbors_classes
end

@inline function MetaGraphs.inneighbors(graph::PharmGraph,
                                        class::PharmClass;
                                        normalization::Bool = true)
    if normalization
        return _inneighbors(graph, normalize(class))
    else
        return _inneighbors(graph, class)
    end
end

@inline function _inneighbors(graph::PharmGraph, class::PharmClass)
    vert_int = _class_to_vertex_integer_nocreate(graph, class)
    inneighbors_vert_ints = MetaGraphs.inneighbors(graph.g, vert_int)
    inneighbors_classes = _vertex_integers_to_class(graph, inneighbors_vert_ints)
    unique!(inneighbors_classes)
    sort!(inneighbors_classes)
    return inneighbors_classes
end

@inline function MetaGraphs.all_neighbors(graph::PharmGraph,
                                          class::PharmClass;
                                          normalization::Bool = true)
    if normalization
        return _all_neighbors(graph, normalize(class))
    else
        return _all_neighbors(graph, class)
    end
end

@inline function _all_neighbors(graph::PharmGraph, class::PharmClass)
    vert_int = _class_to_vertex_integer_nocreate(graph, class)
    allneighbors_vert_ints = MetaGraphs.all_neighbors(graph.g, vert_int)
    allneighbors_classes = _vertex_integers_to_class(graph, allneighbors_vert_ints)
    unique!(allneighbors_classes)
    sort!(allneighbors_classes)
    return allneighbors_classes
end

@inline function _locations_of_nonzero_entries(entries::AbstractVector{I}) where I <: Integer
    n::I = length(entries)
    bools = BitVector(undef, n)
    for i = 1:n
        entry = entries[i]
        if entry > 0
            bools[i] = true
        else
            bools[i] = false
        end
    end
    one_to_n::UnitRange{I} = I(1):I(n)
    result::Vector{I} = one_to_n[bools]
    return result
end

@inline function parents(graph::PharmGraph,
                         class::PharmClass;
                         normalization::Bool = true)
    if normalization
        return _parents(graph, normalize(class))
    else
        return _parents(graph, class)
    end
end

@inline function _parents(graph::PharmGraph, class::PharmClass)
    vert_int = _class_to_vertex_integer_nocreate(graph, class)
    _parents = LightGraphs.bfs_parents(graph.g.graph, vert_int; dir = :in)
    parent_vert_ints = _locations_of_nonzero_entries(_parents)
    classes = _vertex_integers_to_class(graph, parent_vert_ints)
    return classes
end

@inline function children(graph::PharmGraph,
                          class::PharmClass;
                          normalization::Bool = true)
    if normalization
        return _children(graph, normalize(class))
    else
        return _children(graph, class)
    end
end

@inline function _children(graph::PharmGraph, class::PharmClass)
    vert_int = _class_to_vertex_integer_nocreate(graph, class)
    _parents = LightGraphs.bfs_parents(graph.g.graph, vert_int; dir = :out)
    parent_vert_ints = _locations_of_nonzero_entries(_parents)
    classes = _vertex_integers_to_class(graph, parent_vert_ints)
    return classes
end

@inline function equivalent(graph::PharmGraph,
                            class::PharmClass;
                            normalization::Bool = true)
    if normalization
        return _equivalent(graph, normalize(class))
    else
        return _equivalent(graph, class)
    end
end

@inline function _equivalent(graph::PharmGraph, class::PharmClass)
    class_parents = _parents(graph, class)
    class_children = _children(graph, class)
    class_equivalent = intersect(class_parents, class_children)
    unique!(class_equivalent)
    sort!(class_equivalent)
    return class_equivalent
end


text:
# Empirical estimation of CDF and PDF

## Empirical CDF

function ecdf{T<:Real}(X::RealVector{T})
    Xs = sort(X)
    n = length(X)

    ef(x::Real) = searchsortedlast(Xs, x) / n

    function ef(v::RealVector)
        ord = sortperm(v)
        m = length(v)
        r = Array(T, m)
        r0 = 0

        i = 1
        for x in Xs
            while i <= m && x > v[ord[i]]
                r[ord[i]] = r0
                i += 1
            end
            r0 += 1
            if i > m
            	break
            end
        end
        while i <= m
            r[ord[i]] = n
            i += 1
        end
        return r / n
    end

    return ef
end

#######################################
#
#  Kernel density estimation
#
#######################################

# Silverman's rule of thumb for KDE bandwidth selection
function bandwidth(data::Vector{Float64}, alpha::Float64 = 0.9)
    # Determine length of data
    ndata = length(data)

    # Calculate width using variance and IQR
    var_width = std(data)
    q25, q75 = quantile(data, [0.25, 0.75])
    quantile_width = (q75 - q25) / 1.34

    # Deal with edge cases with 0 IQR or variance
    width = min(var_width, quantile_width)
    if width == 0.0
        if var_width == 0.0
            width = 1.0
        else
            width = var_width
        end
    end

    # Set bandwidth using Silverman's rule of thumb
    return alpha * width * ndata^(-0.2)
end

# Store both grid and density for KDE over the real line
immutable UnivariateKDE
    x::Vector{Float64}
    density::Vector{Float64}
end

# Store both grid and density for KDE over R2
immutable BivariateKDE
    x::Vector{Float64}
    y::Vector{Float64}
    density::Matrix{Float64}
end

# Algorithm AS 176 for calculating univariate KDE
function kde(data::RealVector; width::Float64=NaN, npoints::Integer=2048)
    # Determine length of data
    ndata = length(data)

    # Check that the width is a positive constant
    if isnan(width)
    	width = bandwidth(data)
    elseif width <= 0.0
        error("Window must be positive")
    end

    # Find interval that will contain almost all mass
    dlo = minimum(data) - 3 * width
    dhi = maximum(data) + 3 * width

    # Check that interval for estimation is valid
    if dlo >= dhi
        error("Interval must be specified as lower bound, upper bound")
    end

    # Set up a grid for discretized data
    grid = zeros(Float64, npoints)

    # Define some more constants
    step = (dhi - dlo) / npoints
    ainc = 1.0 / (ndata * step)
    npoints2 = fld(npoints, 2)
    hw = width / step
    fac1 = 32.0 * (atan(1.0) * hw / npoints)^2

    # Discretize the data using a histogram
    dlo1 = dlo - step
    for i in 1:ndata
        j = iround(fld(data[i] - dlo1, step))
        if j >= 1 && j <= npoints
            grid[j] = grid[j] + ainc
        end
    end

    # Transform to Fourier basis
    ft = rfft(grid)

    # Find transform of KDE by convolving grid with the
    # Fourier transform of a Gaussian kernel
    for j = 2:length(ft)
        ft[j] *= exp(-fac1 * (j-1)^2)
    end

    # Invert the Fourier transform to get the KDE
    density = irfft(ft, npoints)

    # Fix any noise that crept in
    for j in 1:npoints
        if density[j] < 0.0
            density[j] = 0.0
        end
    end

    # Expand the grid over which KDE was calculated
    x = [dlo:step:(dhi - step)]

    return UnivariateKDE(x, density)
end

# Algorithm from MASS Chapter 5 for calculating 2D KDE
function kde(x::RealVector, y::RealVector; width::Float64=NaN, resolution::Int=25)
    n = length(x)

    if length(y) != n
        error("x and y must have the same length")
    end

    if isnan(width)
        h1 = bandwidth(x)
        h2 = bandwidth(y)
    else
        h1 = width
        h2 = width
    end

    min_x, max_x = extrema(x)
    min_y, max_y = extrema(y)

    grid_x = [min_x:((max_x - min_x) / (resolution - 1)):max_x]
    grid_y = [min_y:((max_y - min_y) / (resolution - 1)):max_y]

    mx = Array(Float64, resolution, n)
    my = Array(Float64, resolution, n)
    for i in 1:resolution
        for j in 1:n
            mx[i, j] = pdf(Normal(), (grid_x[i] - x[j]) / h1)
            my[i, j] = pdf(Normal(), (grid_y[i] - y[j]) / h2)
        end
    end

    z = A_mul_Bt(mx, my)
    for i in 1:(resolution^2)
        z[i] /= (n * h1 * h2)
    end

    return BivariateKDE(grid_x, grid_y, z)
end


text:
"""
Module ParallelKernel

Enables writing parallel high-performance kernels and whole applications that can be deployed on both GPUs and CPUs.

# Usage
    using ParallelStencil.ParallelKernel

# Primary macros
- [`@init_parallel_kernel`](@ref)
- [`@parallel`](@ref)
- [`@hide_communication`](@ref)
- [`@zeros`](@ref)
- [`@ones`](@ref)
- [`@rand`](@ref)
!!! note "Advanced"
    - [`@parallel_indices`](@ref)
    - [`@parallel_async`](@ref)
    - [`@synchronize`](@ref)

# Macros available for [`@parallel_indices`](@ref) kernels
- [`@pk_show`](@ref)
- [`@pk_println`](@ref)
!!! note "Advanced"
    - [`@gridDim`](@ref)
    - [`@blockIdx`](@ref)
    - [`@blockDim`](@ref)
    - [`@threadIdx`](@ref)
    - [`@sync_threads`](@ref)
    - [`@sharedMem`](@ref)

# Modules generated in caller
- [`Data`](@ref)

To see a description of a macro or module type `?<macroname>` (including the `@`) or `?<modulename>`, respectively.
"""
module ParallelKernel

## Alphabetical include of submodules.
include("Data.jl");
include("Exceptions.jl");
using .Exceptions

## Include of constant parameters, types and syntax sugar shared in ParallelKernel module only
include("shared.jl")

## Alphabetical include of function files
include("allocators.jl")
include("hide_communication.jl")
include("init_parallel_kernel.jl")
include("kernel_language.jl")
include("parallel.jl")
include("reset_parallel_kernel.jl")

## Exports
export @init_parallel_kernel, @parallel, @hide_communication, @parallel_indices, @parallel_async, @synchronize, @zeros, @ones, @rand
export @gridDim, @blockIdx, @blockDim, @threadIdx, @sync_threads, @sharedMem, @pk_show, @pk_println
export PKNumber

end # Module ParallelKernel


text:
# Note that this script can accept some limited command-line arguments, run
# `julia build_tarballs.jl --help` to see a usage message.
using BinaryBuilder

name = "LibPQ"
version = v"12.0.0+0"

# Collection of sources required to build LibPQ
sources = [
    "https://ftp.postgresql.org/pub/source/v12.0/postgresql-12.0.tar.gz" =>
    "15c7f267b476d764c79401d7f61f39c76222314951f77e6893a5854db26b6616",

    "https://data.iana.org/time-zones/releases/tzdata2019c.tar.gz" =>
    "79c7806dab09072308da0e3d22c37d3b245015a591891ea147d3b133b60ffc7c",

    "https://data.iana.org/time-zones/releases/tzcode2019c.tar.gz" =>
    "f6ebd3668e02d5ed223d3b7b1947561bf2d2da2f4bd1db61efefd9e06c167ed4",

]

# Bash recipe for building across all platforms
script = raw"""
cd $WORKSPACE/srcdir
make CC=$BUILD_CC
export ZIC=$WORKSPACE/srcdir/zic
cd postgresql-12.0/
./configure --prefix=$prefix --host=$target --with-includes=$prefix/include --with-libraries=$prefix/lib --without-readline --without-zlib --with-openssl
make -C src/interfaces/libpq install
rm -rf ${prefix}/logs
exit

"""

# These are the platforms we will build for by default, unless further
# platforms are passed in on the command line
platforms = [
    Linux(:i686, libc=:glibc),
    Linux(:x86_64, libc=:glibc),
    Linux(:aarch64, libc=:glibc),
    Linux(:armv7l, libc=:glibc, call_abi=:eabihf),
    Linux(:powerpc64le, libc=:glibc),
    Linux(:i686, libc=:musl),
    Linux(:x86_64, libc=:musl),
    Linux(:aarch64, libc=:musl),
    Linux(:armv7l, libc=:musl, call_abi=:eabihf),
    MacOS(:x86_64),
    FreeBSD(:x86_64),
    Windows(:i686),
    Windows(:x86_64)
]

# The products that we will ensure are always built
products(prefix) = [
    LibraryProduct(prefix, "libpq", :LIBPQ_HANDLE)
]

# Dependencies that must be installed before this package can be built
dependencies = [
    "https://gist.githubusercontent.com/giordano/9b876f8ae84421f204f803e7dc16eaf1/raw/b389179bca8062c96b6131a9be5c7ef269a1678d/build_OpenSSL.v1.1.1+c.jl"
]

# Build the tarballs, and possibly a `build.jl` as well.
build_tarballs(ARGS, name, version, sources, script, platforms, products, dependencies)


text:
"""
    eigenvalues_Roe(u,c1,c2,λ)

The eigenvalues of the Roe matrix with a 1D velocity `u` are c₂u, c₂u + √Δ and c₂u - √Δ
with the discriminant Δ = λc₁ - c₂(c₁-c₂)u.
"""
function eigenvalues_Roe(u,c1,c2,λ)
    Δ = (c2^2 - c1*c2) * u^2 + λ*c1
    eval_0 = c2*u
    eval_p = c2*u + sqrt(Δ)
    eval_m = c2*u - sqrt(Δ)
    return eval_p, eval_m, eval_0
end

"""
    abs_Roe_matrix(ρl,ρr,ul,ur,vl,vr,c1,c2,λ)

Return the absolute value of the Roe matrix for the 1D Riemannn problem with respective left and right densities
`(ρl,ul,vl)` and `(ρr,ur,vr)`. The eigenvalues are increased to avoid sonic rarefaction wave (LLF method).
"""
function abs_Roe_matrix(ρl,ρr,ul,ur,vl,vr,c1,c2,λ)

    #### Roe average
    um = (sqrt(ρl)*ul + sqrt(ρr)*ur) / (sqrt(ρl) + sqrt(ρr))
    vm = (sqrt(ρl)*vl + sqrt(ρr)*vr) / (sqrt(ρl) + sqrt(ρr))

    #### Eigenvalues and eigenvectors
    eval_p, eval_m, eval_0 = eigenvalues_Roe(um,c1,c1,λ)
    z_p = (c1*c2*um*vm - c2*vm*eval_p) / (c2*um - eval_p)
    z_m = (c1*c2*um*vm - c2*vm*eval_m) / (c2*um - eval_m)
    detP = c1 * (eval_m - eval_p)

    #### Increase the eigenvalues
    eval_l = eigenvalues_Roe(ul,c1,c2,λ)
    eval_r = eigenvalues_Roe(ur,c1,c2,λ)
    eval_p_fix = max(abs(eval_l[1]),abs(eval_r[1]))
    eval_m_fix = max(abs(eval_l[2]),abs(eval_r[2]))
    eval_0_fix = max(abs(eval_l[3]),abs(eval_r[3]))

    #### Compute Roe matrix A = P*D*P^{-1}
    a11 = c1*(eval_p_fix*eval_m - eval_m_fix*eval_p)/detP
    a12 = c1^2*(eval_m_fix - eval_p_fix)/detP
    a13 = 0.
    a21 = eval_m*eval_p*(eval_p_fix - eval_m_fix)/detP
    a22 = c1*(eval_m_fix*eval_m - eval_p_fix*eval_p)/detP
    a23 = 0.
    a31 = (-eval_m_fix*eval_p*z_m + eval_p_fix*eval_m*z_p + eval_0_fix*(eval_p*z_m - eval_m*z_p))/detP
    a32 = c1*(eval_m_fix*z_m - eval_p_fix*z_p + eval_0_fix*(z_p - z_m))/detP
    a33 = eval_0_fix
    return a11,a12,a13,a21,a22,a23,a31,a32,a33
end

"""
    flux_x(ρl,ρr,ul,ur,vl,vr,c1,c2,λ,method)

Return the flux along the x-axis for the 1D Riemannian problem given by the respective left and right
densities `(ρl,ul,vl)` and `(ρr,ur,vr)`. The chosen method `method` can be either `"Roe"` or `"HLLE"`.
"""
function flux_x(ρl,ρr,ul,ur,vl,vr,c1,c2,λ,method)
    if ρl<1e-9 && ρr<1e-9
        return 0.,0.,0.
    end
    if method == "Roe"
        a11,a12,a13,a21,a22,a23,a31,a32,a33 = abs_Roe_matrix(ρl,ρr,ul,ur,vl,vr,c1,c2,λ)
        avg = 0.5 .* (c1*ρl*ul + c1*ρr*ur,
                    c2*(ρl*ul^2 + ρr*ur^2) + λ*(ρl + ρr),
                    c2*(ρl*ul*vl + ρr*ur*vr))
        Roe_term = 0.5 .* (a11*(ρr - ρl) + a12*(ρr*ur - ρl*ul) + a13*(ρr*vr - ρl*vl),
                        a21*(ρr - ρl) + a22*(ρr*ur - ρl*ul) + a23*(ρr*vr - ρl*vl),
                        a31*(ρr - ρl) + a32*(ρr*ur - ρl*ul) + a33*(ρr*vr - ρl*vl))
        return avg .- Roe_term
    elseif method == "HLLE"
        um = (sqrt(ρl)*ul + sqrt(ρr)*ur) / (sqrt(ρl) + sqrt(ρr))
        vp_l = eigenvalues_Roe(ul,c1,c2,λ)
        vp_r = eigenvalues_Roe(ur,c1,c2,λ)
        vp_Roe = eigenvalues_Roe(um,c1,c2,λ)
        s_l = minimum(min(vp_l,vp_Roe))
        s_r = maximum(max(vp_r,vp_Roe))
        s_lm = min(s_l,0.)
        s_rp = max(s_r,0.)

        f_l = (c1*ρl*ul, c2*ρl*ul^2 + λ*ρl, c2*ρl*ul*vl)
        f_r = (c1*ρr*ur, c2*ρr*ur^2 + λ*ρr, c2*ρr*ur*vr)
        U_l = (ρl,ρl*ul,ρl*vl)
        U_r = (ρr,ρr*ur,ρr*vr)
        return ((s_rp.*f_l .- s_lm.*f_r) .+ (s_rp*s_lm) .* (U_r .- U_l)) ./ (s_rp - s_lm)
    else
        error("Method not defined!")
    end

end


text:
#deprecated example of creating custom types

using Equations
import Equations: simplify, *, dot #extending * breaks functionality...

mutable struct Exp<:Component #these are already implemented
	x
end
simplify(c::Exp)=isa(simplify(c.x),Number) ? exp(simplify(c.x)) : c
mutable struct Cos<:Component
	x
end
simplify(c::Cos)=isa(simplify(c.x),Number) ? cos(simplify(c.x)) : c
mutable struct Sin<:Component
	x
end
simplify(c::Sin)=isa(simplify(c.x),Number) ? sin(simplify(c.x)) : c
e1=Vec([1,0,0]);e2=Vec([0,1,0]);e3=Vec([0,0,1])

mutable struct Differentiable <: Component
	fun
	der::Dict
end
#getargs(d::Differentiable)=(d.fun,)
#Equations.maketype(d::Differentiable,fun)=Differentiable(fun(d.fun),d.der)
Differentiable(a,b)=Differentiable(a,[:x=>b])
#ed=Vec(Differentiable(1,0),Differentiable(0,1),0) #this causes infinite recursion...
mutable struct DerOp #<: Operator
	dx
end
DerOp()=DerOp(:x)
*(d::DerOp,f::Differentiable)=haskey(f.der,d.dx) ? f.der[d.dx] : Der(f,d.dx)
*(d::DerOp,ex::Factor)=Der(ex,d.dx)
mutable struct Grad <: Operator
	v
end
Grad()=Grad(Vec([DerOp(:x),DerOp(:y),DerOp(:z)]))
Base.print(io::IO,g::Grad)=print(io,"∇")
function dot(v1::Grad,v2::Differentiable)
	ex=Expression(Term[Factor[(v1.v[1]*v2)[1]]])
	for t in 2:length(v1.v)
		push!(ex,Factor[(v1.v[t]*v2)[t]])
	end
	return simplify(ex)
end

eq1=Cross(:j,:B)≖Grad()*:p
eq2=Cross(Grad(),:B)≖:μ0*:j
eq2p=Dot(Grad(),Cross(Grad(),:B))≖:μ0*Dot(Grad(),:j)
eq3=Dot(Grad(),:B)≖0

eqc=eq1≖0


text:
maximum(f, x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS, TimeType} = threads ? hp_maximum(f, x) : stat_maximum(f, x)
maximum(f, x) = Base.maximum(f, x)
maximum(x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS, TimeType}  = threads ? hp_maximum(identity, x) : stat_maximum(identity, x)
maximum(x) = Base.maximum(x)

minimum(f, x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS, TimeType} = threads ? hp_minimum(f, x) : stat_minimum(f, x)
minimum(f, x) = Base.minimum(f, x)
minimum(x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS, TimeType}= threads ? hp_minimum(identity, x) : stat_minimum(identity, x)
minimum(x) = Base.minimum(x)
# not optimised for simd
maximum(f, x::AbstractVector{Union{Missing, T}}) where T <: AbstractString = mapreduce(f, _stat_max_fun, x)
minimum(f, x::AbstractVector{Union{Missing, T}}) where T <: AbstractString = mapreduce(f, _stat_min_fun, x)
maximum(x::AbstractVector{Union{Missing, T}}) where T <: AbstractString = maximum(identity, x)
minimum(x::AbstractVector{Union{Missing, T}}) where T <: AbstractString = minimum(identity, x)

sum(f, x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS} = threads ? hp_sum(f, x) : stat_sum(f, x)
sum(f, x)=Base.sum(f, x)
sum(x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS} = threads ? hp_sum(identity, x) : stat_sum(identity, x)
sum(x) = Base.sum(x)

Statistics.mean(f, x::AbstractArray{Union{T,Missing},1}) where T <: Union{INTEGERS, FLOATS} = stat_mean(f, x)
Statistics.mean(x::AbstractArray{Union{T,Missing},1}) where T <: Union{INTEGERS, FLOATS} = stat_mean(x)

wsum(f, x::AbstractVector, w::AbstractVector) = stat_wsum(f, x, w)
wsum(x::AbstractVector, w::AbstractVector) = stat_wsum(identity, x, w)

wmean(f, x::AbstractVector, w::AbstractVector) = stat_wmean(f, x, w)
wmean(x::AbstractVector, w::AbstractVector) = stat_wmean(identity, x, w)

Statistics.var(f, x::AbstractArray{Union{T,Missing},1}, dof = true) where T <: Union{INTEGERS, FLOATS}= stat_var(f, x, dof)
Statistics.var(x::AbstractArray{Union{T,Missing},1}, dof = true) where T <: Union{INTEGERS, FLOATS}= stat_var(x, dof)

std(f, x::AbstractArray{Union{T,Missing},1}, dof = true) where T <: Union{INTEGERS, FLOATS}= stat_std(f, x, dof)
std(x::AbstractArray{Union{T,Missing},1}, dof = true) where T <: Union{INTEGERS, FLOATS}= stat_std(x, dof)

Statistics.median(x::AbstractArray{Union{T,Missing},1}) where T = stat_median(x)
Statistics.median!(x::AbstractArray{Union{T,Missing},1}) where T = stat_median!(x)

extrema(f, x::AbstractArray{Union{Missing, T},1}; threads = false) where T <: Union{INTEGERS, FLOATS, TimeType} = threads ? (hp_minimum(f, x), hp_maximum(f, x)) : (stat_minimum(f, x), stat_maximum(f, x))
extrema(f, x) = Base.extrema(f, x)
extrema(x::AbstractArray{Union{Missing, T},1}; threads = false) where T  <: Union{INTEGERS, FLOATS, TimeType}= threads ? (hp_minimum(identity, x), hp_maximum(identity, x)) : (stat_minimum(identity, x), stat_maximum(identity, x))
extrema(x) = Base.extrema(x)

# when by is a function the following functions find argmax/min(by.(x))
argmax(x::AbstractArray{Union{Missing, T},1}; by = identity) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmax(by, x)[2]
argmax(x) = Base.argmax(x)
argmin(x::AbstractArray{Union{Missing, T},1}; by = identity) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmin(by, x)[2]
argmin(x) = Base.argmin(x)

findmax(f, x::AbstractArray{Union{Missing, T},1}) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmax(f, x)
findmax(f, x) = Base.findmax(f, x)
findmax(x::AbstractArray{Union{Missing, T},1}) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmax(x)
findmax(x) = Base.findmax(x)
findmin(f, x::AbstractArray{Union{Missing, T},1}) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmin(f, x)
findmin(f, x) = Base.findmin(f, x)
findmin(x::AbstractArray{Union{Missing, T},1}) where T <: Union{INTEGERS, FLOATS, TimeType, AbstractString} = stat_findmin(x)
findmin(x) = Base.findmin(x)

function cumsum(x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS}
    if missings == :ignore
        stat_cumsum_ignore(x)
    elseif missings == :skip
        stat_cumsum_skip(x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
cumsum(x) = Base.cumsum(x)
function cumsum!(outx::AbstractVector, x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS}
    if missings == :ignore
        stat_cumsum!_ignore(outx, x)
    elseif missings == :skip
        stat_cumsum!_skip(outx, x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
cumsum!(x,y) = Base.cumsum!(x,y)
function cumprod(x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS}
    if missings == :ignore
        stat_cumprod_ignore(x)
    elseif missings == :skip
        stat_cumprod_skip(x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
cumprod(x)=Base.cumprod(x)
function cumprod!(outx::AbstractVector, x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS}
    if missings == :ignore
        stat_cumprod!_ignore(outx, x)
    elseif missings == :skip
        stat_cumprod!_skip(outx, x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
cumprod!(x,y) = Base.cumprod!(x,y)
function cummin(x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS, TimeType}
    if missings == :ignore
        stat_cummin_ignore(x)
    elseif missings == :skip
        stat_cummin_skip(x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
function cummin!(outx, x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS, TimeType}
    if missings == :ignore
        stat_cummin!_ignore(outx, x)
    elseif missings == :skip
        stat_cummin!_skip(outx, x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
function cummax(x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS, TimeType}
    if missings == :ignore
        stat_cummax_ignore(x)
    elseif missings == :skip
        stat_cummax_skip(x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end
function cummax!(outx, x::AbstractArray{Union{Missing, T},1}; missings = :ignore) where T <: Union{INTEGERS, FLOATS, TimeType}
    if missings == :ignore
        stat_cummax!_ignore(outx, x)
    elseif missings == :skip
        stat_cummax!_skip(outx, x)
    else
        throw(ArgumentError("`missings` must be either `:ignore` or `:skip`"))
    end
end


text:
#=
---------------------------------------------------------------------
This function computes the data structures required for a CG or DG method.
Examples include: the coord, intma, and periodicity arrays

Written by F.X. Giraldo on 7/14/2021
           Department of Applied Mathematics
           Naval Postgraduate School
           Monterey; CA 93943-5216
---------------------------------------------------------------------
=#

function create_CGDG_Storage(space_method,Npoin_CG,Npoin_DG,coord_CG,intma_CG,periodicity_CG,Ne,Np,DFloat)

    if (space_method == "CG")
        Npoin=Npoin_CG
    elseif (space_method == "DG")
        Npoin=Npoin_DG
    end
    DG_to_CG=zeros(Int64,Npoin)

    #Initialize Global Arrays
    coord=zeros(DFloat,2,Npoin)
    intma=zeros(Int64,Np,Np,Ne)
    periodicity=zeros(Int64,Npoin,1)

    if (space_method == "CG")
        coord=copy(coord_CG)
        intma=copy(intma_CG)
        periodicity = copy(periodicity_CG)
        for i=1:Npoin_CG
            DG_to_CG[i]=i
        end

    elseif (space_method == "DG")
        Npoin=Npoin_DG
        i_DG=0

        #Get INTMA
        for e=1:Ne
            for j=1:Np, i=1:Np
                i_DG=i_DG + 1
                i_CG=intma_CG[i,j,e]
                intma[i,j,e]=i_DG
                DG_to_CG[i_DG]=i_CG
            end
        end

        if (i_DG != Npoin_DG)
            println("ERROR in CREATE_CGDG_STORAGE: i_DG != Npoin_DG = ",i_DG,", ",Npoin_DG)
            exit
        end

        #Get COORD
        for i_DG=1:Npoin_DG
            i_CG=DG_to_CG[i_DG]
            coord[1,i_DG]=coord_CG[1,i_CG]
            coord[2,i_DG]=coord_CG[2,i_CG]
        end

        #Get periodicity
        for i_DG=1:Npoin_DG
            periodicity[i_DG]=i_DG
        end

    end #if

    return(coord,intma,periodicity,DG_to_CG,Npoin)
end