### 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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using BasisFunctions
using MoreBasisFunctions
using Test

include("vandermonde_tests.jl")
include("bernstein_tests.jl")
include("lagrange_tests.jl")


using DiffEqBayes, OrdinaryDiffEq, ParameterizedFunctions, RecursiveArrayTools
using Stan, Mamba, Test

ProjDir = dirname(@__FILE__)

#=
Experimental link to DiffEqBayes examples

Possible issues to investigate:

1. Sampling results vary too much. chains often get stuck
2. @ode_def_nohes needs to be kept outside local scope block
3. Minimize string replacements in Stan language model to prevent recompilations

=#

f1 = @ode_def_nohes LotkaVolterraTest1 begin
  dx = a*x - b*x*y
  dy = -c*y + d*x*y
end a=>1.5 b=1.0 c=3.0 d=1.0

cd(ProjDir) do

  isdir("tmp") && rm("tmp", recursive=true)
  isfile("LotkaVolterra1-1.pdf") &&
    rm("LotkaVolterra1-1.pdf")

    println("One parameter case")
    u0 = [1.0,1.0]
    tspan = (0.0,10.0)
    prob1 = ODEProblem(f1,u0,tspan)
    sol = solve(prob1,Tsit5())
    t = collect(range(1, stop=10, length=10))
    randomized = VectorOfArray([(sol(t[i]) + .01randn(2)) for i in 1:length(t)])
    data = convert(Array,randomized)
    priors = [Truncated(Normal(1.5,0.1),0,2)]

    bayesian_result = stan_inference(prob1,t,data,priors;num_samples=300,
      num_warmup=500,likelihood=Normal,vars =("StanODEData",InverseGamma(3,2)))

    global sim1 = bayesian_result.chain_results[1:end,["theta.1", "sigma1.1", "sigma1.2"], :]

  ## Plotting
  p = plot(sim1, [:trace, :mean, :density, :autocor], legend=true);
  draw(p, ncol=4, filename="LotkaVolterra1", fmt=:pdf)

  theta1 = bayesian_result.chain_results[:,["theta.1"],:]
  @test mean(theta1.value) ≈ 1.5 atol=3e-1
end


# Motion parameter utilities

export derivative1d_linop, derivative1d_motionpars_linop, interpolation1d_linop, interpolation1d_motionpars_linop

function derivative1d_linop(t::AbstractVector{T}, order::Integer) where {T<:Real}
    (order != 1) && (order != 2) && error("Order not supported (only 1 or 2)")
    nt = length(t)
    Δt = t[2:end]-t[1:end-1]
    if order == 1
        d_0 = [-T(1)./Δt; T(0)]
        d_p1 = T(1)./Δt;
        return spdiagm(nt, nt, 0 => d_0, 1 => d_p1)
    elseif order == 2
        Δt_m1 = Δt[1:end-1]
        Δt_p1 = Δt[2:end]
        Δt_mean = (Δt_m1+Δt_p1)/T(2)
        d_m1 = T(1)./(Δt_m1.*Δt_mean)
        d_p1 = [T(0); T(1)./(Δt_p1.*Δt_mean)]
        d_0  = [T(0); -d_m1-d_p1[2:end]]
        return spdiagm(nt, nt, -1 => d_m1, 0 => d_0, 1 => d_p1)
    end
end

function derivative1d_motionpars_linop(t::AbstractVector{T}, order::Integer; pars::NTuple{6,Bool}=(true, true, true, true, true, true)) where {T<:Real}
    D = derivative1d_linop(t, order)
    Id = sparse(I, length(t), length(t))
    A = []
    for i = 1:6
        pars[i] ? push!(A, D) : push!(A, Id)
    end
    return cat(A...; dims=(1,2))
end

function interpolation1d_linop(t::AbstractVector{T}, ti::AbstractVector{T}; interp::Symbol=:linear) where {T<:Real}
    (interp != :linear) && error("Only linear interpolation supported")
    nt = length(t)
    nti = length(ti)
    I = repeat(reshape(1:nti, :, 1); outer=(1,2))
    J = Array{Int64,2}(undef, nti, 2)
    V = Array{T,2}(undef, nti, 2)
    for i = 1:nti
        if     ti[i] < t[1]
            J[i, :] .= [1; 1]
            V[i, :] .= [T(0); T(0)]
        elseif ti[i] > t[end]
            J[i, :] .= [1; 1]
            V[i, :] .= [T(0); T(0)]
        elseif ti[i] == t[1]
            J[i, :] .= [1; 2]
            V[i, :] .= [T(1); T(0)]
        elseif ti[i] == t[end]
            J[i, :] .= [nt-1; nt]
            V[i, :] .= [T(0); T(1)]
        else
            idx_ = findall(t .< ti[i])
            if length(idx_) != 0
                idx = maximum(idx_)
                J[i,:] .= [idx; idx+1]
                Δt = t[idx+1]-t[idx]
                V[i,:] .= [(t[idx+1]-ti[i])/Δt; (ti[i]-t[idx])/Δt]
            else
                J[i, :] .= [1; 1]
                V[i, :] .= [T(0); T(0)]
            end
        end
    end
    return sparse(vec(I), vec(J), vec(V), nti, nt)
end

function interpolation1d_motionpars_linop(t::NTuple{6,AbstractVector{T}}, ti::NTuple{6,AbstractVector{T}}; interp::Symbol=:linear) where {T<:Real}
    Ip = []
    for i=1:6
        push!(Ip, interpolation1d_linop(t[i], ti[i]; interp=interp))
    end
    return cat(Ip...; dims=(1,2))
end

interpolation1d_motionpars_linop(t::AbstractVector{T}, ti::NTuple{6,AbstractVector{T}}; interp::Symbol=:linear) where {T<:Real} = interpolation1d_motionpars_linop((t,t,t,t,t,t), ti; interp=interp)

interpolation1d_motionpars_linop(t::NTuple{6,AbstractVector{T}}, ti::AbstractVector{T}; interp::Symbol=:linear) where {T<:Real} = interpolation1d_motionpars_linop(t, (ti,ti,ti,ti,ti,ti); interp=interp)

interpolation1d_motionpars_linop(t::AbstractVector{T}, ti::AbstractVector{T}; interp::Symbol=:linear) where {T<:Real} = interpolation1d_motionpars_linop((t,t,t,t,t,t), (ti,ti,ti,ti,ti,ti); interp=interp)


# by lazarusA # HIDE
#using GLMakie, Random # HIDE
using GLMakie
GLMakie.activate!() #HIDE
let
    x = LinRange(-2, 0.5, 501)
    y = LinRange(-2, 2, 501)
    # objective function
    z = 100 .* (y' .- x .^ 2) .^ 2 .+ (1 .- x) .^ 2
    # build constraints
    zin = copy(z)
    zout1 = copy(z)
    zout2 = copy(z)
    for (i,x) in enumerate(x), (j,y) in enumerate(y)
        if x*y >=1
            zin[i,j] = NaN
            zout1[i,j] = z[i,j]
            zout2[i,j] = NaN
        elseif (x + y^2) >=1
            zin[i,j] = NaN
            zout1[i,j] = NaN
            zout2[i,j] = z[i,j]
        else
            zin[i,j] = z[i,j]
            zout1[i,j] = NaN
            zout2[i,j] = NaN
        end
    end
    fig = Figure(resolution = (1200,900), fontsize = 20)
    ax = Axis3(fig, aspect = (1,1,1), perspectiveness = 0.5, elevation = π/9,
        azimuth = 0.2π, zgridcolor = :grey,ygridcolor=:grey,xgridcolor = :grey)
    pltobj = surface!(ax, x, y, zin, colormap = Reverse(:inferno))
    surface!(ax, x, y, zout1, colormap = (:orangered,:red))
    surface!(ax, x, y, zout2, colormap = (:dodgerblue,:blue))
    cbar = Colorbar(fig, pltobj, label = "xy <=1 & x + y^2<=1",
        height = Relative(0.5), width = 20 )
    fig[1,1] = ax
    fig[1,2] = cbar
    fig
    save(joinpath(@__DIR__, "output", "surfaceConstraints.png"), fig, px_per_unit = 2.0) # HIDE
end
using Pkg # HIDE
Pkg.status(["GLMakie"]) # HIDE


to_discrete_position(pos, pathfinder) =
    floor.(Int, pos ./ pathfinder.dims .* size(pathfinder.walkmap)) .+ 1
to_continuous_position(pos, pathfinder) =
    pos ./ size(pathfinder.walkmap) .* pathfinder.dims .-
    pathfinder.dims ./ size(pathfinder.walkmap) ./ 2.
sqr_distance(from, to, pathfinder::AStar{D,true}) where {D} =
    sum(min.(abs.(from .- to), pathfinder.dims .- abs.(from .- to)) .^ 2)
sqr_distance(from, to, pathfinder::AStar{D,false}) where {D} =
    sum((from .- to) .^ 2)

"""
    find_continuous_path(pathfinder, from, to)
Functions like `find_path`, but uses the output of `find_path` and converts it to the coordinate
space used by the corresponding `ContinuousSpace`. Performs checks on the last two waypoints
in the discrete path to ensure continuous path is optimal.
"""
function find_continuous_path(
    pathfinder::AStar{D},
    from::NTuple{D,Float64},
    to::NTuple{D,Float64},
) where {D}
    discrete_from = to_discrete_position(from, pathfinder)
    discrete_to = to_discrete_position(to, pathfinder)
    discrete_path = find_path(pathfinder, discrete_from, discrete_to)
    # find_path returns nothing if no path exists
    isnothing(discrete_path) && return
    # if discrete_path is empty, `from` and `to` are in the same grid cell,
    # so `to` is the only waypoint
    isempty(discrete_path) && return Path{D,Float64}(to)

    cts_path = Path{D,Float64}()
    for pos in discrete_path
        push!(cts_path, to_continuous_position(pos, pathfinder))
    end

    # Handles an edge case to prevent backtracking for a fraction of a grid cell
    # Consider case where each grid cell is (1., 1.), and the path is to be calculated from
    # (0.5, 0.5) to (0.5, 1.2). Without this, the resultant path would look like
    # [(0.5, 0.5), (0.5, 1.5), (0.5, 1.2)], causing the agent to go to the last waypoint and
    # then backtrack to the target
    last_pos = last(cts_path)
    pop!(cts_path)
    # It's possible there's only one waypoint in the path, in which case the second last
    # position is the starting position
    second_last_pos = isempty(cts_path) ? from : last(cts_path)
    last_to_end = sqr_distance(last_pos, to, pathfinder)
    second_last_to_end = sqr_distance(second_last_pos, to, pathfinder)
    if last_to_end < second_last_to_end
        push!(cts_path, last_pos)
    end
    # If `to` is already at the center of a grid cell, there's no need
    # to push it to the path
    last_to_end ≈ 0. || push!(cts_path, to)
    return cts_path
end

function Agents.plan_route!(
    agent::A,
    dest::NTuple{D,Float64},
    pathfinder::AStar{D,P,M,Float64},
) where {A<:AbstractAgent,D,P,M}
    path = find_continuous_path(pathfinder, agent.pos, dest)
    isnothing(path) && return
    pathfinder.agent_paths[agent.id] = path
end

function Agents.plan_best_route!(
    agent::A,
    dests,
    pathfinder::AStar{D,P,M,Float64};
    condition::Symbol = :shortest,
) where {A<:AbstractAgent,D,P,M}
    @assert condition ∈ (:shortest, :longest)
    compare = condition == :shortest ? (a, b) -> a < b : (a, b) -> a > b
    best_path = Path{D,Float64}()
    best_target = nothing
    for target in dests
        path = find_continuous_path(pathfinder, agent.pos, target)
        isnothing(path) && continue
        if isempty(best_path) || compare(length(path), length(best_path))
            best_path = path
            best_target = target
        end
    end

    isnothing(best_target) && return
    pathfinder.agent_paths[agent.id] = best_path
    return best_target
end

"""
    move_along_route!(agent, model::ABM{<:ContinuousSpace{D}}, pathfinder::AStar{D}, speed, dt = 1.0)
Move `agent` for one step along the route toward its target set by
[`plan_route!`](@ref) at the given `speed` and timestep `dt`.

For pathfinding in models with [`ContinuousSpace`](@ref)

If the agent does not have a precalculated path or the path is empty, it remains stationary.
"""
function Agents.move_along_route!(
    agent::A,
    model::ABM{<:ContinuousSpace{D},A},
    pathfinder::AStar{D},
    speed::Float64,
    dt::Real = 1.0,
) where {D,A<:AbstractAgent}
    isempty(agent.id, pathfinder) && return
    from = agent.pos
    next_pos = agent.pos
    while true
        next_waypoint = first(pathfinder.agent_paths[agent.id])
        dir = get_direction(from, next_waypoint, model)
        dist_to_target = norm(dir)
        # edge case
        if dist_to_target ≈ 0.
            from = next_waypoint
            popfirst!(pathfinder.agent_paths[agent.id])
            if isempty(agent.id, pathfinder)
                next_pos = next_waypoint
                break
            end
            continue
        end
        dir = dir ./ dist_to_target
        next_pos = from .+ dir .* (speed * dt)
        # overshooting means we reached the waypoint
        dist_to_next = edistance(from, next_pos, model)
        if dist_to_next > dist_to_target
            # change from and dt so it appears we're coming from the waypoint just skipped, instead
            # of directly where the agent was. E.g:
            # agent.pos = (0, 0)
            # pathfinder.agent_paths[agent.id] = [(1, 0), (1, 1)]
            # speed = 1
            # dt = 1.2
            # without this, agent would end up at (0.85, 0.85) instead of (1, 0.2)
            from = next_waypoint
            dt -= dist_to_target / speed
            popfirst!(pathfinder.agent_paths[agent.id])
            # if the path is now empty, go directly to the end
            if isempty(agent.id, pathfinder)
                next_pos = next_waypoint
                break
            end
        else
            break
        end
    end

    move_agent!(agent, next_pos, model)
end

function random_walkable(model::ABM{<:ContinuousSpace{D}}, pathfinder::AStar{D}) where {D}
    discrete_pos = Tuple(rand(
        model.rng,
        filter(x -> pathfinder.walkmap[x], CartesianIndices(pathfinder.walkmap))
    ))
    half_cell_size = model.space.extent ./ size(pathfinder.walkmap) ./ 2.
    return to_continuous_position(discrete_pos, pathfinder) .+
        Tuple(rand(model.rng, D) .- 0.5) .* half_cell_size
end

walkable_cells_in_radius(pos, r, pathfinder::AStar{D,false}) where {D} =
    Iterators.filter(
        x -> all(1 .<= x .<= size(pathfinder.walkmap)) &&
            pathfinder.walkmap[x...] &&
            sum(((x .- pos) ./ r) .^ 2) <= 1,
        Iterators.product([(pos[i]-r[i]):(pos[i]+r[i]) for i in 1:D]...)
    )

walkable_cells_in_radius(pos, r, pathfinder::AStar{D,true}) where {D} =
    Iterators.map(
        x -> mod1.(x, size(pathfinder.walkmap)),
            Iterators.filter(
                x -> pathfinder.walkmap[mod1.(x, size(pathfinder.walkmap))...] && sum(((x .- pos) ./ r) .^ 2) <= 1,
                Iterators.product([(pos[i]-r[i]):(pos[i]+r[i]) for i in 1:D]...)
            )
    )

"""
    Pathfinding.random_walkable(pos, model::ABM{<:ContinuousSpace{D}}, pathfinder::AStar{D}, r = 1.0)
Return a random position within radius `r` of `pos` which is walkable, as specified by `pathfinder`.
Return `pos` if no such position exists.
"""
function random_walkable(
    pos,
    model::ABM{<:ContinuousSpace{D}},
    pathfinder::AStar{D},
    r = 1.0,
) where {D}
    discrete_r = to_discrete_position(r, pathfinder) .- 1
    discrete_pos = to_discrete_position(pos, pathfinder)
    options = collect(walkable_cells_in_radius(discrete_pos, discrete_r, pathfinder))
    isempty(options) && return pos
    discrete_rand = rand(
        model.rng,
        options
    )
    half_cell_size = model.space.extent ./ size(pathfinder.walkmap) ./ 2.
    cts_rand = to_continuous_position(discrete_rand, pathfinder) .+
        Tuple(rand(model.rng, D) .- 0.5) .* half_cell_size
    dist = edistance(pos, cts_rand, model)
    dist > r && (cts_rand = mod1.(
        pos .+ get_direction(pos, cts_rand, model) ./ dist .* r,
        model.space.extent
    ))
    return cts_rand
end


struct Linear <: Degree{1} end  # boundary conditions not supported

"""
    Linear()

Indicate that the corresponding axis should use linear interpolation.

# Extended help

Assuming uniform knots with spacing 1, the `i`th piece of linear b-spline
implemented here is defined as follows.

    y_i(x) = c p(x) + cp p(1-x)

where

    p(δx) = x

and the values `cX` for `X ∈ {_, p}` are the coefficients.

Linear b-splines are naturally interpolating, and require no prefiltering;
there is therefore no need for boundary conditions to be provided.

Also, although the implementation is slightly different in order to re-use
the framework built for general b-splines, the resulting interpolant is just
a piecewise linear function connecting each pair of neighboring data points.
"""
Linear

function positions(::Linear, ax::AbstractUnitRange{<:Integer}, x)
    f = floor(x)
    # When x == last(ax) we want to use the x-1, x pair
    f = ifelse(x == last(ax), f - oneunit(f), f)
    fi = fast_trunc(Int, f)
    return fi, x-f
end

value_weights(::Linear, δx) = (1-δx, δx)
gradient_weights(::Linear, δx) = (-oneunit(δx), oneunit(δx))
hessian_weights(::Linear, δx) = (zero(δx), zero(δx))

padded_axis(ax::AbstractUnitRange, ::BSpline{Linear}) = ax


##############################################################################
#
# REFERENCES: Wasserman, "All of Statistics"
#
# NOTES: Fails CDF/Quantile matching test
#
##############################################################################

immutable Bernoulli <: DiscreteUnivariateDistribution
    p0::Float64
    p1::Float64

    function Bernoulli(p::Real)
        zero(p) <= p <= one(p) || error("prob must be in [0,1]")
        new(1.0 - p, float64(p))
    end
end

Bernoulli() = Bernoulli(0.5)

min(d::Bernoulli) = 0
max(d::Bernoulli) = 1

cdf(d::Bernoulli, q::Real) = q >= zero(q) ? (q >= one(q) ? 1.0 : d.p0) : 0.

function entropy(d::Bernoulli)
    p0 = d.p0
    p1 = d.p1
    p0 == 0. || p0 == 1. ? 0. : -(p0 * log(p0) + p1 * log(p1))
end

insupport(::Bernoulli, x::Real) = (x == zero(x)) || (x == one(x))
insupport(::Type{Bernoulli}, x::Real) = (x == zero(x)) || (x == one(x))

mean(d::Bernoulli) = d.p1

var(d::Bernoulli) = d.p0 * d.p1

skewness(d::Bernoulli) = (d.p0 - d.p1) / sqrt(d.p0 * d.p1)

kurtosis(d::Bernoulli) = 1.0 / (d.p0 * d.p1) - 6.0

median(d::Bernoulli) = d.p1 < 0.5 ? 0.0 : 1.0

mgf(d::Bernoulli, t::Real) = d.p0 + d.p1 * exp(t)

cf(d::Bernoulli, t::Real) = d.p0 + d.p1 * exp(im * t)

mode(d::Bernoulli) = d.p1 > 0.5 ? 1 : 0

function modes(d::Bernoulli)
    d.p1 < 0.5 ? [0] :
    d.p1 > 0.5 ? [1] : [0, 1]
end

pdf(d::Bernoulli, x::Real) = x == zero(x) ? d.p0 : x == one(x) ? d.p1 : 0.0

quantile(d::Bernoulli, p::Real) = zero(p) <= p <= one(p) ? (p <= d.p0 ? 0 : 1) : NaN

rand(d::Bernoulli) = rand() > d.p1 ? 0 : 1

## MLE fitting

immutable BernoulliStats <: SufficientStats
    cnt0::Float64
    cnt1::Float64

    BernoulliStats(c0::Real, c1::Real) = new(float64(c0), float64(c1))
end

fit_mle(::Type{Bernoulli}, ss::BernoulliStats) = Bernoulli(ss.cnt1 / (ss.cnt0 + ss.cnt1))

function suffstats{T<:Integer}(::Type{Bernoulli}, x::Array{T})
    n0 = 0
    n1 = 0
    for xi in x
        if xi == 0
            n0 += 1
        elseif xi == 1
            n1 += 1
        else
            throw(DomainError())
        end
    end
    BernoulliStats(n0, n1)
end

function suffstats{T<:Integer}(::Type{Bernoulli}, x::Array{T}, w::Array{Float64})
    n = length(x)
    if length(w) != n
        throw(ArgumentError("Inconsistent argument dimensions."))
    end

    n0 = 0.0
    n1 = 0.0
    for i = 1:n
        xi = x[i]
        if xi == 0
            n0 += w[i]
        elseif xi == 1
            n1 += w[i]
        else
            throw(DomainError())
        end
    end
    BernoulliStats(n0, n1)
end


using MATLAB

function patch(Γ)

    V = vertexarray(Γ)
    F = cellarray(Γ)

    @mput V F
    mat"""
        hold('on')
        patch('Vertices', V, 'Faces', F, 'FaceColor', 'red')
    """

    mat"axis equal"
    mat"colorbar"
    mat"view(3)"

end

function patch(Γ, C)

    V = vertexarray(Γ)
    F = cellarray(Γ)

    @mput V F C
    mat"""
        hold('on')
        patch('Vertices', V, 'Faces', F, 'FaceVertexCData', C, 'FaceColor', 'flat')
    """

    mat"axis equal"
    mat"colorbar"
    mat"view(3)"
end

function jmatlab_quiver(mesh, facecurs)
    els = elements(mesh)
    cts = center.(els)
    C = [ct[i] for ct in cts, i in 1:3]
    N = [fc[i] for fc in facecurs, i in 1:3]
    @mput C N
    mat"quiver3(C(:,1),C(:,2),C(:,3), N(:,1),N(:,2),N(:,3))"
end

function jmatlab_quiver(m)
    els = [chart(m, cells(m,i)) for i in 1:numcells(m)]
    C = zeros(numcells(m),3)
    N = zeros(numcells(m),3)
    for i in 1:size(C,1)
        s = els[i]
        p = neighborhood(s, [1,1]/3)
        n = normal(p)
        c = cartesian(p)
        C[i,:] = [c[1],c[2],c[3]]
        N[i,:] = [n[1],n[2],n[3]]
    end
    @mput C N
    mat"quiver3(C(:,1),C(:,2),C(:,3), N(:,1),N(:,2),N(:,3))"
end

function jmatlab_plot(x,y)
    @mput x
    @mput y
    mat"plot(x,y)"
end

# function quiver(Γ, fcr)
#
#   D = length(eltype(fcr))
#   V = vertexarray(Γ)
#   F = cellarray(Γ)
#
#   C = [cartesian(center(chart(Γ,c))) for c in cells(Γ)]
#   C = [C[i][j] for i in eachindex(C), j in 1:3]
#
#   Q = [fcr[i][j] for i in eachindex(fcr), j in 1:3]
#
#   @show size(C)
#   @show size(Q)
#
#   @mput V F C Q
#   mat"""
#   patch('Vertices', V, 'Faces', F, 'FaceColor', 'red')
#   quiver3(C(:,1),C(:,2),C(:,3),Q(:,1),Q(:,2),Q(:,3))
#   """
# end


function coreVIfull(logl::Function, μarray::Array{Array{Float64,1},1}, Σarray::Array{Array{Float64, 2},1}; gradlogl = gradlogl, seed = 1, S = 100, optimiser=Optim.LBFGS(), iterations=1, numerical_verification = false, Stest=0, show_every=-1, inititerations=0)

    D = length(μarray[1])

    @assert(D == size(Σarray[1], 1) == size(Σarray[1], 2))

    @assert(length(μarray) == length(Σarray))

    @printf("Running VI with full covariance: seed=%d, S=%d, Stest=%d, D=%d for %d iterations\n", seed, S, Stest, D, iterations)

    #----------------------------------------------------
    # generate latent variables
    #----------------------------------------------------

    Ztrain = generatelatentZ(S = S, D = D, seed = seed)

    Ztest  = generatelatentZ(S = Stest, D = D, seed = seed+1)

    #----------------------------------------------------
    function unpack(param)
    #----------------------------------------------------

        @assert(length(param) == D+D*D)

        local μ = param[1:D]

        local C = reshape(param[D+1:D+D*D], D, D)

        return μ, C

    end

    #----------------------------------------------------
    function minauxiliary(param)
    #----------------------------------------------------

        local μ, C = unpack(param)

        return -1.0 * elbo(μ, C, Ztrain)

    end

    #----------------------------------------------------
    function minauxiliary_grad(param)
    #----------------------------------------------------

        local μ, C = unpack(param)

        return -1.0 * elbo_grad(μ, C, Ztrain)

    end

    #----------------------------------------------------
    function getcov(C)
    #----------------------------------------------------

        return C*C'

    end

    #----------------------------------------------------
    function getcovroot(C)
    #----------------------------------------------------

        return C

    end

    #----------------------------------------------------
    function elbo(μ, C, Z)
    #----------------------------------------------------

        mean(map(z -> logl(μ .+ C*z), Z)) + ApproximateVI.entropy(C)

    end

    #----------------------------------------------------
    function elbo_grad(μ, C, Z)
    #----------------------------------------------------

        local Σroot = getcovroot(C)
        local gradC = (Σroot\I)' # entropy contribution
        local gradμ = zeros(eltype(μ), D)
        local S     = length(Z)

        for s=1:S
            g = gradlogl(μ .+ Σroot*Z[s])
            gradC += (1/S)*g*Z[s]'
            gradμ += g/S
        end

        return [vec(gradμ); vec(gradC)]

    end

    gradhelper(storage, param) = copyto!(storage, minauxiliary_grad(param))

    #----------------------------------------------------
    # Numerically verify gradient
    #----------------------------------------------------

    if numerical_verification

        local C = Matrix(cholesky(Σarray[1]).L)
        adgrad = ForwardDiff.gradient(minauxiliary, [μarray[1]; vec(C)])
        angrad = minauxiliary_grad([μarray[1];vec(C)])
        @printf("gradient from AD vs analytical gradient\n")
        display([vec(adgrad) vec(angrad)])
        @printf("maximum absolute difference is %f\n", maximum(abs.(vec(adgrad) - vec(angrad))))

    end

    #----------------------------------------------------
    # Evaluate initial solutions for few iterations
    #----------------------------------------------------

    initoptimise(μ, Σ) = Optim.optimize(minauxiliary, gradhelper, [μ; vec(Matrix(cholesky(Σ).L))], optimiser, Optim.Options(iterations = inititerations))

    results = if inititerations>0
        @showprogress "Initial search with random start " map(initoptimise, μarray, Σarray)
    else
        map(initoptimise, μarray, Σarray)
    end

    bestindex = argmin(map(r -> r.minimum, results))

    bestinitialsolution = results[bestindex].minimizer

    #----------------------------------------------------
    # Call optimiser
    #----------------------------------------------------

    options = Optim.Options(extended_trace = false, store_trace = false, show_trace = show_every > 0, show_every = show_every, iterations = iterations, g_tol = 1e-6)

    result  = Optim.optimize(minauxiliary, gradhelper, bestinitialsolution, optimiser, options)

    μopt, Copt = unpack(result.minimizer)

    #----------------------------------------------------
    # Return results
    #----------------------------------------------------

    Σopt = getcov(Copt)

    return MvNormal(μopt, Σopt), elbo(μopt, Copt, generatelatentZ(S = 10*S, D = D, seed = seed+2))

end


using NonlinearEigenproblemsTest
using NonlinearEigenproblems
using Test
using LinearAlgebra

@testset "Interpolation" begin

#########################################################################################
@bench @testset "Random dep" begin
nep = nep_gallery("dep0")
λ1,x1 = newton(nep,logger=displaylevel,maxit=40, λ=-0.75, v=ones(size(nep,1)));
@test compute_resnorm(nep,λ1,x1) < eps()*100

# Newton on interpolated dep (interpolating pep of degree 2)
intpoints = [λ1-1, λ1, λ1+1]
pep = interpolate(nep, intpoints)
λ2,x2 = newton(pep,logger=displaylevel,maxit=40, λ=-0.75, v=ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*100
@test abs(λ1-λ2)/abs(λ1) < eps()*1000

# Newton on interpolated dep (interpolating pep of degree 8)
intpoints = [λ1-5, λ1-1, λ1, λ1+5, λ1+1, λ1+5im, λ1+1im, λ1-5im, λ1-1im]
pep = interpolate(nep, intpoints)
λ2,x2 = newton(pep,logger=displaylevel,maxit=40, λ=-0.75, v=ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*100
@test abs(λ1-λ2)/abs(λ1) < eps()*1000
end

#########################################################################################
@bench @testset "Random sparse dep" begin
nep = nep_gallery("dep0_sparse", 30)

λ1,x1 = newton(nep,logger=displaylevel,maxit=40, λ=0.2, v=ones(size(nep,1)),armijo_factor=0.5);
@test compute_resnorm(nep,λ1,x1) < eps()*100

# Newton on interpolated sparse dep
intpoints = [λ1-.5, λ1, λ1+.5]
pep = interpolate(nep, intpoints)
λ2,x2 = newton(pep,logger=displaylevel,maxit=40, λ=λ1+0.1, v=x1+0.1*ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*1000000
@test abs(λ1-λ2)/abs(λ1) < eps()*1000

end

#########################################################################################
@bench @testset "Random pep (original pep of degree 2)" begin
nep = nep_gallery("pep0")
λ1,x1 = newton(nep,logger=displaylevel,maxit=40, λ=1, v=ones(size(nep,1)));
@test compute_resnorm(nep,λ1,x1) < eps()*100

#Newton on interpolated pep (interpolation of degree 2)
intpoints = [λ1-1, λ1, λ1+1]
pep = interpolate(nep, intpoints)
λ2,x2 = newton(pep,logger=displaylevel,maxit=40, λ=1, v=ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*100
@test abs(λ1-λ2)/abs(λ1) < eps()*100

@testset "Coefficient matrix differences (monomes) deg 2" begin
for i = 1:3
    @test opnorm(nep.A[i]-pep.A[i])/opnorm(nep.A[i]) < eps()*100
end
end

#Newton on interpolated pep (interpolation of degree 4) in real arithmetics
intpoints = [λ1-3, λ1-1, λ1, λ1+1, λ1+3]

pep = interpolate(Float64, nep, intpoints)
λ2,x2 = newton(pep,logger=displaylevel,maxit=40, λ=1, v=ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*100
@test abs(λ1-λ2)/abs(λ1) < eps()*100

@testset "Coefficient matrix differences (monomes) deg 4" begin
for i = 1:3
    @test opnorm(nep.A[i]-pep.A[i])/opnorm(nep.A[i]) < eps()*100
end
for i = 4:5
    @test opnorm(pep.A[i]) < eps()*100
end
end

end

#########################################################################################
@bench @testset "Random sparse pep (original pep of degree 2)" begin
nep=nep_gallery("pep0_sparse")
λ1,x1 =newton(nep,logger=displaylevel,maxit=40, λ=-0.75, v=ones(size(nep,1)));
@test compute_resnorm(nep,λ1,x1) < eps()*1000

#Newton on interpolated sparse pep
intpoints = [λ1-1, λ1, λ1+1.5]
pep = interpolate(nep, intpoints)
λ2,x2 =newton(pep,logger=displaylevel,maxit=40, λ=-0.75, v=ones(size(nep,1)));
@test compute_resnorm(pep,λ2,x2) < eps()*1000
@test abs(λ1-λ2)/abs(λ1) < eps()*100

@testset "Coefficient matrix differences (monomes) deg 2" begin
for i = 1:3
 @test opnorm(nep.A[i]-pep.A[i],Inf)/opnorm(nep.A[i],Inf) < eps()*100
end
end

end

end


using DrWatson
@quickactivate :Continuation

using Revise
using SparseArrays, LinearAlgebra, DiffEqOperators, Setfield, Parameters
using BifurcationKit
using Plots
const BK = BifurcationKit

N = 1000
Δx = 0.1
p = @dict ϵ=1.16 ν=1
params = GenericParams(N, Δx, p)

# initial condition
x = collect((1:N) * Δx);
x0 = x[500]
σ = 1
sol0 = @. cos(x) * exp(-((x - x0) / (2 * σ))^2)

plot(x, sol0)
# ok

# newton!
p = @set p[:ϵ] -= 0.1
X2 = newton(X2, RHS_SHE1D, Jacobian_SHE1D, p, N, Δx)
plot(x, X2)

S = read_state(params)

xs = getXs(S)
heatmap(xs, xs, S.u)

S = @set S.flags.useFFT = false

V = RHS_SHE(S);
heatmap(xs, xs, V)
L = sum(abs.(V)) / params.N^2
m = maximum(abs.(V))

X = Array(vec(transpose(S.u)))
Y = RHS_SHE(X, S.gp.p, N, Δx)

J = Jacobian_SHE(X, S.gp.p, N, Δx)

X2 = newton(X2, RHS_SHE, Jacobian_SHE, S.gp.p, N, Δx)

S = @set S.gp.p[:ϵ] = 1.175
heatmap(xs, xs, transpose(reshape(X2, (N, N))))


function standardDeviationSD(x)

    sqrt(varianceSD(x))

end


# bogaczki shampine
struct BogaczkiShampine23 <: AbstractAdaptiveExplicitSolver
end

function order(::BogaczkiShampine23)::Int
    return 3
end

function n_intermediates(::BogaczkiShampine23)
    return 4
end

function n_temporaries(::BogaczkiShampine23)
    return 2
end

function stepper!(ode_solver::BogaczkiShampine23, ode_fcn, stepper_options, stepper_temp::StepperTemporaries)

    # renaming for better readability
    t          = stepper_temp.t_old;
    x          = stepper_temp.x_old;
    dt         = stepper_temp.dt;
    n_prob_dim = n_dim(stepper_temp)

    # views into the temporaries
    k1 = view(stepper_temp.der_x, 1:n_prob_dim, 1)
    k2 = view(stepper_temp.der_x, 1:n_prob_dim, 2)
    k3 = view(stepper_temp.der_x, 1:n_prob_dim, 3)
    k4 = view(stepper_temp.der_x, 1:n_prob_dim, 4)

    x2 = view(stepper_temp.x_tmp, 1:n_prob_dim, 1)
    x3 = view(stepper_temp.x_tmp, 1:n_prob_dim, 2)

    # intermediate steps
    t2 = t + 1/2 * dt
    @. x2 = x + 1/2 * dt * k1
    ode_fcn(k2, t2, x2)

    t3 = t + 3/4 * dt
    @. x3 = x + 3/4 * dt * k2
    ode_fcn(k3, t3, x3)

    # new step
    stepper_temp.t_new = t + dt
    @. stepper_temp.x_new = x + dt * (2/9 * k1 + 1/3 * k2 + 4/9 * k3)
    ode_fcn(k4, stepper_temp.t_new, stepper_temp.x_new)

    # error estimation
    @. stepper_temp.x_est = x + dt * (7/24 * k1 + 1/4 * k2 + 1/3 * k3 + 1/8 * k4)
end


"""
    SpinBasis(n)

Basis for spin-n particles.

The basis can be created for arbitrary spinnumbers by using a rational number,
e.g. `SpinBasis(3//2)`. The Pauli operators are defined for all possible
spin numbers.
"""
struct SpinBasis{S,T} <: Basis
    shape::Vector{T}
    spinnumber::Rational{T}
    function SpinBasis{S}(spinnumber::Rational{T}) where {S,T<:Int}
        @assert isa(S, Rational{Int})
        n = numerator(spinnumber)
        d = denominator(spinnumber)
        @assert d==2 || d==1
        @assert n > 0
        N = numerator(spinnumber*2 + 1)
        new{spinnumber,T}([N], spinnumber)
    end
end
SpinBasis(spinnumber::Rational{Int}) = SpinBasis{spinnumber}(spinnumber)
SpinBasis(spinnumber::Int) = SpinBasis(convert(Rational{Int}, spinnumber))

==(b1::SpinBasis, b2::SpinBasis) = b1.spinnumber==b2.spinnumber

"""
    sigmax(b::SpinBasis)

Pauli ``σ_x`` operator for the given Spin basis.
"""
function sigmax(b::SpinBasis)
    N = length(b)
    diag = ComplexF64[complex(sqrt(real((b.spinnumber + 1)*2*a - a*(a+1)))) for a=1:N-1]
    data = spdiagm(1 => diag, -1 => diag)
    SparseOperator(b, data)
end

"""
    sigmay(b::SpinBasis)

Pauli ``σ_y`` operator for the given Spin basis.
"""
function sigmay(b::SpinBasis)
    N = length(b)
    diag = ComplexF64[1im*complex(sqrt(real((b.spinnumber + 1)*2*a - a*(a+1)))) for a=1:N-1]
    data = spdiagm(-1 => diag, 1 => -diag)
    SparseOperator(b, data)
end

"""
    sigmaz(b::SpinBasis)

Pauli ``σ_z`` operator for the given Spin basis.
"""
function sigmaz(b::SpinBasis)
    N = length(b)
    diag = ComplexF64[complex(2*m) for m=b.spinnumber:-1:-b.spinnumber]
    data = spdiagm(0 => diag)
    SparseOperator(b, data)
end

"""
    sigmap(b::SpinBasis)

Raising operator ``σ_+`` for the given Spin basis.
"""
function sigmap(b::SpinBasis)
    N = length(b)
    S = (b.spinnumber + 1)*b.spinnumber
    diag = ComplexF64[complex(sqrt(float(S - m*(m+1)))) for m=b.spinnumber-1:-1:-b.spinnumber]
    data = spdiagm(1 => diag)
    SparseOperator(b, data)
end

"""
    sigmam(b::SpinBasis)

Lowering operator ``σ_-`` for the given Spin basis.
"""
function sigmam(b::SpinBasis)
    N = length(b)
    S = (b.spinnumber + 1)*b.spinnumber
    diag = [complex(sqrt(float(S - m*(m-1)))) for m=b.spinnumber:-1:-b.spinnumber+1]
    data = spdiagm(-1 => diag)
    SparseOperator(b, data)
end

"""
    spinup(b::SpinBasis)

Spin up state for the given Spin basis.
"""
spinup(b::SpinBasis) = basisstate(b, 1)

"""
    spindown(b::SpinBasis)

Spin down state for the given Spin basis.
"""
spindown(b::SpinBasis) = basisstate(b, b.shape[1])


using DataFrames,DataArrays,Distributions,StatsBase,MixedModels

"""
Create a DataFrame with crossed grouping factors, `S` and `I`
with `M` and `N` levels, respectively.  The condition, `C`,
should be a vector of length 2.  The effect of all the `rep` calls
is similar to `expand.grid` in R.
"""
function mkdata(C::Vector=[zero(Int8),one(Int8)],N=10,M=100)
    l = length(C)
    DataFrame(y = zeros(l*N*M),
              C = rep(C,N*M),
              I = rep(rep(compact(pool(collect(1:N))),fill(l,N)),M),
              S = rep(compact(pool(collect(1:M))),fill(l*N,M)))
end

"""
Simulate a single response vector for a linear mixed model with
fixed-effects parameter β, and independent random effects for intercept
and condition, `C`, by subject and by item.  The function modifies the
first argument, `dat`, replacing dat[:y].
"""
function sim1!(dat::DataFrame,β::Vector,σ=600.,σRS=300.,σRI=30.)
    y,C,I,S = dat[:y],dat[:C],dat[:I],dat[:S]
    length(β) == length(unique(C)) == 2 || throw(DimensionMismatch())
    # vectors of random effects (subject-int,subject-C,item-int,item-C)
    M,N = length(S.pool),length(I.pool)
    re = Vector{Float64}[σRS*randn(M),σRS*randn(M)*0.5,
                         σRI*randn(N),σRI*randn(N)*0.5]
    I,S = I.refs,S.refs
    for ii in eachindex(y)
        yi = β[1] + re[1][S[ii]] + re[3][I[ii]] + σ*randn()
        if (cc = C[ii]) ≠ 0
            yi += cc*(β[2]+re[2][S[ii]]+re[4][I[ii]])
        end
        y[ii] = yi
    end
    dat
end

const dat = mkdata();
const Xs = hcat(ones(2000),dat[:C]);
const Xst = Xs'
srand(1234321);

sim1!(dat,[2000.,0.]);
ms1 = fit!(lmm(y ~ 1+C + (1+C|S) + (1+C|I),dat));
#ms2 = fit!(lmm(y ~ 1+C + (1+C|S) + (1|I),dat))


abstract type AbstractKernelEmbedding end

measure(ke::AbstractKernelEmbedding) = ke.measure

struct KernelEmbedding{Tk<:Kernel,Tm,Tσ<:Real,Tl} <: AbstractKernelEmbedding
    kernel::Tk # Kernel function
    σ::Tσ # Kernel variance
    l::Tl # Kernel lengthscale
    measure::Tm # Measure
end

function KernelEmbedding(bquad::AbstractBQ, prior)
    return KernelEmbedding(bquad.kernel, bquad.σ, bquad.l, prior)
end

scale(ke::KernelEmbedding) = ke.σ

@doc raw"""
    kernel_mean(ke::KernelEmbedding{Kernel,Measure}, samples::AbstractVector)

Compute the kernel mean of the kernel embedding `ke` for each one of
the `samples` $$x_i$$:
```math
    z_i = \int k(x, x_i)d\mu(x)
```
"""
kernel_mean

function kernel_mean(ke::KernelEmbedding{<:SqExponentialKernel,<:AbstractMvNormal}, samples::AbstractVector)
    z = samples .- Ref(mean(measure(ke)))
    B = Λ(ke.l)
    return scale(ke) / sqrt(det(inv(B) * cov(measure(ke)) + I)) *
           exp.(- PDMats.invquad.(Ref(PDMat(B + cov(measure(ke)))), z) / 2)
end

@doc raw"""
    kernel_variance(ke::KernelEmbedding{Kernel,Measure})

Compute the kernel variance of the given kernel embedding:
```math
    C = \int\int k(x,x')d\mu(x)d\mu(x')
```
"""
kernel_variance

function kernel_variance(ke::KernelEmbedding{<:SqExponentialKernel,<:AbstractMvNormal})
    B = Λ(ke.l)
    return scale(ke) / sqrt(det(2 * inv(B) * cov(measure(ke)) + I))
end

# Turn the lengthscale into a Diagonal matrix of noise
Λ(l::Real) = abs2(l) * I
Λ(l::AbstractVector) = Diagonal(abs2.(l))
Λ(l::LowerTriangular) = Cholesky(l, :L, 1)
Λ(l::AbstractMatrix) = l * l'

# Turns a transform into a lengthscale
transform_param(t::ScaleTransform) = inv(first(t.s))
transform_param(t::ARDTransform) = inv.(t.v)
transform_param(t::LinearTransform) = inv(t.A)