### A Pluto.jl notebook ###
# v0.17.2

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 iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 8d4298e0-6c44-11eb-3406-31a39340d1aa
begin
	#try
		using PlutoUI
		using Plots
		using LinearAlgebra
		using SparseArrays
		using SpecialFunctions
		using StatsBase

		using Random
		md"""
		# Packages

		All needed Packages available :) """
	#=catch
		using Pkg;
		Pkg.activate(mktempdir())
		Pkg.add("PlutoUI")
		Pkg.add("Plots")
		#Pkg.add("LinearAlgebra")
		#Pkg.add("SparseArrays")
		Pkg.add("SpecialFunctions")
		Pkg.add("StatsBase")
		using PlutoUI
		using Plots
		using LinearAlgebra
		using SparseArrays
		using SpecialFunctions
		using StatsBase

		using Random
		md"""
		# Packages

		Some Package sources not added, this will take approx. 3 minutes"""
	end
	=#
	# TODO: add a GIF of the golden Nautilus wheel spinning and maybe a link to the derivation of the proabilities and to the video
end

# ╔═╡ 9637fe40-6c44-11eb-1df3-eb0b0d36db34
md"""

# **The golden Nautilus wheel** - _Distributions and the law of large numbers_

As you can watch in video below 👇, Captain Venn wants to find out the chances for the four tasks of the day. Help him with doing the experiments, press **_START_** and count the results in the **histogram** below. The red bullets 📍 indicate the intrinsic probabilities times the number of experiments $P_i \cdot \mathcal{N}$.

"""

# ╔═╡ f3f80e2e-51d2-4fce-b7df-b2cdd3a0cad5
html"""

"""

# ╔═╡ e03eeec0-8299-11eb-119b-5dfd576a2811
md"""
Choose the **random seed** $\,\,$ 👉 $(@bind seed Slider(1:100, default=1, show_value = true))
"""

# ╔═╡ 8d180f60-829e-11eb-36a3-6fb2eab9c846
md"""
Start the **measurements** (don't forget to stop the clock at some point!):  $(@bind n_sample_size Clock(0.2,true,false))
"""

#md"""
#Change the number of measurements 👉 $\mathcal{N} =$ $(@bind n_sample_size #Scrubbable(1:1:200, default=10))  (*drag the red number with the mouse* 🖱)
#"""

# ╔═╡ 32d0d060-82b1-11eb-37a0-4bcd4881e6e1
md"""
##### Wages
Captain Venn trusts you a lot, you can even **change the wages** Captain Venn pays for the four tasks (just drag the numbers with the mouse).

_sailing_ ⛵ $(@bind sailing Scrubbable(6)) pennies,

_fishing_ 🐬$(@bind fishing Scrubbable(4)) pennies,

_shrubbing the deck_ 🧹 $(@bind shrubbing Scrubbable(3)) pennies,

_free day_ 🏝 $(@bind free_day Scrubbable(1)) penny.
"""

# ╔═╡ b7a7b220-6c56-11eb-1e6c-6f7595f02439
md"""
In order to see the variation of the sample mean we **repeat the measurements with fixed sample size** ($\mathcal{N}$) $n$ times. This way we can **sample the distribution of the sample mean value** (which is a random variable in contrast to the intrinsic mean $\langle X \rangle$ = 0.1056. Use the slider to change the number of repeated measurements $N$ of samples of size $n$.

"""

# ╔═╡ 8b8cd540-6d86-11eb-339e-7b8a2df774f9
md"""
Choose the **number of repetitions** 👉 n =   $(@bind rep Scrubbable(100:100:2000)) of sampling samples with **sample size**
👉 N = $(@bind n_sam_size Scrubbable(500:1000:12000))
"""

# ╔═╡ b1515d9a-8039-4410-bbd2-54890804fda8
md"""
Learn more about the law of large numbers and how to evaluate the standard error in the lesson below 👇 or in the [`IMOOX course`](https://imoox.at/mooc/local/landingpage/course.php?shortname=bayes&lang=en) equipped with many quizzes."""

# ╔═╡ 936054d5-2863-4f69-9a12-c865d6dd37b0
html"""

"""

# ╔═╡ 3cc85769-13d1-44d9-acd8-20353c848829
md"""
## About the creators

This notebook was created by **Gerhard Dorn** in the context of the course **Bayesian probability theory**.

The course is a free massive open online course (MOOC) available on the platform [`IMOOX`](https://imoox.at/mooc/local/landingpage/course.php?shortname=bayes&lang=en)

$(Resource("https://raw.githubusercontent.com/Captain-Bayes/images/main/adventure_map.gif"))
"""

# ╔═╡ a2541e60-8356-11eb-1b16-6fb19948ed1c
md"""
# Some code below
In order to have a look at the code download the notebook or load it on binder (takes 6-10 minutes)

To do so, **press the binder button** in the right top corner ↗
"""

# ╔═╡ 48f7c480-6d84-11eb-1f24-e16d7f649c9c
begin
	phi = (1+sqrt(5))/2 #golden ratio
	prob = phi.^(0:3)./sum(phi.^(0:3)) # intrinsic probabilities of the four sectors
	prob_string = [string(round(prob[i]*100, digits=1)) * "%, " for i in 1:4]
md"""
##### Calculating the intrinsic probabilities of the golden Nautilus wheel

 $P$ :   [$(prob_string[:]) ]

	"""
end

# ╔═╡ 70209d20-8350-11eb-2da3-9f99a25eb4f9
begin
	rng = MersenneTwister(20)

	local sample_temp = 5 .- sum(rand(rng, 1,rep*n_sam_size) .<= cumsum(prob), dims=1)
	local sample_array = reshape(sample_temp, (n_sam_size,:))
	X = sum(sample_array .== 1,dims=1)/n_sam_size

	md"""
	##### sampling the sample means
	"""
end

# ╔═╡ 5523cb40-82b6-11eb-3e29-81d50ff29db6
begin

	histogram(X[:], nbins=20,
	xlim=[0.06, 0.15], label=:none, xlabel="Sample mean", ylabel="Frequency"
	)

	plot!([0.1, 0.1], [0, rep/6], line=(3.2, 1.0, :line), linestyle=:dash, color = :red, label="10% limit")
	plot!([0.1056, 0.1056], [0, rep/6], line=(3.2, 1.0, :line), linestyle=:dash, color = :green, label="intrinsic mean")
end

# ╔═╡ 86962d00-8449-11eb-31f9-dfb502c7e132
md"""
Captain Venn's statement of having at least 10% free days is **wrong** in **$(round(100*sum(X.<0.1)/length(X), digits=1)) %** of all $(n_sam_size) day-long sailing turns *(a contract for $(round(n_sam_size/365, digits=1)) years)*"""

# ╔═╡ 8cc9f417-ad45-48d2-8960-6a8d465cb9a5
begin

		bayes = "https://raw.githubusercontent.com/Captain-Bayes/images/main/bayes_50px.gif"
	bernoulli = "https://raw.githubusercontent.com/Captain-Bayes/images/main/bernoulli_50px.gif"

	venn = "https://raw.githubusercontent.com/Captain-Bayes/images/main/venn_50px1.gif"

	md"""Images"""
end

# ╔═╡ 67f01600-82ae-11eb-2c1a-c906e8a6a892
begin
md"""
$(Resource(bayes, :width=>180))
$(Resource(venn, :width=>180))

**Captain Bayes and Captain Venn** are investigating the **golden Nautilus wheel** and are wondering how one can make reliable predictions about the minimum free days ratio.

A **rough rule of thumb** says that for most distributions the standard deviation can be used as a measure of how much probability mass is **spread around the mean value**, namely that within the following intervals one can find:

``P(X \in [μ - σ, μ + σ])  ≈ 67\%,``

``P(X ∈ [μ - 2σ, μ + 2σ]) ≈ 95\%,``

``P(X ∈ [μ - 3σ, μ + 3σ]) ≈ 99\%.``

We examine the standard error which is the standard deviation of the sample mean 	(since we want to find an estimate for the sample mean ``\overline{\boldsymbol{X}}``).

"""

end

# ╔═╡ 59607242-82b0-11eb-35cf-cb7b33cddf21
x_rand_var = [free_day, shrubbing, fishing, sailing]

# ╔═╡ d6519370-825a-11eb-1ae5-ed41bc0c343f
begin
	function take_sample(N_sample_size, seed)
	rng = MersenneTwister(seed)
	sample = 5 .- sum(rand(rng, 1,N_sample_size) .<= cumsum(prob), dims=1)
	return sample
	end

	md"""
	##### Function for taking a sample of size N using a seed
	"""
end

# ╔═╡ 5e0bfb10-6d8a-11eb-06ef-9335eb90064c
begin
	n_size = minimum([n_sample_size, 200])
	sample_1 = take_sample(n_size, seed)
	sample_count = [count(==(i), sample_1) for i = 1:4]

	plot(x_rand_var, sample_count,
    line = (1.0 , 2.0 , :bar),
    normalize = false,
	bar_width = 0.6,
    marker = (6, 0.5, :none),
    markerstrokewidth = 5.,
    color = :steelblue,
    fill = 0.9,
    orientation = :v,
    title = "Histogram with total sample size: " * string(n_size),
	ylabel = "frequency N",
	xlabel = "Pennies (random variable X)",
	label = :none,
	ylim =[0,maximum([20,n_size/2])])

	plot!(
    x_rand_var,prob.*n_size,
    line = ( 1, 0., :path),
    normalize = false,
    marker = (5, 1.0, :o),
    markerstrokewidth = 3.,
    color = :red,
    fill = 0.,
    orientation = :v,
	label = "expected value",
	legend = :topleft
	)
end

# ╔═╡ 4171c850-82a6-11eb-1031-73a99aff7207
begin

	free_days = [(i>1) ? md"🔴" : md"🟢" for i in sample_1]
	free_days_table = reshape([free_days [md"" for i in 1:1, j in (1:(7-Integer(length(free_days)- floor(length(free_days)/7)*7)))]], (7,:))
	fd_table = permutedims(free_days_table, [2,1])
md"""
#### Working days:
 $(Resource(bernoulli, :width=>180))

	Bernoulli made a calender 📅 of all working 🔴 and free 🟢 days:

	"""

end

# ╔═╡ 3f47bb1e-8350-11eb-2cca-cb42ea1105d2
	fd_table

# ╔═╡ 26395db0-82b4-11eb-3c44-d18d737d9e53
md"""
The estimated free days ratio from the data is: $(round(100*sum(sample_1 .==1)/length(sample_1))) %
"""

# ╔═╡ 2293a98e-d6a8-443a-9f84-47b632a199f4
TableOfContents()

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

[compat]
Plots = "~1.24.0"
PlutoUI = "~0.7.20"
SpecialFunctions = "~2.0.0"
StatsBase = "~0.33.13"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "0fa77022fe4b511826b39c894c90daf5fce3334a"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "0.9.17"

[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "8a954fed8ac097d5be04921d595f741115c1b2ad"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+0"

[[Hyperscript]]
deps = ["Test"]
git-tree-sha1 = "8d511d5b81240fc8e6802386302675bdf47737b9"
uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91"
version = "0.0.4"

[[HypertextLiteral]]
git-tree-sha1 = "2b078b5a615c6c0396c77810d92ee8c6f470d238"
uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
version = "0.9.3"

[[IOCapture]]
deps = ["Logging", "Random"]
git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
version = "0.2.2"

[[IniFile]]
deps = ["Test"]
git-tree-sha1 = "098e4d2c533924c921f9f9847274f2ad89e018b8"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.0"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[InverseFunctions]]
deps = ["Test"]
git-tree-sha1 = "a7254c0acd8e62f1ac75ad24d5db43f5f19f3c65"
uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
version = "0.1.2"

[[IrrationalConstants]]
git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.1"

[[IterTools]]
git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.3.0"

[[IteratorInterfaceExtensions]]
git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
uuid = "82899510-4779-5014-852e-03e436cf321d"
version = "1.0.0"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.3.0"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "8076680b162ada2a031f707ac7b4953e30667a37"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.2"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.0+0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "f2355693d6778a178ade15952b7ac47a4ff97996"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.3.0"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"]
git-tree-sha1 = "a8f4f279b6fa3c3c4f1adadd78a621b13a506bce"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.9"

[[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 = "0b4a5d71f3e5200a7dff793393e09dfc2d874290"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+1"

[[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"
version = "2.35.0+0"

[[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"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["ChainRulesCore", "ChangesOfVariables", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "be9eef9f9d78cecb6f262f3c10da151a6c5ab827"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.5"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "3d3e902b31198a27340d0bf00d6ac452866021cf"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.9"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

[[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"
version = "1.1.10+0"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "ae4bbcadb2906ccc085cf52ac286dc1377dceccc"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.1.2"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.15"

[[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", "UnicodeFun"]
git-tree-sha1 = "02a083caba3f73e42decb810b2e0740783022978"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.24.0"

[[PlutoUI]]
deps = ["AbstractPlutoDingetjes", "Base64", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "1e0cb51e0ccef0afc01aab41dc51a3e7f781e8cb"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.20"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[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"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
git-tree-sha1 = "6bf3f380ff52ce0832ddd3a2a7b9538ed1bcca7d"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.2.1"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[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 = "e08890d19787ec25029113e88c34ec20cac1c91e"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.0.0"

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

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

[[StatsAPI]]
git-tree-sha1 = "0f2aa8e32d511f758a2ce49208181f7733a0936a"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.1.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "2bb0cb32026a66037360606510fca5984ccc6b75"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.13"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

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

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

[[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"

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

[[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"

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

[[UnicodeFun]]
deps = ["REPL"]
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[[XML2_jll]]
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version = "1.6.38+0"

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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"
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[[xkbcommon_jll]]
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git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─9637fe40-6c44-11eb-1df3-eb0b0d36db34
# ╟─f3f80e2e-51d2-4fce-b7df-b2cdd3a0cad5
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# ╟─8d180f60-829e-11eb-36a3-6fb2eab9c846
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# ╟─8cc9f417-ad45-48d2-8960-6a8d465cb9a5
# ╟─59607242-82b0-11eb-35cf-cb7b33cddf21
# ╟─d6519370-825a-11eb-1ae5-ed41bc0c343f
# ╠═2293a98e-d6a8-443a-9f84-47b632a199f4
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002


module TestProject

using StaticArrays

function dot(x)
    v = SVector(x...)
    return v'v
end

end # module


@doc raw"""
    y = prox_distance(M,λ,f,x [, p=2])
    prox_distance!(M, y, λ, f, x [, p=2])

compute the proximal map ``\operatorname{prox}_{λ\varphi}`` with
parameter λ of ``φ(x) = \frac{1}{p}d_{\mathcal M}^p(f,x)``.
For the mutating variant the computation is done in place of `y`.

# Input
* `M` – a [Manifold](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.Manifold) ``\mathcal M``
* `λ` – the prox parameter
* `f` – a point ``f ∈ \mathcal M`` (the data)
* `x` – the argument of the proximal map

# Optional argument
* `p` – (`2`) exponent of the distance.

# Ouput
* `y` – the result of the proximal map of ``φ``
"""
function prox_distance(M::AbstractManifold, λ, f, x, p::Int=2)
    d = distance(M, f, x)
    if p == 2
        t = λ / (1 + λ)
    elseif p == 1
        t = (λ < d) ? λ / d : 1.0
    else
        throw(
            ErrorException(
                "Proximal Map of distance(M,f,x) not implemented for p=$(p) (requires p=1 or 2)",
            ),
        )
    end
    return exp(M, x, log(M, x, f), t)
end
function prox_distance!(M::AbstractManifold, y, λ, f, x, p::Int=2)
    d = distance(M, f, x)
    if p == 2
        t = λ / (1 + λ)
    elseif p == 1
        t = (λ < d) ? λ / d : 1.0
    else
        throw(
            ErrorException(
                "Proximal Map of distance(M,f,x) not implemented for p=$(p) (requires p=1 or 2)",
            ),
        )
    end
    return exp!(M, y, x, log(M, x, f), t)
end

@doc raw"""
    [y1,y2] = prox_TV(M, λ, [x1,x2] [,p=1])
    prox_TV!(M, [y1,y2] λ, [x1,x2] [,p=1])

Compute the proximal map ``\operatorname{prox}_{λ\varphi}`` of
``φ(x,y) = d_{\mathcal M}^p(x,y)`` with
parameter `λ`.

# Input

* `M` – a [Manifold](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.Manifold)
* `λ` – a real value, parameter of the proximal map
* `(x1,x2)` – a tuple of two points,

# Optional

(default is given in brackets)
* `p` – (1) exponent of the distance of the TV term

# Ouput

* `(y1,y2)` – resulting tuple of points of the ``\operatorname{prox}_{λφ}(```(x1,x2)```)``.
  The result can also be computed in place.
"""
function prox_TV(M::AbstractManifold, λ::Number, x::Tuple{T,T}, p::Int=1) where {T}
    d = distance(M, x[1], x[2])
    if p == 1
        t = min(0.5, λ / d)
    elseif p == 2
        t = λ / (1 + 2 * λ)
    else
        throw(
            ErrorException(
                "Proximal Map of TV(M,x1,x2,p) not implemented for p=$(p) (requires p=1 or 2)",
            ),
        )
    end
    return (exp(M, x[1], log(M, x[1], x[2]), t), exp(M, x[2], log(M, x[2], x[1]), t))
end
function prox_TV!(M::AbstractManifold, y, λ::Number, x::Tuple{T,T}, p::Int=1) where {T}
    d = distance(M, x[1], x[2])
    if p == 1
        t = min(0.5, λ / d)
    elseif p == 2
        t = λ / (1 + 2 * λ)
    else
        throw(
            ErrorException(
                "Proximal Map of TV(M,x1,x2,p) not implemented for p=$(p) (requires p=1 or 2)",
            ),
        )
    end
    X1 = log(M, x[1], x[2])
    X2 = log(M, x[2], x[1])
    exp!(M, y[1], x[1], X1, t)
    exp!(M, y[2], x[2], X2, t)
    return y
end
@doc raw"""
    ξ = prox_TV(M,λ,x [,p=1])

compute the proximal maps ``\operatorname{prox}_{λ\varphi}`` of
all forward differences occurring in the power manifold array, i.e.
``\varphi(xi,xj) = d_{\mathcal M}^p(xi,xj)`` with `xi` and `xj` are array
elemets of `x` and `j = i+e_k`, where `e_k` is the ``k``th unitvector.
The parameter `λ` is the prox parameter.

# Input
* `M` – a [Manifold](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.Manifold)
* `λ` – a real value, parameter of the proximal map
* `x` – a point.

# Optional
(default is given in brackets)
* `p` – (1) exponent of the distance of the TV term

# Ouput
* `y` – resulting  point containing with all mentioned proximal
  points evaluated (in a cyclic order). The computation can also be done in place
"""
function prox_TV(M::PowerManifold, λ, x, p::Int=1)
    y = deepcopy(x)
    power_size = power_dimensions(M)
    R = CartesianIndices(Tuple(power_size))
    d = length(power_size)
    maxInd = last(R).I
    for k in 1:d # for all directions
        ek = CartesianIndex(ntuple(i -> (i == k) ? 1 : 0, d)) #k th unit vector
        for l in 0:1
            for i in R # iterate over all pixel
                if (i[k] % 2) == l
                    J = i.I .+ ek.I #i + e_k is j
                    if all(J .<= maxInd) # is this neighbor in range?
                        j = CartesianIndex(J...) # neigbbor index as Cartesian Index
                        (y[i], y[j]) = prox_TV(M.manifold, λ, (y[i], y[j]), p) # Compute TV on these
                    end
                end
            end # i in R
        end # even odd
    end # directions
    return y
end
function prox_TV!(M::PowerManifold, y, λ, x, p::Int=1)
    power_size = power_dimensions(M)
    R = CartesianIndices(Tuple(power_size))
    d = length(power_size)
    copyto!(M, y, x)
    maxInd = last(R).I
    for k in 1:d # for all directions
        ek = CartesianIndex(ntuple(i -> (i == k) ? 1 : 0, d)) #k th unit vector
        for l in 0:1
            for i in R # iterate over all pixel
                if (i[k] % 2) == l # even/odd splitting
                    J = i.I .+ ek.I #i + e_k is j
                    if all(J .<= maxInd) # is this neighbor in range?
                        j = CartesianIndex(J...) # neigbbor index as Cartesian Index
                        prox_TV!(M.manifold, [y[i], y[j]], λ, (y[i], y[j]), p) # Compute TV on these
                    end
                end
            end # i in R
        end # even odd
    end # directions
    return y
end
@doc raw"""
    y = prox_parallel_TV(M, λ, x [,p=1])
    prox_parallel_TV!(M, y, λ, x [,p=1])

compute the proximal maps ``\operatorname{prox}_{λφ}`` of
all forward differences occurring in the power manifold array, i.e.
``φ(x_i,x_j) = d_{\mathcal M}^p(x_i,x_j)`` with `xi` and `xj` are array
elements of `x` and `j = i+e_k`, where `e_k` is the ``k``th unit vector.
The parameter `λ` is the prox parameter.

# Input
* `M`     – a `PowerManifold` manifold
* `λ`     – a real value, parameter of the proximal map
* `x`     – a point

# Optional
(default is given in brackets)
* `p` – (`1`) exponent of the distance of the TV term

# Ouput
* `y`  – resulting Array of points with all mentioned proximal
  points evaluated (in a parallel within the arrays elements).
  The computation can also be done in place.

*See also* [`prox_TV`](@ref)
"""
function prox_parallel_TV(M::PowerManifold, λ, x::AbstractVector, p::Int=1)
    R = CartesianIndices(x[1])
    d = ndims(x[1])
    if length(x) != 2 * d
        throw(
            ErrorException(
                "The number of inputs from the array ($(length(x))) has to be twice the data dimensions ($(d)).",
            ),
        )
    end
    maxInd = Tuple(last(R))
    # create an array for even/odd splitted proxes along every dimension
    y = deepcopy(x)
    yV = reshape(y, d, 2)
    xV = reshape(x, d, 2)
    for k in 1:d # for all directions
        ek = CartesianIndex(ntuple(i -> (i == k) ? 1 : 0, d)) #k th unit vector
        for l in 0:1 # even odd
            for i in R # iterate over all pixel
                if (i[k] % 2) == l
                    J = i.I .+ ek.I #i + e_k is j
                    if all(J .<= maxInd) # is this neighbor in range?
                        j = CartesianIndex(J...) # neigbbor index as Cartesian Index
                        # parallel means we apply each (direction even/odd) to a seperate copy of the data.
                        (yV[k, l + 1][i], yV[k, l + 1][j]) = prox_TV(
                            M.manifold, λ, (xV[k, l + 1][i], xV[k, l + 1][j]), p
                        ) # Compute TV on these
                    end
                end
            end # i in R
        end # even odd
    end # directions
    return y
end
function prox_parallel_TV!(
    M::PowerManifold, y::AbstractVector, λ, x::AbstractVector, p::Int=1
)
    R = CartesianIndices(x[1])
    d = ndims(x[1])
    if length(x) != 2 * d
        throw(
            ErrorException(
                "The number of inputs from the array ($(length(x))) has to be twice the data dimensions ($(d)).",
            ),
        )
    end
    maxInd = Tuple(last(R))
    # init y
    for i in 1:length(x)
        copyto!(M, y[i], x[i])
    end
    yV = reshape(y, d, 2)
    xV = reshape(x, d, 2)
    for k in 1:d # for all directions
        ek = CartesianIndex(ntuple(i -> (i == k) ? 1 : 0, d)) #k th unit vector
        for l in 0:1 # even odd
            for i in R # iterate over all pixel
                if (i[k] % 2) == l
                    J = i.I .+ ek.I #i + e_k is j
                    if all(J .<= maxInd) # is this neighbor in range?
                        j = CartesianIndex(J...) # neigbbor index as Cartesian Index
                        # parallel means we apply each (direction even/odd) to a seperate copy of the data.
                        prox_TV!(
                            M.manifold,
                            [yV[k, l + 1][i], yV[k, l + 1][j]],
                            λ,
                            (xV[k, l + 1][i], xV[k, l + 1][j]),
                            p,
                        ) # Compute TV on these in place of y
                    end
                end
            end # i in R
        end # even odd
    end # directions
    return y
end

@doc raw"""
    (y1,y2,y3) = prox_TV2(M,λ,(x1,x2,x3),[p=1], kwargs...)
    prox_TV2!(M, y, λ,(x1,x2,x3),[p=1], kwargs...)

Compute the proximal map ``\operatorname{prox}_{λ\varphi}`` of
``\varphi(x_1,x_2,x_3) = d_{\mathcal M}^p(c(x_1,x_3),x_2)`` with
parameter `λ`>0, where ``c(x,z)`` denotes the mid point of a shortest
geodesic from `x1` to `x3` that is closest to `x2`.
The result can be computed in place of `y`.

# Input

* `M`          – a manifold
* `λ`          – a real value, parameter of the proximal map
* `(x1,x2,x3)` – a tuple of three points

* `p` – (`1`) exponent of the distance of the TV term

# Optional
`kwargs...` – parameters for the internal [`subgradient_method`](@ref)
    (if `M` is neither `Euclidean` nor `Circle`, since for these a closed form
    is given)

# Output
* `(y1,y2,y3)` – resulting tuple of points of the proximal map.
  The computation can also be done in place.
"""
function prox_TV2(
    M::AbstractManifold,
    λ,
    x::Tuple{T,T,T},
    p::Int=1;
    stopping_criterion::StoppingCriterion=StopAfterIteration(10),
    kwargs...,
) where {T}
    (p != 1) && throw(
        ErrorException(
            "Proximal Map of TV2(M, λ, x, p) not implemented for p=$(p) (requires p=1) on general manifolds.",
        ),
    )
    PowX = SVector(x)
    PowM = PowerManifold(M, NestedPowerRepresentation(), 3)
    xR = PowX
    F(M, x) = 1 / 2 * distance(M, PowX, x)^2 + λ * costTV2(M, x)
    ∂F(PowM, x) = log(PowM, x, PowX) + λ * grad_TV2(PowM, x)
    subgradient_method!(PowM, F, ∂F, xR; stopping_criterion=stopping_criterion, kwargs...)
    return (xR...,)
end
function prox_TV2!(
    M::AbstractManifold,
    y,
    λ,
    x::Tuple{T,T,T},
    p::Int=1;
    stopping_criterion::StoppingCriterion=StopAfterIteration(10),
    kwargs...,
) where {T}
    (p != 1) && throw(
        ErrorException(
            "Proximal Map of TV2(M, λ, x, p) not implemented for p=$(p) (requires p=1) on general manifolds.",
        ),
    )
    PowX = SVector(x)
    PowM = PowerManifold(M, NestedPowerRepresentation(), 3)
    copyto!(M, y, PowX)
    F(M, x) = 1 / 2 * distance(M, PowX, x)^2 + λ * costTV2(M, x)
    ∂F!(M, y, x) = log!(M, y, x, PowX) + λ * grad_TV2!(M, y, x)
    subgradient_method!(
        PowM,
        F,
        ∂F!,
        y;
        stopping_criterion=stopping_criterion,
        evaluation=MutatingEvaluation(),
        kwargs...,
    )
    return y
end
@doc raw"""
    y = prox_TV2(M, λ, x[, p=1])
    prox_TV2!(M, y, λ, x[, p=1])

compute the proximal maps ``\operatorname{prox}_{λ\varphi}`` of
all centered second order differences occuring in the power manifold array, i.e.
``\varphi(x_k,x_i,x_j) = d_2(x_k,x_i.x_j)``, where ``k,j`` are backward and forward
neighbors (along any dimension in the array of `x`).
The parameter `λ` is the prox parameter.

# Input
* `M` – a [Manifold](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.Manifold)
* `λ` – a real value, parameter of the proximal map
* `x` – a points.

# Optional
(default is given in brackets)
* `p` – (`1`) exponent of the distance of the TV term

# Output
* `y` – resulting point with all mentioned proximal points evaluated (in a cylic order).
  The computation can also be done in place.
"""
function prox_TV2(M::PowerManifold{N,T}, λ, x, p::Int=1) where {N,T}
    y = deepcopy(x)
    return prox_TV2!(M, y, λ, x, p)
end
function prox_TV2!(M::PowerManifold{N,T}, y, λ, x, p::Int=1) where {N,T}
    power_size = power_dimensions(M)
    R = CartesianIndices(power_size)
    d = length(size(x))
    minInd = first(R).I
    maxInd = last(R).I
    copyto!(M, y, x)
    for k in 1:d # for all directions
        ek = CartesianIndex(ntuple(i -> (i == k) ? 1 : 0, d)) #k th unit vector
        for l in 0:2
            for i in R # iterate over all pixel
                if (i[k] % 3) == l
                    JForward = i.I .+ ek.I #i + e_k
                    JBackward = i.I .- ek.I # i - e_k
                    all(JForward .<= maxInd) &&
                        all(JBackward .>= minInd) &&
                        prox_TV2!(
                            M.manifold,
                            [y[M, JBackward...], y[M, i.I...], y[M, JForward...]],
                            λ,
                            (y[M, JBackward...], y[M, i.I...], y[M, JForward...]),
                            p,
                        )
                end # if mod 3
            end # i in R
        end # for mod 3
    end # directions
    return y
end
@doc raw"""
    project_collaborative_TV(M, λ, x, Ξ[, p=2,q=1])
    project_collaborative_TV!(M, Θ, λ, x, Ξ[, p=2,q=1])

compute the projection onto collaborative Norm unit (or α-) ball, i.e. of the function

```math
F^q(x) = \sum_{i∈\mathcal G}
  \Bigl( \sum_{j∈\mathcal I_i}
    \sum_{k=1^d} \lVert X_{i,j}\rVert_x^p\Bigr)^\frac{q/p},
```

where ``\mathcal G`` is the set of indices for ``x∈\mathcal M`` and ``\mathcal I_i``
is the set of its forward neighbors.
The computation can also be done in place of `Θ`.

This is adopted from the paper by Duran, Möller, Sbert, Cremers:
_Collaborative Total Variation: A General Framework for Vectorial TV Models_
(arxiv: [1508.01308](https://arxiv.org/abs/1508.01308)), where the most inner
norm is not on a manifold but on a vector space, see their Example 3 for
details.
"""
function project_collaborative_TV(N::PowerManifold, λ, x, Ξ, p=2.0, q=1.0, α=1.0)
    pdims = power_dimensions(N)
    if length(pdims) == 1
        d = 1
        s = 1
        iRep = (1,)
    else
        d = pdims[end]
        s = length(pdims) - 1
        if s != d
            throw(
                ErrorException(
                    "the last dimension ($(d)) has to be equal to the number of the previous ones ($(s)) but its not.",
                ),
            )
        end
        iRep = (Integer.(ones(d))..., d)
    end
    if q == 1 # Example 3 case 2
        if p == 1
            normΞ = norm.(Ref(N.manifold), x, Ξ)
            return max.(normΞ .- λ, 0.0) ./ ((normΞ .== 0) .+ normΞ) .* Ξ
        end
        if p == 2 # Example 3 case 3
            norms = sqrt.(sum(norm.(Ref(N.manifold), x, Ξ) .^ 2; dims=d + 1))
            if length(iRep) > 1
                norms = repeat(norms; inner=iRep)
            end
            # if the norm is zero add 1 to avoid division by zero, also then the
            # nominator is already (max(-λ,0) = 0) so it stays zero then
            return max.(norms .- λ, 0.0) ./ ((norms .== 0) .+ norms) .* Ξ
        end
        throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
    elseif q == Inf
        if p == 2
            norms = sqrt.(sum(norm.(Ref(N.manifold), x, Ξ) .^ 2; dims=d + 1))
            if length(iRep) > 1
                norms = repeat(norms; inner=iRep)
            end
        elseif p == 1
            norms = sum(norm.(Ref(N.manifold), x, Ξ); dims=d + 1)
            if length(iRep) > 1
                norms = repeat(norms; inner=iRep)
            end
        elseif p == Inf
            norms = norm.(Ref(N.manifold), x, Ξ)
        else
            throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
        end
        return (α .* Ξ) ./ max.(Ref(α), norms)
    end # end q
    return throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
end
function project_collaborative_TV(N::PowerManifold, λ, x, Ξ, p::Int, q::Float64=1.0, α=1.0)
    return project_collaborative_TV(N, λ, x, Ξ, Float64(p), q, α)
end
function project_collaborative_TV(N::PowerManifold, λ, x, Ξ, p::Float64, q::Int, α=1.0)
    return project_collaborative_TV(N, λ, x, Ξ, p, Float64(q), α)
end
function project_collaborative_TV(N::PowerManifold, λ, x, Ξ, p::Int, q::Int, α=1.0)
    return project_collaborative_TV(N, λ, x, Ξ, Float64(p), Float64(q), α)
end

function project_collaborative_TV!(N::PowerManifold, Θ, λ, x, Ξ, p=2.0, q=1.0, α=1.0)
    pdims = power_dimensions(N)
    if length(pdims) == 1
        d = 1
        s = 1
        iRep = (1,)
    else
        d = pdims[end]
        s = length(pdims) - 1
        if s != d
            throw(
                ErrorException(
                    "the last dimension ($d) has to be equal to the number of the previous ones ($s) but its not.",
                ),
            )
        end
        iRep = (Integer.(ones(d))..., d)
    end
    if q == 1 # Example 3 case 2
        if p == 1
            normΞ = norm.(Ref(N.manifold), x, Ξ)
            Θ .= max.(normΞ .- λ, 0.0) ./ ((normΞ .== 0) .+ normΞ) .* Ξ
            return Θ
        elseif p == 2 # Example 3 case 3
            norms = sqrt.(sum(norm.(Ref(N.manifold), x, Ξ) .^ 2; dims=d + 1))
            if length(iRep) > 1
                norms = repeat(norms; inner=iRep)
            end
            # if the norm is zero add 1 to avoid division by zero, also then the
            # nominator is already (max(-λ,0) = 0) so it stays zero then
            Θ .= max.(norms .- λ, 0.0) ./ ((norms .== 0) .+ norms) .* Ξ
            return Θ
        else
            throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
        end
    elseif q == Inf
        if p == 2
            norms = sqrt.(sum(norm.(Ref(N.manifold), x, Ξ) .^ 2; dims=d + 1))
            (length(iRep) > 1) && (norms = repeat(norms; inner=iRep))
        elseif p == 1
            norms = sum(norm.(Ref(N.manifold), x, Ξ); dims=d + 1)
            (length(iRep) > 1) && (norms = repeat(norms; inner=iRep))
        elseif p == Inf
            norms = norm.(Ref(N.manifold), x, Ξ)
        else
            throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
        end
        Θ .= (α .* Ξ) ./ max.(Ref(α), norms)
        return Θ
    end # end q
    return throw(ErrorException("The case p=$p, q=$q is not yet implemented"))
end
function project_collaborative_TV!(
    N::PowerManifold, Θ, λ, x, Ξ, p::Int, q::Float64=1.0, α=1.0
)
    return project_collaborative_TV!(N, Θ, λ, x, Ξ, Float64(p), q, α)
end
function project_collaborative_TV!(N::PowerManifold, Θ, λ, x, Ξ, p::Float64, q::Int, α=1.0)
    return project_collaborative_TV!(N, Θ, λ, x, Ξ, p, Float64(q), α)
end
function project_collaborative_TV!(N::PowerManifold, Θ, λ, x, Ξ, p::Int, q::Int, α=1.0)
    return project_collaborative_TV!(N, Θ, λ, x, Ξ, Float64(p), Float64(q), α)
end


using OrdinaryDiffEq, ParameterizedFunctions, Plots, ODE, ODEInterfaceDiffEq, LSODA, DiffEqDevTools, Sundials
using LinearAlgebra
LinearAlgebra.BLAS.set_num_threads(1)

gr() #gr(fmt=:png)

f = @ode_def Hires begin
  dy1 = -1.71*y1 + 0.43*y2 + 8.32*y3 + 0.0007
  dy2 = 1.71*y1 - 8.75*y2
  dy3 = -10.03*y3 + 0.43*y4 + 0.035*y5
  dy4 = 8.32*y2 + 1.71*y3 - 1.12*y4
  dy5 = -1.745*y5 + 0.43*y6 + 0.43*y7
  dy6 = -280.0*y6*y8 + 0.69*y4 + 1.71*y5 -
           0.43*y6 + 0.69*y7
  dy7 = 280.0*y6*y8 - 1.81*y7
  dy8 = -280.0*y6*y8 + 1.81*y7
end

u0 = zeros(8)
u0[1] = 1
u0[8] = 0.0057
prob = ODEProblem(f,u0,(0.0,321.8122))

sol = solve(prob,Rodas5(),abstol=1/10^14,reltol=1/10^14)
test_sol = TestSolution(sol)
abstols = 1.0 ./ 10.0 .^ (4:11)
reltols = 1.0 ./ 10.0 .^ (1:8);

plot(sol)

plot(sol,tspan=(0.0,5.0))

#sol = solve(prob,ode23s()); println("Total ODE.jl steps: $(length(sol))")
#using GeometricIntegratorsDiffEq
#try
#    sol = solve(prob,GIRadIIA3(),dt=1/10)
#catch e
#    println(e)
#end

setups = [
#Dict(:alg=>ROCK2()),
#Dict(:alg=>ROCK4())
#Dict(:alg=>ESERK5())
]

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Rodas3()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>rodas()),
          Dict(:alg=>radau()),
          Dict(:alg=>RadauIIA5()),
          Dict(:alg=>ROS34PW1a()),
          Dict(:alg=>lsoda()),
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;dense = false,verbose=false,
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Kvaerno3()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>KenCarp3()),
    # Dict(:alg=>SDIRK2()), # Removed because it's bad
          Dict(:alg=>radau())]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;dense = false,verbose=false,
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>KenCarp5()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>KenCarp3()),
          Dict(:alg=>ARKODE(order=5)),
          Dict(:alg=>ARKODE()),
          Dict(:alg=>ARKODE(order=3))]
names = ["Rosenbrock23" "KenCarp5" "KenCarp4" "KenCarp3" "ARKODE5" "ARKODE4" "ARKODE3"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      names=names,save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;dense = false,verbose=false,
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>ImplicitEulerExtrapolation()),
          #Dict(:alg=>ImplicitDeuflhardExtrapolation()), # Diverges
          #Dict(:alg=>ImplicitHairerWannerExtrapolation()), # Diverges
          Dict(:alg=>ABDF2()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>Exprb43()),
          Dict(:alg=>Exprb32()),
]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

abstols = 1.0 ./ 10.0 .^ (7:13)
reltols = 1.0 ./ 10.0 .^ (4:10)

setups = [Dict(:alg=>GRK4A()),
          Dict(:alg=>Rodas4P()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>ddebdf()),
          Dict(:alg=>Rodas5()),
          Dict(:alg=>rodas()),
          Dict(:alg=>radau()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>RadauIIA5()),
]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;verbose=false,
                      dense=false,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

setups = [
          Dict(:alg=>Rodas5()),
          Dict(:alg=>Kvaerno5()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>radau())]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;verbose=false,
                      dense=false,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      appxsol=test_sol,maxiters=Int(1e5),error_estimate=:L2)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>KenCarp5()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>KenCarp3()),
          Dict(:alg=>ARKODE(order=5)),
          Dict(:alg=>ARKODE()),
          Dict(:alg=>ARKODE(order=3))]
names = ["Rosenbrock23" "KenCarp5" "KenCarp4" "KenCarp3" "ARKODE5" "ARKODE4" "ARKODE3"]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;
                      names=names,save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
plot(wp)

wp = WorkPrecisionSet(prob,abstols,reltols,setups;verbose=false,
                      dense=false,appxsol=test_sol,maxiters=Int(1e5),error_estimate=:l2)
plot(wp)

#setups = [#Dict(:alg=>Hairer4()),
          #Dict(:alg=>Hairer42()),
          #Dict(:alg=>Rodas3()),
          #Dict(:alg=>Kvaerno4()),
          #Dict(:alg=>KenCarp5()),
          #Dict(:alg=>Cash4())
#]
#wp = WorkPrecisionSet(prob,abstols,reltols,setups;
#                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5))
#plot(wp)

using DiffEqBenchmarks
DiffEqBenchmarks.bench_footer(WEAVE_ARGS[:folder],WEAVE_ARGS[:file])


# Neighborhood Strength Driven Label Propagation Algorithm
function nsdlpa{V,T<:Real}(g::AbstractGraph{V}, λ::T=1.0, CN::Vector{Int}=triangle(g))
    @graph_requires g edge_map incidence_list vertex_map

    # calculate number of common neighbors
    N = num_vertices(g)
    membership = [1:N]
    label_counters = zeros(T, N)
    dominant_labels = Int[]
    nonzero_labels = Int[]

    running = true
    STEP = 0
    while running
        STEP += 1
        running = false

        # Shuffle the node ordering vector
        X = shuffle(collect(vertices(g)))

        # In the prescribed order, loop over the vertices and reassign labels
        for v in X
            v_idx = vertex_index(v, g)

            # Clear dominant_labels and nonzero_labels
            dominant_labels = Int[]
            nonzero_labels = Int[]

            # recount
            max_count = zero(T)

            for w in out_edges(v, g)
                k = membership[vertex_index(target(w, g), g)]

                # skip if it has no label yet
                k != 0 || continue

                was_zero = label_counters[k] == zero(T)
                label_counters[k] += 1 + λ*CN[edge_index(w, g)]
                if was_zero && label_counters[k] != zero(T)
                    # counter just became nonzero
                    push!(nonzero_labels, k)
                end
                if max_count < label_counters[k]
                    max_count = label_counters[k]
                    resize!(dominant_labels, 1)
                    dominant_labels[1] = k
                elseif max_count == label_counters[k]
                    push!(dominant_labels, k)
                end
            end

            if !isempty(dominant_labels)

                # Select randomly from the dominant labels
                k = dominant_labels[rand(1:length(dominant_labels))]

                # Check if the current label of the node is also dominant
                if label_counters[membership[v_idx]] != max_count
                    # Nope, we need at least one more iteration
                    running = true
                end

                # Update label of the current node
                membership[v_idx] = k
            end
            # Clear the nonzero elements in label_counters
            for i in nonzero_labels
                label_counters[i] = zero(T)
            end
        end
    end
    permute_labels!(membership), STEP
end

function nsdlpa1{V,T<:Real}(g::AbstractGraph{V}, λ::T=1.0, CN::Vector{Int}=triangle(g), DS::Vector{Int}=quadrangle(g))
    @graph_requires g edge_map incidence_list vertex_map

    # calculate number of common neighbors
    N = num_vertices(g)
    membership = [1:N]
    label_counters = zeros(T, N)
    proximity_sum = zeros(T, N)

    running = true
    STEP = 0
    while running
        STEP += 1
        running = false

        # Shuffle the node ordering vector
        X = shuffle(collect(vertices(g)))

        # In the prescribed order, loop over the vertices and reassign labels
        for v in X
            v_idx = vertex_index(v, g)

            # Clear dominant_labels and nonzero_labels
            dominant_labels = Int[]
            nonzero_labels = Int[]
            similarity = Float64[]

            # recount
            max_count = zero(T)

            for w in out_edges(v, g)
                k = membership[vertex_index(target(w, g), g)]

                # skip if it has no label yet
                k != 0 || continue

                was_zero = label_counters[k] == zero(T)
                label_counters[k] += 1 + λ*CN[edge_index(w, g)]
                proximity_sum[k] += DS[edge_index(w, g)]
                if was_zero && label_counters[k] != zero(T)
                    # counter just became nonzero
                    push!(nonzero_labels, k)
                end
                if max_count < label_counters[k]
                    max_count = label_counters[k]
                    resize!(dominant_labels, 1)
                    resize!(similarity, 1)
                    dominant_labels[1] = k
                    similarity[1] = proximity_sum[k]
                elseif max_count == label_counters[k]
                    push!(dominant_labels, k)
                    push!(similarity, proximity_sum[k])
                end
            end

            if !isempty(dominant_labels)

                max_similarity = maximum(similarity)
                dominant_labels = dominant_labels[similarity.<=max_similarity]

                # Select randomly from the dominant labels
                k = dominant_labels[rand(1:length(dominant_labels))]

                # Check if the current label of the node is also dominant
                if label_counters[membership[v_idx]] != max_count
                    # Nope, we need at least one more iteration
                    running = true
                end

                # Update label of the current node
                membership[v_idx] = k
            end
            # Clear the nonzero elements in label_counters
            for i in nonzero_labels
                label_counters[i] = zero(T)
                proximity_sum[i] = zero(T)
            end
        end
    end
    permute_labels!(membership), STEP
end


##
# Represent an m x n rank-r matrix
# A = U V^T
##

balance!(U::Matrix{T},V::Matrix{T},m::Int,n::Int,r::Int) where {T<:Union{Integer,Rational}} = U,V
function balance!(U::Matrix{T},V::Matrix{T},m::Int,n::Int,r::Int) where T
    for k=1:r
        uk = zero(T)
        for i=1:m
            @inbounds uk += abs2(U[i,k])
        end
        vk = zero(T)
        for j=1:n
            @inbounds vk += abs2(V[j,k])
        end
        uk,vk = sqrt(uk),sqrt(vk)
        σk = sqrt(uk*vk)
        if abs2(uk) ≥ eps(T)^2 && abs2(vk) ≥ eps(T)^2
            uk,vk = σk/uk,σk/vk
            for i=1:m
                @inbounds U[i,k] *= uk
            end
            for j=1:n
                @inbounds V[j,k] *= vk
            end
        end
    end
    U,V
end

# constructors

function pad!(L::LowRankMatrix,n::Integer,::Colon)
    L.U=pad(L.U,n,:)
    L
end
function pad!(L::LowRankMatrix,::Colon,m::Integer)
    L.V=pad(L.V,m,:)
    L
end
pad!(L::LowRankMatrix,n::Integer,m::Integer) = pad!(pad!(L,n,:),:,m)