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

using Markdown
using InteractiveUtils

# ╔═╡ 4b348c51-c844-4e64-85e4-a4f7ea952fc9
using Pkg, DrWatson

# ╔═╡ 75ce54b9-b801-412a-b70e-ab5222df137f
begin
	using Random
	using StatsBase
	using Distributions
	using StatsPlots
	using StatsFuns
	using Logging

	using CSV
	using DataFrames
	using Optim

	using MCMCChains

	using Turing
	using StatisticalRethinking
end

# ╔═╡ e803d172-2b8a-43b4-b99d-307e64cf29d5
md"# Chapter 9. Markov Chain Monte Carlo."

# ╔═╡ 77a70403-fa95-4b13-b886-415d1a91f15a
begin
	default(labels=false)
	Logging.disable_logging(Logging.Warn);
end

# ╔═╡ f9ff23ea-fdf2-4d48-bb84-98f40db8603e
md"## 9.1 Good King Markov and his island kingdom."

# ╔═╡ 821bcd65-4ab5-43e8-837e-8bdf96f24edd
md"### Code 9.1"

# ╔═╡ 0ccba2cd-c10c-4713-80e1-77e4a3e9c85c
begin
	Random.seed!(1)
	num_weeks = 10^5
	positions = []
	current = 10
end;

# ╔═╡ b71b59cb-b9ed-4a18-a713-400d8fb48517
for i ∈ 1:num_weeks
    # record current position
    push!(positions, current)
    # flip coin to generate proposal
    proposal = current + sample([-1, 1])
    # handle loops around
    proposal < 1 && (proposal = 10)
    proposal > 10 && (proposal = 1)
    # move?
    prob_move = proposal / current
    rand() < prob_move && (current = proposal)
end

# ╔═╡ 62852231-aebe-43bc-a591-ff3c79651739
md"### Code 9.2"

# ╔═╡ 2f603bc2-938d-45b4-a9ed-60daa5847af5
scatter(positions[1:100], xlab="week", ylab="island")

# ╔═╡ 3f745e69-690e-4ab9-934f-b05f24f20a98
md"### Code 9.3"

# ╔═╡ a88970a9-341c-4e92-891b-6057dbac1e01
histogram(positions, xlab="island", ylab="number of weeks")

# ╔═╡ e7569635-07ab-4b69-8014-f26d42955fce
md"## 9.2 Metropolis algorithms"

# ╔═╡ d4991ad9-4341-47e8-aee4-9c35cbe28e35
md"### Code 9.4"

# ╔═╡ 9322d404-9ebd-4dd8-985d-8322f4f77571
begin
	D = 10
	T = 1000
	Y = rand(MvNormal(zeros(D), ones(D)), T)
	Rd = sqrt.(sum.(eachcol(Y.^2)))
	density(Rd)
end

# ╔═╡ 7d94a7f1-7ce5-4f6a-ba72-0a8a9cde44e9
md"## 9.3 Hamiltonian Monte Carlo"

# ╔═╡ 19d13362-93a0-4a93-b3bb-6e30d3d934a3
md"### Code 9.5"

# ╔═╡ 29a1a866-522b-4354-866e-9f121ba378b9
begin
	Random.seed!(7)

	x = rand(Normal(), 50)
	y = rand(Normal(), 50)
	x = standardize(ZScoreTransform, x)
	y = standardize(ZScoreTransform, y);
end;

# ╔═╡ e3cc237b-9997-41b6-8298-1b79e4444662
function U(q::Vector{Float64}; a=0, b=1, k=0, d=1)::Float64
    μy, μx = q
    U = sum(normlogpdf.(μy, 1, y)) + sum(normlogpdf.(μx, 1, x))
    U += normlogpdf(a, b, μy) + normlogpdf(k, d, μx)
    -U
end

# ╔═╡ cb41155d-43a6-4fef-932f-13cabcdf907a
md"### Code 9.6"

# ╔═╡ 7d29cbaf-2b27-4b4e-8dee-74a049408fba
function ∇U(q::Vector{Float64}; a=0, b=1, k=0, d=1)::Vector{Float64}
    μy, μx = q
    G₁ = sum(y .- μy) + (a - μy) / b^2  # ∂U/∂μy
    G₂ = sum(x .- μx) + (k - μx) / d^2  # ∂U/∂μx
    [-G₁, -G₂]
end

# ╔═╡ 9d674865-3fed-48a6-a970-d997ae9d7443
md"### Codes 9.8 - 9.10 (before 9.7 to define HMC2 function)"

# ╔═╡ aa9be028-c3dc-4e4a-9405-86be642540ca
function HMC2(U, ∇U, ϵ::Float64, L::Int, current_q::Vector{Float64})
    q = current_q
    p = rand(Normal(), length(q))  # random flick - p is momentum
    current_p = p

    # make a half step for momentum at the beginning
    p -= ϵ .* ∇U(q) ./ 2

    # initialize bookkeeping - saves trajectory
    qtraj = [q]
    ptraj = [p]

    # Alternate full steps for position and momentum
    for i ∈ 1:L
        q += @. ϵ * p  # full step for the position
        # make a full step for the momentum except at the end of trajectory
        if i != L
            p -= ϵ * ∇U(q)
            push!(ptraj, p)
        end
        push!(qtraj, q)
    end

    # Make a half step for momentum at the end
    p -= ϵ * ∇U(q) / 2
    push!(ptraj, p)

    # negate momentum at the end of trajectory to make the proposal symmetric
    p = -p

    # evaluate potential and kinetic energies at the start and the end of trajectory
    current_U = U(current_q)
    current_K = sum(current_p.^2)/2
    proposed_U = U(q)
    proposed_K = sum(p.^2)/2

    # accept or reject the state at the end of trajectory, returning either
    # the position at the end of the trajectory or the initial position
    accept = (rand() < exp(current_U - proposed_U + current_K - proposed_K))

    if accept
        current_q = q
    end

    (q=current_q, traj=qtraj, ptraj=ptraj, accept=accept)
end

# ╔═╡ e61d81ed-761a-450b-9ea0-1398a9ac8e22
md"### Code 9.7"

# ╔═╡ a74b283d-febc-40b1-9654-3d63b7d014b2
begin
	Random.seed!(1)
	Q = (q=[-0.1, 0.2],)
	pr = 0.3
	step1 = 0.03
	L = 11
	n_samples = 4
	p = scatter([Q.q[1]], [Q.q[2]], xlab="μx", ylab="μy")

	for i ∈ 1:n_samples
	    Q = HMC2(U, ∇U, step1, L, Q.q)
	    if n_samples < 10
	        cx, cy = [], []
	        for j ∈ 1:L
	            K0 = sum(Q.ptraj[j].^2)/2
	            plot!(
	                [Q.traj[j][1], Q.traj[j+1][1]],
	                [Q.traj[j][2], Q.traj[j+1][2]],
	                lw=1+2*K0,
	                c=:black,
	                alpha=0.5
	            )
	            push!(cx, Q.traj[j+1][1])
	            push!(cy, Q.traj[j+1][2])
	        end
	        scatter!(cx, cy, c=:white, ms=3)
	    end
	    scatter!([Q.q[1]], [Q.q[2]], shape=(Q.accept ? :circle : :rect), c=:blue)
	end
	p
end

# ╔═╡ 4dc8aa1b-7544-4a80-9109-a2c9040d9321
md"## 9.4 Easy HMC: ulam"

# ╔═╡ e6a3d465-57d2-40aa-b71d-8cca5db234d9
md"### Code 9.11"

# ╔═╡ ff957f53-dbc7-4885-9797-61d10b8dac15
begin
	d = CSV.read(sr_datadir("rugged.csv"), DataFrame)
	dd = d[completecases(d, :rgdppc_2000),:]
	dd[:,:log_gdp] = log.(dd.rgdppc_2000);
	dd[:,:log_gdp_std] = dd.log_gdp / mean(dd.log_gdp)
	dd[:,:rugged_std] = dd.rugged / maximum(dd.rugged)
	dd[:,:cid] = @. ifelse(dd.cont_africa == 1, 1, 2);
end;

# ╔═╡ 06d5c8b7-7a87-4ddb-9c30-d6d623a71814
md"### Code 9.12"

# ╔═╡ e29edd1a-890d-4e21-a221-d4311bd845d3
r̄ = mean(dd.rugged_std);

# ╔═╡ 200f0e88-b30f-4aac-a733-a25f5a52c5d8
@model function model_m8_3(rugged_std, cid,  log_gdp_std)
    σ ~ Exponential()
    a ~ MvNormal([1, 1], 0.1)
    b ~ MvNormal([0, 0], 0.3)
    μ = @. a[cid] + b[cid] * (rugged_std - r̄)
    log_gdp_std ~ MvNormal(μ, σ)
end

# ╔═╡ fbcda425-2e88-4a49-b24b-17894231642e
begin
	m8_3 = optimize(model_m8_3(dd.rugged_std, dd.cid,  dd.log_gdp_std), MAP())
	m8_3_df = DataFrame(sample(m8_3, 1000))
	PRECIS(m8_3_df)
end

# ╔═╡ f67d530a-e56d-408c-8f21-d4e37e8dab42
md"### Code 9.13"

# ╔═╡ 60a185df-0656-4a46-b301-388407073ab6
md"#### For Turing this is not needed"

# ╔═╡ 9d6afd05-6570-44e0-8b49-5e4bc7e8a76f
begin
	dat_slim = dd[!,[:log_gdp_std, :rugged_std, :cid]]
	describe(dat_slim)
end

# ╔═╡ cfe67e9f-41bb-419f-96ba-282926de31a2
md"### Code 9.14"

# ╔═╡ 8409b9f4-7fa5-4f38-9b38-ae34e4c60e83
@model function model_m9_1(rugged_std, cid,  log_gdp_std)
    σ ~ Exponential()
    a ~ MvNormal([1, 1], 0.1)
    b ~ MvNormal([0, 0], 0.3)
    μ = @. a[cid] + b[cid] * (rugged_std - r̄)
    log_gdp_std ~ MvNormal(μ, σ)
end

# ╔═╡ 4e94cf59-81eb-4b99-b048-502f54faee15
md"#### One chain will be produced by default"

# ╔═╡ 76782d05-9e32-424c-a206-8081e0fdd146
m9_1 = sample(model_m8_3(dd.rugged_std, dd.cid,  dd.log_gdp_std), NUTS(), 1000);

# ╔═╡ e8dd780e-b94d-413c-9b0b-0476817b2094
md"### Code 9.15"

# ╔═╡ b80e5ff3-445a-465c-9665-697f5b0b897e
PRECIS(DataFrame(m9_1))

# ╔═╡ b6057787-3948-482c-be7e-69c4231386b6
md"### Code 9.16"

# ╔═╡ d438bddd-9138-410e-ab98-40f7c90e780f
md"#### For this to use multiple cores, julia has to be started with `--threads 4` parameter, otherwise chains will be sampled sequentially"

# ╔═╡ 3f4a3b7c-2db7-4d7b-800a-c14968a02855
m9_1_4 = sample(model_m8_3(dd.rugged_std, dd.cid,  dd.log_gdp_std), NUTS(),
	MCMCThreads(), 1000, 4);

# ╔═╡ 2d2aa3ef-8fb7-4c69-8869-42e1939dc950
md"### Code 9.17"

# ╔═╡ 61b5e6a8-3baa-4101-9a21-5d49af43bfe7
md"#### This shows combined chains statistics. To get information about individual chains, use `m9_1[:,:,1]`"

# ╔═╡ 35f25d1e-234b-481f-9303-3c5973ab1228
m9_1

# ╔═╡ 4e20ad94-3011-4b89-b29e-8a793e733e03
md"### Code 9.18"

# ╔═╡ ed026592-6cb3-40a3-9f0a-565e14038114
PRECIS(DataFrame(m9_1[:,:,1]))

# ╔═╡ b25c0b7a-4469-4945-be28-bfb2c22103ca
md"### Code 9.19"

# ╔═╡ 8a0295cc-4f0c-475a-a8c6-ccf2b99ed00f
@df DataFrame(m9_1) corrplot(cols(1:5), seriestype=:scatter, ms=0.2, size=(950, 800), bins=30, grid=false)

# ╔═╡ 580db5a8-550a-42d1-9aef-c95f4099a3fa
md"### Code 9.20"

# ╔═╡ 0bd7dbfe-0689-40d4-8fd0-1fd372b2edc2
traceplot(m9_1)

# ╔═╡ 534f0c53-da8b-4da5-81d4-c0ab3ceb270b
md"### Code 9.21"

# ╔═╡ a28ee7bb-d9d7-43ea-833f-6460a2cf7579
histogram(m9_1)

# ╔═╡ 4535c71f-be7c-41e8-940b-57289c83e509
md"## 9.5 Care and feeding of your Markov chain."

# ╔═╡ f39cd55f-ed8f-487e-885e-f1e16b78d9dd
md"### Codes 9.22 - 9.23"

# ╔═╡ cd971faa-315a-4cea-a11a-0cb28f3e3997
let
	# To make it diverting with Turing, it was needed to increase exp() argument.
	Random.seed!(1)
	y = [-1., 1.]

	@model function model_m9_2(y)
	    α ~ Normal(0, 1000)
	    σ ~ Exponential(1/0.0001)
	    y ~ Normal(α, σ)
	end

	global m9_2 = sample(model_m9_2(y), NUTS(), 1000)
	m9_2_df = DataFrame(m9_2)
	PRECIS(m9_2_df)
end

# ╔═╡ a9febef3-06d0-4552-8590-2f165ce36547
md"### Code 9.23"

# ╔═╡ 7b2588d2-d86d-44fb-9895-d6452875bc47
plot(
    traceplot(m9_2),
    histogram(m9_2),
    size=(900, 500)
)

# ╔═╡ cc97f4d9-ad00-4d2c-b520-2071b9cf86f6
md"### Code 9.24"

# ╔═╡ 631bb03b-00a5-466a-be0d-36f86a9721aa
Random.seed!(2)

# ╔═╡ 95dc66d6-660a-4b0c-b89d-487a22dfec82
@model function model_m9_3(y)
    α ~ Normal(1, 10)
    σ ~ Exponential(1)
    y ~ Normal(α, σ)
end

# ╔═╡ c5570bf1-c04d-41e3-b8ae-6e6dac45cb70
begin
	m9_3 = sample(model_m9_3(y), NUTS(), 1000)
	m9_3_df = DataFrame(m9_3)
	PRECIS(m9_3_df)
end

# ╔═╡ 8f945f6c-2561-433b-aedb-3448d2c4b419
ess_rhat(m9_3)

# ╔═╡ ecd23fa3-fdd2-4ffb-a0e9-ad1feee04d22
md"### Code 9.25 - 9.26"

# ╔═╡ f10e310f-165b-4b4b-b3e0-66af6430ba4f
let
	Random.seed!(41)
	y = rand(Normal(), 100)

	Random.seed!(384)

	@model function model_m9_4(y)
	    a1 ~ Normal(0, 1000)
	    a2 ~ Normal(0, 1000)
	    σ ~ Exponential(1)
	    μ = a1 + a2
	    y ~ Normal(μ, σ)
	end

	global m9_4 = sample(model_m9_4(y), NUTS(), 1000)
	m9_4_df = DataFrame(m9_4)
	PRECIS(m9_4_df)

end

# ╔═╡ 4c09eda3-8f4c-4fa2-91ea-99b92cfbb422
ess_rhat(m9_4)

# ╔═╡ 1f2c5378-bbcd-4fb7-9130-9c6cebb4c6d4
plot(m9_4)

# ╔═╡ 517d31ba-5843-40a7-ad06-96e7a8ea8845
md"### Code 9.27"

# ╔═╡ 1cc0a6c1-3fcf-4df3-a8d9-ba3454c0a37b
Random.seed!(384)

# ╔═╡ 9a76593b-c9a9-45cc-aca5-9877432baffd
@model function model_m9_5(y)
    a1 ~ Normal(0, 10)
    a2 ~ Normal(0, 10)
    σ ~ Exponential(1)
    μ = a1 + a2
    y ~ Normal(μ, σ)
end

# ╔═╡ 6a7ea8be-bea2-41c4-89ab-9a6bbcfba116
begin
	m9_5 = sample(model_m9_5(y), NUTS(), 1000)
	m9_5_df = DataFrame(m9_5)
	PRECIS(m9_5_df)
end

# ╔═╡ 942e3f19-a5c1-430e-a3be-097c53bc7394
ess_rhat(m9_5)

# ╔═╡ 6caa4b75-9ac1-496b-a4b2-38b72b203e59
plot(m9_5)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
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DrWatson = "634d3b9d-ee7a-5ddf-bec9-22491ea816e1"
Logging = "56ddb016-857b-54e1-b83d-db4d58db5568"
MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d"
Optim = "429524aa-4258-5aef-a3af-852621145aeb"
Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
StatisticalRethinking = "2d09df54-9d0f-5258-8220-54c2a3d4fbee"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
StatsFuns = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
Turing = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"

[compat]
CSV = "~0.9.11"
DataFrames = "~1.2.2"
Distributions = "~0.25.34"
DrWatson = "~2.7.5"
MCMCChains = "~5.0.1"
Optim = "~1.5.0"
StatisticalRethinking = "~4.4.3"
StatsBase = "~0.33.13"
StatsFuns = "~0.9.14"
StatsPlots = "~0.14.29"
Turing = "~0.19.0"
"""

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

[[ANSIColoredPrinters]]
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deps = ["LeftChildRightSiblingTrees", "Logging", "Markdown", "Printf", "ProgressLogging", "UUIDs"]
git-tree-sha1 = "62846a48a6cd70e63aa29944b8c4ef704360d72f"
uuid = "5d786b92-1e48-4d6f-9151-6b4477ca9bed"
version = "0.1.5"

[[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"

[[TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "216b95ea110b5972db65aa90f88d8d89dcb8851c"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.6"

[[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"

[[TreeViews]]
deps = ["Test"]
git-tree-sha1 = "8d0d7a3fe2f30d6a7f833a5f19f7c7a5b396eae6"
uuid = "a2a6695c-b41b-5b7d-aed9-dbfdeacea5d7"
version = "0.3.0"

[[Tullio]]
deps = ["ChainRulesCore", "DiffRules", "LinearAlgebra", "Requires"]
git-tree-sha1 = "0288b7a395fc412952baf756fac94e4f28bfec65"
uuid = "bc48ee85-29a4-5162-ae0b-a64e1601d4bc"
version = "0.3.2"

[[Turing]]
deps = ["AbstractMCMC", "AdvancedHMC", "AdvancedMH", "AdvancedPS", "AdvancedVI", "BangBang", "Bijectors", "DataStructures", "Distributions", "DistributionsAD", "DocStringExtensions", "DynamicPPL", "EllipticalSliceSampling", "ForwardDiff", "Libtask", "LinearAlgebra", "MCMCChains", "NamedArrays", "Printf", "Random", "Reexport", "Requires", "SciMLBase", "SpecialFunctions", "Statistics", "StatsBase", "StatsFuns", "Tracker", "ZygoteRules"]
git-tree-sha1 = "c40eac01a46b8efc8af9481982a618d6e54b506d"
uuid = "fce5fe82-541a-59a6-adf8-730c64b5f9a0"
version = "0.19.0"

[[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"

[[UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[VectorizationBase]]
deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "Hwloc", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static"]
git-tree-sha1 = "17e5847bb36730d90801170ecd0ce4041a3dde86"
uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f"
version = "0.21.22"

[[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"]
git-tree-sha1 = "66d72dc6fcc86352f01676e8f0f698562e60510f"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.23.0+0"

[[WeakRefStrings]]
deps = ["DataAPI", "InlineStrings", "Parsers"]
git-tree-sha1 = "c69f9da3ff2f4f02e811c3323c22e5dfcb584cfa"
uuid = "ea10d353-3f73-51f8-a26c-33c1cb351aa5"
version = "1.4.1"

[[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"
"""

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module SimpleLife
export life_step, life_step!, random_life

"""
`_get(A,i,j,wrap=fasle)` returns `A[i,j]` or `0` if subscripts are out of bounds.
With `wrap==true` then the indices are taken modulo the number of rows/columns
as appropriate.
"""
function _get(A::Matrix{T}, i::Int, j::Int, wrap::Bool=false)::T where T<:Number
    r,c = size(A)
    if (1<=i<=r) && (1<=j<=c)
        return @inbounds A[i,j]
    end

    if !wrap
        return zero(T)
    end

    i = mod(i,r)
    j = mod(j,c)
    if i==0
        i=r
    end
    if j==0
        j=c
    end
    return @inbounds A[i,j]
end

"""
`_neighbor_count(A,i,j,wrap=false)` counts how many neighbors of entry `i,j`
of matrix `A` are nonzero.
"""
function _neighbor_count(A::Matrix{Int},i::Int,j::Int,wrap::Bool=false)::Int
    return sum(_get(A,p,q,wrap) for p=i-1:i+1 for q=j-1:j+1 if (p,q) != (i,j))
end

"""
`_new_entry(A,i,j)` computes the new entry in cell `i,j`
if the current state is given by matrix `A`.
"""
function _new_entry(A::Matrix{Int},i::Int,j::Int,wrap::Bool=false)
    nc = _neighbor_count(A,i,j,wrap)
    if A[i,j]==0
        if nc==3
            return 1
        else
            return 0
        end
    end
    # so A[i,j] is 1
    if nc==2 || nc==3
        return 1
    else
        return 0
    end
end

"""
`life_step(A,wrap=false)` gives the next generation after state `A`
in Conway's *Game of Life*. With `wrap==true` treats the board toroidally.
"""
function life_step(A::Matrix{Int},wrap::Bool=false)::Matrix{Int}
    r,c = size(A)
    B = zeros(Int,r,c)
    for i=1:r
        for j=1:c
            @inbounds B[i,j] = _new_entry(A,i,j,wrap)
        end
    end
    return B
end

"""
`life_step!(A,wrap=false)` overwrites `A` with `life_step(A,wrap)`.
This function returns `true` if there has been a change to `A`;
otherwise, returns `false` if there was no change.
"""
function life_step!(A::Matrix{Int}, wrap::Bool=false)
    r,c = size(A)
    B = life_step(A,wrap)
    change = A!=B
    for i=1:r
        for j=1:c
            @inbounds A[i,j] = B[i,j]
        end
    end
    return change
end

"""
`random_life(r,c,p=0.5)` creates a random `r`-by-`c` instance of a life board
with density `p`.

`random_life(n,p=0.5)` is equivalent to `random_life(n,n,p)`.
"""
function random_life(r::Int, c::Int, p::AbstractFloat=0.5)
    M = rand(r,c) .< p
    return Int.(M)
end

random_life(n::Int,p::AbstractFloat=0.5) = random_life(n,n,p)

"""
`_life_check(A)` determines if the integer matrix `A` only contains
zeros and ones.
"""
function _life_check(A::Matrix{Int})
    return 0 <= minimum(A) <= maximum(A) <= 1
end

include("visualize.jl")

end  # module SimpleLife


# Test sample functions

using Distributions
using Base.Test

function est_p(x, K)
	h = zeros(Int, K)
	for xi in x
		h[xi] += 1
	end
	p = h / length(x)
end

#### sample with replacement

n = 10^5
x = sample([10,20,30], n)
@test isa(x, Vector{Int})
@test length(x) == n

h = [sum(x .== 10), sum(x .== 20), sum(x .== 30)]
@test sum(h) == n
ph = h / n
p0 = fill(1/3, 3)
@test_approx_eq_eps ph p0 0.02

#### sample without replacement

# case: K == 2

n = 10^5
x = zeros(Int, 2, n)
for i = 1:n
	v = sample(11:15, 2; replace=false)
	@assert v[1] != v[2]
	x[:,i] = v
end
@test min(x) == 11
@test max(x) == 15

x[:] -= 10  # brings x to 1:5
p0 = fill(1/5, 5)
@test_approx_eq_eps est_p(x, 5) p0 0.02

# case: K == 4 with moderate a (using Fisher-Yates)

n = 10^5
x = zeros(Int, 4, n)
for i = 1 : n
	v = sample(11:20, 4; replace=false)
	sv = sort(v)
	@assert sv[1] < sv[2] < sv[3] < sv[4]
	x[:,i] = v
end
@test min(x) == 11
@test max(x) == 20

x[:] -= 10
p0 = fill(0.1, 10)
@test_approx_eq_eps est_p(x, 10) p0 0.01

# case: K == 4 with very large a (using self-avoid)

n = 10^4
x = zeros(Int, 4, n)
a = 10^7 + 1
b = 2 * 10^7
for i = 1 : n
	v = sample(a:b, 4; replace=false)
	sv = sort(v)
	@assert sv[1] < sv[2] < sv[3] < sv[4]
	x[:,i] = v
end
@test min(x) >= a
@test max(x) <= b

#### weighted sampling

w = [2., 5., 3.]
n = 10^5
x = wsample([10,20,30], w, n)

h = [sum(x .== 10), sum(x .== 20), sum(x .== 30)]
@test sum(h) == n
p0 = w / sum(w)
ph = h / n
@test_approx_eq_eps ph p0 0.02


# This file is a part of AstroLib.jl. License is MIT "Expat".
# Copyright (C) 2016 Mosè Giordano.

function aitoff(l::T, b::T) where {T<:AbstractFloat}
    l = rem(l, T(360), RoundNearest)
    alpha2 = deg2rad(l/2)
    delta = deg2rad(b)
    r2 = sqrt(T(2))
    f = 2*r2/pi
    sin_alpha2, cos_alpha2 = sincos(alpha2)
    sin_delta, cos_delta = sincos(delta)
    denom = sqrt(1 + cos_delta * cos_alpha2)
    return rad2deg(cos_delta * sin_alpha2 * 2 * r2 / denom / f),
           rad2deg(sin_delta * r2 / denom / f)
end

"""
    aitoff(l, b) -> x, y

### Purpose ###

Convert longitude `l` and latitude `b` to `(x, y)` using an Aitoff projection.

### Explanation ###

This function can be used to create an all-sky map in Galactic coordinates with
an equal-area Aitoff projection.  Output map coordinates are zero longitude
centered.

### Arguments ###

* `l`: longitude, scalar or vector, in degrees.
* `b`: latitude, number of elements as `l`, in degrees.

Coordinates can be given also as a 2-tuple `(l, b)`.

### Output ###

2-tuple `(x, y)`.

* `x`: x coordinate, same number of elements as `l`.  `x` is normalized to be in
  \$[-180, 180]\$.
* `y`: y coordinate, same number of elements as `l`.  `y` is normalized to be in
  \$[-90, 90]\$.

### Example ###

Get \$(x ,y)\$ Aitoff coordinates of Sirius, whose Galactic coordinates are
\$(227.23, -8.890)\$.

```jldoctest
julia> using AstroLib

julia> x, y = aitoff(227.23, -8.890)
(-137.92196683723276, -11.772527357473054)
```

### Notes ###

See AIPS memo No. 46
(ftp://ftp.aoc.nrao.edu/pub/software/aips/TEXT/PUBL/AIPSMEMO46.PS), page 4, for
details of the algorithm.  This version of `aitoff` assumes the projection is
centered at b=0 degrees.

Code of this function is based on IDL Astronomy User's Library.
"""
aitoff(l::Real, b::Real) = aitoff(promote(float(l), float(b))...)

aitoff(lb::Tuple{Real, Real}) = aitoff(lb...)

function aitoff(l::AbstractArray{L}, b::AbstractArray{B}) where {L<:Real,B<:Real}
    @assert length(l) == length(b)
    typel = float(L)
    x = similar(l, typel)
    y = similar(b, typel)
    for i in eachindex(l)
        x[i], y[i] = aitoff(l[i], b[i])
    end
    return x, y
end


@testset "nn" begin
    using LinearAlgebra
    k = NeuralNetworkKernel()
    v1 = rand(3); v2 = rand(3)
    @test k(v1,v2) ≈ asin(v1' * v2 / sqrt((1 + v1' * v1) * (1 + v2' * v2))) atol=1e-5

    # kernelmatrix tests
    m1 = rand(3,4)
    m2 = rand(3,4)
    @test kernelmatrix(k, m1, m1) ≈ kernelmatrix(k, m1) atol=1e-5
    @test_broken kernelmatrix(k, m1, m2) ≈ k(m1, m2) atol=1e-5

    x1 = rand()
    x2 = rand()
    @test kernelmatrix(k, x1*ones(1,1), x2*ones(1,1))[1] ≈ k(x1, x2) atol=1e-5

    @test k(v1, v2) ≈ k(v1, v2) atol=1e-5
    @test typeof(k(v1, v2)) <: Real

    @test_broken size(k(m1, m2)) == (4, 4)
    @test_broken size(k(m1)) == (4, 4)

    A1 = ones(4, 4)
    kernelmatrix!(A1, k, m1, m2)
    @test A1 ≈ kernelmatrix(k, m1, m2) atol=1e-5

    A2 = ones(4, 4)
    kernelmatrix!(A2, k, m1)
    @test A2 ≈ kernelmatrix(k, m1) atol=1e-5

    @test size(kerneldiagmatrix(k, m1)) == (4,)
    A3 = kernelmatrix(k, m1)
    @test kerneldiagmatrix(k, m1) ≈ [A3[i, i] for i in 1:LinearAlgebra.checksquare(A3)] atol=1e-5

    A4 = ones(4)
    kerneldiagmatrix!(A4, k, m1)
    @test kerneldiagmatrix(k, m1) ≈ A4 atol=1e-5

    A5 = ones(4,4)
    @test_throws AssertionError kernelmatrix!(A5, k, m1, m2, obsdim=3)
    @test_throws AssertionError kernelmatrix!(A5, k, m1, obsdim=3)
    @test_throws DimensionMismatch kernelmatrix!(A5, k, ones(4,3), ones(3,4))

    @test k([x1], [x2]) ≈ k(x1, x2) atol=1e-5
    test_ADs(NeuralNetworkKernel, ADs = [:ForwardDiff, :ReverseDiff])
    @test_broken "Zygote uncompatible with BaseKernel"
end


#--- test linear solve method (Foster PhD)
using Plots, Revise, VortexDistributions

pv1 = PointVortex(0.1,0.3,1)
v1 = ScalarVortex(pv1)

x = LinRange(-10,10,256)
y = x

psi = Torus(one.(x.*y') |> complex,x,y)

vortex!(psi,v1)

psiv = psi.ψ

heatmap(x,y,angle.(psiv))

#--- define linear solve (Foster)

function vortex_detect2d(psi)
# Detect vortices in a field psi, on plaquettes of a 2D grid.
#
# First we localize vortices on the grid plaquettes using the phase winding
# algorithm.  After we've found which grid plaquettes contain vortices, we
# estimate the positions more accurately by solving the equations
#
#   real(psi(x,y)) = 0, imag(psi(x,y)) = 0
#
# In principle we could do this using the exact psi as computed via spectral
# interpolation (provided psi is defined in a spectral basis) but that would
# probably be relatively slow.  Instead, we make a linear approximation over
# the plaquette for the real and imaginary parts of psi, and find the position
# where these approximations are mutually zero.  This corresponds to the
# position of the vortex.
#
# The linear approximations are computed from the values of psi at the corners
# of the plaquette using least squares.
#
# Inputs:
#   psi - complex wavefunction in which to detect vortices.
#
# Outputs:
#   positions - Array of vortex positions with shape 2xN where N is the number
#               of vortices:  [ x1  x2  x3 ... xN ]
#                             [ y1  y2  y3 ... yN ]
#   winding - The vortex winding number of the positions array, +1 or -1.

vorticity = phase_winding2d(angle(psi));
[iy,ix] = find(vorticity);
positions = [ix,iy]';

if nargout > 1
    winding = vorticity(vorticity~=0);
end

# % Compute least squares estimator for the linear function coefficients on a
# % plaquette.  Suppose the real part of psi on a plaquette containing a vortex
# % is
# %
# %   [r1 r2;
# %    r3 r4]
# %
# % We want to fit a plane (a*x + b*y + c) to this such that
# %
# %   r1 = a*0 + b*0 + c*1
# %   r2 = a*1 + b*0 + c*1
# %   r3 = a*0 + b*1 + c*1
# %   r4 = a*1 + b*1 + c*1
# %
# % In matrix form, that would be L * [a;b;c] = [r1;r2;r3;r4], which is the
# % origin of the mysterious looking matrix below.
# %
L = [0 0 1;
     1 0 1;
     0 1 1;
     1 1 1];
lsEst = inv(L'*L)*L';
# % (For future reference, lsEst actually turns out to be the following)
# % lsEst = 1/4 * [-2,  2, -2,  2;
# %                -2, -2,  2,  2;
# %                 3,  1,  1, -1];
for i = 1:length(ix)
    # Extract relevant grid plaquette containing the vortex.
    p = psi(iy(i)+[0,1], ix(i)+[0,1]).'
    # Compute linear approximations of real & imag parts of p
    rcoeff = lsEst*real(p[:])
    icoeff = lsEst*imag(p[:])
    # Find position where these are mutually zero
    pos = [rcoeff(1:2), icoeff(1:2)]' \ -[rcoeff(3); icoeff(3)]
    positions[:,i] = positions[:,i] + pos;
    return positions, winding
end


export quadraticFieldmap, stairsFieldmap, polarFieldmap

"""
    quadraticFieldmap(Nx::Int64, Ny::Int64, maxOffresonance::Float64=125.0)

Computes a parabolic fieldmap.
"""
function quadraticFieldmap(Nx::Int64, Ny::Int64, maxOffresonance::Float64=125.0)
  xx = range(-1, stop=1, length=Nx)
  yy = range(-1, stop=1, length=Ny)
  fieldmap = zeros(Nx,Ny, 1)
  for nx=1:Nx
    for ny=1:Ny
      fieldmap[nx,ny,1] = maxOffresonance*(xx[nx]^2 + yy[ny]^2) - maxOffresonance
    end
  end
  return fieldmap
end

"""
Computes a polaroid fieldmap.
"""
function polarFieldmap(Nx::Int64, Ny::Int64,
                        maxOffresonance::Float64=125.0;
                        xCenter::Int64=0,
                        yCenter::Int64=0
                       )
  fieldmap = zeros(Nx,Ny,1)
  xx = range(-1, stop=1, length=Nx)
  yy = range(-1, stop=1, length=Ny)
  for nx=1:Nx
    for ny=1:Ny
      fieldmap[nx,ny,1] = maxOffresonance*sqrt(xx[nx]^2 + yy[ny]^2)- maxOffresonance
    end
  end
  return fieldmap
end

"""
Computes a fieldmap consisting of stairs, which has discrete linear
ascending/descending values
"""
function stairsFieldmap(Nx::Int64, Ny::Int64;
                        numStairs::Int64=8,
                        minOffresonance::Float64=-125.0,
                        maxOffresonance::Float64=125.0,
                        ascending=true)
  # Input check for the number of stairs
  if numStairs < 2 || numStairs > (Nx > Ny ? Ny : Nx)
    error("Number of stairs not used correctly")
  end
  # Allocating output
  fieldmap = zeros(Nx,Ny)
  # Determine the height and width of the stairs (e.g for NxN -> deltaX = deltaY)
  deltaX = Int64(floor((Nx / (numStairs*2) )))
  deltaY = Int64(floor((Ny / (numStairs*2) )))

  if ascending == true
    scaling = range(minOffresonance,stop=maxOffresonance,length=numStairs)
  else
    scaling = -range(minOffresonance,stop=maxOffresonance,length=numStairs)
  end

  # Filling (Quarter) Matrix for constructing the stairs
  partialFieldMap = zeros(Int64(floor((Nx/2))),Int64(floor((Ny/2))))

  for i=1:numStairs
    partialFieldMap[(i-1)*deltaX+1 : end     , (i-1)*deltaY+1 : i*deltaY  ] .= -1 * scaling[i]
    partialFieldMap[(i-1)*deltaX+1 : i*deltaX, (i-1)*deltaY+1 : end ]       .= -1 * scaling[i]
  end

  # Correction if numOfStairs is not divisible by 4 fill the lower right corner of
  # the upper left (quarter) partial matrix
  if numStairs % 4 != 0
    partialFieldMap[numStairs*deltaX:end , numStairs*deltaY:end] .= -1 * scaling[numStairs]
  end

  # If width or height is not divisible by two the middle row or column(or both)
  # must be filled manually
  partSizeX , partSizeY = size(partialFieldMap)
  xRemainder, yRemainder = Nx%2, Ny%2

  # Rotating and mirroring the filled quarter of the matrix to complete the stairs
  fieldmap[1:partSizeX                , 1: partSizeY]                 = partialFieldMap
  fieldmap[1:partSizeX                , partSizeY+1+yRemainder : end] = partialFieldMap[:,end:-1:1]
  fieldmap[partSizeX+1+xRemainder:end , 1: partSizeY]                 = partialFieldMap[end:-1:1,:]
  fieldmap[partSizeX+1+xRemainder:end , partSizeY+1+yRemainder : end] = rot180(partialFieldMap)

  # Filling the row or column,needed when number of rows/columns is odd
  if xRemainder != 0 && yRemainder != 0
    midXIndx, midYIndx  = Int64(ceil(Nx/2)), Int64(ceil(Ny/2))
    fieldmap[midXIndx,: ] = fieldmap[midXIndx-1,:]
    fieldmap[:,midYIndx ] = fieldmap[:,midYIndx-1]
  elseif xRemainder != 0
    midXIndx = Int64(ceil(Nx/2))
    fieldmap[midXIndx,: ] = fieldmap[midXIndx-1,:]
  elseif yRemainder != 0
    midYIndx = Int64(ceil(Ny/2))
    fieldmap[:,midYIndx ] = fieldmap[:,midYIndx-1]
  end

  return reshape(fieldmap,Nx,Ny,1)
end


function r2cf(n1::Integer, n2::Integer)
    ret = Int[]
    while n2 != 0
        n1, (t1, n2) = n2, divrem(n1, n2)
        push!(ret, t1)
    end
    ret
end
r2cf(r::Rational) = r2cf(numerator(r), denominator(r))

function r2cf(n1, n2, maxiter=20)
    ret = Int[]
    while n2 != 0 && maxiter > 0
        n1, (t1, n2) = n2, divrem(n1, n2)
        push!(ret, t1)
        maxiter -= 1
    end
    ret
end

mutable struct NG
    a1::Int
    a::Int
    b1::Int
    b::Int
end

function ingress(ng, n)
    ng.a, ng.a1= ng.a1, ng.a + ng.a1 * n
    ng.b, ng.b1 = ng.b1, ng.b + ng.b1 * n
end

needterm(ng) = ng.b == 0 || ng.b1 == 0 || !(ng.a // ng.b == ng.a1 // ng.b1)

function egress(ng)
    n = ng.a // ng.b
    ng.a, ng.b = ng.b, ng.a - ng.b * n
    ng.a1, ng.b1 = ng.b1, ng.a1 - ng.b1 * n
    r2cf(n)
end

egress_done(ng) = (if needterm(ng) ng.a, ng.b = ng.a1, ng.b1 end; egress(ng))

done(ng) = ng.b == 0 && ng.b1 == 0

function testng()
    data = [["[1;5,2] + 1/2",      [2,1,0,2], [13,11]],
        ["[3;7] + 1/2",        [2,1,0,2], [22, 7]],
        ["[3;7] divided by 4", [1,0,0,4], [22, 7]],
        ["[1;1] divided by sqrt(2)", [0,1,1,0], [1,sqrt(2)]]]

    for d in data
        str, ng, r = d[1], NG(d[2]...), d[3]
        print(rpad(str, 25), "->")
        for n in r2cf(r...)
            if !needterm(ng)
                print(" $(egress(ng))")
            end
            ingress(ng, n)
        end
        while true
            print(" $(egress_done(ng))")
            if done(ng)
                println()
                break
            end
        end
    end
end

testng()


#Code written by Scott Hanna 6/27/2019
#Last Update: 8/14/2019
#Contact: shanna7@jhu.edu
#
#The Modulus type is here defined as the residue class of a modulo n where
#[a] = a + nZ := {a +nz | z ∈ Z}
#the quotient set with respect to Rn, is often denoted by Z/nZ.
#Thus, Z/nZ := {[0], [1], . . . , [n − 1]}.
#
#The set of all classes modulo n is a field only in the case that n is prime.
#i.e. it only guarantees an inverse element if n is prime.  If n is not prime
#it is possible that a particular value, a,has an inverse (that is a*a^-1 = 1),
#but the set is still not considered a field.
module ModulusN

using LinearAlgebra, Random
import Base: convert,promote_rule,+,-,*,/,zero,one,abs,abs2,^,<,<=,>,>=,==,isequal,inv,isnan
import LinearAlgebra: conj
import Random: rand

export getVal,getMod,Modulus,hasInverse

struct Modulus{N} <: Number
    val::Real
    function Modulus{N}(val) where N
        @assert N isa Real
        @assert N != 0
        new{N}(mod(val,N))
    end
    Modulus{N}(x::Modulus{N}) where N = Modulus{N}(x.val)
end

convert(::Modulus{N}, x::M) where {N,M<:Real} = Modulus{N}(x)
promote_rule(::Type{Modulus{N}}, ::Type{M}) where {N,M<:Real} = Modulus{N}
promote_rule(::Type{M}, ::Type{Modulus{N}}) where {N,M<:Real} = Modulus{N}

getVal(x::Modulus{N}) where N = return x.val
getMod(x::Modulus{N}) where N = return N

+(x::Modulus{N},y::Modulus{N}) where N = Modulus{N}(x.val + y.val)
-(x::Modulus{N},y::Modulus{N}) where N = Modulus{N}(x.val - y.val)
*(x::Modulus{N},y::Modulus{N}) where N = Modulus{N}(x.val * y.val)
/(x::Modulus{N},y::Modulus{N}) where N = inv(y) * x
-(x::Modulus{N}) where N = Modulus{N}(-x.val)
zero(x::Modulus{N}) where N = Modulus{N}(0)
one(x::Modulus{N}) where N = Modulus{N}(1)
abs(x::Modulus{N}) where N = Modulus{N}(abs(x.val))
^(x::Modulus{N}, y::Int) where N = Modulus{N}(x.val^y)
abs2(x::Modulus{N}) where N = abs(x)^2

<(x::Modulus{N}, y::Modulus{N}) where N = x.val < y.val
<=(x::Modulus{N}, y::Modulus{N}) where N = x.val <= y.val
>(x::Modulus{N}, y::Modulus{N}) where N = x.val > y.val
>=(x::Modulus{N}, y::Modulus{N}) where N = x.val >= y.val
==(x::Modulus{N}, y::Modulus{M}) where {N,M} = x.val == y.val && N==M
isequal(x::Modulus{N}, y::Modulus{M}) where {N,M} = x.val == y.val && N==M

isnan(x::Modulus{N}) where N = isnan(x.val)
conj(x::Modulus{N}) where N = abs(x)

function inv(x::Modulus{N}) where N
    divs,v = gcdx(x.val,N)
    if divs != 1 || N == 1
        error("No Inverse Exists")
    end
    return Modulus{N}(v)
end

function hasInverse(x::Modulus{N}) where N
    return gcd(x.val,N)==1 && N != 1
end

rand(rng::AbstractRNG, ::Random.SamplerType{Modulus{N}}) where N = Modulus{N}(rand(rng,Int))

end #end module ModulusN


module Radon

export radon
export backprojection, backprojection_anim
export iradon, fitered_backprojection, SART

"""
    radon(f::Matrix{T}; θ=Vector(0:(π/180):π-π/180)::Vector{T}, in_circle=true) where {T<:Real}

return Radon transform of `f`.

# Keyword arguments

- `f`: 2-dimension array.(phantom image)
- `θ`: projection angle.(radian)
- `in_circle`:
"""
function radon(f::Matrix{T}; θ=Vector(0:(π/180):π-π/180)::Vector{T}, in_circle=true) where {T<:Real}
    ℛf = zeros(maximum(size(f)), length(θ))

    slim, θlim = size(ℛf)
    ylim, xlim = size(f)

    if in_circle == true
        L = ylim
    else
        L = round(Int64, hypot(xlim, ylim))
    end

    cos_ = cos.(θ)
    sin_ = sin.(θ)

    s = LinRange(-L/2, L/2, slim)
    t = LinRange(-L/2, L/2, L)

    for sᵢ ∈ 1:slim, θᵢ ∈ 1:θlim
        for l ∈ 1:L
            x = round(Int64, xlim/2 + s[sᵢ]*cos_[θᵢ] + t[l]*sin_[θᵢ])
            y = round(Int64, ylim/2 - t[l]*cos_[θᵢ] + s[sᵢ]*sin_[θᵢ])
            if 1 ≤ x ≤ xlim && 1 ≤ y ≤ ylim
                ℛf[sᵢ, θᵢ] += f[end-y+1, x]
            end
        end
    end

    return ℛf
end

"""
    backprojection(ℛf)(f::Matrix{T}, θ::Vector{T}) where {T<:Real}

return [back projection](https://freshrimpsushi.github.io/posts/back-projection-dual-transform-of-radon-transform/) of `ℛf`

# Keyword arguments

- `ℛf`: 2d array w.r.t. Radon transform of 'f'
- `θ`: projection angle(radian)
"""
function backprojection(ℛf, θ = 0:π/180:π-π/180)
    slim, θlim = size(ℛf)
    reconstructed_image = zeros(slim, slim)

    ylim, xlim = size(reconstructed_image)
    L = round(Int64, hypot(xlim, ylim))

    x= range(-xlim/2+1, xlim/2, xlim)
    y= range(ylim/2, -ylim/2+1, ylim)

    X = x'.*ones(xlim)
    Y = ones(ylim)'.*y

    s = range(-L/2, L/2, length=slim)

    for (value, angle) ∈ zip(eachcol(ℛf), θ)
        S = X.*cos(angle) + Y.*sin(angle)
        interpolation = LinearInterpolation(s, value, extrapolation_bc=Line())
        reconstructed_image .+= interpolation.(S)
    end
    return reconstructed_image
end

"""
    backprojection_anim(f::Matrix{T}, θ::Vector{T}) where {T<:Real}

return animation for calculation backprojection of ℛf

# Keyword arguments

- `ℛf`: 2d array w.r.t. Radon transform of 'f'
- `θ`: projection angle(radian)
"""
function backprojection_anim(ℛf)
    slim, θlim = size(ℛf)
    reconstructed_image = zeros(slim, slim)

    slim, θlim= size(ℛf)
    ylim, xlim = size(reconstructed_image)
    L = round(Int64, hypot(xlim, ylim))

    x= range(-xlim/2+1, xlim/2, xlim)
    y= range(ylim/2, -ylim/2+1, ylim)

    X = x'.*ones(xlim)
    Y = ones(ylim)'.*y

    θ = 0 : π/θlim : π-π/θlim
    s = range(-L/2, L/2, length=slim)

    anim = Animation()
    for (value, angle) ∈ zip(eachcol(ℛf), θ)
        S = X.*cos(angle) + Y.*sin(angle)
        interpolation = LinearInterpolation(s, value, extrapolation_bc=Line())
        reconstructed_image .+= interpolation.(S)
        frame(anim, heatmap(reverse(reconstructed_image, dims=1)))
    end
    for _ ∈ 1:80
        frame(anim, heatmap(reverse(reconstructed_image, dims=1)))
    end
    return anim
end

"""
    iradon(ℛf, method="fbp")

return Radon transform of `f`

# Keyword arguments

- `f`: phantom
- `θ`: projection angle
"""
function iradon(ℛf, method="fbp")

    if method == "fbp"
        reconstructed_f = fitered_backprojection(ℛf)
    end

    return reconstructed_f
end

"""
    fitered_backprojection(ℛff::Matrix{T}, θ::Vector{T}) where {T<:Real}
return f

# Keyword arguments

- `ℛf`: 2d array w.r.t. Radon transform of `f`
- `θ`: projection angle
"""
function fitered_backprojection(ℛf)
    N = size(ℛf)[1]

    ℱℛf = fft(ℛf, [1])

    fourier_filter = abs.(fftfreq(N))
    filterd_ℱℛf = ℱℛf .* fourier_filter

    ℱ⁻¹_filterd_ℱℛf = real.(ifft(filterd_ℱℛf, [1]))

    reconstructed_f = (π/360)*backprojection(ℱ⁻¹_filterd_ℱℛf)
    return reconstructed_f
end

"""
    SART(ℛf)
return Radon transform of `f`

# Keyword arguments

- `ℛf`: 2d array w.r.t. Radon transform of 'f'
- `θ`: projection angle
"""
function SART(ℛf)
end

end # module