### A Pluto.jl notebook ###
# v0.19.40

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

# ╔═╡ d2637322-e08b-11ee-323b-1b12a99d682e
begin
	using CairoMakie
	using Distributions
	import Distributions: pdf
	using WGLMakie
	using PlutoUI
	using Random
	import Random: rand
	using SpecialFunctions
	WGLMakie.activate!()
end

# ╔═╡ bc0eac9d-8958-4290-bbd0-231ba12ae0e7
Random.seed!(42)

# ╔═╡ a8e3b493-4b31-44da-8f26-e197f964e715
md"""
# Introduction / Motivation
1. Why Monte-Carlo methods?
2. Why geneneralized inverse gaussian?
## Monte-Carlo methods

"""

# ╔═╡ fd01d6a8-28e2-4cd5-890d-2e478597c64b
md"## Random Number Generation"

# ╔═╡ 757c3eaa-8803-4a65-9cdf-f0c7cf13ee2f
md"Iterations $(@bind niter NumberField(100:100_000; default = 300))"

# ╔═╡ c7279d6b-9066-414d-80f8-786f93c4f9f3
md"""
## Generalized Inverse Gaussian (GIG)
Used in:
- __finance:__ asset returns
- __geostatics:__ precipitation levels, pollutant concentrations
- __statistical linguistics:__ word lengths
Main benefits:
- able to model __heavy tails__
- able to model __high skewness__ and data __asymmetry__
- __flexibilty__ of the parametric representation $\rightarrow$ special cases include _inverse Gaussian_ (``\lambda \equiv -\tfrac{1}{2}``), _gamma_ (``\chi \equiv 0``) and _inverse gamma_ (``\psi  \equiv 0``)

```math
f(x; \lambda, \psi, \chi) = \frac{(\psi/\chi)^{\lambda/2}}{2 \cdot \operatorname{K}_{p}(\sqrt{\psi\chi})} \cdot x^{(\lambda - 1)} \cdot \exp\left\{ - \frac{1}{2} \left(\frac{\chi}{x} + \psi x\right)\right\}, \quad x > 0
```
"""

# ╔═╡ 26797a7a-4af5-4b26-be6b-1917aedab58c
begin
	WGLMakie.activate!()
	local fig = Figure()
	local slg = SliderGrid(fig[2,1],
		(label = L"\lambda", range = -5:0.01:5, startvalue = 1),
		(label = L"\psi", range = 0:0.01:5, startvalue = 1),
		(label = L"\chi", range = 0:0.01:5, startvalue = 1)
	)
	local λ = slg.sliders[1].value
	local ψ = slg.sliders[2].value
	local χ = slg.sliders[3].value

	local pdf = @lift x -> begin
		    if x > 0
        	ω = sqrt($ψ * $χ)
        return ($ψ / $χ)^($λ / 2) / (2 * besselk($λ, ω)) * x^($λ - 1) * exp(-($ψ * x + $χ / x) / 2)
    	else
        	return 0
    	end
	end
	lines(fig[1,1], 0..10, pdf)
	fig
end

# ╔═╡ 211032c9-18a9-4b51-b733-9cef4532e923
md"""
# Accept-Reject Sampling
- John von Neumann 1951
- Universal sampler $\equiv$ applicable for a wide selection of distributions
"""

# ╔═╡ bbd54830-2abf-4cab-b805-ab0add47b340
md"""
## Example - Beta distribution
- target distribution $\mathop{Beta}(\alpha, \beta)$

```math
	p_0(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{\mathop{B}(\alpha, \beta)},
```
where
```math
	\mathop{B}(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}
```

- proposal distribution $\mathcal{U}_{[0, 1]}$
```math
	\pi(x) = 1
```
"""

# ╔═╡ 8774e552-868f-4aad-a4c7-30e4be56d535
md"""
We need to form an envelope from the proposal distribution for the target distribution...

```math
	L \coloneqq \max_{x \in [0, 1]} p_0(x) = p_0(\tfrac{\alpha - 1}{\alpha + \beta - 2}) = \frac{(\alpha - 1)^{\alpha - 1} (\beta - 1)^{\beta - 1}}{(\alpha + \beta - 2)^{\alpha + \beta - 2} \mathop{B}(\alpha, \beta)}
```
Thus
```math
	L \pi(x) \geq p_0(x) \quad \forall x \in [0,1]
```
"""

# ╔═╡ dbbae56a-a053-4ca8-a217-6ae57aed2af4
md"""
Histogram (nbins = $(@bind nbins_beta Scrubbable(1:100; default = 26))) of $(@bind nsamps_beta NumberField(1:5000; default = 10_000)) samples from ``\mathop{Beta} (\alpha = 5, \beta = 3)``.
"""

# ╔═╡ 14929f52-664d-4cc4-abcc-5bd6a964a858
md"""
## Ratio of Uniforms method
$(LocalResource("/home/kunzaatko/Current_Semestr/studentský_matematický_seminář/RoU.png"))
"""

# ╔═╡ e9f77fcd-d272-412b-bf3a-bb0f8b2fafc7
md"## Sampling the GIG"

# ╔═╡ 7d070844-c122-4701-a2fa-9aed336e8378
begin
	struct GeneralizedInverseGaussian{T<:Real} <: ContinuousUnivariateDistribution
    λ::T
    ψ::T
    χ::T
    GeneralizedInverseGaussian{T}(λ::T, ψ::T, χ::T) where {T<:Real} = new{T}(λ, ψ, χ)
end

function GeneralizedInverseGaussian(λ::T, ψ::T, χ::T; check_args::Bool=true) where {T<:Real}
    if iszero(χ)
        return Gamma(λ, 2 / ψ)
    elseif iszero(ψ)
        return InverseGamma(-λ, χ / 2)
    end
    return GeneralizedInverseGaussian{T}(λ, ψ, χ)
end
GeneralizedInverseGaussian(λ::Real, ψ::Real, χ::Real; check_args::Bool=true) = GeneralizedInverseGaussian(promote(λ, ψ, χ)...; check_args=check_args)
GeneralizedInverseGaussian(λ::Integer, ψ::Integer, χ::Integer; check_args::Bool=true) = GeneralizedInverseGaussian(float(λ), float(ψ), float(χ); check_args=check_args)

	params(d::GeneralizedInverseGaussian) = (d.λ, d.ψ, d.χ)
function Distributions.pdf(d::GeneralizedInverseGaussian{T}, x::Real) where {T<:Real}
    if x > 0
        λ, ψ, χ = params(d)
        ω = sqrt(ψ * χ)
        return (ψ / χ)^(λ / 2) / (2 * besselk(λ, ω)) * x^(λ - 1) * exp(-(ψ * x + χ / x) / 2)
    else
        return zero(T)
    end
end

end

# ╔═╡ 43c34a72-8ade-4cce-b9f4-1154388440da
begin
	WGLMakie.activate!()
	y_0 = 0.1
	ys = [y_0]
	local λ = 4
	for i in 1:niter
		push!(ys, λ * ys[end] * (1 - ys[end]))
	end
	local fig, ax, _  = hist(ys; normalization = :pdf, bins = 100)
	lines!(ax, 0..1, x -> pdf(Beta(0.5, 0.5), x); color = :orange)
	fig
end

# ╔═╡ aba10bfd-edaf-4179-bfb5-e01015d373c7
begin
	WGLMakie.activate!()
	local fig = Figure(;)
	local slg = SliderGrid(fig[2,1],
		(label = L"\alpha", range = 1:0.01:5, startvalue = 1.5),
		(label = L"\beta", range = 1:0.01:5, startvalue = 3.5)
	)
	local dist = @lift Beta($(slg.sliders[1].value), $(slg.sliders[2].value))
	lines(fig[1,1], 0..1, @lift(x -> pdf($dist, x)); axis = (;title = L"p_0(x; \alpha, \beta)",))
	fig
end

# ╔═╡ 890659d8-4568-4c65-909d-0f0cfbec1de6
begin
struct GIGRejectionNonConcaveSampler{T<:Real} <: Sampleable{Univariate,Continuous} # Refection-based for non-concave parametric subset
    # instrument quasi-density parameters
    λ::T
    α::T
    β::T
    g::Function

    m::T # mode
    x_0::T # inflection point
    x_star::T # problematic interval right-bound

    # (0, x_0)
    k_1::T
    A_1::T

    # (x_0, 2/β)
    k_2::T
    A_2::T

    # (x_star, ∞)
    k_3::T
    A_3::T

    A::T
end

function GIGRejectionNonConcaveSampler(d::GeneralizedInverseGaussian{T}) where {T}
    # TODO: Warn for parameters that are not in the domain of the sampler <08-02-24> 
    λ = d.λ             # parameters λ of the instrument quasi-density
    α = sqrt(d.ψ / d.χ) # scaling parameter of the sampling
    β = sqrt(d.ψ * d.χ) # parameters β of the instrument quasi-density

    g(x) = x^(λ - 1) * exp(-(β / 2) * (x + 1 / x)) # quasi-density of Y = α ⋅X

    m = β / ((1 - λ) + sqrt((1 - λ)^2 + β^2))
    x_0 = β / (1 - λ)
    x_star = max(x_0, 2 / β)

    k_1 = g(m)
    A_1 = k_1 * x_0

    local k_2, A_2
    if x_0 < 2 / β
        k_2 = exp(-β)
        A_2 = λ == 0 ? k_2 * log(2 / β^2) : k_2 * ((2 / β)^λ - x_0^λ) / λ
    else
        k_2 = 0
        A_2 = 0
    end

    k_3 = x_star^(λ - 1)
    A_3 = 2k_3 * exp(-x_star * β / 2) / β

    A = A_1 + A_2 + A_3

    return GIGRejectionNonConcaveSampler(T(λ), T(α), T(β), g, T(m), T(x_0), T(x_star), T(k_1), T(A_1), T(k_2), T(A_2), T(k_3), T(A_3), T(A))
end

function Random.rand(rng::AbstractRNG, s::GIGRejectionNonConcaveSampler)
    while true
        U = rand(rng, Uniform(0, 1))
        V = rand(rng, Uniform(0, s.A))
        if V <= s.A_1
            Y = s.x_0 * V / s.A_1
            h = s.k_1
        elseif V <= s.A_1 + s.A_2
            V = V - s.A_1
            Y = s.λ == 0 ? s.β * exp(V * exp(s.β)) : (s.x_0^s.λ + V * s.λ / s.k_2)^(1 / s.λ)
            h = s.k_2 * Y^(s.λ - 1)
        else
            V = V - (s.A_1 + s.A_2)
            Y = -2 / s.β * log(exp(-s.x_star * s.β / 2) - V * s.β / (2 * s.k_3))
            h = s.k_3 * exp(-Y * s.β / 2)
        end
        if U * h <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end

end

# ╔═╡ b2d43deb-a848-4725-9c1e-609d474b25df
md"""
```julia
function rand(rng::AbstractRNG, s::GIGRejectionNonConcaveSampler)
    while true
        U = rand(rng, Uniform(0, 1))
        V = rand(rng, Uniform(0, s.A))
        if V <= s.A_1
            Y = s.x_0 * V / s.A_1
            h = s.k_1
        elseif V <= s.A_1 + s.A_2
            V = V - s.A_1
            Y = s.λ == 0 ? s.β * exp(V * exp(s.β)) : (s.x_0^s.λ + V * s.λ / s.k_2)^(1 / s.λ)
            h = s.k_2 * Y^(s.λ - 1)
        else
            V = V - (s.A_1 + s.A_2)
            Y = -2 / s.β * log(exp(-s.x_star * s.β / 2) - V * s.β / (2 * s.k_3))
            h = s.k_3 * exp(-Y * s.β / 2)
        end
        if U * h <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end
```
"""

# ╔═╡ 918d0f71-e321-4011-99ed-76cf4a10fb39
begin
	struct GIGRoUNoModeShiftSampler{T<:Real} <: Sampleable{Univariate,Continuous} # Ratio-of-Uniforms without mode shift
    α::T
    g::Function
    v_plus::T
    u_plus::T
end

function GIGRoUNoModeShiftSampler(d::GeneralizedInverseGaussian{T}) where {T}
    λ = d.λ             # parameters λ of the instrument quasi-density
    α = sqrt(d.ψ / d.χ) # scaling parameter of the distribution
    β = sqrt(d.ψ * d.χ) # parameters β of the instrument quasi-density

    g(x) = x^(λ - 1) * exp(-(β / 2) * (x + 1 / x)) # quasi-density of Y = α ⋅X

    m = β / ((1 - λ) + sqrt((1 - λ)^2 + β^2)) # mode
    x_plus = ((1 + λ) + sqrt((1 + λ)^2 + β^2)) / 2

    v_plus = sqrt(g(m))
    u_plus = x_plus * sqrt(g(x_plus))

    return GIGRoUNoModeShiftSampler(T(α), g, T(v_plus), T(u_plus))
end

function Random.rand(rng::AbstractRNG, s::GIGRoUNoModeShiftSampler)
    while true
        U = rand(rng, Uniform(0, s.u_plus))
        V = rand(rng, Uniform(0, s.v_plus))
        Y = U / V
        if V^2 <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end
end

# ╔═╡ 12e26a60-28d9-48a4-b8db-189ed3d9b4e8
md"""
```julia
function rand(rng::AbstractRNG, s::GIGRoUNoModeShiftSampler)
    while true
        U = rand(rng, Uniform(0, s.u_plus))
        V = rand(rng, Uniform(0, s.v_plus))
        Y = U / V
        if V^2 <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end
```
"""

# ╔═╡ 95cae5d3-e233-4310-8e0d-cb93d7a205f9
begin
	struct GIGRoUWithModeShiftSampler{T<:Real} <: Sampleable{Univariate,Continuous} # Ratio-of-Uniforms with mode shift
    α::T
    g::Function

    m::T # mode

    v_plus::T
    u_minus::T
    u_plus::T
end

function GIGRoUWithModeShiftSampler(d::GeneralizedInverseGaussian{T}) where {T}
    λ = d.λ             # parameters λ of the instrument quasi-density
    α = sqrt(d.ψ / d.χ) # scaling parameter of the distribution
    β = sqrt(d.ψ * d.χ) # parameters β of the instrument quasi-density

    g(x) = x^(λ - 1) * exp(-(β / 2) * (x + 1 / x)) # quasi-density of Y = α ⋅X
    m = ((λ - 1) + sqrt((λ - 1)^2 + β^2)) / β # mode
    a = -2 * (λ + 1) / β - m
    b = 2 * (λ - 1) * m - 1
    c = m
    p = b - a^2 / 3
    q = 2a^3 / 27 - a * b / 3 + c
    ϕ = acos(-q / 2 * sqrt(-27 / p^3))
    x_minus = sqrt(-4 / 3 * p) * cos(ϕ / 3 + 4π / 3) - a / 3
    x_plus = sqrt(-4 / 3 * p) * cos(ϕ / 3) - a / 3
    v_plus = sqrt(g(m))
    u_minus = (x_minus - m) * sqrt(g(x_minus))
    u_plus = (x_plus - m) * sqrt(g(x_plus))

    GIGRoUWithModeShiftSampler(T(α), g, T(m), T(v_plus), T(u_minus), T(u_plus))
end

function Random.rand(rng::AbstractRNG, s::GIGRoUWithModeShiftSampler)
    while true
        U = rand(rng, Uniform(s.u_minus, s.u_plus))
        V = rand(rng, Uniform(0, s.v_plus))
        Y = U / V + s.m
        if V^2 <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end
end

# ╔═╡ 60a5c466-af9c-4796-980a-7502dc5cf26f
begin
	WGLMakie.activate!()
	local MAX_SAMPS = 100
	local fig = Figure(;)
	local slg = SliderGrid(fig[2,1],
		(label = L"\alpha", range = 1:0.01:5, startvalue = 1.5),
		(label = L"\beta", range = 1:0.01:5, startvalue = 3.5),
		(label = "#samples", range = 0:MAX_SAMPS, startvalue = 0)
	)
	local α = slg.sliders[1].value
	local β = slg.sliders[2].value
	local dist = @lift Beta($α, $β)
	local L = @lift pdf($dist, ($α - 1)/($α + $β - 2))
	
	local ax = Makie.Axis(fig[1,1], title = @lift(string(round(100/$L; sigdigits = 3)) * "%"))
	lines!(ax, 0..1, @lift(x -> pdf($dist, x)); label = L"p_0(x)")
	hlines!(ax, @lift([$L]); color = :orange, label = L"L\pi(x)")
	axislegend(ax)
	
	local band_xs = 0:0.01:1
	band!(ax, band_xs, fill(0, length(band_xs)), @lift(pdf.($dist, band_xs)); alpha = 0.1, color = :green) # Acceptance band
	band!(ax, band_xs, @lift(pdf.($dist, band_xs)), @lift(fill($L, length(band_xs))); alpha = 0.1, color = :red) # Rejection band

	# samples
	local nsamps = slg.sliders[3].value
	local samps = rand(Uniform(0, 1), MAX_SAMPS)
	local acc_rej = rand(Uniform(0, 1), MAX_SAMPS)
	local acc_rej_thresh = @lift pdf.($dist, samps)
	local green_blue = @lift(map(ar -> ar[1] > ar[2] ? :red : :green, zip(acc_rej .* $L, $acc_rej_thresh)))
	local alpha = @lift(vcat(fill(1.0, $nsamps), fill(0.0, MAX_SAMPS - 		$nsamps)))
	local color = @lift(collect(zip($green_blue, $alpha)))
	scatter!(ax, samps, @lift(acc_rej.* $L); marker = :x, color) 

	fig
end

# ╔═╡ b6684689-bafa-470e-b2f3-bac20d895e6e
begin
	local α = 5
	local β = 3
	CairoMakie.activate!()
	local L = pdf(Beta(α, β), (α - 1)/(α + β - 2))
	efficiency_beta = 1/L
	local samps = []
	nsamps_rej = 0
	while length(samps) < nsamps_beta
		X_tilde = rand(Uniform(0,1))
		P_acc_rej = rand(Uniform(0,L))
		if P_acc_rej <= pdf(Beta(α, β), X_tilde)
			push!(samps, X_tilde)
		else
			global nsamps_rej += 1
		end
	end
	local fig, ax,_ = lines(0..1, x -> pdf(Beta(α, β), x); color = :orange)
	hist!(ax, samps; bins = nbins_beta, normalization = :pdf, alpha = 0.5)
	fig
end

# ╔═╡ 58412e62-51d8-496a-b128-0f887a6b73ba
md"Number of rejected samples: **$nsamps_rej** (theoretical effieciency = **$(round(efficiency_beta * 100; sigdigits = 3))%**)."

# ╔═╡ 803b98a5-0f06-4da6-80d8-804efaa89de5
md"""
```julia
function rand(rng::AbstractRNG, s::GIGRoUWithModeShiftSampler)
    while true
        U = rand(rng, Uniform(s.u_minus, s.u_plus))
        V = rand(rng, Uniform(0, s.v_plus))
        Y = U / V + s.m
        if V^2 <= s.g(Y)
            X = Y / s.α # rescaling
            return X
        end
    end
end
```
"""

# ╔═╡ e26123ee-86c6-4a9e-bec8-e37fb66bb6ab
begin
	function sampler(d::GeneralizedInverseGaussian)
    if d.χ == 0 # No rescaling is possible for the case of χ = 0... In this case X is distributed as Gamma(λ, 2/ψ)
        return sampler(Gamma(d.λ, 2 / d.ψ)) # NOTE: If χ ≡ 0, then ψ > 0 by parameter domain of the GIG distribution  
    end
    λ = d.λ
    β = sqrt(d.ψ * d.χ)
    if (0 <= λ < 1) && (0 < β <= ((2 / 3) * sqrt(1 - λ)))
        return GIGRejectionNonConcaveSampler(d)
    elseif (0 <= λ <= 1) && (min(1 / 2, (2 / 3) * sqrt(1 - λ)) <= β <= 1)
        return GIGRoUNoModeShiftSampler(d)
    elseif (λ > 1) && (β > 1)
        return GIGRoUWithModeShiftSampler(d)
    else
        return DomainError((λ, β), "Not yet implemeted for these parameters")
    end
end
end

# ╔═╡ 5c9b9eb5-b29b-45a7-a368-bca7d4285f21
md"""
```julia
function sampler(d::GeneralizedInverseGaussian)
    if d.χ == 0 # No rescaling is possible for the case of χ = 0... In this case X is distributed as Gamma(λ, 2/ψ)
        return sampler(Gamma(d.λ, 2 / d.ψ)) # NOTE: If χ ≡ 0, then ψ > 0 by parameter domain of the GIG distribution  
    end
    λ = d.λ
    β = sqrt(d.ψ * d.χ)
    if (0 <= λ < 1) && (0 < β <= ((2 / 3) * sqrt(1 - λ)))
        return GIGRejectionNonConcaveSampler(d)
    elseif (0 <= λ <= 1) && (min(1 / 2, (2 / 3) * sqrt(1 - λ)) <= β <= 1)
        return GIGRoUNoModeShiftSampler(d)
    elseif (λ > 1) && (β > 1)
        return GIGRoUWithModeShiftSampler(d)
    else
        return DomainError((λ, β), "Not yet implemeted for these parameters")
    end
end

```
"""

# ╔═╡ 8fb59b21-90a6-4b1b-9e6b-6e6f62594f82
md"""
λ $(@bind λ PlutoUI.Slider(-1:0.01:1; default = 0.5))

ψ $(@bind ψ PlutoUI.Slider(0.1:0.01:5; default = 0.34))

χ $(@bind χ PlutoUI.Slider(0.1:0.01:5; default = 0.6))
"""

# ╔═╡ a2b16097-9a85-4fff-ab22-2c7eedc7cee2
md"""
λ = $(λ)

ψ = $(ψ)

χ = $(χ)
"""

# ╔═╡ d9411bcc-9b6e-4ceb-85fb-3927b4160714
gig = GeneralizedInverseGaussian(λ, ψ, χ);

# ╔═╡ cf6815eb-a5a0-48bb-84a9-46914d73c930
md"GIG(λ = $(gig.λ), ψ = $(gig.ψ), χ = $(gig.χ))"

# ╔═╡ da3020f9-9b70-473a-acdb-4e0d3f60b215
samp = sampler(gig)

# ╔═╡ 7bd4c9c3-0562-479f-ae6c-666904b41695
md"Number of samples $(@bind nsamps_gig NumberField(100:100_000; default = 500))"

# ╔═╡ ca5d5741-d689-4bd2-a956-c6b5e41945bb
samps = rand(samp, nsamps_gig)

# ╔═╡ c84e5c6b-58b0-4eb7-a983-c26215deccaa
begin
	local fig, ax,_ = hist(samps; normalization = :pdf, bins = round(Int,length(samps)/100))
	lines!(ax, 0:0.01:maximum(samps), x -> pdf(gig, x); color = :orange)
	fig
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
SpecialFunctions = "276daf66-3868-5448-9aa4-cd146d93841b"
WGLMakie = "276b4fcb-3e11-5398-bf8b-a0c2d153d008"

[compat]
CairoMakie = "~0.11.9"
Distributions = "~0.25.107"
PlutoUI = "~0.7.58"
SpecialFunctions = "~2.3.1"
WGLMakie = "~0.9.8"
"""

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

julia_version = "1.10.2"
manifest_format = "2.0"
project_hash = "2724be00d4ae69b6157576d5199490322a07dc02"

[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.5.0"
weakdeps = ["ChainRulesCore", "Test"]

    [deps.AbstractFFTs.extensions]
    AbstractFFTsChainRulesCoreExt = "ChainRulesCore"
    AbstractFFTsTestExt = "Test"

[[deps.AbstractLattices]]
git-tree-sha1 = "222ee9e50b98f51b5d78feb93dd928880df35f06"
uuid = "398f06c4-4d28-53ec-89ca-5b2656b7603d"
version = "0.3.0"

[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "0f748c81756f2e5e6854298f11ad8b2dfae6911a"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.3.0"

[[deps.AbstractTrees]]
git-tree-sha1 = "2d9c9a55f9c93e8887ad391fbae72f8ef55e1177"
uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
version = "0.4.5"

[[deps.Adapt]]
deps = ["LinearAlgebra", "Requires"]
git-tree-sha1 = "cea4ac3f5b4bc4b3000aa55afb6e5626518948fa"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "4.0.3"
weakdeps = ["StaticArrays"]

    [deps.Adapt.extensions]
    AdaptStaticArraysExt = "StaticArrays"

[[deps.Animations]]
deps = ["Colors"]
git-tree-sha1 = "e81c509d2c8e49592413bfb0bb3b08150056c79d"
uuid = "27a7e980-b3e6-11e9-2bcd-0b925532e340"
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[[deps.ArgTools]]
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[[deps.ArrayInterface]]
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git-tree-sha1 = "44691067188f6bd1b2289552a23e4b7572f4528d"
uuid = "4fba245c-0d91-5ea0-9b3e-6abc04ee57a9"
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    [deps.ArrayInterface.extensions]
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    ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices"
    ArrayInterfaceCUDAExt = "CUDA"
    ArrayInterfaceChainRulesExt = "ChainRules"
    ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore"
    ArrayInterfaceReverseDiffExt = "ReverseDiff"
    ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore"
    ArrayInterfaceTrackerExt = "Tracker"

    [deps.ArrayInterface.weakdeps]
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    GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527"
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    StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"

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

[[deps.Automa]]
deps = ["PrecompileTools", "TranscodingStreams"]
git-tree-sha1 = "588e0d680ad1d7201d4c6a804dcb1cd9cba79fbb"
uuid = "67c07d97-cdcb-5c2c-af73-a7f9c32a568b"
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[[deps.AxisAlgorithms]]
deps = ["LinearAlgebra", "Random", "SparseArrays", "WoodburyMatrices"]
git-tree-sha1 = "01b8ccb13d68535d73d2b0c23e39bd23155fb712"
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[[deps.AxisArrays]]
deps = ["Dates", "IntervalSets", "IterTools", "RangeArrays"]
git-tree-sha1 = "16351be62963a67ac4083f748fdb3cca58bfd52f"
uuid = "39de3d68-74b9-583c-8d2d-e117c070f3a9"
version = "0.4.7"

[[deps.Base64]]
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[[deps.BenchmarkTools]]
deps = ["JSON", "Logging", "Printf", "Profile", "Statistics", "UUIDs"]
git-tree-sha1 = "f1dff6729bc61f4d49e140da1af55dcd1ac97b2f"
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[[deps.BitFlags]]
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uuid = "d1d4a3ce-64b1-5f1a-9ba4-7e7e69966f35"
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[[deps.Bonito]]
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git-tree-sha1 = "e4bb5188989c011e065adebea9be379e44aa5632"
uuid = "824d6782-a2ef-11e9-3a09-e5662e0c26f8"
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[[deps.Bzip2_jll]]
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git-tree-sha1 = "9e2a6b69137e6969bab0152632dcb3bc108c8bdd"
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[[deps.CEnum]]
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[[deps.CRC32c]]
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[[deps.CRlibm_jll]]
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git-tree-sha1 = "e329286945d0cfc04456972ea732551869af1cfc"
uuid = "4e9b3aee-d8a1-5a3d-ad8b-7d824db253f0"
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[[deps.Cairo]]
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git-tree-sha1 = "d0b3f8b4ad16cb0a2988c6788646a5e6a17b6b1b"
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[[deps.CairoMakie]]
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git-tree-sha1 = "6dc1bbdd6a133adf4aa751d12dbc2c6ae59f873d"
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[[deps.Cairo_jll]]
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git-tree-sha1 = "a4c43f59baa34011e303e76f5c8c91bf58415aaf"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
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[[deps.Calculus]]
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[[deps.ChainRulesCore]]
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    [deps.ChainRulesCore.extensions]
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[[deps.CodecBzip2]]
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git-tree-sha1 = "9b1ca1aa6ce3f71b3d1840c538a8210a043625eb"
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[[deps.CodecZlib]]
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[[deps.ColorBrewer]]
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[[deps.ColorSchemes]]
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git-tree-sha1 = "67c1f244b991cad9b0aa4b7540fb758c2488b129"
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[[deps.ColorTypes]]
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[[deps.ColorVectorSpace]]
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    [deps.ColorVectorSpace.extensions]
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[[deps.Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "fc08e5930ee9a4e03f84bfb5211cb54e7769758a"
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[[deps.Combinatorics]]
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[[deps.CommonSubexpressions]]
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[[deps.Compat]]
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    [deps.Compat.extensions]
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[[deps.CompilerSupportLibraries_jll]]
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uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
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[[deps.ConcurrentUtilities]]
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[[deps.Contour]]
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[[deps.DelaunayTriangulation]]
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[[deps.Deno_jll]]
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[[deps.Distributions]]
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[[deps.FFMPEG_jll]]
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[[deps.FFTW_jll]]
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[[deps.FiniteDiff]]
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[[deps.Xorg_libXext_jll]]
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git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
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[[deps.x264_jll]]
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[[deps.x265_jll]]
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version = "3.5.0+0"
"""

# ╔═╡ Cell order:
# ╠═d2637322-e08b-11ee-323b-1b12a99d682e
# ╠═bc0eac9d-8958-4290-bbd0-231ba12ae0e7
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# ╟─757c3eaa-8803-4a65-9cdf-f0c7cf13ee2f
# ╠═43c34a72-8ade-4cce-b9f4-1154388440da
# ╟─c7279d6b-9066-414d-80f8-786f93c4f9f3
# ╟─26797a7a-4af5-4b26-be6b-1917aedab58c
# ╟─211032c9-18a9-4b51-b733-9cef4532e923
# ╟─bbd54830-2abf-4cab-b805-ab0add47b340
# ╟─aba10bfd-edaf-4179-bfb5-e01015d373c7
# ╟─8774e552-868f-4aad-a4c7-30e4be56d535
# ╟─60a5c466-af9c-4796-980a-7502dc5cf26f
# ╟─dbbae56a-a053-4ca8-a217-6ae57aed2af4
# ╟─b6684689-bafa-470e-b2f3-bac20d895e6e
# ╟─58412e62-51d8-496a-b128-0f887a6b73ba
# ╟─14929f52-664d-4cc4-abcc-5bd6a964a858
# ╟─e9f77fcd-d272-412b-bf3a-bb0f8b2fafc7
# ╟─7d070844-c122-4701-a2fa-9aed336e8378
# ╟─890659d8-4568-4c65-909d-0f0cfbec1de6
# ╟─b2d43deb-a848-4725-9c1e-609d474b25df
# ╟─918d0f71-e321-4011-99ed-76cf4a10fb39
# ╟─12e26a60-28d9-48a4-b8db-189ed3d9b4e8
# ╟─95cae5d3-e233-4310-8e0d-cb93d7a205f9
# ╟─803b98a5-0f06-4da6-80d8-804efaa89de5
# ╟─e26123ee-86c6-4a9e-bec8-e37fb66bb6ab
# ╟─5c9b9eb5-b29b-45a7-a368-bca7d4285f21
# ╟─8fb59b21-90a6-4b1b-9e6b-6e6f62594f82
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