### A Pluto.jl notebook ###
# v0.17.1

using Markdown
using InteractiveUtils

# ╔═╡ 9c2b64cf-c236-46aa-a4d6-02a22c5b7faa
using Distributions

# ╔═╡ c5ef7103-4aca-49dc-b49b-4cf5ac157701
using Random.Random

# ╔═╡ f126a1f8-afc6-4b52-9021-39e946dd0ea2
md"""
# Cross-Entropy Method

STMO Exam Project by **Ceri-Anne Laureyssens**
"""

# ╔═╡ c629d020-2c17-477c-8517-68c39f46fbcb
md"""
### Cross-entropy

As the title states the method has to do with the cross-entropy. It's a metric used to measure the distance between two probability distributions. These two distributions, as you will read later on, are in fact the original distribution ($$f$$) and the optimized distribution based on elite samples ($$g$$). The distance used to define cross-entropy is called the Kullback-Leibler (KL) distance and measures how one probability distribution is different from a second, reference probability distribution. This KL distance or relative entropy can be easily used to derive the cross-entropy itself and is defined as follows:

```math 
\begin{equation}
D_{KL}(f,g) = E_f[\log\frac{f(X)}{g(X)}]
= \int_{x\in χ} f(x)\log f(x)dx - \int_{x\in χ} f(x)\log g(x)dx
\end{equation}
```

With $$D$$ being the KL distance, $$E$$ the expectation, and $$X$$ a random variable with support $$χ$$. 

Minimizing this KL distance in between distribution $$f$$ and $$g$$ (parameterized by $$θ$$) is equivalent to choosing $$θ$$ that minimizes the cross-entropy:

```math 
\begin{equation}
H(f,g) = H(f) + D_{KL}(f,g)
= -E_f[\log g(X)]
= -\int_{x\in χ} f(x)\log g(x|θ)dx
\end{equation}
```

With $$H$$($$f$$) being the entropy of distribution $$f$$.

This assumes that $$f$$ and $$g$$ share the support $$χ$$ and are continuous with respect to $$x$$. The minimization problem becomes the following:

```math 
\begin{equation}
minimize_θ - \int_{x\in χ} f(x)\log g(x|θ)dx
\end{equation}
```

Finding this minimum is the goal of the method explained in the next section.
"""

# ╔═╡ 2c923610-52c4-11ec-367b-2565c12740ab
md"""
### Method description and implementation

The cross-entropy or CE method is a Monte Carlo method for importance sampling and optimization.
Monte-Carlo algorithms rely on repeated sampling to obtain numerical results and use randomness as underlying concept to solve the given problem.

The CE method can be used for both combinatorial, such as the travelling salesman problem and the knapsack problem, as continuous problems, with either a static or noisy objective, and is also a valuable asset in rare-event simulation. In the examples called before a brute-force algorithm is not tractable. As such specialized algorithms, like the CE method, can be used because they quickly rule out large parts of the search space.

The simple two-step process involves generating a random data sample according to a specified mechanism and updating the parameters of the mechanism based on the data to produce a better sample in the next iteration as can be seen in the figure underneath.
"""

# ╔═╡ 2588e834-f2ee-41d1-b2c3-dbc43a0cda63
md"""
![](https://github.com/CeriAnneLaureyssens/CrossEntropy.jl/blob/master/notebook/figures/CE_visual.jpg?raw=true)
"""

# ╔═╡ 9b4da26a-6136-4be7-b3d8-ad573a89c7f9
md"""
We start with a specified distribution in the left panel from which a fixed amount of samples are randomly derived. These randomly derived samples can be seen in the middle panel. The cross-entropy method makes use of an objective, or loss (as mentioned further down), function ($$S$$) which minimizes the cross-entropy between the known distribution $$f$$ and a proposal distribution $$g$$ parameterized by $$θ$$.

```math 
\begin{equation}
θ_g^* = \text{argmin}_{θ_g} - \int_{x\in χ} I_{(S(x)≥γ)}g(x|θ) \log g(x|θ_g)dx
= \text{argmin}_{θ_g} - E_θ[I_{(S(x)≥γ)} \log g(x|θ_g)]
\end{equation}
```

With $$I$$ being the indicator function, and $$γ$$ a threshold with which elite samples are determined.

The grey dots indicate the elite samples. One becomes the elite samples by sorting the outcome of the objective function (done for every sample in the distribution). These elite samples are then selected as the samples of the new distribution, which can be seen in the third panel, and the whole process, starting from the left panel, is repeated until the distribution is optimized.
"""

# ╔═╡ aba389ff-77b4-4683-9e41-35e8dc429797
md"""
The optimized distribution parameter $θ_g^*$ is estimated iteratively via the algorithm underneath which runs `max_iter` times and is used here for optimizing multivariate time series distributions.

Suppose we have a timeseries `X = {x₁, ..., xₙ}` where each `xᵢ` is a vector of dimension `m`. The `cross_entropy_method` function can handle two different scenarios:

1. The time series is sampled IID from a single distribution `p`: `xᵢ ~ p(x)`. In this case, the distribution is represented as a `Dict{Symbol, Tuple{Sampleable, Int64}}`. The dictionary will contain `m` symbols, one for each variable in the series. The `Sampleable` object represents `p` and the integer is the length of the timeseries (`N`).
2. The time series is sampled from a different distribution at each timestep `pᵢ`: `xᵢ ~ pᵢ(x)`. In this case, the distribution is also represented as a `Dict{Symbol, Tuple{Sampleable, Int64}}`.

The parameters $θ_{iteration'}$ are defined based on the parameter $θ_{iteration}$. The threshold $γ_{iteration}$ becomes smaller than its initial value, artificially making events less rare under $$X$$ ~ $$g$$($$x$$|$θ_{ iteration}$).
"""

# ╔═╡ cb36a92a-0888-4993-ba11-ebbc7b05a3f9
md"""
The `cross_entropy_method` function takes the following input:
- `loss`: The loss or objective function.
- `d_in`: The starting sampling distribution.
- `max_iter`: Maximum number of iterations.
- `N`: Population size (number of samples).
- `elite_thresh`: The threshold below which a sample will be considered elite. To have a fixed number of elite samples set this to `-Inf` and use the `min_elite_samples`.
- `min_elite_samples`: The minimum number of elite samples.
- `max_elite_samples`: The maximum number of allowed elite samples.
- `weight_fn`: A function that specifies the weight of each sample. Use the likelihood ratio when trying to perform importance sampling.
- `rng::AbstractRNG`: The random number generator used.
- `verbose`: Whether or not to print progress.
- `show_progress`: Whether or not to show the progress meter.
- `batched`: Indicates batched loss evaluation (loss function must return an array containing loss values for each sample).
- `add_entropy`: A function that transforms the sampling distribution after fitting. Use it to enforce a maximum level of entropy if converging too quickly.

And provides you with an output `d`, which is the optimized distribution.
"""

# ╔═╡ 4115dcd8-9931-4871-bc32-5e6e08450242
md"""
### Fun little example

Now the CE method itself is implemented we will try it out on the importance sampling technique.

**Importance sampling**

Importance sampling is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest.

We start off with choosing a base distribution and a starting sampling distribution. Following this a random time series is sampled from the starting distribution.
"""

# ╔═╡ d799b586-f338-4614-9799-89ad9d04e695
md"""
The next two code chunks provide the functions for determining the loss function and determining the likelihood ratio weighting function. In the third code chunk underneath you can see the output of the weighting funtion.
"""

# ╔═╡ d77ca772-6b3b-43fb-a3a1-459e04bc37fc
# Define a loss function that identifies the values of 3,4,5 as those of interest
function l(d, s)
    v = s[:x][1]
    -(v in [3,4,5])
end

# ╔═╡ f9a0105c-b570-46b2-93a6-af1982fd4966
md"""
Now we can run the `cross_entropy_method` function with the determined loss function on the sampling distribution. This will return the optimized distribution and can then be plotted in the second code chunk underneath. The plot contains both the base distribution, from which we started, and the optimal distribution as the `cross_entropy_method` funtion returned. `Random.seed!(1234)` is used so the results stay the same each time the code is run. This is done because the CE method uses randomness as underlying concept and thus might provide slightly different results each time its run (sometimes even a vector with different length in comparison to the base distribution, which will provide problems in the underlying plot).
"""

# ╔═╡ e4cf6503-a9e1-4328-bdd5-d32b4330a6a0
md"""
### References

Kloek, T., and van Dijk, H.K. (1978). *Bayesian Estimates of Equation System Parameters: An Application of Integration by Monte Carlo.* Econometrica, 46(1), 1-19.

Moss, R.J. (2020). *Cross-Entropy Variants for Optimization.*

Rubinstein, R.Y., and Kroese, D.P. (2004). *The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning.* Springer-Verlag, New York.

Wikipedia (2021). *Combinatorial optimization.* Checked on the 31 January 2022 via https://en.wikipedia.org/wiki/Combinatorial_optimization
"""

# ╔═╡ 04ef2558-ff4b-4a2d-9f59-2f8b6edde3f0
md"""
# Appendices

Underneath you can find all functions this notebook relies on. Please leave these as they are ☺.
"""

# ╔═╡ abb2b875-0439-49f8-accd-5475ae388a67
function Distributions.logpdf(d::Dict{Symbol, Vector{Sampleable}}, x, i)
    sum([logpdf(d[k][i], x[k][i]) for k in keys(d)])
end

# ╔═╡ dbcf3ddc-f4ad-40ed-8c0f-3d7ace8b5069
function Distributions.logpdf(d::Dict{Symbol, Vector{Sampleable}}, x)
    sum([logpdf(d, x, i) for i=1:length(first(x)[2])])
end

# ╔═╡ aeb1f33e-1b4b-4808-bcee-79e8c5e4438f
function Distributions.logpdf(d::Dict{Symbol, Tuple{Sampleable, Int64}}, x, i)
    sum([logpdf(d[k][1], x[k][i]) for k in keys(d)])
end

# ╔═╡ fcb9f3ce-c23a-41ee-9ad2-eaea4699d5bd
function Distributions.logpdf(d::Dict{Symbol, Tuple{Sampleable, Int64}}, x)
    sum([logpdf(d, x, i) for i=1:length(first(x)[2])])
end

# ╔═╡ c81f3bd0-b576-467c-8621-21e75fa587a5
function Base.rand(rng::AbstractRNG, d::Dict{Symbol, Vector{Sampleable}})
    Dict(k => rand.(Ref(rng), d[k]) for k in keys(d))
end

# ╔═╡ 452fb99f-349e-45b0-bdf8-ada524b6d0db
function Base.rand(rng::AbstractRNG, d::Dict{Symbol, Tuple{Sampleable, Int64}})
    Dict(k => rand(rng, d[k][1], d[k][2]) for k in keys(d))
end

# ╔═╡ 05cf58ae-b9f7-4f69-b7e7-64586f1e27a8
function Base.rand(rng::AbstractRNG, d::Dict{Symbol, Vector{Sampleable}}, N::Int)
    [rand(rng, d) for i=1:N]
end

# ╔═╡ 88ea50a3-02bd-4a90-8d72-cf571273ba7f
function Base.rand(rng::AbstractRNG, d::Dict{Symbol, Tuple{Sampleable, Int64}}, N::Int)
    [rand(rng, d) for i=1:N]
end

# ╔═╡ b0054103-d45f-401c-ba20-7b36729fc53e
begin
	# Actual distibution
	base_dist = Categorical([0.74899, 0.2, 0.05, 0.001, 0.00001])
	
	# Starting sampling distribution
	is_dist_0 = Dict{Symbol, Vector{Sampleable}}(:x => [Categorical(5)])
	
	# Sample a random time series from the starting distribution
	rand(is_dist_0)
end

# ╔═╡ 7427a6ac-44f1-4fe6-8d0d-e16088391764
# Define the likelihood ratio weighting function
function w(d, s)
    v = s[:x][1]
    exp(logpdf(base_dist, v) - logpdf(d, s))
end

# ╔═╡ 0160a7ed-7418-4061-bb80-cac0d93c0ac8
# Show what the weighting function does
begin
	s = Dict(:x => [5])
	w(is_dist_0, s)
end

# ╔═╡ 028f3359-5335-4a13-9b3d-f23084087b8e
function Distributions.fit(d::Dict{Symbol, Vector{Sampleable}}, samples, weights; add_entropy = (x) -> x)
    N = length(samples)
    new_d = Dict{Symbol, Vector{Sampleable}}()
    for s in keys(d)
        dtype = typeof(d[s][1])
        m = length(d[s])
        new_d[s] = [add_entropy(fit(dtype, [samples[j][s][i] for j=1:N], weights)) for i=1:m]
    end
    new_d
end

# ╔═╡ 199c2691-aeb2-4d0a-b30e-2dd130493038
function Distributions.fit(d::Dict{Symbol, Tuple{Sampleable, Int64}}, samples, weights; add_entropy = (x)->x)
    N = length(samples)
    new_d = Dict{Symbol, Tuple{Sampleable, Int64}}()
    for s in keys(d)
        dtype = typeof(d[s][1])
        m = d[s][2]
        all_samples = vcat([samples[j][s][:] for j=1:N]...)
        all_weights = vcat([fill(weights[j], length(samples[j][s][:])) for j=1:N]...)
        new_d[s] = (add_entropy(fit(dtype, all_samples, all_weights)), m)
    end
    new_d
end

# ╔═╡ 6259dd44-2c35-4faf-b340-c69070e7fc48
# This version uses a vector of distributions for sampling
# N is the number of samples taken
# m is the length of the vector

# If batched is set to true, loss function must return an array containing loss values for each sample   
function cross_entropy_method(loss,
                              d_in;
                              max_iter,
                              N=100,
                              elite_thresh = -0.99,
                              min_elite_samples = floor(Int, 0.1*N),
                              max_elite_samples = typemax(Int64),
                              weight_fn = (d,x) -> 1.,
                              rng::AbstractRNG = Random.GLOBAL_RNG,
                              verbose = false,
                              show_progress = false,
                              batched = false,
                              add_entropy = (x)->x
                             )
    d = deepcopy(d_in)
    show_progress ? progress = Progress(max_iter) : nothing

    for iteration in 1:max_iter
        # Get samples -> Nxm
        samples = rand(rng, d, N)

        # Sort the samples by loss and select elite number
        if batched
            losses = loss(d,samples)
        else
            losses = [loss(d, s) for s in samples]    
        end
        
        order = sortperm(losses)
        losses = losses[order]
        N_elite = losses[end] < elite_thresh ? N : findfirst(losses .> elite_thresh) - 1
        N_elite = min(max(N_elite, min_elite_samples), max_elite_samples)

        verbose && println("iteration ", iteration, " of ", max_iter, " N_elite: ", N_elite)

        # Update based on elite samples
        elite_samples = samples[order[1:N_elite]]
        weights = [weight_fn(d, s) for s in elite_samples]
        if all(abs.(weights) .< 1e8)
            println("Warning: all weights are zero")
        end
        d = fit(d, elite_samples, weights, add_entropy = add_entropy)
        show_progress && next!(progress)
    end
    d
end

# ╔═╡ 06af1708-5e0a-4c5c-a8a5-6d3ba97f2690
begin
	Random.seed!(1234)
	
	# Run the optimization
	is_dist_opt = cross_entropy_method(l, is_dist_0; max_iter = 4, N=10000, weight_fn = w, verbose = true)
	
	# See the final distribution
	is_dist_opt[:x][1].p
end

# ╔═╡ 0902be86-d083-4c30-815b-f1aed4e92102
# Plot the base distribution and the optimized distribution
begin
	using Plots
	x = 1:5
	y = [0.74899, 0.2, 0.05, 0.001, 0.00001] # Base distribution
	plot(x, y, title = "Importance sampling", label = "Base distribution")
	
	z = is_dist_opt[:x][1].p # Optimized distribution
	plot!(x, z, label = "Optimal distribution")
end

# ╔═╡ d60f308c-72c0-4eff-8b21-23ef6803be8d
# Gets a function that adds entropy h to categorical distribution c
function add_categorical_entropy(h::Vector{Float64})
    function add_entropy(c::Categorical)
        p = h
        p[1:length(c.p)] .+= c.p
        Categorical(p ./ sum(p))
    end
end

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git-tree-sha1 = "d8a578692e3077ac998b50c0217dfd67f21d1e5f"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.0+0"

[[FillArrays]]
deps = ["LinearAlgebra", "Random", "SparseArrays", "Statistics"]
git-tree-sha1 = "8756f9935b7ccc9064c6eef0bff0ad643df733a3"
uuid = "1a297f60-69ca-5386-bcde-b61e274b549b"
version = "0.12.7"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "0c603255764a1fa0b61752d2bec14cfbd18f7fe8"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.5+1"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "RelocatableFolders", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "4a740db447aae0fbeb3ee730de1afbb14ac798a1"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.63.1"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "aa22e1ee9e722f1da183eb33370df4c1aeb6c2cd"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.63.1+0"

[[GeometryBasics]]
deps = ["EarCut_jll", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"]
git-tree-sha1 = "58bcdf5ebc057b085e58d95c138725628dd7453c"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.4.1"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "a32d672ac2c967f3deb8a81d828afc739c838a06"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.68.3+2"

[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+1"

[[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 = "fa6287a4469f5e048d763df38279ee729fbd44e5"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.4.0"

[[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 = "e5718a00af0ab9756305a0392832c8952c7426c1"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.6"

[[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 = "f755f36b19a5116bb580de457cda0c140153f283"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.6"

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

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+1"

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

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "648107615c15d4e09f7eca16307bc821c1f718d8"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.13+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"

[[PDMats]]
deps = ["LinearAlgebra", "SparseArrays", "SuiteSparse"]
git-tree-sha1 = "ee26b350276c51697c9c2d88a072b339f9f03d73"
uuid = "90014a1f-27ba-587c-ab20-58faa44d9150"
version = "0.11.5"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "92f91ba9e5941fc781fecf5494ac1da87bdac775"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.2.0"

[[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 = "6f1b25e8ea06279b5689263cc538f51331d7ca17"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.1.3"

[[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", "Unzip"]
git-tree-sha1 = "7e4920a7d4323b8ffc3db184580598450bde8a8e"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.25.7"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "2cf929d64681236a2e074ffafb8d568733d2e6af"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.3"

[[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"

[[QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "78aadffb3efd2155af139781b8a8df1ef279ea39"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.4.2"

[[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 = "37c1631cb3cc36a535105e6d5557864c82cd8c2b"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.5.0"

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

[[RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "cdbd3b1338c72ce29d9584fdbe9e9b70eeb5adca"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "0.1.3"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[Rmath]]
deps = ["Random", "Rmath_jll"]
git-tree-sha1 = "bf3188feca147ce108c76ad82c2792c57abe7b1f"
uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa"
version = "0.7.0"

[[Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "68db32dff12bb6127bac73c209881191bf0efbb7"
uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
version = "0.3.0+0"

[[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 = "2884859916598f974858ff01df7dfc6c708dd895"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.3.3"

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

[[StatsAPI]]
git-tree-sha1 = "d88665adc9bcf45903013af0982e2fd05ae3d0a6"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.2.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "51383f2d367eb3b444c961d485c565e4c0cf4ba0"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.14"

[[StatsFuns]]
deps = ["ChainRulesCore", "InverseFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "bedb3e17cc1d94ce0e6e66d3afa47157978ba404"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.14"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "d21f2c564b21a202f4677c0fba5b5ee431058544"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.4"

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

[[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 = "bb1064c9a84c52e277f1096cf41434b675cd368b"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.1"

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

[[Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[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"

[[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"

[[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 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

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

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

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

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

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

# ╔═╡ Cell order:
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# ╟─2588e834-f2ee-41d1-b2c3-dbc43a0cda63
# ╟─9b4da26a-6136-4be7-b3d8-ad573a89c7f9
# ╟─aba389ff-77b4-4683-9e41-35e8dc429797
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# ╠═6259dd44-2c35-4faf-b340-c69070e7fc48
# ╟─4115dcd8-9931-4871-bc32-5e6e08450242
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# ╠═d77ca772-6b3b-43fb-a3a1-459e04bc37fc
# ╠═7427a6ac-44f1-4fe6-8d0d-e16088391764
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# ╠═0902be86-d083-4c30-815b-f1aed4e92102
# ╟─e4cf6503-a9e1-4328-bdd5-d32b4330a6a0
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# ╠═9c2b64cf-c236-46aa-a4d6-02a22c5b7faa
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# ╠═c81f3bd0-b576-467c-8621-21e75fa587a5
# ╠═452fb99f-349e-45b0-bdf8-ada524b6d0db
# ╠═05cf58ae-b9f7-4f69-b7e7-64586f1e27a8
# ╠═88ea50a3-02bd-4a90-8d72-cf571273ba7f
# ╠═028f3359-5335-4a13-9b3d-f23084087b8e
# ╠═199c2691-aeb2-4d0a-b30e-2dd130493038
# ╠═d60f308c-72c0-4eff-8b21-23ef6803be8d
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
