experiments.jl

### A Pluto.jl notebook ###
# v0.19.42

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

# ╔═╡ a1c1ef76-64d9-4f7b-b6b8-b70e180fc0ff
using RxInfer, Distributions, PlutoUI, PyPlot, PGFPlotsX, LaTeXStrings, Random, LinearAlgebra

# ╔═╡ 7c5873ce-a0d5-46f2-9d59-188ffd09cb5b
begin
	using SpecialFunctions
	@rule Categorical(:p, Marginalisation) (m_out::Categorical, q_out::PointMass) = begin
	    @logscale -SpecialFunctions.logfactorial(length(probvec(q_out)))
	    return Dirichlet(probvec(q_out) .+ one(eltype(probvec(q_out))))
	end

	function ReactiveMP.constrain_form(
		::PointMassFormConstraint, 
		distribution::Categorical
	) 
		k = mode(distribution)
		v = zeros(length(distribution.support))
		v[k] = 1
		return PointMass(v)
	end

	struct EnforceMarginalFunctionalDependency <: ReactiveMP.FunctionalDependencies
	    edge :: Symbol
	end

	function ReactiveMP.collect_functional_dependencies(
		::MixtureNode, enforce::EnforceMarginalFunctionalDependency
	)
		return enforce
	end

	function ReactiveMP.functional_dependencies(enforce::EnforceMarginalFunctionalDependency, factornode, interface, iindex)
		message_dependencies, default = functional_dependencies(
			ReactiveMP.collect_functional_dependencies(factornode, nothing), 
			factornode, 
			interface, 
			iindex
		)

		index = ReactiveMP.findnext(
			i -> ReactiveMP.name(i) === enforce.edge, 
			ReactiveMP.getinterfaces(factornode), 
			1
		)
		
	    if index === iindex 
	        return message_dependencies, default
	    end

		extra_localmarginal = ReactiveMP.FactorNodeLocalMarginal(enforce.edge)
	    vmarginal = ReactiveMP.getmarginal(
			ReactiveMP.getvariable(ReactiveMP.getinterfaces(factornode)[index]), ReactiveMP.IncludeAll()
		)
		extra_stream = ReactiveMP.MarginalObservable()
		ReactiveMP.connect!(extra_stream, vmarginal)
		ReactiveMP.setmarginal!(extra_localmarginal, extra_stream)
		
	    # Find insertion position (probably might be implemented more efficiently)
	    insertafter = sum(first(el) < iindex ? 1 : 0 for el in default; init = 0)
	    marginal_dependencies = ReactiveMP.TupleTools.insertafter(
			default, insertafter, (extra_localmarginal, )
		)

		return message_dependencies, marginal_dependencies
	end

	# function for using hard switching
function ReactiveMP.functional_dependencies(::EnforceMarginalFunctionalDependency, factornode::MixtureNode{N}, interface, iindex::Int) where {N}
    message_dependencies = if iindex === 1
        # output depends on:
        (factornode.inputs,)
    elseif iindex === 2
        # switch depends on:
        (factornode.out, factornode.inputs)
    elseif 2 < iindex <= N + 2
        # k'th input depends on:
        (factornode.out, )
    else
        error("Bad index in functional_dependencies for SwitchNode")
    end

    marginal_dependencies = if iindex === 1
        # output depends on:
        (factornode.switch,)
    elseif iindex == 2
        #  switch depends on
        ()
    elseif 2 < iindex <= N + 2
        # k'th input depends on:
        (factornode.switch,)
    else
        error("Bad index in function_dependencies for SwitchNode")
    end
    # println(marginal_dependencies)
    return message_dependencies, marginal_dependencies
end

# create an observable that is used to compute the switch with pipeline constraints
function ReactiveMP.collect_latest_messages(::EnforceMarginalFunctionalDependency, factornode::MixtureNode{N}, messages::Tuple{ReactiveMP.NodeInterface, NTuple{N, ReactiveMP.IndexedNodeInterface}}) where {N}
    switchinterface  = messages[1]
    inputsinterfaces = messages[2]

    msgs_names = Val{(ReactiveMP.name(switchinterface), ReactiveMP.name(inputsinterfaces[1]))}()
    msgs_observable =
    combineLatest((ReactiveMP.messagein(switchinterface), combineLatest(map((input) -> ReactiveMP.messagein(input), inputsinterfaces), PushNew())), PushNew()) |>
        map_to((ReactiveMP.messagein(switchinterface), ReactiveMP.ManyOf(map((input) -> ReactiveMP.messagein(input), inputsinterfaces))))
    return msgs_names, msgs_observable
end

# create an observable that is used to compute the output with pipeline constraints
function ReactiveMP.collect_latest_messages(::EnforceMarginalFunctionalDependency, factornode::MixtureNode{N}, messages::Tuple{NTuple{N, ReactiveMP.IndexedNodeInterface}}) where {N}
    inputsinterfaces = messages[1]

    msgs_names = Val{(ReactiveMP.name(inputsinterfaces[1]), )}()
    msgs_observable =
    combineLatest(map((input) -> ReactiveMP.messagein(input), inputsinterfaces), PushNew()) |>
        map_to((ReactiveMP.ManyOf(map((input) -> ReactiveMP.messagein(input), inputsinterfaces)),))
    return msgs_names, msgs_observable
end

# create an observable that is used to compute the input with pipeline constraints
function ReactiveMP.collect_latest_messages(::EnforceMarginalFunctionalDependency, factornode::MixtureNode{N}, messages::Tuple{ReactiveMP.NodeInterface}) where {N}
    outputinterface = messages[1]

    msgs_names = Val{(ReactiveMP.name(outputinterface), )}()
    msgs_observable = combineLatestUpdates((ReactiveMP.messagein(outputinterface), ), PushNew())
    return msgs_names, msgs_observable
end

function ReactiveMP.collect_latest_marginals(::EnforceMarginalFunctionalDependency, factornode::MixtureNode{N}, marginals::Tuple{ReactiveMP.NodeInterface}) where {N}
    switchinterface = marginals[1]

    marginal_names       = Val{(ReactiveMP.name(switchinterface), )}()
    marginals_observable = combineLatestUpdates((getmarginal(ReactiveMP.getvariable(switchinterface), IncludeAll()), ), PushNew())

    return marginal_names, marginals_observable
end

	# import Distributions: Dirichlet
	# using SpecialFunctions: loggamma
	# function Dirichlet{T}(alpha::AbstractVector{T}; check_args::Bool=true) where T
 #        alpha0 = sum(alpha)
 #        lmnB = sum(loggamma, alpha) - loggamma(alpha0)
 #        Dirichlet{T,typeof(alpha),typeof(lmnB)}(alpha, alpha0, lmnB)
 #    end
	
end

# ╔═╡ d2910ba3-6f0c-4905-b10a-f32ad8239ab6
md"""
# Online Structure Learning with Dirichlet Processes through Message Passing
"""

# ╔═╡ c30f10ee-7bb7-4b12-8c90-8c1947ce0bc9
begin
	push!(PGFPlotsX.CUSTOM_PREAMBLE, raw"\usepackage{amssymb}");
	push!(PGFPlotsX.CUSTOM_PREAMBLE, raw"\usepackage{bm}");
	push!(PGFPlotsX.CUSTOM_PREAMBLE, raw"\usepgfplotslibrary{statistics}");
	push!(PGFPlotsX.CUSTOM_PREAMBLE, raw"\pgfplotsset{compat=1.5.1}");
	function plot_ellipse(means::Vector, cov::Matrix)
		@assert length(means) == 2
		@assert size(cov) == (2,2)
		rads = eigvals(cov)
		vecs = eigvecs(cov)
		xrad = rads[1]
		yrad = rads[2]
		rot = atand(vecs[2,1], vecs[1,2])
		z = " [black] (axis cs: $(means[1]), $(means[2])) ellipse [rotate=$(rot), x radius=$(xrad), y radius=$(yrad)];"
		return raw"\draw" * z
	end
end;

# ╔═╡ 2bfcbece-2b63-443d-95d9-de7479ded607
md"""
### Generate data
"""

# ╔═╡ fa56df5b-e46b-4ed7-a9a7-7ae1d8697250
md"""number of samples: $(@bind nr_samples Slider(1:2_000; default=1500, show_value=true))"""

# ╔═╡ 1ee92290-71ac-41ce-9666-241478bc04cb
begin
	means = (rand(MersenneTwister(123), 8, 2).-0.5)*40
	means_vec = [means[k,:] for k in 1:8]
	dist = MixtureModel(
		map(m -> MvNormal(m, I) ,means_vec),
		Categorical([0.1, 0.3, 0.2, 0.05, 0.15, 0.1, 0.05, 0.05])
	)
end;

# ╔═╡ b9550341-a928-4274-b9da-40ac84d2a991
data = rand(MersenneTwister(123), dist, nr_samples);

# ╔═╡ d6fbfce8-c87b-4e0e-b11c-4d9fdce25b51
begin
	plt.figure()
	plt.scatter(data[1,:], data[2,:], alpha=0.1)
	plt.xlabel(L"y_1")
	plt.ylabel(L"y_2")
	plt.xlim(-20, 20)
	plt.ylim(-20, 20)
	plt.gcf()
end

# ╔═╡ 4eee819f-f099-4d4d-b72b-434be3077f99
md"""
### Model specification
"""

# ╔═╡ 01187b6e-8bf2-45eb-b918-73b767cf94b8
md"""upper bound number of components: $(@bind nr_components Slider(1:20; default=10, show_value=true))"""

# ╔═╡ e9c3f636-0049-4b42-b0f9-cc42bee61360
@model function model_dirichlet_process(y, α, μ_θ, Λ_θ)
    # `y` is specify experimental outcomes
    # `α`, `μ_θ` and `Λ_θ` are updatable parameters
	local θk 
	
    # specify initial distribution over clusters
    π ~ Dirichlet(α)
		
	# prior over model selection variable
	z ~ Categorical(π) where { 
		dependencies = EnforceMarginalFunctionalDependency(:out) 
	}

	# specify prior models over θ
	for k in 1:nr_components
		θk[k] ~ MvNormalMeanPrecision(μ_θ[k], Λ_θ[k])
	end

	# specify mixture distribution
	θ ~ Mixture(switch = z, inputs = θk) where { 
		dependencies = EnforceMarginalFunctionalDependency(:switch) 
	}

	# specify observation noise
	y ~ MvNormalMeanPrecision(θ, diagm(ones(2)))
end

# ╔═╡ 5fd5ee6b-3208-42fe-b8b2-14e95ffd08b5
md"""
### Constraint specification
"""

# ╔═╡ 07b1d74f-d203-475c-834e-ae83459d714a
@constraints function constraints_dirichlet_process()
    q(z) :: PointMassFormConstraint()
end;

# ╔═╡ 9bc23d63-8693-4eb8-ae90-29ddc4ec4997
md"""
### Probabilistic inference
"""

# ╔═╡ 16e37d4b-523f-4d1b-ab71-b54ba81364b1
@bind alpha Slider(-5:5; default=-1, show_value=false)

# ╔═╡ 6293a919-a314-4063-a679-c70008b12b2f
md"""alpha = 1e$(alpha)"""

# ╔═╡ 5f6aaef7-1deb-4812-9888-fc52009bdc5f
base_measure = MvNormalMeanPrecision(zeros(2), 0.1*diagm(ones(2)));

# ╔═╡ 37897bf5-c864-456f-87c0-1251ad532010
begin
	function update_alpha_vector(α_prev)
		ind = findfirst(x -> isapprox(1e-10,x;rtol=0.1), α_prev)
		if isnothing(ind)
			α_new = α_prev
			@error "upper bound reached"
		elseif ind == 2 && α_prev[1] != 10.0^alpha
			α_new = α_prev
			α_new[ind-1] = 1
			α_new[ind] = 10.0^alpha
		elseif ind > 2 && α_prev[ind-1] ≈ 1+10.0^alpha
			α_new = α_prev
			α_new[ind-1] = 1
			α_new[ind] = 10.0^alpha
		else
			α_new = α_prev
		end
		return α_new
	end
	function update_alpha(dist)
		return update_alpha_vector(probvec(dist))
	end
	function broadcast_mean_precision(dist)
		tmp = mean_precision.(dist)
		return first.(tmp), last.(tmp)
	end
	@rule Mixture((:inputs, k), Marginalisation) (m_out::Any, q_switch::PointMass) = begin
	
	    # check whether mean is one-hot
	    p = mean(q_switch)
	    @assert sum(p) ≈ 1 "The selector variable connected to the Mixture node is not normalized."
	    @assert all(x -> x == 1 || x == 0, p) "The selector variable connected to the Mixture node is not one-hot encoded."
	
	    # get selected cluster
	    kmax = argmax(p)
	
	    if k == kmax
	        @logscale 0
	        return m_out
	    else
	        @logscale missing
	        return missing
	    end
	end
end;

# ╔═╡ aec5408c-aab9-4fba-9bf2-0bace3c2c29f
autoupdates_dirichlet_process = @autoupdates begin
    α = update_alpha(q(π))
	μ_θ, Λ_θ = broadcast_mean_precision(q(θk))
end;

# ╔═╡ 5f3b9e1f-2ffc-403b-95df-3e48504399bc
 function run_dirichlet_process(data)
	 alpha_start = 1e-10*ones(nr_components)
	 alpha_start[1] = 10.0^alpha
	 return infer(
		model          = model_dirichlet_process(),
		data           = (y = [data[:,k] for k=1:size(data,2)], ),
		constraints    = constraints_dirichlet_process(),
		autoupdates    = autoupdates_dirichlet_process,
		initialization = @initialization(begin 
			q(π) = Dirichlet(alpha_start; check_args=false)
			q(θk) = base_measure
		end),
		returnvars     = (:π, :θk),
		keephistory    = size(data,2),
		historyvars    = (z = KeepLast(), π = KeepLast(), θk = KeepLast()),
		autostart      = true,
		addons         = AddonLogScale()
	)
 end;

# ╔═╡ 1b2c3587-b4b6-4d5b-a54a-430fc478dcd4
md"""
### Results
"""

# ╔═╡ 2bfa1683-86c3-4b9d-b7e3-3890bb32c645
results_dirichlet_process = run_dirichlet_process(data)

# ╔═╡ 075cbfdd-698b-4a38-8ae3-557d39acb5d2
md"""sample index: $(@bind N Slider(1:nr_samples; default=nr_samples, show_value=true))"""

# ╔═╡ e2ed2836-5a4d-4762-ab59-d77277b47f39
begin
	plt.figure()
	plt.scatter(data[1,1:N], data[2,1:N], alpha=0.1, c=argmax.(mean.(results_dirichlet_process.history[:z][1:N])))
	for k in findall(x -> x >= 1, probvec(results_dirichlet_process.history[:π][N]))
		plt.scatter(mean(results_dirichlet_process.history[:θk][N][k])[1], mean(results_dirichlet_process.history[:θk][N][k])[2], marker="x", color="black")
	end
	plt.xlabel(L"y_1")
	plt.ylabel(L"y_2")
	plt.xlim(-20, 20)
	plt.ylim(-20, 20)
	plt.grid()
	plt.gcf()
end

# ╔═╡ ef365ff1-9a53-4e7e-96ce-a68baeb6b67b
begin
	data_1500 = rand(MersenneTwister(123), dist, 1500)
	results_dirichlet_process_1500 = run_dirichlet_process(data_1500)
	N1 = 5
	N2 = 25
	N3 = 250
	N4 = 1500

	C1 = isnothing(findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N1]))) ? nr_components : findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N1])) - 2
	C2 = isnothing(findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N2]))) ? nr_components : findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N2])) - 2	
	C3 = isnothing(findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N3]))) ? nr_components : findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N3])) - 2	
	C4 = isnothing(findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N4]))) ? nr_components : findfirst(x -> isapprox(1e-10,x;rtol=0.1), probvec(results_dirichlet_process_1500.history[:π][N4])) - 2	

	fig_tikz = @pgf GroupPlot(

		# group plot options
		{
			group_style = {
				group_size = "2 by 4",
				horizontal_sep = "2cm"
			},
			label_style={font="\\footnotesize"},
			ticklabel_style={font="\\scriptsize",},
	        grid = "major",
		},

		# row 1 column 1
		{
			xlabel="\$y_1\$",
			ylabel_style={align="center"},
			ylabel = "\$\\bm{N=$(N1)}\$ \\\\ \\\\ \$y_2\$",
			xmin = -20,
			ymin = -20,
			xmax = 20,
			ymax = 20,
			width = "2in",
			height = "2in",
			title = "\\textbf{Assignments and clusters}",
			axis_equal,
		},
		Plot({ 
				scatter,
				only_marks,
				opacity=0.2,
            	scatter_src="explicit"
	        },
			Table(
				{
	                meta = "label"
	            },
	            x = data_1500[1,1:N1],
	            y = data_1500[2,1:N1],
	            label = argmax.(mean.(results_dirichlet_process_1500.history[:z][1:N1]))
			)
    	),
		Plot({ 
				only_marks,
				mark_size="4pt",
				mark_color="black",
				mark="x",
				very_thick,
	        },
			Table(hcat(mean.(results_dirichlet_process_1500.history[:θk][N1][1:C1])...)')
    	),
		plot_ellipse.(mean.(results_dirichlet_process_1500.history[:θk][N1])[1:C1], map(x->x.+diagm(ones(2)), cov.(results_dirichlet_process_1500.history[:θk][N1])[1:C1])),

		# row 1 column 2
		{
			ybar,
			bar_width="10pt",
			ylabel = "\$\\alpha_k\$",
			xlabel = "\$k\$",
			ymin = 0,
			width = "4in",
			height = "2in",
			title = "\\textbf{Posterior concentration parameters}"
	    },
	    Plot({ 
				fill="blue",
	        },
	        Table(collect(1:nr_components), probvec(results_dirichlet_process_1500.history[:π][N1]))
	    ),

		# row 2 column 1
		{
			xlabel="\$y_1\$",
			ylabel_style={align="center"},
			ylabel = "\$\\bm{N=$(N2)}\$ \\\\ \\\\ \$y_2\$",
			xmin = -20,
			ymin = -20,
			xmax = 20,
			ymax = 20,
			width = "2in",
			height = "2in",
			axis_equal,
		},
		Plot({ 
				scatter,
				only_marks,
				opacity=0.2,
            	scatter_src="explicit"
	        },
			Table(
				{
	                meta = "label"
	            },
	            x = data_1500[1,1:N2],
	            y = data_1500[2,1:N2],
	            label = argmax.(mean.(results_dirichlet_process_1500.history[:z][1:N2]))
			)
    	),
		Plot({ 
				only_marks,
				mark_size="4pt",
				mark_color="black",
				mark="x",
				very_thick,
	        },
			Table(hcat(mean.(results_dirichlet_process_1500.history[:θk][N2][1:C2])...)')
    	),
		plot_ellipse.(mean.(results_dirichlet_process_1500.history[:θk][N2])[1:C2], map(x->x.+diagm(ones(2)), cov.(results_dirichlet_process_1500.history[:θk][N2])[1:C2])),

		# row 2 column 2
		{
			ybar,
			bar_width="10pt",
			ylabel = "\$\\alpha_k\$",
			xlabel = "\$k\$",
			ymin = 0,
			width = "4in",
			height = "2in",
	    },
	    Plot({ 
				fill="blue",
	        },
	        Table(collect(1:nr_components), probvec(results_dirichlet_process_1500.history[:π][N2]))
	    ),

		# row 3 column 1
		{
			xlabel="\$y_1\$",
			ylabel_style={align="center"},
			ylabel = "\$\\bm{N=$(N3)}\$ \\\\ \\\\ \$y_2\$",
			xmin = -20,
			ymin = -20,
			xmax = 20,
			ymax = 20,
			width = "2in",
			height = "2in",
			axis_equal,
		},
		Plot({ 
				scatter,
				only_marks,
				opacity=0.2,
         		scatter_src="explicit",
	        },
			Table(
				{
	                meta = "label"
	            },
	            x = data_1500[1,1:N3],
	            y = data_1500[2,1:N3],
	            label = argmax.(mean.(results_dirichlet_process_1500.history[:z][1:N3]))
			)
    	),
		Plot({ 
				only_marks,
				mark_size="4pt",
				mark_color="black",
				mark="x",
				very_thick,
	        },
			Table(hcat(mean.(results_dirichlet_process_1500.history[:θk][N3][1:C3])...)')
    	),
		plot_ellipse.(mean.(results_dirichlet_process_1500.history[:θk][N3])[1:C3], map(x->x.+diagm(ones(2)), cov.(results_dirichlet_process_1500.history[:θk][N3])[1:C3])),

		# row 3 column 2
		{
			ybar,
			bar_width="10pt",
			ylabel = "\$\\alpha_k\$",
			xlabel = "\$k\$",
			ymin = 0,
			width = "4in",
			height = "2in",
	    },
	    Plot({ 
				fill="blue",
	        },
	        Table(collect(1:nr_components), probvec(results_dirichlet_process_1500.history[:π][N3]))
	    ),

		# row 4 column 1
		{
			xlabel="\$y_1\$",
			ylabel_style={align="center"},
			ylabel = "\$\\bm{N=$(N4)}\$ \\\\ \\\\ \$y_2\$",
			xmin = -20,
			ymin = -20,
			xmax = 20,
			ymax = 20,
			width = "2in",
			height = "2in",
			axis_equal,
		},
		Plot({ 
				scatter,
				only_marks,
				opacity=0.2,
            	scatter_src="explicit"
	        },
			Table(
				{
	                meta = "label"
	            },
	            x = data_1500[1,1:N4],
	            y = data_1500[2,1:N4],
	            label = argmax.(mean.(results_dirichlet_process_1500.history[:z][1:N4]))
			)
    	),
		Plot({ 
				only_marks,
				mark_size="4pt",
				mark_color="black",
				mark="x",
				very_thick,
	        },
			Table(hcat(mean.(results_dirichlet_process_1500.history[:θk][N4][1:C4])...)')
    	),
		plot_ellipse.(mean.(results_dirichlet_process_1500.history[:θk][N4])[1:C4], map(x->x.+diagm(ones(2)), cov.(results_dirichlet_process_1500.history[:θk][N4])[1:C4])),

		# row 4 column 2
		{
			ybar,
			bar_width="10pt",
			ylabel = "\$\\alpha_k\$",
			xlabel = "\$k\$",
			ymin = 0,
			width = "4in",
			height = "2in",
	    },
	    Plot({ 
				fill="blue",
	        },
	        Table(collect(1:nr_components), probvec(results_dirichlet_process_1500.history[:π][N4]))
	    ),
			
	)
end

# ╔═╡ b6ecbc4d-6bc2-4ec5-ad5b-78ae441fbf49
begin
	mkdir("exports")
	pgfsave("exports/experiments_dirichlet.tikz", fig_tikz)
	pgfsave("exports/experiments_dirichlet.png", fig_tikz)
	pgfsave("exports/experiments_dirichlet.pdf", fig_tikz)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
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git-tree-sha1 = "66e0a8e672a0bdfca2c3f5937efb8538b9ddc085"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.2.1"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
version = "1.10.0"

[[deps.SpecialFunctions]]
deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "2f5d4697f21388cbe1ff299430dd169ef97d7e14"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.4.0"

    [deps.SpecialFunctions.extensions]
    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"

    [deps.SpecialFunctions.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"

[[deps.Static]]
deps = ["IfElse"]
git-tree-sha1 = "d2fdac9ff3906e27f7a618d47b676941baa6c80c"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.8.10"

[[deps.StaticArrayInterface]]
deps = ["ArrayInterface", "Compat", "IfElse", "LinearAlgebra", "PrecompileTools", "Requires", "SparseArrays", "Static", "SuiteSparse"]
git-tree-sha1 = "5d66818a39bb04bf328e92bc933ec5b4ee88e436"
uuid = "0d7ed370-da01-4f52-bd93-41d350b8b718"
version = "1.5.0"
weakdeps = ["OffsetArrays", "StaticArrays"]

    [deps.StaticArrayInterface.extensions]
    StaticArrayInterfaceOffsetArraysExt = "OffsetArrays"
    StaticArrayInterfaceStaticArraysExt = "StaticArrays"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"]
git-tree-sha1 = "9ae599cd7529cfce7fea36cf00a62cfc56f0f37c"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.9.4"

    [deps.StaticArrays.extensions]
    StaticArraysChainRulesCoreExt = "ChainRulesCore"
    StaticArraysStatisticsExt = "Statistics"

    [deps.StaticArrays.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StaticArraysCore]]
git-tree-sha1 = "36b3d696ce6366023a0ea192b4cd442268995a0d"
uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
version = "1.4.2"

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deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.10.0"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1ff449ad350c9c4cbc756624d6f8a8c3ef56d3ed"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.7.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "5cf7606d6cef84b543b483848d4ae08ad9832b21"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.34.3"

[[deps.StatsFuns]]
deps = ["HypergeometricFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "cef0472124fab0695b58ca35a77c6fb942fdab8a"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "1.3.1"

    [deps.StatsFuns.extensions]
    StatsFunsChainRulesCoreExt = "ChainRulesCore"
    StatsFunsInverseFunctionsExt = "InverseFunctions"

    [deps.StatsFuns.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

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

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "7.2.1+1"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

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

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits"]
git-tree-sha1 = "cb76cf677714c095e535e3501ac7954732aeea2d"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.11.1"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

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

[[deps.ThreadingUtilities]]
deps = ["ManualMemory"]
git-tree-sha1 = "eda08f7e9818eb53661b3deb74e3159460dfbc27"
uuid = "8290d209-cae3-49c0-8002-c8c24d57dab5"
version = "0.5.2"

[[deps.TinyHugeNumbers]]
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uuid = "783c9a47-75a3-44ac-a16b-f1ab7b3acf04"
version = "1.0.2"

[[deps.TranscodingStreams]]
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uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.10.9"
weakdeps = ["Random", "Test"]

    [deps.TranscodingStreams.extensions]
    TestExt = ["Test", "Random"]

[[deps.Tricks]]
git-tree-sha1 = "eae1bb484cd63b36999ee58be2de6c178105112f"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.8"

[[deps.TupleTools]]
git-tree-sha1 = "41d61b1c545b06279871ef1a4b5fcb2cac2191cd"
uuid = "9d95972d-f1c8-5527-a6e0-b4b365fa01f6"
version = "1.5.0"

[[deps.URIs]]
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uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

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

[[deps.Unrolled]]
deps = ["MacroTools"]
git-tree-sha1 = "6cc9d682755680e0f0be87c56392b7651efc2c7b"
uuid = "9602ed7d-8fef-5bc8-8597-8f21381861e8"
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[[deps.VectorizationBase]]
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git-tree-sha1 = "e863582a41c5731f51fd050563ae91eb33cf09be"
uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f"
version = "0.21.68"

[[deps.VersionParsing]]
git-tree-sha1 = "58d6e80b4ee071f5efd07fda82cb9fbe17200868"
uuid = "81def892-9a0e-5fdd-b105-ffc91e053289"
version = "1.3.0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"
"""

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