random_walks.jl

### A Pluto.jl notebook ###
# v0.19.45

#> [frontmatter]
#> chapter = 2
#> video = "https://www.youtube.com/watch?v=14hHtGJ4s-g"
#> image = "https://user-images.githubusercontent.com/6933510/136196563-f4b5b44c-5252-4e67-8c82-c550de891c55.png"
#> section = 5
#> order = 5
#> title = "Random Walks"
#> layout = "layout.jlhtml"
#> youtube_id = "14hHtGJ4s-g"
#> description = ""
#> tags = ["lecture", "module2", "programming", "track_julia", "plotting", "structure", "type", "interactive", "random", "statistics", "track_math", "track_data"]

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

# ╔═╡ 97e807b2-9237-11eb-31ef-6fe0d4cc94d3
using Plots, PlutoUI, BenchmarkTools

# ╔═╡ 5f0d7a44-91e0-11eb-10ae-d73156f965e6
TableOfContents(aside=true)

# ╔═╡ 9647147a-91ab-11eb-066f-9bc190368fb2
md"""
# Julia concepts

- Benchmarking: BenchmarkTools.jl
- Plotting in a loop
- Generic programming
- Mutable vs immutable structs
- Vectors of vectors
- Aliasing of memory 
- `cumsum`
"""

# ╔═╡ ff1aca1e-91e7-11eb-343e-0f89d9570b06
md"""
# Motivation: Dynamics of hard discs
"""

# ╔═╡ 66a2f510-9232-11eb-3be9-131febc0039f
md"""
Brown observed **Brownian motion** in 1827: Large particles like sand or pollen in water move around seemingly at random. Einstein explained this in 1905 as repeated impacts with water molecules.

We can visualise that with a simulation of hard discs bouncing off one another. Even though the dynamics is not random -- each disc follows Newton's laws -- if we just look at a single one of them it *looks* random.
"""

# ╔═╡ bd3170e6-91ae-11eb-06f8-ebb6b2e7869f
md"""
## Visualising random walks
"""

# ╔═╡ a304c842-91df-11eb-3fac-6dd63087f6de
md"""
A **random walk** models random motion in time and space. At each time step an object moves in a random direction.

Let's visualise the result in 2 dimensions.
"""

# ╔═╡ 798507d6-91db-11eb-2e4a-3ba02f12ba65
md"""
N = $(@bind N Slider(1:6, show_value=true, default=1))
"""

# ╔═╡ 3504168a-91de-11eb-181d-1d580d5dc071
md"""
t = $(@bind t Slider(1:10^N, show_value=true, default=1))
"""

# ╔═╡ b62c4af8-9232-11eb-2f66-dd27dcb87d20
md"""
We see that the dynamics closely resembles, at least qualitatively, that of the hard disc.
"""

# ╔═╡ 905379ce-91ad-11eb-295d-8354ecf5c5b1
md"""
# Why use random walks?

#### Why should we model with random processes?

- Either we need to get a lot of data to precisely characterise a system, or we assume it's random

- It may well be that *can* have all the details, but by making a simplifying assumption we actually gain more *understanding*.


#### Examples:


- Stock price going up and down


- Pollutants getting dispersed in the air


- Neutral genes moving through a population


"""

# ╔═╡ 5c4f0f26-91ad-11eb-033b-2bd221f0bdba
md"""
# Simple random walk

The simplest version is called **simple random walk**: one object jumps on the integers: at each time step it jumps left or right. 

Each such step is a new random variable, taking values $\pm 1$ at each step, each with probability $1/2$ (in the simplest case). 
We can think of this as a scaling of a Bernoulli random variable.

In order to simulate this, we need to know how to generate the jumps.
"""

# ╔═╡ da98b676-91e0-11eb-0d97-57b8a8aadf2a
md"""
## Julia: Benchmarking
"""

# ╔═╡ e0b607c0-91e0-11eb-10aa-53ec33570e59
md"""
There are various ways we could generate random values $\pm 1$. Let's use this as an opportunity to learn about **benchmarking** in Julia, i.e. measuring run time to compare performance. In this case we will do "micro-benchmarks", which try to compare tiny pieces of code which we intend to run many millions of times, so that small differences in run time can be significant.

The `BenchmarkTools.jl` package provides relatively easy-to-use tools for measuring this, by running the code many times and calculating statistics. The `@btime` macro is a simple way to estimate actual running time. Each option should be enclosed in a function.
"""

# ╔═╡ fa1635d4-91e3-11eb-31bd-cf61c502ad35
md"""
Here are a few different ways we could generate random steps:
"""

# ╔═╡ f7f9e4c6-91e3-11eb-1a56-8b98f0b09b46
begin
	step1() = rand( (-1, +1) )
	
	step2() = 2 * (rand() < 0.5) - 1
	
	step3() = 2 * rand(Bool) - 1
	
	step4() = sign(randn())
end

# ╔═╡ 5da7b076-91b4-11eb-3eba-b3f5849efabb
with_terminal() do
	@btime step1()
	@btime step2()
	@btime step3()
	@btime step4()
end

# ╔═╡ ea9e77e2-91b1-11eb-185d-cd006db11f60
md"""
## Trajectory of a random walk
"""

# ╔═╡ 12b4d528-9239-11eb-2824-8ddb5e2ba892
md"""
We can now calculate the **trajectory** of a 1D random walk as it takes several steps.
It starts at an initial position, for example, 0, and takes consecutive steps:
"""

# ╔═╡ 2f525796-9239-11eb-1865-9b01eadcf548
function walk1D(N)
	x = 0
	xs = [x]
	
	for i in 1:N
		x += step1()
		push!(xs, x)
	end
	
	return xs
end

# ╔═╡ 51abfe6e-9239-11eb-362a-259570250663
begin
	plot()
	
	for i in 1:10
		plot!(walk1D(100), leg=false, size=(500, 300), lw=2, alpha=0.5)
	end
	
	plot!()
end

# ╔═╡ b847b5ca-9239-11eb-02fe-db4d9625bc5f
md"""
# Making it more general: Random walks using types
"""

# ╔═╡ c2deb090-9239-11eb-0739-a74379c15ce6
md"""
Now suppose we want to think about more general random walks, for example moving around in 2D. Then we need to *generalise* the above function.
		
Based on our experience from last time, you should suspect that a good way to do this is with *types*. We will define 
"""

# ╔═╡ d420d492-91d9-11eb-056d-33cc8f0aed74
abstract type Walker end

# ╔═╡ ad2d4dd8-91d5-11eb-27af-6f0c6e61a86a
struct Walker1D <: Walker
	pos::Int
end

# ╔═╡ d0f81f28-91d9-11eb-2e79-61461ef5b132
position(w::Walker) = w.pos

# ╔═╡ b8f2c508-91d5-11eb-31b5-61810f171270
step(w::Walker1D) = rand( (-1, +1) )

# ╔═╡ 23b84ce2-91da-11eb-01f8-c308ac4d1c7a
struct Walker2D <: Walker
	x::Int
	y::Int
end

# ╔═╡ 537f952a-91da-11eb-33cf-6be2fd3bc45c
position(w::Walker2D) = (w.x, w.y)

# ╔═╡ 3c3971e2-91da-11eb-384c-01c627318bdc
update(w::W, step) where {W <: Walker} = W(position(w) + step)

# ╔═╡ 5b972296-91da-11eb-29b1-074f3926181e
step(w::Walker2D) = rand( [ [1, 0], [0, 1], [-1, 0], [0, -1] ] )

# ╔═╡ 3ad5a93c-91db-11eb-3227-c96bf8fd2206
update(w::Walker2D, step::Vector) = Walker2D(w.x + step[1], w.y + step[2])

# ╔═╡ cb0ef266-91d5-11eb-314b-0545c0c817d0
function trajectory(w::W, N) where {W}   # W is a type parameter
	ws = [position(w)]

	for i in 1:N
		pos = position(w)
		w = update(w, step(w))
		
		push!(ws, position(w))
	end
	
	return ws
end

# ╔═╡ 048fac02-91da-11eb-0d26-4f258b4cd043
trajectory(Walker1D(0), 10)

# ╔═╡ 74182fe0-91da-11eb-219a-01f13b86406d
traj = trajectory(Walker2D(0, 0), 10^N)

# ╔═╡ 4c8d8294-91db-11eb-353d-c3696c615b3d
begin
	plot(traj[1:t], ratio=1, leg=false, alpha=0.5, lw=2)
	scatter!([ traj[1], traj[t] ], c=[:red, :green])
	
	xlims!(minimum(first.(traj)) - 1, maximum(first.(traj)) + 1)
	ylims!(minimum(last.(traj)) - 1, maximum(last.(traj)) + 1)
	
end

# ╔═╡ 57972a32-91e5-11eb-1d62-fbc22c494db9
md"""
## Random walks as sum of random variables
"""

# ╔═╡ 63122dd0-91e5-11eb-34c3-b1b5c87809b8
md"""
We can connect up with the discussion from last lecture about the sum of random variables: The position $S_n$ of a random walk at time $n$ is a random variable that is the sum of $n$ random variables (which are often independent and id

$$S_n = X_1 + \cdots + X_n$$

where $X_i$ is the random variable modelling the $i$th step of the walk.

Often each $X_i$ will be independent random variables of the same type, in which case we talk about an **independent and identically distributed** (IID) collection.

In principle we could use the method from last lecture to model this, but there is a difficulty since we are interested in all intermediate steps: The values $S_{n-1}$ and $S_n$, for example, are *not* independent (even though the $X_i$ are). For example, if $S_{n-1}$ is large, then $S_n$ is also about as large, so they are highly correlated.
"""

# ╔═╡ bd013582-91e7-11eb-2d79-e18e45f6d639
md"""
## Cumulative sum
"""

# ╔═╡ c092ab72-91e7-11eb-2bd1-13c8c8bb30e4
md"""
Suppose we generate steps $X_i$, e.g.
"""

# ╔═╡ cfb0f9ba-91e7-11eb-26b5-5f0f59d03cff
steps = rand( (-1, +1), 10 )

# ╔═╡ d6edb326-91e7-11eb-03b1-d93e3bc83ca6
md"""
The trajectory, or sample path, of the random walk, is the collection of positions over time. These are the **partial sums**

$$\begin{align}
S_1 & = X_1 \\
S_2 &= X_1 + X_2 \\
S_3 &= X_1 + X_2 + X_3
\end{align}$$

etc.
"""

# ╔═╡ 4ef42cec-91e8-11eb-2976-0950ffe5de6c
md"""
We can calculate all of these values using the `cumsum` ("cumulative sum", also called "prefix sum" or "scan") function:
"""

# ╔═╡ ace8658e-91e8-11eb-0b9d-4b759635e417
cumsum(steps)

# ╔═╡ b049ff58-91e8-11eb-203b-4f4b5ee5f01f
md"""
Let's plot this:
"""

# ╔═╡ b6775d3a-91e8-11eb-0187-618cb538d142
plot(cumsum(steps), m=:o, leg=false, size=(500, 300))

# ╔═╡ 8b1441b6-91e4-11eb-16b2-d7eadd3fd69c
md"""
# Trajectories vs evolving probability distributions
"""

# ╔═╡ d1315f94-91e4-11eb-1076-81156e24d2f1
md"""
So far we have looked at single trajectories of random walks. We can think of this as the equivalent of sampling using `rand`. Suppose that we were to (conceptually) run millions or billions of trajectories of our random walk. At each *time* step, we can ask for the probability distribution in *space*, averaged over all those walks. 
Note that we now have *two* variables. We could do this by literally running many walks and computing histograms. Instead we can look at how probabilities move around.

Let's call $p_i^t$ the probability of being at site $i$ at time $t$. We can then ask what the probability of being at site $i$ at time $t+1$. It must have been in one of the neighbouring sites at time $t$, so

$$p_i^{t+1} = \textstyle \frac{1}{2} (p_{i-1}^{t} + p_{i+1}^{t}).$$

This is sometimes called the **master equation** (even though that is rather a useless, non-descriptive name). It describes how the probability distribution 
"""

# ╔═╡ 97c958c2-91eb-11eb-17cb-410acc7f3e49
md"""
At time $t$ we have a whole probability distribution, which we can think of as a vector $\mathbf{p}^t$.

Let's write a function to **evolve** the vector to the next time step.
"""

# ╔═╡ b7a98dba-91eb-11eb-3c78-074aa835b5fb
function evolve(p)
	p′ = similar(p)   # make a vector of the same length and type
	                  # to store the probability vector at the next time step
	
	for i in 2:length(p)-1
		p′[i] = 0.5 * (p[i-1] + p[i+1])
	end
	
	p′[1] = 0
	p′[end] = 0
	
	return p′
	
end

# ╔═╡ f7f17c0c-91eb-11eb-3fa5-3bd90bb7044e
md"""
Wait... Do you recognise this?

We have seen this before! This is just a **convolution**, like the blurring kernel from image processing (except that in our particular model `p[i]` itself does not contribute to the value of `new_p[i]`.)
"""

# ╔═╡ f0aed302-91eb-11eb-13fb-d9418ef327a8
md"""
Note that just as we did with images, we have a problem at the edges of the system. We need to specify **boundary conditions**. For now we have put 0s at the boundary. This corresponds to probability *escaping* at the boundary: any probability that arrives at the boundary (i.e. the first or last cell) at a given time step does not stay in the system. We can think of the probability as analogous to a chemical that disappears 
"""

# ╔═╡ c1062e00-922b-11eb-1f31-ddbd03f8f986
md"""
We also need to specify an *initial* condition $\mathbf{p}_0$. This tells us where our walker is at time $0$. Suppose that all walkers start at $0$. We will place this in the middle of our vector. Then the probability at that location is 1, while the probability elsewhere is 0:
"""

# ╔═╡ 547188ea-9233-11eb-1a89-5ff9468b31f7
function initial_condition(n)
	
	p0 = zeros(n)
	p0[n ÷ 2 + 1] = 1
	
	return p0
end

# ╔═╡ 2920abfe-91ec-11eb-19bc-935fa1ba0a96
md"""
Now let's try and visualise the time evolution.
"""

# ╔═╡ 3fadf88c-9236-11eb-19fa-d191ac5a6191
function time_evolution(p0, N)
	ps = [p0]
	p = p0
	
	for i in 1:N
		p = evolve(p)
		push!(ps, copy(p))
	end
	
	return ps
end

# ╔═╡ 58653a70-9236-11eb-3dae-47adc2a77cb4
p0 = initial_condition(101)

# ╔═╡ 5d02f21e-9236-11eb-26ea-6593aa80a2eb
ps = time_evolution(p0, 100)

# ╔═╡ 863caf0a-9236-11eb-1013-ab7d83f3fc0c
ps[2]

# ╔═╡ b803406e-9236-11eb-3aad-056b7f2c9b4b
md"""
t = $(@bind tt Slider(1:length(ps), show_value=true, default=1))
"""

# ╔═╡ cc7aaeea-9236-11eb-3fad-2b5ad3962ec1
plot(ps[tt], ylim=(0, 1), leg=false, size=(500, 300))

# ╔═╡ dabb5766-9236-11eb-3be9-9b33ba5af68a
ps[tt]

# ╔═╡ 6cde6ef4-9236-11eb-219a-4d20adaf9988
M = reduce(hcat, ps)'

# ╔═╡ 7e8c1a2a-9236-11eb-20e9-57f6601f5472
heatmap(M, yflip=true)

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git-tree-sha1 = "8e9441ee83492030ace98f9789a654a6d0b1f643"
uuid = "2702e6a9-849d-5ed8-8c21-79e8b8f9ee43"
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[[deps.ExceptionUnwrapping]]
deps = ["Test"]
git-tree-sha1 = "dcb08a0d93ec0b1cdc4af184b26b591e9695423a"
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[[deps.Expat_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
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[[deps.FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
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[[deps.FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "PCRE2_jll", "Zlib_jll", "libaom_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "466d45dc38e15794ec7d5d63ec03d776a9aff36e"
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[[deps.FileWatching]]
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[[deps.FixedPointNumbers]]
deps = ["Statistics"]
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[[deps.Fontconfig_jll]]
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[[deps.Format]]
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[[deps.FreeType2_jll]]
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[[deps.FriBidi_jll]]
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[[deps.GLFW_jll]]
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[[deps.GR]]
deps = ["Artifacts", "Base64", "DelimitedFiles", "Downloads", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Preferences", "Printf", "Random", "Serialization", "Sockets", "TOML", "Tar", "Test", "p7zip_jll"]
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[[deps.GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "FreeType2_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Qt6Base_jll", "Zlib_jll", "libpng_jll"]
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[[deps.Gettext_jll]]
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git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
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[[deps.Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE2_jll", "Zlib_jll"]
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[[deps.Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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[[deps.Grisu]]
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[[deps.HTTP]]
deps = ["Base64", "CodecZlib", "ConcurrentUtilities", "Dates", "ExceptionUnwrapping", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "OpenSSL", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
git-tree-sha1 = "d1d712be3164d61d1fb98e7ce9bcbc6cc06b45ed"
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[[deps.HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
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[[deps.Hyperscript]]
deps = ["Test"]
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[[deps.HypertextLiteral]]
deps = ["Tricks"]
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[[deps.IOCapture]]
deps = ["Logging", "Random"]
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[[deps.InteractiveUtils]]
deps = ["Markdown"]
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[[deps.IrrationalConstants]]
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[[deps.JLFzf]]
deps = ["Pipe", "REPL", "Random", "fzf_jll"]
git-tree-sha1 = "a53ebe394b71470c7f97c2e7e170d51df21b17af"
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[[deps.JLLWrappers]]
deps = ["Artifacts", "Preferences"]
git-tree-sha1 = "7e5d6779a1e09a36db2a7b6cff50942a0a7d0fca"
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[[deps.JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
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[[deps.JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c84a835e1a09b289ffcd2271bf2a337bbdda6637"
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[[deps.LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
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[[deps.LERC_jll]]
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[[deps.LLVMOpenMP_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "d986ce2d884d49126836ea94ed5bfb0f12679713"
uuid = "1d63c593-3942-5779-bab2-d838dc0a180e"
version = "15.0.7+0"

[[deps.LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "70c5da094887fd2cae843b8db33920bac4b6f07d"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
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[[deps.LaTeXStrings]]
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[[deps.Latexify]]
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    [deps.Latexify.extensions]
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    SymEngineExt = "SymEngine"

    [deps.Latexify.weakdeps]
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[[deps.LibCURL]]
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[[deps.LibCURL_jll]]
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[[deps.LibGit2]]
deps = ["Base64", "LibGit2_jll", "NetworkOptions", "Printf", "SHA"]
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[[deps.LibGit2_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll"]
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[[deps.LibSSH2_jll]]
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[[deps.Libdl]]
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[[deps.Libffi_jll]]
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git-tree-sha1 = "0b4a5d71f3e5200a7dff793393e09dfc2d874290"
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[[deps.Libgcrypt_jll]]
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git-tree-sha1 = "9fd170c4bbfd8b935fdc5f8b7aa33532c991a673"
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[[deps.Libglvnd_jll]]
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git-tree-sha1 = "6f73d1dd803986947b2c750138528a999a6c7733"
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[[deps.Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "fbb1f2bef882392312feb1ede3615ddc1e9b99ed"
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[[deps.Libiconv_jll]]
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git-tree-sha1 = "f9557a255370125b405568f9767d6d195822a175"
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[[deps.Libmount_jll]]
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git-tree-sha1 = "0c4f9c4f1a50d8f35048fa0532dabbadf702f81e"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
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[[deps.Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "LERC_jll", "Libdl", "XZ_jll", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "2da088d113af58221c52828a80378e16be7d037a"
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[[deps.Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "5ee6203157c120d79034c748a2acba45b82b8807"
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[[deps.LinearAlgebra]]
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[[deps.LogExpFunctions]]
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    [deps.LogExpFunctions.extensions]
    LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"

    [deps.LogExpFunctions.weakdeps]
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[[deps.Logging]]
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[[deps.LoggingExtras]]
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[[deps.MIMEs]]
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[[deps.MacroTools]]
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[[deps.Markdown]]
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[[deps.MbedTLS]]
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[[deps.MbedTLS_jll]]
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[[deps.Measures]]
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[[deps.Missings]]
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[[deps.Mmap]]
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[[deps.MozillaCACerts_jll]]
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[[deps.NaNMath]]
deps = ["OpenLibm_jll"]
git-tree-sha1 = "0877504529a3e5c3343c6f8b4c0381e57e4387e4"
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[[deps.NetworkOptions]]
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[[deps.Ogg_jll]]
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git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
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[[deps.OpenBLAS_jll]]
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[[deps.OpenLibm_jll]]
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[[deps.OpenSSL]]
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git-tree-sha1 = "38cb508d080d21dc1128f7fb04f20387ed4c0af4"
uuid = "4d8831e6-92b7-49fb-bdf8-b643e874388c"
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[[deps.OpenSSL_jll]]
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git-tree-sha1 = "a028ee3cb5641cccc4c24e90c36b0a4f7707bdf5"
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[[deps.Opus_jll]]
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git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
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[[deps.OrderedCollections]]
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[[deps.PCRE2_jll]]
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version = "10.42.0+1"

[[deps.Parsers]]
deps = ["Dates", "PrecompileTools", "UUIDs"]
git-tree-sha1 = "8489905bcdbcfac64d1daa51ca07c0d8f0283821"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
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[[deps.Pipe]]
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[[deps.Pixman_jll]]
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git-tree-sha1 = "35621f10a7531bc8fa58f74610b1bfb70a3cfc6b"
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[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e678132f07ddb5bfa46857f0d7620fb9be675d3b"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+0"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a68c9655fbe6dfcab3d972808f1aafec151ce3f8"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.43.0+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.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"

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

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "d7015d2e18a5fd9a4f47de711837e980519781a4"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.43+1"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[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"

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

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

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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