abelsiqueira/tulipa-fitting.jl

## tulipa-fitting.jl
### A Pluto.jl notebook ###
# v0.19.32

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

# ╔═╡ 027aeb5e-81fb-11ee-3849-256f23c3d04b
using PlutoUI, Plots, Roots, Random, Statistics

# ╔═╡ a49f5ee7-55a0-44ef-aba1-30554d696885
md"""
Given a list of $m$ numbers $1 \geq p_1 \geq p_2 \geq \cdots \geq p_m \geq 0$, and a target value $\mu \in (0, 1)$, we are looking for a solution to

$$\frac{1}{m}\sum_{i=1}^m p_i^x = \mu.$$

Questions:

1. When is it possible to find such $x$?
2. If such $x$ exists, how do we find it?
3. What to do when it is not possible to find such $x$?
"""

# ╔═╡ 55a72735-a6e1-445f-ade9-41f369c02770
md"""
Since both $p_i$ and $x$ can be zero, $p_i^x$ can be undefined. [In Julia, and other programming languages](https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero), $0^0$ is commonly defined as 1, but I don't think it makes sense here, since the $p_i$ values are fixed, and $x$ is variable. I suggest defining $\sigma:[0,1]\times[0,\infty)$ as

$$\sigma(p, x) = \left\{\begin{array}{ll}p^x, \ & \text{if } p > 0, \\ 0, \ & \text{otherwise}.\end{array}\right.$$

This allows $x \mapsto \sigma(p, x)$ to be continuous for any value $p \in [0, 1]$.

Therefore, the problem can be redefined as finding $x$ such that

$$\frac{1}{m}\sum_{i=1}^m \sigma(p_i, x) = \mu.$$
"""

# ╔═╡ 5ecf5d15-d05e-417d-a105-c869ed1e8a57
md"""
However, to avoid this notation, we can assume, without loss of generality, that there are $r$ non-zero $p_i$, i.e., $p_1 \geq \cdots \geq p_r > 0$, and $p_{r+1} = \cdots = p_m = 0$.

This simplifies the problem to finding $x$ such that

$$\frac{1}{m} \sum_{i=1}^r p_i^x = \mu.$$

Let's define $S(x) = \frac{1}{m}\sum_{i=1}^r p_i^x$ to help us with the notation.
"""

# ╔═╡ 646e6bc3-f8fd-4e0e-b430-3babce82b3ba
begin
	Random.seed!(0)

	m = 50
	divm4 = div(m, 4)
	P1 = [0.0; 1.0; rand(m - 2)] |> sort |> reverse
	P2 = [zeros(divm4); ones(divm4); rand(m - 2divm4)] |> sort |> reverse
	P3 = 0.2 * P2 |> sort |> reverse
	P4 = 1.0 .- 0.2 * P2 |> sort |> reverse

	σ(p, x) = p > 0 ? p^x : 0.0
	S(P, x) = mean(σ.(P, x))
end

# ╔═╡ 01a10c06-3886-410b-8843-ee8eeaaebaa6
md"""
To visualize the problem, we are going to use four sets:

- a set with a 0 and a 1
- a set with a lot of 0s and 1s (25% each)
- a set that only contains numbers from 0 to $\alpha < 0.5$.
- a set that only contains numbers from $1 - \alpha$ to 1.

First, let's visualize the effect of $p_i^x$ on these sets:
"""

# ╔═╡ 111bebf3-5c8e-421c-a4cb-4c840787277c
md"""
x = $(@bind ui_x_1 Slider(2.0 .^ (-10:0.1:10), show_value=true, default=1.0))
"""

# ╔═╡ 17136806-1dc9-4218-9b7d-bfefdf10b930
let
	plt = plot(layout = grid(2, 2), size=(800, 600), leg=false)
	for (i, P) in enumerate([P1, P2, P3, P4])
		scatter!(plt[i], P, c=:lightblue)
		plot!(plt[i], [0, m + 1], mean(P) * ones(2), c=:blue)
		x = ui_x_1
		r = count(P .> 0)
		n = count(P .== 1)
		if x != 1
			scatter!(plt[i], σ.(P, x), c=:pink)
			plot!(plt[i], [0, m + 1], mean(σ.(P, x)) * ones(2), c=:red)
		end
		ylims!(plt[i], -0.05, 1.05)
		title!(plt[i], "Set $i")
	end
	plt
end

# ╔═╡ 877d1f60-4c7a-4d64-98b0-c7daa1d75691
md"""
Furthermore, let's take a look at the plot of $S$ for each of the sets.
"""

# ╔═╡ 89fcc338-55d5-4bbf-ad89-835c7fa7f389
let
	plt = plot(size=(800, 600))
	for (i, P) in enumerate([P1, P2, P3, P4])
		plot!(plt, x -> S(P, x), 0, 2^5, lab="Set $i")
	end
	plot!(plt, ones(2), [0, 1], c=:gray, l=:dash, lab="x = 1")
	ylims!(0, 1)
	plt
end

# ╔═╡ 78133169-d63f-4c3b-9db0-d0c816e5adf7
md"""
**Question 1**: Since $p^0 = 1$ for positive $p$, then $S(0) = \frac{r}{m}$.
$S$ is non-increasing since

$$S'(x) = \frac{1}{m} \sum_{i=1}^r p_i^x \ln p_i \leq 0.$$

And assuming that there are $n$ values such that $p_i = 1$, then

$$\lim_{x \to \infty} S(x) = \frac{n}{m}.$$

This means that there is a solution to the problem if $\dfrac{n}{m} < \mu \leq \dfrac{r}{m}$.
"""

# ╔═╡ 7134c7ef-6b93-4682-bf76-f1e383a81966
md"""
**Question 2**:
Assuming $S$ decreasing and $\mu$ in range, we can solve the problem by looking for an interval $[a, b]$ such that $S(x) - \mu$ changes sign.
This can be done by creating an increasing sequence of points $v_1, v_2, \dots$, such that $v_1 = 0$, and $v_{i+1} > v_i$ with $\lim_{i \to \infty} v_i = \infty$.
For instance, the sequence $0, 1, 2, 4, 8, \dots$.

Since $S$ is decreasing, and $\mu$ is in range, then either $S(v_i) = \mu$ for some $v_i$, or $S(x) - \mu$ will change sign in some interval $[v_i, v_{i+1}]$.

Given the interval, we perform a bisection search or similar.
"""

# ╔═╡ 0498587c-959b-40f2-a314-256d862544d6
md"""
**Question 3:** If $\mu > \dfrac{r}{m}$, then $x = 0$ will yield $S(0) = \dfrac{r}{m}$, which is the closest to $\mu$.

Alternatively, if $\mu \leq \dfrac{n}{m}$, then $S(x) \to \dfrac{n}{m}$ as $x \to \infty$. In this case, we just select a reasonably large value for $x$.

"""

# ╔═╡ d5430df1-dbc3-416e-ab53-1cb47cab83c5
md"""
μ = $(@bind ui_μ Slider(0.0:0.01:1.0, show_value=true, default=0.5))
$br
show solution = $(@bind ui_show_solution CheckBox(default=false))
"""

# ╔═╡ c1640e97-0049-4df2-a541-cc6f1215a811
begin
	"""
		(a, b) = find_search_interval(f)

	Returns an interval such that ``f(a) × f(b) ≯ 0``.
	It could be 0 for either endpoint, but it is not positive,
	ensuring that there is a root in `[a, b]`.
	"""
	function find_search_interval(f)
		a = 0.0
		b = 1.0
		fa, fb = f(a), f(b)
		while fa * fb > 0
			δ = b - a
			a, fa = b, fb
			b += 2δ
			fb = f(b)
		end
		return a, b
	end
end

# ╔═╡ e00b48ee-5c75-4d19-822c-079373cdb668
begin
	"""
		x = find_solution(P, μ)

	Finds a points such that ``S(x) = μ``, if possible, where

	```math
	S(x) = \\frac{1}{|P|} \\sum_{p ∈ P:\\ p > 0} p^x.
	```

	If not possible, return either 0 or 1000, depending on what is most appropriate.
	"""
	function find_solution(P, μ)
		m = length(P)
		r = count(P .> 0)
		n = count(P .== 1)
		if μ > r / m
			return 0.0
		elseif μ > n / m
			a, b = find_search_interval(x -> S(P, x) - μ)
			return Roots.find_zero(x -> S(P, x) - μ, (a, b))
		else
			return 1000.0
		end
	end
end

# ╔═╡ 158ab3f5-4ec4-4654-a8ff-25570d2dc842
let
	plt = plot(layout = grid(4, 2), size=(800, 1000), leg=false)
	for (i, P) in enumerate([P1, P2, P3, P4])
		μ = ui_μ
		r = count(P .> 0)
		n = count(P .== 1)

		scatter!(plt[i,1], P, c=:lightblue)
		plot!(plt[i,1], [0, m + 1], mean(P) * ones(2), c=:blue)
		plot!(plt[i,1], [0, m + 1], μ * ones(2), c=:magenta)

		if n < μ * m ≤ r
			ℓ, u = find_search_interval(x -> S(P, x) - μ)
			plot!(plt[i,2], x -> S(P, x), 0, 2u, lab="Set $i")
			plot!(plt[i,2], [0, 2u], μ * ones(2), c=:magenta)
			plot!(plt[i,2], [ℓ, ℓ], [0, 1], c=:red, lab="ℓ")
			plot!(plt[i,2], [u, u], [0, 1], c=:red, lab="u")
		else
			plot!(plt[i,2], x -> S(P, x), 0, 15, lab="Set $i")
			plot!(plt[i,2], [0, 15], μ * ones(2), c=:magenta)
		end

		if ui_show_solution
			x = find_solution(P, μ)
			plot!(plt[i,1], [0, m + 1], S(P, x) * ones(2), c=:red, lw=2, l=:dash)
		end

		ylims!(plt[i,1], -0.05, 1.05)
		ylims!(plt[i,2], -0.05, 1.05)
		title!(plt[i,1], "Set $i")
	end
	plt
end

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version = "3.4.0+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", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[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"
"""

# ╔═╡ Cell order:
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