### 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 # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" Roots = "f2b01f46-fcfa-551c-844a-d8ac1e96c665" Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [compat] Plots = "~1.39.0" PlutoUI = "~0.7.53" Roots = "~2.0.20" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised julia_version = "1.10.0-rc1" manifest_format = "2.0" project_hash = "8a58ab81a206a44b71c07903807067bda46cabd2" [[deps.AbstractPlutoDingetjes]] deps = ["Pkg"] git-tree-sha1 = "91bd53c39b9cbfb5ef4b015e8b582d344532bd0a" uuid = "6e696c72-6542-2067-7265-42206c756150" 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= ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "47cf33e62e138b920039e8ff9f9841aafe1b733e" uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09" version = "0.35.1+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", "Pkg"] git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" 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: # ╠═027aeb5e-81fb-11ee-3849-256f23c3d04b # ╟─a49f5ee7-55a0-44ef-aba1-30554d696885 # ╟─55a72735-a6e1-445f-ade9-41f369c02770 # ╟─5ecf5d15-d05e-417d-a105-c869ed1e8a57 # ╟─646e6bc3-f8fd-4e0e-b430-3babce82b3ba # ╟─01a10c06-3886-410b-8843-ee8eeaaebaa6 # ╟─111bebf3-5c8e-421c-a4cb-4c840787277c # ╟─17136806-1dc9-4218-9b7d-bfefdf10b930 # ╟─877d1f60-4c7a-4d64-98b0-c7daa1d75691 # ╠═89fcc338-55d5-4bbf-ad89-835c7fa7f389 # ╟─78133169-d63f-4c3b-9db0-d0c816e5adf7 # ╟─7134c7ef-6b93-4682-bf76-f1e383a81966 # ╟─0498587c-959b-40f2-a314-256d862544d6 # ╟─d5430df1-dbc3-416e-ab53-1cb47cab83c5 # ╟─158ab3f5-4ec4-4654-a8ff-25570d2dc842 # ╟─c1640e97-0049-4df2-a541-cc6f1215a811 # ╟─e00b48ee-5c75-4d19-822c-079373cdb668 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002