### A Pluto.jl notebook ### # v0.17.1 using Markdown using InteractiveUtils # ╔═╡ 6975f10e-2e06-11eb-34a5-f360be7dc49c using Plots # ╔═╡ b862ae08-2e2a-11eb-3471-c1e89f113b62 md"# Notebook 16 -- Math 2121, Fall 2021 In today's class we discussed how to compute a line of best fit by finding a least-squares solution to a certain linear system. Here we explore the problem of finding a **polynomial curve of best fit** that passes through some given datapoints, which imagine as having the form $(x,f(x))$ for some unknown function $f$." # ╔═╡ 015278ce-2e22-11eb-2710-596cdf9dac71 md"Suppose xvalues = [x1, x2, ..., xn]. The following method constructs the matrix $A = \left[\begin{array}{cccc} 1 & x_1 & x_1^2 & \dots & x_1^{m-1} \\ 1 & x_2 & x_2^2 & \dots & x_2^{m-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \dots & x_n^{m-1} \end{array}\right].$ This is sometimes called a **Vandermonde matrix**. If $f:\mathbb{R} \to \mathbb{R}$ is a function and we have $A\left[\begin{array}{c} c_0 \\ c_1 \\ \vdots \\ c_{m-1}\end{array}\right] = \left[\begin{array}{c} f(x_1) \\ f(x_2) \\ \vdots \\ f(x_n)\end{array}\right] \in \mathbb{R}^n$ then $p(x) = c_0 + c_1 x + c_2 x^2 + \dots + c_{m-1} x^{m-1}$ is a polynomial satisfying $p(x_i) = f(x_i)$ for each $i=1,2,\dots,n$." # ╔═╡ 826afaa6-2e06-11eb-1388-4f44b4c3ca88 function vandermonde(xvalues, m=0) n = length(xvalues) m = m == 0 ? n : m A = zeros(n, m) for i=1:n for j=1:m A[i, j] = xvalues[i]^(j - 1) end end return A end # ╔═╡ e519748e-2e06-11eb-2cba-ebf5f74892d2 vandermonde([1, 3, 4, 5], 4) # ╔═╡ d70f2b1a-2e22-11eb-1ffb-dd93e5423d51 md"Any square Vandermonde matrix is invertible. Hence, there is a unique polynomial of degree $d+1$ passing through any given $d$ datapoints $(x_i, y_i)$ with $x_1= 2 @assert b > a delta = (b - a) / (n - 1) return [i for i=a:delta:b] end # ╔═╡ b422d966-2e14-11eb-0511-03d646321184 equally_spaced_xvalues(-1, 1, 9) # ╔═╡ c2725a5c-2e14-11eb-09a2-9b5505007e04 md"The following method returns the **polynomial of best fit** (in the sense of least squares) that agrees in the input function fn at the given xvalues. If not given, the degree of the returned polynomial is the length of xvalues." # ╔═╡ 13e70d26-2e07-11eb-2249-87f0f6a78e9c function fit_polynomial(fn, xvalues, degree=0) b = [fn(x) for x=xvalues] A = vandermonde(xvalues, degree) coefficients = A \ b return x -> sum([x^(i - 1) * coefficients[i] for i=1:length(coefficients)]) end # ╔═╡ 2c8c6dae-2e15-11eb-07ca-5fd8c4abaf5a md"Below are some methods to plot our interpolations." # ╔═╡ 9342973e-2e07-11eb-0481-3986bf6792c2 function compare_plots(fun, pol, xvalues, title="") d = 0.001 rng = xvalues[1]:d:xvalues[end] p = plot(fun, rng, label="function", legend = :outertopright, title=title) plot!(pol, rng, label="polynomial") scatter!(xvalues, zeros(length(xvalues), 1), label=false, color=:grey) return p end # ╔═╡ 55fba216-2e08-11eb-2f88-1de5db0d81a0 function polynomial_iterpolation_anim(maxpoints, f, a, b, xvalues, degree=0) anim = @animate for npoints=[minimum([i maxpoints]) for i=2:maxpoints + 10] xvals = xvalues(a, b, npoints) compare_plots(f, fit_polynomial(f, xvals, degree), xvals, "Degree$(degree == 0 ? npoints - 1 : degree) interpolation with $(npoints) datapoints") end every 1 gif(anim, "anim_fps15.gif", fps = 2) end # ╔═╡ 428532a8-2e15-11eb-3190-7706ebd1e984 md"Here are methods to measure the error of our interpolations." # ╔═╡ 7f2670a0-2e09-11eb-144e-cf0b9ed5f43f function error(f, g, a, b) delta = (b - a) / 10000 sum([abs(f(i) - g(i))^2 * delta for i=a:delta:b]) end # ╔═╡ a965a8ce-2e09-11eb-1bac-2b748a87b4c7 function plot_log_error(maxpoints, f, a, b, xvalues, degree=0) plot([ log(error(f, fit_polynomial(f, xvalues(a, b, npoints), degree), a, b)) for npoints = 2:maxpoints ], legend=false, title="Log error in successive polynomial interpolations", xlabel="degree of interpolation", ylabel="error") end # ╔═╡ 73eed7b0-2e15-11eb-3340-631a9da2d80e md"Consider the function$f(x) = \frac{1}{1+25 x^2}$." # ╔═╡ 472693fa-2e07-11eb-02fc-798fd0984b03 f = x -> 1 / (1 + 25 * x^2) # ╔═╡ b503692c-2e07-11eb-3721-8591f77c8927 begin npoints = 9 a, b = -1, 1 equally_spaced = equally_spaced_xvalues(a, b, npoints) interpolation = fit_polynomial(f, equally_spaced) compare_plots( f, interpolation, equally_spaced, "Polynomial interpolation with$(npoints) datapoints") end # ╔═╡ 26f92248-2e23-11eb-3dda-195e49a1f203 md"A major problem involved in polynomial interpolation is **overfitting**. We see this for the preceding function when we sample evenly spaced points in the interval $[-1,1]$. As we add more datapoints and compute higher degree polynomials that fit the data, the 'fit' of these polynomials actually becomes worse! On other intervals, say $[-1, 0]$, this problem is less dramatic." # ╔═╡ a03e0238-2e08-11eb-235c-7b30b3d50de7 polynomial_iterpolation_anim(20, f, a, b, equally_spaced_xvalues) # ╔═╡ 0b1b5ec4-2e0a-11eb-01c2-41fc79082e0a plot_log_error(40, f, a, b, equally_spaced_xvalues) # ╔═╡ 91258612-2e3d-11eb-308f-57404de0272b polynomial_iterpolation_anim(12, f, -1, 0, equally_spaced_xvalues) # ╔═╡ 9ff14e4a-2e3d-11eb-28ee-d7528104b28f plot_log_error(20, f, -1, 0, equally_spaced_xvalues) # ╔═╡ 8ed5afde-2e15-11eb-10bf-15893fbb9812 md"One solution to the problem of overfitting is to fix the degree of our polynomial interpolation. This gives a better error that won't explode as we add more datapoints. On the other hand, the error may have a nontrivial lower bound and may not tend to zero as we add more data. If the number of datapoints $n$ is greater than our fixed degree $m$, then it may be impossible to construct a polynomial that passes through every datapoint. In this case the polynomial of best fit corresponds to a least-squares (approximate) solution to the linear system $A\left[\begin{array}{c} c_0 \\ c_1 \\ \vdots \\ c_m\end{array}\right] = \left[\begin{array}{c} f(x_1) \\ f(x_2) \\ \vdots \\ f(x_n)\end{array}\right]$ where $A$ is a Vandermonde matrix as defined above." # ╔═╡ aa6d33b6-2e13-11eb-01b7-4f47746ed88f fixed_degree = 20 # ╔═╡ a22bf464-2e13-11eb-06ee-c77315099327 polynomial_iterpolation_anim(45, f, a, b, equally_spaced_xvalues, fixed_degree) # ╔═╡ a350847c-2e13-11eb-0dc5-3bd49d4c0fe4 plot_log_error(60, f, a, b, equally_spaced_xvalues, fixed_degree) # ╔═╡ d90370dc-2e15-11eb-111e-83697cb9ed4d md"Another solution to the overfitting problem is to sample our input function at different $x$-values." # ╔═╡ d79b246a-2e0b-11eb-0475-63269816ed30 function chebyshev_spaced_xvalues(a, b, n) @assert n >= 2 @assert b > a cheb = [cos((2 * k - 1) / (2 * n) * pi) for k=n:-1:1] return [a + (i + 1) * (b - a) / 2 for i=cheb] end # ╔═╡ d2e19b90-2e13-11eb-20e8-b38226d4d76a md"The $x$-values returned by this method are not equally spaced between $a$ and $b$. However, if we sample our function at these points, the resulting polynomial interpolation is much more accurate, and we avoid overfitting as the degree of our interpolation increases:" # ╔═╡ 451b9054-2e0c-11eb-26e7-1be699377bc3 polynomial_iterpolation_anim(30, f, a, b, chebyshev_spaced_xvalues) # ╔═╡ c6d097cc-2e0c-11eb-24dc-b79e505e4dfd plot_log_error(30, f, a, b, chebyshev_spaced_xvalues) # ╔═╡ 0cea8658-2e14-11eb-0783-71ed809fef75 md"The error here is better than what we get by fixing the degree of our interpolation and using least-squares." # ╔═╡ f9808f2c-2e13-11eb-1f63-858268d8ccf5 md"What are the numbers chebyshev_spaced_xvalues(a, b, n)?" # ╔═╡ 56be592e-2e11-11eb-0d09-a9062e21a97c md"The **Chebyshev polynomials $T_n(x)$ of the first kind** are defined by the recurrence $T_0(x) = 1,$ $T_1(x) = x,$ $T_{n+1}(x) = 2x T_n(x) - T_{n-1}(x).$ These are the unique polynomials that satisfy $T_n(\cos(\theta)) = \cos(n\theta)$. " # ╔═╡ 973470ce-2e11-11eb-263c-cb3662b4080f function chebyshev(n) if n == 0 return x -> 1 elseif n == 1 return x -> x else return x -> 2 * x * chebyshev(n - 1)(x) - chebyshev(n - 2)(x) end end # ╔═╡ cc0a6c10-2e11-11eb-3f9e-07a9d7c03059 begin p = plot(title="Chebyshev polynomials of the first kind") for i=1:5 plot!(chebyshev(i), -1, 1, label="T_$(i)(x)", legend=:outerright) end plot(p) end # ╔═╡ 277520f6-2e13-11eb-1671-2ba29088898e md"The values returned by chebyshev_spaced_xvalues(-1, 1, n) are the roots of$T_n(x)$." # ╔═╡ af955bdc-2e12-11eb-0327-e35870887820 function plot_chebyshev(n) q = plot(chebyshev(n), -1, 1, title="Plot of T_$(n)(x) and Chebyshev-spaced x-values", legend=:outerright) xvalues = chebyshev_spaced_xvalues(-1, 1, n) scatter!(xvalues, zeros(n, 1), legend=false, color=:grey) plot(q) end # ╔═╡ 145d56a0-2e13-11eb-0d82-15c569b2e437 plot_chebyshev(20) # ╔═╡ 6cbb2d44-2e14-11eb-3855-a1bb833192c8 md"It can be shown that using chebyshev_spaced_xvalues(a, b, n) to sample our input function yields the best possible exact polynomial interpolation of degree $n$. See for more information." # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" [compat] Plots = "~1.23.6" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised [[Adapt]] deps = ["LinearAlgebra"] git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7" uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" version = "3.3.1" [[ArgTools]] uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" [[Artifacts]] uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" [[Base64]] uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" [[Bzip2_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2" uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0" version = "1.0.8+0" [[Cairo_jll]] deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", 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[[IterTools]] git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18" uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e" version = "1.3.0" [[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 = "be9eef9f9d78cecb6f262f3c10da151a6c5ab827" uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" version = "0.3.5" [[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 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb" uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3" version = "0.3.5" [[NetworkOptions]] uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908" [[Ogg_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f" uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051" version = "1.3.5+0" [[OpenSSL_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a" uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95" version = "1.1.10+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" [[Parsers]] deps = ["Dates"] git-tree-sha1 = "ae4bbcadb2906ccc085cf52ac286dc1377dceccc" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.1.2" [[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 = "b084324b4af5a438cd63619fd006614b3b20b87b" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" version = "1.0.15" [[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"] git-tree-sha1 = "0d185e8c33401084cab546a756b387b15f76720c" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.23.6" [[Preferences]] deps = ["TOML"] git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a" uuid = "21216c6a-2e73-6563-6e65-726566657250" version = "1.2.2" [[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" [[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 = "44a75aa7a527910ee3d1751d1f0e4148698add9e" uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01" version = "1.1.2" [[RecipesPipeline]] deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"] git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8" uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c" version = "0.4.1" [[Reexport]] git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" uuid = "189a3867-3050-52da-a836-e630ba90ab69" version = "1.2.2" [[Requires]] deps = ["UUIDs"] git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621" uuid = "ae029012-a4dd-5104-9daa-d747884805df" version = "1.1.3" [[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" [[StaticArrays]] deps = ["LinearAlgebra", "Random", "Statistics"] git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da" uuid = "90137ffa-7385-5640-81b9-e52037218182" version = "1.2.13" [[Statistics]] deps = ["LinearAlgebra", "SparseArrays"] uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [[StatsAPI]] git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510" uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0" version = "1.0.0" [[StatsBase]] deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] git-tree-sha1 = "eb35dcc66558b2dda84079b9a1be17557d32091a" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" version = "0.33.12" [[StructArrays]] deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"] git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3" uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" version = "0.6.3" [[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 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0" uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" version = "1.6.0" [[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" [[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", "Wayland_jll"] git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670" uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91" version = "1.18.0+4" [[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 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb" uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" version = "1.3.7+0" [[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: # ╟─b862ae08-2e2a-11eb-3471-c1e89f113b62 # ╠═6975f10e-2e06-11eb-34a5-f360be7dc49c # ╟─015278ce-2e22-11eb-2710-596cdf9dac71 # ╠═826afaa6-2e06-11eb-1388-4f44b4c3ca88 # ╠═e519748e-2e06-11eb-2cba-ebf5f74892d2 # ╟─d70f2b1a-2e22-11eb-1ffb-dd93e5423d51 # ╠═b0233b20-2e06-11eb-0da9-1be5d4f4f1ef # ╠═b422d966-2e14-11eb-0511-03d646321184 # ╟─c2725a5c-2e14-11eb-09a2-9b5505007e04 # ╠═13e70d26-2e07-11eb-2249-87f0f6a78e9c # ╟─2c8c6dae-2e15-11eb-07ca-5fd8c4abaf5a # ╠═9342973e-2e07-11eb-0481-3986bf6792c2 # ╠═55fba216-2e08-11eb-2f88-1de5db0d81a0 # ╟─428532a8-2e15-11eb-3190-7706ebd1e984 # ╠═7f2670a0-2e09-11eb-144e-cf0b9ed5f43f # ╠═a965a8ce-2e09-11eb-1bac-2b748a87b4c7 # ╟─73eed7b0-2e15-11eb-3340-631a9da2d80e # ╠═472693fa-2e07-11eb-02fc-798fd0984b03 # ╠═b503692c-2e07-11eb-3721-8591f77c8927 # ╟─26f92248-2e23-11eb-3dda-195e49a1f203 # ╠═a03e0238-2e08-11eb-235c-7b30b3d50de7 # ╠═0b1b5ec4-2e0a-11eb-01c2-41fc79082e0a # ╠═91258612-2e3d-11eb-308f-57404de0272b # ╠═9ff14e4a-2e3d-11eb-28ee-d7528104b28f # ╟─8ed5afde-2e15-11eb-10bf-15893fbb9812 # ╠═aa6d33b6-2e13-11eb-01b7-4f47746ed88f # ╠═a22bf464-2e13-11eb-06ee-c77315099327 # ╠═a350847c-2e13-11eb-0dc5-3bd49d4c0fe4 # ╟─d90370dc-2e15-11eb-111e-83697cb9ed4d # ╠═d79b246a-2e0b-11eb-0475-63269816ed30 # ╟─d2e19b90-2e13-11eb-20e8-b38226d4d76a # ╠═451b9054-2e0c-11eb-26e7-1be699377bc3 # ╠═c6d097cc-2e0c-11eb-24dc-b79e505e4dfd # ╟─0cea8658-2e14-11eb-0783-71ed809fef75 # ╟─f9808f2c-2e13-11eb-1f63-858268d8ccf5 # ╟─56be592e-2e11-11eb-0d09-a9062e21a97c # ╠═973470ce-2e11-11eb-263c-cb3662b4080f # ╠═cc0a6c10-2e11-11eb-3f9e-07a9d7c03059 # ╟─277520f6-2e13-11eb-1671-2ba29088898e # ╠═af955bdc-2e12-11eb-0327-e35870887820 # ╠═145d56a0-2e13-11eb-0d82-15c569b2e437 # ╟─6cbb2d44-2e14-11eb-3855-a1bb833192c8 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002