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pluto_whatisdata.jl
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### A Pluto.jl notebook ###
# v0.19.36
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
# ╔═╡ 8f3a2089-c44c-450d-8d10-d355c9503ed7
begin
using PlutoUI
using LaTeXStrings
using Plots
using Colors
using ImageFiltering
using Images
using ImageMagick
using TestImages
using ImageTransformations
using LinearAlgebra
using LowRankApprox
using StatsBase
using Statistics
using Formatting
using DataFrames
using CSV
using Dates
using FreqTables
end
# ╔═╡ 6b1ce739-aca6-47d6-842a-701712b2cc61
md"""
### データとは何か
2022年12月5日(月) 8:30-
千原浩之
- 8:30-9:30 Zoom での講義
- 高等学校 情報I におけるデータ活用
- 線形代数という数ベクトルや行列を扱う数学
- n次元データ, n=1,2,3,4
- データ処理は線形代数の基本的演算である
- 9:30-10:00 WebClass の試験機能を利用した演習
- サイズの小さいデータに関する選択式問題若干
- この資料
- Julia Programming Language の Pluto.jl を使っている.
- 下記を一体化している:
- LaTeX による数式を含む文書
- プログラムのコードと出力
- 見てもらう資料ではなく使ってもらう資料である.
- 使いやすいが商用の MATLAB でも同様のことができる.
"""
# ╔═╡ 47930bf6-8594-413d-8f2d-86ad500aa66a
md"""
### 高等学校 情報I におけるデータ活用
- 2022年度に導入された新科目で必修である.
- 従来の情報リテラシーに簡単な情報活用が追加されている.
- 公開データの活用
- データの可視化
- Julia 言語を使って公開データの活用と可視化をやってみよう.
"""
# ╔═╡ a1dc1236-a3d4-456a-ab2b-5a3f2bbac38a
md"""
## 1. Singular Value Decomposition
Let $\mathbb{K}$ denote $\mathbb{R}$ or $\mathbb{C}$, and let $A \in \mathbb{K}^{m{\times}n}$. Set $r=\operatorname{rank}(A)$ for short. We state well-known facts.
1. $\operatorname{rank}(A^\ast{A})=\operatorname{rank}(AA^\ast)=\operatorname{rank}(A).$
2. $A^\ast{A}\geqq0.$
3. There exist positive numbers $\sigma_1 \geqq \dotsc \geqq \sigma_r >0$ such that
$\det(zE_n-A^\ast{A})=
(z-\sigma_1^2)\dotsc(z-\sigma_r^2)z^{n-r}.$
4. There exist an orthonormal system $\{u_1,\dotsc,u_m\}$ of $\mathbb{K}^m$ and an orthonormal system $\{v_1,\dotsc,v_n\}$ of $\mathbb{K}^n$ such that
$A
=
U{\Sigma}V^\ast
=
\sum_{l=1}^r
\sigma_l u_l v_l^\ast.$
5. The above decomposition of a matrix $A$ is said to be the singular value decomposition of $A$, where $U$ and $V$ are unitary matrices and $\Sigma$ is a rectangular diagonal matrix of the form
$U=[u_1,\dotsc,u_m],
\quad
V=[v_1,\dotsc,v_n],
\quad
\Sigma
=
\begin{bmatrix}
\sigma_1 & & &
\\
& \ddots & &
\\
& & \sigma_r &
\\
& & &
\end{bmatrix}.$
"""
# ╔═╡ 1e72b8d9-b161-4f31-92e1-b90a25a02a35
md"""
## 2. Low rank approximation
Let $A \in \mathbb{K}^{m{\times}n}$, and let $r=\operatorname{rank}(A)$.
Suppose that the SVD of $A$ is given by
$A=\displaystyle\sum_{l=1}^r\sigma_l u_l v_l^\ast.$
Set
$A_k:=\sum_{l=1}^k\sigma_l u_l v_l^\ast,
\quad
k=1,\dotsc,r.$
Then $\operatorname{rank}(A_k)=k$. It is well-known that $A_k$ is the best approximation of $A$ among $m{\times}n$ matrices with rank $k$. More precisely we have the Eckart-Young inequality of the form
$\lVert{A-B}\rVert
\geqq
\lVert{A-A_k}\rVert,
\quad
B\in\mathbb{K}^{m{\times}n},
\
\operatorname{rank}(B)=k,$
where $\lVert\cdot\rVert$ is the spectral norm of matrices defined by
$\lVert{B}\rVert
:=
\max_{\substack{x \in \mathbb{K}^n \\ x\ne0}}
\frac{\lvert{Bx}\rvert}{\lvert{x}\rvert},$
and $\lvert\cdot\rvert$ is the standard norm of numeric vectors. The same inequlity holds if we replace the spectral norm by the Frobenius norm or the nuclear norm.
"""
# ╔═╡ 2b8cafe5-0efb-4892-957c-055c63bf23ed
begin
I=load("./material/char_kway_teow.jpg");
X=imresize(I, ratio=1/4);
(p,q)=size(X);
A=channelview(X);
R=Array{Float64}(A[1,:,:]);
G=Array{Float64}(A[2,:,:]);
B=Array{Float64}(A[3,:,:]);
RU, RS, RV=psvd(R);
GU, GS, GV=psvd(G);
BU, BS, BV=psvd(B);
rank=60;
DR=zeros(p,q,rank);
DG=zeros(p,q,rank);
DB=zeros(p,q,rank);
for r=1:rank
DR[:,:,r]=sum(RS[n]*RU[1:p,n]*(RV[1:q,n])' for n=1:r);
DG[:,:,r]=sum(GS[n]*GU[1:p,n]*(GV[1:q,n])' for n=1:r);
DB[:,:,r]=sum(BS[n]*BU[1:p,n]*(BV[1:q,n])' for n=1:r);
end
end
# ╔═╡ dd7e95e0-8d80-44f0-8ad6-5c9dece4d18f
begin
Y=zeros(3,p,q,rank+16);
Z=zeros(p,q)
for r=1:4
Y[1,:,:,r]=R;
Y[2,:,:,r]=G;
Y[3,:,:,r]=B;
end
for r=5:8
Y[1,:,:,r]=Z;
Y[2,:,:,r]=Z;
Y[3,:,:,r]=Z;
end
for r=9:rank+8
Y[1,:,:,r]=DR[:,:,r-8];
Y[2,:,:,r]=DG[:,:,r-8];
Y[3,:,:,r]=DB[:,:,r-8];
end
for r=rank+9:rank+12
Y[1,:,:,r]=Z;
Y[2,:,:,r]=Z;
Y[3,:,:,r]=Z;
end
for r=rank+13:rank+16
Y[1,:,:,r]=R;
Y[2,:,:,r]=G;
Y[3,:,:,r]=B;
end
end
# ╔═╡ f29dc05b-8209-42a1-8f37-4f608c216459
begin
W=zeros(3,p,q,rank);
for r=1:rank
W[1,:,:,r]=DR[:,:,r];
W[2,:,:,r]=DG[:,:,r];
W[3,:,:,r]=DB[:,:,r];
end
end
# ╔═╡ 33e36dbc-6682-4de6-a911-7eb2a0ec4f05
md"""
rank = $(@bind r Slider(1:rank, show_value=true))
"""
# ╔═╡ a9f38308-8483-4b3a-9160-f2622027c7e2
begin
p1=plot(colorview(RGB,W[:,:,:,r]),
title="Low Rank Approx 1-60/384",
xaxis=false,
xticks=false,
yaxis=false,
yticks=false,
grid=false);
p2=plot(colorview(RGB,X),
title="Original RGB Image",
xaxis=false,
xticks=false,
yaxis=false,
yticks=false,
grid=false);
plot(p1,p2)
end
# ╔═╡ b1e6cc97-a684-4cc2-8a53-5514544b453b
size(X)
# ╔═╡ 33649745-0f75-4d96-a9b5-b8abe8655f24
md"""
#### Singularity and Information
"""
# ╔═╡ 8cd077eb-d458-46f5-acfc-e412a1691876
begin
G5=zeros(100,100)
H5=ones(100,100);
Z5=zeros(100,100);
for i=1:100
for j=1:100
G5[i,j]=j/100;
end
end
G5=[Z5 G5];
H5=[Z5 H5];
for i=1:1
end
end
# ╔═╡ 2204dff1-faaf-4bdf-b28c-83734030b2e2
begin
p5=heatmap(G5,
title="ReLU",
xaxis=L"x",
xticks=([1 100 200],[-1,0,1]),
guidefont=(16);
xtickfont=font(12),
ytickfont=font(12),
yaxis=L"y",
yticks=([1 100],[0,1]),
legend=false,
size=(300,100),
grid=false);
q5=heatmap(H5,
title="Heaviside=ReLU'",
xaxis=L"x",
xticks=([1 100 200],[-1,0,1]),
xtickfont=font(12),
ytickfont=font(12),
guidefont=(15),
yaxis=L"y",
yticks=([1 100],[0,1]),
legend=false,
size=(300,100),
grid=false);
plot(p5,q5,size=(600,200))
end
# ╔═╡ 5d62a85b-37d0-4be2-8b20-b45867574684
begin
r5=surface(G5,
grid=false,
title="ReLU",
xaxis=L"x",
xticks=([1 100 200],[-1,0,1]),
guidefont=(16);
xtickfont=font(12),
ytickfont=font(12),
#yaxis=L"y",
yticks=false,
zticks=false,
camera = (-30, 20),
legend=:none)
s5=surface(H5,
grid=false,
title="Heaviside=ReLU'",
xaxis=L"x",
xticks=([1 100 200],[-1,0,1]),
guidefont=(16);
xtickfont=font(12),
ytickfont=font(12),
#yaxis=L"y",
yticks=false,
zticks=false,
camera = (-30, 20),
legend=false)
plot(r5,s5)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CSV = "336ed68f-0bac-5ca0-87d4-7b16caf5d00b"
Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0"
Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
Formatting = "59287772-0a20-5a39-b81b-1366585eb4c0"
FreqTables = "da1fdf0e-e0ff-5433-a45f-9bb5ff651cb1"
ImageFiltering = "6a3955dd-da59-5b1f-98d4-e7296123deb5"
ImageMagick = "6218d12a-5da1-5696-b52f-db25d2ecc6d1"
ImageTransformations = "02fcd773-0e25-5acc-982a-7f6622650795"
Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
LowRankApprox = "898213cb-b102-5a47-900c-97e73b919f73"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
[compat]
CSV = "~0.10.4"
Colors = "~0.12.8"
DataFrames = "~1.3.6"
Formatting = "~0.4.2"
FreqTables = "~0.4.5"
ImageFiltering = "~0.7.2"
ImageMagick = "~1.2.2"
ImageTransformations = "~0.9.5"
Images = "~0.25.2"
LaTeXStrings = "~1.3.0"
LowRankApprox = "~0.5.2"
Plots = "~1.34.3"
PlutoUI = "~0.7.43"
StatsBase = "~0.33.21"
TestImages = "~1.7.0"
"""
# ╔═╡ 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"
manifest_format = "2.0"
project_hash = "6c4f8e3ee79a27d41a04233a2ac8e6e80379e7c5"
[[deps.AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.5.0"
weakdeps = ["ChainRulesCore", "Test"]
[deps.AbstractFFTs.extensions]
AbstractFFTsChainRulesCoreExt = "ChainRulesCore"
AbstractFFTsTestExt = "Test"
[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "793501dcd3fa7ce8d375a2c878dca2296232686e"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.2.2"
[[deps.Adapt]]
deps = ["LinearAlgebra", "Requires"]
git-tree-sha1 = "cde29ddf7e5726c9fb511f340244ea3481267608"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.7.2"
weakdeps = ["StaticArrays"]
[deps.Adapt.extensions]
AdaptStaticArraysExt = "StaticArrays"
[[deps.ArgTools]]
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