### A Pluto.jl notebook ###
# v0.17.5

using Markdown
using InteractiveUtils

# ╔═╡ c7edc362-1d55-4556-94e2-5ccae0fce152
begin
	using LinearAlgebra
	using Statistics
	using PlutoUI
	using Plots
	gr()
end

# ╔═╡ e91395c4-e4dd-4bf7-8efa-c48d05418bb4
using Polynomials

# ╔═╡ f16d832c-67ad-11ec-3474-d9f0f2241ff9
html"""
	<div>Happy holiday! Remember to take care of yourself and your loved ones!</div>
<div id="snow"></div>
<style>
	body:not(.disable_ui):not(.more-specificity) {
        background-color:#e9ecff;
    }
	pluto-output{
		border-radius: 0px 8px 0px 0px;
        background-color:#e9ecff;
	}
	#snow {
        position: fixed;
    	top: 0;
    	left: 0;
    	right: 0;
    	bottom: 0;
    	pointer-events: none;
    	z-index: 1000;
	}
</style>
<script src="https://cdn.jsdelivr.net/particles.js/2.0.0/particles.min.js"></script>
<script>
        setTimeout(() => window.particlesJS("snow", {
            "particles": {
                "number": {
                    "value": 70,
                    "density": {
                        "enable": true,
                        "value_area": 800
                    }
                },
                "color": {
                    "value": "#ffffff"
                },
                "opacity": {
                    "value": 0.7,
                    "random": false,
                    "anim": {
                        "enable": false
                    }
                },
                "size": {
                    "value": 5,
                    "random": true,
                    "anim": {
                        "enable": false
                    }
                },
                "line_linked": {
                    "enable": false
                },
                "move": {
                    "enable": true,
                    "speed": 5,
                    "direction": "bottom",
                    "random": true,
                    "straight": false,
                    "out_mode": "out",
                    "bounce": false,
                    "attract": {
                        "enable": true,
                        "rotateX": 300,
                        "rotateY": 1200
                    }
                }
            },
            "interactivity": {
                "events": {
                    "onhover": {
                        "enable": false
                    },
                    "onclick": {
                        "enable": false
                    },
                    "resize": false
                }
            },
            "retina_detect": true
        }), 3000);
	</script>
"""


# ╔═╡ f16d834c-67ad-11ec-05db-f726c8f8c643
md""" # 機械学習帳
ここを参考にした.[機械学習帳](https://chokkan.github.io/mlnote/regression/01sra.html)
## 1. 単回帰
### (続き)1.4. 平均二乗誤差は$a$,$b$の関数
"""

# ╔═╡ 6ce3e88a-e10d-45c2-9722-aeed933931d9
begin
	D = [
		1 3
		3 6
		6 5
		8 7]

	X = D[:,1]
	Y = D[:,2]
end;

# ╔═╡ bc133d95-c122-49be-8789-01dd2b87ac0a
function calc_msr(a, b, D)
	xm = mean(D[:,1])
	ym = mean(D[:,2])
	x2m = mean(D[:,1].^2) 
	y2m = mean(D[:,2].^2) 
	xym = mean(D[:,1] .* D[:,2])
	return @. x2m * a^2 + b^2 + 2*xm *a *b - 2*xym *a - 2*ym*b + y2m
end

# ╔═╡ 86db1e1c-5164-4e3f-b877-f9c6f1a535c7
begin
	N = 100
	aa = range(-10,10, N)
	bb = range(-10,10, N)
end

# ╔═╡ c47e13d4-f0e9-46b0-b746-73563344487e
J = calc_msr(aa, bb', D)

# ╔═╡ f1896c70-c8ce-4d7f-832e-11a3eede0883
let
	plot(xlim=(-10,10), ylim=(-10,10), framestyle=:box, aspect_ratio=1)
	plot!(xticks=-10:5:10, yticks=-10.0:2.5:10.0)
	plot!(xlabel="a", ylabel="b", zlabel="L̂_D(a,b)")
	heatmap!(aa, bb, J'; colorbar_scale=:log10, clim=(10^(-1),10^4),
		fillcolor=cgrad(:viridis) )
	J_levels = [1., 10.0, 100.0, 1000.0]
	contour!(aa, bb, J'; lc=:red, lw=0.5, ls=:dash, levels= J_levels)
end

# ╔═╡ 02c50675-5c7e-4662-84fd-56cf5e34e666
md""" ### 1.5. 平均二乗残差を最小化する
単回帰の目的関数(平均二乗残差)は
```math
L̂_D(a,b) = \frac{N}{1} \sum_{i=1}^{N} ϵ_i^2 = \frac{1}{N} \sum_{i=1}^{N} (y_i - ŷ_i)^2 = \frac{1}{N} \sum_{i=1}^{N} (y_i - ax_i - b)^2
```
である。
上式を $b$ で偏微分すると
```math
b = ȳ - ax̄
```
となる。 単回帰の傾き $a$ を求める式
```math
a = \frac{ \overline{xy} - (x̄)(ȳ)}{ \overline{x^2} - (x̄)^2} 
	= \frac{  \rm{Cov[} \it{X, Y} \rm{]} } {\rm{Cov[} \it{X, X} \rm{]}}
    = \frac{  \rm{Cov[} \it{X, Y} \rm{]} } {\rm{Var[} \it{X} \rm{]}} 
```

"""

# ╔═╡ 0367c8ba-1c50-43ea-828f-fb0cbaefe691
function set_plot()
	plot(size=(500,500), xlim=(0,10), ylim=(0,10), framestyle=:box, legend=:topleft)
	plot!(xlabel="x", ylabel="y", gridalpha=0.5)
end

# ╔═╡ 4765fa67-4e96-4cc0-85fa-266128fb8fae
function plot_y(X, Y, colormap)
	scatter!(X, Y; c=colormap,  markerstrokewidth=0, label="")
end

# ╔═╡ 8f419e32-93f9-4067-9461-e0e2f69a723a
function plot_ŷ(a, b, offset, colormap)

	offset_ŷ = offset

	a, b = a, b
	f(x, a, b) = a*x + b
	Ŷ = f.(X, a, b)
	x = [0, 10]

	a = round(a; digits=3)
	b = round(b; digits=3)
	
	label = (a == 1) ? "x + $(b))" : "$(a)x + $(b)" 
	scatter!(D[:,1], Ŷ; c=colormap, shape=:star5, ms=6, markerstrokewidth=0, label="")
	plot!(x -> f(x, a, b); c=:black, lw=2, label=label)
	
	texts_ŷ = ["(x₁,ŷ₁)", "(x₂,ŷ₂)", "(x₃,ŷ₃)", "(x₄,ŷ₄)"]
	for i in 1:length(colormap)
		annotate!([(X[i],Ŷ[i]+offset_ŷ[i],text(texts_ŷ[i], 10,
			:center, colormap[i]))])
	end
	plot!()
end

# ╔═╡ 59bd7898-2dcf-4c23-815b-abf13a6954d0
function plot_error(a, b, offset, colormap)

	f(x, a, b) = a*x + b
	Ŷ = f.(X, a, b)

	offset_ϵ = offset
	x_ϵ = X + offset_ϵ
	y_ϵ = (Y + Ŷ) / 2
	texts_ϵ = ["ϵ₁", "ϵ₂", "ϵ₃", "ϵ₄"]
		
	for i in 1:length(colormap)
		plot!([(X[i], Ŷ[i]), (X[i], Y[i])]; c=colormap[i], ls=:dash, label="")
		annotate!([(x_ϵ[i],y_ϵ[i],text(texts_ϵ[i], 10, :center, colormap[i]))])
	end
	plot!()
end

# ╔═╡ e776c006-777f-4a6c-8a3e-a273820a0ad8
const colormap = [:red, :blue, :green, :orange]

# ╔═╡ 75364a95-c8ef-40d3-8373-6176540ceaf8
let
	# パラメータ a, b の計算
	x̄ = D[:,1] |> mean
	ȳ = D[:,2] |> mean

	Cov_XY = 1 / length(X) * sum( (X .- x̄) .* (Y .- ȳ) )
	Var_X =  1 / length(X) * sum( (X .- x̄).^2 )

	a = Cov_XY / Var_X
	b = ȳ - a* x̄

	# 可視化
	offset_y_new = [-1., 1.8, -1.2, 0.7]
	offset_ϵ_new = [0.3, -0.3, 0.3, -0.3]

	set_plot()
	plot_y(X, Y, colormap)
	plot_ŷ(a, b, offset_y_new, colormap)
	plot_error(a, b, offset_ϵ_new, colormap)
	
end

# ╔═╡ 792c62f0-222b-4048-ad6a-9655db74f080
md""" ### 1.6. 回帰直線の性質
"""

# ╔═╡ 0f1d5e27-f04c-47f4-bcff-bc9deea6eda2
function plot_means(X, Y)
	x̄ = D[:,1] |> mean
	ȳ = D[:,2] |> mean

	vline!([x̄]; c=:black, lw=1.5, ls=:dash, label="x̄")
	hline!([ȳ]; c=:black, lw=1.5, ls=:dot, label="ȳ")
	
end

# ╔═╡ 4c2841e2-2e5e-47ca-987a-0e4c5557f023
function plot_regions(X, Y)
	x̄ = D[:,1] |> mean
	ȳ = D[:,2] |> mean

	rectangle(x, y, w, h) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])
	plot!(rectangle(0, 0, x̄, ȳ), color=:red, alpha=0.1, label="")
	plot!(rectangle(x̄, ȳ, 10-x̄, 10-ȳ), color=:red, alpha=0.1, label="")
	
	plot!(rectangle(0, ȳ, x̄, ȳ), color=:blue, alpha=0.1, label="")
	plot!(rectangle(x̄, 0, 10-x̄, ȳ), color=:blue, alpha=0.1, label="")
	
end

# ╔═╡ dde67d9a-5c75-4477-bbd0-3eefc7b5ef13
let
	set_plot()
	plot_y(X, Y, colormap)
	plot_means(X, Y)
	plot_regions(X, Y)
end

# ╔═╡ c0a1bf99-d350-47ee-9a01-763acdcdb63a
function plot_regions2(X, Y, colormap)
	x̄ = D[:,1] |> mean
	ȳ = D[:,2] |> mean

	rectangle(x, y, w, h) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])

	xs = [D[1,1], D[2,1], x̄, x̄]
	ys = [D[1,2], D[2,2], D[3,2], ȳ]
	ws = [x̄-D[1,1], x̄-D[2,1], D[3,1]-x̄, D[4,1]-x̄]
	hs = [ȳ-D[1,2], ȳ-D[2,2], ȳ-D[3,2], D[4,2]-ȳ]

	for i in 1:length(xs)
		plot!(rectangle(xs[i], ys[i], ws[i], hs[i]), color=colormap[i],
			alpha=0.1, label="")
	end
	plot!()
end

# ╔═╡ 407a1c1b-bb2d-4188-997b-17565ef4431d
let
	set_plot()
	plot_y(X, Y, colormap)
	plot_means(X, Y)
	plot_regions2(X, Y, colormap)
end

# ╔═╡ cc5a8abc-ff3e-4693-94fe-79a73aaf2a19
md""" #### 1.6.3. 回帰直線は観測データの重心を通る
"""

# ╔═╡ 6da0bfa7-bd9e-4463-9057-bd40c56019eb
function plot_slope(x̄, ȳ, a, b)

	Cov_XY = 1 / length(X) * sum( (X .- x̄) .* (Y .- ȳ) )
	Var_X =  1 / length(X) * sum( (X .- x̄).^2 )

	a = Cov_XY / Var_X
	b = ȳ - a* x̄
	
	# horizontal
	plot!([0, x̄],[b, b]; lw=2, color=:brown, label="")
	annotate!([(x̄/2,b-0.3,text("x̄", 10, :center, color=:brown))])
	# vertical
	plot!([x̄, x̄], [b, ȳ]; lw=2, color=:brown, label="")
	annotate!([(x̄+0.3, b+(ȳ-b)/2,text("ax̄", 10, :center, color=:brown))])
end

# ╔═╡ f4215be2-c8af-4f7e-92cb-76b338714f1a
function plot_ŷ_only(a, b)

	a, b = a, b
	f(x, a, b) = a*x + b
	Ŷ = f.(X, a, b)
	x = [0, 10]

	a = round(a; digits=3)
	b = round(b; digits=3)
	
	label = (a == 1) ? "x + $(b))" : "$(a)x + $(b)" 
	plot!(x -> f(x, a, b); c=:black, lw=2, label=label)
	
end

# ╔═╡ c1635018-bff4-4245-91ff-278fda46f3a0
let

	# パラメータ a, b の計算
	x̄ = D[:,1] |> mean
	ȳ = D[:,2] |> mean

	Cov_XY = 1 / length(X) * sum( (X .- x̄) .* (Y .- ȳ) )
	Var_X =  1 / length(X) * sum( (X .- x̄).^2 )

	a = Cov_XY / Var_X
	b = ȳ - a* x̄

	# ----- 可視化 -----
	# 初期化
	set_plot()
	# データ点をプロット
	plot_y(X, Y, colormap)
	# xとyの平均の直線をプロット
	plot_means(X, Y)
	# +(赤)、 -(青)になる領域をプロット
	plot_regions(X, Y)
	# スロープ部分(茶色)をプロット
	plot_slope(x̄, ȳ, a, b)
	# 推定値 ŷ をプロット
	plot_ŷ_only(a, b)
end

# ╔═╡ fcd91d11-e8e0-4419-a992-47f82b39b393
md""" #### 1.8. 単回帰の実施例
"""

# ╔═╡ e2ae904b-c4aa-4f3c-ad18-58b0fabb6877
begin
	mmt = [
		9.1, 11.2, 12.3, 18.9, 22.2, 26. , 30.9, 31.2, 28.8, 23. , 18.3,
       11.1,  8.3,  9.1, 12.5, 18.5, 23.6, 24.8, 30.1, 33.1, 29.8, 23. ,
       16.3, 11.2,  9.6, 10.3, 16.4, 19.2, 24.1, 26.5, 31.4, 33.2, 28.8,
	   23. , 17.4, 12.1, 10.6,  9.8, 14.5, 19.6, 24.7, 26.9, 30.5, 31.2,
	   26.9, 23. , 17.4, 11. , 10.4, 10.4, 15.5, 19.3, 26.4, 26.4, 30.1,
	   30.5, 26.4, 22.7, 17.8, 13.4, 10.6, 12.2, 14.9, 20.3, 25.2, 26.3,
	   29.7, 31.6, 27.7, 22.6, 15.5, 13.8, 10.8, 12.1, 13.4, 19.9, 25.1,
	   26.4, 31.8, 30.4, 26.8, 20.1, 16.6, 11.1,  9.4, 10.1, 16.9, 22.1,
	   24.6, 26.6, 32.7, 32.5, 26.6, 23. , 17.7, 12.1, 10.3, 11.6, 15.4,
	   19. , 25.3, 25.8, 27.5, 32.8, 29.4, 23.3, 17.7, 12.6, 11.1, 13.3,
	   16. , 18.2, 24. , 27.5, 27.7, 34.1, 28.1, 21.4, 18.6, 12.3
	]
	
	spend = [
	  463.,  360.,  380.,  584.,  763.,  886., 1168., 1325.,  847.,
      542.,  441.,  499.,  363.,  327.,  414.,  545.,  726.,  847.,
     1122., 1355.,  916.,  571.,  377.,  465.,  377.,  362.,  518.,
      683.,  838., 1012., 1267., 1464., 1000.,  629.,  448.,  466.,
      404.,  343.,  493.,  575.,  921., 1019., 1149., 1303.,  805.,
      739.,  587.,  561.,  486.,  470.,  564.,  609.,  899.,  946.,
     1295., 1325.,  760.,  667.,  564.,  633.,  478.,  450.,  567.,
      611.,  947.,  962., 1309., 1307.,  930.,  668.,  496.,  650.,
      506.,  423.,  531.,  672.,  871.,  986., 1368., 1319.,  924.,
      716.,  651.,  708.,  609.,  535.,  717.,  890., 1054., 1077.,
     1425., 1378.,  900.,  725.,  554.,  542.,  561.,  459.,  604.,
      745., 1105.,  973., 1263., 1533., 1044.,  821.,  621.,  601.,
      549.,  572.,  711.,  819., 1141., 1350., 1285., 1643., 1133.,
      784.,  682.,  587.
	]
end;

# ╔═╡ 37558fb1-0ea2-4aa5-bdae-d1b54d237812
function plot_temp_vs_spend()
	plot(title="MMT vs. Spend for Icecream", framestyle=:box)
	plot!(xlabel="month mean maximum temperature [̇Degree]", ylabel="spend [Yen]")
	plot!(xlim=(0,35), ylim=(-250,2000), yticks=-250:250:2000, gridalpha=0.5)
	scatter!(mmt, spend; marker=:dot, ms=3, label="")
end

# ╔═╡ 569e9185-fb79-4a0b-b32d-b6e20dee29e8
let
	ŷ = fit(mmt, spend, 1)
	x = range(0, 35, length=100)
	plot_temp_vs_spend()
	plot!(x, ŷ.(x); lw=2, color=:red, label="")
end

# ╔═╡ e802792b-052f-4bd5-976a-2f5983fe7e06


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Polynomials = "f27b6e38-b328-58d1-80ce-0feddd5e7a45"
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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 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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