PCM20230424_ISLR2_2.1.1_What_Is_Statistical_Learning.jl

### A Pluto.jl notebook ###
# v0.19.25

using Markdown
using InteractiveUtils

# ╔═╡ 849dbc25-8dd1-416b-b706-672e826eec6a
using CSV, Plots, GLM, LsqFit, Statistics,StatsPlots, HypothesisTests

# ╔═╡ faef2569-8983-4704-9e1a-7080fb16dd06
md"
=====================================================================================
#### ISLR2_2.1.1 What Is Statistical Learning ?
##### file: PCM20230424\_ISLR2\_2.1.1\_What\_Is\_Statistical\_Learning.jl
##### Julia/Pluto (1.9.0/0.19.25) by PCM *** 2023/05/15 ***
=====================================================================================
"

# ╔═╡ abc89755-f470-4958-916a-fec048e28189
md"
*Statistical learning* to me means describing data using *adaptive, parsimonious* mathematical expressions, *inductively* determining the *generative* or *causal* mechanisms that produces s*sample* of data from well-defined *populations*. Of lesser importance to me are *deep learning* methods that analyze *large* data sets that define populations by themselves.
"

# ╔═╡ 622d27ff-e36f-4b69-949d-1fa78a8413ce
md"
---
#### 1. The *Advertising* Data Set
##### 1.1 Dataframe
"

# ╔═╡ 7b047630-b637-47ee-9aee-6d685dd3e773
advertisingDataFrame = CSV.File("C:/Users/claus/Nextcloud/PCM_Book/PCM2023_ISLR_Data/AdvertisingData/AdvertisingData.csv")

# ╔═╡ f698b886-e349-436f-932d-657bf7ff1eb5
md"
---
##### 1.2 ISLR2, Fig.2.2
###### 1.2.1 2D-Plot $Sales \times TV\text{-}Advertising$
"

# ╔═╡ de0c31cf-a954-41fb-afca-a03cf28364dc
let
	#-------------------------------------------------------------------------------
	plot(title="Sales X TV-Advertising (ISLR2e, Fig. 2.1 - left)", xlims=(-10,320), ylims=(0,32))
	plot!(advertisingDataFrame.TV, advertisingDataFrame.Sales, seriestype=:scatter, label="(TV, Sales)", xlabel="advertisingDataFrame.TV", ylabel="advertisingDataFrame.Sales")
	#-------------------------------------------------------------------------------
end # let

# ╔═╡ b852829d-0da2-44db-b7ed-9bf390858882
md"
---
###### 1.2.2 Linear Univariate OLS-Regression Model $f(X)=\beta_0 + \beta_1X_1$
where $Y=Sales$ and $X_1=TV\text{-}Advertising$
"

# ╔═╡ 009a4323-0593-4900-b0d9-e2c4fc7a8756
let
	ols_lin = lm(@formula(Sales ~ TV), advertisingDataFrame)
	yHat = predict(ols_lin)
	rxy = round(cor(advertisingDataFrame.TV, advertisingDataFrame.Sales), digits=3)
	ryyHat   = round(cor(advertisingDataFrame.Sales, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Sales -> TV-Advertising (ISLR2e, Fig. 2.1 - left)", xlims=(-10,320), ylims=(0,32))
	plot!(advertisingDataFrame.TV, advertisingDataFrame.Sales, seriestype=:scatter, label="(TV, Sales)", xlabel="advertisingDataFrame.TV", ylabel="advertisingDataFrame.Sales")
	plot!(advertisingDataFrame.TV, yHat, seriestype=:line, width=2, label="(TV, Sales^)")
	#--------------------------------------------------------------------------------
	annotate!(72, 22.5, "r(Sales, TV) = $rxy", 10)
	annotate!(62, 21, "r(Sales, Sales^) = $ryyHat", 10)
	annotate!(56, 19.5, "r^2(Sales, Sales^) = $ryyHatsQ", 10)
end # let

# ╔═╡ e08f52e5-6dfc-4efd-8363-cb0e02b98796
md"
---
###### 1.2.3 Linear Univariate OLS-Regression Model $f(X)=\beta_0 + \beta_1X_1$
where $Y=Sales$ and $X_1=Radio\text{-}Advertising$
"

# ╔═╡ 4d9fc2b5-3ee4-4acf-be66-d2881f9f994a
let
	ols_lin = lm(@formula(Sales ~ Radio), advertisingDataFrame)
	yHat = predict(ols_lin)
	rxy = round(cor(advertisingDataFrame.Sales, advertisingDataFrame.Radio), digits=3)
	rxyHat   = round(cor(advertisingDataFrame.Sales, yHat), digits=3)
	rxyHatsQ = round(rxyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Sales -> Radio-Advertising (ISLR2e, Fig. 2.1 - middle)", xlims=(-3,55), ylims=(0,32))
	plot!(advertisingDataFrame.Radio, advertisingDataFrame.Sales, seriestype=:scatter, label="(x,y)", xlabel="advertisingDataFrame.Radio", ylabel="advertisingDataFrame.Sales")
	plot!(advertisingDataFrame.Radio, yHat, seriestype=:line, width=2, label="(x, y^)")
	#--------------------------------------------------------------------------------
	annotate!(11, 22.5, "r(Sales, Radio) = $rxy", 10)
	annotate!(10.4, 21, "r(Sales, Sales^) = $rxyHat", 10)
	annotate!(09.3, 19.5, "r^2(Sales, Sales^) = $rxyHatsQ", 10)
end # let

# ╔═╡ fddf7996-2a6c-48db-8ebb-15e6807c216c
md"
---
###### 1.2.4 Linear Univariate OLS-Regression Model $f(X)=\beta_0 + \beta_1X_1$
where $Y=Sales$ and $X_1=Newspaper\text{-}Advertising$
"

# ╔═╡ da5a6876-ef9f-4d73-bd3b-1369c70bc103
let
	ols_lin = lm(@formula(Sales ~ Newspaper), advertisingDataFrame)
	yHat = predict(ols_lin)
	rxy = round(cor(advertisingDataFrame.Sales, advertisingDataFrame.Newspaper), digits=3)
	ryyHat   = round(cor(yHat, advertisingDataFrame.Sales), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Sales -> Newsp-Advertising (ISLR2e, Fig. 2.1 - right)", xlims=(-5,125), ylims=(-2,30))
	plot!(advertisingDataFrame.Newspaper, advertisingDataFrame.Sales, seriestype=:scatter, label="(x, y)", xlabel="advertisingDataFrame.Newspaper", ylabel="advertisingDataFrame.Sales")
	plot!(advertisingDataFrame.Newspaper, yHat, seriestype=:line, width=2, label="(x, y^)")
	#--------------------------------------------------------------------------------
	annotate!(103.2, 29, "r(Sales, Newsp) = $rxy", 10)
	annotate!(102.8, 27.5, "r(Sales, Sales^) = $ryyHat", 10)
	annotate!(100.4, 26, "r^2(Sales, Sales^) = $ryyHatsQ", 10)
end # let

# ╔═╡ 24576004-b1d2-43e1-a861-eba45d1fda0e
md"
---
#### 2. The *Income1* Data Set

Both *Income* data sets are generated from numerical simulations. The *true* model function $f(Education, Seniority)$ is depicted in Figure 2.3 (ISLR2, 2021, p.18) as a *blue* surface. Its mathematical form is *not* published by the authors. So we have to estimate $f$ by various approaches.
"

# ╔═╡ 1feee341-401b-4afd-8970-7c2804fc3f89
md"
---
##### 2.1 Dataframe
"

# ╔═╡ 8f6de19f-c4ff-46dd-a120-269ebf647578
income1DataFrame = CSV.File("C:/Users/claus/Nextcloud/PCM_Book/PCM2023_ISLR_Data/IncomeData/Income1Data.csv")

# ╔═╡ d800b01d-477b-4776-bd2c-848fff8a5da9
md"
---
##### 2.2 ISLR2, Fig.2.2
###### 2.2.1 2D-Plot $Income \times Education$
"

# ╔═╡ 7ad28ec0-74ad-4f15-8235-259f091622a6
let
	xdata = income1DataFrame.Education
	ydata = income1DataFrame.Income
	plot(title="Income X Education (ISLR2e, Fig. 2.2 - left)", xlims=(9,23), ylims=(5,95))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income1DataFrame.Education", ylabel="income1DataFrame.Income")
end # let

# ╔═╡ 55a0b4d7-890f-4d32-a10f-002631906bda
md"
---
###### 2.2.2 Linear Univariate OLS-Regression Model $f(X)=\beta_0 + \beta_1X_1$
where: $Y=Income$, $X_1=Education$, and #parms$=2$
"

# ╔═╡ bfcdfaf5-50c1-452a-946b-941e5c4c1041
let
	xdata = income1DataFrame.Education
	ydata = income1DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin = lm(@formula(Income ~ 1 + Education), income1DataFrame)
	yHat = predict(ols_lin)
	e = residuals(ols_lin)
	sse = round(e'e, digits=2)
	rxy    = round(cor(ydata, xdata), digits=3)
	ryyHat = round(cor(ydata, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Income -> Education (ISLR2e, Fig. 2.2 - left)", xlims=(9,23), ylims=(5,95))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income1DataFrame.Education", ylabel="income1DataFrame.Income")
	plot!(xdata, yHat, seriestype=:line, width=2, label="(x, y^)")
	plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#--------------------------------------------------------------------------------
	annotate!((21.6,28,"r(y, x) = $rxy", 10))
	annotate!((21.4,22,"r(y, y^) = $ryyHat", 10))
	annotate!((21.15,16,"r^2(y, y^) = $ryyHatsQ", 10))
	annotate!((19.2,10,"SSE = $sse (= error sum of squares)", 10))
end # let

# ╔═╡ 5479bd68-aafa-47c6-bce5-d5a8817c131a
md"
---
###### 2.2.3 Nonlinear Univariate OLS-Regression with the [Generalized Logistic Function](https://en.wikipedia.org/wiki/Generalised_logistic_function)

$f(x) = L + (U-L) \cdot \frac{1}{1+e^{-k\left(x - x_{k_{max}}\right)}}$

$\;$

where:
- #parms $= 4$
- **$U$** = Upper (right) asymptote of $X$
- **$L$** = Lower (left) asymptote of $X$
- **$x_{k_{max}}$** = $\underset{X}{\operatorname{argmax}\;k} \;\; \text{(= }x\text{ is the inflection point of } k)$ 
- **$k_{max}$** = $max(k)$ = max. growth rate *k* of $X$ in the simple *standard* logistic function

The above function $f(X)$ is a solution of *differential equation* describing a [*(logistic) growth process*](https://en.wikipedia.org/wiki/Logistic_function). We derive the differential equation for the simple logistic function $(0 < f_{simple}(X) < 1)$ in the appendix:

$f_{simple}(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}.$ 

"

# ╔═╡ 92ea0bb5-d318-4758-83c9-ac5960918e12
let
	#--------------------------------------------------------------------------------
	xdata = income1DataFrame.Education
	ydata = income1DataFrame.Income
	#--------------------------------------------------------------------------------
	myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
	parms0 = [18.0, 78.0, 0.5, 16.0]                    # initial estimates of parms
	myModel(xdata, parms0)
	fit = curve_fit(myModel, xdata, ydata, parms0)
	e = fit.resid
	sse = round(e'e, digits=2)
	parms_hat = coef(fit)
	L  = round(parms_hat[1], digits=2)                    # greatest Lower bound (gLb)
	U  = round(parms_hat[2], digits=2)                    # least Upper bound (lUb)
	k  = round(parms_hat[3], digits=2)                    # max. growth rate k
	x_k_max = round(parms_hat[4], digits=2)               # point x of k_max
	yHat = myModel(xdata, parms_hat)
	rxy    = round(cor(ydata, xdata), digits=3)
	ryyHat = round(cor(ydata, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	Plots.plot(title="Income -> Education (ISLR2e, Fig. 2.2 - right)", xlims=(9,23), ylims=(5,95))
	Plots.plot!(xdata, ydata, seriestype=:scatter, xlabel="income1DataFrame.Education", ylabel="income1DataFrame.Income", label="(x, y)")
	Plots.plot!(xdata, yHat, seriestype=:line, label="(x, y^)", w=2)
	Plots.plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#--------------------------------------------------------------------------------
	annotate!((21.7,52.2,"r(y, x) = $rxy", 10))
	annotate!((21.7,46,"r(y,y^) = $ryyHat ", 10))
	annotate!((21.3,40,"r^2(y,y^) = $ryyHatsQ", 10))
	annotate!((21.4,34,"x_k_max = $x_k_max", 10))
	annotate!((19.4,28,"U=$U (= Upper (right) asymptote)", 10))
	annotate!((19.6,22,"L=$L (= Lower (left) asymptote)", 10))
	annotate!((19.2,16,"SSE = $sse ( = error sum of squares)", 10))
	annotate!((17.45,10,"k=$k (= max. growth rate of standard logistic function)", 10))
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ e5a0b52d-f482-43fa-9efc-84ecd0711794
md"""

The derivative $f'(x)$ is the description of the logistic growth process and is called [*logistic differential equation*](https://en.wikipedia.org/wiki/Logistic_function) with boundary condition $f(0)= \frac{1}{2}$.


This can be simplified to the product of two terms:


$$f'(x) = \frac{e^x}{(e^x+1)^2} = \frac{e^x}{(e^x+1)(e^x+1)} = \frac{e^x}{(e^x+1)} \cdot \frac{1}{(e^x+1)} = f(x)(1-f(x)).$$


"""

# ╔═╡ 5adf8b34-2d41-4b15-8c13-0cbf84ed0cce
md"
---
#### 3. The *Income2* Data Set
##### 3.1 Dataframe

"

# ╔═╡ ec30ce62-3699-40f8-88c3-8e394d6d826e
income2DataFrame = CSV.File("C:/Users/claus/Nextcloud/PCM_Book/PCM2023_ISLR_Data/IncomeData/Income2Data.csv")

# ╔═╡ c54e5513-f30a-4bc1-afe0-102ba43a6f87
md"
---
##### 3.2 ISLR2, Figs. 2.3 - 2.6 $Income \rightarrow Education$
###### 3.2.1 2D-Plot $Income \times Education$
" 

# ╔═╡ a559d47a-bdf9-4229-b363-d33ff4994de6
let
	xdata = income2DataFrame.Education
	ydata = income2DataFrame.Income
	plot(title="Income X Education (ISLR2e, cf. Fig. 2.3)", xlims=(9,23), ylims=(5,120))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income2DataFrame.Education", ylabel="income2DataFrame.Income")
end # let

# ╔═╡ a0e54499-eb79-4381-81eb-acfd8826511e
md"
---
###### 3.2.2 Linear Univariate Regression Model $f(X)=\beta_0 + \beta_1X_1$
where $Y=Income$, $X_1=Education$, and #parms$=2$
"

# ╔═╡ 7be41329-e7b8-4647-8eb3-a791f8669f50
let
	xdata = income2DataFrame.Education
	ydata  = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_linX = lm(@formula(Income ~ 1 + Education), income2DataFrame)
	yHat   = predict(ols_linX)
	eX    = residuals(ols_linX)
	sseX  = round(eX'eX, digits=2)
	rxy    = round(cor(ydata, xdata), digits=3)
	rxyHat = round(cor(ydata, yHat), digits=3)
	rxyHatsQ = round(rxyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Linear OLS: Income -> Education (ISLR2e, cf. Fig.2.3)", xlims=(9,23), ylims=(5,120))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income2DataFrame.Education", ylabel="income2DataFrame.Income")
	plot!(xdata, yHat, seriestype=:line, width=2, label="(x, y^)")
	plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#--------------------------------------------------------------------------------
	annotate!((21,28,"r(y, x) = $rxy", 10))
	annotate!((20.9,22,"r(x,y^) = $rxyHat", 10))
	annotate!((20.6,16,"r^2(x,y^) = $rxyHatsQ", 10))
	annotate!((18.4,10,"SSE = $sseX ( = error sum of squares)", 10))
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ d1982304-7008-4486-a16d-5d4064935ebc
md"
---
###### 3.2.3 Nonlinear Univariate OLS-Regressions using the [Generalized Logistic Function](https://en.wikipedia.org/wiki/Generalised_logistic_function)

$f(x) = L + (U-L) \cdot \frac{1}{1+e^{-k\left(x - x_{k_{max}}\right)}}$

$\;$

where:
- #parms $= 4$
- **$U$** = Upper (right) asymptote of $X$
- **$L$** = Lower (left) asymptote of $X$
- **$x_{k_{max}}$** = $\underset{X}{\operatorname{argmax}\;k} \;\; \text{(= }x\text{ is the inflection point of rate } k)$ 
- **$k_{max}$** = $max(k)$ = max. growth rate *k* of $X$ in the simple *standard* logistic function

$\;$

The above function $f(X)$ is a solution of *differential equation* describing a [*(logistic) growth process*](https://en.wikipedia.org/wiki/Logistic_function). We derive the differential equation for the *simple* logistic function $(0 < f_{simple}(x) < 1)$ in the appendix:

$f_{simple}(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}.$ 

$\;$

$\;$


"

# ╔═╡ 8ad1be9a-ffdd-4c43-b6ff-edd49550b6d2
let
	xdata = income2DataFrame.Education
	ydata = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
	parms0 = [28.0, 88.0, 0.9, 16.0]                      # initial estimates of parms
	myModel(xdata, parms0)
	fit = curve_fit(myModel, xdata, ydata, parms0)
	parmsHat = coef(fit)
	yHat  = myModel(xdata, parmsHat)
	e   = fit.resid
	sse = round(e'e, digits=2)
	rxy    = round(cor(ydata, xdata), digits=3)
	ryyHat = round(cor(ydata, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	L  = round(parmsHat[1], digits=2)                    # greatest Lower bound (gLb)
	U  = round(parmsHat[2], digits=2)                    # least Upper bound (lUb)
	k  = round(parmsHat[3], digits=2)                    # max. growth rate k
	x_k_max = round(parmsHat[4], digits=2)               # point x of k_max
	#--------------------------------------------------------------------------------
	plot(title="Logistic OLS: Income -> Education (ISLR2e, cf.Fig.2.3)", xlims=(9,22), ylims=(5,120))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income2DataFrame.Education", ylabel="income2DataFrame.Income")
	plot!(xdata, yHat, seriestype=:line, label="(x, yHat)", w=2)
	plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#--------------------------------------------------------------------------------
	annotate!((20.85,58,"r(y, x) = $rxy", 10))
	annotate!((20.9,52,"r(y,y^) = $ryyHat ", 10))
	annotate!((20.45,46,"r^2(y,y^) = $ryyHatsQ", 10))
	annotate!((20.55,40,"x_k_max = $x_k_max", 10))
	annotate!((19.5,34,"k=$k (= max. growth rate)", 10))
	annotate!((18.7,28,"U=$U (= Upper (right) asymptote)", 10))
	annotate!((19.0,22,"L=$L (= Lower (left) asymptote)", 10))
	annotate!((18.4,16,"SSE = $sse ( = error sum of squares)", 10))
	annotate!((16.9,10,"k=$k (= max. growth rate of standard logistic function)", 10))
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ df58736c-ebd1-4020-915c-b32291985f3f
md"
---
##### 3.3 ISLR2, Fig.2.3 $Income \rightarrow Seniority$
###### 3.3.1 2D-Plot $Income \times Seniority$
"

# ╔═╡ 8e78b3e0-e97e-4b8d-9342-a180b26bbf03
let
	xdata = income2DataFrame.Seniority
	ydata = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	plot(title="Income X Seniority (ISLR2e, cf. Fig. 2.3)", xlims=(9,200), ylims=(5,120))
	plot!(xdata, ydata, seriestype=:scatter, label="(y,z)", 	
	xlabel="income2DataFrame.Seniority", ylabel="income2DataFrame.Income")
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 2771a9f1-353f-4bfa-b4d9-a0deda6e031a
md"
---
###### 3.3.2 Linear Univariate OLS-Regression Model $f(X)=\beta_0 + \beta_2X_2$
where: $Y=Income$, $X_2=Seniority$, and #parms$= 2$
"

# ╔═╡ 84c06bd1-4dbf-411f-96c4-97554146ca21
let
	xdata = income2DataFrame.Seniority
	ydata = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin = lm(@formula(Income ~ 1 + Seniority), income2DataFrame)
	yHat = predict(ols_lin)
	e    = residuals(ols_lin)
	sse  = round(e'e, digits=2)
	rxy = round(cor(ydata, xdata), digits=3)
	ryyHat = round(cor(ydata, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Linear OLS: Income -> Seniority (ISLR2e, cf. Fig.2.3)", xlims=(9,200), ylims=(5,120))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income2DataFrame.Seniority", ylabel="income2DataFrame.Income")
	plot!(xdata, yHat, seriestype=:line, width=2, label="(x, y^)")
	plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#--------------------------------------------------------------------------------
	annotate!((178,28,"r(y,x) = $rxy", 10))
	annotate!((175.8,22,"r(y,y^) = $ryyHat", 10))
	annotate!((172,16,"r^2(y,y^) = $ryyHatsQ", 10))
	annotate!((140,10,"SSE = $sse ( = error sum of squares)", 10))
end # let

# ╔═╡ baa67da7-d394-4a1d-9eee-3a982c0f74c2
md"
---
###### 3.3.3 Nonlinear Univariate Regression using the [Generalized Logistic Function](https://en.wikipedia.org/wiki/Generalised_logistic_function)

$f(x) = L + \frac{U-L}{1+e^{-k\left(x - x_{k_{max}}\right)}}$

where:
- #parms $= 4$
- **$U$** = Upper (right) asymptote of $X$
- **$L$** = Lower (left) asymptote of $X$
- **$x_{k_{max}}$** = $\underset{X}{\operatorname{argmax}\;k} \;\; \text{(= }x\text{ is the inflection point of rate } k)$ 
- **$k_{max}$** = $max(k)$ = max. growth rate *k* of $X$ in the simple *standard* logistic function

$\;$

The above function $f(x)$ is a solution of *differential equation* describing a [*(logistic) growth process*](https://en.wikipedia.org/wiki/Logistic_function). We derive the differential equation for the *simple* logistic function $f_{simple}(x)$ in the appendix:

$f_{simple}(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}.$ 

$\;$

$\;$
"

# ╔═╡ f3025a7a-0256-46c7-b296-4cc3a36c5372
let
	xdata = income2DataFrame.Seniority
	ydata = income2DataFrame.Income
	#------------------------------------------------------------------------------
	myModel(x, p) = p[1] .+ (p[2] .-p[1]) ./((1 .+ exp.(-p[3] .*(x .- p[4]))))
	parms0 = [40.0, 78.0, 4.0, 70.0]                      # initial estimates of parms
	myModel(xdata, parms0)
	fit = curve_fit(myModel, xdata, ydata, parms0)
	e = fit.resid
	sse = round(e'e, digits=2)
	parms_hat = coef(fit)
	yHat = myModel(xdata, parms_hat)
	rxy = round(cor(xdata, ydata), digits=3)
	ryyHat = round(cor(ydata, yHat), digits=3)
	ryyHatsQ = round(ryyHat^2, digits=3)
	#--------------------------------------------------------------------------------
	L  = round(parms_hat[1], digits=2)                  # greatest Lower bound (gLb)
	U  = round(parms_hat[2], digits=2)                  # least Upper bound (lUb)
	k  = round(parms_hat[3], digits=2)                  # max. growth rate k
	y_k_max = round(parms_hat[4], digits=2)             # point x of k_max
	#------------------------------------------------------------------------------
	plot(title="Logistic OLS: Income -> Seniority (ISLR2e, cf. Fig.2.3)", xlims=(9,200), ylims=(5,125))
	plot!(xdata, ydata, seriestype=:scatter, label="(x, y)", xlabel="income2DataFrame.Seniority", ylabel="income2DataFrame.Income")
	plot!(xdata, yHat, seriestype=:line, label="(x, y^)", w=2)
	plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
	#------------------------------------------------------------------------------
	annotate!((40.7,102,"r(y, x) = $rxy", 10))
	annotate!((38.5, 96,"r(y, y^) = $ryyHat", 10))
	annotate!((36.14,90,"r^(y, y^) = $ryyHatsQ", 10))
	annotate!((155.0,30,"Ly=$L (= Lower (left) asymptote)", 10))
	annotate!((152.4,24,"Uy=$U (= Upper (right) asymptote)", 10))
	annotate!((147.9,18,"sseY=$sse ( = error sum of squares)", 10))
	annotate!((127.2,12,"ky=$k (= max. growth rate of standard logistic function)", 10))
	#------------------------------------------------------------------------------
end # let

# ╔═╡ e31097bf-1cbf-4146-91b3-841b91aa11b3
md"
---
##### 3.4 ISLR2, Fig.2.3 $Education \times Seniority$
###### 3.4.1 2D-Plot $Education \times Seniority$
"

# ╔═╡ f98e88e4-6abf-41c6-86c4-60f61523e1bb
let
	#-------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	rxy = round(cor(xs, ys), digits=3)
	#-------------------------------------------------------------------------------
	plot(title="Income2 Data: Split at Inflection Points of Rate k", xlabel="Education", ylabel="Seniority")
	plot!(xs, ys, seriestype=:scatter, label="(x,y)")
	plot!([(16.0, 0),  (16, 220)], legend=:none)     # vertical line at x_k_max = 16
	plot!([( 9.5, 70), (22,  70)], legend=:none)     # horizontal line at y_k_max = 70
	#-------------------------------------------------------------------------------
	annotate!(20.7,  60, "y_k_max=70", 10)
	annotate!(14.7, 210, "x_k_max=16", 10)
	annotate!(12.4, 165, "r(Education, Seniority) = $rxy", 10)
	#-------------------------------------------------------------------------------
end # let

# ╔═╡ 1e5f39c1-b6e0-4088-b0dd-5e6162602bed
md"
---
###### 3.4.2 Group-based Frequencies and Means
"

# ╔═╡ 4da9634c-41ab-469f-a25b-dc618e65eacb
let
	xs  = income2DataFrame.Education
	ys  = income2DataFrame.Seniority 
	zs  = income2DataFrame.Income
	#-------------------------------------------------------------------
	lx_ly =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x <= 16.0 && y <= 70.0 ? z : 0, xs, ys, zs)))
	mean_lx_ly   = mean(lx_ly)
	length_lx_ly = length(lx_ly)
	#---------------------------------------------------------
	lx_hy =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x <= 16.0 && y > 70.0 ? z : 0, xs, ys, zs)))
	mean_lx_hy   = mean(lx_hy)
	length_lx_hy = length(lx_hy)
	#---------------------------------------------------------
	hx_ly =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x > 16.0 && y <= 70.0 ? z : 0, xs, ys, zs)))
	mean_hx_ly   = mean(hx_ly)
	length_hx_ly = length(hx_ly)
	#---------------------------------------------------------
	hx_hy =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x > 16.0 && y > 70.0 ? z : 0, xs, ys, zs)))
	mean_hx_hy   = mean(hx_hy)
	length_hx_hy = length(hx_hy)
	#-------------------------------------------------------------------
	plot(title="Income2 Data: Split at Inflection Points of Rate k", seriestype=:scatter, legend=:none, xlabel="Education", ylabel="Seniority")
	plot!([(16.0, 0),  (16, 220)])                # vertical line at x_k_max = 16
	plot!([( 9.5, 70), (22,  70)])                # horizontal line at y_k_max = 70
	#-------------------------------------------------------------------------------
	annotate!(20.7,  60, "y_k_max=70", 10)
	annotate!(14.7, 210, "x_k_max=16", 10)
	#-------------------------------------------------------------------------------
	annotate!(11.5,  40, "N=7", 10)
	annotate!(11.5,  25, "mean(Income)=24.1", 10)
	annotate!(11.5,  10, "group=lxly", 10)
	#----------------------------------------
	annotate!(11.5, 140, "N=4", 10)
	annotate!(11.5, 125, "mean(Income)=43.8", 10)
	annotate!(11.5, 110, "group=lxhy", 10)
	#----------------------------------------
	annotate!(19.0,  40, "N=4", 10)
	annotate!(19.0,  25, "mean(Income)=69.0", 10)
	annotate!(19.0,  10, "group=hxly", 10)
	#----------------------------------------
	annotate!(19.0, 140, "N=15", 10)
	annotate!(19.0, 125, "mean(Income)=84.1", 10)
	annotate!(19.0, 110, "group=hxhy", 10)
	#-------------------------------------------------------------------
	# (N_lx_ly=length_lx_ly, N_lx_hy=length_lx_hy, N_hx_ly=length_hx_ly, N_hx_hy=length_hx_hy, mean_lx_ly=mean_lx_ly, mean_lx_hy=mean_lx_hy, mean_hx_ly=mean_hx_ly, mean_hx_hy=mean_hx_hy)
	#-------------------------------------------------------------------------------
end # let

# ╔═╡ f76d77bf-eb40-4f9b-8fca-4b12994d8aef
md"
---
##### 3.5 ISLR2, Fig.2.3 $Income \rightarrow (Education \times Seniority)$
###### 3.5.1 2D-Group-Histogram
"

# ╔═╡ cedc8274-24c4-4fab-bae8-70c5f9276e49
let
	#-------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#-------------------------------------------------------------------------------
	lx_ly =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x <= 16.0 && y <= 70.0 ? (z, :lxly) : 0, xs, ys, zs)))
	#---------------------------------------------------------
	lx_hy =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x <= 16.0 && y > 70.0 ? (z, :lxhy) : 0, xs, ys, zs)))
	#---------------------------------------------------------
	hx_ly =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x > 16.0 && y <= 70.0 ? (z, :hxly) : 0, xs, ys, zs)))
	#---------------------------------------------------------
	hx_hy =
		filter(z -> z !== 0, 
			(map((x, y, z) -> 
				x > 16.0 && y > 70.0 ? (z, :hxhy) : 0, xs, ys, zs)))
	#-------------------------------------------------------------------------------
	gs  = cat(lx_ly, lx_hy, hx_ly, hx_hy; dims=1)
    gzs = [gsi[1] for gsi in gs] 
	ggs = [gsi[2] for gsi in gs]
	groupedhist(gzs, group = ggs, title="Income2 Data: Split at Inflection Points of Rate k", xlabel="Income", bins=10)
	#-------------------------------------------------------------------------------
end # let

# ╔═╡ 467c6c7c-3acc-4919-99d9-4f197a067174
md"""
---
###### 3.5.2 Frequency Table and Subgroup Means
The results of the 'Test for nonzero correlation' shows that we can stick to the *Nullhypothesis* $H_0: \rho=0$. This means that we can assume that the *population* correlation coeffizient $\rho = 0$. So we can assume that the *variables* $Education$ and $Seniority$ are *stochastic linear independent*.
"""

# ╔═╡ b6dedd81-c882-469d-8bf7-c6abbd8ce724
let
	xs  = income2DataFrame.Education
	ys  = income2DataFrame.Seniority 
	zs  = income2DataFrame.Income
	rXY   = cor(xs, ys)
	CorrelationTest(xs, ys)
end # let

# ╔═╡ 4ea401b0-4604-4ab1-a5c4-cbdf4dc3195e
md"
---
###### 3.5.3 Model-free Data Analysis and Modelling Consequences

From cross-tables and histograms we can see that the rank order in $Income$ is from groups $lxly$ over $lxhy$ and $hxly$ to $hxhy$. So we see that a low value in $Education$ *cannot* be compensated by $Seniority$ though a higher $Education$ or $Seniority$ *alone* are favorable influences on $Income$ (at least in this *simulated* data set).

So it is a good idea to model the univariate influences by a *monotonic ascending* function. We used linear and nonlinear (*sigmoid* or *logistic*) functions. In case of the logistic functions we have the advantage that these are solutions of simple *growth processes* formalizable as linear differential equations.

Despite the fact that the frequencies of the two extreme groups $lxly$ and $hxhy$ are higher than in the two other groups the variables $Education$ and $Seniority$ are correlationally *independent*. The corresponding *product-moment correlation* coefficient is $r = 0.19$ (as can be seen above). This gives us the opportunity to construct a nonlinear 3D-regression surface for the multivariate regression model $Income \rightarrow Education \times Seniority$ by *multiplying* the marginal regression functions of the two marginal nonlinear regression models $Income \rightarrow Education$ and $Income \rightarrow Seniority$.

"

# ╔═╡ 2db54260-2d77-40a1-834f-d6d0e4164e44
md"
---
##### 3.6 ISLR2, Fig.2.3 $Income \rightarrow Education \times Seniority$
###### 3.6.1 3D Plot $Income \times Education \times Seniority$
"

# ╔═╡ 8beec881-7371-40dc-948d-deb61ad2d99f
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	plot(title="Income X Education X Seniority (ISLR2e, Fig. 2.3 - (new))", xlims=(10,20), ylims=(25,180), zlims=(0,100), windowsize=(600*2, 400*2))
	plot!(xs, ys, zs, seriestype=:scatter, label="(x,y,z)", xlabel = "Education", ylabel="Seniority", zlabel="Income", markersize=8, grid=true, gridstyle=:solid, gridlinewidth=5, camera=(10,10))              # camera(horizontal, vertical)
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 01cc3ff4-e7a8-4610-b990-088eedbfcd9a
md"
---
###### 3.6.2 Linear Multiple Regression $f(X)=\beta_0 + \beta_1X_1 + \beta_2 X_2 + \beta_3(X_1 \cdot X_2)$
where $Y=Income, X_1=Education$, and $X_2=Seniority$ 

This model has $4$ parameters $\beta_0, \beta_1, \beta_2, \beta_3$.

"

# ╔═╡ 347ae10c-0991-4cd7-b0d2-d87de4b43f17
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin = lm(@formula(Income ~ 1 + Education * Seniority), income2DataFrame)
	zhat = predict(ols_lin)
	multipleR   = round(cor(zs, zhat), digits=3)
	multipleRsQ = round(multipleR^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Linear Multiple Regression: Income -> f(Education, Seniority, Education * Seniority) (ISLR2e, cf.Fig.2.3)", xlims=(10,20), ylims=(25,180), zlims=(0,100), windowsize=(600*2, 400*2))
	plot!(map((x, y, z, zh) -> ([x, x], [y, y], [z, zh]), xs, ys, zs, zhat), label="", linewidth=3)
	plot!(xs, ys, zs, seriestype=:scatter, label="(x,y,z)", xlabel = "Education", ylabel="Seniority", zlabel="Income", markersize=8, grid=true, gridstyle=:solid, gridlinewidth=5, camera=(10,10))
	plot!(xs, ys, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
	#--------------------------------------------------------------------------------
	annotate!(22,160,40, "multiple R = $multipleR")
	annotate!(21.6,160,34, "multiple R^2 = $multipleRsQ")
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 55f36e93-0c4e-4eaf-962e-e6ff7cf3e321
md"
---
###### 3.6.3 Linear Multiple OLS-Regression $f(X)=\beta_0 + \beta_1X_1 + \beta_2X_2$
where $Y=Income, X_1=Education$, and $X_2=Seniority$ 

This model has $3$ parameters $\beta_0, \beta_1, \beta_2$.
"

# ╔═╡ fcbbb2c6-55ca-4302-bd7c-638ba47ac9de
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin = lm(@formula(Income ~ 1 + Education + Seniority), income2DataFrame)
	zhat = predict(ols_lin)
	multipleR   = round(cor(zs, zhat), digits=3)
	multipleRsQ = round(multipleR^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Linear Regression: Income -> Education + Seniority (ISLR2e, cf. Fig. 2.3)", xlims=(10,20), ylims=(25,180), zlims=(0,100), windowsize=(600*2, 400*2))
	plot!(map((x, y, z, zh) -> ([x, x], [y, y], [z, zh]), xs, ys, zs, zhat), label="", linewidth=3)
	plot!(xs, ys, zs, seriestype=:scatter, label="(x,y,z)", xlabel = "Education", ylabel="Seniority", zlabel="Income", markersize=8, grid=true, gridstyle=:solid, gridlinewidth=5, camera=(10,10))          # camera(horizontal, vertical)
	plot!(xs, ys, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
	#--------------------------------------------------------------------------------
	annotate!(12,160,90, "multiple R = $multipleR")
	annotate!(11.6,160,84, "multiple R^2 = $multipleRsQ")
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ cf103b63-ee7e-4970-9f85-b6b9ae72e025
md"
---
###### 3.6.4 Linear Univariate Regression $f(X)=\beta_0 + \beta_1(X_1 \cdot X_2)$
where $Y=Income, X_1=Education$, and $X_2=Seniority$ 

This model has $2$ parameters $\beta_0, \beta_1$.
"

# ╔═╡ b276aaa9-ee61-444c-a4fa-c09f62f792b8
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin = lm(@formula(Income ~ 1 + Education & Seniority), income2DataFrame)
	zhat = predict(ols_lin)
	multipleR   = round(cor(zs, zhat), digits=3)
	multipleRsQ = round(multipleR^2, digits=3)
	#--------------------------------------------------------------------------------
	plot(title="Linear Regression: Income -> (Education & Seniority) (ISLR2e, cf. Fig. 2.3)", xlims=(10,20), ylims=(25,180), zlims=(0,100), windowsize=(600*2, 400*2))
	plot!(map((x, y, z, zh) -> ([x, x], [y, y], [z, zh]), xs, ys, zs, zhat), label="", linewidth=3)
	plot!(xs, ys, zs, seriestype=:scatter, label="(x,y,z)", xlabel = "Education", ylabel="Seniority", zlabel="Income", markersize=8, grid=true, gridstyle=:solid, gridlinewidth=5, camera=(10,10))          # camera(horizontal, vertical)
	plot!(xs, ys, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
	#--------------------------------------------------------------------------------
	annotate!(22,160,40, "multiple R = $multipleR")
	annotate!(21.6,160,34, "multiple R^2 = $multipleRsQ")
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ c94b50d3-2e18-4f61-921c-ad8d34b28185
md"
---
###### 3.6.5 Comparison of Regression Models 3.6.2-3.6.4 and Interpretation

The *full model* is the the 4-parameter model with interaction term. The other two models are *nested* special cases obtained by *restrictions* on the parameters of the full model. There is a statistical test (*F-test*) to evaluate the increase of the SSE (= sum of error squares) going from the *full* to the *restricted* models. 

"

# ╔═╡ b0ea196b-fd17-46b1-a480-9eb0c8dc605b
md"
---
###### 3.6.5.1 Full Model 3.6.2 vs Restricted Model 3.6.3: Significance of Interaction ?

The result of the F-Test full model 3.6.2 versus restricted model 3.6.3 demonstrates that we *may* drop the *interaction* term without loss of accounted variance.

"

# ╔═╡ 2e43b039-725d-4e31-b169-a7029d96b19e
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin_3_6_2 = lm(@formula(Income ~ 1 + Education * Seniority), income2DataFrame)
	ols_lin_3_6_3 = lm(@formula(Income ~ 1 + Education + Seniority), income2DataFrame)                        # no interaction term
	#--------------------------------------------------------------------------------
    ftest(ols_lin_3_6_2.model, ols_lin_3_6_3.model)
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 7260981e-2c46-4691-a07a-1aa8cc27d3dd
md"
---
###### 3.6.5.2 Full Model 3.6.2 vs Restricted Model 3.6.4: Significance of Main Effects ?

The result of the F-Test full model 3.6.2 versus restricted model 3.6.4 demonstrates that we may *not* drop the *main effects* without loss of accounted variance.

"

# ╔═╡ bd1fe6b8-262b-4202-89c9-4476a672ea34
let
	#--------------------------------------------------------------------------------
	xs = income2DataFrame.Education
	ys = income2DataFrame.Seniority
	zs = income2DataFrame.Income
	#--------------------------------------------------------------------------------
	ols_lin_3_6_2 = lm(@formula(Income ~ 1 + Education * Seniority), income2DataFrame)
	ols_lin_3_6_4 = lm(@formula(Income ~ 1 + Education & Seniority), income2DataFrame)                        # no interaction term
	#--------------------------------------------------------------------------------
    ftest(ols_lin_3_6_2.model, ols_lin_3_6_4.model)
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 3a4ce2bc-6387-4581-acd2-517b834d8301
md"
---
##### 3.7 Summary
The highest accounted variance with a *linear* parameter-parsimoneous model is $93.4\% = R^2*100$. This achieved with the $3$ parameter model 3.6.3. This has *no* interaction term.
"

# ╔═╡ e13cda08-84ec-48c3-ade8-fd61a3c5ac17
md"
---
##### References
- **James, G., Witten, D., Hastie, T. & Tibshirani, R.**; *An Introduction to Statistical Learning, Heidelberg: Springer, 2021, 2/e, 
- **Wikipedia**; *Logistic Function*; [https://en.wikipedia.org/wiki/Logistic_function](https://en.wikipedia.org/wiki/Logistic_function); last visit 2023/05/07
- **Wikipedia**; *The Generalized Logistic Function*, [https://en.wikipedia.org/wiki/Generalised_logistic_function](https://en.wikipedia.org/wiki/Generalised_logistic_function); last visit 2023/05/04
"

# ╔═╡ 75c6cefe-7be7-4086-a5c6-0e8312b8b6ee
md"
---
##### end of ch. 2.1.1
"

# ╔═╡ 6566d57d-d090-4bed-87e8-be8b516e86e1
md"
====================================================================================

This is a **draft** under the Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)** license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

====================================================================================
"

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PLUTO_PROJECT_TOML_CONTENTS = """
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HypothesisTests = "09f84164-cd44-5f33-b23f-e6b0d136a0d5"
LsqFit = "2fda8390-95c7-5789-9bda-21331edee243"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"

[compat]
CSV = "~0.10.9"
GLM = "~1.8.3"
HypothesisTests = "~0.10.13"
LsqFit = "~0.13.0"
Plots = "~1.38.11"
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[[deps.JLFzf]]
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[[deps.Random]]
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[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
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[[deps.Requires]]
deps = ["UUIDs"]
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[[deps.Rmath]]
deps = ["Random", "Rmath_jll"]
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[[deps.Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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[[deps.Roots]]
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    [deps.Roots.extensions]
    RootsForwardDiffExt = "ForwardDiff"
    RootsIntervalRootFindingExt = "IntervalRootFinding"

    [deps.Roots.weakdeps]
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[[deps.SHA]]
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[[deps.Showoff]]
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[[deps.SimpleBufferStream]]
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[[deps.SnoopPrecompile]]
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[[deps.Sockets]]
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[[deps.SortingAlgorithms]]
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[[deps.SpecialFunctions]]
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[[deps.StaticArrays]]
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[[deps.StaticArraysCore]]
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[[deps.Statistics]]
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[[deps.StatsAPI]]
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[[deps.StatsBase]]
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[[deps.StatsFuns]]
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    StatsFunsInverseFunctionsExt = "InverseFunctions"

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[[deps.StatsModels]]
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[[deps.StatsPlots]]
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[[deps.SuiteSparse_jll]]
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[[deps.TOML]]
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[[deps.TableOperations]]
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[[deps.Tables]]
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[[deps.Tar]]
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[[deps.TensorCore]]
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[[deps.TranscodingStreams]]
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[[deps.UUIDs]]
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[[deps.Unicode]]
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[[deps.XSLT_jll]]
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[[deps.Xorg_libXcursor_jll]]
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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 = "5.7.0+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.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.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[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 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
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