PCM20230517_ISLR2_2.1.4_Neural_Network_Models

### A Pluto.jl notebook ###
# v0.19.26

using Markdown
using InteractiveUtils

# ╔═╡ 0de0b5c9-ec94-4d19-936d-b2cbb5f3b3b8
using CSV, DataFrames, Statistics, GLM,  Plots

# ╔═╡ 2174f7b1-3da2-44bb-b6c0-c7aa26760111
using Flux

# ╔═╡ 64c23eb4-4664-4573-ae57-c2815f01148f
using Flux: train! 

# ╔═╡ f80ea5a0-f57d-11ed-2f72-530e4436b981
md"
==================================================================================
#### ISLR2\_2.1.4\_Neural Network Models
##### file: PCM20230517\_ISLR2\_2.1.4\_Neural\_Network\_Models
##### code: (1.9.0/0.19.25) by PCM *** 2023/05/29 ***
==================================================================================
"


# ╔═╡ e3eaba32-50a8-4cb9-877b-1bbfa3b3e8e1
md"
---
##### 1. [Fitting a straight line](http://fluxml.ai/Flux.jl/stable/models/overview/)
"

# ╔═╡ 8237996f-fde7-40b9-a285-cc614f98dc1a
let
	# =============================================================================
	regressionModel(x) = 4x + 2
	estimatedModel(x, w, b) = w .* x .+ b
	# =============================================================================
	# construct training and test data for predictor x
	x_train, x_test = hcat(0:5...), hcat(6:10...)
	# construct training and test data for criterium y
	y_train, y_test = regressionModel.(x_train), regressionModel.(x_test)
	#----------------------------------------------------------------------
	# beware ! in 'train!' 'data' is expected
	#          in 'loss' 'x_train, y_train' is expected
	data = [(x_train, y_train)] 
	# =============================================================================
	# construct neural net with 1 input, 0 latent, and 1 output node
	neuralRegressionModel = Dense(1 => 1)
	#------------------------------------------------------------------------------
	# definition of mean square error loss function
	loss(model, x, y) = mean(abs2.(model(x) .- y))
	epsilon = 10f-6
	#------------------------------------------------------------------------------
	# initial loss and loss after one training epoch
	lossHistory = Array{Float32}(undef, 0)                          # new array
	weight0, bias0 = neuralRegressionModel.weight, neuralRegressionModel.bias 
	push!(lossHistory, loss(neuralRegressionModel, x_train, y_train))
	lossOld = last(lossHistory)
	optStrategy = Descent()
	train!(loss, neuralRegressionModel, data, optStrategy)	
	push!(lossHistory, loss(neuralRegressionModel, x_train, y_train))
	lossNew = last(lossHistory)
	#------------------------------------------------------------------------------
	nOfEpochs = 0
	while abs(lossOld - lossNew) > epsilon
		nOfEpochs += 1
		lossOld = last(lossHistory)
		train!(loss, neuralRegressionModel, data, optStrategy)
		lossNew = loss(neuralRegressionModel, x_train, y_train)
		push!(lossHistory, lossNew)
	end # while
	weight, bias = neuralRegressionModel.weight, neuralRegressionModel.bias 
	#------------------------------------------------------------------------------
	plot(title="Data and Regression Model E(Y|X=x) = 4x + 2")
	plot!(x_train, y_train, seriestype=:scatter, color=:cornflowerblue, legend=:none, xlabel="X", ylabel="Y")
	yHat = estimatedModel(0:0.1:5, weight, bias)
	plot!(0:0.1:5, yHat, seriestype=:line, linecolor=:red)
	lossNewRounded = round(lossNew, digits=4)
	annotate!(4.5, 5, "mse=$lossNewRounded")
	# (epochs=nOfEpochs, lossHistory=lossHistory, weight=weight, bias=bias)
	# =============================================================================
end # let

# ╔═╡ 2ccc41a0-45d0-4de5-ae71-6ad572a2525b
md"
---
##### 2. Fitting a Linear Univariate Regression
(inspired by the corresponding [Flux-tutorial](http://fluxml.ai/Flux.jl/stable/tutorials/linear_regression/)) but with some modifications :)
"

# ╔═╡ f6b48792-9c03-42e8-9327-44b616198bd3
md"
---
###### 2.1 Generating Function, Randomization of $Y$, and Classical OLS-Regression

$f(x) = 3x + 2$

$\;$

$E(Y|x) = \beta_0 + \beta_1x$

$\;$

$Y = \beta_0 + \beta_1 x + E$

$\;$

$\hat{\beta} = (X'X)^{-1}X'ys$

$\;$

where $E$ = error variable with constant variance $\sigma_{Y|x}.$
"

# ╔═╡ d8b92235-7c8a-4efd-8151-a258cf64cfef
let # in contrast to tutorial slightly modified code
	# ================================================================================
	# definition of generating function
	f(x) = 3.0 .* x .+ 2.0
	nObs = 61                                      # number of observations
	#---------------------------------------------------------------------------------
	# generation of predictor data
	x = [z for z in -3.0:0.1:3.0]
	y = f(x)
	ys = y .* rand(nObs)                           # the ys are randomized 
	rxy = round(cor(x, ys), digits=3)
	#---------------------------------------------------------------------------------
	# estimation of classical linear univariate regression model in pure Julia
    X = hcat(ones(nObs, 1), x)  # predictor matrix X
	b = (X'X)^-1*(X'ys)
	yHat = X*b
	ryHaty = round(cor(yHat, ys), digits=3)        # correlation(yhat, y)
	bHat   = round(b[2], digits=2)
	cHat   = round(b[1], digits=2)
	#---------------------------------------------------------------------------------
	# definition of mean square error loss function
	mse = round(mean(abs2.(yHat .- y)), digits=3)
	# ================================================================================
	plot(title = "Generating Function, Data, and Class. OLS-Regression", xlims=(-3.5, 3.5), ylims=(-9, 13))
	#--------------------------------------------------------------------------------
	# plot of generating function
	plot!(x, y, linestyle = :dash, lw = 2, label = "f(x) = 3.0 .* x .+ 2.0 (Generating Function)",  xlabel = "x", ylabel= "y")
	#--------------------------------------------------------------------------------
	# plot of data
	plot!(x, ys, seriestype = :scatter, label = " (x, y) = Data",  xlabel = "x", ylabel= "y")
	#---------------------------------------------------------------------------------
	# plot of classical linear univariate regression line
	plot!(x, yHat, seriestype=:line, linewidth=2, label ="E(Y|x) = $bHat .* x .+ $cHat (Classical OLS)")
	#---------------------------------------------------------------------------------
	annotate!(2.3, -5, "r(X, Y) = $rxy", 10)
	annotate!(2.07, -6.2, "r(E(Y|x), Y) = $ryHaty", 10)
	annotate!(2.4, -7.5, "mse = $mse", 10)
	# ================================================================================
end # let

# ╔═╡ d83382df-7e37-430c-b2b6-cc8a25257a3c
md"
---
###### 2.2 As 2.1 plus GLM-OLS-Regression

$f(x) = 3x + 2$

$\;$

$E(Y|x) = \beta_0 + \beta_1x$

$\;$

$Y = \beta_0 + \beta_1 x + E$

$\;$

$\hat{\beta} = (X'X)^{-1}X'ys$

$\;$

where $E$ = error variable with constant variance $\sigma_{Y|x}.$
"

# ╔═╡ 1b05d618-e9b5-4027-a99e-04cf7f1f32f6
let # in contrast to tutorial slightly modified code
	# ================================================================================
	# definition of generating function
	f(x) = 3.0 .* x .+ 2.0
	nObs = 61                                      # number of observations
	#---------------------------------------------------------------------------------
	# generation of predictor data
	x = [z for z in -3.0:0.1:3.0]
	y = f(x)
	ys = y .* rand(nObs)                           # the ys are randomized 
	rxy = round(cor(x, ys), digits=3)
	#---------------------------------------------------------------------------------
	# estimation of classical linear univariate regression model in pure Julia
    X = hcat(ones(nObs, 1), x)  # predictor matrix X
	b = (X'X)^-1*(X'ys)
	yHat = X*b
	bHat   = round(b[2], digits=2)
	cHat   = round(b[1], digits=2)
	# ================================================================================
	# construction of dataframe for use in GLM's lm
	dataFr = DataFrame(X=x, Ys=ys) 
	#---------------------------------------------------------------------------------
	# classical OLS-regression with GLM
	univRegOLS = lm(@formula(Ys ~ 1 + X), dataFr)  # regression object
	yHatLm = predict(univRegOLS)                     # predictions of y
	ryHaty = round(cor(yHatLm, ys), digits=3)        # correlation(yhat, y)
	cHatLm, bHatLm,  =                                 # estimated parameters
		round(coef(univRegOLS)[1], digits=2), round(coef(univRegOLS)[2], digits=2)
	#---------------------------------------------------------------------------------
	# definition of mean square error loss function
	mse = round(mean(abs2.(yHat .- y)), digits=3)
	# ================================================================================
	plot(title = "As 2.1 plus GLM-Regression", xlims=(-3.5, 3.5), ylims=(-9, 13))
	#--------------------------------------------------------------------------------
	# plot of generating function
	plot!(x, y, linestyle = :dash, lw = 2, label = "f(x) = 3.0 .* x .+ 2.0 (Generating Function)",  xlabel = "x", ylabel= "y")
	#--------------------------------------------------------------------------------
	# plot of data
	plot!(x, ys, seriestype = :scatter, label = " (x, y) = Data",  xlabel = "x", ylabel= "y")
	#---------------------------------------------------------------------------------
	# plot of classical linear univariate regression line
	plot!(x, yHat, seriestype=:line, linewidth=2, label ="E(Y|x) = $bHat .* x .+ $cHat (Classical OLS)")
	#--------------------------------------------------------------------------------
	# plot of GLM-regression line
	plot!(x, yHatLm, seriestype=:line, linestyle = :dash, linewidth=2, label ="E(Y|x) = $bHat .* x .+ $cHat (GLM-Regression)")
	#---------------------------------------------------------------------------------
	annotate!(2.3, -5, "r(X, Y) = $rxy", 10)
	annotate!(2.07, -6.2, "r(E(Y|x), Y) = $ryHaty", 10)
	annotate!(2.35, -7.4, "mse = $mse", 10)
	# ===========3====================================================================
end # let

# ╔═╡ 244c6a02-0150-4669-b319-bf2dd12c23df
md"
---
###### 2.3 As 2.2 plus Gradient-Descent OLS-Regression

$f(x) = 3x + 2$

$\;$

$E(Y|x) = \beta_0 + \beta_1x$

$\;$

$Y = \beta_0 + \beta_1 x + E$
$\;$

where $E$ = error variable with constant variance $\sigma_{Y|x}.$
"

# ╔═╡ 56f5a82c-770e-404e-bfdd-d36d8ec70b34
let # in contrast to tutorial slightly modified code
	# ================================================================================
	# definition of generating function
	f(x) = 3.0 .* x .+ 2.0
	nObs = 61                                      # number of observations
	#---------------------------------------------------------------------------------
	# generation of predictor data
	x = [z for z in -3.0:0.1:3.0]
	y = f(x)
	ys = y .* rand(nObs)                           # the ys are randomized 
	rxy = round(cor(x, ys), digits=3)
	#---------------------------------------------------------------------------------
	# estimation of classical linear univariate regression model in pure Julia
    X = hcat(ones(nObs, 1), x)  # predictor matrix X
	b = (X'X)^-1*(X'ys)
	yHat = X*b
	bHat   = round(b[2], digits=2)
	cHat   = round(b[1], digits=2)
	# ================================================================================
	# construction of dataframe for use in GLM's lm
	dataFr = DataFrame(X=x, Ys=ys) 
	#---------------------------------------------------------------------------------
	# classical OLS-regression with GLM
	univRegOLS = lm(@formula(Ys ~ 1 + X), dataFr)  # regression object
	yHat = predict(univRegOLS)                     # predictions of y
	cHat, bHat =                                   # estimated parameters
		round(coef(univRegOLS)[1], digits=2), round(coef(univRegOLS)[2], digits=2)
	# ================================================================================
	# definition of regression model for estimation of parameters by gradient descent
	regressionModel(W, b, x) = @. W*x + b          # makes predictions yHat
	lossHistory = Array{Float64}(undef, 0)         # new array for documenting loss
	#---------------------------------------------------------------------------------
	# definition of mean square error loss function
	mseLoss(model, W, b, x, y) = mean(abs2.(model(W, b, x) .- y))
	#---------------------------------------------------------------------------------
	# initialization and first trial
	W = rand(Float64, 1, 1)       # initial regression coefficient w0 in Float64
	W0 = W
	b = [0.0e0]                   # initial bias b0 in Float64
	b0 = b
	regressionModel(W, b, x) |> size == (61, 1)  # ==> true --> :)
	regressionModel(W, b, x)[1], y[1]            # first trial, ok ?
	push!(lossHistory, mseLoss(regressionModel, W, b, x, ys))  # first MSE
	lossOld = last(lossHistory)                  # first MSE			
	#---------------------------------------------------------------------------------
	# Train classical regression model
	# define gradient(mseLoss, regressionModel, W, b, x, y)
	_, dLdW, dLdb, _, _ = gradient(mseLoss, regressionModel, W, b, x, ys)
	#--------------------------------------------------------------------------------
	# initial parameter update
	W = W .- 0.1 .* dLdW
	b = b .- 0.1 .* dLdb
	push!(lossHistory, mseLoss(regressionModel, W, b, x, ys))  # 2nd MSE
	lossOld = last(lossHistory)                                # 2nd MSE	
	#--------------------------------------------------------------------------------
	# definition of one step training
	function trainRegressionModel()
           _, dLdW, dLdb, _, _ = gradient(mseLoss, regressionModel, W, b, x, ys)
           @. W = W - 0.1 * dLdW
		   @. b = b - 0.1 * dLdb
	end # function trainRegressionModel
	#--------------------------------------------------------------------------------
	# one step training
	trainRegressionModel()
	lossNew = mseLoss(regressionModel, W, b, x, ys)
	W, b, lossOld, lossNew
	#--------------------------------------------------------------------------------
	nOfEpochs = 0
	epsilon = 1.0E-7
	# while !(lossOld ≈ lossNew)
	while abs((lossOld - lossNew) > epsilon)
		nOfEpochs += 1
		lossOld = last(lossHistory)
		trainRegressionModel()
		lossNew = mseLoss(regressionModel, W, b, x, ys)
		push!(lossHistory, lossNew)
	end # while
	yHatGrD = regressionModel(W, b, x)
	ryHaty = round(cor(yHatGrD, ys)[1,1], digits=3)        # correlation(yhat, y)
	mse = round(lossNew, digits=3)
	slope, constant = round(W[1,1], digits=2), round(b[1], digits=2)
	nOfEpochs, W0, b0, W, b, mse
	# ================================================================================
	plot(title = "As 2.2 plus Gradient-Descent OLS-Regression", xlims=(-3.5, 3.5), ylims=(-9, 13))
	#--------------------------------------------------------------------------------
	# plot of generating function
	plot!(x, y, linestyle = :dash, linewidth = 2, label = "f(x) = 3 .* x .+ 2 (Generating Function)",  xlabel = "x", ylabel= "y")
	#--------------------------------------------------------------------------------
	# plot of data
	plot!(x, ys, seriestype = :scatter, label = " (x, y) = Data",  xlabel = "x", ylabel= "y")
	#---------------------------------------------------------------------------------
	# plot of classical linear univariate regression line
	plot!(x, yHat, seriestype=:line, linewidth=2, label ="E(Y|x) = $bHat .* x .+ $cHat (Classical OLS)")
	#--------------------------------------------------------------------------------
	# plot of GLM-regression line
	plot!(x, yHat, seriestype=:line, linestyle=:dash, linewidth = 2, label ="E(Y|x) = $bHat .* x .+ $cHat (GLM-Regression)")
	#--------------------------------------------------------------------------------
	# plot of gradient-descent OLS-regression line
	plot!(x, regressionModel(W, b, x), seriestype=:line, linestyle=:dash, linewidth = 2,label ="E(Y|x) = $slope .* x .+ $constant (Gradient Descent)")
	#---------------------------------------------------------------------------------
	annotate!(2.3, -5, "r(X, Y) = $rxy", 10)
	annotate!(2.07, -6.2, "r(E(Y|x), Y) = $ryHaty", 10)
	annotate!(2.4, -7.4, "mse = $mse", 10)
	# ================================================================================
end # let

# ╔═╡ defa354b-394f-4c90-8077-f05a631b29f8
md"
---
###### 2.4 As 2.3 plus Flux-Neural-Net OLS-Regression

$f(x) = E(Y|x) = \beta_0 + \beta_1x$

$\;$

$Y = \beta_0 + \beta_1 x + E$
$\;$

where $E$ = error variable with constant variance $\sigma_{Y|x}.$
"

# ╔═╡ 15ce3433-4f38-43a9-877c-9d2e57407732
let # in contrast to tutorial slightly modified code
	# ================================================================================
	# definition of generating function
	f(x) = 3.0 .* x .+ 2.0
	nObs = 61                                      # number of observations
	#---------------------------------------------------------------------------------
	# generation of predictor data
	x = [z for z in -3.0:0.1:3.0]
	y = f(x)
	ys = y .* rand(nObs)                           # the ys are randomized 
	rxy = round(cor(x, ys), digits=3)
	#---------------------------------------------------------------------------------
	# estimation of classical linear univariate regression model in pure Julia
    X = hcat(ones(nObs, 1), x)  # predictor matrix X
	b = (X'X)^-1*(X'ys)
	yHat = X*b
	bHat   = round(b[2], digits=2)
	cHat   = round(b[1], digits=2)
	# ================================================================================
	# construction of dataframe for use in GLM's lm
	dataFr = DataFrame(X=x, Ys=ys) 
	#---------------------------------------------------------------------------------
	# classical OLS-regression with GLM
	univRegOLS = lm(@formula(Ys ~ 1 + X), dataFr)  # regression object
	yHat = predict(univRegOLS)                     # predictions of y
	cHat, bHat =                                   # estimated parameters
		round(coef(univRegOLS)[1], digits=2), round(coef(univRegOLS)[2], digits=2)
	# ================================================================================
	# construction of dataframe for use in GLM's lm
	dataFr = DataFrame(X=x, Ys=ys) 
	#---------------------------------------------------------------------------------
	# classical OLS-regression with GLM
	univRegOLS = lm(@formula(Ys ~ 1 + X), dataFr)  # regression object
	yHat = predict(univRegOLS)                     # predictions of y
	ryHaty = round(cor(yHat, ys), digits=4)        # correlation(yhat, y)
	cHat, bHat =                                   # estimated parameters
		round(coef(univRegOLS)[1], digits=2), round(coef(univRegOLS)[2], digits=2)
	# ================================================================================
	# definition of regression model for estimation of parameters by gradient descent
	regressionModel(W, b, x) = @. W*x + b          # makes predictions yHat
	lossHistory = Array{Float64}(undef, 0)         # new array for documenting loss
	#---------------------------------------------------------------------------------
	# definition of mean square error loss function
	mseLoss(model, W, b, x, y) = mean(abs2.(model(W, b, x) .- y))
	#---------------------------------------------------------------------------------
	# initialization and first trial
	W = rand(Float64, 1, 1)       # initial regression coefficient w0 in Float64
	W0 = W
	b = [0.0e0]                   # initial bias b0 in Float64
	b0 = b
	regressionModel(W, b, x) |> size == (61, 1)  # ==> true --> :)
	regressionModel(W, b, x)[1], y[1]            # first trial, ok ?
	push!(lossHistory, mseLoss(regressionModel, W, b, x, ys))  # first MSE
	lossOld = last(lossHistory)                  # first MSE			
	#---------------------------------------------------------------------------------
	# Train classical regression model
	# define gradient(mseLoss, regressionModel, W, b, x, y)
	_, dLdW, dLdb, _, _ = gradient(mseLoss, regressionModel, W, b, x, ys)
	#--------------------------------------------------------------------------------
	# initial parameter update
	W = W .- 0.1 .* dLdW
	b = b .- 0.1 .* dLdb
	push!(lossHistory, mseLoss(regressionModel, W, b, x, ys))  # 2nd MSE
	lossOld = last(lossHistory)                                # 2nd MSE	
	#--------------------------------------------------------------------------------
	# definition of one step training
	function trainRegressionModel()
           _, dLdW, dLdb, _, _ = gradient(mseLoss, regressionModel, W, b, x, ys)
           @. W = W - 0.1 * dLdW
		   @. b = b - 0.1 * dLdb
	end # function trainRegressionModel
	#--------------------------------------------------------------------------------
	# one step training the OLS-regression model
	trainRegressionModel()
	lossOld = last(lossHistory)                                 # 2nd MSE	
	lossNew = mseLoss(regressionModel, W, b, x, ys)             # 3rd MSE
	W, b, lossOld, lossNew
	#--------------------------------------------------------------------------------
	# n-step training process
	nOfEpochs = 0
	epsilon = 1.0E-7
	# while !(lossOld ≈ lossNew)
	while abs((lossOld - lossNew) > epsilon)
		nOfEpochs += 1
		lossOld = last(lossHistory)
		trainRegressionModel()
		lossNew = mseLoss(regressionModel, W, b, x, ys)
		push!(lossHistory, lossNew)
	end # while
	mse = round(lossNew, digits=3)
	slope, constant = round(W[1,1], digits=2), round(b[1], digits=2)
	nOfEpochs, W0, b0, W, b, mse
	# ================================================================================
	# definition of FLUX 1-0-1 neural net regression model
	fluxRegressionModel = Dense(1 => 1)
	#------------------------------------------------------------------------
	# definition of mean square error loss function
	fluxMseLoss(fluxRegressionModel, x, y) = Flux.mse(fluxRegressionModel(x), y)
	fluxLossHistory = Array{Float32}(undef, 0)       # new array for documenting loss
	#---------------------------------------------------------------------------------
	# initial guesses and first trial
	x32   = convert(Vector{Float32}, x)
	ys32  = convert(Vector{Float32}, ys)
	#--------------------------------------------------------------------------------
	# initial guesses, first trial, initial loss
	weight, bias = fluxRegressionModel.weight, fluxRegressionModel.bias
	# test, whether correct size of input 
	fluxRegressionModel(x32') |> size == (1, 61)  # ==> true --> :)
	# first trial
	fluxRegressionModel(x32'), ys32
	# initial loss
	fluxLossOld = fluxMseLoss(fluxRegressionModel, x32', ys32')    # 1st MSE
	push!(fluxLossHistory, fluxLossOld)                            # 1st MSE
	#--------------------------------------------------------------------------------
	# comparison: is final OLS-regression loss approx eq to initial Flux-loss ?
	mseLoss(regressionModel, W, b, x, ys) ≈    # '≈' = is approximately equal ?
		fluxMseLoss(fluxRegressionModel, x32', ys32') ? ":)" : ":("
	#--------------------------------------------------------------------------------
	# definition of one step Flux-training
	function trainFluxRegressionModel()
		dLdm, _, _ = gradient(fluxMseLoss, fluxRegressionModel, x32', ys32')
        @. fluxRegressionModel.weight = 
			fluxRegressionModel.weight - 0.001 * dLdm.weight
        @. fluxRegressionModel.bias = 
			fluxRegressionModel.bias - 0.001 * dLdm.bias
	end # function trainFluxRegressionModel
	#--------------------------------------------------------------------------------
	# one step training the Flux-OLS-regression model
	trainFluxRegressionModel()
	fluxLossNew = fluxMseLoss(fluxRegressionModel, x32', ys32')     # 2nd MSE
	push!(fluxLossHistory, fluxLossNew)                             # 2nd MSE 
	fluxWeight0 = fluxRegressionModel.weight
	fluxBias0   = fluxRegressionModel.bias
	fluxWeight0, fluxBias0, fluxLossHistory
	#--------------------------------------------------------------------------------
	# n-step training process
	nOfFluxEpochs = 0
	fluxEpsilon = 1.0E-8
	while abs(fluxLossOld - fluxLossNew) > fluxEpsilon
		nOfFluxEpochs += 1
		fluxLossOld = last(fluxLossHistory)
		trainFluxRegressionModel()
		fluxLossNew = fluxMseLoss(fluxRegressionModel, x32', ys32')
		push!(fluxLossHistory, fluxLossNew)
	end # while
	yHatFNN = fluxRegressionModel(x32')
	ryHaty = round(cor(yHatFNN', ys)[1,1], digits=3)        # correlation(yhat, y)
	fluxMse = round(fluxLossNew, digits=3)
	fluxWeight, fluxBias = 
		round(fluxRegressionModel.weight[1,1], digits=convert(Int32, 2)), 
		round(fluxRegressionModel.bias[1], digits=convert(Int32, 2))
	nOfFluxEpochs, fluxWeight0, fluxBias0, fluxWeight, fluxBias, fluxLossHistory, mseLoss(regressionModel, W, b, x, ys) 
	#--------------------------------------------------------------------------------
	# comparison: is final OLS-regression loss approx eq to initial Flux-loss ?
	mseLoss(regressionModel, W, b, x, ys) ≈    # '≈' = is approximately equal ?
		fluxMseLoss(fluxRegressionModel, x32', ys32') ? ":)" : ":("
	# ================================================================================
	plot(title = "As 2.3 plus Flux-Neural-Net OLS-Regression", xlims=(-4, 4), ylims=(-9, 14))
	#--------------------------------------------------------------------------------
	# plot of generating function
	plot!(x, y, linestyle = :dash, linewidth = 2, label = "f(x) = 3.0 .* x .+ 2.0 (Generating Function)",  xlabel = "x", ylabel= "y")
	#--------------------------------------------------------------------------------
	# plot of data
	plot!(x, ys, seriestype = :scatter, label = " (x, y) = Data",  xlabel = "x", ylabel= "y")
	#---------------------------------------------------------------------------------
	# plot of classical linear univariate regression line
	plot!(x, yHat, seriestype=:line, linewidth=2, label ="E(Y|x) = $bHat .* x .+ $cHat (Classical OLS)")
	#--------------------------------------------------------------------------------
	# plot of GLM-regression line
	plot!(x, yHat, seriestype=:line, linestyle=:dash, linewidth = 2, label ="E(Y|x) = $bHat .* x .+ $cHat (GLM-Regression)")
	#--------------------------------------------------------------------------------
	# plot of gradient-descent OLS-regression line
	plot!(x, regressionModel(W, b, x), seriestype=:line, linestyle=:dash, linewidth = 2,label ="E(Y|x) = $slope .* x .+ $constant (Gradient Descent)")
	#---------------------------------------------------------------------------------
	# plot of neural-net OLS-regression line
	plot!(x, fluxRegressionModel(x32')', linewidth = 2, linestyle=:dash, label ="E(Y|x) = $fluxWeight .* x .+ $fluxBias (FluxNeuralNet)")
	#---------------------------------------------------------------------------------
	annotate!(2.3, -5.8, "r(X, Y) = $rxy", 10)
	annotate!(2.07, -7, "r(E(Y|x), Y) = $ryHaty", 10)
	annotate!(2.4, -8.2, "mse = $mse", 10)
	# ================================================================================
end # let

# ╔═╡ 0d3b93ca-e5cd-44db-a537-d027336fae3f
md"
---
##### References
- **Flux Overview: Fitting A Straight Line**; [http://fluxml.ai/Flux.jl/stable/models/overview/](http://fluxml.ai/Flux.jl/stable/models/overview/); last visit 2023/0529
- **Flux Tutorial: Linear Regression**; [http://fluxml.ai/Flux.jl/stable/tutorials/linear_regression/](http://fluxml.ai/Flux.jl/stable/tutorials/linear_regression/); last visit 2023/05/29
"

# ╔═╡ ab48cb8c-da35-4de3-8d2b-c977d6b7e773
md"
---
##### end of ch. 2.1.4
"

# ╔═╡ a1274c46-2010-498a-bd8b-ddc2b3cab34f
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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uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+6"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "1544b926975372da01227b382066ab70e574a3ec"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.10.1"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TimerOutputs]]
deps = ["ExprTools", "Printf"]
git-tree-sha1 = "f548a9e9c490030e545f72074a41edfd0e5bcdd7"
uuid = "a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f"
version = "0.5.23"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "9a6ae7ed916312b41236fcef7e0af564ef934769"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.13"

[[deps.Transducers]]
deps = ["Adapt", "ArgCheck", "BangBang", "Baselet", "CompositionsBase", "DefineSingletons", "Distributed", "InitialValues", "Logging", "Markdown", "MicroCollections", "Requires", "Setfield", "SplittablesBase", "Tables"]
git-tree-sha1 = "25358a5f2384c490e98abd565ed321ffae2cbb37"
uuid = "28d57a85-8fef-5791-bfe6-a80928e7c999"
version = "0.4.76"

[[deps.URIs]]
git-tree-sha1 = "074f993b0ca030848b897beff716d93aca60f06a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.2"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.UnsafeAtomics]]
git-tree-sha1 = "6331ac3440856ea1988316b46045303bef658278"
uuid = "013be700-e6cd-48c3-b4a1-df204f14c38f"
version = "0.2.1"

[[deps.UnsafeAtomicsLLVM]]
deps = ["LLVM", "UnsafeAtomics"]
git-tree-sha1 = "ea37e6066bf194ab78f4e747f5245261f17a7175"
uuid = "d80eeb9a-aca5-4d75-85e5-170c8b632249"
version = "0.1.2"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.WeakRefStrings]]
deps = ["DataAPI", "InlineStrings", "Parsers"]
git-tree-sha1 = "b1be2855ed9ed8eac54e5caff2afcdb442d52c23"
uuid = "ea10d353-3f73-51f8-a26c-33c1cb351aa5"
version = "1.4.2"

[[deps.WorkerUtilities]]
git-tree-sha1 = "cd1659ba0d57b71a464a29e64dbc67cfe83d54e7"
uuid = "76eceee3-57b5-4d4a-8e66-0e911cebbf60"
version = "1.6.1"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "93c41695bc1c08c46c5899f4fe06d6ead504bb73"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.10.3+0"

[[deps.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"

[[deps.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"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.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"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.Zygote]]
deps = ["AbstractFFTs", "ChainRules", "ChainRulesCore", "DiffRules", "Distributed", "FillArrays", "ForwardDiff", "GPUArrays", "GPUArraysCore", "IRTools", "InteractiveUtils", "LinearAlgebra", "LogExpFunctions", "MacroTools", "NaNMath", "PrecompileTools", "Random", "Requires", "SparseArrays", "SpecialFunctions", "Statistics", "ZygoteRules"]
git-tree-sha1 = "ebac1ae9f048c669317ad48c9bed815790a468d8"
uuid = "e88e6eb3-aa80-5325-afca-941959d7151f"
version = "0.6.61"

    [deps.Zygote.extensions]
    ZygoteColorsExt = "Colors"
    ZygoteDistancesExt = "Distances"
    ZygoteTrackerExt = "Tracker"

    [deps.Zygote.weakdeps]
    Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
    Distances = "b4f34e82-e78d-54a5-968a-f98e89d6e8f7"
    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"

[[deps.ZygoteRules]]
deps = ["ChainRulesCore", "MacroTools"]
git-tree-sha1 = "977aed5d006b840e2e40c0b48984f7463109046d"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.3"

[[deps.cuDNN]]
deps = ["CEnum", "CUDA", "CUDNN_jll"]
git-tree-sha1 = "ec954b59f6b0324543f2e3ed8118309ac60cb75b"
uuid = "02a925ec-e4fe-4b08-9a7e-0d78e3d38ccd"
version = "1.0.3"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.7.0+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:
# ╟─f80ea5a0-f57d-11ed-2f72-530e4436b981
# ╠═0de0b5c9-ec94-4d19-936d-b2cbb5f3b3b8
# ╠═2174f7b1-3da2-44bb-b6c0-c7aa26760111
# ╠═64c23eb4-4664-4573-ae57-c2815f01148f
# ╟─e3eaba32-50a8-4cb9-877b-1bbfa3b3e8e1
# ╟─8237996f-fde7-40b9-a285-cc614f98dc1a
# ╟─2ccc41a0-45d0-4de5-ae71-6ad572a2525b
# ╟─f6b48792-9c03-42e8-9327-44b616198bd3
# ╟─d8b92235-7c8a-4efd-8151-a258cf64cfef
# ╟─d83382df-7e37-430c-b2b6-cc8a25257a3c
# ╟─1b05d618-e9b5-4027-a99e-04cf7f1f32f6
# ╟─244c6a02-0150-4669-b319-bf2dd12c23df
# ╟─56f5a82c-770e-404e-bfdd-d36d8ec70b34
# ╟─defa354b-394f-4c90-8077-f05a631b29f8
# ╟─15ce3433-4f38-43a9-877c-9d2e57407732
# ╟─0d3b93ca-e5cd-44db-a537-d027336fae3f
# ╟─ab48cb8c-da35-4de3-8d2b-c977d6b7e773
# ╟─a1274c46-2010-498a-bd8b-ddc2b3cab34f
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002