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PCM20230508_ISLR2_2.1.2_Nonlinear_Model_Functions.jl
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PCM20230508_ISLR2_2.1.2_Nonlinear_Model_Functions.jl
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### A Pluto.jl notebook ###
# v0.19.25
using Markdown
using InteractiveUtils
# ╔═╡ 4f55d011-4a61-47ce-a2a4-7b4502df7733
using CSV, DataFrames,Plots, LsqFit, GLM, Statistics, Symbolics
# ╔═╡ 9e28d02c-2279-4e46-8d53-99009af90d6a
md"
==================================================================================
#### ISLR2\_2.1.2\_Nonlinear Model Functions
##### file: PCM20230508\_ISLR2\_2.1.2\_Nonlinear\_Model\_Functions
##### code: (1.9.0/0.19.25) by PCM *** 2023/05/17 ***
==================================================================================
"
# ╔═╡ a1aa0b52-e7cc-435a-b430-5230e9882b43
md"
---
#### 3. The *Income2* Data Set
##### 3.1 Dataframe
"
# ╔═╡ 88236f93-ab5f-411e-a27d-6c224ee81dba
income2DataFrame = CSV.File("C:/Users/claus/Nextcloud/PCM_Book/PCM2023_ISLR_Data/IncomeData/Income2Data.csv")
# ╔═╡ 860745cc-8672-4322-ab5f-a19cb090d0c1
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:
$f_{simple}(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}.$
$\;$
$\;$
"
# ╔═╡ 39651169-8720-48f6-92ff-358d4d2ac7e3
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Income
#--------------------------------------------------------------------------------
# Logistic transformation of xdata = income2DataFrame.Education
#
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
parmsHatMyModelX = [28.1998, 88.1722, 0.973715, 15.9852] # estimates of parms X
yHat = myModel(xdata, parmsHatMyModelX)
rxy = round(cor(ydata, yHat), digits=3)
ryyHat = round(cor(ydata, yHat), digits=3)
ryyHatsQ = round(ryyHat^2, digits=3)
#--------------------------------------------------------------------------------
Lx = round(parmsHatMyModelX[1], digits=2) # greatest Lower bound (gLb)
Ux = round(parmsHatMyModelX[2], digits=2) # least Upper bound (lUb)
kx = round(parmsHatMyModelX[3], digits=2) # max. growth rate k
x_k_max = round(parmsHatMyModelX[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, y^)", w=2)
plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
#--------------------------------------------------------------------------------
annotate!((21., 52,"r(y, x) = $rxy", 10))
annotate!((20.85, 46,"r(y, y^) = $ryyHat", 10))
annotate!((20.45, 40,"r^2(y, y^) = $ryyHatsQ", 10))
annotate!((20.6,34,"x_k_max = $x_k_max", 10))
annotate!((19.5,28,"kx=$kx (= max. growth rate)", 10))
annotate!((18.7,22,"Ux=$Ux (= Upper (right) asymptote)", 10))
annotate!((19.0,16,"Lx=$Lx (= Lower (left) asymptote)", 10))
annotate!((16.9,10,"kx=$kx (= max. growth rate of standard logistic function)", 10))
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 29d37cfe-9173-4c1b-a5aa-7ca74508c6b3
md"
---
###### 3.3.3 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:
$f_{simple}(x) = \frac{1}{1+e^{-x}} = \frac{e^x}{e^x+1}.$
$\;$
$\;$
"
# ╔═╡ 9dcf8a1b-1b41-42f3-81da-717c25544121
let
xdata = income2DataFrame.Seniority
ydata = income2DataFrame.Income
#--------------------------------------------------------------------------------
# Logistic transformation of ydata = income2DataFrame.Seniority
#
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
parmsHatMyModel = [40.4476, 75.6536, 4.0, 70.0] # estimates of parms Y
yHat = myModel(xdata, parmsHatMyModel)
rxy = round(cor(ydata, xdata), digits=3)
ryyHat = round(cor(ydata, yHat), digits=3)
ryyHatsQ = round(ryyHat^2, digits=3)
#--------------------------------------------------------------------------------
Ly = round(parmsHatMyModel[1], digits=2) # greatest Lower bound (gLb)
Uy = round(parmsHatMyModel[2], digits=2) # least Upper bound (lUb)
ky = round(parmsHatMyModel[3], digits=2) # max. growth rate k
y_k_max = round(parmsHatMyModel[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="(y,z)", xlabel="income2DataFrame.Seniority", ylabel="income2DataFrame.Income")
plot!(xdata, yHat, seriestype=:line, label="(y,zYhat)", w=2)
plot!(map(((x, y, yh) -> ([x, x],[y, yh])), xdata, ydata, yHat), label="")
#--------------------------------------------------------------------------------
annotate!((45.1, 98,"r(y, x) = $rxy", 10))
annotate!((42.9, 92,"r(y, y^) = $ryyHat", 10))
annotate!((39.0, 86,"r^2(y, y^) = $ryyHatsQ", 10))
annotate!((178.35,34,"y_k_max = $y_k_max", 10))
annotate!((161.5,28,"ky=$ky (= max. growth rate)", 10))
annotate!((148.7,22,"Uy=$Uy (= Upper (right) asymptote)", 10))
annotate!((151.0,16,"Ly=$Ly (= Lower (left) asymptote)", 10))
annotate!((122.9,10,"ky=$ky (= max. growth rate of standard logistic function)", 10))
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 49300f6a-82dc-47b7-a502-871b42529dd9
md"
---
##### 3.8 Two-Step Nonlinear-Linear Multiple OLS-Regression Models
We use a *two-step* modeling approach. That reduces the total computational costs. In the *first* step we *reuse* the nonlinear univariate OLS-regressions 3.2.3 and 3.3.3 and generate *predictions*
$\hat{I}_{sigmoid}(E) = \text{ predicted }\textit{Income}\text{ on basis of }\textit{Education},$
$\;$
$\hat{I}_{sigmoid}(S) = \text{ predicted }\textit{Income}\text{ on basis of }\textit{Seniority}.$
$\;$
$\;$
In the *second* step we plug in these predictions into *linear multiple* OLS-regression models:
$\;$
$E_1(Z | X, Y, X \cdot Y) = \beta_0 + \beta_1X + \beta_2Y + \beta_3X \cdot Y$
$\;$
$E_2(Z | X, Y) = \beta_0 + \beta_1X + \beta_2Y$
$\;$
$E_3(Z | X \cdot Y) = \beta_0 + \beta_3X \cdot Y$
$\;$
where
$Z = Income, X = \hat{I}_{sigmoid}(E) \text{, and } Y = \hat{I}_{sigmoid}(S).$
$\;$
"
# ╔═╡ eeadec6a-758e-49fd-b40d-ed298552f18e
md"
---
###### 3.7.1 3D Plot Income X Education X Seniority
"
# ╔═╡ d54cdff6-b137-4b50-947e-b60cb548dba8
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
# ╔═╡ 4589ad67-9e68-4343-9cfc-e4ec80c4447a
md"
---
###### 3.8.2 Linear Multiple OLS-Regression with 4-Parameters
$E_1(Z | \hat{I}(E), \hat{I}(S)) = \beta_0 + \beta_1\hat{I}(E) + \beta_2\hat{I}(S) + \beta_3\hat{I}(E)\hat{I}(S)$
$\;$
The $p$-values of predictors reveal that the interaction term $Education\&Seniority$ is *not* necessary at least in the context of the other predictors.
"
# ╔═╡ 7575eb49-97d4-4bbc-8a4a-9a51cea0be08
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Seniority
zdata = income2DataFrame.Income
#--------------------------------------------------------------------------------
logigisticDataFrame = DataFrame()
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
#--------------------------------------------------------------------------------
parmsX0 = [28.0, 88.0, 0.9, 16.0] # initial estimates of parms
myModel(xdata, parmsX0)
fitX = curve_fit(myModel, xdata, zdata, parmsX0)
parmsX_hat = coef(fitX)
xhat = myModel(xdata, parmsX_hat)
#--------------------------------------------------------------------------------
parmsY0 = [40.0, 78.0, 4.0, 70.0] # initial estimates of parms
myModel(ydata, parmsY0)
fitY = curve_fit(myModel, ydata, zdata, parmsY0)
parmsY_hat = coef(fitY)
yhat = myModel(ydata, parmsY_hat)
#--------------------------------------------------------------------------------
logigisticDataFrame[:, :logEducation] = xhat
logigisticDataFrame[:, :logSeniority] = yhat
logigisticDataFrame[:, :Income] = income2DataFrame.Income
#--------------------------------------------------------------------------------
ols_lin_reg =
lm(@formula(Income ~ 1 + logEducation * logSeniority), logigisticDataFrame)
zhat = predict(ols_lin_reg)
#--------------------------------------------------------------------------------
e = residuals(ols_lin_reg)
sse = round(e'e, digits=2)
multipleR = round(cor(zdata, zhat), digits=3)
multipleRsQ = round(multipleR^2, digits=3)
#--------------------------------------------------------------------------------
plot(title="Income -> logEducation '*' logSeniority (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]), xdata, ydata, zdata, zhat), label="", linewidth=3)
#------------------------------------------------------------------------
plot!(xdata, ydata, zdata, 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!(xdata, ydata, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
#--------------------------------------------------------------------------------
annotate!(23.7,160,40, "multiple R = $multipleR", 12)
annotate!(23.5,150,30, "multiple R^2 = $multipleRsQ", 12)
annotate!(21.65,140,20, "SSE = $sse ( = error sum of squares)", 12)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 533da37d-b74f-4b05-9544-3363577d0cbd
md"
---
###### 3.8.3 Linear Multiple OLS-Regression with 3-Parameters
$E_1(Z | \hat{I}(E), \hat{I}(S)) = \beta_0 + \beta_1\hat{I}(E) + \beta_2\hat{I}(S)$
$\;$
Dropping the *interaction* term $Education\&Seniority$ has *no* negative effect on the percentage of accounted variance in the criterium. It remains constant $95.6\%.$
"
# ╔═╡ ad053ad8-701b-409a-95b1-5dc8b0a1bffc
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Seniority
zdata = income2DataFrame.Income
#--------------------------------------------------------------------------------
logigisticDataFrame = DataFrame()
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
#--------------------------------------------------------------------------------
parmsX0 = [28.0, 88.0, 0.9, 16.0] # initial estimates of parms
myModel(xdata, parmsX0)
fitX = curve_fit(myModel, xdata, zdata, parmsX0)
parmsX_hat = coef(fitX)
xhat = myModel(xdata, parmsX_hat)
#--------------------------------------------------------------------------------
parmsY0 = [40.0, 78.0, 4.0, 70.0] # initial estimates of parms
myModel(ydata, parmsY0)
fitY = curve_fit(myModel, ydata, zdata, parmsY0)
parmsY_hat = coef(fitY)
yhat = myModel(ydata, parmsY_hat)
#--------------------------------------------------------------------------------
logigisticDataFrame[:, :logEducation] = xhat
logigisticDataFrame[:, :logSeniority] = yhat
logigisticDataFrame[:, :Income] = income2DataFrame.Income
#--------------------------------------------------------------------------------
ols_lin_reg =
lm(@formula(Income ~ 1 + logEducation + logSeniority), logigisticDataFrame)
zhat = predict(ols_lin_reg)
#--------------------------------------------------------------------------------
e = residuals(ols_lin_reg)
sse = round(e'e, digits=2)
multipleR = round(cor(zdata, zhat), digits=3)
multipleRsQ = round(multipleR^2, digits=3)
#--------------------------------------------------------------------------------
plot(title="Income -> logEducation + logSeniority (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]), xdata, ydata, zdata, zhat), label="", linewidth=3)
#------------------------------------------------------------------------
plot!(xdata, ydata, zdata, 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!(xdata, ydata, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
#--------------------------------------------------------------------------------
annotate!(23.7,160,40, "multiple R = $multipleR", 12)
annotate!(23.5,150,30, "multiple R^2 = $multipleRsQ", 12)
annotate!(21.65,140,20, "SSE = $sse ( = error sum of squares)", 12)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ d19dcb44-b913-4285-9451-5cb155d8bb70
md"
---
###### 3.8.4 Linear Multiple OLS-Regression with 2-Parameters
$E_3(Z | \hat{I}(S)) = \beta_3\hat{I}(E)\hat{I}(S)$
$\;$
Dropping the *main effects* has an effect on the accounted criterium variance. This $95.6\% - 89.5\% = 6.1\%$. Wether this statistical significant has to be tested by an appropriate F-Test.
"
# ╔═╡ 67c06dd8-eca9-4afe-996b-786f0aa48066
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Seniority
zdata = income2DataFrame.Income
#--------------------------------------------------------------------------------
logigisticDataFrame = DataFrame()
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
#--------------------------------------------------------------------------------
parmsX0 = [28.0, 88.0, 0.9, 16.0] # initial estimates of parms
myModel(xdata, parmsX0)
fitX = curve_fit(myModel, xdata, zdata, parmsX0)
parmsX_hat = coef(fitX)
xhat = myModel(xdata, parmsX_hat)
#--------------------------------------------------------------------------------
parmsY0 = [40.0, 78.0, 4.0, 70.0] # initial estimates of parms
myModel(ydata, parmsY0)
fitY = curve_fit(myModel, ydata, zdata, parmsY0)
parmsY_hat = coef(fitY)
yhat = myModel(ydata, parmsY_hat)
#--------------------------------------------------------------------------------
logigisticDataFrame[:, :logEducation] = xhat
logigisticDataFrame[:, :logSeniority] = yhat
logigisticDataFrame[:, :Income] = income2DataFrame.Income
#--------------------------------------------------------------------------------
ols_lin_reg =
lm(@formula(Income ~ 1 + logEducation & logSeniority), logigisticDataFrame)
zhat = predict(ols_lin_reg)
#--------------------------------------------------------------------------------
e = residuals(ols_lin_reg)
sse = round(e'e, digits=2)
multipleR = round(cor(zdata, zhat), digits=3)
multipleRsQ = round(multipleR^2, digits=3)
#--------------------------------------------------------------------------------
plot(title="Income -> logEducation '&' logSeniority (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]), xdata, ydata, zdata, zhat), label="", linewidth=3)
#------------------------------------------------------------------------
plot!(xdata, ydata, zdata, 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!(xdata, ydata, zhat, seriestype=:surface, color=:viridis, legend=:none, colorbar=false, alpha=0.5)
#--------------------------------------------------------------------------------
annotate!(23.8,160,40, "multiple R = $multipleR", 12)
annotate!(23.6,150,30, "multiple R^2 = $multipleRsQ", 12)
annotate!(21.6,140,20, "SSE = $sse ( = error sum of squares)", 12)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ f28799e0-4221-4bf9-b048-d393e138f65f
md"
Further simplifications of models to linear-nonlinear univariate models are not necessary because intercepts $\beta_0$ become $0$ and regression cofficients $\beta_1$ or $\beta_2$ become $1.0$. Percentages of variance accounted are the same as in 3.2.3 ($88.4\%$) and in 3.3.3 ($40.8\%$).
"
# ╔═╡ 7f9d135b-ba6d-48a2-9867-3e5934543646
md"
---
##### 3.8.5 Comparison of Regression Models 3.8.2-3.8.4 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.
"
# ╔═╡ 2009e053-f96f-49c6-aa3d-b22453544bac
md"
---
###### 3.8.5.1 Full Model 3.8.2 vs Restricted Model 3.8.3: Significance of Interaction ?
The result of the F-Test full model 3.8.2 versus restricted model 3.8.3 demonstrates that we may drop the interaction term without loss of accounted variance.
"
# ╔═╡ 9f3f6c40-685b-4e94-b805-55f25db476fa
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Seniority
zdata = income2DataFrame.Income
#--------------------------------------------------------------------------------
logigisticDataFrame = DataFrame()
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
#--------------------------------------------------------------------------------
parmsX0 = [28.0, 88.0, 0.9, 16.0] # initial estimates of parms
myModel(xdata, parmsX0)
fitX = curve_fit(myModel, xdata, zdata, parmsX0)
parmsX_hat = coef(fitX)
xhat = myModel(xdata, parmsX_hat)
#--------------------------------------------------------------------------------
parmsY0 = [40.0, 78.0, 4.0, 70.0] # initial estimates of parms
myModel(ydata, parmsY0)
fitY = curve_fit(myModel, ydata, zdata, parmsY0)
parmsY_hat = coef(fitY)
yhat = myModel(ydata, parmsY_hat)
#--------------------------------------------------------------------------------
logigisticDataFrame[:, :logEducation] = xhat
logigisticDataFrame[:, :logSeniority] = yhat
logigisticDataFrame[:, :Income] = income2DataFrame.Income
#--------------------------------------------------------------------------------
ols_lin_reg_3_8_2 =
lm(@formula(Income ~ 1 + logEducation * logSeniority), logigisticDataFrame)
#--------------------------------------------------------------------------------
ols_lin_reg_3_8_3 =
lm(@formula(Income ~ 1 + logEducation + logSeniority), logigisticDataFrame)
#--------------------------------------------------------------------------------
#=
ols_lin_reg_3_8_4 =
lm(@formula(Income ~ 1 + logEducation & logSeniority), logigisticDataFrame)
=#
#--------------------------------------------------------------------------------
ftest(ols_lin_reg_3_8_2.model, ols_lin_reg_3_8_3.model)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 92a345e4-fb65-4dc9-9803-0ba55e905ce0
md"
---
###### 3.8.5.2 Full Model 3.8.2 vs Restricted Model 3.8.4: Significance of Main Effects ?
The result of the F-Test full model 3.8.2 versus restricted model 3.8.3 demonstrates that we may *not* drop the main terms without loss of accounted variance.
"
# ╔═╡ ba8e0f69-fe65-4224-ba3a-7401e9e5c0cf
let
xdata = income2DataFrame.Education
ydata = income2DataFrame.Seniority
zdata = income2DataFrame.Income
#--------------------------------------------------------------------------------
logigisticDataFrame = DataFrame()
myModel(x, p) = p[1] .+ (p[2]-p[1]) .* (1 ./ ((1 .+ exp.(-p[3] .* (x .- p[4])))))
#--------------------------------------------------------------------------------
parmsX0 = [28.0, 88.0, 0.9, 16.0] # initial estimates of parms
myModel(xdata, parmsX0)
fitX = curve_fit(myModel, xdata, zdata, parmsX0)
parmsX_hat = coef(fitX)
xhat = myModel(xdata, parmsX_hat)
#--------------------------------------------------------------------------------
parmsY0 = [40.0, 78.0, 4.0, 70.0] # initial estimates of parms
myModel(ydata, parmsY0)
fitY = curve_fit(myModel, ydata, zdata, parmsY0)
parmsY_hat = coef(fitY)
yhat = myModel(ydata, parmsY_hat)
#--------------------------------------------------------------------------------
logigisticDataFrame[:, :logEducation] = xhat
logigisticDataFrame[:, :logSeniority] = yhat
logigisticDataFrame[:, :Income] = income2DataFrame.Income
#--------------------------------------------------------------------------------
ols_lin_reg_3_8_2 =
lm(@formula(Income ~ 1 + logEducation * logSeniority), logigisticDataFrame)
#--------------------------------------------------------------------------------
#=
ols_lin_reg_3_8_3 =
lm(@formula(Income ~ 1 + logEducation + logSeniority), logigisticDataFrame)
=#
#--------------------------------------------------------------------------------
ols_lin_reg_3_8_4 =
lm(@formula(Income ~ 1 + logEducation & logSeniority), logigisticDataFrame)
#--------------------------------------------------------------------------------
ftest(ols_lin_reg_3_8_2.model, ols_lin_reg_3_8_4.model)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 18ea4995-9f44-4057-b1b5-32752736fe21
md"
---
##### 3.9 Discussion
We discuss the results under the two aspects *prediction precision* and *parameter stability*:
- As the main result we achieved to *predict* the criteriums variable $Income$ to a very high degree though we did not know the underlying latent data generation process. The best model in this respect was *model 3.8.3* which was able to account $95.6%$ of criterium's variance. This model is a *two-step* model. In the *first* step we fitted *nonlinear univariate* OLS-regressions. In the *second* step we plugged in the nonlinear univariate model functions into the *linear multiple* OLS-regessions. The *first* two nonlinear univariate regressions needed $2 \cdot 4 = 8$ parameter and the *second* linear multiple regression had $3$ parameters. So the total two-step model implements $11$ parameters.
- Nearly the same precision is achieved by the full linear multiple *model 3.6.3*. It accounts $93.3\%$ of the criterium's variance with only $3$ parameters. Under the *statistical* perspective of parameter stability we prefer this model. It promises to be stable when fitting this model to other samples.
"
# ╔═╡ 7c695228-a9de-4dc3-aad4-dfd11680f898
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
"
# ╔═╡ bfa5c7d5-626d-4ed5-8584-29e78733b288
md"
---
##### end of ch. 2.1.2
"
# ╔═╡ 85161f10-dccd-43d6-bbda-af5becd725c8
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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