PCM20230508_ISLR2_2.1.3_Nonlinear_Model_Functions.jl

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

using Markdown
using InteractiveUtils

# ╔═╡ 4f55d011-4a61-47ce-a2a4-7b4502df7733
using Plots, Symbolics, DifferentialEquations

# ╔═╡ 9e28d02c-2279-4e46-8d53-99009af90d6a
md"
==================================================================================
#### ISLR2\_2.1.3\_Nonlinear Model Functions 
##### file: PCM20230508\_ISLR2\_2.1.3\_Nonlinear\_Model\_Functions
##### code: (1.8.5/0.1\9.14) by PCM *** 2023/05/17 ***
==================================================================================
"

# ╔═╡ cd71c970-252d-4a36-b56e-816459392339
md"
---
##### 3.10 Appendix: Derivatives of Sigmoid (= Logistic) Functions

###### 3.10.1 Derivatives of the simple [*Sigmoid* (= *Logistic*) function](https://en.wikipedia.org/wiki/Logistic_function)
"

# ╔═╡ 4d3a4552-69d2-4117-ab4e-84a9dcf8e3f6
md"

The simple *sigmoid* (= *logistic*) model function $(0 < f(x) < 1)$ is:

$f(x) = \frac{1}{1+e^{-x}}=\frac{e^x}{e^x(1+e^{-x})}=\frac{e^x}{e^x+e^0}=\frac{e^x}{e^x+1},$ 


and


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

$\;$

$\;$

Applying the *quotient rule* the derivation is either

$f_1'(x) = \frac{0+e^{-x}}{(1+e^{-x})^2}=\frac{e^{-x}}{(1+e^{-x})^2}$

or

$f_2'(x) = \frac{d}{dx}f(x) = \frac{e^x(e^x+1)-e^xe^x}{(e^x+1)^2} = \frac{e^{2x}+e^x-e^{2x}}{(e^x+1)^2} = \frac{e^x}{(e^x+1)^2}.$

$\;$

$\;$

$\;$


The *second* case $f_2'(x)$ can be simplified to the product of two terms:



$f_2'(x) = \frac{e^x}{(e^x+1)^2} = \frac{e^x}{(e^x+1)(e^x+1)} = \frac{e^x}{(e^x+1)} \cdot \frac{1}{(e^x+1)} = f(x)(1-f(x)).$

$\;$

$\;$

$\;$

Another alternative is known as (WolframAlpha, 2023):

$f_3'(x) = \frac{1}{(e^x+1)} - \frac{1}{(e^x+1)^2}.$

$\;$

$\;$

The decomposition $f_2'(x) = f(x)(1-f(x))$ is known as the [*logistic differential equation*](https://en.wikipedia.org/wiki/Logistic_function) with boundary conditions

$\;$

$\;$

$f(0)= \frac{1}{2} \text{ and } f'(0) = \frac{1}{4}.$

$\;$

$\;$

The differential equation is used for the description of simple *growth* or *learning* processes with saturation.

"

# ╔═╡ db9ebcc0-a894-4eab-9560-aa5a1b35f4da
let
	x_k_max = 0
	L = 0
	plot(title="Simple Logistic Function f(x) = 1/(1 + exp(-x))")
	f(x) = exp(x-x_k_max)/(1+exp(x-x_k_max))
	plot!([x for x in -10:0.1:10], [f(x) for x in -10:0.1:10])
	plot!([x_k_max, x_k_max],[L, f(x_k_max)], legend=:none)        # vertical line
	plot!([-10, 0],[f(x_k_max), f(x_k_max)], legend=:none)         # horizntal line
end # let

# ╔═╡ 9ee7df29-b3d5-4943-8a8a-d75578fce41a
let
	plot(title="Derivative f'_1(x) of the Simple Logistic Function")
	fDeriv1(x) = exp(-x)/(1+exp(-x))^2
	fDeriv2(x) = exp(x)/(1+exp(x))^2
	fDeriv3(x) = 1/(exp(x)+1) - 1/(exp(x)+1)^2
	plot!([x for x in -10:0.1:10], [fDeriv1(x) for x in -10:0.1:10])
	plot!([0,0], [0,fDeriv1(0)])
end # let

# ╔═╡ 30e2428a-e5ba-4f34-bbf5-bcbe36be0c66
let
	plot(title="Derivative f'_2(x) of the Simple Logistic Function")
	fDeriv1(x) = exp(-x)/(1+exp(-x))^2
	fDeriv2(x) = exp(x)/(1+exp(x))^2
	fDeriv3(x) = 1/(exp(x)+1) - 1/(exp(x)+1)^2
	plot!([x for x in -10:0.1:10], [fDeriv2(x) for x in -10:0.1:10])
	plot!([0,0], [0,fDeriv2(0)])
end # let

# ╔═╡ b29aca1f-2b87-4878-8b9d-f4ba57e190e8
let
	plot(title="Derivative f'_3(x) of the Simple Logistic Function")
	fDeriv1(x) = exp(-x)/(1+exp(-x))^2
	fDeriv2(x) = exp(x)/(1+exp(x))^2
	fDeriv3(x) = 1/(exp(x)+1) - 1/(exp(x)+1)^2
	plot!([x for x in -10:0.1:10], [fDeriv3(x) for x in -10:0.1:10])
	plot!([0,0], [0,fDeriv3(0)])
end # let

# ╔═╡ f2902e87-2ce1-40a5-8c0d-a0c3f1e326b2
md"
---
###### 3.10.2 Derivative of the [*Generalized Sigmoid* (= *Logistic*) function](https://en.wikipedia.org/wiki/Generalised_logistic_function)

"

# ╔═╡ 258e9fb8-e087-43ed-8f44-7430c6adde6c
###### 3.7.2 Derivative of our [Generalized Logistic Function](https://en.wikipedia.org/wiki/Generalised_logistic_function) 

md"
---

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

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

$\;$

The original output of $expand\_derivatives(Dx(myModel))$ is:

$f'(x)=\frac{d}{dx}f(x)=0.58\left(\frac{59.36}{(1+e^{-0.58(-15.6+x)})^2} \right)\cdot e^{-0.58(-15.6+x)}.$

$\;$

$\;$

$\;$

When we rearrange terms the similarity with $f_2'(x)$ (above) becomes obvious.

$\;$

$f'(x)=\frac{d}{dx}f(x)=(0.58\cdot 59.36)\left(\frac{e^{-0.58(-15.6+x)}}{(1+e^{-0.58(-15.6+x)})^2} \right)$

$\;$

$\;$

$\;$

$=34.4288\left(\frac{e^{-0.58(-15.6+x)}}{(1+e^{-0.58(-15.6+x)})^2} \right)$

$\;$

$\;$

$\;$

with *boundary conditions*:

$\;$

$f(x_{k_{max}}) = 18.89 + (78.25-18.89)\cdot\frac{1}{2} = 48.57 \text{ and } f'(x_{k_{max}}) = 34.4288\cdot\frac{1}{4} = 8.6072$
"

# ╔═╡ d5958a90-bbe3-4adf-a7d4-6a1d71b73cd4
md"
The abstracted form of $f'(x)$ is:

$f'(x) = k \cdot (U-L)\cdot \left(\frac{e^{-k\left(x-x_{k_{max}}\right)}}{\left(1+e^{-k\left(x-x_{k_{max}}\right)}\right)^2} \right)$

$\;$

$\;$

with *boundary conditions*

$\;$

$\;$

$f(x_{k_{max}}) = L + (U-L)\cdot\frac{1}{2} \text{ and } f'(x_{k_{max}}) = k\cdot (U-L)\cdot\frac{1}{4}$

"

# ╔═╡ 79e83f54-ab82-4eb6-abc6-961d436b186c
let
	x_k_max = 15.65
	k = 0.58
	U = 78.25
	L = 18.89
	plot(title="Gen.Log.Fun. f_g(x) = L+(U-L)/(1 + exp(-k(x-x_k_max))")
	f(x) = L + (U-L)/(1 + exp(-k*(x - x_k_max)))
	plot!([x for x in 0:0.1:30], [f(x) for x in 0:0.1:30], legend=:none)
	plot!([x_k_max, x_k_max],[L, f(x_k_max)], legend=:none)        # vertical line
	plot!([0, x_k_max],[f(x_k_max), f(x_k_max)], legend=:none)     # horizntal line
	annotate!(26, 20, "Income1 Data")
end # let

# ╔═╡ 8d541667-ac3d-45d4-af22-869312c73185
let
	#--------------------------------------------------------------------------------
	plot(title="Derivative f'_g (= Logistic Diff. Equation)")
	@variables x
	L = 18.89        # Lower left asymptote
	U = 78.25        # Upper right asymptote
	k =  0.58        # max rate k
	x_k_max = 15.65  # arg_max k = that x for max k = x|max(k)
	#--------------------------------------------------------------------------------
	myModel = L + (U - L)*(1/((1 + exp(-k*(x - x_k_max)))))
	Dx = Differential(x)
	dmyModelToX = Dx(myModel)
	dglx = expand_derivatives(Dx(myModel))
	expExpression(x) = exp(-k*(-15.65+x))
	dgl_myModel(x) = 0.58*(59.36/(1+expExpression(x))^2)*expExpression(x)
	#--------------------------------------------------------------------------------
	plot!([x for x in 0.65:0.1:15.65*2], [dgl_myModel(x) for x in 0.65:0.1:15.65*2])
	plot!([x_k_max, x_k_max], [0, dgl_myModel(x_k_max)])
	annotate!(27.5, 1, "Income1 Data")
	#--------------------------------------------------------------------------------
end # let

# ╔═╡ 630eee61-efdf-4694-80b5-6ee12b9e6d00
md"
---
##### 3.11 Numeric Solutions of Logistic ODEs
###### 3.11.1 [Solution](https://docs.sciml.ai/DiffEqDocs/stable/getting_started/) of ODE Logistic Differential Equation

The solution was obtained by package [DifferentialEquations.jl](DifferentialEquations.jl).
"

# ╔═╡ b7c9d2da-5883-4332-a92a-04cb054cb912
let
	fLogistic = x -> 1/(1 + exp(-x))
	fDeriv(y, p, x) = exp(-x)/(1+exp(-x))^2
	y0 = 0.0
	tspan = (-10.0, 10.0)
	problem = ODEProblem(fDeriv, y0, tspan)
	solution = solve(problem, Tsit5(), reltol = 1e-8, abstol = 1e-8)
	#----------------------------------------------------------------------
	plot(solution, linewidth = 3, title = "Solution to the Logistic ODE", xaxis = "Time (t)", yaxis = "x(t) (in μm)", label = "Solution of Logistic ODE") 
	plot!(solution.t, fLogistic, lw = 3, ls = :dash, label = "True Solution!")
end # let

# ╔═╡ 017b7454-4c72-4c31-973d-53d6bab8f94c
md"
---
###### 3.11.2 [Solution](https://docs.sciml.ai/DiffEqDocs/stable/getting_started/) of ODE General Logistic Differential Equation

The solution was obtained by package [DifferentialEquations.jl](DifferentialEquations.jl).
"

# ╔═╡ ad019b34-3824-4048-a489-9e8254bdfbcd
let
	x_k_max = 15.65
	k = 0.58
	U = 78.25
	L = 18.89
	y0 = 18.89
	fLogistic = x -> L + (U-L)/(1 + exp(-k*(x - x_k_max)))
	expExpression(x) = exp(-k*(-15.65+x))
	dgl_myModel(y, p, x) = 0.58*(59.36/(1+expExpression(x))^2)*expExpression(x)
	y0 = 18.89
	tspan = (5.0, 25.0)
	problem = ODEProblem(dgl_myModel, y0, tspan)
	solution = solve(problem, Tsit5(), reltol = 1e-8, abstol = 1e-8)
	#----------------------------------------------------------------------
	plot(solution, linewidth = 3, title = "Solution to the General Logistic ODE", xaxis = "Time (t)", yaxis = "x(t) (in μm)", label = "Solution of General Logistic ODE") 
	plot!(solution.t, fLogistic, lw = 3, ls = :dash, label = "True Solution!")
end # let

# ╔═╡ bad16089-458c-46dd-882b-434bbd2ed449
md"
---
##### 3.12 Discussion

We could demonstrate that we 
- *estimated* by the ordinary least squares (OLS) parameter estimation technique parameters of nonlinear general logistic model functions 
- *derived* symbolically the ordinary differential equation (ODE) of the general logistic model function to obtain the growth model of $y(x) = Income(Education)$
- *integrated* the ODE numerically and got a numeric solution which is identical to the original model equation
- *used* packages $Symbolics.jl$ and $Differential Equations.jl$
 
"

# ╔═╡ 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
- **WolframAlpha**; Version 1.4.19.2022041167; last visit 23/05/04
"

# ╔═╡ bfa5c7d5-626d-4ed5-8584-29e78733b288
md"
---
##### end of ch. 2.1.3
"

# ╔═╡ 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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[[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"
"""

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