### A Pluto.jl notebook ###
# v0.19.22

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ c8f935c4-1afb-4ab8-a2c9-e1eea5040ab4
using Distributions, LinearAlgebra, Plots, PlotlyJS

# ╔═╡ c6719da9-a3f7-49ba-8f9c-800487a742a2
using AbstractGPs,Optim,StatsFuns

# ╔═╡ 86ad3341-6df1-4ac1-a114-647d33abd675
begin
	using PlutoUI
md"# Gaussian Process Kernels
GP with $\mu(x)=0$ and 
```math
K_{\text{SE}}(x,x')=\nu \exp\left( -\frac{1}{\ell}|| x-x'||^2 \right)+\sigma\\
```

Sensitivity to hyper-parameters:
- kernel $K()$: $(@bind Kf Select([SqExponentialKernel,Matern32Kernel]))
- prior scale $ν=$ :  $(@bind ν Slider((-5:5)))
- length-scale $\ell=$:  $(@bind ℓ Slider((-5:5)))
- noise variance $σ=$:  $(@bind σ Slider((-10:5)))
	"
end

# ╔═╡ 31917785-9e11-4a5c-81e3-a94966041ec1
html"<button onclick='present()'>present</button>"

# ╔═╡ 4d3bf546-4dab-42bb-8f2d-4a76917e42fd
html"""
<style>
	main {
		margin: 0 auto;
		max-width: 2000px;
    	padding-left: max(60px, 10%);
    	padding-right: max(60px, 10%);
	}
</style>
"""

# ╔═╡ 94dafc99-e291-4dd4-bbb7-620adf5928cb
md"
# Toy Problem #1
"

# ╔═╡ 1628bac8-62a2-44b9-b01d-5cf2bbc2c16b
begin g(x) = x.^2-2x.+2.0+σₑ*e[1:length(x)]; xg=-1:0.01:4; Plots.plotlyjs();e=randn(100);σₑ=0.3; α=1e-10;end;

# ╔═╡ 9c19426c-3bff-4fcc-ae4d-6f5eee5cd3b0
xl0=[0.,2.,3,1.,4,-1]; 

# ╔═╡ 388d5f34-9be8-4f7a-9614-eaa22fccc341
@bind dispfit MultiCheckBox(["LS", "Bayes"])

# ╔═╡ 95aa04f1-f614-4f23-b4ba-0145a8cf1f07
begin 
	y = g(xl0);
	plt=Plots.scatter(xl0,y,xlim=[-1,4],ylim=[-1,15],xlabel="x",ylabel="f(x)"); 
	if "LS" in dispfit
		Xlf(xl0) = hcat(xl0.^2,xl0,ones(size(xl0)));
		Xl = Xlf(xl0);
		Sₙ= inv(Xl'*Xl+α*I)
		θ  = Sₙ*Xl'*y;
		yₚ = Xlf(xg)*θ;
		plt=Plots.plot!(xg,yₚ)
	end
	if "Bayes" in dispfit
		stl=[1e-1 .+ (Xlf(x)*Sₙ*Xlf(x)')[1] for x in xg]
		plt=Plots.plot!(xg,yₚ,ribbon=stl, fillalpha=0.5)
	end
	plt
end

# ╔═╡ 087fb715-0dce-4f82-b102-d8630cfc74be
md" ## Polynomial fit
- From theory we know that any function can be locally approximated by a Polynomial

Fit by a quadratic function: 
```math 
\begin{array}{lcll}
{\color{green}y_{1}} & =a{\color{red}x_{1}^2} & +b{\color{red}x_1}+c{\color{red}1}, & +e_{1}\\
{\color{green}y_{2}} & =a{\color{red}x_{2}^2} & +b{\color{red}x_2}+c{\color{red}1} & +e_{2},\\
\vdots & \vdots & \quad\vdots & \quad\vdots
\end{array}
```
In matrix notation $\theta=[a,b,c]^{T}$:
```math
{\color{green}\mathbf{y}}={\color{red}X}\theta+\mathbf{e},
```
Minimize $\sum_{i}e_{i}^{2}=\mathbf{e}^{T}\mathbf{e}$
```math 
\begin{array}{r}
\frac{d(\mathbf{e}^{T}\mathbf{e})}{d\theta}	=0.\\
\frac{d}{d\theta}((\mathbf{y}-X\theta)^{T}(\mathbf{y}-X\theta))	=0\\
\frac{d}{d\theta}(\mathbf{y}^{T}\mathbf{y}-\theta^{T}X^{T}\mathbf{y}-\mathbf{y}^{T}X\theta+\theta^{T}X^{T}X\theta)	=0\\
-X^{T}\mathbf{y}+X^{T}X\theta	=0\\
\hat\theta = (X^TX)X^Ty\\
\end{array}
```"

# ╔═╡ 7a2335a8-c34a-4d4c-99f0-c9fac07d39d4
md"## Trust Region idea

- Polynom is valid only locally
    - The model is fitted only to data within a selected region
    - Points outside this region are penalized (or ignored)
- How to define the region
    - adaptive rules
    - enlarge if the model fits
    - shrink if the model does not fit
    - restarts at different initial points

- Levenberg–Marquardt algorithm
"

# ╔═╡ 49d14876-cb2a-11ed-0efb-95a6f4cc8ffa
begin f(x) =cos.(x*4.3.-0.1)./(abs.(x).+1);  end;

# ╔═╡ 5fa0263b-ce1d-4718-a523-3a4a9263c3fa
md"# Human-based optimization
Find minimum of a function given by point-wise evaluation"

# ╔═╡ 22276226-73fa-4e2d-98cf-72fc696bd68c
x0=[-1.,0.,1.,2.,3.,4.];

# ╔═╡ d97ff6a8-b9d7-426b-a2d6-e095c2164ffb
md"Add points to x0 by interaction with audience..."

# ╔═╡ a2476a4c-d89d-4537-b273-69efa94f97dc
xh=[x0...,1.3,0.8,0.7,0.5,-0.7,2.5,3.5,-0.5,-0.8];

# ╔═╡ 1e65ccc0-8e41-4913-bde6-23ce30abcbd2


# ╔═╡ af211146-587f-421e-a53a-a4f846951b22
Plots.scatter(xh,f(xh),ylims=[-0.7,1.1])

# ╔═╡ 9452b609-463d-4d1b-96fa-66991b6fbac0
md"# How fast it was?
Evolution of the best minima as a function of iterations "

# ╔═╡ 4dcf9ed0-c352-44bc-b8c1-e0207bd6ae08
function extractmins(x)
	minv=[];
	for i=1:length(x)
		push!(minv,minimum(f(x[1:i])))
	end
	minv
end;

# ╔═╡ fb715215-0c90-424f-9106-cfb1979d926b
Plots.plot(extractmins(xh[6:end]),xlabel="Iteration count",ylabel="minimum at iteration",label="humans")

# ╔═╡ badbf2d7-b535-47be-8fa9-7daa7b32309b
md"# Fast optimization as Decision making

Decision
: variable that we are free to select, $x$

Knowledge
: data acquired so far $D=[X,Y],X=[x_1,\ldots,x_n],Y=[y_1,\ldots,y_n],$

*Uncertainty*
: function value at $x$, $f(x)$

*Utility*
: useful outcome is when $f(x)<\min(X)$


Solution:
```math
x^* = \arg\max_{x\in\mathcal{X}} \int \chi(f(x)<\min(X)) p(f(x)|D) df(x)
```

"

# ╔═╡ 19b98088-31a6-4c14-9e75-52d671b6c061
md"# Gaussian process: distribution over functions
A time continuous function $\{f(x);x\in \mathcal{X}\}$ is a Gaussian process if for every finite set of indeces $\{x_1,\ldots,x_n\}$ are Gaussian distributed, given functions 
```math
\begin{aligned}
\text{mean:}\quad&\mu(x)\\
\text{kernel:}\quad&k(x,x')\\
\end{aligned}
```


Consider function values $f(x),f(x')$, for two points $x,x'$ with choices $\mu(x)=0$, $k(x,x)=\nu$, then:
```math
\begin{equation}
p(f(x),f(x')|\theta)=\mathcal{N}\left(
\begin{bmatrix}0\\0\end{bmatrix},
\begin{bmatrix}\nu+\sigma & k_\theta(x,x')\\
k_\theta(x,x') & \nu+\sigma\end{bmatrix}
\right),
\end{equation}
```



Gaussian distribution has very nice analytical properties:
```math
\begin{equation}
\begin{aligned}
p(f(x)|x,x_1,f(x_1))&=\mathcal{N}\left(\mu_{x_1}(x),\sigma_{x_1}(x)\right),\\
& \mu_{x_1}(x)= k(x,x_1)f(x_1)\\
& \sigma_{x_1}(x)= (\nu+\sigma-k(x,x_1)^{2}/(\nu+\sigma))\\

p(f(x))&=\mathcal{N}(0,\nu)
\end{aligned}
\end{equation}
```

Read:
- Rasmussen, Carl Edward, and Christopher KI Williams. Gaussian processes for machine learning. Vol. 1. Cambridge, MA: MIT press, 2006.
"


# ╔═╡ 8d62893a-bf53-4d97-aa8c-739c340446e1
function buildgp(x,params,Kf=SqExponentialKernel)
	ν,ℓ,σ = exp.(params)
	kernel = ν * (Kf() ∘ ScaleTransform(ℓ))
	f = GP(kernel)
	f(x, σ+1e-9)
end;

# ╔═╡ 7f4cc792-5cea-4216-871f-6310293ca4a6
begin xt=[x0...]; yt=f(xt); end;

# ╔═╡ 69b6467d-c92b-4807-8d16-05b792032ab7
testGP=buildgp(xt,[ν,ℓ,σ],Kf);

# ╔═╡ 929369b0-8ba6-44c8-a07c-4def5a1212b2
md"# Estimating hyper-parameters
The fit before was done for known valued of hyperparameters $\theta$, i.e. 
```math
\begin{equation}
p(f(x)|x_1,y_1,x,\theta)
\end{equation}
```
Since it is a proper likelihood, we can compute best parameter value $\hat{\theta}$ for given points
```math
\begin{equation}
 \hat\theta=\arg\min -\log p(Y|X,\theta) \end{equation}
```
where $X=[x_1,\ldots,x_n],Y=[y_1,\ldots,y_n]$

- poor but simple!
"

# ╔═╡ f8317740-5c8d-4edc-92ca-51cb131a9946
θ0 = [1.0,1.,-5];

# ╔═╡ 83439886-7e84-483d-9d27-24492377c51b
opt = Optim.optimize(p->-logpdf(buildgp(x0,p),f(x0)), θ0, LBFGS());

# ╔═╡ d367026a-b207-492c-a37d-92b029610f6c
gpf=posterior(buildgp(x0,opt.minimizer),f(x0));

# ╔═╡ d9bae228-8994-4cfb-bd0d-45246e743809
md"# Where it is most likely that the function has a minimum? 
In each point $x'$ the function value $f(x')$ has a Gaussian distribution 
```math
\begin{equation}
f(x)\sim G(\mu(x),\sigma(x))
\end{equation}
```

Probability of a value being lower than a threshold is an analytical formula:
```math
\begin{aligned}
P(f(x)<\min(X)|Y,X)&=\int_{-\inf}^{\min(X) } G(\mu_X(x),\sigma_X(x)) df(x)\\
& 	=(\min(X)-\mu_X(x))\text{cdf}(Z).+σ_X(x) \text{pdf}(Z)\\
Z & = (\min(X)-\mu_X(x))/σ_X(x)
\end{aligned}
```
"

# ╔═╡ 8392a879-0697-475c-af1c-c911b6ab7a54
begin ξ=-0.01; N01=Normal(); function EI(gp,xg,xb)
	μ,v=mean_and_var(gp,xg)
	σ=.√v
	Z=(-μ.+xb.+ξ)./σ
	EI=(-μ.+xb.+ξ).*cdf(N01,Z).+σ.*pdf.(N01,Z)
end; end

# ╔═╡ f1b69665-5e9e-48ed-9e72-37ffef263d84
begin 
	function plotBO(gpf,xv,plotEI=true)
		Plots.plot(xg, gpf; label=false, ribbon_scale=2)
		if plotEI
			acq=EI(gpf,xg,minimum(f(xv)));
			Plots.plot!(xg,0.5acq./maximum(acq).-1.0,label="E(f(x)<ymin)")
		end
		Plots.scatter!(xv,f(xv),label="data")
	end; 
	plotBO(gpf,x0,false)
end

# ╔═╡ df50a8bd-bb26-459b-9f92-7cf878cc36d2
plotBO(posterior(testGP,yt),xt,false)

# ╔═╡ 6959198f-6aa5-4e91-a6d7-6e85a3159eeb
plotBO(gpf,x0,true)

# ╔═╡ d7f47c33-1cbc-4fb9-8d54-64eae6bb63df
md"- Evaluate the point with highest probability of occurence of lower value."

# ╔═╡ e48fc2ff-fd6d-42c9-8743-f9190664e926
md"# Repeating the procedure for each point"

# ╔═╡ 0f1ecdd5-2adc-4881-a9ff-3faff6a3f7e3
function nextx(x,p0=[1.0,1.0,-5.])
	opt = Optim.optimize(p->-logpdf(buildgp(x,p),f(x)), p0, LBFGS())
	gpf = posterior(buildgp(x,opt.minimizer),f(x))
	xg  = -1:0.01:4
	poi = EI(gpf,xg,minimum(f(x)))
	xg[argmax(poi)], opt.minimizer, gpf
end

# ╔═╡ dfb08742-f5fe-447e-88bc-be71924c4f14
begin nx,p,ngp=nextx(x0); xv=[x0...,nx];X=[xv];P=[p];G=[ngp]; 
for i=1:20
	nx,p,ngp=nextx(X[i],P[i])
	xv=[X[i]...,nx]
	push!(X,xv)
	push!(P,p)
	push!(G,ngp)
end
end;

# ╔═╡ 2cbba696-4ec4-4028-b15c-24645051f85a
md"Step of GP: $(@bind st Slider(2:20))"

# ╔═╡ 706b99a1-8b08-4330-8bfc-7d81cbdaac80
plotBO(G[st],X[st-1])

# ╔═╡ f1af7a97-a3e6-4b91-9525-66b7f435ef38
md"# Is AI smarter than us? "

# ╔═╡ acca9fa4-df30-492c-b4eb-7b6d71c4d982
begin
	Plots.plot(extractmins(xh[6:end]),xlabel="Iteration count",ylabel="minimum at iteration",label="human")
	Plots.plot!(extractmins(X[end][6:end]),label="bayes")
end

# ╔═╡ 243adef4-4adf-4f22-a83b-3c37a5806d53
function plotParam(x,y,allY)
	p=Plots.scatter(x,y)
	μ=mean(allY);
	σ=std(allY);
	p=Plots.plot!(xg,μ)
	p=Plots.plot!(xg,μ.+5σ,linestyle=:dot)
	p=Plots.plot!(xg,μ.-5σ,linestyle=:dot)
	vY=var(allY);
	p=Plots.plot!(xg,vY./maximum(vY).-4)
end
		

# ╔═╡ 00000000-0000-0000-0000-000000000001
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StatsFuns = "4c63d2b9-4356-54db-8cca-17b64c39e42c"

[compat]
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+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.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+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.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[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 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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