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cost_function.jl
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### A Pluto.jl notebook ###
# v0.20.5
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)
#! format: off
return 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
#! format: on
end
# ╔═╡ 10c7a648-1488-11eb-0f9c-c3bfe711a449
md" ### Nomenclatura"
# ╔═╡ 10170496-1488-11eb-0882-df72e258d63c
md"Seja $X$ uma matriz com dimensões $n x m$ com $n$ linhas e $m$ colunas que contém nossos exemplos. $X_i^{(j)}$ corresponde ao elemento de $X$ em que $i$ é uma linha da matriz (exemplo) e $j$ uma coluna da matriz (atributo), cujo acesso na linguagem representamos por x[i,j]. Para pegarmos um vetor linha desta matriz (exemplo) representamos formalmente por $X_i$ e na linguagem por x[i,:]."
# ╔═╡ 3996c40e-1489-11eb-266a-95abbb9cd199
md"Seja $\theta$ um vetor linha com $m$ elementos tal que $m$ corresponde ao número de atributos mais um. $\theta = [\theta^{(0)},\theta^{(1)}, ...,\theta^{(m)}]$ em que cada elemento corresponde ao j-ésimo parâmetro associado ao j-ésimo atributo. Na nossa lingugaem acessamos um elemento por $\theta[i]$ e o vetor por $\theta$"
# ╔═╡ 391d5e2a-1489-11eb-15d4-77bf4828cfcd
md"Seja $y$ nosso vetor alvo em que representamos o i-ésimo elemento por $y_i$. Na nossa linguagem representamos por $y[i]$."
# ╔═╡ 794e1306-1487-11eb-203c-37766708b1a2
md" ### Hipótese e modelo linear"
# ╔═╡ 7882a5fe-1487-11eb-3f45-fd54863cdfbd
md"$H_{\theta}(X_i) = \sum_{j}^{m}{x_{i}^{(j)} \theta^{(j)}} = X_i\theta^{T}$"
# ╔═╡ 272f710e-1f99-11eb-21f0-d98d7f2445c8
md"Exemplificando o produto de vetores, nossa hipótese faria o seguinte cálculo para um exemplo $X_i = \begin{bmatrix} 1 & 500 \end{bmatrix}$ e um vetor de parâmetros $\theta = \begin{bmatrix}
0.5 \\
1.0
\end{bmatrix}$:"
# ╔═╡ 2821e75e-1f99-11eb-2f10-831050151433
md"e.g. $H_{\theta}(X_i) = \begin{bmatrix}
1 & 500
\end{bmatrix}\begin{bmatrix}
0.5 \\
1.0
\end{bmatrix} = 500.5$"
# ╔═╡ 295c3c20-1f99-11eb-20df-1b793c23e91f
md"Note que o resultado é um escalar, pois estamos estimando um valor contínuo (problema de regressão)."
# ╔═╡ f6213c24-1486-11eb-0981-ad6addc2a8ef
md" ### Função de custo"
# ╔═╡ d8c3e0d8-1485-11eb-3eab-a11e3d31269f
md"$J(\theta)= \frac{1}{2n}\sum_{i=1}^{n}{(H_{\theta}(X_i) - y_i)²} = \frac{1}{2n}\sum_{i=1}^{n}{(X_i\theta^T - y_i)²} = J_2$"
# ╔═╡ be81b7b2-148b-11eb-222c-135499b083cf
md"que pode ser expandido ainda mais na forma: "
# ╔═╡ d16ba126-148b-11eb-388b-ed88f5dc2099
md"$= \frac{1}{2n}\sum_{i=1}^{n}{(\sum_{j=1}^{m}X_{i}^{(j)}\theta_{j} - y_i)²} = J_1$"
# ╔═╡ 1980e48e-148d-11eb-282a-155aa6620659
md"Podemos ainda calcular toda a função de custo apenas por operações de matrizes e vetores sem um laço de repetição explícito da seguinte forma:"
# ╔═╡ b3ed93fc-1406-11eb-34d4-31acb0c7d641
begin
import Statistics: mean
import BenchmarkTools: @benchmark, @btime
import StaticArrays: @SMatrix
μ = mean
end
# ╔═╡ 35a275a0-1402-11eb-3ce4-fba2f21b24d6
function J₁(Θ,χ,γ)
n,m = size(χ)
∑ = 0.0
for i=1:n
h = 0.0
for j=1:m
h += Θ[j] * χ[i,j]
end
∑ += (h - γ[i])^2
end
.5∑/n
end
# ╔═╡ 1c4d435e-140d-11eb-1585-476688f1ec0c
function J₂(Θ,χ,γ)
n,m = size(χ)
∑ = 0.0
for (x,y) in zip(eachrow(χ),γ)
∑ += (Θ*x .- y).^2 ...
end
.5∑/n
end
# ╔═╡ 329221e4-1498-11eb-0569-d9bde9e62e1a
md" É possível ainda vetorizar todo o cálculo sem precisar fazer um loop explícito. Desta maneira, geramos em uma multiplicação de matrizes, um vetor de predições com a mesma dimensão da saída. Após isto, subtraímos elemento a elemento. Após isto, elevamos cada elemento ao quadrado e depois tiramos uma média."
# ╔═╡ 19392bd6-1405-11eb-3022-bbaecf23c544
J₃(Θ,χ,γ) = .5μ((χ*Θ' .- γ).^2)
# ╔═╡ 7b5e9f88-1498-11eb-2acf-a139d4c8e8fb
md" Abaixo temos um exemplo de teste da função de custo."
# ╔═╡ 48f06c34-1405-11eb-067e-cbe31a8e3cf3
begin
Θ = @SMatrix [0. 4.]
χ = @SMatrix [1 50.;1 60.;1 100.; 1 200.]
γ = @SMatrix [200 ; 240 ; 400; 800]
J₁(Θ,χ,γ),J₂(Θ,χ,γ),J₃(Θ,χ,γ)
end
# ╔═╡ 1c69de24-1499-11eb-1151-bbc0bf07a89c
md"### Medindo o tempo computacional"
# ╔═╡ 93db1528-1498-11eb-035d-efef99b051ed
md"Podemos medir o tempo de cada função a partir de um benchmark. É importante ressaltar que diversas otimizações podem ser testadas para melhorar o tempo. Uma forma seria trabalhar com a matriz $X$ transposta e $\theta$ como vetor coluna. Isto porque Julia representa uma matriz em ordem de coluna (como Matlab, R, Fortran, ...) devido à compatibilidade com libs de fortran dierentemente de outras linguagens como C, C++, ... Há ainda otimizações com vetores estáticos, macros como @inbounds, ... tudo isto pode acelerar ainda mais seu código."
# ╔═╡ cdfe994c-1415-11eb-01c0-13fd220aaa92
@benchmark J₁(Θ,χ,γ) seconds=1 gctrial=true
#@benchmark J₂(Θ,χ,γ) seconds=1 gctrial=true
#@benchmark J₃(Θ,χ,γ) seconds=1 gctrial=true
# ╔═╡ 7ee07dd4-149d-11eb-1f6d-2d565ee61a12
md" ### E se fosse você escolhesse os valores de $\theta$ ?"
# ╔═╡ 7fff1162-149d-11eb-2ec6-c3ad197a6d4c
# ╔═╡ 6aa7e186-14a2-11eb-287d-35f60c88cc0b
@bind θ₀ html"<input type='range' step='0.1' min='-10' max='10'>"
# ╔═╡ d357e744-14a2-11eb-2652-c35400acb3cf
@bind θ₁ html"<input type='range' step='0.1' min='-10' max='10'>"
# ╔═╡ 150cec5a-14aa-11eb-2be7-390b12287032
begin
using Plots;
x = range(1,stop=300,length=300)
n = length(x)
y = zeros(n)
for i=1:n
y[i] = θ₀ + θ₁*x[i]
#y[i] = θ₀ + θ₁^2*x[i]
#y[i] = θ₀ + θ₁*x[i]^2
end
plot(x,y,xlims = (0,300),ylims = (0,900),xlabel="Metros",ylabel="Preço",label="estimado");
scatter!(χ[:,2],γ[:],xlims = (0,300),ylims = (0,900),label="exemplo");
end
# ╔═╡ 8ef513d8-14a2-11eb-3f6e-5b2f0a09363e
θ = [θ₀ θ₁]
# ╔═╡ 7c888c54-14a5-11eb-315f-3d32578eaf32
md"##### Erro : $(J₁([θ₀ θ₁],χ,γ))"
# ╔═╡ bfd7693a-14a5-11eb-3479-057e03ef43f4
md" ### Gráfico do modelo linear"
# ╔═╡ f17bfc22-14b3-11eb-319a-47522cda243c
md"### Curva de erro"
# ╔═╡ 3ac52d04-14b4-11eb-17cf-1b2f8531335d
begin
e = []
for i=-1:1:10
for j=1:1:10
append!(e,J₁([i j],χ,γ))
end
end
println("e = ",e)
scatter(1:1:10,e,xlabel="θ₁",ylabel="J",label="erro") # apenas com relação à θ₁. para fazer com θ₀ teríamos outro eixo e teríamos uma superfície.
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
BenchmarkTools = "6e4b80f9-dd63-53aa-95a3-0cdb28fa8baf"
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StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
[compat]
BenchmarkTools = "~1.6.0"
Plots = "~1.40.11"
StaticArrays = "~1.9.13"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
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