⚡ Pluto.jl ⚡

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using Plots

1.8 ms

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using LinearAlgebra

3.5 ms

A Brief Introduction to Markov Chains

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md"""
[A Brief Introduction to Markov Chains](https://office.fxrobot.kr/Products/Files/DocEditor.aspx?fileid=2258)
"""

32.6 μs

3×1 Adjoint{Float64,Array{Float64,2}}:
 0.0
 0.0
 1.0

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# Finite state-space markov chain example
​
# Transition Operator
#  S  F  R
#  U  O  A
#  N  G  I
#  N  G  N
#  Y  Y  Y
​
begin
    P = [0.8 0.15 0.05;  # Sunny
         0.4 0.5 0.1;    # Foggy
         0.1 0.3 0.6]'   # Rainy
    nWeeks = 25
​
    # Initial state is rainy
    X = [ .0  .0  1.]'
end

800 ns

3×3 Adjoint{Float64,Array{Float64,2}}:
 0.8   0.4  0.1
 0.15  0.5  0.3
 0.05  0.1  0.6

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P

0.0 ns

Float641

0.596489

0.263159

0.140353

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# Run markov chain
begin
    Xₖ = Array{Float64,2}(undef,(3,nWeeks))
    Xₖ[:,1] = X
    for k in 2:nWeeks
        Xₖ[:,k]=P*Xₖ[:,k-1]
    end
    Xₖ[:,end]
end
​

41.2 μs

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# Display
begin
    plot(1:nWeeks,Xₖ[1,:],linecolor=:red,label="Sunny",linewidth=2,
            xaxis="Week", yaxis="p(Weather)")
    plot!(1:nWeeks,Xₖ[2,:],linecolor=:black,label="Foggy",linewidth=2)
    plot!(1:nWeeks,Xₖ[3,:],linecolor=:blue,label="Rainy",linewidth=2)
    plot!([15, 15],[0, 1],linecolor=:green, linestyle=:dash,label="Burn In",linewidth=2)
end

3.3 ms