INTRODUCTION_TO_DYNAMICS/Tools_Techniques (#270)

* First_go_(the dictionaries_in_graphs_are formated)

* Apply formatting to Markdown files

* First_go_issue_250

* Apply formatting to Markdown files

* changed_myst

* ran_julia_formatter

* changed_format_file

* ran_for_zeta_changes

* added_to_myst_file_some_formatting

* ran_formatting

* removed_''

* empty_push

* Apply formatting to Markdown files

* added_epislon_to_unicode

* add_lambda_and_rho

* missed_changes

---------

Co-authored-by: GitHub Action <[email protected]>

Author: Jonah-Heyl

format_myst.jl

@@ -49,14 +49,15 @@ function format_myst(input_file_path, output_file_path, extra_replacements = fal
             if ignore_errors
                 return m # i.e., don't change anything
             else
-                throw(e)
+                throw(e) 
             end
         end
     end
 
     # Additional replacements are optional.  This may be useful when replacing variable names to make it easier to type in ascii
     replacements = Dict("α" => "alpha", "β" => "beta", "γ" => "gamma", "≤" => "<=",
-    "≥" => ">=", "Σ" => "Sigma", "σ" => "sigma","μ"=>"mu") 
+    "≥" => ">=", "Σ" => "Sigma", "σ" => "sigma","μ"=>"mu","ϕ"=>"phi","ψ"=>"psi","ϵ"=>"epsilon",
+    "δ"=>"delta","θ" => "theta","ζ"=>"zeta","X̄" => "X_bar","p̄" => "p_bar","x̂" => "x_hat","λ"=>"lambda","ρ"=>"rho") 
 
     # Replace the code blocks in the content and handle exceptions
     try

lectures/introduction_dynamics/ar1_processes.md

@@ -149,7 +149,7 @@ c = 0.5
 
 # initial conditions mu_0, v_0
 mu = -3.0
-v = 0.6 
+v = 0.6
 ```
 
 Here's the sequence of distributions:
@@ -165,7 +165,7 @@ for t in 1:sim_length
     mu = a * mu + b
     v = a^2 * v + c^2
     dist = Normal(mu, sqrt(v))
-    plot!(plt, x_grid, pdf.(dist, x_grid), label=L"\psi_{%$t}", linealpha=0.7)
+    plot!(plt, x_grid, pdf.(dist, x_grid), label = L"\psi_{%$t}", linealpha = 0.7)
 end
 plt
 ```
@@ -177,15 +177,15 @@ Notice that, in the figure above, the sequence $\{ \psi_t \}$ seems to be conver
 This is even clearer if we project forward further into the future:
 
 ```{code-cell} julia
-function plot_density_seq(mu_0=-3.0, v_0=0.6; sim_length=60)
+function plot_density_seq(mu_0 = -3.0, v_0 = 0.6; sim_length = 60)
     mu = mu_0
     v = v_0
     plt = plot()
     for t in 1:sim_length
         mu = a * mu + b
         v = a^2 * v + c^2
         dist = Normal(mu, sqrt(v))
-        plot!(plt, x_grid, pdf.(dist, x_grid), label=nothing, linealpha=0.5)
+        plot!(plt, x_grid, pdf.(dist, x_grid), label = nothing, linealpha = 0.5)
     end
     return plt
 end
@@ -236,7 +236,7 @@ plt = plot_density_seq(3.0)
 mu_star = b / (1 - a)
 std_star = sqrt(c^2 / (1 - a^2))  # square root of v_star
 psi_star = Normal(mu_star, std_star)
-plot!(plt, x_grid, psi_star, color = :black, label=L"\psi^*", linewidth=2)
+plot!(plt, x_grid, psi_star, color = :black, label = L"\psi^*", linewidth = 2)
 plt
 ```
 
@@ -447,7 +447,7 @@ c = 0.5
 mu_star = b / (1 - a)
 std_star = sqrt(c^2 / (1 - a^2))  # square root of v_star
 
-function sample_moments_ar1(k, m=100_000, mu_0=0.0, sigma_0=1.0; seed=1234)
+function sample_moments_ar1(k, m = 100_000, mu_0 = 0.0, sigma_0 = 1.0; seed = 1234)
     Random.seed!(seed)
     sample_sum = 0.0
     x = mu_0 + sigma_0 * randn()
@@ -480,8 +480,8 @@ for (k_idx, k) in enumerate(k_vals)
 end
 
 plt = plot()
-plot!(plt, k_vals, true_moments, label="true moments")
-plot!(plt, k_vals, sample_moments, label="sample moments")
+plot!(plt, k_vals, true_moments, label = "true moments")
+plot!(plt, k_vals, sample_moments, label = "sample moments")
 plt
 ```
 
@@ -498,13 +498,13 @@ end
 ```
 
 ```{code-cell} julia
-function plot_kde(ϕ, n, plt, idx; x_min=-0.2, x_max=1.2)
-    x_data = rand(ϕ, n)
+function plot_kde(phi, n, plt, idx; x_min = -0.2, x_max = 1.2)
+    x_data = rand(phi, n)
     x_grid = range(-0.2, 1.2, length = 100)
     c = std(x_data)
-    h = 1.06 * c * n^(-1/5)
-    plot!(plt[idx], x_grid, f.(x_grid, Ref(x_data), h), label="estimate")
-    plot!(plt[idx], x_grid, pdf.(ϕ, x_grid), label="true density")
+    h = 1.06 * c * n^(-1 / 5)
+    plot!(plt[idx], x_grid, f.(x_grid, Ref(x_data), h), label = "estimate")
+    plot!(plt[idx], x_grid, pdf.(phi, x_grid), label = "true density")
     return plt
 end
 ```
@@ -513,8 +513,8 @@ end
 n = 500
 parameter_pairs = [(2, 2), (2, 5), (0.5, 0.5)]
 plt = plot(layout = (3, 1))
-for (idx, (α, β)) in enumerate(parameter_pairs)
-    plot_kde(Beta(α, β), n, plt, idx)
+for (idx, (alpha, beta)) in enumerate(parameter_pairs)
+    plot_kde(Beta(alpha, beta), n, plt, idx)
 end
 plt
 ```
@@ -530,14 +530,14 @@ Here is our solution:
 a = 0.9
 b = 0.0
 c = 0.1
-μ = -3
+mu = -3
 s = 0.2
 
-μ_next = a * μ + b
+mu_next = a * mu + b
 s_next = sqrt(a^2 * s^2 + c^2)
 
-ψ = Normal(μ, s)
-ψ_next = Normal(μ_next, s_next)
+psi = Normal(mu, s)
+psi_next = Normal(mu_next, s_next)
 ```
 
 ```{code-cell} julia
@@ -550,17 +550,18 @@ end
 
 ```{code-cell} julia
 n = 2000
-x_draws = rand(ψ, n)
+x_draws = rand(psi, n)
 x_draws_next = a * x_draws .+ b + c * randn(n)
 c = std(x_draws_next)
-h = 1.06 * c * n^(-1/5)
+h = 1.06 * c * n^(-1 / 5)
 
-x_grid = range(μ - 1, μ + 1, length = 100)
+x_grid = range(mu - 1, mu + 1, length = 100)
 plt = plot()
 
-plot!(plt, x_grid, pdf.(ψ, x_grid), label=L"$\psi_t$")
-plot!(plt, x_grid, pdf.(ψ_next, x_grid), label=L"$\psi_{t+1}$")
-plot!(plt, x_grid, f.(x_grid, Ref(x_draws_next), h), label=L"estimate of $\psi_{t+1}$")
+plot!(plt, x_grid, pdf.(psi, x_grid), label = L"$\psi_t$")
+plot!(plt, x_grid, pdf.(psi_next, x_grid), label = L"$\psi_{t+1}$")
+plot!(plt, x_grid, f.(x_grid, Ref(x_draws_next), h),
+      label = L"estimate of $\psi_{t+1}$")
 
 plt
 ```

lectures/introduction_dynamics/finite_markov.md

@@ -256,7 +256,7 @@ function mc_sample_path(P; init = 1, sample_size = 1000)
     X[1] = init # set the initial state
 
     for t in 2:sample_size
-        dist = dists[X[t-1]] # get discrete RV from last state's transition distribution
+        dist = dists[X[t - 1]] # get discrete RV from last state's transition distribution
         X[t] = rand(dist) # draw new value
     end
     return X
@@ -291,7 +291,7 @@ Random.seed!(42);  # for result reproducibility
 ```{code-cell} julia
 P = [0.4 0.6; 0.2 0.8]
 X = mc_sample_path(P, sample_size = 100_000); # note 100_000 = 100000
-μ_1 = count(X .== 1)/length(X) # .== broadcasts test for equality. Could use mean(X .== 1)
+mu_1 = count(X .== 1) / length(X) # .== broadcasts test for equality. Could use mean(X .== 1)
 ```
 
 ```{code-cell} julia
@@ -321,7 +321,7 @@ Random.seed!(42);  # For reproducibility
 P = [0.4 0.6; 0.2 0.8];
 mc = MarkovChain(P)
 X = simulate(mc, 100_000);
-μ_2 = count(X .== 1)/length(X) # or mean(x -> x == 1, X)
+mu_2 = count(X .== 1) / length(X) # or mean(x -> x == 1, X)
 ```
 
 ```{code-cell} julia
@@ -331,7 +331,7 @@ tags: [remove-cell]
 @testset "QE Sample Path Test" begin
     @test P ≈ [0.4 0.6; 0.2 0.8] # Make sure the primitive doesn't change.
     @test X[1:5] == [2, 2, 2, 2, 2]
-    #test μ_1 ≈ μ_2 atol = 1e-2
+    #test mu_1 ≈ mu_2 atol = 1e-2
 end
 ```
 
@@ -704,9 +704,9 @@ As seen in {eq}`fin_mc_fr`, we can shift probabilities forward one unit of time
 Some distributions are invariant under this updating process --- for example,
 
 ```{code-cell} julia
-P = [.4 .6; .2 .8];
-ψ = [0.25, 0.75];
-ψ' * P
+P = [0.4 0.6; 0.2 0.8];
+psi = [0.25, 0.75];
+psi' * P
 ```
 
 Such distributions are called **stationary**, or **invariant**.
@@ -801,7 +801,7 @@ Hence we need to impose the restriction that the solution must be a probability
 A suitable algorithm is implemented in [QuantEcon.jl](http://quantecon.org/quantecon-jl) --- the next code block illustrates
 
 ```{code-cell} julia
-P = [.4 .6; .2 .8];
+P = [0.4 0.6; 0.2 0.8];
 mc = MarkovChain(P);
 stationary_distributions(mc)
 ```
@@ -824,7 +824,7 @@ P = [0.971 0.029 0.000
      0.145 0.778 0.077
      0.000 0.508 0.492] # stochastic matrix
 
-ψ = [0.0 0.2 0.8] # initial distribution
+psi = [0.0 0.2 0.8] # initial distribution
 
 t = 20 # path length
 x_vals = zeros(t)
@@ -833,18 +833,18 @@ z_vals = similar(x_vals)
 colors = [repeat([:red], 20); :black] # for plotting
 
 for i in 1:t
-    x_vals[i] = ψ[1]
-    y_vals[i] = ψ[2]
-    z_vals[i] = ψ[3]
-    ψ = ψ * P # update distribution
+    x_vals[i] = psi[1]
+    y_vals[i] = psi[2]
+    z_vals[i] = psi[3]
+    psi = psi * P # update distribution
 end
 
 mc = MarkovChain(P)
-ψ_star = stationary_distributions(mc)[1]
-x_star, y_star, z_star = ψ_star # unpack the stationary dist
+psi_star = stationary_distributions(mc)[1]
+x_star, y_star, z_star = psi_star # unpack the stationary dist
 plt = scatter([x_vals; x_star], [y_vals; y_star], [z_vals; z_star], color = colors,
               gridalpha = 0.5, legend = :none)
-plot!(plt, camera = (45,45))
+plot!(plt, camera = (45, 45))
 ```
 
 ```{code-cell} julia
@@ -1234,20 +1234,20 @@ Random.seed!(42);  # For reproducibility
 ```
 
 ```{code-cell} julia
-α = 0.1 # probability of getting hired
-β = 0.1 # probability of getting fired
+alpha = 0.1 # probability of getting hired
+beta = 0.1 # probability of getting fired
 N = 10_000
-p̄ = β / (α + β) # steady-state probabilities
-P = [1 - α   α
-     β   1 - β] # stochastic matrix
+p_bar = beta / (alpha + beta) # steady-state probabilities
+P = [1-alpha alpha
+     beta 1-beta] # stochastic matrix
 mc = MarkovChain(P)
 labels = ["start unemployed", "start employed"]
 y_vals = Array{Vector}(undef, 2) # sample paths holder
 
 for x0 in 1:2
     X = simulate_indices(mc, N; init = x0) # generate the sample path
-    X̄ = cumsum(X .== 1) ./ (1:N) # compute state fraction. ./ required for precedence
-    y_vals[x0] = X̄ .- p̄ # plot divergence from steady state
+    X_bar = cumsum(X .== 1) ./ (1:N) # compute state fraction. ./ required for precedence
+    y_vals[x0] = X_bar .- p_bar # plot divergence from steady state
 end
 
 plot(y_vals, color = [:blue :green], fillrange = 0, fillalpha = 0.1,