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using Plots # for plotting trajectory
using Distributions # for fitting a normal distribution
function dynamics!(params) # ! notation tells us that arguments will be modified
(x_change_per_step, num_steps, num_points) = params
x = zeros(num_points) # array to store x positions after the random walks
for i in 1:num_points
# determine the direction of the step
# set starting point to x
x[i] = 0.0
# simulate the random walk
for _ in 1:num_steps
if rand() < 0.5
# go left, negative
dx = -x_change_per_step
x[i] += dx
else
# go right, positive
dx = x_change_per_step
x[i] += dx
end
end
end
μ = sum(x) / length(x)
σ = sqrt(sum((x .- μ).^2) / length(x))
normal = Normal(μ, σ)
println("μ: $μ, σ: $σ, n: $(length(x))")
return x
end
function plot_frequencies(x)
# make a map each positions to it's Frequency
freqs = Dict()
for xi in x
if haskey(freqs, xi)
freqs[xi] += 1
else
freqs[xi] = 1
end
end
num_points = length(x)
plt = scatter(
collect(keys(freqs)), collect(values(freqs)),
label="Frequency", xlabel="Position", ylabel="Frequency",
title="Freq of Positions ($num_points points)",
msw=0, color=:green, legend=false)
return plt
end
function plot_histogram(x)
# make a histogram of the random walk
num_points = length(x)
plt = histogram(x, xlabel="Position",
ylabel="Frequency", legend=false,
title="Histogram of Random Walk")
return plt
end
function fit_normal(x)
# fit a normal distribution to the random walk
μ = sum(x) / length(x)
σ = sqrt(sum((x .- μ).^2) / length(x))
normal = Normal(μ, σ)
# get the pdf's over the min and max values found
x_min = minimum(x)
x_max = maximum(x)
bigger = max(abs(x_min), abs(x_max)) # center the plot
x_range = range(-bigger, bigger, length=100)
pdf_vals = pdf.(normal, x_range)
# plot the normal distribution
return plot(x_range, pdf_vals, lw=2, color=:red, title="Normal Fit", legend=false)
end
function demo_1(num_points=1000)
# simulation parameters
x_change_per_step = 1.0 # step size
num_steps = 100 # number of steps in the random walk
params = (x_change_per_step, num_steps, num_points)
x = dynamics!(params) # run the random walk
# plot only the frequencies
p = plot_frequencies(x)
savefig(p, "figs/rw-demo1.png")
end
function demo_3()
# simulation parameters
x_change_per_step = 1.0 # step size
num_steps = 100 # number of steps in the random walk
# run the random walk with a smaller number of points
num_points = 15 # number of random walks
params = (x_change_per_step, num_steps, num_points)
x_1 = dynamics!(params) # run the random walk
# run a second random walk with 1000 points
num_points = 100
params = (x_change_per_step, num_steps, num_points)
x_2 = dynamics!(params) # run the random walk
# run a third random walk with 10000 points
num_points = 10000
params = (x_change_per_step, num_steps, num_points)
x_3 = dynamics!(params) # run the random walk
# plot the results
p0_1 = plot_frequencies(x_1)
p1_1 = plot_histogram(x_1)
p2_1 = fit_normal(x_1)
p0_2 = plot_frequencies(x_2)
p1_2 = plot_histogram(x_2)
p2_2 = fit_normal(x_2)
p0_3 = plot_frequencies(x_3)
p1_3 = plot_histogram(x_3)
p2_3 = fit_normal(x_3)
p = plot(
p0_1, p1_1, p2_1,
p0_2, p1_2, p2_2,
p0_3, p1_3, p2_3,
layout=(3, 3),
size=(1000, 1000),
)
savefig(p, "figs/rw-demo3.png")
end
println("Starting Random Speed Simulations...\n")
# demo_1(1000000) # plot a frequncy plot of param n points
demo_3()
println("\nRandom Walk Simulations Complete!")
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