add files via upload

7 files changed, 216 insertions, 0 deletions

diff --git a/figs/rs-1dvels.png b/figs/rs-1dvels.png
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index 0000000..c4c06d4
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+++ b/figs/rs-1dvels.png
diff --git a/figs/rs-3dvels.png b/figs/rs-3dvels.png
new file mode 100644
index 0000000..8dae989
--- /dev/null
+++ b/figs/rs-3dvels.png
diff --git a/figs/rs-speeds.png b/figs/rs-speeds.png
new file mode 100644
index 0000000..6110329
--- /dev/null
+++ b/figs/rs-speeds.png
diff --git a/figs/rw-demo1.png b/figs/rw-demo1.png
new file mode 100644
index 0000000..7e64a49
--- /dev/null
+++ b/figs/rw-demo1.png
diff --git a/figs/rw-demo3.png b/figs/rw-demo3.png
new file mode 100644
index 0000000..09e8623
--- /dev/null
+++ b/figs/rw-demo3.png
diff --git a/random-speed.jl b/random-speed.jl
new file mode 100644
index 0000000..0af9121
--- /dev/null
+++ b/random-speed.jl
@@ -0,0 +1,69 @@
+using Plots
+using Distributions
+
+num_velocities = 1000
+
+println("Starting Random Speed Simualtions...\n")
+
+function make_random_velocity()
+    # pull 3 nums randomly from normal distribution
+    N = Normal(0, 1)
+    return (rand(N), rand(N), rand(N))
+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 ($num_points points)",
+        msw=0, color=:green, legend=false)
+    return plt
+end
+
+
+# make velocities
+velocities = [make_random_velocity() for _ in 1:num_velocities]
+
+# plot these velocities on 1d plots
+px = histogram([v[1] for v in velocities], bins=30, title="X Velocity")
+py = histogram([v[2] for v in velocities], bins=30, title="Y Velocity")
+pz = histogram([v[3] for v in velocities], bins=30, title="Z Velocity")
+
+# print the mean and standard deviation of each velocity distribution
+println("X->\tμ:$(mean([v[1] for v in velocities])), σ:$(std([v[1] for v in velocities])), n:$(length([v[1] for v in velocities]))")
+println("Y->\tμ:$(mean([v[2] for v in velocities])), σ:$(std([v[2] for v in velocities])), n:$(length([v[2] for v in velocities]))")
+println("Z->\tμ:$(mean([v[3] for v in velocities])), σ:$(std([v[3] for v in velocities])), n:$(length([v[3] for v in velocities]))")
+
+p = plot(px, py, pz, layout=(3, 1), size=(800, 800))
+savefig(p, "figs/rs-1dvels.png")
+
+# plot these velocities in 3d space
+p = Plots.scatter(
+    [v[1] for v in velocities], 
+    [v[2] for v in velocities], 
+    [v[3] for v in velocities], 
+    title="Velocities")
+savefig(p, "figs/rs-3dvels.png")
+
+# plot their speeds
+speeds = [sqrt(v[1]^2 + v[2]^2 + v[3]^2) for v in velocities]
+p = histogram(
+    speeds, title="Randomly Generated Speeds (n=$num_velocities)", 
+    legend=false, xlabel="Speed", ylabel="Frequency")
+savefig(p, "figs/rs-speeds.png")
+# print the mean and standard deviation of the speed distribution
+println("\nSpeed->\tμ:$(mean(speeds)), σ:$(std(speeds)), n:$(length(speeds))")
+
+
+println("\nRandom Speed Simualtions Complete!")
\ No newline at end of file
diff --git a/random-walk.jl b/random-walk.jl
new file mode 100644
index 0000000..51e58b8
--- /dev/null
+++ b/random-walk.jl
@@ -0,0 +1,147 @@
+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!")