Edits to MC lecture

[jstac]

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[jstac]

@HumphreyYang , could you please use the logic in

ψ = (0.0, 0.2, 0.8)        # Initial condition

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')

ax.set(xlim=(0, 1), ylim=(0, 1), zlim=(0, 1),
       xticks=(0.25, 0.5, 0.75),
       yticks=(0.25, 0.5, 0.75),
       zticks=(0.25, 0.5, 0.75))

x_vals, y_vals, z_vals = [], [], []
for t in range(20):
    x_vals.append(ψ[0])
    y_vals.append(ψ[1])
    z_vals.append(ψ[2])
    ψ = ψ @ P

ax.scatter(x_vals, y_vals, z_vals, c='r', s=60)
ax.view_init(30, 210)

mc = qe.MarkovChain(P)
ψ_star = mc.stationary_distributions[0]
ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=60)

plt.show()

in the two code cells that follow it --- see https://63fbdf37cab3e348c2918816--taupe-gaufre-c4e660.netlify.app/markov_chains.html

Perhaps you can unify these three code cells at least partially by writing a function that updates according to $\psi = \psi P$ and returns the time series of distributions $\psi, \psi P, \psi P^2, \ldots$?

Then you can extract coordinates for the plots.

Thanks.

[HumphreyYang]

Many thanks @jstac for your review.

I have integrated these code blocks with a function for simulating $\psi s$ and $\psi_0 s$

Please let me know if it needs further changes.

[jstac]

@HumphreyYang I suggest we avoid the use of the word "simulate" here and in the name of the function, since there is no randomness involved. (We are just iterating on the deterministic vector difference equation $\psi_{t+1} = \psi_t P$.)

[HumphreyYang]

Many thanks @jstac for picking this up. I have changed the function name to iterate_ψ.

[jstac]

Thanks @HumphreyYang for your quick response and nice update. Please see the comment above.

[jstac]

Nice work @HumphreyYang . Merging.

[HumphreyYang]

Hi @jstac,

Thanks for reminding me about the consistency in notation. It helps a lot to keep the code clear.

I just noticed that you were avoiding the use of ψ_0 in this function for initial distribution. I can see why this is preferred given no other $\psi$s are used, so _0 is redundant, but I also think it would be good if the notation for initial distributions is treated the same throughout this lecture. Please kindly let me know what you think about changing all initial distributions to ψ_0.

Many thanks in advance.

[jstac]

Hi @HumphreyYang , I changed that one to be consistent with the mathematics above.

But you are right --- the code is now inconsistent with code that follows. Shall we change it back and then change the maths above it to use $\psi_0$ instead of $\psi$?

[jstac]

BTW, I appreciate you raising this. Getting all the small things right is essential to creating a great lecture.

[HumphreyYang]

Hi @HumphreyYang , I changed that one to be consistent with the mathematics above.

But you are right --- the code is now inconsistent with code that follows. Shall we change it back and then change the maths above it to use ψ0 instead of ψ?

Many thanks @jstac; I think changing to $ψ_0$ would be great. I will create a PR to check and change the math notation.