[markov_chain_I] Update animation for different initial distributions (#515)

* update markov_chain_animation

Dear John,

This pull request updates the Markov chain animation for different initial distributions.

In addition, this pull request also updates the descriptions of the code.

The previous pull request #492  can be closed as this one has no merge conflict.

Best,
Longye

* Revert "update markov_chain_animation"

This reverts commit f699174.

* Update markov_chains_I.md

* Update markov_chains_I.md

* Update markov_chains_I.md

Author: longye-tian

lectures/markov_chains_I.md

@@ -812,7 +812,7 @@ P = np.array([[0.971, 0.029, 0.000],
 P @ P
 ```
 
-Let's pick an initial distribution $\psi_0$ and trace out the sequence of distributions $\psi_0 P^t$ for $t = 0, 1, 2, \ldots$
+Let's pick an initial distribution $\psi_1, \psi_2, \psi_3$ and trace out the sequence of distributions $\psi_i P^t$ for $t = 0, 1, 2, \ldots$, for $i=1, 2, 3$.
 
 First, we write a function to iterate the sequence of distributions for `ts_length` period
 
@@ -829,26 +829,46 @@ def iterate_ψ(ψ_0, P, ts_length):
 Now we plot the sequence
 
 ```{code-cell} ipython3
-ψ_0 = (0.0, 0.2, 0.8)        # Initial condition
+:tags: [hide-input]
+
+ψ_1 = (0.0, 0.0, 1.0)
+ψ_2 = (1.0, 0.0, 0.0)
+ψ_3 = (0.0, 1.0, 0.0)                   # Three initial conditions
+colors = ['blue','red', 'green']   # Different colors for each initial point
+
+# Define the vertices of the unit simplex
+v = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]])
+
+# Define the faces of the unit simplex
+faces = [
+    [v[0], v[1], v[2]],
+    [v[0], v[1], v[3]],
+    [v[0], v[2], v[3]],
+    [v[1], v[2], v[3]]
+]
 
 fig = plt.figure()
 ax = fig.add_subplot(projection='3d')
 
-def update(n):
-    ψ_t = iterate_ψ(ψ_0, P, n+1)
-    
+def update(n):    
     ax.clear()
     ax.set_xlim([0, 1])
     ax.set_ylim([0, 1])
     ax.set_zlim([0, 1])
-    ax.view_init(30, 210)
+    ax.view_init(45, 45)
     
-    for i, point in enumerate(ψ_t):
-        ax.scatter(point[0], point[1], point[2], color='r', s=60, alpha=(i+1)/len(ψ_t))
+    simplex = Poly3DCollection(faces, alpha=0.03)
+    ax.add_collection3d(simplex)
     
+    for idx, ψ_0 in enumerate([ψ_1, ψ_2, ψ_3]):
+        ψ_t = iterate_ψ(ψ_0, P, n+1)
+        
+        for i, point in enumerate(ψ_t):
+            ax.scatter(point[0], point[1], point[2], color=colors[idx], s=60, alpha=(i+1)/len(ψ_t))
+            
     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)
+    ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='yellow', s=60)
     
     return fig,
 
@@ -860,9 +880,9 @@ HTML(anim.to_jshtml())
 Here
 
 * $P$ is the stochastic matrix for recession and growth {ref}`considered above <mc_eg2>`.
-* The highest red dot is an arbitrarily chosen initial marginal probability distribution  $\psi_0$, represented as a vector in $\mathbb R^3$.
-* The other red dots are the marginal distributions $\psi_0 P^t$ for $t = 1, 2, \ldots$.
-* The black dot is $\psi^*$.
+* The red, blue and green dots are initial marginal probability distributions  $\psi_1, \psi_2, \psi_3$, each of which is represented as a vector in $\mathbb R^3$.
+* The transparent dots are the marginal distributions $\psi_i P^t$ for $t = 1, 2, \ldots$, for $i=1,2,3.$.
+* The yellow dot is $\psi^*$.
 
 You might like to try experimenting with different initial conditions.
 
@@ -899,6 +919,8 @@ We can see similar phenomena in higher dimensions.
 The next figure illustrates this for a periodic Markov chain with three states.
 
 ```{code-cell} ipython3
+:tags: [hide-input]
+
 ψ_1 = (0.0, 0.0, 1.0)
 ψ_2 = (0.5, 0.5, 0.0)
 ψ_3 = (0.25, 0.25, 0.5)