Revert "update markov_chain_animation"

This reverts commit f699174.

Author: longye-tian

lectures/markov_chains_I.md

@@ -812,48 +812,43 @@ P = np.array([[0.971, 0.029, 0.000],
 P @ P
 ```
 
-Let's pick some initial distributions $\psi_1, \psi_2, \psi_3$ and trace out the sequence of distributions $\psi_iP^t$ for $t = 0, 1, 2, \ldots$ and $i=1, 2, 3$.
+Let's pick an initial distribution $\psi_0$ and trace out the sequence of distributions $\psi_0 P^t$ for $t = 0, 1, 2, \ldots$
 
+First, we write a function to iterate the sequence of distributions for `ts_length` period
 
 ```{code-cell} ipython3
-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
+def iterate_ψ(ψ_0, P, ts_length):
+    n = len(P)
+    ψ_t = np.empty((ts_length, n))
+    ψ_t[0 ]= ψ_0
+    for t in range(1, ts_length):
+        ψ_t[t] = ψ_t[t-1] @ P
+    return ψ_t
+```
 
-# Define the vertices of the unit simplex
-v = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1], [0, 0, 0]])
+Now we plot the sequence
 
-# 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]]
-]
+```{code-cell} ipython3
+ψ_0 = (0.0, 0.2, 0.8)        # Initial condition
 
 fig = plt.figure()
 ax = fig.add_subplot(projection='3d')
 
-def update(n):    
+def update(n):
+    ψ_t = iterate_ψ(ψ_0, P, n+1)
+    
     ax.clear()
     ax.set_xlim([0, 1])
     ax.set_ylim([0, 1])
     ax.set_zlim([0, 1])
-    ax.view_init(45, 45)
+    ax.view_init(30, 210)
     
-    simplex = Poly3DCollection(faces, alpha=0.03)
-    ax.add_collection3d(simplex)
+    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))
     
-    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='yellow', s=60)
+    ax.scatter(ψ_star[0], ψ_star[1], ψ_star[2], c='k', s=60)
     
     return fig,
 
@@ -865,9 +860,9 @@ HTML(anim.to_jshtml())
 Here
 
 * $P$ is the stochastic matrix for recession and growth {ref}`considered above <mc_eg2>`.
-* The red, blue and green dots are initial marginal probability distribution  $\psi_1, \psi_2, \psi_3$, 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 $\psi^*$.
+* 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 $\psi^*$.
 
 You might like to try experimenting with different initial conditions.