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[markov_chain_I] Update animation using multiple initial conditions #492

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[markov_chain_I] Update animation using multiple initial conditions #492

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36 changes: 28 additions & 8 deletions lectures/markov_chains_I.md
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Original file line number Diff line number Diff line change
Expand Up @@ -58,6 +58,8 @@ import numpy as np
import networkx as nx
from matplotlib import cm
import matplotlib as mpl
from matplotlib.patches import Polygon
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
from mpl_toolkits.mplot3d import Axes3D
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
Expand Down Expand Up @@ -810,26 +812,44 @@ 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
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,

Expand Down