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Random Walk

Here, we will study a few examples of Random Walks.

1-Dimensional​

yt=yt−1+εt,y_t=y_{t-1}+\varepsilon_t,

where y0=0y_0=0, and εt\varepsilon_t is random variable which takes value 11 or −1-1 with equal probability.

import numpy as np
import random
import imageio
import os
import matplotlib.pyplot as plt

random.seed(2609)

# Initialize an empty numpy array
y = np.array([])

# Set initial value of y
y_0 = 0
y = np.append(y,y_0)

# Number of iterations for the loop
n_iterations = 1000

# Loop to generate y_t values
for t in range(1, n_iterations):
    # Generate epsilon_t which takes value 1 and -1 with equal probability
    epsilon_t = random.choice([-1, 1])
    # Calculate y_t
    y_t = y[t-1] + epsilon_t
    # Append y_t to the array
    y = np.append(y, y_t)

# Create frames for each step
x=np.arange(0,n_iterations)
filenames = []
for i in range(1, n_iterations + 1):
    plt.figure(figsize=(8, 6))
    plt.plot(x[:i], y[:i], color='blue')
    plt.plot(x[i-1], y[i-1], color='r',marker='o')
    plt.xlim(min(x)-10, max(x)+10)
    plt.ylim(min(y)-10, max(y)+10)
    plt.title(f'Step {i}/{n_iterations}')
    plt.xlabel("t")
    plt.ylabel("$y_t$")
    plt.grid()
    
    # Save frame
    filename = f'frames/step_{i}.png'
    filenames.append(filename)
    plt.savefig(filename)
    plt.close()

# Create GIF
with imageio.get_writer('random_walk.gif', mode='I', duration=0.1) as writer:
    for filename in filenames:
        image = imageio.imread(filename)
        writer.append_data(image)

# Clean up the frames
for filename in filenames:
    os.remove(filename)
Fig. 1

2-Dimensional​

(xt,yt)=(xt−1,yt−1)+εt,(x_t,y_t)=(x_{t-1},y_{t-1})+\varepsilon_t,

where (x0,y0)=(0,0)(x_0,y_0)=(0,0), and εt\varepsilon_t is random variable which takes value (0,1),(0,−1),(1,0)(0,1), (0,-1), (1,0) or (−1,0)(-1,0) with equal probability.

import matplotlib.pyplot as plt
import random
import imageio
import os

random.seed(2609)
# Function to create a 2D random walk
def random_walk_2d(n_steps):
    x, y = 0, 0
    walk_x, walk_y = [x], [y]

    for i in range(n_steps):
        dx, dy = random.choice([(0,1), (0,-1), (1,0), (-1,0)])
        x, y = x + dx, y + dy
        walk_x.append(x)
        walk_y.append(y)
    
    return walk_x, walk_y

# Generate a random walk
n_steps = 1000
walk_x, walk_y = random_walk_2d(n_steps)

# Create directory for frames
os.makedirs('frames', exist_ok=True)

# Create frames for each step
filenames = []
for i in range(1, n_steps + 1):
    plt.figure(figsize=(8, 6))
    plt.plot(walk_x[:i], walk_y[:i], color='blue')
    plt.plot(walk_x[i-1], walk_y[i-1], color='r',marker='o')
    plt.xlim(min(walk_x)-1, max(walk_x)+1)
    plt.ylim(min(walk_y)-1, max(walk_y)+1)
    plt.title(f'Step {i}/{n_steps}')
    
    # Save frame
    filename = f'frames/step_{i}.png'
    filenames.append(filename)
    plt.savefig(filename)
    plt.close()

# Create GIF
with imageio.get_writer('random_walk.gif', mode='I', duration=0.1) as writer:
    for filename in filenames:
        image = imageio.imread(filename)
        writer.append_data(image)

# Clean up the frames
for filename in filenames:
    os.remove(filename)
Fig. 2