Monte Carlo control Python sample code snippets

Monte Carlo Integration Example

python
import numpy as np
import matplotlib.pyplot as plt

# Define the function to integrate
def f(x):
    return x * np.cos(71 * x) + np.sin(13 * x)

# Generate random points
n = 100
reps = 1000
x = f(np.random.random((n, reps)))

# Calculate the Monte Carlo estimate
y = 1 / np.arange(1, n + 1)[:, None] * np.cumsum(x, axis=0)
upper, lower = np.percentile(y, [2.5, 97.5], axis=1)

# Plot the results
plt.plot(np.arange(1, n + 1), y, c='grey', alpha=0.02)
plt.plot(np.arange(1, n + 1), y[:, 0], c='red', linewidth=1)
plt.plot(np.arange(1, n + 1), upper, 'b', np.arange(1, n + 1), lower, 'b')
plt.show()

Monte Carlo Variance Reduction Example

python
import numpy as np
import matplotlib.pyplot as plt

# Define the function to integrate
def f(x):
    return x * np.cos(71 * x) + np.sin(13 * x)

# Generate random points
n = 100
reps = 1000
x = f(np.random.random((n, reps)))

# Calculate the Monte Carlo estimate
y = 1 / np.arange(1, n + 1)[:, None] * np.cumsum(x, axis=0)
upper, lower = np.percentile(y, [2.5, 97.5], axis=1)

# Plot the results
plt.plot(np.arange(1, n + 1), y, c='grey', alpha=0.02)
plt.plot(np.arange(1, n + 1), y[:, 0], c='red', linewidth=1)
plt.plot(np.arange(1, n + 1), upper, 'b', np.arange(1, n + 1), lower, 'b')
plt.show()

Monte Carlo Control Variate Example

python
import numpy as np
import matplotlib.pyplot as plt

# Define the functions to integrate
def f0(x):
    return x * np.cos(71 * x) + np.sin(13 * x)

def f1(x):
    return x * np.cos(71 * x) + np.sin(13 * x)

# Generate random points
n = 100
reps = 1000
x0 = f0(np.random.random((n, reps)))
x1 = f1(np.random.random((n, reps)))

# Calculate the Monte Carlo estimate
y0 = 1 / np.arange(1, n + 1)[:, None] * np.cumsum(x0, axis=0)
y1 = 1 / np.arange(1, n + 1)[:, None] * np.cumsum(x1, axis=0)

# Calculate the control variate estimate
cv_mean = y0.mean() + (y1.mean() - y0.mean()) * (x1.mean() - x0.mean()) / (x1.mean() - x0.mean())

# Plot the results
plt.plot(np.arange(1, n + 1), y0, c='grey', alpha=0.02)
plt.plot(np.arange(1, n + 1), y1, c='grey', alpha=0.02)
plt.plot(np.arange(1, n + 1), cv_mean, 'b')
plt.show()