Karhunen Loeve Expansion 2 Dimesional

In this example, the KL Expansion is used to generate 2 dimensional stochastic fields from a prescribed Autocorrelation Function. This example illustrates how to use the KarhunenLoeveExpansion2D class for a 2 dimensional random field and compare the statistics of the generated random field with the expected values.

Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to import the KarhunenLoeveExpansionTwoDimension class from the stochastic_processes module of UQpy.

from matplotlib import pyplot as plt
from UQpy.stochastic_process import KarhunenLoeveExpansion2D
import numpy as np

The input parameters necessary for the generation of the stochastic processes are given below:

n_samples = 100  # Num of samples
nx, nt = 20, 10
dx, dt = 0.05, 0.1

x = np.linspace(0, (nx - 1) * dx, nx)
t = np.linspace(0, (nt - 1) * dt, nt)
xt_list = np.meshgrid(x, x, t, t, indexing='ij')  # R(t_1, t_2, x_1, x_2)

# Defining the Autocorrelation Function.

R = np.exp(-(xt_list[0] - xt_list[1]) ** 2 - (xt_list[2] - xt_list[3]) ** 2)
# R(x_1, x_2, t_1, t_2) = exp(-(x_1 - x_2) ** 2 -(t_1 - t_2) ** 2)

KLE_Object = KarhunenLoeveExpansion2D(n_samples=n_samples, correlation_function=R, time_intervals=np.array([dt, dx]),
                                      thresholds=[4, 5], random_state=128)

# Simulating the samples.

samples = KLE_Object.samples

# Plotting a sample of the stochastic field.

fig = plt.figure()
plt.title('Realisation of the Karhunen Loeve Expansion for a 2D stochastic field')
plt.imshow(samples[0, 0])
plt.ylabel('t (Time)')
plt.xlabel('x (Space)')
plt.show()

print('The mean of the samples is ', np.mean(samples), 'whereas the expected mean is 0.000')
print('The variance of the samples is ', np.var(samples), 'whereas the expected variance is 1.000')

The mean of the samples is  0.06238506034183266 whereas the expected mean is 0.000
The variance of the samples is  0.873477875144356 whereas the expected variance is 1.000