Karhunen Loeve Expansion

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

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

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

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

n_sim = 10000  # Num of samples

m = 400 + 1
T = 1000
dt = T / (m - 1)
t = np.linspace(0, T, m)

Defining the Autocorrelation Function.

# Target Covariance(ACF)
R = np.zeros([m, m])
for i in range(m):
    for j in range(m):
        R[i, j] = 2 * np.exp(-((t[j] - t[i]) / 281) ** 2)  # var = 2

KLE_Object = KarhunenLoeveExpansion(n_samples=n_sim, correlation_function=R, time_interval=dt)
samples = KLE_Object.samples

fig, ax = plt.subplots()
plt.title('Realisation of the Karhunen Loeve Expansion')
plt.plot(t, samples[0, 0])
ax.yaxis.grid(True)
ax.xaxis.grid(True)
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 2.000')

/home/docs/checkouts/readthedocs.org/user_builds/uqpyproject/envs/latest/lib/python3.9/site-packages/UQpy/stochastic_process/KarhunenLoeveExpansion.py:56: ComplexWarning: Casting complex values to real discards the imaginary part
  self.lam = lam.astype(np.float64)
The mean of the samples is  0.020575704620114856 whereas the expected mean is 0.000
The variance of the samples is  2.0371742868489147 whereas the expected variance is 2.000