<gh_stars>0
"""

N-dimensional & multiple-variables
=================================================================

In this example, the Spectral Representation Method is used to generate stochastic processes from a prescribed Power
Spectrum. This example illustrates how to use the SRM class for 'n' dimensional and one variable case and compare the
statistics of the generated stochastic processes with the expected values.

"""

#%% md
#
# Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to
# import the :class:`.SpectralRepresentation` class from the :py:mod:`stochastic_processes` module of UQpy.

#%%

from UQpy.stochastic_process import SpectralRepresentation
import numpy as np
import matplotlib.pyplot as plt
from pylab import *
plt.style.use('seaborn')

#%% md
#
# The input parameters necessary for the generating of the stochastic processes are given below:

#%%

n_sim = 1000 # Num of samples

n = 2 # Num of dimensions

m = 1 # Num of variables

T = 10
nt = 200
dt = T/nt
t = np.linspace(0, T-dt, nt)

# Frequency
W = np.array([1.0, 1.5])
nw = 100
dw = W / nw
x_list = [np.linspace(0, W[i] - dw[i], nw) for i in range(n)]
xy_list = np.array(np.meshgrid(*x_list, indexing='ij'))

#%% md
#
# Defining the Power Spectral Density(S)

#%%

S_nd_1v = 125 / 4 * np.linalg.norm(xy_list, axis=0) ** 2 * np.exp(-5 * np.linalg.norm(xy_list, axis=0))

#%% md
#
# Make sure that the input parameters are in order to prevent aliasing

#%%

t_u = 2*np.pi/2/W

if dt>t_u.all():
    print('Error')

SRM_object = SpectralRepresentation(n_sim, S_nd_1v, [dt, dt], dw, [nt, nt], [nw, nw], random_state=128)
samples_nd_1v = SRM_object.samples

t_list = [t for _ in range(n)]
tt_list = np.array(np.meshgrid(*t_list, indexing='ij'))

fig1 = plt.figure()
plt.title('2d random field with a prescribed Power Spectrum')
pcm = pcolor(tt_list[0], tt_list[1], samples_nd_1v[0, 0], cmap='RdBu_r', vmin=-6, vmax=6)
plt.colorbar(pcm, extend='both', orientation='vertical')
plt.xlabel('$X_{1}$')
plt.ylabel('$X_{2}$')
plt.show()

print('The mean of the samples is ', np.mean(samples_nd_1v), 'whereas the expected mean is 0.000')
print('The variance of the samples is ', np.var(samples_nd_1v), 'whereas the expected variance is ',
      np.sum(S_nd_1v)*np.prod(dw)*(2**n))
"""

N-dimensional & multiple-variables
=================================================================

In this example, the Spectral Representation Method is used to generate stochastic processes from a prescribed Power
Spectrum. This example illustrates how to use the SRM class for 'n' dimensional and one variable case and compare the
statistics of the generated stochastic processes with the expected values.

"""

#%% md
#
# Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to
# import the :class:`.SpectralRepresentation` class from the :py:mod:`stochastic_processes` module of UQpy.

#%%

from UQpy.stochastic_process import SpectralRepresentation
import numpy as np
import matplotlib.pyplot as plt
from pylab import *
plt.style.use('seaborn')

#%% md
#
# The input parameters necessary for the generating of the stochastic processes are given below:

#%%

n_sim = 1000 # Num of samples

n = 2 # Num of dimensions

m = 1 # Num of variables

T = 10
nt = 200
dt = T/nt
t = np.linspace(0, T-dt, nt)

# Frequency
W = np.array([1.0, 1.5])
nw = 100
dw = W / nw
x_list = [np.linspace(0, W[i] - dw[i], nw) for i in range(n)]
xy_list = np.array(np.meshgrid(*x_list, indexing='ij'))

#%% md
#
# Defining the Power Spectral Density(S)

#%%

S_nd_1v = 125 / 4 * np.linalg.norm(xy_list, axis=0) ** 2 * np.exp(-5 * np.linalg.norm(xy_list, axis=0))

#%% md
#
# Make sure that the input parameters are in order to prevent aliasing

#%%

t_u = 2*np.pi/2/W

if dt>t_u.all():
    print('Error')

SRM_object = SpectralRepresentation(n_sim, S_nd_1v, [dt, dt], dw, [nt, nt], [nw, nw], random_state=128)
samples_nd_1v = SRM_object.samples

t_list = [t for _ in range(n)]
tt_list = np.array(np.meshgrid(*t_list, indexing='ij'))

fig1 = plt.figure()
plt.title('2d random field with a prescribed Power Spectrum')
pcm = pcolor(tt_list[0], tt_list[1], samples_nd_1v[0, 0], cmap='RdBu_r', vmin=-6, vmax=6)
plt.colorbar(pcm, extend='both', orientation='vertical')
plt.xlabel('$X_{1}$')
plt.ylabel('$X_{2}$')
plt.show()

print('The mean of the samples is ', np.mean(samples_nd_1v), 'whereas the expected mean is 0.000')
print('The variance of the samples is ', np.var(samples_nd_1v), 'whereas the expected variance is ',
      np.sum(S_nd_1v)*np.prod(dw)*(2**n))
N-dimensional & multiple-variables =================================================================  In this example, the Spectral Representation Method is used to generate stochastic processes from a prescribed Power Spectrum. This example illustrates how to use the SRM class for 'n' dimensional and one variable case and compare the statistics of the generated stochastic processes with the expected values. #%% md # # Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to # import the :class:`.SpectralRepresentation` class from the :py:mod:`stochastic_processes` module of UQpy. #%% #%% md # # The input parameters necessary for the generating of the stochastic processes are given below: #%% # Num of samples # Num of dimensions # Num of variables # Frequency #%% md # # Defining the Power Spectral Density(S) #%% #%% md # # Make sure that the input parameters are in order to prevent aliasing #%%