"""

Translation
=================================================================

In this example, a Gaussian stochastic processes is translated into a stochastic processes of a number of distributions.
This example illustrates how to use the :class:`.Correlate` class to translate from Gaussian to other probability
distributions and compare how the statistics of the translated stochastic processes change along with distributions.

"""

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

# %%

from UQpy.stochastic_process import Translation, SpectralRepresentation
import numpy as np
import matplotlib.pyplot as plt

plt.style.use('seaborn')

# %% md
#
# Firstly we generate Gaussian Stochastic Processes using the Spectral Representation Method.

# %%

n_sim = 10000  # Num of samples
T = 100  # Time(1 / T = dw)
nt = 256  # Num.of Discretized Time
F = 1 / T * nt / 2  # Frequency.(Hz)
nw = 128  # Num of Discretized Freq.
dt = T / nt
t = np.linspace(0, T - dt, nt)
dw = F / nw
w = np.linspace(0, F - dw, nw)
S = 125 * w ** 2 * np.exp(-2 * w)

SRM_object = SpectralRepresentation(n_sim, S, dt, dw, nt, nw, random_state=128)
samples = SRM_object.samples


def S_to_R(S, w, t):
    dw = w[1] - w[0]
    fac = np.ones(len(w))
    fac[1: len(w) - 1: 2] = 4
    fac[2: len(w) - 2: 2] = 2
    fac = fac * dw / 3
    R = np.zeros(len(t))
    for i in range(len(t)):
        R[i] = 2 * np.dot(fac, S * np.cos(w * t[i]))
    return R


R_g = S_to_R(S, w, t)
r_g = R_g / R_g[0]

# %% md
#
# We translate the samples to be Uniform samples from :math:`1` to :math:`2`

# %%

from UQpy.distributions import Lognormal

distribution = Lognormal(0.5)
samples = samples.flatten()[:, np.newaxis]

Translate_object = Translation(distributions=distribution, time_interval=dt, frequency_interval=dw,
                               n_time_intervals=nt, n_frequency_intervals=nw,
                               correlation_function_gaussian=R_g, samples_gaussian=samples)
samples_ng = Translate_object.samples_non_gaussian

R_ng = Translate_object.scaled_correlation_function_non_gaussian
r_ng = Translate_object.correlation_function_non_gaussian

# %% md
#
# Plotting the actual and translated autocorrelation functions

# %%

fig1 = plt.figure()
plt.plot(r_g, label='Gaussian')
plt.plot(r_ng, label='non-Gaussian')
plt.title('Correlation Function (r)')
plt.legend()
plt.show()

"""

Translation
=================================================================

In this example, a Gaussian stochastic processes is translated into a stochastic processes of a number of distributions.
This example illustrates how to use the :class:`.Correlate` class to translate from Gaussian to other probability
distributions and compare how the statistics of the translated stochastic processes change along with distributions.

"""

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

# %%

from UQpy.stochastic_process import Translation, SpectralRepresentation
import numpy as np
import matplotlib.pyplot as plt

plt.style.use('seaborn')

# %% md
#
# Firstly we generate Gaussian Stochastic Processes using the Spectral Representation Method.

# %%

n_sim = 10000  # Num of samples
T = 100  # Time(1 / T = dw)
nt = 256  # Num.of Discretized Time
F = 1 / T * nt / 2  # Frequency.(Hz)
nw = 128  # Num of Discretized Freq.
dt = T / nt
t = np.linspace(0, T - dt, nt)
dw = F / nw
w = np.linspace(0, F - dw, nw)
S = 125 * w ** 2 * np.exp(-2 * w)

SRM_object = SpectralRepresentation(n_sim, S, dt, dw, nt, nw, random_state=128)
samples = SRM_object.samples


def S_to_R(S, w, t):
    dw = w[1] - w[0]
    fac = np.ones(len(w))
    fac[1: len(w) - 1: 2] = 4
    fac[2: len(w) - 2: 2] = 2
    fac = fac * dw / 3
    R = np.zeros(len(t))
    for i in range(len(t)):
        R[i] = 2 * np.dot(fac, S * np.cos(w * t[i]))
    return R


R_g = S_to_R(S, w, t)
r_g = R_g / R_g[0]

# %% md
#
# We translate the samples to be Uniform samples from :math:`1` to :math:`2`

# %%

from UQpy.distributions import Lognormal

distribution = Lognormal(0.5)
samples = samples.flatten()[:, np.newaxis]

Translate_object = Translation(distributions=distribution, time_interval=dt, frequency_interval=dw,
                               n_time_intervals=nt, n_frequency_intervals=nw,
                               correlation_function_gaussian=R_g, samples_gaussian=samples)
samples_ng = Translate_object.samples_non_gaussian

R_ng = Translate_object.scaled_correlation_function_non_gaussian
r_ng = Translate_object.correlation_function_non_gaussian

# %% md
#
# Plotting the actual and translated autocorrelation functions

# %%

fig1 = plt.figure()
plt.plot(r_g, label='Gaussian')
plt.plot(r_ng, label='non-Gaussian')
plt.title('Correlation Function (r)')
plt.legend()
plt.show()

Translation =================================================================  In this example, a Gaussian stochastic processes is translated into a stochastic processes of a number of distributions. This example illustrates how to use the :class:`.Correlate` class to translate from Gaussian to other probability distributions and compare how the statistics of the translated stochastic processes change along with distributions. # %% md # # Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to # import the class:`.Correlate` class from the :py:mod:`stochastic_processes` module of UQpy. # %% # %% md # # Firstly we generate Gaussian Stochastic Processes using the Spectral Representation Method. # %% # Num of samples # Time(1 / T = dw) # Num.of Discretized Time # Frequency.(Hz) # Num of Discretized Freq. # %% md # # We translate the samples to be Uniform samples from :math:`1` to :math:`2` # %% # %% md # # Plotting the actual and translated autocorrelation functions # %%