added example file and documentation for the method along with anothe…

…r test for KLE 2d code

docs/code/stochastic_processes/karhunen_loeve/plot_karhunen_loeve_2d.py

@@ -0,0 +1,72 @@
+"""
+
+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 :class:`.KarhunenLoeveExpansion2D` class for a 2 dimensional
+random field and compare the statistics of the generated random field 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:`.KarhunenLoeveExpansionTwoDimension` class from the :py:mod:`stochastic_processes` module of UQpy.
+
+# %%
+from matplotlib import pyplot as plt
+from UQpy.stochastic_process import KarhunenLoeveExpansion2D
+import numpy as np
+
+plt.style.use('seaborn')
+
+# %% md
+#
+# 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)
+
+# %% md
+
+# 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)
+
+# %% md
+
+# Simulating the samples.
+
+# %%
+
+samples = KLE_Object.samples
+
+# %% md
+
+# Plotting a sample of the stochastic field.
+
+# %%
+
+fig = plt.figure()
+plt.title('Realisation of the Karhunen Loeve Expansion for a 2D stochastic field')
+plt.plot(samples[0, 0])
+plt.xlabel('t (Time)')
+plt.ylabel('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')

docs/source/stochastic_process/karhunen_loeve_2d.rst

@@ -0,0 +1,32 @@
+Karhunen Loève Expansion for Multi-Dimensional Fields
+----------------------------
+
+The Karhunen Loève Expansion expands the stochastic field as follows:
+
+.. math:: A(x, t) = \sum_{n=1}^{\infty} \sum_{k=1}^{\infty}\eta_{nk}(\theta) \sqrt{\lambda_n(x)} f_n(t, x) \sqrt{\mu_{nk}} g_{nk}(x)
+
+where :math:`\eta_{nk}(\theta)` are uncorrelated standardized normal random variables and :math:`\lambda_n(x)` and :math:`f_n(x, t)` are the eigenvalues and eigenvectors repsectively of the "quasi" one dimensional covariance function :math:`C(x, t_1, t_2)`. :math:`\mu_{nk}` and :math:`g_{nk}(x)` are the eigenvalues and eigenvectors of the derived "one" dimensional covariance function :math:`H(x_1, x_2)`. Additional details regarding the simulation formula can be found at https://www.sciencedirect.com/science/article/abs/pii/S0045782516318692.
+
+KarhunenLoeve2D Class
+^^^^^^^^^^^^^^^^^^^^
+
+The :class:`.KarhunenLoeve2D` class is imported using the following command:
+
+>>> from UQpy.stochastic_process.KarhunenLoeveExpansionTwoDimension2D import KarhunenLoeveExpansion
+
+Methods
+"""""""
+.. autoclass:: UQpy.stochastic_process.KarhunenLoeveExpansion2D
+    :members: run
+
+Attributes
+""""""""""
+.. autoattribute:: UQpy.stochastic_process.KarhunenLoeveExpansion2D.samples
+.. autoattribute:: UQpy.stochastic_process.KarhunenLoeveExpansion2D.xi
+
+Examples
+""""""""""
+
+.. toctree::
+
+   Karhunen Loeve Examples <../auto_examples/stochastic_processes/karhunen_loeve_2d/index>

src/UQpy/stochastic_process/KarhunenLoeveExpansion2D.py

@@ -112,7 +112,7 @@ def run(self, n_samples, random_variables=None):
 
         if self.samples is None:
             self.samples = samples
-            self.random_variables = random_variables
+            self.xi = random_variables
         else:
             self.samples = np.concatenate((self.samples, samples), axis=0)
-            self.random_variables = np.concatenate((self.random_variables, random_variables), axis=2)
+            self.xi = np.concatenate((self.xi, random_variables), axis=2)

tests/unit_tests/stochastic_process/test_karhunen_loeve_2d.py

@@ -23,3 +23,9 @@ def test_samples_shape():
 
 def test_samples_values():
     assert np.isclose(samples[13, 0, 13, 6], -1.4385717324103362, rtol=0.01)
+
+
+def test_run_method():
+    nsamples_second_run = 100
+    KLE_Object.run(n_samples=nsamples_second_run)
+    assert samples.shape == (n_samples + nsamples_second_run, 1, len(x), len(t))