Added examples Cramér-von Mises sensitivity

docs/code/sensitivity/cramer_von_mises/local_exponential.py

@@ -0,0 +1,20 @@
+"""
+
+Auxiliary file
+==============================================
+
+"""
+
+import numpy as np
+
+
+def evaluate(X: np.array) -> np.array:
+    r"""A non-linear function that is used to demonstrate sensitivity index.
+
+    .. math::
+        f(x) = \exp(x_1 + 2*x_2)
+    """
+
+    Y = np.exp(X[:, 0] + 2 * X[:, 1])
+
+    return Y

docs/code/sensitivity/cramer_von_mises/local_sobol_func.py

@@ -0,0 +1,42 @@
+"""
+
+Auxiliary file
+==============================================
+
+"""
+
+import numpy as np
+import copy
+
+
+def evaluate(X, a_values):
+
+    dims = len(a_values)
+    g = 1
+
+    for i in range(dims):
+        g_i = (np.abs(4 * X[:, i] - 2) + a_values[i]) / (1 + a_values[i])
+        g *= g_i
+
+    return g
+
+
+def sensitivities(a_values):
+
+    dims = len(a_values)
+
+    Total_order = np.zeros((dims, 1))
+
+    V_i = (3 * (1 + a_values) ** 2) ** (-1)
+
+    total_variance = np.prod(1 + V_i) - 1
+
+    First_order = V_i / total_variance
+
+    for i in range(dims):
+
+        rem_First_order = copy.deepcopy(V_i)
+        rem_First_order[i] = 0
+        Total_order[i] = V_i[i] * np.prod(rem_First_order + 1) / total_variance
+
+    return First_order.reshape(-1, 1), Total_order

docs/code/sensitivity/cramer_von_mises/plot_cvm_exponential.py

@@ -0,0 +1,58 @@
+"""
+
+Exponential function
+==============================================
+
+.. math::
+    f(x) := \exp(x_1 + 2x_2), \quad x_1, x_2 \sim \mathcal{N}(0, 1)
+
+"""
+
+# %%
+from UQpy.run_model.RunModel import RunModel
+from UQpy.run_model.model_execution.PythonModel import PythonModel
+from UQpy.distributions import Normal
+from UQpy.distributions.collection.JointIndependent import JointIndependent
+from UQpy.sensitivity.cramer_von_mises import CramervonMises as cvm
+
+# %%
+# Create Model object
+model = PythonModel(
+    model_script="local_exponential.py",
+    model_object_name="evaluate",
+    var_names=[r"$X_1$", "$X_2$"],
+    delete_files=True,
+)
+
+runmodel_obj = RunModel(model=model)
+
+# Define distribution object
+dist_object = JointIndependent([Normal(0, 1)] * 2)
+
+# %% [markdown]
+# Compute Cramer-von Mises indices
+
+# %%
+# create cvm object
+SA = cvm(runmodel_obj, dist_object)
+
+# Compute Sobol indices using the pick and freeze algorithm
+computed_indices = SA.run(n_samples=20_000, estimate_sobol_indices=True)
+
+# %% [markdown]
+# Cramer-von Mises sensitivity analysis
+#
+# Expected value of the sensitivity indices:
+#
+# $S^1_{CVM} = \frac{6}{\pi} \operatorname{arctan}(2) - 2 \approx 0.1145$
+#
+# $S^2_{CVM} = \frac{6}{\pi} \operatorname{arctan}(\sqrt{19}) - 2 \approx 0.5693$
+
+# %%
+computed_indices["CVM_i"]
+
+# %%
+computed_indices["sobol_i"]
+
+# %%
+computed_indices["sobol_total_i"]

docs/code/sensitivity/cramer_von_mises/plot_cvm_sobol_func.py

@@ -0,0 +1,70 @@
+r"""
+
+Sobol function
+==============================================
+
+.. math::
+
+    g(x_1, x_2, \ldots, x_D) := \prod_{i=1}^{D} \frac{|4x_i - 2| + a_i}{1 + a_i},
+
+where,
+
+.. math::
+    x_i \sim \mathcal{U}(0, 1), \quad a_i \in \mathbb{R}.
+
+"""
+
+# %%
+import numpy as np
+
+from UQpy.run_model.RunModel import RunModel
+from UQpy.run_model.model_execution.PythonModel import PythonModel
+from UQpy.distributions import Uniform
+from UQpy.distributions.collection.JointIndependent import JointIndependent
+from UQpy.sensitivity.cramer_von_mises import CramervonMises as cvm
+
+# %%
+# Create Model object
+num_vars = 6
+a_vals = np.array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0])
+
+model = PythonModel(
+    model_script="local_sobol_func.py",
+    model_object_name="evaluate",
+    var_names=[r"$X_1$", "$X_2$"],
+    delete_files=True,
+    a_values=a_vals,
+)
+
+runmodel_obj = RunModel(model=model)
+
+# Define distribution object
+dist_object = JointIndependent([Uniform(0, 1)] * num_vars)
+
+# %%
+SA = cvm(runmodel_obj, dist_object)
+
+# Compute Sobol indices using the pick and freeze algorithm
+computed_indices = SA.run(n_samples=20_000, estimate_sobol_indices=True)
+
+# %%
+computed_indices["CVM_i"]
+
+# %% [markdown]
+# Sobol indices computed analytically
+#
+# $S_1$ = 0.46067666
+#
+# $S_2$ = 0.20474518
+#
+# $S_3$ = 0.11516917
+#
+# $S_4$ = 0.07370827
+#
+# $S_5$ = 0.0511863
+#
+# $S_6$ = 0.03760626
+#
+
+# %%
+computed_indices["sobol_i"]

docs/source/conf.py

@@ -92,6 +92,7 @@
         "../code/transformations/nataf",
         "../code/sensitivity/morris",
         "../code/sensitivity/sobol",
+        "../code/sensitivity/cramer_von_mises",
         "../code/stochastic_processes/bispectral",
         "../code/stochastic_processes/karhunen_loeve",
         "../code/stochastic_processes/spectral",
@@ -127,6 +128,7 @@
         "auto_examples/transformations/nataf",
         "auto_examples/sensitivity/morris",
         "auto_examples/sensitivity/sobol",
+        "auto_examples/sensitivity/cramer_von_mises",
         "auto_examples/stochastic_processes/bispectral",
         "auto_examples/stochastic_processes/karhunen_loeve",
         "auto_examples/stochastic_processes/spectral",