CircleCI build openturns 16723

openturns/master/_downloads/e3fd5d734a0eba35d5c47a04fa2a07ac/plot_chaos_cleaning_strategy.py

@@ -11,35 +11,31 @@
 # Polynomial chaos expansion
 # --------------------------
 #
-# Let :math:`g : \mathcal{X} \rightarrow \mathbb{R}` be a function
-# where :math:`\mathcal{X} \subseteq \mathbb{R}^p` is the domain of :math:`g`.
-# Let :math:`f` be a probability density function on :math:`\mathcal{X}`.
-# Let :math:`T` be the iso-probabilistic transformation from the physical
-# space :math:`\mathcal{X}` to the standard space :math:`\mathcal{\bar{X}}`:
+# Let :math:`g : \mathcal{D} \rightarrow \mathbb{R}` be a function
+# where :math:`\mathcal{D} \subseteq \mathbb{R}^p` is the domain of :math:`g`.
+# We consider :math:`\vect{X}` a random vector which
+# probability density function is denoted by :math:`f`.
+# We assume that :math:`g(\vect{X})` has a finite second order moment.
+# Let :math:`T` be an iso-probabilistic transformation such that :math:`\vect{Z} = T(\vect{X})`
+# follows a distribution uniquely defined by all its moments.
+# Let :math:`h` be the function defined by:
 #
 # .. math::
 #
-#   \xi = T(\boldsymbol{x}) \in \mathcal{\bar{X}}
+#   h = g \circ T^{-1}.
 #
-# for any :math:`\boldsymbol{x} \in \mathcal{X}`.
-# Let :math:`h` be the function defined by the equation :
+# The polynomial chaos decomposition of :math:`h` with respect to the measure of
+# :math:`\vect{Z}` is (see [blatman2009]_ page 73) :
 #
 # .. math::
 #
-#   h(\boldsymbol{\xi}) = \left(g \circ T^{-1}\right)(\boldsymbol{\xi})
+#   h(\vect{z}) = \sum_{\vect{\alpha} \in \mathbb{N}^p}
+#   a_{\vect{\alpha}} \psi_{\vect{\alpha}}(\vect{z})
 #
-# for any :math:`\boldsymbol{\xi} \in \mathcal{\bar{X}}`.
-# The polynomial chaos decomposition of :math:`h` is ([blatman2009]_ page 73) :
-#
-# .. math::
-#
-#   h(\boldsymbol{\xi}) = \sum_{\boldsymbol{\alpha} \in \mathbb{N}^p}
-#   a_{\boldsymbol{\alpha}} \psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi}) + \epsilon
-#
-# where :math:`\boldsymbol{\alpha} = (\alpha_1, ..., \alpha_p) \in \mathbb{N}^p`
-# is a multiindex, :math:`a_{\boldsymbol{\alpha}} \in \mathbb{R}` is the
-# coefficient,  :math:`\psi_{\boldsymbol{\alpha}} : \mathcal{\bar{X}} \rightarrow \mathbb{R}`
-# is a multivariate polynomial and :math:`\epsilon` is a random variable.
+# where :math:`\vect{\alpha} = (\alpha_1, ..., \alpha_p) \in \mathbb{N}^p`
+# is a multiindex, :math:`a_{\vect{\alpha}} \in \mathbb{R}` is the
+# coefficient,  :math:`\psi_{\vect{\alpha}} : \mathcal{\bar{X}} \rightarrow \mathbb{R}`
+# is a multivariate polynomial.
 
 # %%
 #
@@ -56,28 +52,28 @@
 #
 # .. math::
 #
-#   \mathcal{A}^{d} = \left\{ \boldsymbol{\alpha} \in \mathbb{N}^p
-#   \; | \; \|\boldsymbol{\alpha}\|_1 \leq d\right\}
+#   \mathcal{A}^{d} = \left\{ \vect{\alpha} \in \mathbb{N}^p
+#   \; | \; \|\vect{\alpha}\|_1 \leq d\right\}
 #
 # where
 #
 # .. math::
 #
-#   \|\boldsymbol{\alpha}\|_d = \alpha_1 + ... + \alpha_p
+#   \|\vect{\alpha}\|_d = \alpha_1 + ... + \alpha_p
 #
-# is the 1-norm of the multi-index :math:`\boldsymbol{\alpha}`.
+# is the 1-norm of the multi-index :math:`\vect{\alpha}`.
 # Therefore, the truncated polynomial chaos expansion is:
 #
 # .. math::
 #
-#   h(\boldsymbol{\xi}) = \sum_{\boldsymbol{\alpha} \in \mathcal{A}^{d}}
-#   a_{\boldsymbol{\alpha}} \psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi}) + \epsilon.
+#   \widetilde{h}(\vect{z}) = \sum_{\vect{\alpha} \in \mathcal{A}^{d}}
+#   a_{\vect{\alpha}} \psi_{\vect{\alpha}}(\vect{z}).
 #
 # In order to ensure a low error, we may choose a large value of the
 # parameter :math:`P`. This, however, leads to a large number of
-# coefficients :math:`\boldsymbol{\alpha} \in \mathcal{A}^{d}` to
+# coefficients :math:`\vect{\alpha} \in \mathcal{A}^{d}` to
 # estimate. More precisely, the number of coefficients to estimate
-# is ([blatman2009]_ page 73) :
+# is (see [blatman2009]_ page 73) :
 #
 # .. math::
 #
@@ -90,31 +86,31 @@
 #
 # Low-rank polynomial chaos expansion
 # -----------------------------------
-# For any :math:`\boldsymbol{\alpha} \in \mathbb{N}^p`, let
-# :math:`\|\boldsymbol{\alpha}\|_0` be the rank of the multiindex, that is,
+# For any :math:`\vect{\alpha} \in \mathbb{N}^p`, let
+# :math:`\|\vect{\alpha}\|_0` be the rank of the multiindex, that is,
 # the number of nonzero components:
 #
 # .. math::
 #
-#   \|\boldsymbol{\alpha}\|_0 = \sum_{i = 1}^p \boldsymbol{1}_{\alpha_i > 0}
+#   \|\vect{\alpha}\|_0 = \sum_{i = 1}^p \vect{1}_{\alpha_i > 0}
 #
-# where :math:`\boldsymbol{1}` is the indicator function.
+# where :math:`\vect{1}` is the indicator function.
 # The multiindex set of maximum total degree :math:`d \in \mathbb{N}`
 # and maximum rank :math:`j \in \mathbb{N}` is ([blatman2009]_ page 74):
 #
 # .. math::
 #
-#   \mathcal{A}^{d,j} = \left\{ \boldsymbol{\alpha} \in \mathbb{N}^p
-#   \; | \; \|\boldsymbol{\alpha}\|_1 \leq d, \;
-#   \; \|\boldsymbol{\alpha}\|_0 \leq j\right\}.
+#   \mathcal{A}^{d,j} = \left\{ \vect{\alpha} \in \mathbb{N}^p
+#   \; | \; \|\vect{\alpha}\|_1 \leq d, \;
+#   \; \|\vect{\alpha}\|_0 \leq j\right\}.
 #
 # Therefore, the rank-`j` polynomial chaos expansion is:
 #
 # .. math::
 #
-#   h(\boldsymbol{\xi}) = \sum_{\boldsymbol{\alpha} \in
-#   \mathcal{A}^{d,j}} a_{\boldsymbol{\alpha}}
-#   \psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi}) + \epsilon.
+#   \widetilde{h}(\vect{z}) = \sum_{\vect{\alpha} \in
+#   \mathcal{A}^{d,j}} a_{\vect{\alpha}}
+#   \psi_{\vect{\alpha}}(\vect{z}).
 #
 # The rank is now a hyperparameter of the model: [blatman2009]_ suggests
 # to use :math:`j = 2, 3, 4`. An example of low-rank PCE for the G-Sobol'
@@ -129,7 +125,7 @@
 # If :math:`\textrm{card}\left(\mathcal{A}^{d}\right)` is large, many coefficients
 # may be poorly estimated, which may reduce the quality of the metamodel. We may
 # want to select a subset of the coefficients which best predict the output.
-# In other words, we may compute a subset
+# In other words, we may compute a subset:
 #
 # .. math::
 #
@@ -139,41 +135,40 @@
 #
 # .. math::
 #
-#   h(\boldsymbol{\xi}) = \sum_{\boldsymbol{\alpha} \in \mathcal{A}}
-#   a_{\boldsymbol{\alpha}} \psi_{\boldsymbol{\alpha}}(\boldsymbol{\xi})
-#   + \epsilon.
+#   \widetilde{h}(\vect{z}) = \sum_{\vect{\alpha} \in \mathcal{A}}
+#   a_{\vect{\alpha}} \psi_{\vect{\alpha}}(\vect{z})
 #
 # An enumeration rule is a function from the set of integers :math:`k` to
-# the corresponding set of multiindices :math:`\boldsymbol{\alpha}`. More
+# the corresponding set of multiindices :math:`\vect{\alpha}`. More
 # precisely, let :math:`r : \mathbb{N} \rightarrow \mathbb{N}^p` be the
 # function such that :
 #
 # .. math::
 #
-#   r(k) = \boldsymbol{\alpha}
+#   r(k) = \vect{\alpha}
 #
 # for any :math:`k \geq 0`.
 # Let :math:`K \in \mathbb{N}` be a parameter representing the number of
 # coefficients considered in the selection. Given an enumeration rule for
-# the multiindices :math:`\boldsymbol{\alpha}`, at most :math:`K` multiindices
+# the multiindices :math:`\vect{\alpha}`, at most :math:`K` multiindices
 # will be considered. Let :math:`\mathcal{A}_K` be the corresponding multiindex set :
 #
 # .. math::
 #
-#   \mathcal{A}_K = \left\{ \boldsymbol{\alpha}
-#   \; | \; r^{-1}(\boldsymbol{\alpha}) = k \leq K \right\}.
+#   \mathcal{A}_K = \left\{ \vect{\alpha}
+#   \; | \; r^{-1}(\vect{\alpha}) = k \leq K \right\}.
 #
 #
 # Let :math:`\epsilon > 0` be a parameter representing the minimum relative
-# value of a significant coefficient :math:`a_{\boldsymbol{\alpha}}`.
+# value of a significant coefficient :math:`a_{\vect{\alpha}}`.
 # The :class:`~openturns.CleaningStrategy` uses the following criteria to select the coefficients :
 #
 # .. math::
 #
 #   \mathcal{A}_\epsilon =
 #   \left\{
-#   |a_{\boldsymbol{\alpha}}| \geq \epsilon \max_{ a_{\boldsymbol{\alpha}}
-#   \in \mathcal{A}_K } |a_{\boldsymbol{\alpha}}| \right\}
+#   |a_{\vect{\alpha}}| \geq \epsilon \max_{ a_{\vect{\alpha}}
+#   \in \mathcal{A}_K } |a_{\vect{\alpha}}| \right\}
 #
 # where :math:`\epsilon` is the significance factor, which by default is
 # :math:`\epsilon = 10^{-4}`. This rule selects only the coefficients which
@@ -191,8 +186,8 @@
 #
 # .. math::
 #
-#   d := \textrm{max}_{\boldsymbol{\alpha} \in \mathcal{A}}
-#   \|\boldsymbol{\alpha}\|_1.
+#   d := \textrm{max}_{\vect{\alpha} \in \mathcal{A}}
+#   \|\vect{\alpha}\|_1.
 #
 # The index of sparsity of :math:`\mathcal{A}` is ([blatman2009]_  eq. 4.42 page 86) :
 #
@@ -477,7 +472,7 @@ def draw_polynomial_chaos_validation(
 #
 # The `CleaningStrategy` has the following algorithm. On input, it considers
 # only the first `maximumConsideredTerms` coefficients
-# :math:`a_{\boldsymbol{\alpha}}`. On output it selects the `mostSignificant`
+# :math:`a_{\vect{\alpha}}`. On output it selects the `mostSignificant`
 # most significant coefficients. To do this, it uses the
 # `significanceFactor` parameter.
 #
@@ -643,7 +638,7 @@ def compute_cleaning_PCE(
 # - `currentVectorIndex_` : the current value of the index in the full multiindex set, according to the enumeration rule.
 #
 # Each time the selection method is called, it is passed a
-# coefficient :math:`a_{\boldsymbol{\alpha}}` which is a new candidate to be
+# coefficient :math:`a_{\vect{\alpha}}` which is a new candidate to be
 # considered by the algorithm. The first time the method is evaluated, the
 # active multiindex set is empty, so that it must be filled with the first
 # coefficients in the multiindex set, according to the enumeration rule. The

openturns/master/_sources/auto_calibration/bayesian_calibration/plot_gibbs.rst.txt

@@ -283,7 +283,7 @@ Let us plot the posterior density.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 12.155 seconds)
+   **Total running time of the script:** (0 minutes 12.547 seconds)
 
 
 .. _sphx_glr_download_auto_calibration_bayesian_calibration_plot_gibbs.py:

openturns/master/_sources/auto_calibration/bayesian_calibration/plot_rwmh_python_distribution.rst.txt

@@ -425,7 +425,7 @@ Plot posterior marginal plots only as prior cannot be drawn meaningfully.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 5.424 seconds)
+   **Total running time of the script:** (0 minutes 5.440 seconds)
 
 
 .. _sphx_glr_download_auto_calibration_bayesian_calibration_plot_rwmh_python_distribution.py:

openturns/master/_sources/auto_calibration/bayesian_calibration/sg_execution_times.rst.txt

@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:22.316** total execution time for 7 files **from auto_calibration/bayesian_calibration**:
+**00:22.751** total execution time for 7 files **from auto_calibration/bayesian_calibration**:
 
 .. container::
 
@@ -33,23 +33,23 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_gibbs.py` (``plot_gibbs.py``)
-     - 00:12.155
+     - 00:12.547
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_rwmh_python_distribution.py` (``plot_rwmh_python_distribution.py``)
-     - 00:05.424
+     - 00:05.440
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_bayesian_calibration_flooding.py` (``plot_bayesian_calibration_flooding.py``)
-     - 00:01.287
+     - 00:01.286
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_bayesian_calibration.py` (``plot_bayesian_calibration.py``)
-     - 00:01.160
+     - 00:01.163
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_gibbs_simus.py` (``plot_gibbs_simus.py``)
-     - 00:01.111
+     - 00:01.112
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_ackley_distribution.py` (``plot_ackley_distribution.py``)
-     - 00:00.902
+     - 00:00.921
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_bayesian_calibration_plot_imh_python_distribution.py` (``plot_imh_python_distribution.py``)
-     - 00:00.277
+     - 00:00.282
      - 0.0

openturns/master/_sources/auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_chaboche.rst.txt

@@ -1927,7 +1927,7 @@ Reset default settings
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 6.716 seconds)
+   **Total running time of the script:** (0 minutes 7.101 seconds)
 
 
 .. _sphx_glr_download_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_chaboche.py:

openturns/master/_sources/auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_deflection_tube.rst.txt

@@ -896,7 +896,7 @@ Reset default settings
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 3.787 seconds)
+   **Total running time of the script:** (0 minutes 4.120 seconds)
 
 
 .. _sphx_glr_download_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_deflection_tube.py:

openturns/master/_sources/auto_calibration/least_squares_and_gaussian_calibration/plot_calibration_flooding.rst.txt

@@ -1695,7 +1695,7 @@ Reset default settings
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 8.623 seconds)
+   **Total running time of the script:** (0 minutes 8.970 seconds)
 
 
 .. _sphx_glr_download_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_flooding.py:

openturns/master/_sources/auto_calibration/least_squares_and_gaussian_calibration/sg_execution_times.rst.txt

@@ -6,7 +6,7 @@
 
 Computation times
 =================
-**00:20.353** total execution time for 8 files **from auto_calibration/least_squares_and_gaussian_calibration**:
+**00:21.512** total execution time for 8 files **from auto_calibration/least_squares_and_gaussian_calibration**:
 
 .. container::
 
@@ -33,26 +33,26 @@ Computation times
      - Time
      - Mem (MB)
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_flooding.py` (``plot_calibration_flooding.py``)
-     - 00:08.623
+     - 00:08.970
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_chaboche.py` (``plot_calibration_chaboche.py``)
-     - 00:06.716
+     - 00:07.101
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_deflection_tube.py` (``plot_calibration_deflection_tube.py``)
-     - 00:03.787
+     - 00:04.120
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_logistic.py` (``plot_calibration_logistic.py``)
-     - 00:00.702
+     - 00:00.774
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_quickstart.py` (``plot_calibration_quickstart.py``)
-     - 00:00.295
+     - 00:00.308
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_calibration_withoutobservedinputs.py` (``plot_calibration_withoutobservedinputs.py``)
-     - 00:00.086
+     - 00:00.090
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_generate_chaboche.py` (``plot_generate_chaboche.py``)
-     - 00:00.073
+     - 00:00.077
      - 0.0
    * - :ref:`sphx_glr_auto_calibration_least_squares_and_gaussian_calibration_plot_generate_flooding.py` (``plot_generate_flooding.py``)
-     - 00:00.069
+     - 00:00.072
      - 0.0

openturns/master/_sources/auto_data_analysis/distribution_fitting/plot_estimate_conditional_quantile.rst.txt

@@ -525,6 +525,11 @@ Our estimated conditional quantile is a good approximate and should be better th
 
 
 
+.. rst-class:: sphx-glr-timing
+
+   **Total running time of the script:** (0 minutes 2.022 seconds)
+
+
 .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_conditional_quantile.py:
 
 .. only:: html

openturns/master/_sources/auto_data_analysis/distribution_fitting/plot_estimate_gev_fremantle.rst.txt

@@ -1061,7 +1061,7 @@ improvements with respect to model tested before.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 5.397 seconds)
+   **Total running time of the script:** (0 minutes 5.720 seconds)
 
 
 .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_gev_fremantle.py:

openturns/master/_sources/auto_data_analysis/distribution_fitting/plot_estimate_gev_pirie.rst.txt

@@ -947,7 +947,7 @@ the threshold :math:`c_{\alpha}` or if the p-value is less than the Type I error
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 2.042 seconds)
+   **Total running time of the script:** (0 minutes 2.145 seconds)
 
 
 .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_gev_pirie.py:

openturns/master/_sources/auto_data_analysis/distribution_fitting/plot_estimate_gev_racetime.rst.txt

@@ -1108,7 +1108,7 @@ quadratic model explains even better a large variation in the data.
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 10.552 seconds)
+   **Total running time of the script:** (0 minutes 11.135 seconds)
 
 
 .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_gev_racetime.py:

openturns/master/_sources/auto_data_analysis/distribution_fitting/plot_estimate_multivariate_distribution.rst.txt

@@ -419,7 +419,7 @@ We build joint distribution from marginal distributions and dependency structure
 
 .. rst-class:: sphx-glr-timing
 
-   **Total running time of the script:** (0 minutes 6.356 seconds)
+   **Total running time of the script:** (0 minutes 6.237 seconds)
 
 
 .. _sphx_glr_download_auto_data_analysis_distribution_fitting_plot_estimate_multivariate_distribution.py: