===========================
User manual - Data analyses
===========================

The interface allows one to perform the following types of data analysis:
  - Data analyses to get moments, minimum, maximum, PDF, etc.
  - Marginals inferences
  - Dependence inferences
  - Metamodels creation

1- Data analysis
================

1-1 Creation
''''''''''''

A new sample analysis can be created through:
  - the context menu of the **Definition** item of the data model

    .. image:: /user_manual/graphical_interface/data_analysis/dataAnalysisdefContextMenu.png
        :align: center

  - the **Data analysis** box of the model diagram

    .. image:: /user_manual/graphical_interface/data_analysis/dataModelDiagramBoxes.png
        :align: center

When the analysis is required, a new item is added in the study tree below the data model item.

Its context menu has the following actions:
  - **Rename**: Rename the analysis
  - **Remove**: Remove the analysis from the study

This item is associated with a window showing a
progress bar and Run/Stop buttons, to launch or stop the analysis.

.. image:: /user_manual/graphical_interface/data_analysis/dataAnalysisWindow.png
    :align: center

.. _dataanalysisresult:

1-2 Results
'''''''''''

When the analysis is finished or stopped, the following window appears.

.. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_summary.png
    :align: center

The window shows numerous tabs, some of which are interactively linked (**Table**,
**Parallel coordinates plot**, **Plot matrix** and **Scatter plot** tabs):
when the user selects points on one of these representations, the same points are
automatically selected in the other tabs.

- The **Summary** tab summarizes the results of the analysis, for a selected variable (left column):
  sample size, elapsed time, moment estimates, empirical quantiles, minimum/maximum values, input
  values at extremum.

- The **PDF/CDF** tab presents the PDF/CDF of the variables
  together with a `kernel smoothing <http://openturns.github.io/openturns/latest/theory/data_analysis/kernel_smoothing.html>`_ representation.

  - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters and
    select the graphic type: PDF (default) or CDF
  - Graph interactivity:
      - Left-click to translate the graph
      - Mouse wheel up/down to zoom in/zoom out

  .. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_PDF.png
      :align: center

- The **Box plots** tab presents the `box plot <https://commons.wikimedia.org/w/index.php?curid=14524285>`_
  of the variables. They are rescaled for each variable (:math:`x`), using mean (:math:`\mu`) and standard deviation (:math:`\sigma`): :math:`y = (x - \mu)/\sigma`

  - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
  - Graph interactivity:
      - Left-click to translate the graph
      - Mouse wheel up/down to zoom in/zoom out

  .. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_boxplot.png
      :align: center

.. _dependenceTab:


- The **Dependence** tab displays the `Spearman's matrix <http://openturns.github.io/openturns/latest/theory/data_analysis/spearman_coefficient.html>`_ estimate.

    - The cells are colored according to the value of the Spearman's coefficient.
    - Its context menu allows one to export the table in a CSV file or as a PNG image.
    - Select cells and Press Ctrl+C to copy values in the clipboard

  .. image:: /user_manual/graphical_interface/deterministic_analysis/doe_dependence.png
      :align: center

- The **Table** tab shows the input/output samples. The table can be exported (**Export** button).

  - Table interactivity:
      - Left-click on lines to select them (optional: + Ctrl for adding to current selection, + Shift for contiguous selection)
      - Left-click on column header to sort values in ascending or descending order
      - Left-click on a column header and drag it in another place to change columns order
      - Right-click on table to export or copy to the clipboard (shortcut Ctrl+C) the current selection
      - Ctrl+A selects all lines

  .. image:: /user_manual/graphical_interface/deterministic_analysis/designOfExperimentTable.png
      :align: center


- The **Parallel coordinates plot** tab displays the sample points.

  - Use the :ref:`Graph settings <firstgraphsettings>` window to set up graphical parameters.
  - Graph interactivity:
      - Left-click on columns to select curves (multiple selection possible)

  .. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_Cobweb.png
      :align: center


- The **Plot matrix** tab: histograms of the distribution of each variable (diagonal) and
  scatter plots between each couple of input/output variables (off-diagonal).

  - Use the :ref:`Graph settings <firstgraphsettings>` window to set up graphical parameters.
  - Graph interactivity:
      - Right-click to select points
      - Left-click to translate the graph
      - Mouse wheel up/down to zoom in/zoom out

  .. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_plotmatrixYX.png
      :align: center

- The **Scatter plots** tab displays the scatter plot of two parameters.

  - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters and
    select the variables to plot on X-axis and Y-axis (default: first output versus first input)
  - Graph interactivity:
      - Right-click to select points
      - Left-click to translate the graph
      - Mouse wheel up/down to zoom in/zoom out

  .. image:: /user_manual/graphical_interface/data_analysis/data_model_analysis_scatterplot.png
      :align: center

.. _inferenceAnalysis:

2- Marginals inference
======================

The inference analysis allows one to perform a Bayesian Information Criterion
(`BIC <http://openturns.github.io/openturns/latest/theory/data_analysis/bic.html>`_) and
either a `Kolmogorov-Smirnov <http://openturns.github.io/openturns/latest/theory/data_analysis/kolmogorov_test.html>`_ or `Lilliefors <http://openturns.github.io/openturns/latest/auto_data_analysis/statistical_tests/plot_kolmogorov_test.html#case-2-the-distribution-parameters-are-estimated-from-the-sample>`_ goodness-of-fit tests for 1-d continuous distributions.

New marginals inference can be created thanks to:
  - the context menu of the **Definition** item of the data model

    .. image:: /user_manual/graphical_interface/data_analysis/dataAnalysisdefContextMenu.png
        :align: center

  - the **Marginals inference** box of the model diagram

    .. image:: /user_manual/graphical_interface/data_analysis/dataModelDiagramBoxes.png
        :align: center

.. _marginalsinferencewizard:

2-1 Definition
''''''''''''''

.. image:: /user_manual/graphical_interface/data_analysis/inference_wizard.png
    :align: center

When an analysis is required, a window appears, in order to set up:
  - the variables of interest (default: all variables are analysed) by checking off
    the corresponding line in the first table
  - the list of distributions to infer for each variable (default: Normal distribution):
      - The list of distributions can be different for each variable.
      - Click on **Apply the list of distributions to all variables**
        in the context menu of a variable to set up the same list of distributions
        to the other checked variables.

        .. image:: /user_manual/graphical_interface/data_analysis/inference_wizard_applyToAll.png
            :align: center

      - To add a distribution, click on the **Add** combo box and select a distribution of the list
        which appears (or all of them with the **All** item):

        - the distribution is added in the table
        - the distribution is removed from the combo box

        .. image:: /user_manual/graphical_interface/data_analysis/inference_wizard_distributions_list.png
            :align: center

      - To remove a distribution, select it in the table and click on **Remove**.
        Press the Ctrl or Shift key to select multiple lines.
  - the Kolmogorov-Smirnov/Lilliefors level such that :math:`\alpha = 1 - {\rm level}` is the risk of
    committing a Type I error, that is an incorrect rejection of a true null hypothesis
    (default: 0.05., expected: float in the range :math:`]0, 1[`)
  - **Advanced paramters** are as follows:

    - Require an estimation of the tested distributions parameters confidence interval at a specified level
    - Fine-tune Lilliefors parameters (precision, min/max sampling sizes)



2-2 Launch
''''''''''

When the analysis is required, a new item is added in the study tree below the data model item.

Its context menu has two actions:
  - **Rename**: Rename the analysis
  - **Modify**: Reopen the setting window to change the analysis parameters
  - **Remove**: Remove the analysis from the study

This item is associated with a window displaying the list of the parameters, a
progress bar and Run/Stop buttons, to launch or stop the analysis.

.. image:: /user_manual/graphical_interface/data_analysis/inferenceWindow.png
    :align: center

.. _marginalsinferenceresult:

2-3 Results
'''''''''''

When the analysis is finished or stopped, a window appears.

.. image:: /user_manual/graphical_interface/data_analysis/inference_resultWindow_tab_summary_PDF.png
    :align: center

The results window gathers:

- The **Summary** tab includes, for a selected variable (left column):
   - a table of all the tested distributions, the associated Bayesian Information Criterion value and the p-value.
       - The last column indicates whether the distribution is accepted or not according to the given level.
       - The distributions are sorted in increasing order of BIC values.
   - for the selected distribution:
      - The **PDF/CDF** tab presents the PDF/CDF of the sample
        together with the distribution PDF.

        - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters and
          select the graphic type: PDF (default) or CDF
        - Graph interactivity:
            - Left-click to translate the graph
            - Mouse wheel up/down to zoom in/zoom out

      - The **Q-Q plot** tab presents the `Q-Q plot <http://openturns.github.io/openturns/latest/theory/data_analysis/graphical_fitting_test.html>`_
        which opposes the data quantiles to the quantiles of the tested distribution.

        .. image:: /user_manual/graphical_interface/data_analysis/inference_resultWindow_tab_summary_QQplot.png
           :align: center

        - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
        - Graph interactivity:
            - Left-click to translate the graph
            - Mouse wheel up/down to zoom in/zoom out

      - The **Parameters** tab includes a table with the moments of the selected distribution
        and the values
        `estimate <http://openturns.github.io/openturns/latest/theory/data_analysis/parametric_estimation.html>`_
        of its native parameters.

        .. image:: /user_manual/graphical_interface/data_analysis/inference_resultWindow_tab_summary_parameters.png
           :align: center

        *failed* in the **Acceptation** column means that an error occurred when building a distribution
        with the given sample. Then, the **Parameters** tab shows the error message.

        .. image:: /user_manual/graphical_interface/data_analysis/inference_resultWindow_tab_summary_parameters_error_message.png
            :align: center


The result can be used in the :ref:`Probabilistic model window <inferenceresultwizard>`.

.. _dependenceInference:

3- Dependence inference
=======================

The dependence inference allows one to infer copulas on the sample of the data model.

This analysis can be created thanks to:
  - the context menu of the **Definition** item of the relevant data model

    .. image:: /user_manual/graphical_interface/data_analysis/dataAnalysisdefContextMenu.png
        :align: center

  - the **Dependence inference** box of the model diagram

    .. image:: /user_manual/graphical_interface/data_analysis/dataModelDiagramBoxes.png
        :align: center

.. _dependenceinferencewizard:

3-1 Definition
''''''''''''''

When an analysis is required, a window appears:

.. image:: /user_manual/graphical_interface/data_analysis/dependenceInference_wizard.png
    :align: center

The windows allows one to set up:
  - the groups of variables to test:
      - Select at least two variables of the model (left table):
          - Refer to the estimate of the :ref:`Spearman's matrix <dependenceTab>` in
            the data analysis result window to create groups
          - For convenience, the list of groups may be set by default thanks to this estimate
            (if correlation between variables exists)
      - Click on the right arrow:
          - the group is added in the second table
          - a third table appears with the default item *Normal*

  .. image:: /user_manual/graphical_interface/data_analysis/dependenceInference_wizardOneGroup.png
    :align: center

  - the copulas to infer on the groups:
    - Click on the **Add** combo box
    - Select a copula in the list (or all of them with the **All** item):

        - *For a pair of variables* : bivariate
          `copulas <http://openturns.github.io/openturns/latest/user_manual/_generated/openturns.Copula.html>`_
          are available (Ali-Mikhail-Haq, Clayton, Farlie-Gumbel-Morgenstern, Frank, Gumbel, Normal)
        - *For a group with more than two variables*: only the Normal copula is available
          (**Add** and **Remove** buttons are then disabled)

    .. image:: /user_manual/graphical_interface/data_analysis/dependenceInference_wizard_copulaList.png
      :align: center

To remove a group:
  - Select a group in the second table
  - Click on the left arrow

3-2 Launch
'''''''''''

When the analysis is required, a new item is added in the study tree below the data model item.

Its context menu has the following actions:
  - **Rename**: Rename the analysis;
  - **Modify**: Reopen the setting window to change the analysis parameters;
  - **Remove**: Remove the analysis from the study.

This item is associated with a window displaying the list of the parameters, a
progress bar and Run/Stop buttons, to launch or stop the analysis.

.. image:: /user_manual/graphical_interface/data_analysis/copulaInferenceWindow.png
    :align: center

.. _dependenceinferenceresult:

3-3 Results
'''''''''''

When the analysis is finished or stopped, a window appears:

.. image:: /user_manual/graphical_interface/data_analysis/copulaInference_resultWindow_tab_summary_PDF.png
    :align: center

The window gathers:

- The **Summary** tab includes, for a selected set of variables:
    - a table of all the tested copulas
    - for the selected copula:
        - the **PDF/CDF** tab presents, for each pair of variables, the PDF/CDF of the sample
          together with the distribution PDF.

            - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters and
              select the graphic type: PDF (default) or CDF
            - Graph interactivity:
                - Left-click to translate the graph
                - Mouse wheel up/down to zoom in/zoom out

        - the **Kendall plot** tab presents a visual fitting test for each pair of variables using the
          `Kendall plot <http://openturns.github.io/openturns/latest/theory/data_analysis/graphical_fitting_test.html>`_.
          This plot can be interpreted as a QQ-plot (for marginals): the more the curve fits
          the diagonal, the more adequate the dependence model is.

            - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
            - Graph interactivity:
                - Left-click to translate the graph
                - Mouse wheel up/down to zoom in/zoom out

        .. image:: /user_manual/graphical_interface/data_analysis/copulaInference_resultWindow_tab_summary_Kendall.png
            :align: center

        - the **Parameters** tab includes the parameters `estimate <http://openturns.github.io/openturns/latest/theory/data_analysis/parametric_estimation.html>`_ of the selected copula.

            .. image:: /user_manual/graphical_interface/data_analysis/copulaInference_resultWindow_tab_summary_parameters.png
                :align: center

            - *For the Gaussian copula*: the tab displays the `Spearman's coefficients <http://openturns.github.io/openturns/latest/theory/data_analysis/spearman_coefficient.html>`_.


            - *'-'* in the **BIC** column means that an error occurred when building a copula
              with the given sample. Then, the **Parameters** tab shows the error message.

            .. image:: /user_manual/graphical_interface/data_analysis/copulaInference_resultWindow_tab_summary_parameters_ErrorMessage.png
                :align: center

The result can be used in the :ref:`Probabilistic model window <dependenceinferenceresultwizard>`.

4- Metamodel creation
======================

To perform this analysis, the data model or the design of experiments must contain an output sample.

A new metamodel can be created in 4 different ways:
  - the context menu of a design of experiments item

    .. image:: /user_manual/graphical_interface/data_analysis/doe_eval_ContextMenu.png
        :align: center

  - the **Metamodel creation** box of a physical model diagram

    .. image:: /user_manual/graphical_interface/data_analysis/physicalModel_Diagram_metamodelBox.png
        :align: center

  - the context menu of the **Definition** item of a data model

    .. image:: /user_manual/graphical_interface/data_analysis/dataAnalysisdefContextMenu.png
        :align: center

  - the **Metamodel creation** box of a data model diagram

    .. image:: /user_manual/graphical_interface/data_analysis/dataModelDiagramBoxes.png
        :align: center

.. _metamodelwizard:

4-1 Definition
''''''''''''''

When an analysis is required, a window appears, in order to set up:
  - the outputs of interest (**Select outputs** - default: all outputs are analyzed)
  - the method: polynomial regression (default), functional chaos or kriging

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_wizard.png
    :align: center

4-1-1 Linear regression
~~~~~~~~~~~~~~~~~~~~~~~~~~~

The **Linear regression** window allows one to define:
  - **Parameters**: polynomial degree (default: 1, expected: integer in [1, 2]), interaction terms (if degree>1 only)

Refer to :class:`~persalsys.PolynomialRegressionAnalysis` for implementation details.

4-1-2 Functional chaos
~~~~~~~~~~~~~~~~~~~~~~

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_functional_chaos_wizard.png
    :align: center

The **Functional chaos parameters** window allows one to define:
  - **Parameters**: chaos degree (default: 2, expected: integer greater or equal to 1)
  - **Advanced Parameters** (default: hidden): sparse chaos (default: not sparse)

Refer to :class:`~persalsys.FunctionalChaosAnalysis` for implementation details.

4-1-3 Kriging
~~~~~~~~~~~~~

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_kriging_wizard.png
    :align: center

The **Kriging parameters** window allows one to define:
  - **Parameters**:
     - *The type of covariance model*: Squared exponential (default), Absolute exponential,
       Generalized exponential, Matérn model
     - *Parameters of the covariance model* (default: hidden, visible if a model is chosen):
         - **Generalized exponential**: parameter **p**,
           exponent of the euclidean norm (default: 1., positive float expected)

         .. image:: /user_manual/graphical_interface/data_analysis/kriging_p_parameter.png
              :align: center

         - **Matérn**: coefficient **nu** (default: 1.5, positive float expected)

         .. image:: /user_manual/graphical_interface/data_analysis/kriging_nu_parameter.png
              :align: center

     - *The type of the trend basis*: Constant (default), Linear or Quadratic
  - **Advanced Parameters** are accessible for model covariance optimization (default: hidden):
     - Optimize the covariance model parameters (default: checked)
     - Scales for each input (default: 1): To edit the scales, click on the "**...**" button to
       generate the input variables table and their scale through a wizard.

     .. image:: /user_manual/graphical_interface/data_analysis/kriging_scale_wizard.png
          :align: center

     - Amplitude of the process (default: 1., positive float expected)

Refer to :class:`~persalsys.KrigingAnalysis` for implementation details.

4-1-3 Validation
~~~~~~~~~~~~~~~~

In the following window, the generated metamodel can be validated, with three different methods:
  - *Analytically* (default): This method corresponds to an approximation of the Leave-one-out method result.
      - For more information about Kriging, see O. Dubrule, Cross Validation of Kriging in a Unique Neighborhood,
        Mathematical Geology,1983.
      - For more information about Functional chaos, see G. Blatman, Adaptive sparse polynomial chaos
        expansions for uncertainty propagation and sensitivity analysis.,
        PhD thesis. Blaise Pascal University-Clermont II, France, 2009.
  - *Using a test sample*: The data sample is divided into two subsamples, by picking points randomly (default seed = 1):
    training sample (default: 80% of the sample points) and test sample
    (default: 20% of the sample points).
    A new metamodel is built with the training sample and is validated with the test sample.
  - *Using the* `K-Fold <http://openturns.github.io/openturns/latest/theory/meta_modeling/cross_validation.html>`_ *method*:
    Define the number of folds (default: 5, expected: integer greater than 1) and specify how the
    folds are generated (default seed:1).

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_validation_page.png
    :align: center

4-2 Results
'''''''''''

When the window is validated, a new element appears in the study tree below the
data model item or the design of experiments item.

The context menu of this item contains these actions:
  - **Rename**: Rename the analysis
  - **Modify**: Reopen the setting window to change the analysis parameters
  - **Convert metamodel into physical model** (default: disabled, enabled when the analysis
    is successfully finished): Add the metamodel in the study tree
  - **Remove**: Remove the analysis from the study

This item is associated with a window displaying the list of the parameters, a
progress bar and Run/Stop buttons, to launch or stop the analysis.

.. image:: /user_manual/graphical_interface/data_analysis/metaModelWindow.png
    :align: center

.. _functionalchaosresult:

4-2-1 Functional chaos
~~~~~~~~~~~~~~~~~~~~~~

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_plot.png
    :align: center

The results window gathers:

- The **Metamodel** tab shows different information about the selected output (left column):
    - Number of points
    - Relative error: :math:`\displaystyle rel = \frac{\sum_{i=0}^N (y_i - \hat{y_i})^2}{\sum_{i=0}^N {(y_i - \bar{y})^2}}`
    - Residual: :math:`\displaystyle res = \frac{\sqrt{\sum_{i=0}^N (y_i - \hat{y_i})^2}}{N}`.
    - The fitting curve between the physical model output values (**Real otput values**) and the
      metamodel values (**Prediction**).
      The reference diagonal (in black) is built with the physical model output values.

        - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
        - Graph interactivity:
           - Left-click to translate the graph
           - Mouse wheel up/down to zoom in/zoom out

- The **Results** tab presents different parameter, for a selected output (left column):
    - first and second order moments
    - polynomial basis: dimension, maximum degree, full/truncated size
    - part of variance explained by each polynom

    .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_moments.png
        :align: center

- The **Sobol indices** tab includes, for a selected output (left column):

  - The graphic representation of the first and total order indices for each variable.
    Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
  - A summary table with the first and total order indices.

      - Table interactivity:
          - Select cells and Press Ctrl+C to copy values in the clipboard
          - Left-click on column header to sort values in ascending or descending order.
            Sorting the table will automatically sort the indices on the graph.

  - The index corresponding to the interactions (below the table).

  .. |attentionButton| image:: /user_manual/graphical_interface/probabilistic_analysis/task-attention.png

  If the Sobol's indices estimates are incoherent, an |attentionButton| will appear in the table.
  It is advised to refer to the associated warning message (tooltip of the |attentionButton|).

  .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_sobol_indices.png
    :align: center

- The **Validation** tab (default: hidden; visible if a metamodel validation is required) shows for each method and selected output:
    - The metamodel predictivity coefficient: :math:`\displaystyle Q2 = 1 - \frac{\sum_{i=0}^N (y_i - \hat{y_i})^2}{\sum_{i=0}^N {(\bar{y} - y_i)^2}}`
    - The residual: :math:`\displaystyle res = \frac{\sqrt{\sum_{i=0}^N (y_i - \hat{y_i})^2}}{N}`.
    - *K-Fold* and *Test sample*: A plot showing the relation between the output values (physical
      model) and the predicted metamodel values. The relation is compared to a reference
      diagonal built with the physical model output values.

        - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
        - Graph interactivity:
           - Left-click to translate the graph
           - Mouse wheel up/down to zoom in/zoom out

      .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_LOO_plot.png
         :align: center

    - *Analytical*: the Q2 value

      .. image:: /user_manual/graphical_interface/data_analysis/FC_analyticalValidation.png
         :align: center


- The **Parameters** tab summarizes the parameters of the metamodel creation.

  .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_parameters.png
      :align: center

.. _krigingresult:

4-2-2 Kriging
~~~~~~~~~~~~~

.. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_kriging_plot.png
    :align: center

The results window gathers:

- The **Metamodel** tab shows for a selected output the graphic relation between output values from
  the physical model (**Real output values**) and metamodel values (**Prediction**).
  The reference diagonal (in black) is built with the physical model output values.

  - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
  - Graph interactivity:
         - Left-click to translate the graph
         - Mouse wheel up/down to zoom in/zoom out

- The **Results** tab presents the optimized covariance model parameters and the trend coefficients.

  .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_kriging_results.png
      :align: center

- If a metamodel validation is required, a **Validation** tab appears for the selected method and output:
    - The residual: :math:`\displaystyle res = \frac{\sqrt{\sum_{i=0}^N (y_i - \hat{y_i})^2}}{N}`.
    - The metamodel predictivity coefficient: :math:`\displaystyle Q2 = 1 - \frac{\sum_{i=0}^N (y_i - \hat{y_i})^2}{\sum_{i=0}^N {(\bar{y} - y_i)^2}}`
    - A plot showing the relation between the output values (physical
      model) and the predicted metamodel values. The relation is compared to a reference
      diagonal built with the physical model output values.

      - Use the :ref:`Graph settings <secondgraphsettings>` window to set up graphical parameters.
      - Graph interactivity:
           - Left-click to translate the graph
           - Mouse wheel up/down to zoom in/zoom out

      .. image:: /user_manual/graphical_interface/data_analysis/metaModel_result_window_LOO_plot.png
         :align: center

- The **Parameters** tab summarizes the parameters of the metamodel creation.
