.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/inference/plot_oneway_synthetic.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_inference_plot_oneway_synthetic.py>`
        to download the full example code. or to run this example in your browser via Binder

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_inference_plot_oneway_synthetic.py:


One-way functional ANOVA with synthetic data
============================================

This example shows how to perform a functional one-way ANOVA test with
synthetic data.

.. GENERATED FROM PYTHON SOURCE LINES 8-12

.. code-block:: Python


    # Author: David García Fernández
    # License: MIT








.. GENERATED FROM PYTHON SOURCE LINES 13-30

**One-way ANOVA** (analysis of variance) is a test that can be used to
compare the means of different samples of data.
Let :math:`X_{ij}(t), j=1, \dots, n_i` be trajectories corresponding to
:math:`k` independent samples :math:`(i=1,\dots,k)` and let
:math:`E(X_i(t)) = m_i(t)`. Thus, the null hypothesis in the statistical
test is:

.. math::

    H_0: m_1(t) = \dots = m_k(t)

In this example we will explain the nature of ANOVA method and its behavior
under certain conditions simulating data. Specifically, we will generate
three different trajectories, for each one we will simulate a stochastic
process by adding to them white noise. The main objective of the
test is to illustrate the differences in the results of the ANOVA method
when the covariance function of the brownian processes changes.

.. GENERATED FROM PYTHON SOURCE LINES 33-34

First, the means for the future processes are drawn.

.. GENERATED FROM PYTHON SOURCE LINES 34-58

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from skfda.representation import FDataGrid

    n_samples = 10
    n_features = 100
    n_groups = 3
    start = 0
    stop = 1

    t = np.linspace(start, stop, n_features)

    m1 = t * (1 - t) ** 5
    m2 = t ** 2 * (1 - t) ** 4
    m3 = t ** 3 * (1 - t) ** 3

    FDataGrid(
        [m1, m2, m3],
        dataset_name="Means to be used in the simulation",
    ).plot()
    plt.show()




.. image-sg:: /auto_examples/inference/images/sphx_glr_plot_oneway_synthetic_001.png
   :alt: Means to be used in the simulation
   :srcset: /auto_examples/inference/images/sphx_glr_plot_oneway_synthetic_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 59-61

A total of ``n_samples`` trajectories will be created for each mean, so an
array of labels is created to identify them when plotting.

.. GENERATED FROM PYTHON SOURCE LINES 61-66

.. code-block:: Python


    groups = np.full(n_samples * n_groups, "Sample 1")
    groups[10:20] = "Sample 2"
    groups[20:] = "Sample 3"








.. GENERATED FROM PYTHON SOURCE LINES 67-70

First simulation uses a low :math:`\sigma^2 = 0.01` value. In this case the
differences between the means of each group should be clear, and the
p-value for the test should be near to zero.

.. GENERATED FROM PYTHON SOURCE LINES 70-110

.. code-block:: Python


    from skfda.datasets import make_gaussian_process
    from skfda.inference.anova import oneway_anova
    from skfda.misc.covariances import WhiteNoise

    sigma2 = 0.01
    cov = WhiteNoise(variance=sigma2)

    fd1 = make_gaussian_process(
        n_samples,
        mean=m1,
        cov=cov,
        n_features=n_features,
        random_state=1,
        start=start,
        stop=stop,
    )
    fd2 = make_gaussian_process(
        n_samples,
        mean=m2,
        cov=cov,
        n_features=n_features,
        random_state=2,
        start=start,
        stop=stop,
    )
    fd3 = make_gaussian_process(
        n_samples,
        mean=m3,
        cov=cov,
        n_features=n_features,
        random_state=3,
        start=start,
        stop=stop,
    )
    stat, p_val = oneway_anova(fd1, fd2, fd3, random_state=4)
    print(f"Statistic: {stat:.3f}")
    print(f"p-value: {p_val:.3f}")






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Statistic: 0.082
    p-value: 0.073




.. GENERATED FROM PYTHON SOURCE LINES 111-115

In the following, the same process will be followed incrementing sigma
value, this way the differences between the averages of each group will be
lower and the p-values will increase (the null hypothesis will be harder to
refuse).

.. GENERATED FROM PYTHON SOURCE LINES 117-118

Plot for :math:`\sigma^2 = 0.1`:

.. GENERATED FROM PYTHON SOURCE LINES 118-154

.. code-block:: Python

    sigma2 = 0.1
    cov = WhiteNoise(variance=sigma2)

    fd1 = make_gaussian_process(
        n_samples,
        mean=m1,
        cov=cov,
        n_features=n_features,
        random_state=1,
        start=t[0],
        stop=t[-1],
    )
    fd2 = make_gaussian_process(
        n_samples,
        mean=m2,
        cov=cov,
        n_features=n_features,
        random_state=2,
        start=t[0],
        stop=t[-1],
    )
    fd3 = make_gaussian_process(
        n_samples,
        mean=m3,
        cov=cov,
        n_features=n_features,
        random_state=3,
        start=t[0],
        stop=t[-1],
    )

    stat, p_val = oneway_anova(fd1, fd2, fd3, random_state=4)
    print(f"Statistic: {stat:.3f}")
    print(f"p-value: {p_val:.3f}")






.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Statistic: 0.653
    p-value: 0.329




.. GENERATED FROM PYTHON SOURCE LINES 155-156

Plot for :math:`\sigma^2 = 1`:

.. GENERATED FROM PYTHON SOURCE LINES 156-191

.. code-block:: Python


    sigma2 = 1
    cov = WhiteNoise(variance=sigma2)

    fd1 = make_gaussian_process(
        n_samples,
        mean=m1,
        cov=cov,
        n_features=n_features,
        random_state=1,
        start=t[0],
        stop=t[-1],
    )
    fd2 = make_gaussian_process(
        n_samples,
        mean=m2,
        cov=cov,
        n_features=n_features,
        random_state=2,
        start=t[0],
        stop=t[-1],
    )
    fd3 = make_gaussian_process(
        n_samples,
        mean=m3,
        cov=cov,
        n_features=n_features,
        random_state=3,
        start=t[0],
        stop=t[-1],
    )

    stat, p_val = oneway_anova(fd1, fd2, fd3, random_state=4)
    print(f"Statistic: {stat:.3f}")
    print(f"p-value: {p_val:.3f}")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Statistic: 6.304
    p-value: 0.394





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.106 seconds)


.. _sphx_glr_download_auto_examples_inference_plot_oneway_synthetic.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/GAA-UAM/scikit-fda/develop?filepath=examples/inference/plot_oneway_synthetic.py
        :alt: Launch binder
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_oneway_synthetic.ipynb <plot_oneway_synthetic.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_oneway_synthetic.py <plot_oneway_synthetic.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_oneway_synthetic.zip <plot_oneway_synthetic.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_