.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/simulation/plot_karhunen.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_simulation_plot_karhunen.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_simulation_plot_karhunen.py:


Simulation using Karhunen-Loève decomposition
=============================================

.. GENERATED FROM PYTHON SOURCE LINES 6-17

.. code-block:: Python


    # Author: Steven Golovkine <steven_golovkine@icloud.com>
    # License: MIT

    # Load packages
    import numpy as np

    from FDApy.representation import DenseArgvals
    from FDApy.simulation import KarhunenLoeve
    from FDApy.visualization import plot








.. GENERATED FROM PYTHON SOURCE LINES 18-36

The simulation of univariate functional data
:math:`X: \mathcal{T} \rightarrow \mathbb{R}` is based on the truncated
Karhunen-Loève representation of :math:`X`. Consider that the representation
is truncated at :math:`K` components, then, for a particular realization
:math:`i` of the process :math:`X`:

.. math::
      X_i(t) = \mu(t) + \sum_{k = 1}^K c_{i, k}\phi_k(t),
      \quad t \in \mathcal{T}

with a common mean function :math:`\mu(t)` and eigenfunctions
:math:`\phi_k, k = 1, \cdots, K`. The scores :math:`c_{i, k}` are the
projection of the curves :math:`X_i` onto the eigenfunctions :math:`\phi_k`.
These scores are random variables with mean :math:`0` and variance
:math:`\lambda_k`, which are the eigenvalues associated to each
eigenfunctions and that decreases toward :math:`0` when :math:`k` goes to
infinity. This representation is valid for domains of arbitrary dimension,
such as images (:math:`\mathcal{T} = \mathbb{R}^2`).

.. GENERATED FROM PYTHON SOURCE LINES 36-47

.. code-block:: Python



    # Set general parameters
    rng = 42
    n_obs = 10

    # Parameters of the basis
    name = "fourier"
    n_functions = 25
    argvals = DenseArgvals({"input_dim_0": np.arange(0, 10.01, 0.01)})








.. GENERATED FROM PYTHON SOURCE LINES 48-57

Simulation for one-dimensional curve
------------------------------------

**First example**
---
We simulate :math:`N = 10` curves on the one-dimensional observation grid
:math:`\{0, 0.1, 0.2, \cdots, 1\}` (default), based on the first
:math:`K = 25` Fourier basis functions on :math:`[0, 1]` and the variance of
the scores random variables equal to :math:`1` (default).

.. GENERATED FROM PYTHON SOURCE LINES 57-64

.. code-block:: Python

    kl = KarhunenLoeve(
        basis_name=name, n_functions=n_functions, random_state=rng
    )
    kl.new(n_obs=n_obs)

    _ = plot(kl.data)




.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_karhunen_001.png
   :alt: plot karhunen
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_karhunen_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 65-71

**Second example**
---
We simulate :math:`N = 10` curves on the one-dimensional observation grid
:math:`\{0, 0.01, 0.02, \cdots, 10\}`, based on the first
:math:`K = 25` Fourier basis functions on :math:`[0, 10]` and the variance of
the scores random variables equal to :math:`1` (default).

.. GENERATED FROM PYTHON SOURCE LINES 71-78

.. code-block:: Python

    kl = KarhunenLoeve(
        basis_name=name, argvals=argvals, n_functions=n_functions, random_state=rng
    )
    kl.new(n_obs=n_obs)

    _ = plot(kl.data)




.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_karhunen_002.png
   :alt: plot karhunen
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_karhunen_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 79-85

**Third example**
---
We simulate :math:`N = 10` curves on the one-dimensional observation grid
:math:`\{0, 0.01, 0.02, \cdots, 10\}` (default), based on the first
:math:`K = 25` Fourier basis functions on :math:`[0, 1]` and the decreasing
of the variance of the scores is exponential.

.. GENERATED FROM PYTHON SOURCE LINES 85-92

.. code-block:: Python

    kl = KarhunenLoeve(
        basis_name=name, argvals=argvals, n_functions=n_functions, random_state=rng
    )
    kl.new(n_obs=n_obs, clusters_std="exponential")

    _ = plot(kl.data)




.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_karhunen_003.png
   :alt: plot karhunen
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_karhunen_003.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 93-106

Simulation for two-dimensional curve (image)
--------------------------------------------

For the simulation on a two-dimensional domain, we construct an
two-dimensional eigenbasis based on tensor products of univariate eigenbasis.

**First example**
---
We simulate :math:`N = 1` image on the two-dimensional observation grid
:math:`\{0, 0.01, 0.02, \cdots, 10\} \times \{0, 0.01, 0.02, \cdots, 10\}`
(default), based on the tensor product of the first :math:`K = 25` Fourier
basis functions on :math:`[0, 10] \times [0, 10]` and the variance of
the scores random variables equal to :math:`1` (default).

.. GENERATED FROM PYTHON SOURCE LINES 106-121

.. code-block:: Python


    # Parameters of the basis
    name = ("fourier", "fourier")
    n_functions = (5, 5)
    argvals = DenseArgvals(
        {"input_dim_0": np.arange(0, 10.01, 0.01), "input_dim_1": np.arange(0, 10.01, 0.01)}
    )

    kl = KarhunenLoeve(
        basis_name=name, argvals=argvals, n_functions=n_functions, random_state=rng
    )
    kl.new(n_obs=1)

    _ = plot(kl.data)




.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_karhunen_004.png
   :alt: plot karhunen
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_karhunen_004.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 122-129

**Second example**
---
We simulate :math:`N = 1` image on the two-dimensional observation grid
:math:`\{0, 0.01, 0.02, \cdots, 1\} \times \{0, 0.01, 0.02, \cdots, 1\}`
(default), based on the tensor product of the first :math:`K = 25` Fourier
basis functions on :math:`[0, 1] \times [0, 1]` and the decreasing
of the variance of the scores is linear.

.. GENERATED FROM PYTHON SOURCE LINES 129-135

.. code-block:: Python

    kl = KarhunenLoeve(
        basis_name=name, argvals=argvals, n_functions=n_functions, random_state=rng
    )
    kl.new(n_obs=1, clusters_std="linear")

    _ = plot(kl.data)



.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_karhunen_005.png
   :alt: plot karhunen
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_karhunen_005.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.854 seconds)


.. _sphx_glr_download_auto_examples_simulation_plot_karhunen.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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