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        :ref:`Go to the end <sphx_glr_download_auto_examples_basis_plot_basis_1d.py>`
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.. _sphx_glr_auto_examples_basis_plot_basis_1d.py:


One-dimensional Basis
=====================

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.. code-block:: Python


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

    # Load packages
    import numpy as np

    from FDApy.representation import Basis
    from FDApy.representation import DenseArgvals
    from FDApy.visualization import plot








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The package include different basis functions to represent functional data. In this section, we are showing the building blocks of the representation of basis functions. To define a :class:`~FDApy.representation.Basis` object, we need to specify the name of the basis, the number of functions in the basis and the sampling points. The sampling points are defined as a :class:`~FDApy.representation.DenseArgvals`.

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We will show the basis functions for the Fourier, B-splines and Wiener basis. The number of functions in the basis is set to :math:`5` and the sampling points are defined as a :class:`~FDApy.representation.DenseArgvals` object with a hundred points between :math:`0` and :math:`1`.

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.. code-block:: Python


    # Parameters
    n_functions = 5
    argvals = DenseArgvals({"input_dim_0": np.linspace(0, 1, 101)})









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Fourier basis
-------------
First, we will show the basis functions for the Fourier basis. The basis functions consist of the sine and cosine functions with a frequency that increases with the number of the function. Note that the first function is a constant function. This basis may be used to represent periodic functions.

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.. code-block:: Python


    basis = Basis(name="fourier", n_functions=n_functions, argvals=argvals)

    _ = plot(basis)




.. image-sg:: /auto_examples/basis/images/sphx_glr_plot_basis_1d_001.png
   :alt: plot basis 1d
   :srcset: /auto_examples/basis/images/sphx_glr_plot_basis_1d_001.png
   :class: sphx-glr-single-img





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B-splines basis
---------------
Second, we will show the basis functions for the B-splines basis. The basis functions are piecewise polynomials that are smooth at the knots. The number of knots is equal to the number of functions in the basis minus :math:`2`. This basis may be used to represent smooth functions.

.. GENERATED FROM PYTHON SOURCE LINES 42-47

.. code-block:: Python


    basis = Basis(name="bsplines", n_functions=n_functions, argvals=argvals)

    _ = plot(basis)




.. image-sg:: /auto_examples/basis/images/sphx_glr_plot_basis_1d_002.png
   :alt: plot basis 1d
   :srcset: /auto_examples/basis/images/sphx_glr_plot_basis_1d_002.png
   :class: sphx-glr-single-img





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Wiener basis
------------
Third, we will show the basis functions for the Wiener basis. The basis functions are the eigenfunctions of a Brownian process. This basis may be used to represent rough functions.

.. GENERATED FROM PYTHON SOURCE LINES 51-55

.. code-block:: Python


    basis = Basis(name="wiener", n_functions=n_functions, argvals=argvals)

    _ = plot(basis)



.. image-sg:: /auto_examples/basis/images/sphx_glr_plot_basis_1d_003.png
   :alt: plot basis 1d
   :srcset: /auto_examples/basis/images/sphx_glr_plot_basis_1d_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.272 seconds)


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