.. _sphx_glr_auto_examples_plot_effect_of_noise.py:


Effect of measurement noise on accuracy of fit
==============================================

Experimental noise often corrupts concentration profiles. When the diffusion
matrix is determined from noisy concentration profiles, the accuracy of the
estimation decreases when the intensity of the noise increases. Furthermore,
minor eigenvectors and eigenvalues are more affected by the noise than the
dominant eigenvector.



.. code-block:: python


    import numpy as np
    import matplotlib.pyplot as plt
    from multidiff import compute_diffusion_matrix, create_diffusion_profiles







Our diffusion system has three components (e.g. three oxides). The diffusion
 matrix has two eigenvalues with quite different magnitudes. Exchange vectors
 correspond to the exchange of the three possible pairs of components
 (meaning that each endmember is enriched in one component and poorer in
 another, compared to the other endmember).



.. code-block:: python


    diags = np.array([0.5, 5.])
    P = np.matrix([[1, 1], [-1, 0]])
    xpoints_exp1 = np.linspace(-10, 10, 100)
    x_points = [xpoints_exp1] * 3


    exchange_vectors = np.array([[0, 1, 1],
                                 [1, -1, 0],
                                 [-1, 0, -1]])


    # Initialization for the diffusion matrix
    diags_init = np.array([1, 2])
    P_init = np.matrix([[1.5, 0.9], [-1, -0.7]])







We now iterate over increasing values of measurement Gaussian noise. For
each value of the noise, synthetic diffusion profiles are created, and the
diffusion matrix is computed from the profiles. We plot the relative error
on the eigenvalues, defined as
abs((lambda_computed - lambda_real)/lambda_real).
The relative error is consistently larger for the smaller eigenvalue, since
errors on the larger (dominant) eigenvalue and eigenvectors result in errors
of comparable magnitude on the smaller eigenvalue(s) and eigenvector(s).

Note that for the dominant eigenvalue, the relative error is smaller than the
intensity of the noise. Indeed, we have enough experiments to overconstrain
the system, plus a large number of measurement points.



.. code-block:: python


    noises = np.arange(0, 0.2, 0.02)
    errors = np.empty((len(noises), 2))

    for i, noise in enumerate(noises):
        concentration_profiles = create_diffusion_profiles((diags, P), x_points,
                                        exchange_vectors, noise_level=noise)
        diags_res, eigvecs, _, _, _ = (
                compute_diffusion_matrix((diags_init, P_init), x_points,
                                            concentration_profiles, plot=False))
        errors[i, :] = np.abs(np.sort(diags_res) - diags) / diags


    plt.figure()
    plt.plot(noises, errors[:, 0], 'co--',
            label='relative error on minor eigenvalue')
    plt.plot(noises, errors[:, 1], 'rs-', ms=10,
             label='relative error on major eigenvalue')
    plt.xlabel('intensity of Gaussian noise', fontsize=16)
    plt.title('Relative error on fitted eigenvalues')
    plt.legend(loc='best')

    plt.show()



.. image:: /auto_examples/images/sphx_glr_plot_effect_of_noise_001.png
    :align: center




**Total running time of the script:** ( 0 minutes  1.760 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

    `Generated by Sphinx-Gallery <http://sphinx-gallery.readthedocs.io>`_