.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/migration/plot_03_israel_rosenthal_wei.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_credit_migration_plot_03_israel_rosenthal_wei.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_credit_migration_plot_03_israel_rosenthal_wei.py:


Israel, Rosenthal and Wei: finding a valid generator
====================================================

Going from an annual transition matrix :math:`P` to a generator is not
as simple as :math:`\log P`. Israel, Rosenthal and Wei (2001) showed that
for empirical rating matrices the principal logarithm usually has
negative off-diagonal entries, which would be negative migration
intensities. Their fix: set those entries to zero and absorb the change
in the diagonal. A weighted variant spreads it over the row instead.
Both leave :math:`e^{G}` within a few basis points of :math:`P`.

.. GENERATED FROM PYTHON SOURCE LINES 15-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import RATINGS, creditmetrics_matrix, generator_from_transition, is_valid_generator, regularize_generator, transition_from_generator
    from financialkit.credit.visualizers import plot_transition_matrix

    P = creditmetrics_matrix()
    L = generator_from_transition(P)
    off = L - np.diag(np.diag(L))
    print(f"log(P) is a valid generator: {is_valid_generator(L)}; {np.sum(off < 0)} negative off-diagonal entries, most negative {off.min():.2e}")
    ax = plot_transition_matrix(L, labels=RATINGS, fmt="{:.2f}")
    ax.set_title("log(P) in % per year: negative intensities in red")




.. image-sg:: /api/gallery/credit/migration/images/sphx_glr_plot_03_israel_rosenthal_wei_001.png
   :alt: log(P) in % per year: negative intensities in red
   :srcset: /api/gallery/credit/migration/images/sphx_glr_plot_03_israel_rosenthal_wei_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    log(P) is a valid generator: False; 7 negative off-diagonal entries, most negative -3.10e-04

    Text(0.5, 1.0, 'log(P) in % per year: negative intensities in red')



.. GENERATED FROM PYTHON SOURCE LINES 29-31

The two adjustments
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-42

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    for ax, method in zip(axes, ("diagonal", "weighted"), strict=True):
        G = regularize_generator(L, method)
        err = transition_from_generator(G) - P
        plot_transition_matrix(err, labels=RATINGS, ax=ax, fmt="{:.2f}", scale=1e4)
        ax.set_title(f"{method}: exp(G) - P in bp (max {1e4 * np.abs(err).max():.1f})")
        print(f"{method:8s}: valid {is_valid_generator(G)}, max |exp(G) - P| = {np.abs(err).max():.2e}")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/credit/migration/images/sphx_glr_plot_03_israel_rosenthal_wei_002.png
   :alt: diagonal: exp(G) - P in bp (max 2.4), weighted: exp(G) - P in bp (max 2.4)
   :srcset: /api/gallery/credit/migration/images/sphx_glr_plot_03_israel_rosenthal_wei_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    diagonal: valid True, max |exp(G) - P| = 2.39e-04
    weighted: valid True, max |exp(G) - P| = 2.39e-04





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

   **Total running time of the script:** (0 minutes 0.191 seconds)


.. _sphx_glr_download_api_gallery_credit_migration_plot_03_israel_rosenthal_wei.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/credit/migration/plot_03_israel_rosenthal_wei.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_israel_rosenthal_wei.ipynb <plot_03_israel_rosenthal_wei.ipynb>`

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

      :download:`Download Python source code: plot_03_israel_rosenthal_wei.py <plot_03_israel_rosenthal_wei.py>`

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

      :download:`Download zipped: plot_03_israel_rosenthal_wei.zip <plot_03_israel_rosenthal_wei.zip>`


.. only:: html

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

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