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.. "auto_tutorials/multi_fidelity/plot_multioutput_acv.py"
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.. only:: html

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        :ref:`Go to the end <sphx_glr_download_auto_tutorials_multi_fidelity_plot_multioutput_acv.py>`
        to download the full example code

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

.. _sphx_glr_auto_tutorials_multi_fidelity_plot_multioutput_acv.py:


Multioutput Approximate Control Variates
========================================

This tutorial demonstrates how computing statistics for multiple outputs simultaneoulsy can improve the accuracy of ACV estimates of individual statistics when compared to ACV applied to each output separately [DWBG2024]_.

The optimal control variate weights are obtained by minimizing the estimator covariance [RM1985]_.

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.. code-block:: default

    import numpy as np
    import matplotlib.pyplot as plt

    from pyapprox import multifidelity as mf
    from pyapprox.benchmarks import setup_benchmark
    from pyapprox.util.visualization import mathrm_labels, mathrm_label

    np.random.seed(1)
    benchmark = setup_benchmark("multioutput_model_ensemble")
    costs = np.array([1, 0.01, 0.001])
    nmodels = 3

    cov = benchmark.covariance

    labels = ([r"$(f_{0})_{%d}$" % (ii+1) for ii in range(benchmark.nqoi)] +
              [r"$(f_{2})_{%d}$" % (ii+1) for ii in range(benchmark.nqoi)] +
              [r"$(f_{2})_{%d}$" % (ii+1) for ii in range(benchmark.nqoi)])
    ax = plt.subplots(1, 1, figsize=(8, 6))[1]
    _ = mf.plot_correlation_matrix(
        mf.get_correlation_from_covariance(cov), ax=ax, model_names=labels,
        label_fontsize=20)

    target_cost = 10
    stat = mf.multioutput_stats["mean"](benchmark.nqoi)
    stat.set_pilot_quantities(cov)
    est = mf.get_estimator("gmf", stat, costs)
    est.allocate_samples(target_cost)

    # get covariance of just first qoi
    qoi_idx = [0]
    cov_0 = stat.get_pilot_quantities_subset(
        nmodels, benchmark.nqoi, [0, 1, 2], qoi_idx)[0]
    stat_0 = mf.multioutput_stats["mean"](benchmark.nqoi)
    stat_0.set_pilot_quantities(cov_0)
    est_0 = mf.get_estimator("gmf", stat_0, costs)
    est_0.allocate_samples(target_cost)

    est_labels = mathrm_labels(["MOACV", "SOACV"])

    # only works if qoi_idx = [0]
    from pyapprox.multifidelity.factory import ComparisonCriteria
    class CustomComparisionCriteria(ComparisonCriteria):
        def __call__(self, est_covariance, est):
            return est_covariance[0, 0]


    ax = plt.subplots(1, 1, figsize=(8, 6))[1]
    _ = mf.plot_estimator_variance_reductions(
        [est, est_0], est_labels, ax, criteria=CustomComparisionCriteria())





.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /auto_tutorials/multi_fidelity/images/sphx_glr_plot_multioutput_acv_001.png
         :alt: plot multioutput acv
         :srcset: /auto_tutorials/multi_fidelity/images/sphx_glr_plot_multioutput_acv_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /auto_tutorials/multi_fidelity/images/sphx_glr_plot_multioutput_acv_002.png
         :alt: plot multioutput acv
         :srcset: /auto_tutorials/multi_fidelity/images/sphx_glr_plot_multioutput_acv_002.png
         :class: sphx-glr-multi-img





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Video
-----
Click on the image below to view a video tutorial on multi-output approximate control variate Monte Carlo quadrature

.. image:: ../../figures/multi-output-acv-thumbnail.png
   :target: https://youtu.be/astvKKFh2yA?si=8vgmKRbjdhJYeUfq

.. GENERATED FROM PYTHON SOURCE LINES 69-72

References
----------
.. [RM1985] `Reuven Y. Rubinstein and Ruth Marcus. Efficiency of multivariate control variates in monte carlo simulation. Operations Research, 33(3):661–677, 1985. <https://doi.org/10.48550/arXiv.2310.00125>`_


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   **Total running time of the script:** ( 0 minutes  0.263 seconds)


.. _sphx_glr_download_auto_tutorials_multi_fidelity_plot_multioutput_acv.py:

.. only:: html

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




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

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

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

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


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

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