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.. "examples/05_kriging/11_intrinsic_collocated_cokriging.py"
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.. only:: html

    .. note::
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        :ref:`Go to the end <sphx_glr_download_examples_05_kriging_11_intrinsic_collocated_cokriging.py>`
        to download the full example code.

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

.. _sphx_glr_examples_05_kriging_11_intrinsic_collocated_cokriging.py:


Intrinsic Collocated Cokriging
-------------------------------

Intrinsic Collocated Cokriging (ICCK) improves variance estimation
compared to Simple Collocated Cokriging.

This example demonstrates the correlogram-based API using MarkovModel1.

The variance formula is:

.. math:: \sigma^2_{ICCK} = (1 - \rho_0^2) \cdot \sigma^2_{SK}

Example
^^^^^^^

Here we compare Simple Kriging with Intrinsic Collocated Cokriging using the
MarkovModel1 correlogram.

.. GENERATED FROM PYTHON SOURCE LINES 20-34

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from gstools import Gaussian, MarkovModel1, krige
    from gstools.cokriging import IntrinsicCollocated

    # condtions
    np.random.seed(4)
    cond_pos = np.array([0.5, 2.1, 3.8, 6.2, 13.5])
    cond_val = np.array([0.8, 1.2, 1.8, 2.1, 1.4])
    gridx = np.linspace(0.0, 15.0, 151)
    model = Gaussian(dim=1, var=0.5, len_scale=2.0)








.. GENERATED FROM PYTHON SOURCE LINES 35-36

Generate correlated secondary data

.. GENERATED FROM PYTHON SOURCE LINES 36-46

.. code-block:: Python


    sec_pos = np.linspace(0, 15, 31)
    primary_trend = np.interp(sec_pos, cond_pos, cond_val)
    gap_feature = -1.6 * np.exp(-(((sec_pos - 10.0) / 2.0) ** 2))
    gap_feature2 = -0.95 * np.exp(-(((sec_pos - 4.0) / 2.0) ** 2))
    sec_val = 0.99 * primary_trend + gap_feature + gap_feature2

    sec_grid = np.interp(gridx, sec_pos, sec_val)
    sec_at_primary = np.interp(cond_pos, sec_pos, sec_val)








.. GENERATED FROM PYTHON SOURCE LINES 47-48

Simple Kriging and Intrinsic Collocated Cokriging

.. GENERATED FROM PYTHON SOURCE LINES 48-74

.. code-block:: Python


    sk = krige.Simple(model, cond_pos=cond_pos, cond_val=cond_val, mean=1.0)
    sk_field, sk_var = sk(gridx, return_var=True)

    # Compute cross-correlation from data
    cross_corr = np.corrcoef(cond_val, sec_at_primary)[0, 1]

    # Create MarkovModel1 correlogram
    correlogram = MarkovModel1(
        primary_model=model,
        cross_corr=cross_corr,
        secondary_var=np.var(sec_val),
        primary_mean=1.0,
        secondary_mean=np.mean(sec_val),
    )

    # Intrinsic Collocated Cokriging
    icck = IntrinsicCollocated(
        correlogram,
        cond_pos=cond_pos,
        cond_val=cond_val,
        secondary_cond_pos=cond_pos,
        secondary_cond_val=sec_at_primary,
    )
    icck_field, icck_var = icck(gridx, secondary_data=sec_grid, return_var=True)








.. GENERATED FROM PYTHON SOURCE LINES 75-96

.. code-block:: Python


    fig, ax = plt.subplots(1, 2, figsize=(10, 3.5))

    ax[0].scatter(cond_pos, cond_val, color="red", label="Primary data")
    ax[0].scatter(
        cond_pos,
        sec_at_primary,
        color="blue",
        marker="s",
        label="Secondary at primary",
    )
    ax[0].plot(sec_pos, sec_val, "b-", alpha=0.6, label="Secondary data")
    ax[0].legend()

    ax[1].plot(gridx, sk_field, label="Simple Kriging")
    ax[1].plot(gridx, icck_field, label="Intrinsic Collocated Cokriging")
    ax[1].scatter(cond_pos, cond_val, color="k", zorder=10, label="Conditions")
    ax[1].legend()

    plt.tight_layout()
    plt.show()



.. image-sg:: /examples/05_kriging/images/sphx_glr_11_intrinsic_collocated_cokriging_001.png
   :alt: 11 intrinsic collocated cokriging
   :srcset: /examples/05_kriging/images/sphx_glr_11_intrinsic_collocated_cokriging_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.192 seconds)


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