.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/06_denoise/06_04_recursive_smoothing_spline.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_06_denoise_06_04_recursive_smoothing_spline.py>`
        to download the full example code. or to run this example in your browser via JupyterLite or Binder

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

.. _sphx_glr_auto_examples_06_denoise_06_04_recursive_smoothing_spline.py:


Recursive Smoothing Spline
==========================

We plot a recursive smoothing spline with different parameters.

.. GENERATED FROM PYTHON SOURCE LINES 13-15

Imports
-------

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python


    import math
    import numpy as np
    import matplotlib.pyplot as plt
    from splineops.denoise.smoothing_spline import smoothing_spline
    from splineops.denoise.smoothing_spline import recursive_smoothing_spline








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Recursive Smoothing Spline
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-67

.. code-block:: Python


    # Example signal: A noisy sine wave
    x = np.linspace(0, np.pi, 100)
    signal = np.sin(x) + 0.1 * np.random.normal(size=x.shape)

    # Different values for the smoothing parameter in recursive smoothing spline
    lam_values = [0.005, 0.05, 0.1]  # You can try smaller or larger values

    # Apply fractional smoothing spline as a baseline for comparison
    lambda_ = 0.1  # Regularization parameter for fractional method
    m = 1          # No upsampling
    gamma = 0.6    # Spline order parameter
    _, smoothed_fractional = smoothing_spline(signal, lambda_, m, gamma)

    # Compute MSE values for different recursive smoothing spline parameters
    mse_values = []

    # Plot results
    plt.figure(figsize=(12, 8))
    plt.plot(x, signal, label="Noisy Signal", linestyle="--", color="gray")
    plt.plot(x, smoothed_fractional, label="Fractional Smoothing Spline", color="red")

    # Apply and plot recursive smoothing spline for each lambda value
    for lam_recursive in lam_values:
        smoothed_recursive = recursive_smoothing_spline(signal, lamb=lam_recursive)
    
        # Compute MSE
        mse = np.mean((smoothed_recursive - smoothed_fractional) ** 2)
        mse_values.append(mse)
    
        plt.plot(x, smoothed_recursive, label=f"Recursive Smoothing (λ={lam_recursive})")

    # Print MSE values
    print("\nMean Squared Error (MSE) between Recursive and Fractional Smoothing Spline:")
    for lam, mse in zip(lam_values, mse_values):
        print(f"λ={lam:.3f}: MSE = {mse:.6f}")

    plt.legend()
    plt.xlabel("x")
    plt.ylabel("Signal Value")
    plt.title("Comparison of Recursive Smoothing with Different λ Values")
    plt.show()



.. image-sg:: /auto_examples/06_denoise/images/sphx_glr_06_04_recursive_smoothing_spline_001.png
   :alt: Comparison of Recursive Smoothing with Different λ Values
   :srcset: /auto_examples/06_denoise/images/sphx_glr_06_04_recursive_smoothing_spline_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Mean Squared Error (MSE) between Recursive and Fractional Smoothing Spline:
    λ=0.005: MSE = 0.000035
    λ=0.050: MSE = 0.001070
    λ=0.100: MSE = 0.003683





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

   **Total running time of the script:** (0 minutes 0.114 seconds)


.. _sphx_glr_download_auto_examples_06_denoise_06_04_recursive_smoothing_spline.py:

.. only:: html

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