.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/fpca/plot_fpca_1d.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_fpca_plot_fpca_1d.py: FPCA of 1-dimensional data =========================== .. GENERATED FROM PYTHON SOURCE LINES 6-19 .. code-block:: Python # Author: Steven Golovkine # License: MIT # Load packages import matplotlib.pyplot as plt import numpy as np from FDApy.representation import DenseArgvals from FDApy.simulation import KarhunenLoeve from FDApy.preprocessing import UFPCA from FDApy.visualization import plot .. GENERATED FROM PYTHON SOURCE LINES 20-21 In this section, we are showing how to perform a functional principal component on one-dimensional data using the :class:`~FDApy.preprocessing.UFPCA` class. We will compare two methods to perform the dimension reduction: the decomposition of the covariance operator and the decomposition of the inner-product matrix. We will use the first :math:`K = 5` principal components to reconstruct the curves. .. GENERATED FROM PYTHON SOURCE LINES 21-33 .. code-block:: Python # Set general parameters rng = 42 n_obs = 50 # Parameters of the basis name = "fourier" n_functions = 25 argvals = DenseArgvals({"input_dim_0": np.linspace(0, 1, 101)}) .. GENERATED FROM PYTHON SOURCE LINES 34-35 We simulate :math:`N = 50` curves on the one-dimensional observation grid :math:`\{0, 0.01, 0.02, \cdots, 1\}`, based on the first :math:`K = 25` Fourier basis functions on :math:`[0, 1]` and the variance of the scores random variables decreasing exponentially. .. GENERATED FROM PYTHON SOURCE LINES 35-46 .. code-block:: Python kl = KarhunenLoeve( n_functions=n_functions, basis_name=name, argvals=argvals, random_state=rng ) kl.new(n_obs=n_obs, clusters_std="exponential") kl.add_noise(noise_variance=0.05) data = kl.noisy_data _ = plot(data) plt.show() .. image-sg:: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_001.png :alt: plot fpca 1d :srcset: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 47-51 Estimation of the eigencomponents --------------------------------- The :class:`~FDApy.preprocessing.UFPCA` class requires two parameters: the number of components to estimate and the method to use. The method parameter can be either `covariance` or `inner-product`. The first method estimates the eigenfunctions by decomposing the covariance operator, while the second method estimates the eigenfunctions by decomposing the inner-product matrix. .. GENERATED FROM PYTHON SOURCE LINES 51-60 .. code-block:: Python # First, we perform a univariate FPCA using a decomposition of the covariance operator. ufpca_cov = UFPCA(n_components=5, method="covariance") ufpca_cov.fit(data) # Plot the eigenfunctions using the decomposition of the covariance operator. _ = plot(ufpca_cov.eigenfunctions) plt.show() .. image-sg:: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_002.png :alt: plot fpca 1d :srcset: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 62-71 .. code-block:: Python # Second, we perform a univariate FPCA using a decomposition of the inner-product matrix. ufpca_innpro = UFPCA(n_components=5, method="inner-product") ufpca_innpro.fit(data) # Plot the eigenfunctions using the decomposition of the inner-product matrix. _ = plot(ufpca_innpro.eigenfunctions) plt.show() .. image-sg:: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_003.png :alt: plot fpca 1d :srcset: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 72-76 Estimation of the scores ------------------------ Once the eigenfunctions are estimated, we can compute the scores using numerical integration, the PACE algorithm or the eigenvectors from the decomposition of the inner-product matrix. The :func:`~FDApy.preprocessing.UFPCA.transform` method requires the data as argument and the method to use. The method parameter can be either `NumInt`, `PACE` or `InnPro`. Note that, when using the eigenvectors from the decomposition of the inner-product matrix, new data can not be passed as argument of the :func:`~FDApy.preprocessing.UFPCA.transform` method because the estimation is performed using the eigenvectors of the inner-product matrix. .. GENERATED FROM PYTHON SOURCE LINES 76-89 .. code-block:: Python scores_numint = ufpca_cov.transform(data, method="NumInt") scores_pace = ufpca_cov.transform(data, method="PACE") scores_innpro = ufpca_innpro.transform(method="InnPro") # Plot of the scores plt.scatter(scores_numint[:, 0], scores_numint[:, 1], label="NumInt") plt.scatter(scores_pace[:, 0], scores_pace[:, 1], label="PACE") plt.scatter(scores_innpro[:, 0], scores_innpro[:, 1], label="InnPro") plt.legend() plt.show() .. image-sg:: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_004.png :alt: plot fpca 1d :srcset: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 90-94 Comparison of the methods ------------------------- Finally, we compare the methods by reconstructing the curves using the first :math:`K = 5` principal components. We plot a sample of curves and their reconstruction. .. GENERATED FROM PYTHON SOURCE LINES 94-127 .. code-block:: Python data_recons_numint = ufpca_cov.inverse_transform(scores_numint) data_recons_pace = ufpca_cov.inverse_transform(scores_pace) data_recons_innpro = ufpca_innpro.inverse_transform(scores_innpro) colors_numint = np.array([[0.9, 0, 0, 1]]) colors_pace = np.array([[0, 0.9, 0, 1]]) colors_innpro = np.array([[0.9, 0, 0.9, 1]]) fig, axes = plt.subplots(nrows=5, ncols=2, figsize=(16, 16)) for idx_plot, idx in enumerate(np.random.choice(n_obs, 10)): temp_ax = axes.flatten()[idx_plot] temp_ax = plot(kl.data[idx], ax=temp_ax, label="True") plot( data_recons_numint[idx], colors=colors_numint, ax=temp_ax, label="Reconstruction NumInt", ) plot( data_recons_pace[idx], colors=colors_pace, ax=temp_ax, label="Reconstruction PACE", ) plot( data_recons_innpro[idx], colors=colors_innpro, ax=temp_ax, label="Reconstruction InnPro", ) temp_ax.legend() plt.show() .. image-sg:: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_005.png :alt: plot fpca 1d :srcset: /auto_examples/fpca/images/sphx_glr_plot_fpca_1d_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.196 seconds) .. _sphx_glr_download_auto_examples_fpca_plot_fpca_1d.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_fpca_1d.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_fpca_1d.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_fpca_1d.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_