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    {
      "cell_type": "code",
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      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n=========================================================\nPrincipal components analysis (PCA)\n=========================================================\n\nThese figures aid in illustrating how a point cloud\ncan be very flat in one direction--which is where PCA\ncomes in to choose a direction that is not flat.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
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      "source": [
        "print(__doc__)\n\n# Authors: Gael Varoquaux\n#          Jaques Grobler\n#          Kevin Hughes\n# License: BSD 3 clause\n\nfrom sklearn.decomposition import PCA\n\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy import stats\n\n\n# #############################################################################\n# Create the data\n\ne = np.exp(1)\nnp.random.seed(4)\n\n\ndef pdf(x):\n    return 0.5 * (stats.norm(scale=0.25 / e).pdf(x)\n                  + stats.norm(scale=4 / e).pdf(x))\n\ny = np.random.normal(scale=0.5, size=(30000))\nx = np.random.normal(scale=0.5, size=(30000))\nz = np.random.normal(scale=0.1, size=len(x))\n\ndensity = pdf(x) * pdf(y)\npdf_z = pdf(5 * z)\n\ndensity *= pdf_z\n\na = x + y\nb = 2 * y\nc = a - b + z\n\nnorm = np.sqrt(a.var() + b.var())\na /= norm\nb /= norm\n\n\n# #############################################################################\n# Plot the figures\ndef plot_figs(fig_num, elev, azim):\n    fig = plt.figure(fig_num, figsize=(4, 3))\n    plt.clf()\n    ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=elev, azim=azim)\n\n    ax.scatter(a[::10], b[::10], c[::10], c=density[::10], marker='+', alpha=.4)\n    Y = np.c_[a, b, c]\n\n    # Using SciPy's SVD, this would be:\n    # _, pca_score, V = scipy.linalg.svd(Y, full_matrices=False)\n\n    pca = PCA(n_components=3)\n    pca.fit(Y)\n    pca_score = pca.explained_variance_ratio_\n    V = pca.components_\n\n    x_pca_axis, y_pca_axis, z_pca_axis = 3 * V.T\n    x_pca_plane = np.r_[x_pca_axis[:2], - x_pca_axis[1::-1]]\n    y_pca_plane = np.r_[y_pca_axis[:2], - y_pca_axis[1::-1]]\n    z_pca_plane = np.r_[z_pca_axis[:2], - z_pca_axis[1::-1]]\n    x_pca_plane.shape = (2, 2)\n    y_pca_plane.shape = (2, 2)\n    z_pca_plane.shape = (2, 2)\n    ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane)\n    ax.w_xaxis.set_ticklabels([])\n    ax.w_yaxis.set_ticklabels([])\n    ax.w_zaxis.set_ticklabels([])\n\n\nelev = -40\nazim = -80\nplot_figs(1, elev, azim)\n\nelev = 30\nazim = 20\nplot_figs(2, elev, azim)\n\nplt.show()"
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