94 lines
2.2 KiB
Python
94 lines
2.2 KiB
Python
# -*- coding: utf-8 -*-
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"""
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=========================================================
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Principal components analysis (PCA)
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=========================================================
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These figures aid in illustrating how a point cloud
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can be very flat in one direction--which is where PCA
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comes in to choose a direction that is not flat.
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"""
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# Authors: Gael Varoquaux
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# Jaques Grobler
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# Kevin Hughes
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# License: BSD 3 clause
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from sklearn.decomposition import PCA
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import numpy as np
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import matplotlib.pyplot as plt
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from scipy import stats
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# #############################################################################
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# Create the data
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e = np.exp(1)
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np.random.seed(4)
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def pdf(x):
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return 0.5 * (stats.norm(scale=0.25 / e).pdf(x) + stats.norm(scale=4 / e).pdf(x))
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y = np.random.normal(scale=0.5, size=(30000))
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x = np.random.normal(scale=0.5, size=(30000))
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z = np.random.normal(scale=0.1, size=len(x))
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density = pdf(x) * pdf(y)
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pdf_z = pdf(5 * z)
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density *= pdf_z
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a = x + y
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b = 2 * y
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c = a - b + z
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norm = np.sqrt(a.var() + b.var())
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a /= norm
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b /= norm
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# #############################################################################
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# Plot the figures
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def plot_figs(fig_num, elev, azim):
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fig = plt.figure(fig_num, figsize=(4, 3))
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plt.clf()
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ax = fig.add_subplot(111, projection="3d", elev=elev, azim=azim)
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ax.set_position([0, 0, 0.95, 1])
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ax.scatter(a[::10], b[::10], c[::10], c=density[::10], marker="+", alpha=0.4)
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Y = np.c_[a, b, c]
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# Using SciPy's SVD, this would be:
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# _, pca_score, Vt = scipy.linalg.svd(Y, full_matrices=False)
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pca = PCA(n_components=3)
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pca.fit(Y)
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V = pca.components_.T
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x_pca_axis, y_pca_axis, z_pca_axis = 3 * V
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x_pca_plane = np.r_[x_pca_axis[:2], -x_pca_axis[1::-1]]
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y_pca_plane = np.r_[y_pca_axis[:2], -y_pca_axis[1::-1]]
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z_pca_plane = np.r_[z_pca_axis[:2], -z_pca_axis[1::-1]]
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x_pca_plane.shape = (2, 2)
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y_pca_plane.shape = (2, 2)
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z_pca_plane.shape = (2, 2)
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ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane)
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ax.w_xaxis.set_ticklabels([])
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ax.w_yaxis.set_ticklabels([])
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ax.w_zaxis.set_ticklabels([])
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elev = -40
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azim = -80
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plot_figs(1, elev, azim)
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elev = 30
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azim = 20
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plot_figs(2, elev, azim)
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plt.show()
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