137 lines
5.6 KiB
Python
137 lines
5.6 KiB
Python
"""
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========================================================================
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Concentration Prior Type Analysis of Variation Bayesian Gaussian Mixture
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========================================================================
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This example plots the ellipsoids obtained from a toy dataset (mixture of three
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Gaussians) fitted by the ``BayesianGaussianMixture`` class models with a
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Dirichlet distribution prior
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(``weight_concentration_prior_type='dirichlet_distribution'``) and a Dirichlet
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process prior (``weight_concentration_prior_type='dirichlet_process'``). On
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each figure, we plot the results for three different values of the weight
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concentration prior.
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The ``BayesianGaussianMixture`` class can adapt its number of mixture
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componentsautomatically. The parameter ``weight_concentration_prior`` has a
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direct link with the resulting number of components with non-zero weights.
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Specifying a low value for the concentration prior will make the model put most
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of the weight on few components set the remaining components weights very close
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to zero. High values of the concentration prior will allow a larger number of
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components to be active in the mixture.
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The Dirichlet process prior allows to define an infinite number of components
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and automatically selects the correct number of components: it activates a
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component only if it is necessary.
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On the contrary the classical finite mixture model with a Dirichlet
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distribution prior will favor more uniformly weighted components and therefore
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tends to divide natural clusters into unnecessary sub-components.
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"""
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# Author: Thierry Guillemot <thierry.guillemot.work@gmail.com>
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# License: BSD 3 clause
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import numpy as np
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import matplotlib as mpl
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import matplotlib.pyplot as plt
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import matplotlib.gridspec as gridspec
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from sklearn.mixture import BayesianGaussianMixture
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print(__doc__)
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def plot_ellipses(ax, weights, means, covars):
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for n in range(means.shape[0]):
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eig_vals, eig_vecs = np.linalg.eigh(covars[n])
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unit_eig_vec = eig_vecs[0] / np.linalg.norm(eig_vecs[0])
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angle = np.arctan2(unit_eig_vec[1], unit_eig_vec[0])
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# Ellipse needs degrees
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angle = 180 * angle / np.pi
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# eigenvector normalization
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eig_vals = 2 * np.sqrt(2) * np.sqrt(eig_vals)
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ell = mpl.patches.Ellipse(means[n], eig_vals[0], eig_vals[1],
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180 + angle, edgecolor='black')
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ell.set_clip_box(ax.bbox)
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ell.set_alpha(weights[n])
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ell.set_facecolor('#56B4E9')
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ax.add_artist(ell)
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def plot_results(ax1, ax2, estimator, X, y, title, plot_title=False):
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ax1.set_title(title)
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ax1.scatter(X[:, 0], X[:, 1], s=5, marker='o', color=colors[y], alpha=0.8)
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ax1.set_xlim(-2., 2.)
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ax1.set_ylim(-3., 3.)
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ax1.set_xticks(())
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ax1.set_yticks(())
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plot_ellipses(ax1, estimator.weights_, estimator.means_,
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estimator.covariances_)
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ax2.get_xaxis().set_tick_params(direction='out')
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ax2.yaxis.grid(True, alpha=0.7)
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for k, w in enumerate(estimator.weights_):
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ax2.bar(k, w, width=0.9, color='#56B4E9', zorder=3,
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align='center', edgecolor='black')
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ax2.text(k, w + 0.007, "%.1f%%" % (w * 100.),
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horizontalalignment='center')
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ax2.set_xlim(-.6, 2 * n_components - .4)
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ax2.set_ylim(0., 1.1)
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ax2.tick_params(axis='y', which='both', left='off',
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right='off', labelleft='off')
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ax2.tick_params(axis='x', which='both', top='off')
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if plot_title:
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ax1.set_ylabel('Estimated Mixtures')
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ax2.set_ylabel('Weight of each component')
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# Parameters of the dataset
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random_state, n_components, n_features = 2, 3, 2
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colors = np.array(['#0072B2', '#F0E442', '#D55E00'])
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covars = np.array([[[.7, .0], [.0, .1]],
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[[.5, .0], [.0, .1]],
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[[.5, .0], [.0, .1]]])
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samples = np.array([200, 500, 200])
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means = np.array([[.0, -.70],
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[.0, .0],
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[.0, .70]])
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# mean_precision_prior= 0.8 to minimize the influence of the prior
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estimators = [
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("Finite mixture with a Dirichlet distribution\nprior and "
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r"$\gamma_0=$", BayesianGaussianMixture(
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weight_concentration_prior_type="dirichlet_distribution",
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n_components=2 * n_components, reg_covar=0, init_params='random',
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max_iter=1500, mean_precision_prior=.8,
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random_state=random_state), [0.001, 1, 1000]),
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("Infinite mixture with a Dirichlet process\n prior and" r"$\gamma_0=$",
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BayesianGaussianMixture(
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weight_concentration_prior_type="dirichlet_process",
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n_components=2 * n_components, reg_covar=0, init_params='random',
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max_iter=1500, mean_precision_prior=.8,
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random_state=random_state), [1, 1000, 100000])]
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# Generate data
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rng = np.random.RandomState(random_state)
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X = np.vstack([
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rng.multivariate_normal(means[j], covars[j], samples[j])
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for j in range(n_components)])
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y = np.concatenate([j * np.ones(samples[j], dtype=int)
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for j in range(n_components)])
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# Plot results in two different figures
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for (title, estimator, concentrations_prior) in estimators:
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plt.figure(figsize=(4.7 * 3, 8))
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plt.subplots_adjust(bottom=.04, top=0.90, hspace=.05, wspace=.05,
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left=.03, right=.99)
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gs = gridspec.GridSpec(3, len(concentrations_prior))
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for k, concentration in enumerate(concentrations_prior):
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estimator.weight_concentration_prior = concentration
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estimator.fit(X)
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plot_results(plt.subplot(gs[0:2, k]), plt.subplot(gs[2, k]), estimator,
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X, y, r"%s$%.1e$" % (title, concentration),
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plot_title=k == 0)
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plt.show()
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