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# Concentration Prior Type Analysis of Variation Bayesian Gaussian Mixture

This example plots the ellipsoids obtained from a toy dataset (mixture of three Gaussians) fitted by the `BayesianGaussianMixture` class models with a Dirichlet distribution prior (`weight_concentration_prior_type='dirichlet_distribution'`) and a Dirichlet process prior (`weight_concentration_prior_type='dirichlet_process'`). On each figure, we plot the results for three different values of the weight concentration prior.

The `BayesianGaussianMixture` class can adapt its number of mixture components automatically. The parameter `weight_concentration_prior` has a direct link with the resulting number of components with non-zero weights. Specifying a low value for the concentration prior will make the model put most of the weight on few components set the remaining components weights very close to zero. High values of the concentration prior will allow a larger number of components to be active in the mixture.

The Dirichlet process prior allows to define an infinite number of components and automatically selects the correct number of components: it activates a component only if it is necessary.

On the contrary the classical finite mixture model with a Dirichlet distribution prior will favor more uniformly weighted components and therefore tends to divide natural clusters into unnecessary sub-components.

    # Author: Thierry Guillemot <thierry.guillemot.work@gmail.com>
    # License: BSD 3 clause

    import matplotlib as mpl
    import matplotlib.gridspec as gridspec
    import matplotlib.pyplot as plt
    import numpy as np

    from sklearn.mixture import BayesianGaussianMixture

    def plot_ellipses(ax, weights, means, covars):
        for n in range(means.shape[0]):
            eig_vals, eig_vecs = np.linalg.eigh(covars[n])
            unit_eig_vec = eig_vecs[0] / np.linalg.norm(eig_vecs[0])
            angle = np.arctan2(unit_eig_vec[1], unit_eig_vec[0])
            # Ellipse needs degrees
            angle = 180 * angle / np.pi
            # eigenvector normalization
            eig_vals = 2 * np.sqrt(2) * np.sqrt(eig_vals)
            ell = mpl.patches.Ellipse(
                means[n], eig_vals[0], eig_vals[1], angle=180 + angle, edgecolor="black"
            )
            ell.set_clip_box(ax.bbox)
            ell.set_alpha(weights[n])
            ell.set_facecolor("#56B4E9")
            ax.add_artist(ell)

    def plot_results(ax1, ax2, estimator, X, y, title, plot_title=False):
        ax1.set_title(title)
        ax1.scatter(X[:, 0], X[:, 1], s=5, marker="o", color=colors[y], alpha=0.8)
        ax1.set_xlim(-2.0, 2.0)
        ax1.set_ylim(-3.0, 3.0)
        ax1.set_xticks(())
        ax1.set_yticks(())
        plot_ellipses(ax1, estimator.weights_, estimator.means_, estimator.covariances_)

        ax2.get_xaxis().set_tick_params(direction="out")
        ax2.yaxis.grid(True, alpha=0.7)
        for k, w in enumerate(estimator.weights_):
            ax2.bar(
                k,
                w,
                width=0.9,
                color="#56B4E9",
                zorder=3,
                align="center",
                edgecolor="black",
            )
            ax2.text(k, w + 0.007, "%.1f%%" % (w * 100.0), horizontalalignment="center")
        ax2.set_xlim(-0.6, 2 * n_components - 0.4)
        ax2.set_ylim(0.0, 1.1)
        ax2.tick_params(axis="y", which="both", left=False, right=False, labelleft=False)
        ax2.tick_params(axis="x", which="both", top=False)

        if plot_title:
            ax1.set_ylabel("Estimated Mixtures")
            ax2.set_ylabel("Weight of each component")

    # Parameters of the dataset
    random_state, n_components, n_features = 2, 3, 2
    colors = np.array(["#0072B2", "#F0E442", "#D55E00"])

    covars = np.array(
        [[[0.7, 0.0], [0.0, 0.1]], [[0.5, 0.0], [0.0, 0.1]], [[0.5, 0.0], [0.0, 0.1]]]
    )
    samples = np.array([200, 500, 200])
    means = np.array([[0.0, -0.70], [0.0, 0.0], [0.0, 0.70]])

    # mean_precision_prior= 0.8 to minimize the influence of the prior
    estimators = [
        (
            "Finite mixture with a Dirichlet distribution\nprior and " r"$\gamma_0=$",
            BayesianGaussianMixture(
                weight_concentration_prior_type="dirichlet_distribution",
                n_components=2 * n_components,
                reg_covar=0,
                init_params="random",
                max_iter=1500,
                mean_precision_prior=0.8,
                random_state=random_state,
            ),
            [0.001, 1, 1000],
        ),
        (
            "Infinite mixture with a Dirichlet process\n prior and" r"$\gamma_0=$",
            BayesianGaussianMixture(
                weight_concentration_prior_type="dirichlet_process",
                n_components=2 * n_components,
                reg_covar=0,
                init_params="random",
                max_iter=1500,
                mean_precision_prior=0.8,
                random_state=random_state,
            ),
            [1, 1000, 100000],
        ),
    ]

    # Generate data
    rng = np.random.RandomState(random_state)
    X = np.vstack(
        [
            rng.multivariate_normal(means[j], covars[j], samples[j])
            for j in range(n_components)
        ]
    )
    y = np.concatenate([np.full(samples[j], j, dtype=int) for j in range(n_components)])

    # Plot results in two different figures
    for title, estimator, concentrations_prior in estimators:
        plt.figure(figsize=(4.7 * 3, 8))
        plt.subplots_adjust(
            bottom=0.04, top=0.90, hspace=0.05, wspace=0.05, left=0.03, right=0.99
        )

        gs = gridspec.GridSpec(3, len(concentrations_prior))
        for k, concentration in enumerate(concentrations_prior):
            estimator.weight_concentration_prior = concentration
            estimator.fit(X)
            plot_results(
                plt.subplot(gs[0:2, k]),
                plt.subplot(gs[2, k]),
                estimator,
                X,
                y,
                r"%s$%.1e$" % (title, concentration),
                plot_title=k == 0,
            )

    plt.show()

Total running time of the script: (0 minutes 4.496 seconds)

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