Gaussian Mixture Model Selection

This example shows that model selection can be performed with Gaussian Mixture Models (GMM) using information-theory criteria. Model selection concerns both the covariance type and the number of components in the model.

In this case, both the Akaike Information Criterion (AIC) and the Bayes Information Criterion (BIC) provide the right result, but we only demo the latter as BIC is better suited to identify the true model among a set of candidates. Unlike Bayesian procedures, such inferences are prior-free.

Data generation

We generate two components (each one containing n_samples) by randomly sampling the standard normal distribution as returned by numpy.random.randn. One component is kept spherical yet shifted and re-scaled. The other one is deformed to have a more general covariance matrix.

import numpy as np

n_samples = 500
np.random.seed(0)
C = np.array([[0.0, -0.1], [1.7, 0.4]])
component_1 = np.dot(np.random.randn(n_samples, 2), C)  # general
component_2 = 0.7 * np.random.randn(n_samples, 2) + np.array([-4, 1])  # spherical

X = np.concatenate([component_1, component_2])

We can visualize the different components:

import matplotlib.pyplot as plt

plt.scatter(component_1[:, 0], component_1[:, 1], s=0.8)
plt.scatter(component_2[:, 0], component_2[:, 1], s=0.8)
plt.title("Gaussian Mixture components")
plt.axis("equal")
plt.show()

Model training and selection

We vary the number of components from 1 to 6 and the type of covariance parameters to use:

• "full": each component has its own general covariance matrix.

• "tied": all components share the same general covariance matrix.

• "diag": each component has its own diagonal covariance matrix.

• "spherical": each component has its own single variance.

We score the different models and keep the best model (the lowest BIC). This is done by using GridSearchCV and a user-defined score function which returns the negative BIC score, as GridSearchCV is designed to maximize a score (maximizing the negative BIC is equivalent to minimizing the BIC).

The best set of parameters and estimator are stored in best_parameters_ and best_estimator_, respectively.

from sklearn.mixture import GaussianMixture
from sklearn.model_selection import GridSearchCV


def gmm_bic_score(estimator, X):
    """Callable to pass to GridSearchCV that will use the BIC score."""
    # Make it negative since GridSearchCV expects a score to maximize
    return -estimator.bic(X)


param_grid = {
    "n_components": range(1, 7),
    "covariance_type": ["spherical", "tied", "diag", "full"],
}
grid_search = GridSearchCV(
    GaussianMixture(), param_grid=param_grid, scoring=gmm_bic_score
)
grid_search.fit(X)

GridSearchCV(estimator=GaussianMixture(),
             param_grid={'covariance_type': ['spherical', 'tied', 'diag',
                                             'full'],
                         'n_components': range(1, 7)},
             scoring=<function gmm_bic_score at 0x7f44c35c4820>)

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Plot the BIC scores

To ease the plotting we can create a pandas.DataFrame from the results of the cross-validation done by the grid search. We re-inverse the sign of the BIC score to show the effect of minimizing it.

import pandas as pd

df = pd.DataFrame(grid_search.cv_results_)[
    ["param_n_components", "param_covariance_type", "mean_test_score"]
]
df["mean_test_score"] = -df["mean_test_score"]
df = df.rename(
    columns={
        "param_n_components": "Number of components",
        "param_covariance_type": "Type of covariance",
        "mean_test_score": "BIC score",
    }
)
df.sort_values(by="BIC score").head()

	Number of components	Type of covariance	BIC score
19	2	full	1046.829429
20	3	full	1084.038689
21	4	full	1114.517272
22	5	full	1148.512281
23	6	full	1179.977890

import seaborn as sns

sns.catplot(
    data=df,
    kind="bar",
    x="Number of components",
    y="BIC score",
    hue="Type of covariance",
)
plt.show()

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

/home/circleci/mambaforge/envs/testenv/lib/python3.9/site-packages/seaborn/_oldcore.py:1498: FutureWarning:

is_categorical_dtype is deprecated and will be removed in a future version. Use isinstance(dtype, CategoricalDtype) instead

In the present case, the model with 2 components and full covariance (which corresponds to the true generative model) has the lowest BIC score and is therefore selected by the grid search.

Plot the best model

We plot an ellipse to show each Gaussian component of the selected model. For such purpose, one needs to find the eigenvalues of the covariance matrices as returned by the covariances_ attribute. The shape of such matrices depends on the covariance_type:

• "full": (n_components, n_features, n_features)

• "tied": (n_features, n_features)

• "diag": (n_components, n_features)

• "spherical": (n_components,)

from matplotlib.patches import Ellipse
from scipy import linalg

color_iter = sns.color_palette("tab10", 2)[::-1]
Y_ = grid_search.predict(X)

fig, ax = plt.subplots()

for i, (mean, cov, color) in enumerate(
    zip(
        grid_search.best_estimator_.means_,
        grid_search.best_estimator_.covariances_,
        color_iter,
    )
):
    v, w = linalg.eigh(cov)
    if not np.any(Y_ == i):
        continue
    plt.scatter(X[Y_ == i, 0], X[Y_ == i, 1], 0.8, color=color)

    angle = np.arctan2(w[0][1], w[0][0])
    angle = 180.0 * angle / np.pi  # convert to degrees
    v = 2.0 * np.sqrt(2.0) * np.sqrt(v)
    ellipse = Ellipse(mean, v[0], v[1], angle=180.0 + angle, color=color)
    ellipse.set_clip_box(fig.bbox)
    ellipse.set_alpha(0.5)
    ax.add_artist(ellipse)

plt.title(
    f"Selected GMM: {grid_search.best_params_['covariance_type']} model, "
    f"{grid_search.best_params_['n_components']} components"
)
plt.axis("equal")
plt.show()