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<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_cluster_comparison.html">Comparing different clustering algorithms on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_linkage_comparison.html">Comparing different hierarchical linkage methods on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_mini_batch_kmeans.html">Comparison of the K-Means and MiniBatchKMeans clustering algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_dbscan.html">Demo of DBSCAN clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_hdbscan.html">Demo of HDBSCAN clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_optics.html">Demo of OPTICS clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_affinity_propagation.html">Demo of affinity propagation clustering algorithm</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_assumptions.html">Demonstration of k-means assumptions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_stability_low_dim_dense.html">Empirical evaluation of the impact of k-means initialization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_digits_agglomeration.html">Feature agglomeration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_feature_agglomeration_vs_univariate_selection.html">Feature agglomeration vs. univariate selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_ward_structured_vs_unstructured.html">Hierarchical clustering: structured vs unstructured ward</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_inductive_clustering.html">Inductive Clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_dict_face_patches.html">Online learning of a dictionary of parts of faces</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_agglomerative_dendrogram.html">Plot Hierarchical Clustering Dendrogram</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_coin_segmentation.html">Segmenting the picture of greek coins in regions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_kmeans_silhouette_analysis.html">Selecting the number of clusters with silhouette analysis on KMeans clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_segmentation_toy.html">Spectral clustering for image segmentation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_digits_linkage.html">Various Agglomerative Clustering on a 2D embedding of digits</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cluster/plot_face_compress.html">Vector Quantization Example</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../covariance/index.html">Covariance estimation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_lw_vs_oas.html">Ledoit-Wolf vs OAS estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_mahalanobis_distances.html">Robust covariance estimation and Mahalanobis distances relevance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_robust_vs_empirical_covariance.html">Robust vs Empirical covariance estimate</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_covariance_estimation.html">Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood</a></li>
<li class="toctree-l2"><a class="reference internal" href="../covariance/plot_sparse_cov.html">Sparse inverse covariance estimation</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../cross_decomposition/index.html">Cross decomposition</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../cross_decomposition/plot_compare_cross_decomposition.html">Compare cross decomposition methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../cross_decomposition/plot_pcr_vs_pls.html">Principal Component Regression vs Partial Least Squares Regression</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets/index.html">Dataset examples</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../datasets/plot_random_multilabel_dataset.html">Plot randomly generated multilabel dataset</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../tree/index.html">Decision Trees</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_tree_regression.html">Decision Tree Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_iris_dtc.html">Plot the decision surface of decision trees trained on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_cost_complexity_pruning.html">Post pruning decision trees with cost complexity pruning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../tree/plot_unveil_tree_structure.html">Understanding the decision tree structure</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../decomposition/index.html">Decomposition</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_ica_blind_source_separation.html">Blind source separation using FastICA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_vs_lda.html">Comparison of LDA and PCA 2D projection of Iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_faces_decomposition.html">Faces dataset decompositions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_varimax_fa.html">Factor Analysis (with rotation) to visualize patterns</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_ica_vs_pca.html">FastICA on 2D point clouds</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_image_denoising.html">Image denoising using dictionary learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_incremental_pca.html">Incremental PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_kernel_pca.html">Kernel PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_vs_fa_model_selection.html">Model selection with Probabilistic PCA and Factor Analysis (FA)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_pca_iris.html">Principal Component Analysis (PCA) on Iris Dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../decomposition/plot_sparse_coding.html">Sparse coding with a precomputed dictionary</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../developing_estimators/index.html">Developing Estimators</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../developing_estimators/sklearn_is_fitted.html"><code class="docutils literal notranslate"><span class="pre">__sklearn_is_fitted__</span></code> as Developer API</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../ensemble/index.html">Ensemble methods</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_categorical.html">Categorical Feature Support in Gradient Boosting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_stack_predictors.html">Combine predictors using stacking</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_hist_grad_boosting_comparison.html">Comparing Random Forests and Histogram Gradient Boosting models</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_random_forest_regression_multioutput.html">Comparing random forests and the multi-output meta estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_regression.html">Decision Tree Regression with AdaBoost</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_early_stopping.html">Early stopping in Gradient Boosting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_importances.html">Feature importances with a forest of trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_feature_transformation.html">Feature transformations with ensembles of trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_hgbt_regression.html">Features in Histogram Gradient Boosting Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_oob.html">Gradient Boosting Out-of-Bag estimates</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_regression.html">Gradient Boosting regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_regularization.html">Gradient Boosting regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_random_forest_embedding.html">Hashing feature transformation using Totally Random Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_isolation_forest.html">IsolationForest example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_monotonic_constraints.html">Monotonic Constraints</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_multiclass.html">Multi-class AdaBoosted Decision Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_ensemble_oob.html">OOB Errors for Random Forests</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_probas.html">Plot class probabilities calculated by the VotingClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_regressor.html">Plot individual and voting regression predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_voting_decision_regions.html">Plot the decision boundaries of a VotingClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_forest_iris.html">Plot the decision surfaces of ensembles of trees on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_gradient_boosting_quantile.html">Prediction Intervals for Gradient Boosting Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_bias_variance.html">Single estimator versus bagging: bias-variance decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="../ensemble/plot_adaboost_twoclass.html">Two-class AdaBoost</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../applications/index.html">Examples based on real world datasets</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_tomography_l1_reconstruction.html">Compressive sensing: tomography reconstruction with L1 prior (Lasso)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_face_recognition.html">Faces recognition example using eigenfaces and SVMs</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_digits_denoising.html">Image denoising using kernel PCA</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_time_series_lagged_features.html">Lagged features for time series forecasting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_model_complexity_influence.html">Model Complexity Influence</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_out_of_core_classification.html">Out-of-core classification of text documents</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_outlier_detection_wine.html">Outlier detection on a real data set</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_prediction_latency.html">Prediction Latency</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_species_distribution_modeling.html">Species distribution modeling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_cyclical_feature_engineering.html">Time-related feature engineering</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_topics_extraction_with_nmf_lda.html">Topic extraction with Non-negative Matrix Factorization and Latent Dirichlet Allocation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/plot_stock_market.html">Visualizing the stock market structure</a></li>
<li class="toctree-l2"><a class="reference internal" href="../applications/wikipedia_principal_eigenvector.html">Wikipedia principal eigenvector</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../feature_selection/index.html">Feature Selection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_f_test_vs_mi.html">Comparison of F-test and mutual information</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_select_from_model_diabetes.html">Model-based and sequential feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_feature_selection_pipeline.html">Pipeline ANOVA SVM</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_rfe_digits.html">Recursive feature elimination</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_rfe_with_cross_validation.html">Recursive feature elimination with cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../feature_selection/plot_feature_selection.html">Univariate Feature Selection</a></li>
</ul>
</details></li>
<li class="toctree-l1 current active has-children"><a class="reference internal" href="index.html">Gaussian Mixture Models</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="plot_concentration_prior.html">Concentration Prior Type Analysis of Variation Bayesian Gaussian Mixture</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_gmm_pdf.html">Density Estimation for a Gaussian mixture</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_gmm_init.html">GMM Initialization Methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_gmm_covariances.html">GMM covariances</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_gmm.html">Gaussian Mixture Model Ellipsoids</a></li>
<li class="toctree-l2"><a class="reference internal" href="plot_gmm_selection.html">Gaussian Mixture Model Selection</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">Gaussian Mixture Model Sine Curve</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../gaussian_process/index.html">Gaussian Process for Machine Learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_noisy.html">Ability of Gaussian process regression (GPR) to estimate data noise-level</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_compare_gpr_krr.html">Comparison of kernel ridge and Gaussian process regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_co2.html">Forecasting of CO2 level on Mona Loa dataset using Gaussian process regression (GPR)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_noisy_targets.html">Gaussian Processes regression: basic introductory example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_iris.html">Gaussian process classification (GPC) on iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_on_structured_data.html">Gaussian processes on discrete data structures</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_xor.html">Illustration of Gaussian process classification (GPC) on the XOR dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpr_prior_posterior.html">Illustration of prior and posterior Gaussian process for different kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc_isoprobability.html">Iso-probability lines for Gaussian Processes classification (GPC)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../gaussian_process/plot_gpc.html">Probabilistic predictions with Gaussian process classification (GPC)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../linear_model/index.html">Generalized Linear Models</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ard.html">Comparing Linear Bayesian Regressors</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_comparison.html">Comparing various online solvers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_bayesian_ridge_curvefit.html">Curve Fitting with Bayesian Ridge Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_multinomial.html">Decision Boundaries of Multinomial and One-vs-Rest Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_early_stopping.html">Early stopping of Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_elastic_net_precomputed_gram_matrix_with_weighted_samples.html">Fitting an Elastic Net with a precomputed Gram Matrix and Weighted Samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_huber_vs_ridge.html">HuberRegressor vs Ridge on dataset with strong outliers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_multi_task_lasso_support.html">Joint feature selection with multi-task Lasso</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_l1_l2_sparsity.html">L1 Penalty and Sparsity in Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_and_elasticnet.html">L1-based models for Sparse Signals</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_lars_ic.html">Lasso model selection via information criteria</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_model_selection.html">Lasso model selection: AIC-BIC / cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_dense_vs_sparse_data.html">Lasso on dense and sparse data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_lasso_lasso_lars_elasticnet_path.html">Lasso, Lasso-LARS, and Elastic Net paths</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic.html">Logistic function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sparse_logistic_regression_mnist.html">MNIST classification using multinomial logistic + L1</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sparse_logistic_regression_20newsgroups.html">Multiclass sparse logistic regression on 20newgroups</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_nnls.html">Non-negative least squares</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgdocsvm_vs_ocsvm.html">One-Class SVM versus One-Class SVM using Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ols.html">Ordinary Least Squares Example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ols_ridge_variance.html">Ordinary Least Squares and Ridge Regression Variance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_omp.html">Orthogonal Matching Pursuit</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ridge_path.html">Plot Ridge coefficients as a function of the regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_iris.html">Plot multi-class SGD on the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_poisson_regression_non_normal_loss.html">Poisson regression and non-normal loss</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_polynomial_interpolation.html">Polynomial and Spline interpolation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_quantile_regression.html">Quantile regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_logistic_path.html">Regularization path of L1- Logistic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ridge_coeffs.html">Ridge coefficients as a function of the L2 Regularization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_robust_fit.html">Robust linear estimator fitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_ransac.html">Robust linear model estimation using RANSAC</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_separating_hyperplane.html">SGD: Maximum margin separating hyperplane</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_penalties.html">SGD: Penalties</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_weighted_samples.html">SGD: Weighted samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_sgd_loss_functions.html">SGD: convex loss functions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_theilsen.html">Theil-Sen Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../linear_model/plot_tweedie_regression_insurance_claims.html">Tweedie regression on insurance claims</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../inspection/index.html">Inspection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_linear_model_coefficient_interpretation.html">Common pitfalls in the interpretation of coefficients of linear models</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_causal_interpretation.html">Failure of Machine Learning to infer causal effects</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_partial_dependence.html">Partial Dependence and Individual Conditional Expectation Plots</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_permutation_importance.html">Permutation Importance vs Random Forest Feature Importance (MDI)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../inspection/plot_permutation_importance_multicollinear.html">Permutation Importance with Multicollinear or Correlated Features</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../kernel_approximation/index.html">Kernel Approximation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../kernel_approximation/plot_scalable_poly_kernels.html">Scalable learning with polynomial kernel approximation</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../manifold/index.html">Manifold learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_compare_methods.html">Comparison of Manifold Learning methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_manifold_sphere.html">Manifold Learning methods on a severed sphere</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_lle_digits.html">Manifold learning on handwritten digits: Locally Linear Embedding, Isomap…</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_mds.html">Multi-dimensional scaling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_swissroll.html">Swiss Roll And Swiss-Hole Reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="../manifold/plot_t_sne_perplexity.html">t-SNE: The effect of various perplexity values on the shape</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../miscellaneous/index.html">Miscellaneous</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_partial_dependence_visualization_api.html">Advanced Plotting With Partial Dependence</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_anomaly_comparison.html">Comparing anomaly detection algorithms for outlier detection on toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_kernel_ridge_regression.html">Comparison of kernel ridge regression and SVR</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_pipeline_display.html">Displaying Pipelines</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_estimator_representation.html">Displaying estimators and complex pipelines</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_outlier_detection_bench.html">Evaluation of outlier detection estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_kernel_approximation.html">Explicit feature map approximation for RBF kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_multioutput_face_completion.html">Face completion with a multi-output estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_set_output.html">Introducing the <code class="docutils literal notranslate"><span class="pre">set_output</span></code> API</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_isotonic_regression.html">Isotonic Regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_metadata_routing.html">Metadata Routing</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_multilabel.html">Multilabel classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_roc_curve_visualization_api.html">ROC Curve with Visualization API</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_johnson_lindenstrauss_bound.html">The Johnson-Lindenstrauss bound for embedding with random projections</a></li>
<li class="toctree-l2"><a class="reference internal" href="../miscellaneous/plot_display_object_visualization.html">Visualizations with Display Objects</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../impute/index.html">Missing Value Imputation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../impute/plot_missing_values.html">Imputing missing values before building an estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="../impute/plot_iterative_imputer_variants_comparison.html">Imputing missing values with variants of IterativeImputer</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../model_selection/index.html">Model Selection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_refit_callable.html">Balance model complexity and cross-validated score</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_likelihood_ratios.html">Class Likelihood Ratios to measure classification performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_randomized_search.html">Comparing randomized search and grid search for hyperparameter estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_successive_halving_heatmap.html">Comparison between grid search and successive halving</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_confusion_matrix.html">Confusion matrix</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_digits.html">Custom refit strategy of a grid search with cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_multi_metric_evaluation.html">Demonstration of multi-metric evaluation on cross_val_score and GridSearchCV</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_det.html">Detection error tradeoff (DET) curve</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_train_error_vs_test_error.html">Effect of model regularization on training and test error</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_roc.html">Multiclass Receiver Operating Characteristic (ROC)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_nested_cross_validation_iris.html">Nested versus non-nested cross-validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cv_predict.html">Plotting Cross-Validated Predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_learning_curve.html">Plotting Learning Curves and Checking Models’ Scalability</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_tuned_decision_threshold.html">Post-hoc tuning the cut-off point of decision function</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cost_sensitive_learning.html">Post-tuning the decision threshold for cost-sensitive learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_precision_recall.html">Precision-Recall</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_roc_crossval.html">Receiver Operating Characteristic (ROC) with cross validation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_text_feature_extraction.html">Sample pipeline for text feature extraction and evaluation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_grid_search_stats.html">Statistical comparison of models using grid search</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_successive_halving_iterations.html">Successive Halving Iterations</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_permutation_tests_for_classification.html">Test with permutations the significance of a classification score</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_underfitting_overfitting.html">Underfitting vs. Overfitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../model_selection/plot_cv_indices.html">Visualizing cross-validation behavior in scikit-learn</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../multiclass/index.html">Multiclass methods</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../multiclass/plot_multiclass_overview.html">Overview of multiclass training meta-estimators</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../multioutput/index.html">Multioutput methods</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../multioutput/plot_classifier_chain_yeast.html">Multilabel classification using a classifier chain</a></li>
</ul>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../neighbors/index.html">Nearest Neighbors</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/approximate_nearest_neighbors.html">Approximate nearest neighbors in TSNE</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_caching_nearest_neighbors.html">Caching nearest neighbors</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_classification.html">Comparing Nearest Neighbors with and without Neighborhood Components Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_dim_reduction.html">Dimensionality Reduction with Neighborhood Components Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_species_kde.html">Kernel Density Estimate of Species Distributions</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_digits_kde_sampling.html">Kernel Density Estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nearest_centroid.html">Nearest Centroid Classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_classification.html">Nearest Neighbors Classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_regression.html">Nearest Neighbors regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_nca_illustration.html">Neighborhood Components Analysis Illustration</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_lof_novelty_detection.html">Novelty detection with Local Outlier Factor (LOF)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_lof_outlier_detection.html">Outlier detection with Local Outlier Factor (LOF)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neighbors/plot_kde_1d.html">Simple 1D Kernel Density Estimation</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_mlp_training_curves.html">Compare Stochastic learning strategies for MLPClassifier</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_rbm_logistic_classification.html">Restricted Boltzmann Machine features for digit classification</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_mlp_alpha.html">Varying regularization in Multi-layer Perceptron</a></li>
<li class="toctree-l2"><a class="reference internal" href="../neural_networks/plot_mnist_filters.html">Visualization of MLP weights on MNIST</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../compose/index.html">Pipelines and composite estimators</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_column_transformer.html">Column Transformer with Heterogeneous Data Sources</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_column_transformer_mixed_types.html">Column Transformer with Mixed Types</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_feature_union.html">Concatenating multiple feature extraction methods</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_transformed_target.html">Effect of transforming the targets in regression model</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_digits_pipe.html">Pipelining: chaining a PCA and a logistic regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="../compose/plot_compare_reduction.html">Selecting dimensionality reduction with Pipeline and GridSearchCV</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_all_scaling.html">Compare the effect of different scalers on data with outliers</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_target_encoder.html">Comparing Target Encoder with Other Encoders</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization_strategies.html">Demonstrating the different strategies of KBinsDiscretizer</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization_classification.html">Feature discretization</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_scaling_importance.html">Importance of Feature Scaling</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_map_data_to_normal.html">Map data to a normal distribution</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_target_encoder_cross_val.html">Target Encoder’s Internal Cross fitting</a></li>
<li class="toctree-l2"><a class="reference internal" href="../preprocessing/plot_discretization.html">Using KBinsDiscretizer to discretize continuous features</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../semi_supervised/index.html">Semi Supervised Classification</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_semi_supervised_versus_svm_iris.html">Decision boundary of semi-supervised classifiers versus SVM on the Iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_self_training_varying_threshold.html">Effect of varying threshold for self-training</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_digits_active_learning.html">Label Propagation digits active learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_digits.html">Label Propagation digits: Demonstrating performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_label_propagation_structure.html">Label Propagation learning a complex structure</a></li>
<li class="toctree-l2"><a class="reference internal" href="../semi_supervised/plot_semi_supervised_newsgroups.html">Semi-supervised Classification on a Text Dataset</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../svm/index.html">Support Vector Machines</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_oneclass.html">One-class SVM with non-linear kernel (RBF)</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_kernels.html">Plot classification boundaries with different SVM Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_iris_svc.html">Plot different SVM classifiers in the iris dataset</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_linearsvc_support_vectors.html">Plot the support vectors in LinearSVC</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_rbf_parameters.html">RBF SVM parameters</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_margin.html">SVM Margins Example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_tie_breaking.html">SVM Tie Breaking Example</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_custom_kernel.html">SVM with custom kernel</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_anova.html">SVM-Anova: SVM with univariate feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_separating_hyperplane.html">SVM: Maximum margin separating hyperplane</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_separating_hyperplane_unbalanced.html">SVM: Separating hyperplane for unbalanced classes</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_weighted_samples.html">SVM: Weighted samples</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_scale_c.html">Scaling the regularization parameter for SVCs</a></li>
<li class="toctree-l2"><a class="reference internal" href="../svm/plot_svm_regression.html">Support Vector Regression (SVR) using linear and non-linear kernels</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../exercises/index.html">Tutorial exercises</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../exercises/plot_cv_diabetes.html">Cross-validation on diabetes Dataset Exercise</a></li>
<li class="toctree-l2"><a class="reference internal" href="../exercises/plot_digits_classification_exercise.html">Digits Classification Exercise</a></li>
<li class="toctree-l2"><a class="reference internal" href="../exercises/plot_iris_exercise.html">SVM Exercise</a></li>
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<li class="toctree-l1 has-children"><a class="reference internal" href="../text/index.html">Working with text documents</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../text/plot_document_classification_20newsgroups.html">Classification of text documents using sparse features</a></li>
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<li class="toctree-l2"><a class="reference internal" href="../text/plot_hashing_vs_dict_vectorizer.html">FeatureHasher and DictVectorizer Comparison</a></li>
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<div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#sphx-glr-download-auto-examples-mixture-plot-gmm-sin-py"><span class="std std-ref">Go to the end</span></a>
to download the full example code. or to run this example in your browser via JupyterLite or Binder</p>
</div>
<section class="sphx-glr-example-title" id="gaussian-mixture-model-sine-curve">
<span id="sphx-glr-auto-examples-mixture-plot-gmm-sin-py"></span><h1>Gaussian Mixture Model Sine Curve<a class="headerlink" href="#gaussian-mixture-model-sine-curve" title="Link to this heading">#</a></h1>
<p>This example demonstrates the behavior of Gaussian mixture models fit on data
that was not sampled from a mixture of Gaussian random variables. The dataset
is formed by 100 points loosely spaced following a noisy sine curve. There is
therefore no ground truth value for the number of Gaussian components.</p>
<p>The first model is a classical Gaussian Mixture Model with 10 components fit
with the Expectation-Maximization algorithm.</p>
<p>The second model is a Bayesian Gaussian Mixture Model with a Dirichlet process
prior fit with variational inference. The low value of the concentration prior
makes the model favor a lower number of active components. This models
“decides” to focus its modeling power on the big picture of the structure of
the dataset: groups of points with alternating directions modeled by
non-diagonal covariance matrices. Those alternating directions roughly capture
the alternating nature of the original sine signal.</p>
<p>The third model is also a Bayesian Gaussian mixture model with a Dirichlet
process prior but this time the value of the concentration prior is higher
giving the model more liberty to model the fine-grained structure of the data.
The result is a mixture with a larger number of active components that is
similar to the first model where we arbitrarily decided to fix the number of
components to 10.</p>
<p>Which model is the best is a matter of subjective judgment: do we want to
favor models that only capture the big picture to summarize and explain most of
the structure of the data while ignoring the details or do we prefer models
that closely follow the high density regions of the signal?</p>
<p>The last two panels show how we can sample from the last two models. The
resulting samples distributions do not look exactly like the original data
distribution. The difference primarily stems from the approximation error we
made by using a model that assumes that the data was generated by a finite
number of Gaussian components instead of a continuous noisy sine curve.</p>
<img src="../../_images/sphx_glr_plot_gmm_sin_001.png" srcset="../../_images/sphx_glr_plot_gmm_sin_001.png" alt="Expectation-maximization, Bayesian Gaussian mixture models with a Dirichlet process prior for $\gamma_0=0.01$., Gaussian mixture with a Dirichlet process prior for $\gamma_0=0.01$ sampled with $2000$ samples., Bayesian Gaussian mixture models with a Dirichlet process prior for $\gamma_0=100$, Gaussian mixture with a Dirichlet process prior for $\gamma_0=100$ sampled with $2000$ samples." class = "sphx-glr-single-img"/><div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="c1"># Authors: The scikit-learn developers</span>
<span class="c1"># SPDX-License-Identifier: BSD-3-Clause</span>
<span class="kn">import</span> <span class="nn">itertools</span>
<span class="kn">import</span> <span class="nn">matplotlib</span> <span class="k">as</span> <span class="nn">mpl</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">linalg</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">mixture</span>
<span class="n">color_iter</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/itertools.html#itertools.cycle" title="itertools.cycle" class="sphx-glr-backref-module-itertools sphx-glr-backref-type-py-function"><span class="n">itertools</span><span class="o">.</span><span class="n">cycle</span></a><span class="p">([</span><span class="s2">"navy"</span><span class="p">,</span> <span class="s2">"c"</span><span class="p">,</span> <span class="s2">"cornflowerblue"</span><span class="p">,</span> <span class="s2">"gold"</span><span class="p">,</span> <span class="s2">"darkorange"</span><span class="p">])</span>
<span class="k">def</span> <span class="nf">plot_results</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">means</span><span class="p">,</span> <span class="n">covariances</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">title</span><span class="p">):</span>
<span class="n">splot</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span> <span class="o">+</span> <span class="n">index</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">covar</span><span class="p">,</span> <span class="n">color</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="nb">zip</span><span class="p">(</span><span class="n">means</span><span class="p">,</span> <span class="n">covariances</span><span class="p">,</span> <span class="n">color_iter</span><span class="p">)):</span>
<span class="n">v</span><span class="p">,</span> <span class="n">w</span> <span class="o">=</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html#scipy.linalg.eigh" title="scipy.linalg.eigh" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">linalg</span><span class="o">.</span><span class="n">eigh</span></a><span class="p">(</span><span class="n">covar</span><span class="p">)</span>
<span class="n">v</span> <span class="o">=</span> <span class="mf">2.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sqrt.html#numpy.sqrt" title="numpy.sqrt" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><span class="mf">2.0</span><span class="p">)</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sqrt.html#numpy.sqrt" title="numpy.sqrt" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span></a><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">/</span> <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.norm.html#scipy.linalg.norm" title="scipy.linalg.norm" class="sphx-glr-backref-module-scipy-linalg sphx-glr-backref-type-py-function"><span class="n">linalg</span><span class="o">.</span><span class="n">norm</span></a><span class="p">(</span><span class="n">w</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># as the DP will not use every component it has access to</span>
<span class="c1"># unless it needs it, we shouldn't plot the redundant</span>
<span class="c1"># components.</span>
<span class="k">if</span> <span class="ow">not</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.any.html#numpy.any" title="numpy.any" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">any</span></a><span class="p">(</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">):</span>
<span class="k">continue</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mf">0.8</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<span class="c1"># Plot an ellipse to show the Gaussian component</span>
<span class="n">angle</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.arctan.html#numpy.arctan" title="numpy.arctan" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">arctan</span></a><span class="p">(</span><span class="n">u</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">/</span> <span class="n">u</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">angle</span> <span class="o">=</span> <span class="mf">180.0</span> <span class="o">*</span> <span class="n">angle</span> <span class="o">/</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="c1"># convert to degrees</span>
<span class="n">ell</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.patches.Ellipse.html#matplotlib.patches.Ellipse" title="matplotlib.patches.Ellipse" class="sphx-glr-backref-module-matplotlib-patches sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mpl</span><span class="o">.</span><span class="n">patches</span><span class="o">.</span><span class="n">Ellipse</span></a><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">v</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">v</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">angle</span><span class="o">=</span><span class="mf">180.0</span> <span class="o">+</span> <span class="n">angle</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<span class="n">ell</span><span class="o">.</span><span class="n">set_clip_box</span><span class="p">(</span><span class="n">splot</span><span class="o">.</span><span class="n">bbox</span><span class="p">)</span>
<span class="n">ell</span><span class="o">.</span><span class="n">set_alpha</span><span class="p">(</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">splot</span><span class="o">.</span><span class="n">add_artist</span><span class="p">(</span><span class="n">ell</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">(</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="mf">4.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">-</span> <span class="mf">6.0</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">(</span><span class="o">-</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="matplotlib.pyplot.xticks" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.yticks.html#matplotlib.pyplot.yticks" title="matplotlib.pyplot.yticks" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">yticks</span></a><span class="p">(())</span>
<span class="k">def</span> <span class="nf">plot_samples</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">,</span> <span class="n">n_components</span><span class="p">,</span> <span class="n">index</span><span class="p">,</span> <span class="n">title</span><span class="p">):</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplot</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">4</span> <span class="o">+</span> <span class="n">index</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">color</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n_components</span><span class="p">),</span> <span class="n">color_iter</span><span class="p">):</span>
<span class="c1"># as the DP will not use every component it has access to</span>
<span class="c1"># unless it needs it, we shouldn't plot the redundant</span>
<span class="c1"># components.</span>
<span class="k">if</span> <span class="ow">not</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.any.html#numpy.any" title="numpy.any" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">any</span></a><span class="p">(</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">):</span>
<span class="k">continue</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html#matplotlib.pyplot.scatter" title="matplotlib.pyplot.scatter" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">scatter</span></a><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">Y</span> <span class="o">==</span> <span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="mf">0.8</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xlim.html#matplotlib.pyplot.xlim" title="matplotlib.pyplot.xlim" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span></a><span class="p">(</span><span class="o">-</span><span class="mf">6.0</span><span class="p">,</span> <span class="mf">4.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">-</span> <span class="mf">6.0</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.ylim.html#matplotlib.pyplot.ylim" title="matplotlib.pyplot.ylim" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span></a><span class="p">(</span><span class="o">-</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">5.0</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.title.html#matplotlib.pyplot.title" title="matplotlib.pyplot.title" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">title</span></a><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.xticks.html#matplotlib.pyplot.xticks" title="matplotlib.pyplot.xticks" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">xticks</span></a><span class="p">(())</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.yticks.html#matplotlib.pyplot.yticks" title="matplotlib.pyplot.yticks" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">yticks</span></a><span class="p">(())</span>
<span class="c1"># Parameters</span>
<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># Generate random sample following a sine curve</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">0</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.zeros.html#numpy.zeros" title="numpy.zeros" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">zeros</span></a><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">step</span> <span class="o">=</span> <span class="mf">4.0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a> <span class="o">/</span> <span class="n">n_samples</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]):</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">i</span> <span class="o">*</span> <span class="n">step</span> <span class="o">-</span> <span class="mf">6.0</span>
<span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>
<span class="n">X</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mf">3.0</span> <span class="o">*</span> <span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.sin.html#numpy.sin" title="numpy.sin" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sin</span></a><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html#numpy.random.normal" title="numpy.random.normal" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.figure.html#matplotlib.pyplot.figure" title="matplotlib.pyplot.figure" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">figure</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">10</span><span class="p">))</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots_adjust.html#matplotlib.pyplot.subplots_adjust" title="matplotlib.pyplot.subplots_adjust" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span></a><span class="p">(</span>
<span class="n">bottom</span><span class="o">=</span><span class="mf">0.04</span><span class="p">,</span> <span class="n">top</span><span class="o">=</span><span class="mf">0.95</span><span class="p">,</span> <span class="n">hspace</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">wspace</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">left</span><span class="o">=</span><span class="mf">0.03</span><span class="p">,</span> <span class="n">right</span><span class="o">=</span><span class="mf">0.97</span>
<span class="p">)</span>
<span class="c1"># Fit a Gaussian mixture with EM using ten components</span>
<span class="n">gmm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mixture</span><span class="o">.</span><span class="n">GaussianMixture</span></a><span class="p">(</span>
<span class="n">n_components</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">covariance_type</span><span class="o">=</span><span class="s2">"full"</span><span class="p">,</span> <span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span>
<span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">plot_results</span><span class="p">(</span>
<span class="n">X</span><span class="p">,</span> <span class="n">gmm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span> <span class="n">gmm</span><span class="o">.</span><span class="n">means_</span><span class="p">,</span> <span class="n">gmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s2">"Expectation-maximization"</span>
<span class="p">)</span>
<span class="n">dpgmm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mixture</span><span class="o">.</span><span class="n">BayesianGaussianMixture</span></a><span class="p">(</span>
<span class="n">n_components</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">covariance_type</span><span class="o">=</span><span class="s2">"full"</span><span class="p">,</span>
<span class="n">weight_concentration_prior</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span>
<span class="n">weight_concentration_prior_type</span><span class="o">=</span><span class="s2">"dirichlet_process"</span><span class="p">,</span>
<span class="n">mean_precision_prior</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span>
<span class="n">covariance_prior</span><span class="o">=</span><span class="mf">1e0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.eye.html#numpy.eye" title="numpy.eye" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">eye</span></a><span class="p">(</span><span class="mi">2</span><span class="p">),</span>
<span class="n">init_params</span><span class="o">=</span><span class="s2">"random"</span><span class="p">,</span>
<span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">plot_results</span><span class="p">(</span>
<span class="n">X</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">means_</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">,</span>
<span class="mi">1</span><span class="p">,</span>
<span class="s2">"Bayesian Gaussian mixture models with a Dirichlet process prior "</span>
<span class="sa">r</span><span class="s2">"for $\gamma_0=0.01$."</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">X_s</span><span class="p">,</span> <span class="n">y_s</span> <span class="o">=</span> <span class="n">dpgmm</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">2000</span><span class="p">)</span>
<span class="n">plot_samples</span><span class="p">(</span>
<span class="n">X_s</span><span class="p">,</span>
<span class="n">y_s</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">n_components</span><span class="p">,</span>
<span class="mi">0</span><span class="p">,</span>
<span class="s2">"Gaussian mixture with a Dirichlet process prior "</span>
<span class="sa">r</span><span class="s2">"for $\gamma_0=0.01$ sampled with $2000$ samples."</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">dpgmm</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture" class="sphx-glr-backref-module-sklearn-mixture sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">mixture</span><span class="o">.</span><span class="n">BayesianGaussianMixture</span></a><span class="p">(</span>
<span class="n">n_components</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span>
<span class="n">covariance_type</span><span class="o">=</span><span class="s2">"full"</span><span class="p">,</span>
<span class="n">weight_concentration_prior</span><span class="o">=</span><span class="mf">1e2</span><span class="p">,</span>
<span class="n">weight_concentration_prior_type</span><span class="o">=</span><span class="s2">"dirichlet_process"</span><span class="p">,</span>
<span class="n">mean_precision_prior</span><span class="o">=</span><span class="mf">1e-2</span><span class="p">,</span>
<span class="n">covariance_prior</span><span class="o">=</span><span class="mf">1e0</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.eye.html#numpy.eye" title="numpy.eye" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">eye</span></a><span class="p">(</span><span class="mi">2</span><span class="p">),</span>
<span class="n">init_params</span><span class="o">=</span><span class="s2">"kmeans"</span><span class="p">,</span>
<span class="n">max_iter</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="n">random_state</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span>
<span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">plot_results</span><span class="p">(</span>
<span class="n">X</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">),</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">means_</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">covariances_</span><span class="p">,</span>
<span class="mi">2</span><span class="p">,</span>
<span class="s2">"Bayesian Gaussian mixture models with a Dirichlet process prior "</span>
<span class="sa">r</span><span class="s2">"for $\gamma_0=100$"</span><span class="p">,</span>
<span class="p">)</span>
<span class="n">X_s</span><span class="p">,</span> <span class="n">y_s</span> <span class="o">=</span> <span class="n">dpgmm</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">2000</span><span class="p">)</span>
<span class="n">plot_samples</span><span class="p">(</span>
<span class="n">X_s</span><span class="p">,</span>
<span class="n">y_s</span><span class="p">,</span>
<span class="n">dpgmm</span><span class="o">.</span><span class="n">n_components</span><span class="p">,</span>
<span class="mi">1</span><span class="p">,</span>
<span class="s2">"Gaussian mixture with a Dirichlet process prior "</span>
<span class="sa">r</span><span class="s2">"for $\gamma_0=100$ sampled with $2000$ samples."</span><span class="p">,</span>
<span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">show</span></a><span class="p">()</span>
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<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="Plot the confidence ellipsoids of a mixture of two Gaussians obtained with Expectation Maximisation (``GaussianMixture`` class) and Variational Inference (``BayesianGaussianMixture`` class models with a Dirichlet process prior)."><img alt="" src="../../_images/sphx_glr_plot_gmm_thumb.png" />
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