.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/covariance/plot_sparse_cov.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via JupyterLite or Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_covariance_plot_sparse_cov.py: ====================================== Sparse inverse covariance estimation ====================================== Using the GraphicalLasso estimator to learn a covariance and sparse precision from a small number of samples. To estimate a probabilistic model (e.g. a Gaussian model), estimating the precision matrix, that is the inverse covariance matrix, is as important as estimating the covariance matrix. Indeed a Gaussian model is parametrized by the precision matrix. To be in favorable recovery conditions, we sample the data from a model with a sparse inverse covariance matrix. In addition, we ensure that the data is not too much correlated (limiting the largest coefficient of the precision matrix) and that there a no small coefficients in the precision matrix that cannot be recovered. In addition, with a small number of observations, it is easier to recover a correlation matrix rather than a covariance, thus we scale the time series. Here, the number of samples is slightly larger than the number of dimensions, thus the empirical covariance is still invertible. However, as the observations are strongly correlated, the empirical covariance matrix is ill-conditioned and as a result its inverse --the empirical precision matrix-- is very far from the ground truth. If we use l2 shrinkage, as with the Ledoit-Wolf estimator, as the number of samples is small, we need to shrink a lot. As a result, the Ledoit-Wolf precision is fairly close to the ground truth precision, that is not far from being diagonal, but the off-diagonal structure is lost. The l1-penalized estimator can recover part of this off-diagonal structure. It learns a sparse precision. It is not able to recover the exact sparsity pattern: it detects too many non-zero coefficients. However, the highest non-zero coefficients of the l1 estimated correspond to the non-zero coefficients in the ground truth. Finally, the coefficients of the l1 precision estimate are biased toward zero: because of the penalty, they are all smaller than the corresponding ground truth value, as can be seen on the figure. Note that, the color range of the precision matrices is tweaked to improve readability of the figure. The full range of values of the empirical precision is not displayed. The alpha parameter of the GraphicalLasso setting the sparsity of the model is set by internal cross-validation in the GraphicalLassoCV. As can be seen on figure 2, the grid to compute the cross-validation score is iteratively refined in the neighborhood of the maximum. .. GENERATED FROM PYTHON SOURCE LINES 52-57 .. code-block:: Python # author: Gael Varoquaux # License: BSD 3 clause # Copyright: INRIA .. GENERATED FROM PYTHON SOURCE LINES 58-60 Generate the data ----------------- .. GENERATED FROM PYTHON SOURCE LINES 60-82 .. code-block:: Python import numpy as np from scipy import linalg from sklearn.datasets import make_sparse_spd_matrix n_samples = 60 n_features = 20 prng = np.random.RandomState(1) prec = make_sparse_spd_matrix( n_features, alpha=0.98, smallest_coef=0.4, largest_coef=0.7, random_state=prng ) cov = linalg.inv(prec) d = np.sqrt(np.diag(cov)) cov /= d cov /= d[:, np.newaxis] prec *= d prec *= d[:, np.newaxis] X = prng.multivariate_normal(np.zeros(n_features), cov, size=n_samples) X -= X.mean(axis=0) X /= X.std(axis=0) .. GENERATED FROM PYTHON SOURCE LINES 83-85 Estimate the covariance ----------------------- .. GENERATED FROM PYTHON SOURCE LINES 85-97 .. code-block:: Python from sklearn.covariance import GraphicalLassoCV, ledoit_wolf emp_cov = np.dot(X.T, X) / n_samples model = GraphicalLassoCV() model.fit(X) cov_ = model.covariance_ prec_ = model.precision_ lw_cov_, _ = ledoit_wolf(X) lw_prec_ = linalg.inv(lw_cov_) .. GENERATED FROM PYTHON SOURCE LINES 98-100 Plot the results ---------------- .. GENERATED FROM PYTHON SOURCE LINES 100-148 .. code-block:: Python import matplotlib.pyplot as plt plt.figure(figsize=(10, 6)) plt.subplots_adjust(left=0.02, right=0.98) # plot the covariances covs = [ ("Empirical", emp_cov), ("Ledoit-Wolf", lw_cov_), ("GraphicalLassoCV", cov_), ("True", cov), ] vmax = cov_.max() for i, (name, this_cov) in enumerate(covs): plt.subplot(2, 4, i + 1) plt.imshow( this_cov, interpolation="nearest", vmin=-vmax, vmax=vmax, cmap=plt.cm.RdBu_r ) plt.xticks(()) plt.yticks(()) plt.title("%s covariance" % name) # plot the precisions precs = [ ("Empirical", linalg.inv(emp_cov)), ("Ledoit-Wolf", lw_prec_), ("GraphicalLasso", prec_), ("True", prec), ] vmax = 0.9 * prec_.max() for i, (name, this_prec) in enumerate(precs): ax = plt.subplot(2, 4, i + 5) plt.imshow( np.ma.masked_equal(this_prec, 0), interpolation="nearest", vmin=-vmax, vmax=vmax, cmap=plt.cm.RdBu_r, ) plt.xticks(()) plt.yticks(()) plt.title("%s precision" % name) if hasattr(ax, "set_facecolor"): ax.set_facecolor(".7") else: ax.set_axis_bgcolor(".7") .. image-sg:: /auto_examples/covariance/images/sphx_glr_plot_sparse_cov_001.png :alt: Empirical covariance, Ledoit-Wolf covariance, GraphicalLassoCV covariance, True covariance, Empirical precision, Ledoit-Wolf precision, GraphicalLasso precision, True precision :srcset: /auto_examples/covariance/images/sphx_glr_plot_sparse_cov_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 149-160 .. code-block:: Python # plot the model selection metric plt.figure(figsize=(4, 3)) plt.axes([0.2, 0.15, 0.75, 0.7]) plt.plot(model.cv_results_["alphas"], model.cv_results_["mean_test_score"], "o-") plt.axvline(model.alpha_, color=".5") plt.title("Model selection") plt.ylabel("Cross-validation score") plt.xlabel("alpha") plt.show() .. image-sg:: /auto_examples/covariance/images/sphx_glr_plot_sparse_cov_002.png :alt: Model selection :srcset: /auto_examples/covariance/images/sphx_glr_plot_sparse_cov_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.609 seconds) .. _sphx_glr_download_auto_examples_covariance_plot_sparse_cov.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/covariance/plot_sparse_cov.ipynb :alt: Launch binder :width: 150 px .. container:: lite-badge .. image:: images/jupyterlite_badge_logo.svg :target: ../../lite/lab/?path=auto_examples/covariance/plot_sparse_cov.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_sparse_cov.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_sparse_cov.py ` .. include:: plot_sparse_cov.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_