Pushing the docs to dev/ for branch: master, commit 67a85b8ed842612d59187e5fdc81a8b86eb50afd

Author: sklearn-ci

dev/_downloads/auto_examples_jupyter.zip

@@ -15,7 +15,7 @@
     }, 
     {
       "source": [
-        "\n==================================================\nAutomatic Relevance Determination Regression (ARD)\n==================================================\n\nFit regression model with Bayesian Ridge Regression.\n\nSee `bayesian_ridge_regression` for more information on the regressor.\n\nCompared to the OLS (ordinary least squares) estimator, the coefficient\nweights are slightly shifted toward zeros, which stabilises them.\n\nThe histogram of the estimated weights is very peaked, as a sparsity-inducing\nprior is implied on the weights.\n\nThe estimation of the model is done by iteratively maximizing the\nmarginal log-likelihood of the observations.\n\n"
+        "\n==================================================\nAutomatic Relevance Determination Regression (ARD)\n==================================================\n\nFit regression model with Bayesian Ridge Regression.\n\nSee `bayesian_ridge_regression` for more information on the regressor.\n\nCompared to the OLS (ordinary least squares) estimator, the coefficient\nweights are slightly shifted toward zeros, which stabilises them.\n\nThe histogram of the estimated weights is very peaked, as a sparsity-inducing\nprior is implied on the weights.\n\nThe estimation of the model is done by iteratively maximizing the\nmarginal log-likelihood of the observations.\n\nWe also plot predictions and uncertainties for ARD\nfor one dimensional regression using polynomial feature expansion.\nNote the uncertainty starts going up on the right side of the plot.\nThis is because these test samples are outside of the range of the training\nsamples.\n\n"
       ], 
       "cell_type": "markdown", 
       "metadata": {}
@@ -69,7 +69,7 @@
     }, 
     {
       "source": [
-        "Plot the true weights, the estimated weights and the histogram of the\nweights\n\n"
+        "Plot the true weights, the estimated weights, the histogram of the\nweights, and predictions with standard deviations\n\n"
       ], 
       "cell_type": "markdown", 
       "metadata": {}
@@ -78,7 +78,7 @@
       "execution_count": null, 
       "cell_type": "code", 
       "source": [
-        "plt.figure(figsize=(6, 5))\nplt.title(\"Weights of the model\")\nplt.plot(clf.coef_, color='darkblue', linestyle='-', linewidth=2,\n         label=\"ARD estimate\")\nplt.plot(ols.coef_, color='yellowgreen', linestyle=':', linewidth=2,\n         label=\"OLS estimate\")\nplt.plot(w, color='orange', linestyle='-', linewidth=2, label=\"Ground truth\")\nplt.xlabel(\"Features\")\nplt.ylabel(\"Values of the weights\")\nplt.legend(loc=1)\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Histogram of the weights\")\nplt.hist(clf.coef_, bins=n_features, color='navy', log=True)\nplt.scatter(clf.coef_[relevant_features], 5 * np.ones(len(relevant_features)),\n            color='gold', marker='o', label=\"Relevant features\")\nplt.ylabel(\"Features\")\nplt.xlabel(\"Values of the weights\")\nplt.legend(loc=1)\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Marginal log-likelihood\")\nplt.plot(clf.scores_, color='navy', linewidth=2)\nplt.ylabel(\"Score\")\nplt.xlabel(\"Iterations\")\nplt.show()"
+        "plt.figure(figsize=(6, 5))\nplt.title(\"Weights of the model\")\nplt.plot(clf.coef_, color='darkblue', linestyle='-', linewidth=2,\n         label=\"ARD estimate\")\nplt.plot(ols.coef_, color='yellowgreen', linestyle=':', linewidth=2,\n         label=\"OLS estimate\")\nplt.plot(w, color='orange', linestyle='-', linewidth=2, label=\"Ground truth\")\nplt.xlabel(\"Features\")\nplt.ylabel(\"Values of the weights\")\nplt.legend(loc=1)\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Histogram of the weights\")\nplt.hist(clf.coef_, bins=n_features, color='navy', log=True)\nplt.scatter(clf.coef_[relevant_features], 5 * np.ones(len(relevant_features)),\n            color='gold', marker='o', label=\"Relevant features\")\nplt.ylabel(\"Features\")\nplt.xlabel(\"Values of the weights\")\nplt.legend(loc=1)\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Marginal log-likelihood\")\nplt.plot(clf.scores_, color='navy', linewidth=2)\nplt.ylabel(\"Score\")\nplt.xlabel(\"Iterations\")\n\n\n# Plotting some predictions for polynomial regression\ndef f(x, noise_amount):\n    y = np.sqrt(x) * np.sin(x)\n    noise = np.random.normal(0, 1, len(x))\n    return y + noise_amount * noise\n\n\ndegree = 10\nX = np.linspace(0, 10, 100)\ny = f(X, noise_amount=1)\nclf_poly = ARDRegression(threshold_lambda=1e5)\nclf_poly.fit(np.vander(X, degree), y)\n\nX_plot = np.linspace(0, 11, 25)\ny_plot = f(X_plot, noise_amount=0)\ny_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True)\nplt.figure(figsize=(6, 5))\nplt.errorbar(X_plot, y_mean, y_std, color='navy',\n             label=\"Polynomial ARD\", linewidth=2)\nplt.plot(X_plot, y_plot, color='gold', linewidth=2,\n         label=\"Ground Truth\")\nplt.ylabel(\"Output y\")\nplt.xlabel(\"Feature X\")\nplt.legend(loc=\"lower left\")\nplt.show()"
       ], 
       "outputs": [], 
       "metadata": {

dev/_downloads/auto_examples_python.zip

@@ -15,6 +15,12 @@
 
 The estimation of the model is done by iteratively maximizing the
 marginal log-likelihood of the observations.
+
+We also plot predictions and uncertainties for ARD
+for one dimensional regression using polynomial feature expansion.
+Note the uncertainty starts going up on the right side of the plot.
+This is because these test samples are outside of the range of the training
+samples.
 """
 print(__doc__)
 
@@ -54,8 +60,8 @@
 ols.fit(X, y)
 
 ###############################################################################
-# Plot the true weights, the estimated weights and the histogram of the
-# weights
+# Plot the true weights, the estimated weights, the histogram of the
+# weights, and predictions with standard deviations
 plt.figure(figsize=(6, 5))
 plt.title("Weights of the model")
 plt.plot(clf.coef_, color='darkblue', linestyle='-', linewidth=2,
@@ -81,4 +87,30 @@
 plt.plot(clf.scores_, color='navy', linewidth=2)
 plt.ylabel("Score")
 plt.xlabel("Iterations")
+
+
+# Plotting some predictions for polynomial regression
+def f(x, noise_amount):
+    y = np.sqrt(x) * np.sin(x)
+    noise = np.random.normal(0, 1, len(x))
+    return y + noise_amount * noise
+
+
+degree = 10
+X = np.linspace(0, 10, 100)
+y = f(X, noise_amount=1)
+clf_poly = ARDRegression(threshold_lambda=1e5)
+clf_poly.fit(np.vander(X, degree), y)
+
+X_plot = np.linspace(0, 11, 25)
+y_plot = f(X_plot, noise_amount=0)
+y_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True)
+plt.figure(figsize=(6, 5))
+plt.errorbar(X_plot, y_mean, y_std, color='navy',
+             label="Polynomial ARD", linewidth=2)
+plt.plot(X_plot, y_plot, color='gold', linewidth=2,
+         label="Ground Truth")
+plt.ylabel("Output y")
+plt.xlabel("Feature X")
+plt.legend(loc="lower left")
 plt.show()

dev/_downloads/plot_ard.ipynb

@@ -15,7 +15,7 @@
     }, 
     {
       "source": [
-        "\n# Bayesian Ridge Regression\n\n\nComputes a Bayesian Ridge Regression on a synthetic dataset.\n\nSee `bayesian_ridge_regression` for more information on the regressor.\n\nCompared to the OLS (ordinary least squares) estimator, the coefficient\nweights are slightly shifted toward zeros, which stabilises them.\n\nAs the prior on the weights is a Gaussian prior, the histogram of the\nestimated weights is Gaussian.\n\nThe estimation of the model is done by iteratively maximizing the\nmarginal log-likelihood of the observations.\n\n"
+        "\n# Bayesian Ridge Regression\n\n\nComputes a Bayesian Ridge Regression on a synthetic dataset.\n\nSee `bayesian_ridge_regression` for more information on the regressor.\n\nCompared to the OLS (ordinary least squares) estimator, the coefficient\nweights are slightly shifted toward zeros, which stabilises them.\n\nAs the prior on the weights is a Gaussian prior, the histogram of the\nestimated weights is Gaussian.\n\nThe estimation of the model is done by iteratively maximizing the\nmarginal log-likelihood of the observations.\n\nWe also plot predictions and uncertainties for Bayesian Ridge Regression\nfor one dimensional regression using polynomial feature expansion.\nNote the uncertainty starts going up on the right side of the plot.\nThis is because these test samples are outside of the range of the training\nsamples.\n\n"
       ], 
       "cell_type": "markdown", 
       "metadata": {}
@@ -69,7 +69,7 @@
     }, 
     {
       "source": [
-        "Plot true weights, estimated weights and histogram of the weights\n\n"
+        "Plot true weights, estimated weights, histogram of the weights, and\npredictions with standard deviations\n\n"
       ], 
       "cell_type": "markdown", 
       "metadata": {}
@@ -78,7 +78,7 @@
       "execution_count": null, 
       "cell_type": "code", 
       "source": [
-        "lw = 2\nplt.figure(figsize=(6, 5))\nplt.title(\"Weights of the model\")\nplt.plot(clf.coef_, color='lightgreen', linewidth=lw,\n         label=\"Bayesian Ridge estimate\")\nplt.plot(w, color='gold', linewidth=lw, label=\"Ground truth\")\nplt.plot(ols.coef_, color='navy', linestyle='--', label=\"OLS estimate\")\nplt.xlabel(\"Features\")\nplt.ylabel(\"Values of the weights\")\nplt.legend(loc=\"best\", prop=dict(size=12))\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Histogram of the weights\")\nplt.hist(clf.coef_, bins=n_features, color='gold', log=True)\nplt.scatter(clf.coef_[relevant_features], 5 * np.ones(len(relevant_features)),\n            color='navy', label=\"Relevant features\")\nplt.ylabel(\"Features\")\nplt.xlabel(\"Values of the weights\")\nplt.legend(loc=\"upper left\")\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Marginal log-likelihood\")\nplt.plot(clf.scores_, color='navy', linewidth=lw)\nplt.ylabel(\"Score\")\nplt.xlabel(\"Iterations\")\nplt.show()"
+        "lw = 2\nplt.figure(figsize=(6, 5))\nplt.title(\"Weights of the model\")\nplt.plot(clf.coef_, color='lightgreen', linewidth=lw,\n         label=\"Bayesian Ridge estimate\")\nplt.plot(w, color='gold', linewidth=lw, label=\"Ground truth\")\nplt.plot(ols.coef_, color='navy', linestyle='--', label=\"OLS estimate\")\nplt.xlabel(\"Features\")\nplt.ylabel(\"Values of the weights\")\nplt.legend(loc=\"best\", prop=dict(size=12))\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Histogram of the weights\")\nplt.hist(clf.coef_, bins=n_features, color='gold', log=True)\nplt.scatter(clf.coef_[relevant_features], 5 * np.ones(len(relevant_features)),\n            color='navy', label=\"Relevant features\")\nplt.ylabel(\"Features\")\nplt.xlabel(\"Values of the weights\")\nplt.legend(loc=\"upper left\")\n\nplt.figure(figsize=(6, 5))\nplt.title(\"Marginal log-likelihood\")\nplt.plot(clf.scores_, color='navy', linewidth=lw)\nplt.ylabel(\"Score\")\nplt.xlabel(\"Iterations\")\n\n\n# Plotting some predictions for polynomial regression\ndef f(x, noise_amount):\n    y = np.sqrt(x) * np.sin(x)\n    noise = np.random.normal(0, 1, len(x))\n    return y + noise_amount * noise\n\n\ndegree = 10\nX = np.linspace(0, 10, 100)\ny = f(X, noise_amount=0.1)\nclf_poly = BayesianRidge()\nclf_poly.fit(np.vander(X, degree), y)\n\nX_plot = np.linspace(0, 11, 25)\ny_plot = f(X_plot, noise_amount=0)\ny_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True)\nplt.figure(figsize=(6, 5))\nplt.errorbar(X_plot, y_mean, y_std, color='navy',\n             label=\"Polynomial Bayesian Ridge Regression\", linewidth=lw)\nplt.plot(X_plot, y_plot, color='gold', linewidth=lw,\n         label=\"Ground Truth\")\nplt.ylabel(\"Output y\")\nplt.xlabel(\"Feature X\")\nplt.legend(loc=\"lower left\")\nplt.show()"
       ], 
       "outputs": [], 
       "metadata": {

dev/_downloads/plot_ard.py

@@ -15,6 +15,12 @@
 
 The estimation of the model is done by iteratively maximizing the
 marginal log-likelihood of the observations.
+
+We also plot predictions and uncertainties for Bayesian Ridge Regression
+for one dimensional regression using polynomial feature expansion.
+Note the uncertainty starts going up on the right side of the plot.
+This is because these test samples are outside of the range of the training
+samples.
 """
 print(__doc__)
 
@@ -51,7 +57,8 @@
 ols.fit(X, y)
 
 ###############################################################################
-# Plot true weights, estimated weights and histogram of the weights
+# Plot true weights, estimated weights, histogram of the weights, and
+# predictions with standard deviations
 lw = 2
 plt.figure(figsize=(6, 5))
 plt.title("Weights of the model")
@@ -77,4 +84,30 @@
 plt.plot(clf.scores_, color='navy', linewidth=lw)
 plt.ylabel("Score")
 plt.xlabel("Iterations")
+
+
+# Plotting some predictions for polynomial regression
+def f(x, noise_amount):
+    y = np.sqrt(x) * np.sin(x)
+    noise = np.random.normal(0, 1, len(x))
+    return y + noise_amount * noise
+
+
+degree = 10
+X = np.linspace(0, 10, 100)
+y = f(X, noise_amount=0.1)
+clf_poly = BayesianRidge()
+clf_poly.fit(np.vander(X, degree), y)
+
+X_plot = np.linspace(0, 11, 25)
+y_plot = f(X_plot, noise_amount=0)
+y_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True)
+plt.figure(figsize=(6, 5))
+plt.errorbar(X_plot, y_mean, y_std, color='navy',
+             label="Polynomial Bayesian Ridge Regression", linewidth=lw)
+plt.plot(X_plot, y_plot, color='gold', linewidth=lw,
+         label="Ground Truth")
+plt.ylabel("Output y")
+plt.xlabel("Feature X")
+plt.legend(loc="lower left")
 plt.show()