1835 lines
54 KiB
Plaintext
1835 lines
54 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Chapter 5 – Support Vector Machines**\n",
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"\n",
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"_This notebook contains all the sample code and solutions to the exercises in chapter 5._"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Setup"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"First, let's import a few common modules, ensure MatplotLib plots figures inline and prepare a function to save the figures. We also check that Python 3.5 or later is installed (although Python 2.x may work, it is deprecated so we strongly recommend you use Python 3 instead), as well as Scikit-Learn ≥0.20."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Python ≥3.5 is required\n",
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"import sys\n",
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"assert sys.version_info >= (3, 5)\n",
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"\n",
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"# Scikit-Learn ≥0.20 is required\n",
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"import sklearn\n",
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"assert sklearn.__version__ >= \"0.20\"\n",
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"\n",
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"# Common imports\n",
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"import numpy as np\n",
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"import os\n",
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"\n",
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"# to make this notebook's output stable across runs\n",
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"np.random.seed(42)\n",
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"\n",
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"# To plot pretty figures\n",
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"%matplotlib inline\n",
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"import matplotlib as mpl\n",
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"import matplotlib.pyplot as plt\n",
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"mpl.rc('axes', labelsize=14)\n",
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"mpl.rc('xtick', labelsize=12)\n",
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"mpl.rc('ytick', labelsize=12)\n",
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"\n",
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"# Where to save the figures\n",
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"PROJECT_ROOT_DIR = \".\"\n",
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"CHAPTER_ID = \"svm\"\n",
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"IMAGES_PATH = os.path.join(PROJECT_ROOT_DIR, \"images\", CHAPTER_ID)\n",
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"os.makedirs(IMAGES_PATH, exist_ok=True)\n",
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"\n",
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"def save_fig(fig_id, tight_layout=True, fig_extension=\"png\", resolution=300):\n",
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" path = os.path.join(IMAGES_PATH, fig_id + \".\" + fig_extension)\n",
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" print(\"Saving figure\", fig_id)\n",
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" if tight_layout:\n",
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" plt.tight_layout()\n",
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" plt.savefig(path, format=fig_extension, dpi=resolution)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Large margin classification"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The next few code cells generate the first figures in chapter 5. The first actual code sample comes after:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"from sklearn.svm import SVC\n",
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"from sklearn import datasets\n",
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"\n",
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"iris = datasets.load_iris()\n",
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"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
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"y = iris[\"target\"]\n",
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"\n",
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"setosa_or_versicolor = (y == 0) | (y == 1)\n",
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"X = X[setosa_or_versicolor]\n",
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"y = y[setosa_or_versicolor]\n",
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"\n",
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"# SVM Classifier model\n",
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"svm_clf = SVC(kernel=\"linear\", C=float(\"inf\"))\n",
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"svm_clf.fit(X, y)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Bad models\n",
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"x0 = np.linspace(0, 5.5, 200)\n",
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"pred_1 = 5*x0 - 20\n",
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"pred_2 = x0 - 1.8\n",
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"pred_3 = 0.1 * x0 + 0.5\n",
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"\n",
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"def plot_svc_decision_boundary(svm_clf, xmin, xmax):\n",
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" w = svm_clf.coef_[0]\n",
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" b = svm_clf.intercept_[0]\n",
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"\n",
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" # At the decision boundary, w0*x0 + w1*x1 + b = 0\n",
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" # => x1 = -w0/w1 * x0 - b/w1\n",
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" x0 = np.linspace(xmin, xmax, 200)\n",
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" decision_boundary = -w[0]/w[1] * x0 - b/w[1]\n",
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"\n",
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" margin = 1/w[1]\n",
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" gutter_up = decision_boundary + margin\n",
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" gutter_down = decision_boundary - margin\n",
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"\n",
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" svs = svm_clf.support_vectors_\n",
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" plt.scatter(svs[:, 0], svs[:, 1], s=180, facecolors='#FFAAAA')\n",
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" plt.plot(x0, decision_boundary, \"k-\", linewidth=2)\n",
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" plt.plot(x0, gutter_up, \"k--\", linewidth=2)\n",
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" plt.plot(x0, gutter_down, \"k--\", linewidth=2)\n",
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"\n",
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"plt.figure(figsize=(12,2.7))\n",
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"\n",
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"plt.subplot(121)\n",
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"plt.plot(x0, pred_1, \"g--\", linewidth=2)\n",
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"plt.plot(x0, pred_2, \"m-\", linewidth=2)\n",
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"plt.plot(x0, pred_3, \"r-\", linewidth=2)\n",
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"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\", label=\"Iris versicolor\")\n",
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"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\", label=\"Iris setosa\")\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.ylabel(\"Petal width\", fontsize=14)\n",
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"plt.legend(loc=\"upper left\", fontsize=14)\n",
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"plt.axis([0, 5.5, 0, 2])\n",
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"\n",
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"plt.subplot(122)\n",
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"plot_svc_decision_boundary(svm_clf, 0, 5.5)\n",
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"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\")\n",
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"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\")\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.axis([0, 5.5, 0, 2])\n",
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"\n",
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"save_fig(\"large_margin_classification_plot\")\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Sensitivity to feature scales"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"Xs = np.array([[1, 50], [5, 20], [3, 80], [5, 60]]).astype(np.float64)\n",
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"ys = np.array([0, 0, 1, 1])\n",
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"svm_clf = SVC(kernel=\"linear\", C=100)\n",
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"svm_clf.fit(Xs, ys)\n",
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"\n",
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"plt.figure(figsize=(12,3.2))\n",
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"plt.subplot(121)\n",
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"plt.plot(Xs[:, 0][ys==1], Xs[:, 1][ys==1], \"bo\")\n",
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"plt.plot(Xs[:, 0][ys==0], Xs[:, 1][ys==0], \"ms\")\n",
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"plot_svc_decision_boundary(svm_clf, 0, 6)\n",
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"plt.xlabel(\"$x_0$\", fontsize=20)\n",
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"plt.ylabel(\"$x_1$ \", fontsize=20, rotation=0)\n",
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"plt.title(\"Unscaled\", fontsize=16)\n",
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"plt.axis([0, 6, 0, 90])\n",
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"\n",
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"from sklearn.preprocessing import StandardScaler\n",
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"scaler = StandardScaler()\n",
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"X_scaled = scaler.fit_transform(Xs)\n",
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"svm_clf.fit(X_scaled, ys)\n",
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"\n",
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"plt.subplot(122)\n",
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"plt.plot(X_scaled[:, 0][ys==1], X_scaled[:, 1][ys==1], \"bo\")\n",
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"plt.plot(X_scaled[:, 0][ys==0], X_scaled[:, 1][ys==0], \"ms\")\n",
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"plot_svc_decision_boundary(svm_clf, -2, 2)\n",
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"plt.xlabel(\"$x_0$\", fontsize=20)\n",
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"plt.title(\"Scaled\", fontsize=16)\n",
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"plt.axis([-2, 2, -2, 2])\n",
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"\n",
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"save_fig(\"sensitivity_to_feature_scales_plot\")\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Sensitivity to outliers"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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"X_outliers = np.array([[3.4, 1.3], [3.2, 0.8]])\n",
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"y_outliers = np.array([0, 0])\n",
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"Xo1 = np.concatenate([X, X_outliers[:1]], axis=0)\n",
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"yo1 = np.concatenate([y, y_outliers[:1]], axis=0)\n",
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"Xo2 = np.concatenate([X, X_outliers[1:]], axis=0)\n",
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"yo2 = np.concatenate([y, y_outliers[1:]], axis=0)\n",
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"\n",
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"svm_clf2 = SVC(kernel=\"linear\", C=10**9)\n",
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"svm_clf2.fit(Xo2, yo2)\n",
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"\n",
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"plt.figure(figsize=(12,2.7))\n",
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"\n",
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"plt.subplot(121)\n",
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"plt.plot(Xo1[:, 0][yo1==1], Xo1[:, 1][yo1==1], \"bs\")\n",
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"plt.plot(Xo1[:, 0][yo1==0], Xo1[:, 1][yo1==0], \"yo\")\n",
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"plt.text(0.3, 1.0, \"Impossible!\", fontsize=24, color=\"red\")\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.ylabel(\"Petal width\", fontsize=14)\n",
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"plt.annotate(\"Outlier\",\n",
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" xy=(X_outliers[0][0], X_outliers[0][1]),\n",
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" xytext=(2.5, 1.7),\n",
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" ha=\"center\",\n",
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" arrowprops=dict(facecolor='black', shrink=0.1),\n",
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" fontsize=16,\n",
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" )\n",
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"plt.axis([0, 5.5, 0, 2])\n",
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"\n",
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"plt.subplot(122)\n",
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"plt.plot(Xo2[:, 0][yo2==1], Xo2[:, 1][yo2==1], \"bs\")\n",
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"plt.plot(Xo2[:, 0][yo2==0], Xo2[:, 1][yo2==0], \"yo\")\n",
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"plot_svc_decision_boundary(svm_clf2, 0, 5.5)\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.annotate(\"Outlier\",\n",
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" xy=(X_outliers[1][0], X_outliers[1][1]),\n",
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" xytext=(3.2, 0.08),\n",
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" ha=\"center\",\n",
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" arrowprops=dict(facecolor='black', shrink=0.1),\n",
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" fontsize=16,\n",
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" )\n",
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"plt.axis([0, 5.5, 0, 2])\n",
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"\n",
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"save_fig(\"sensitivity_to_outliers_plot\")\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Large margin *vs* margin violations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This is the first code example in chapter 5:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"from sklearn import datasets\n",
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"from sklearn.pipeline import Pipeline\n",
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"from sklearn.preprocessing import StandardScaler\n",
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"from sklearn.svm import LinearSVC\n",
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"\n",
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"iris = datasets.load_iris()\n",
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"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
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"y = (iris[\"target\"] == 2).astype(np.float64) # Iris virginica\n",
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"\n",
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"svm_clf = Pipeline([\n",
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" (\"scaler\", StandardScaler()),\n",
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" (\"linear_svc\", LinearSVC(C=1, loss=\"hinge\", random_state=42)),\n",
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" ])\n",
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"\n",
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"svm_clf.fit(X, y)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"svm_clf.predict([[5.5, 1.7]])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now let's generate the graph comparing different regularization settings:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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"scaler = StandardScaler()\n",
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"svm_clf1 = LinearSVC(C=1, loss=\"hinge\", random_state=42)\n",
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"svm_clf2 = LinearSVC(C=100, loss=\"hinge\", random_state=42)\n",
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"\n",
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"scaled_svm_clf1 = Pipeline([\n",
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" (\"scaler\", scaler),\n",
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" (\"linear_svc\", svm_clf1),\n",
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" ])\n",
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"scaled_svm_clf2 = Pipeline([\n",
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" (\"scaler\", scaler),\n",
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" (\"linear_svc\", svm_clf2),\n",
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" ])\n",
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"\n",
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"scaled_svm_clf1.fit(X, y)\n",
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"scaled_svm_clf2.fit(X, y)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Convert to unscaled parameters\n",
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"b1 = svm_clf1.decision_function([-scaler.mean_ / scaler.scale_])\n",
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"b2 = svm_clf2.decision_function([-scaler.mean_ / scaler.scale_])\n",
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"w1 = svm_clf1.coef_[0] / scaler.scale_\n",
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"w2 = svm_clf2.coef_[0] / scaler.scale_\n",
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"svm_clf1.intercept_ = np.array([b1])\n",
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"svm_clf2.intercept_ = np.array([b2])\n",
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"svm_clf1.coef_ = np.array([w1])\n",
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"svm_clf2.coef_ = np.array([w2])\n",
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"\n",
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"# Find support vectors (LinearSVC does not do this automatically)\n",
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"t = y * 2 - 1\n",
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"support_vectors_idx1 = (t * (X.dot(w1) + b1) < 1).ravel()\n",
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"support_vectors_idx2 = (t * (X.dot(w2) + b2) < 1).ravel()\n",
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"svm_clf1.support_vectors_ = X[support_vectors_idx1]\n",
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"svm_clf2.support_vectors_ = X[support_vectors_idx2]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [],
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"source": [
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"plt.figure(figsize=(12,3.2))\n",
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"plt.subplot(121)\n",
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"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\", label=\"Iris virginica\")\n",
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"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\", label=\"Iris versicolor\")\n",
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"plot_svc_decision_boundary(svm_clf1, 4, 6)\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.ylabel(\"Petal width\", fontsize=14)\n",
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"plt.legend(loc=\"upper left\", fontsize=14)\n",
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"plt.title(\"$C = {}$\".format(svm_clf1.C), fontsize=16)\n",
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"plt.axis([4, 6, 0.8, 2.8])\n",
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"\n",
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"plt.subplot(122)\n",
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"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n",
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"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n",
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"plot_svc_decision_boundary(svm_clf2, 4, 6)\n",
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"plt.xlabel(\"Petal length\", fontsize=14)\n",
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"plt.title(\"$C = {}$\".format(svm_clf2.C), fontsize=16)\n",
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"plt.axis([4, 6, 0.8, 2.8])\n",
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"\n",
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"save_fig(\"regularization_plot\")"
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]
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||
},
|
||
{
|
||
"cell_type": "markdown",
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||
"metadata": {
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||
"collapsed": true
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},
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"source": [
|
||
"# Non-linear classification"
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||
]
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},
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||
{
|
||
"cell_type": "code",
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||
"execution_count": 11,
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||
"metadata": {},
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||
"outputs": [],
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||
"source": [
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"X1D = np.linspace(-4, 4, 9).reshape(-1, 1)\n",
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"X2D = np.c_[X1D, X1D**2]\n",
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"y = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n",
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"\n",
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"plt.figure(figsize=(11, 4))\n",
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"\n",
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"plt.subplot(121)\n",
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"plt.grid(True, which='both')\n",
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"plt.axhline(y=0, color='k')\n",
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"plt.plot(X1D[:, 0][y==0], np.zeros(4), \"bs\")\n",
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"plt.plot(X1D[:, 0][y==1], np.zeros(5), \"g^\")\n",
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"plt.gca().get_yaxis().set_ticks([])\n",
|
||
"plt.xlabel(r\"$x_1$\", fontsize=20)\n",
|
||
"plt.axis([-4.5, 4.5, -0.2, 0.2])\n",
|
||
"\n",
|
||
"plt.subplot(122)\n",
|
||
"plt.grid(True, which='both')\n",
|
||
"plt.axhline(y=0, color='k')\n",
|
||
"plt.axvline(x=0, color='k')\n",
|
||
"plt.plot(X2D[:, 0][y==0], X2D[:, 1][y==0], \"bs\")\n",
|
||
"plt.plot(X2D[:, 0][y==1], X2D[:, 1][y==1], \"g^\")\n",
|
||
"plt.xlabel(r\"$x_1$\", fontsize=20)\n",
|
||
"plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n",
|
||
"plt.gca().get_yaxis().set_ticks([0, 4, 8, 12, 16])\n",
|
||
"plt.plot([-4.5, 4.5], [6.5, 6.5], \"r--\", linewidth=3)\n",
|
||
"plt.axis([-4.5, 4.5, -1, 17])\n",
|
||
"\n",
|
||
"plt.subplots_adjust(right=1)\n",
|
||
"\n",
|
||
"save_fig(\"higher_dimensions_plot\", tight_layout=False)\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 12,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.datasets import make_moons\n",
|
||
"X, y = make_moons(n_samples=100, noise=0.15, random_state=42)\n",
|
||
"\n",
|
||
"def plot_dataset(X, y, axes):\n",
|
||
" plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n",
|
||
" plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")\n",
|
||
" plt.axis(axes)\n",
|
||
" plt.grid(True, which='both')\n",
|
||
" plt.xlabel(r\"$x_1$\", fontsize=20)\n",
|
||
" plt.ylabel(r\"$x_2$\", fontsize=20, rotation=0)\n",
|
||
"\n",
|
||
"plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 13,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.datasets import make_moons\n",
|
||
"from sklearn.pipeline import Pipeline\n",
|
||
"from sklearn.preprocessing import PolynomialFeatures\n",
|
||
"\n",
|
||
"polynomial_svm_clf = Pipeline([\n",
|
||
" (\"poly_features\", PolynomialFeatures(degree=3)),\n",
|
||
" (\"scaler\", StandardScaler()),\n",
|
||
" (\"svm_clf\", LinearSVC(C=10, loss=\"hinge\", random_state=42))\n",
|
||
" ])\n",
|
||
"\n",
|
||
"polynomial_svm_clf.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 14,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def plot_predictions(clf, axes):\n",
|
||
" x0s = np.linspace(axes[0], axes[1], 100)\n",
|
||
" x1s = np.linspace(axes[2], axes[3], 100)\n",
|
||
" x0, x1 = np.meshgrid(x0s, x1s)\n",
|
||
" X = np.c_[x0.ravel(), x1.ravel()]\n",
|
||
" y_pred = clf.predict(X).reshape(x0.shape)\n",
|
||
" y_decision = clf.decision_function(X).reshape(x0.shape)\n",
|
||
" plt.contourf(x0, x1, y_pred, cmap=plt.cm.brg, alpha=0.2)\n",
|
||
" plt.contourf(x0, x1, y_decision, cmap=plt.cm.brg, alpha=0.1)\n",
|
||
"\n",
|
||
"plot_predictions(polynomial_svm_clf, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n",
|
||
"\n",
|
||
"save_fig(\"moons_polynomial_svc_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 15,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVC\n",
|
||
"\n",
|
||
"poly_kernel_svm_clf = Pipeline([\n",
|
||
" (\"scaler\", StandardScaler()),\n",
|
||
" (\"svm_clf\", SVC(kernel=\"poly\", degree=3, coef0=1, C=5))\n",
|
||
" ])\n",
|
||
"poly_kernel_svm_clf.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 16,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"poly100_kernel_svm_clf = Pipeline([\n",
|
||
" (\"scaler\", StandardScaler()),\n",
|
||
" (\"svm_clf\", SVC(kernel=\"poly\", degree=10, coef0=100, C=5))\n",
|
||
" ])\n",
|
||
"poly100_kernel_svm_clf.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 17,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"plt.figure(figsize=(11, 4))\n",
|
||
"\n",
|
||
"plt.subplot(121)\n",
|
||
"plot_predictions(poly_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plt.title(r\"$d=3, r=1, C=5$\", fontsize=18)\n",
|
||
"\n",
|
||
"plt.subplot(122)\n",
|
||
"plot_predictions(poly100_kernel_svm_clf, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n",
|
||
"plt.title(r\"$d=10, r=100, C=5$\", fontsize=18)\n",
|
||
"\n",
|
||
"save_fig(\"moons_kernelized_polynomial_svc_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 18,
|
||
"metadata": {
|
||
"scrolled": true
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"def gaussian_rbf(x, landmark, gamma):\n",
|
||
" return np.exp(-gamma * np.linalg.norm(x - landmark, axis=1)**2)\n",
|
||
"\n",
|
||
"gamma = 0.3\n",
|
||
"\n",
|
||
"x1s = np.linspace(-4.5, 4.5, 200).reshape(-1, 1)\n",
|
||
"x2s = gaussian_rbf(x1s, -2, gamma)\n",
|
||
"x3s = gaussian_rbf(x1s, 1, gamma)\n",
|
||
"\n",
|
||
"XK = np.c_[gaussian_rbf(X1D, -2, gamma), gaussian_rbf(X1D, 1, gamma)]\n",
|
||
"yk = np.array([0, 0, 1, 1, 1, 1, 1, 0, 0])\n",
|
||
"\n",
|
||
"plt.figure(figsize=(11, 4))\n",
|
||
"\n",
|
||
"plt.subplot(121)\n",
|
||
"plt.grid(True, which='both')\n",
|
||
"plt.axhline(y=0, color='k')\n",
|
||
"plt.scatter(x=[-2, 1], y=[0, 0], s=150, alpha=0.5, c=\"red\")\n",
|
||
"plt.plot(X1D[:, 0][yk==0], np.zeros(4), \"bs\")\n",
|
||
"plt.plot(X1D[:, 0][yk==1], np.zeros(5), \"g^\")\n",
|
||
"plt.plot(x1s, x2s, \"g--\")\n",
|
||
"plt.plot(x1s, x3s, \"b:\")\n",
|
||
"plt.gca().get_yaxis().set_ticks([0, 0.25, 0.5, 0.75, 1])\n",
|
||
"plt.xlabel(r\"$x_1$\", fontsize=20)\n",
|
||
"plt.ylabel(r\"Similarity\", fontsize=14)\n",
|
||
"plt.annotate(r'$\\mathbf{x}$',\n",
|
||
" xy=(X1D[3, 0], 0),\n",
|
||
" xytext=(-0.5, 0.20),\n",
|
||
" ha=\"center\",\n",
|
||
" arrowprops=dict(facecolor='black', shrink=0.1),\n",
|
||
" fontsize=18,\n",
|
||
" )\n",
|
||
"plt.text(-2, 0.9, \"$x_2$\", ha=\"center\", fontsize=20)\n",
|
||
"plt.text(1, 0.9, \"$x_3$\", ha=\"center\", fontsize=20)\n",
|
||
"plt.axis([-4.5, 4.5, -0.1, 1.1])\n",
|
||
"\n",
|
||
"plt.subplot(122)\n",
|
||
"plt.grid(True, which='both')\n",
|
||
"plt.axhline(y=0, color='k')\n",
|
||
"plt.axvline(x=0, color='k')\n",
|
||
"plt.plot(XK[:, 0][yk==0], XK[:, 1][yk==0], \"bs\")\n",
|
||
"plt.plot(XK[:, 0][yk==1], XK[:, 1][yk==1], \"g^\")\n",
|
||
"plt.xlabel(r\"$x_2$\", fontsize=20)\n",
|
||
"plt.ylabel(r\"$x_3$ \", fontsize=20, rotation=0)\n",
|
||
"plt.annotate(r'$\\phi\\left(\\mathbf{x}\\right)$',\n",
|
||
" xy=(XK[3, 0], XK[3, 1]),\n",
|
||
" xytext=(0.65, 0.50),\n",
|
||
" ha=\"center\",\n",
|
||
" arrowprops=dict(facecolor='black', shrink=0.1),\n",
|
||
" fontsize=18,\n",
|
||
" )\n",
|
||
"plt.plot([-0.1, 1.1], [0.57, -0.1], \"r--\", linewidth=3)\n",
|
||
"plt.axis([-0.1, 1.1, -0.1, 1.1])\n",
|
||
" \n",
|
||
"plt.subplots_adjust(right=1)\n",
|
||
"\n",
|
||
"save_fig(\"kernel_method_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 19,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"x1_example = X1D[3, 0]\n",
|
||
"for landmark in (-2, 1):\n",
|
||
" k = gaussian_rbf(np.array([[x1_example]]), np.array([[landmark]]), gamma)\n",
|
||
" print(\"Phi({}, {}) = {}\".format(x1_example, landmark, k))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 20,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"rbf_kernel_svm_clf = Pipeline([\n",
|
||
" (\"scaler\", StandardScaler()),\n",
|
||
" (\"svm_clf\", SVC(kernel=\"rbf\", gamma=5, C=0.001))\n",
|
||
" ])\n",
|
||
"rbf_kernel_svm_clf.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 21,
|
||
"metadata": {
|
||
"scrolled": true
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVC\n",
|
||
"\n",
|
||
"gamma1, gamma2 = 0.1, 5\n",
|
||
"C1, C2 = 0.001, 1000\n",
|
||
"hyperparams = (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)\n",
|
||
"\n",
|
||
"svm_clfs = []\n",
|
||
"for gamma, C in hyperparams:\n",
|
||
" rbf_kernel_svm_clf = Pipeline([\n",
|
||
" (\"scaler\", StandardScaler()),\n",
|
||
" (\"svm_clf\", SVC(kernel=\"rbf\", gamma=gamma, C=C))\n",
|
||
" ])\n",
|
||
" rbf_kernel_svm_clf.fit(X, y)\n",
|
||
" svm_clfs.append(rbf_kernel_svm_clf)\n",
|
||
"\n",
|
||
"plt.figure(figsize=(11, 7))\n",
|
||
"\n",
|
||
"for i, svm_clf in enumerate(svm_clfs):\n",
|
||
" plt.subplot(221 + i)\n",
|
||
" plot_predictions(svm_clf, [-1.5, 2.5, -1, 1.5])\n",
|
||
" plot_dataset(X, y, [-1.5, 2.5, -1, 1.5])\n",
|
||
" gamma, C = hyperparams[i]\n",
|
||
" plt.title(r\"$\\gamma = {}, C = {}$\".format(gamma, C), fontsize=16)\n",
|
||
"\n",
|
||
"save_fig(\"moons_rbf_svc_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Regression\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 22,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"np.random.seed(42)\n",
|
||
"m = 50\n",
|
||
"X = 2 * np.random.rand(m, 1)\n",
|
||
"y = (4 + 3 * X + np.random.randn(m, 1)).ravel()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import LinearSVR\n",
|
||
"\n",
|
||
"svm_reg = LinearSVR(epsilon=1.5, random_state=42)\n",
|
||
"svm_reg.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 24,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"svm_reg1 = LinearSVR(epsilon=1.5, random_state=42)\n",
|
||
"svm_reg2 = LinearSVR(epsilon=0.5, random_state=42)\n",
|
||
"svm_reg1.fit(X, y)\n",
|
||
"svm_reg2.fit(X, y)\n",
|
||
"\n",
|
||
"def find_support_vectors(svm_reg, X, y):\n",
|
||
" y_pred = svm_reg.predict(X)\n",
|
||
" off_margin = (np.abs(y - y_pred) >= svm_reg.epsilon)\n",
|
||
" return np.argwhere(off_margin)\n",
|
||
"\n",
|
||
"svm_reg1.support_ = find_support_vectors(svm_reg1, X, y)\n",
|
||
"svm_reg2.support_ = find_support_vectors(svm_reg2, X, y)\n",
|
||
"\n",
|
||
"eps_x1 = 1\n",
|
||
"eps_y_pred = svm_reg1.predict([[eps_x1]])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 25,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def plot_svm_regression(svm_reg, X, y, axes):\n",
|
||
" x1s = np.linspace(axes[0], axes[1], 100).reshape(100, 1)\n",
|
||
" y_pred = svm_reg.predict(x1s)\n",
|
||
" plt.plot(x1s, y_pred, \"k-\", linewidth=2, label=r\"$\\hat{y}$\")\n",
|
||
" plt.plot(x1s, y_pred + svm_reg.epsilon, \"k--\")\n",
|
||
" plt.plot(x1s, y_pred - svm_reg.epsilon, \"k--\")\n",
|
||
" plt.scatter(X[svm_reg.support_], y[svm_reg.support_], s=180, facecolors='#FFAAAA')\n",
|
||
" plt.plot(X, y, \"bo\")\n",
|
||
" plt.xlabel(r\"$x_1$\", fontsize=18)\n",
|
||
" plt.legend(loc=\"upper left\", fontsize=18)\n",
|
||
" plt.axis(axes)\n",
|
||
"\n",
|
||
"plt.figure(figsize=(9, 4))\n",
|
||
"plt.subplot(121)\n",
|
||
"plot_svm_regression(svm_reg1, X, y, [0, 2, 3, 11])\n",
|
||
"plt.title(r\"$\\epsilon = {}$\".format(svm_reg1.epsilon), fontsize=18)\n",
|
||
"plt.ylabel(r\"$y$\", fontsize=18, rotation=0)\n",
|
||
"#plt.plot([eps_x1, eps_x1], [eps_y_pred, eps_y_pred - svm_reg1.epsilon], \"k-\", linewidth=2)\n",
|
||
"plt.annotate(\n",
|
||
" '', xy=(eps_x1, eps_y_pred), xycoords='data',\n",
|
||
" xytext=(eps_x1, eps_y_pred - svm_reg1.epsilon),\n",
|
||
" textcoords='data', arrowprops={'arrowstyle': '<->', 'linewidth': 1.5}\n",
|
||
" )\n",
|
||
"plt.text(0.91, 5.6, r\"$\\epsilon$\", fontsize=20)\n",
|
||
"plt.subplot(122)\n",
|
||
"plot_svm_regression(svm_reg2, X, y, [0, 2, 3, 11])\n",
|
||
"plt.title(r\"$\\epsilon = {}$\".format(svm_reg2.epsilon), fontsize=18)\n",
|
||
"save_fig(\"svm_regression_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 26,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"np.random.seed(42)\n",
|
||
"m = 100\n",
|
||
"X = 2 * np.random.rand(m, 1) - 1\n",
|
||
"y = (0.2 + 0.1 * X + 0.5 * X**2 + np.random.randn(m, 1)/10).ravel()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Note**: to be future-proof, we set `gamma=\"scale\"`, as this will be the default value in Scikit-Learn 0.22."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 27,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVR\n",
|
||
"\n",
|
||
"svm_poly_reg = SVR(kernel=\"poly\", degree=2, C=100, epsilon=0.1, gamma=\"scale\")\n",
|
||
"svm_poly_reg.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 28,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVR\n",
|
||
"\n",
|
||
"svm_poly_reg1 = SVR(kernel=\"poly\", degree=2, C=100, epsilon=0.1, gamma=\"scale\")\n",
|
||
"svm_poly_reg2 = SVR(kernel=\"poly\", degree=2, C=0.01, epsilon=0.1, gamma=\"scale\")\n",
|
||
"svm_poly_reg1.fit(X, y)\n",
|
||
"svm_poly_reg2.fit(X, y)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 29,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"plt.figure(figsize=(9, 4))\n",
|
||
"plt.subplot(121)\n",
|
||
"plot_svm_regression(svm_poly_reg1, X, y, [-1, 1, 0, 1])\n",
|
||
"plt.title(r\"$degree={}, C={}, \\epsilon = {}$\".format(svm_poly_reg1.degree, svm_poly_reg1.C, svm_poly_reg1.epsilon), fontsize=18)\n",
|
||
"plt.ylabel(r\"$y$\", fontsize=18, rotation=0)\n",
|
||
"plt.subplot(122)\n",
|
||
"plot_svm_regression(svm_poly_reg2, X, y, [-1, 1, 0, 1])\n",
|
||
"plt.title(r\"$degree={}, C={}, \\epsilon = {}$\".format(svm_poly_reg2.degree, svm_poly_reg2.C, svm_poly_reg2.epsilon), fontsize=18)\n",
|
||
"save_fig(\"svm_with_polynomial_kernel_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Under the hood"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 30,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"iris = datasets.load_iris()\n",
|
||
"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
|
||
"y = (iris[\"target\"] == 2).astype(np.float64) # Iris virginica"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 31,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from mpl_toolkits.mplot3d import Axes3D\n",
|
||
"\n",
|
||
"def plot_3D_decision_function(ax, w, b, x1_lim=[4, 6], x2_lim=[0.8, 2.8]):\n",
|
||
" x1_in_bounds = (X[:, 0] > x1_lim[0]) & (X[:, 0] < x1_lim[1])\n",
|
||
" X_crop = X[x1_in_bounds]\n",
|
||
" y_crop = y[x1_in_bounds]\n",
|
||
" x1s = np.linspace(x1_lim[0], x1_lim[1], 20)\n",
|
||
" x2s = np.linspace(x2_lim[0], x2_lim[1], 20)\n",
|
||
" x1, x2 = np.meshgrid(x1s, x2s)\n",
|
||
" xs = np.c_[x1.ravel(), x2.ravel()]\n",
|
||
" df = (xs.dot(w) + b).reshape(x1.shape)\n",
|
||
" m = 1 / np.linalg.norm(w)\n",
|
||
" boundary_x2s = -x1s*(w[0]/w[1])-b/w[1]\n",
|
||
" margin_x2s_1 = -x1s*(w[0]/w[1])-(b-1)/w[1]\n",
|
||
" margin_x2s_2 = -x1s*(w[0]/w[1])-(b+1)/w[1]\n",
|
||
" ax.plot_surface(x1s, x2, np.zeros_like(x1),\n",
|
||
" color=\"b\", alpha=0.2, cstride=100, rstride=100)\n",
|
||
" ax.plot(x1s, boundary_x2s, 0, \"k-\", linewidth=2, label=r\"$h=0$\")\n",
|
||
" ax.plot(x1s, margin_x2s_1, 0, \"k--\", linewidth=2, label=r\"$h=\\pm 1$\")\n",
|
||
" ax.plot(x1s, margin_x2s_2, 0, \"k--\", linewidth=2)\n",
|
||
" ax.plot(X_crop[:, 0][y_crop==1], X_crop[:, 1][y_crop==1], 0, \"g^\")\n",
|
||
" ax.plot_wireframe(x1, x2, df, alpha=0.3, color=\"k\")\n",
|
||
" ax.plot(X_crop[:, 0][y_crop==0], X_crop[:, 1][y_crop==0], 0, \"bs\")\n",
|
||
" ax.axis(x1_lim + x2_lim)\n",
|
||
" ax.text(4.5, 2.5, 3.8, \"Decision function $h$\", fontsize=16)\n",
|
||
" ax.set_xlabel(r\"Petal length\", fontsize=16, labelpad=10)\n",
|
||
" ax.set_ylabel(r\"Petal width\", fontsize=16, labelpad=10)\n",
|
||
" ax.set_zlabel(r\"$h = \\mathbf{w}^T \\mathbf{x} + b$\", fontsize=18, labelpad=5)\n",
|
||
" ax.legend(loc=\"upper left\", fontsize=16)\n",
|
||
"\n",
|
||
"fig = plt.figure(figsize=(11, 6))\n",
|
||
"ax1 = fig.add_subplot(111, projection='3d')\n",
|
||
"plot_3D_decision_function(ax1, w=svm_clf2.coef_[0], b=svm_clf2.intercept_[0])\n",
|
||
"\n",
|
||
"save_fig(\"iris_3D_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Small weight vector results in a large margin"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 32,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"def plot_2D_decision_function(w, b, ylabel=True, x1_lim=[-3, 3]):\n",
|
||
" x1 = np.linspace(x1_lim[0], x1_lim[1], 200)\n",
|
||
" y = w * x1 + b\n",
|
||
" m = 1 / w\n",
|
||
"\n",
|
||
" plt.plot(x1, y)\n",
|
||
" plt.plot(x1_lim, [1, 1], \"k:\")\n",
|
||
" plt.plot(x1_lim, [-1, -1], \"k:\")\n",
|
||
" plt.axhline(y=0, color='k')\n",
|
||
" plt.axvline(x=0, color='k')\n",
|
||
" plt.plot([m, m], [0, 1], \"k--\")\n",
|
||
" plt.plot([-m, -m], [0, -1], \"k--\")\n",
|
||
" plt.plot([-m, m], [0, 0], \"k-o\", linewidth=3)\n",
|
||
" plt.axis(x1_lim + [-2, 2])\n",
|
||
" plt.xlabel(r\"$x_1$\", fontsize=16)\n",
|
||
" if ylabel:\n",
|
||
" plt.ylabel(r\"$w_1 x_1$ \", rotation=0, fontsize=16)\n",
|
||
" plt.title(r\"$w_1 = {}$\".format(w), fontsize=16)\n",
|
||
"\n",
|
||
"plt.figure(figsize=(12, 3.2))\n",
|
||
"plt.subplot(121)\n",
|
||
"plot_2D_decision_function(1, 0)\n",
|
||
"plt.subplot(122)\n",
|
||
"plot_2D_decision_function(0.5, 0, ylabel=False)\n",
|
||
"save_fig(\"small_w_large_margin_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 33,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVC\n",
|
||
"from sklearn import datasets\n",
|
||
"\n",
|
||
"iris = datasets.load_iris()\n",
|
||
"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
|
||
"y = (iris[\"target\"] == 2).astype(np.float64) # Iris virginica\n",
|
||
"\n",
|
||
"svm_clf = SVC(kernel=\"linear\", C=1)\n",
|
||
"svm_clf.fit(X, y)\n",
|
||
"svm_clf.predict([[5.3, 1.3]])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Hinge loss"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 34,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"t = np.linspace(-2, 4, 200)\n",
|
||
"h = np.where(1 - t < 0, 0, 1 - t) # max(0, 1-t)\n",
|
||
"\n",
|
||
"plt.figure(figsize=(5,2.8))\n",
|
||
"plt.plot(t, h, \"b-\", linewidth=2, label=\"$max(0, 1 - t)$\")\n",
|
||
"plt.grid(True, which='both')\n",
|
||
"plt.axhline(y=0, color='k')\n",
|
||
"plt.axvline(x=0, color='k')\n",
|
||
"plt.yticks(np.arange(-1, 2.5, 1))\n",
|
||
"plt.xlabel(\"$t$\", fontsize=16)\n",
|
||
"plt.axis([-2, 4, -1, 2.5])\n",
|
||
"plt.legend(loc=\"upper right\", fontsize=16)\n",
|
||
"save_fig(\"hinge_plot\")\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Extra material"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## Training time"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 35,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"X, y = make_moons(n_samples=1000, noise=0.4, random_state=42)\n",
|
||
"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"bs\")\n",
|
||
"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"g^\")"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 36,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"import time\n",
|
||
"\n",
|
||
"tol = 0.1\n",
|
||
"tols = []\n",
|
||
"times = []\n",
|
||
"for i in range(10):\n",
|
||
" svm_clf = SVC(kernel=\"poly\", gamma=3, C=10, tol=tol, verbose=1)\n",
|
||
" t1 = time.time()\n",
|
||
" svm_clf.fit(X, y)\n",
|
||
" t2 = time.time()\n",
|
||
" times.append(t2-t1)\n",
|
||
" tols.append(tol)\n",
|
||
" print(i, tol, t2-t1)\n",
|
||
" tol /= 10\n",
|
||
"plt.semilogx(tols, times, \"bo-\")\n",
|
||
"plt.xlabel(\"Tolerance\", fontsize=16)\n",
|
||
"plt.ylabel(\"Time (seconds)\", fontsize=16)\n",
|
||
"plt.grid(True)\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## Linear SVM classifier implementation using Batch Gradient Descent"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 37,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Training set\n",
|
||
"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
|
||
"y = (iris[\"target\"] == 2).astype(np.float64).reshape(-1, 1) # Iris virginica"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 38,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.base import BaseEstimator\n",
|
||
"\n",
|
||
"class MyLinearSVC(BaseEstimator):\n",
|
||
" def __init__(self, C=1, eta0=1, eta_d=10000, n_epochs=1000, random_state=None):\n",
|
||
" self.C = C\n",
|
||
" self.eta0 = eta0\n",
|
||
" self.n_epochs = n_epochs\n",
|
||
" self.random_state = random_state\n",
|
||
" self.eta_d = eta_d\n",
|
||
"\n",
|
||
" def eta(self, epoch):\n",
|
||
" return self.eta0 / (epoch + self.eta_d)\n",
|
||
" \n",
|
||
" def fit(self, X, y):\n",
|
||
" # Random initialization\n",
|
||
" if self.random_state:\n",
|
||
" np.random.seed(self.random_state)\n",
|
||
" w = np.random.randn(X.shape[1], 1) # n feature weights\n",
|
||
" b = 0\n",
|
||
"\n",
|
||
" m = len(X)\n",
|
||
" t = y * 2 - 1 # -1 if t==0, +1 if t==1\n",
|
||
" X_t = X * t\n",
|
||
" self.Js=[]\n",
|
||
"\n",
|
||
" # Training\n",
|
||
" for epoch in range(self.n_epochs):\n",
|
||
" support_vectors_idx = (X_t.dot(w) + t * b < 1).ravel()\n",
|
||
" X_t_sv = X_t[support_vectors_idx]\n",
|
||
" t_sv = t[support_vectors_idx]\n",
|
||
"\n",
|
||
" J = 1/2 * np.sum(w * w) + self.C * (np.sum(1 - X_t_sv.dot(w)) - b * np.sum(t_sv))\n",
|
||
" self.Js.append(J)\n",
|
||
"\n",
|
||
" w_gradient_vector = w - self.C * np.sum(X_t_sv, axis=0).reshape(-1, 1)\n",
|
||
" b_derivative = -C * np.sum(t_sv)\n",
|
||
" \n",
|
||
" w = w - self.eta(epoch) * w_gradient_vector\n",
|
||
" b = b - self.eta(epoch) * b_derivative\n",
|
||
" \n",
|
||
"\n",
|
||
" self.intercept_ = np.array([b])\n",
|
||
" self.coef_ = np.array([w])\n",
|
||
" support_vectors_idx = (X_t.dot(w) + t * b < 1).ravel()\n",
|
||
" self.support_vectors_ = X[support_vectors_idx]\n",
|
||
" return self\n",
|
||
"\n",
|
||
" def decision_function(self, X):\n",
|
||
" return X.dot(self.coef_[0]) + self.intercept_[0]\n",
|
||
"\n",
|
||
" def predict(self, X):\n",
|
||
" return (self.decision_function(X) >= 0).astype(np.float64)\n",
|
||
"\n",
|
||
"C=2\n",
|
||
"svm_clf = MyLinearSVC(C=C, eta0 = 10, eta_d = 1000, n_epochs=60000, random_state=2)\n",
|
||
"svm_clf.fit(X, y)\n",
|
||
"svm_clf.predict(np.array([[5, 2], [4, 1]]))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 39,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"plt.plot(range(svm_clf.n_epochs), svm_clf.Js)\n",
|
||
"plt.axis([0, svm_clf.n_epochs, 0, 100])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 40,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"print(svm_clf.intercept_, svm_clf.coef_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 41,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"svm_clf2 = SVC(kernel=\"linear\", C=C)\n",
|
||
"svm_clf2.fit(X, y.ravel())\n",
|
||
"print(svm_clf2.intercept_, svm_clf2.coef_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 42,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"yr = y.ravel()\n",
|
||
"plt.figure(figsize=(12,3.2))\n",
|
||
"plt.subplot(121)\n",
|
||
"plt.plot(X[:, 0][yr==1], X[:, 1][yr==1], \"g^\", label=\"Iris virginica\")\n",
|
||
"plt.plot(X[:, 0][yr==0], X[:, 1][yr==0], \"bs\", label=\"Not Iris virginica\")\n",
|
||
"plot_svc_decision_boundary(svm_clf, 4, 6)\n",
|
||
"plt.xlabel(\"Petal length\", fontsize=14)\n",
|
||
"plt.ylabel(\"Petal width\", fontsize=14)\n",
|
||
"plt.title(\"MyLinearSVC\", fontsize=14)\n",
|
||
"plt.axis([4, 6, 0.8, 2.8])\n",
|
||
"plt.legend(loc=\"upper left\")\n",
|
||
"\n",
|
||
"plt.subplot(122)\n",
|
||
"plt.plot(X[:, 0][yr==1], X[:, 1][yr==1], \"g^\")\n",
|
||
"plt.plot(X[:, 0][yr==0], X[:, 1][yr==0], \"bs\")\n",
|
||
"plot_svc_decision_boundary(svm_clf2, 4, 6)\n",
|
||
"plt.xlabel(\"Petal length\", fontsize=14)\n",
|
||
"plt.title(\"SVC\", fontsize=14)\n",
|
||
"plt.axis([4, 6, 0.8, 2.8])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 43,
|
||
"metadata": {
|
||
"scrolled": true
|
||
},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.linear_model import SGDClassifier\n",
|
||
"\n",
|
||
"sgd_clf = SGDClassifier(loss=\"hinge\", alpha=0.017, max_iter=1000, tol=1e-3, random_state=42)\n",
|
||
"sgd_clf.fit(X, y.ravel())\n",
|
||
"\n",
|
||
"m = len(X)\n",
|
||
"t = y * 2 - 1 # -1 if t==0, +1 if t==1\n",
|
||
"X_b = np.c_[np.ones((m, 1)), X] # Add bias input x0=1\n",
|
||
"X_b_t = X_b * t\n",
|
||
"sgd_theta = np.r_[sgd_clf.intercept_[0], sgd_clf.coef_[0]]\n",
|
||
"print(sgd_theta)\n",
|
||
"support_vectors_idx = (X_b_t.dot(sgd_theta) < 1).ravel()\n",
|
||
"sgd_clf.support_vectors_ = X[support_vectors_idx]\n",
|
||
"sgd_clf.C = C\n",
|
||
"\n",
|
||
"plt.figure(figsize=(5.5,3.2))\n",
|
||
"plt.plot(X[:, 0][yr==1], X[:, 1][yr==1], \"g^\")\n",
|
||
"plt.plot(X[:, 0][yr==0], X[:, 1][yr==0], \"bs\")\n",
|
||
"plot_svc_decision_boundary(sgd_clf, 4, 6)\n",
|
||
"plt.xlabel(\"Petal length\", fontsize=14)\n",
|
||
"plt.ylabel(\"Petal width\", fontsize=14)\n",
|
||
"plt.title(\"SGDClassifier\", fontsize=14)\n",
|
||
"plt.axis([4, 6, 0.8, 2.8])\n"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# Exercise solutions"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## 1. to 7."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {
|
||
"collapsed": true
|
||
},
|
||
"source": [
|
||
"See appendix A."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# 8."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"_Exercise: train a `LinearSVC` on a linearly separable dataset. Then train an `SVC` and a `SGDClassifier` on the same dataset. See if you can get them to produce roughly the same model._"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's use the Iris dataset: the Iris Setosa and Iris Versicolor classes are linearly separable."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 44,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn import datasets\n",
|
||
"\n",
|
||
"iris = datasets.load_iris()\n",
|
||
"X = iris[\"data\"][:, (2, 3)] # petal length, petal width\n",
|
||
"y = iris[\"target\"]\n",
|
||
"\n",
|
||
"setosa_or_versicolor = (y == 0) | (y == 1)\n",
|
||
"X = X[setosa_or_versicolor]\n",
|
||
"y = y[setosa_or_versicolor]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 45,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVC, LinearSVC\n",
|
||
"from sklearn.linear_model import SGDClassifier\n",
|
||
"from sklearn.preprocessing import StandardScaler\n",
|
||
"\n",
|
||
"C = 5\n",
|
||
"alpha = 1 / (C * len(X))\n",
|
||
"\n",
|
||
"lin_clf = LinearSVC(loss=\"hinge\", C=C, random_state=42)\n",
|
||
"svm_clf = SVC(kernel=\"linear\", C=C)\n",
|
||
"sgd_clf = SGDClassifier(loss=\"hinge\", learning_rate=\"constant\", eta0=0.001, alpha=alpha,\n",
|
||
" max_iter=1000, tol=1e-3, random_state=42)\n",
|
||
"\n",
|
||
"scaler = StandardScaler()\n",
|
||
"X_scaled = scaler.fit_transform(X)\n",
|
||
"\n",
|
||
"lin_clf.fit(X_scaled, y)\n",
|
||
"svm_clf.fit(X_scaled, y)\n",
|
||
"sgd_clf.fit(X_scaled, y)\n",
|
||
"\n",
|
||
"print(\"LinearSVC: \", lin_clf.intercept_, lin_clf.coef_)\n",
|
||
"print(\"SVC: \", svm_clf.intercept_, svm_clf.coef_)\n",
|
||
"print(\"SGDClassifier(alpha={:.5f}):\".format(sgd_clf.alpha), sgd_clf.intercept_, sgd_clf.coef_)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's plot the decision boundaries of these three models:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 46,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"# Compute the slope and bias of each decision boundary\n",
|
||
"w1 = -lin_clf.coef_[0, 0]/lin_clf.coef_[0, 1]\n",
|
||
"b1 = -lin_clf.intercept_[0]/lin_clf.coef_[0, 1]\n",
|
||
"w2 = -svm_clf.coef_[0, 0]/svm_clf.coef_[0, 1]\n",
|
||
"b2 = -svm_clf.intercept_[0]/svm_clf.coef_[0, 1]\n",
|
||
"w3 = -sgd_clf.coef_[0, 0]/sgd_clf.coef_[0, 1]\n",
|
||
"b3 = -sgd_clf.intercept_[0]/sgd_clf.coef_[0, 1]\n",
|
||
"\n",
|
||
"# Transform the decision boundary lines back to the original scale\n",
|
||
"line1 = scaler.inverse_transform([[-10, -10 * w1 + b1], [10, 10 * w1 + b1]])\n",
|
||
"line2 = scaler.inverse_transform([[-10, -10 * w2 + b2], [10, 10 * w2 + b2]])\n",
|
||
"line3 = scaler.inverse_transform([[-10, -10 * w3 + b3], [10, 10 * w3 + b3]])\n",
|
||
"\n",
|
||
"# Plot all three decision boundaries\n",
|
||
"plt.figure(figsize=(11, 4))\n",
|
||
"plt.plot(line1[:, 0], line1[:, 1], \"k:\", label=\"LinearSVC\")\n",
|
||
"plt.plot(line2[:, 0], line2[:, 1], \"b--\", linewidth=2, label=\"SVC\")\n",
|
||
"plt.plot(line3[:, 0], line3[:, 1], \"r-\", label=\"SGDClassifier\")\n",
|
||
"plt.plot(X[:, 0][y==1], X[:, 1][y==1], \"bs\") # label=\"Iris versicolor\"\n",
|
||
"plt.plot(X[:, 0][y==0], X[:, 1][y==0], \"yo\") # label=\"Iris setosa\"\n",
|
||
"plt.xlabel(\"Petal length\", fontsize=14)\n",
|
||
"plt.ylabel(\"Petal width\", fontsize=14)\n",
|
||
"plt.legend(loc=\"upper center\", fontsize=14)\n",
|
||
"plt.axis([0, 5.5, 0, 2])\n",
|
||
"\n",
|
||
"plt.show()"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Close enough!"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"# 9."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"_Exercise: train an SVM classifier on the MNIST dataset. Since SVM classifiers are binary classifiers, you will need to use one-versus-all to classify all 10 digits. You may want to tune the hyperparameters using small validation sets to speed up the process. What accuracy can you reach?_"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"First, let's load the dataset and split it into a training set and a test set. We could use `train_test_split()` but people usually just take the first 60,000 instances for the training set, and the last 10,000 instances for the test set (this makes it possible to compare your model's performance with others): "
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 47,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.datasets import fetch_openml\n",
|
||
"mnist = fetch_openml('mnist_784', version=1, cache=True)\n",
|
||
"\n",
|
||
"X = mnist[\"data\"]\n",
|
||
"y = mnist[\"target\"].astype(np.uint8)\n",
|
||
"\n",
|
||
"X_train = X[:60000]\n",
|
||
"y_train = y[:60000]\n",
|
||
"X_test = X[60000:]\n",
|
||
"y_test = y[60000:]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Many training algorithms are sensitive to the order of the training instances, so it's generally good practice to shuffle them first. However, the dataset is already shuffled, so we do not need to do it."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's start simple, with a linear SVM classifier. It will automatically use the One-vs-All (also called One-vs-the-Rest, OvR) strategy, so there's nothing special we need to do. Easy!\n",
|
||
"\n",
|
||
"**Warning**: this may take a few minutes depending on your hardware."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 48,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"lin_clf = LinearSVC(random_state=42)\n",
|
||
"lin_clf.fit(X_train, y_train)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's make predictions on the training set and measure the accuracy (we don't want to measure it on the test set yet, since we have not selected and trained the final model yet):"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 49,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.metrics import accuracy_score\n",
|
||
"\n",
|
||
"y_pred = lin_clf.predict(X_train)\n",
|
||
"accuracy_score(y_train, y_pred)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Okay, 89.5% accuracy on MNIST is pretty bad. This linear model is certainly too simple for MNIST, but perhaps we just needed to scale the data first:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 50,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"scaler = StandardScaler()\n",
|
||
"X_train_scaled = scaler.fit_transform(X_train.astype(np.float32))\n",
|
||
"X_test_scaled = scaler.transform(X_test.astype(np.float32))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Warning**: this may take a few minutes depending on your hardware."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 51,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"lin_clf = LinearSVC(random_state=42)\n",
|
||
"lin_clf.fit(X_train_scaled, y_train)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 52,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = lin_clf.predict(X_train_scaled)\n",
|
||
"accuracy_score(y_train, y_pred)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"That's much better (we cut the error rate by about 25%), but still not great at all for MNIST. If we want to use an SVM, we will have to use a kernel. Let's try an `SVC` with an RBF kernel (the default)."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"**Note**: to be future-proof we set `gamma=\"scale\"` since it will be the default value in Scikit-Learn 0.22."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 53,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"svm_clf = SVC(gamma=\"scale\")\n",
|
||
"svm_clf.fit(X_train_scaled[:10000], y_train[:10000])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 54,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = svm_clf.predict(X_train_scaled)\n",
|
||
"accuracy_score(y_train, y_pred)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"That's promising, we get better performance even though we trained the model on 6 times less data. Let's tune the hyperparameters by doing a randomized search with cross validation. We will do this on a small dataset just to speed up the process:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 55,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.model_selection import RandomizedSearchCV\n",
|
||
"from scipy.stats import reciprocal, uniform\n",
|
||
"\n",
|
||
"param_distributions = {\"gamma\": reciprocal(0.001, 0.1), \"C\": uniform(1, 10)}\n",
|
||
"rnd_search_cv = RandomizedSearchCV(svm_clf, param_distributions, n_iter=10, verbose=2, cv=3)\n",
|
||
"rnd_search_cv.fit(X_train_scaled[:1000], y_train[:1000])"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 56,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"rnd_search_cv.best_estimator_"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 57,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"rnd_search_cv.best_score_"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"This looks pretty low but remember we only trained the model on 1,000 instances. Let's retrain the best estimator on the whole training set (run this at night, it will take hours):"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 58,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"rnd_search_cv.best_estimator_.fit(X_train_scaled, y_train)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 59,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = rnd_search_cv.best_estimator_.predict(X_train_scaled)\n",
|
||
"accuracy_score(y_train, y_pred)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Ah, this looks good! Let's select this model. Now we can test it on the test set:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 60,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = rnd_search_cv.best_estimator_.predict(X_test_scaled)\n",
|
||
"accuracy_score(y_test, y_pred)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Not too bad, but apparently the model is overfitting slightly. It's tempting to tweak the hyperparameters a bit more (e.g. decreasing `C` and/or `gamma`), but we would run the risk of overfitting the test set. Other people have found that the hyperparameters `C=5` and `gamma=0.005` yield even better performance (over 98% accuracy). By running the randomized search for longer and on a larger part of the training set, you may be able to find this as well."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"## 10."
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"_Exercise: train an SVM regressor on the California housing dataset._"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's load the dataset using Scikit-Learn's `fetch_california_housing()` function:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 61,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.datasets import fetch_california_housing\n",
|
||
"\n",
|
||
"housing = fetch_california_housing()\n",
|
||
"X = housing[\"data\"]\n",
|
||
"y = housing[\"target\"]"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Split it into a training set and a test set:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 62,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.model_selection import train_test_split\n",
|
||
"\n",
|
||
"X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Don't forget to scale the data:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 63,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.preprocessing import StandardScaler\n",
|
||
"\n",
|
||
"scaler = StandardScaler()\n",
|
||
"X_train_scaled = scaler.fit_transform(X_train)\n",
|
||
"X_test_scaled = scaler.transform(X_test)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's train a simple `LinearSVR` first:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 64,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import LinearSVR\n",
|
||
"\n",
|
||
"lin_svr = LinearSVR(random_state=42)\n",
|
||
"lin_svr.fit(X_train_scaled, y_train)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's see how it performs on the training set:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 65,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.metrics import mean_squared_error\n",
|
||
"\n",
|
||
"y_pred = lin_svr.predict(X_train_scaled)\n",
|
||
"mse = mean_squared_error(y_train, y_pred)\n",
|
||
"mse"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Let's look at the RMSE:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 66,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"np.sqrt(mse)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"In this training set, the targets are tens of thousands of dollars. The RMSE gives a rough idea of the kind of error you should expect (with a higher weight for large errors): so with this model we can expect errors somewhere around $10,000. Not great. Let's see if we can do better with an RBF Kernel. We will use randomized search with cross validation to find the appropriate hyperparameter values for `C` and `gamma`:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 67,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"from sklearn.svm import SVR\n",
|
||
"from sklearn.model_selection import RandomizedSearchCV\n",
|
||
"from scipy.stats import reciprocal, uniform\n",
|
||
"\n",
|
||
"param_distributions = {\"gamma\": reciprocal(0.001, 0.1), \"C\": uniform(1, 10)}\n",
|
||
"rnd_search_cv = RandomizedSearchCV(SVR(), param_distributions, n_iter=10, verbose=2, cv=3, random_state=42)\n",
|
||
"rnd_search_cv.fit(X_train_scaled, y_train)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 68,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"rnd_search_cv.best_estimator_"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Now let's measure the RMSE on the training set:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 69,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = rnd_search_cv.best_estimator_.predict(X_train_scaled)\n",
|
||
"mse = mean_squared_error(y_train, y_pred)\n",
|
||
"np.sqrt(mse)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "markdown",
|
||
"metadata": {},
|
||
"source": [
|
||
"Looks much better than the linear model. Let's select this model and evaluate it on the test set:"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 70,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"y_pred = rnd_search_cv.best_estimator_.predict(X_test_scaled)\n",
|
||
"mse = mean_squared_error(y_test, y_pred)\n",
|
||
"np.sqrt(mse)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": null,
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": []
|
||
}
|
||
],
|
||
"metadata": {
|
||
"kernelspec": {
|
||
"display_name": "Python 3",
|
||
"language": "python",
|
||
"name": "python3"
|
||
},
|
||
"language_info": {
|
||
"codemirror_mode": {
|
||
"name": "ipython",
|
||
"version": 3
|
||
},
|
||
"file_extension": ".py",
|
||
"mimetype": "text/x-python",
|
||
"name": "python",
|
||
"nbconvert_exporter": "python",
|
||
"pygments_lexer": "ipython3",
|
||
"version": "3.6.8"
|
||
},
|
||
"nav_menu": {},
|
||
"toc": {
|
||
"navigate_menu": true,
|
||
"number_sections": true,
|
||
"sideBar": true,
|
||
"threshold": 6,
|
||
"toc_cell": false,
|
||
"toc_section_display": "block",
|
||
"toc_window_display": false
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 1
|
||
}
|