6 Non-linear methods.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Non-linear methods"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.datasets.samples_generator import make_circles\n",
    "from sklearn.svm import SVC"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "features, labels = make_circles(100, factor=.1, noise=.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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   "source": [
    "def plot_prediction(features, y_true, y_predicted):\n",
    "    labels = np.unique(y_true)\n",
    "    for i, (label, marker) in enumerate(zip(labels, \"x+*.\")):\n",
    "        class_data = features[y_true == label]\n",
    "        # Color is given by the predicted label\n",
    "        colors = [f\"C{predicted - labels.min()}\" for predicted in y_predicted[y_true == label]]\n",
    "        plt.scatter(class_data[:, 0], class_data[:, 1], marker=marker, label=f\"Class {label}\", c=colors)\n",
    "    plt.legend()\n",
    "    plt.xlabel(\"$x_1$\")\n",
    "    plt.ylabel(\"$x_2$\")\n",
    "    plt.axis(\"equal\")\n",
    "\n",
    "plt.title(\"Labeled data\")\n",
    "plot_prediction(features, labels, labels)\n",
    "x_range = plt.xlim()\n",
    "y_range = plt.ylim()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fit the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_decision_boundary(x_range, y_range, fit, samples=200, color=\"C1\", margin=False):\n",
    "    x, y = np.meshgrid(np.linspace(*x_range, samples), np.linspace(*y_range, samples))\n",
    "    positions = np.stack((x.reshape(-1), y.reshape(-1)), axis=1).reshape(-1, 2)\n",
    "    if margin:\n",
    "        prediction = fit.decision_function(positions)\n",
    "        levels = [-1, 0, 1]\n",
    "        linestyles = ['--', '-', '--']\n",
    "    else:\n",
    "        prediction = fit.predict(positions)\n",
    "        levels = [0]\n",
    "        linestyles = [\"-\"]\n",
    "    cntr = plt.contour(x, y, prediction.reshape(samples, samples),\n",
    "                       colors=color, levels=[-1, 0, 1],\n",
    "                       linestyles=linestyles)\n",
    "    return cntr.legend_elements()[0][1]\n",
    "\n",
    "def plot_prediction_grid(x_range, y_range, fit, samples=100):\n",
    "    x, y = np.meshgrid(np.linspace(*x_range, samples), np.linspace(*y_range, samples))\n",
    "    positions = np.stack((x.reshape(-1), y.reshape(-1)), axis=1).reshape(-1, 2)\n",
    "    prediction = fit.predict_proba(positions)[:, 1]\n",
    "    plt.imshow(prediction.reshape(samples, samples), origin=\"lower\", extent=(x.min(), x.max(), y.min(), y.max()),\n",
    "               cmap=\"coolwarm\", vmin=0, vmax=1, alpha=0.3)\n",
    "    plt.colorbar(label=\"Predicted Probability class -1\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear SVM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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   "source": [
    "svm = SVC(gamma=\"scale\", kernel=\"linear\", C=10)\n",
    "svm.fit(features, labels)\n",
    "\n",
    "plt.title(\"SVM Classifier\")\n",
    "predicted_labels = svm.predict(features)\n",
    "plot_decision_boundary(x_range, y_range, svm, margin=True)\n",
    "plot_prediction(features, labels, predicted_labels)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Use new features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 6,
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   "source": [
    "radius = np.sum(features ** 2, axis=1)\n",
    "phi = np.arctan(features[:, 1] / features[:, 0])\n",
    "\n",
    "mod_features = np.stack([radius, phi], 1)\n",
    "\n",
    "svm = SVC(gamma=\"scale\", kernel=\"linear\", C=10)\n",
    "svm.fit(mod_features, labels)\n",
    "\n",
    "plt.title(\"SVM Classifier\")\n",
    "predicted_labels = svm.predict(mod_features)\n",
    "plot_prediction(mod_features, labels, predicted_labels)\n",
    "plot_decision_boundary(plt.xlim(), plt.ylim(), svm, margin=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Non-linear SVM\n",
    "\n",
    "Use a radial basis function kernel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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   "source": [
    "svm = SVC(gamma=\"scale\", kernel=\"rbf\", C=20)\n",
    "svm.fit(features, labels)\n",
    "\n",
    "plt.title(\"Nonlinear SVM\")\n",
    "predicted_labels = svm.predict(features)\n",
    "plot_decision_boundary(x_range, y_range, svm, margin=True)\n",
    "plot_prediction(features, labels, predicted_labels)\n",
    "plt.scatter(svm.support_vectors_[:, 0], svm.support_vectors_[:, 1],\n",
    "            s=100, linewidth=1, facecolors='none', edgecolors=\"k\", \n",
    "            alpha=0.8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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