Improve 3D dataset and add missing random_state=42 for PCA in exercises
Aurélien Geron
parent
6a4660de6f
commit
3e7fe8e2ff
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@ -125,6 +125,13 @@
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"# PCA"
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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 chapter starts with several figures to explain the concepts of PCA and Manifold Learning. Below is the code to generate these figures. You can skip directly to the [Principal Components](#Principal-Components) section below if you want."
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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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@ -144,20 +151,13 @@
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"\n",
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"np.random.seed(42)\n",
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"m = 60\n",
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"w1, w2 = 0.2, 0.5\n",
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"noise = 0.2\n",
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"angles = np.random.rand(m) * 2 * np.pi * 0.8 + np.pi / 2\n",
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"X = np.empty((m, 3))\n",
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"X[:, 0] = np.cos(angles) + np.sin(angles) / 2 + noise * np.random.randn(m) / 2\n",
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"X[:, 1] = np.sin(angles) * 0.7 + noise * np.random.randn(m) / 2\n",
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"X[:, 2] = X[:, 0] * w1 + X[:, 1] * w2 + noise * np.random.randn(m)"
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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 chapter starts with several figures to explain the concepts of PCA and Manifold Learning. Below is the code to generate these figures. You can skip directly to the [Principal Components](#Principal-Components) section below."
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"angles = (np.random.rand(m) ** 3 + 0.5) * 2 * np.pi\n",
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"X = np.zeros((m, 3))\n",
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"X[:, 0] = np.cos(angles)\n",
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"X[:, 1] = np.sin(angles) * 0.5\n",
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"X += 0.28 * np.random.randn(m, 3)\n",
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"X = rotate_3d(X, -np.pi / 4, np.pi / 30, -np.pi / 20)\n",
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"X += [0.2, 0, 0.2]"
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]
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{
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@ -177,7 +177,9 @@
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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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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"# not in the book\n",
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@ -289,7 +291,7 @@
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"source": [
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"from sklearn.datasets import make_swiss_roll\n",
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"\n",
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"X, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)"
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"X_swiss, t = make_swiss_roll(n_samples=1000, noise=0.2, random_state=42)"
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]
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},
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{
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@ -298,6 +300,8 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"# not in the book\n",
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"\n",
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"from matplotlib.colors import ListedColormap\n",
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"\n",
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"darker_hot = ListedColormap(plt.cm.hot(np.linspace(0, 0.8, 256)))\n",
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@ -307,7 +311,7 @@
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"fig = plt.figure(figsize=(6, 5))\n",
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"ax = fig.add_subplot(111, projection='3d')\n",
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"\n",
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"ax.scatter(X[:, 0], X[:, 1], X[:, 2], c=t, cmap=darker_hot)\n",
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"ax.scatter(X_swiss[:, 0], X_swiss[:, 1], X_swiss[:, 2], c=t, cmap=darker_hot)\n",
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"ax.view_init(10, -70)\n",
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"set_xyz_axes(ax, axes)\n",
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"save_fig(\"swiss_roll_plot\")\n",
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@ -330,14 +334,14 @@
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"plt.figure(figsize=(10, 4))\n",
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"\n",
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"plt.subplot(121)\n",
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"plt.scatter(X[:, 0], X[:, 1], c=t, cmap=darker_hot)\n",
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"plt.scatter(X_swiss[:, 0], X_swiss[:, 1], c=t, cmap=darker_hot)\n",
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"plt.axis(axes[:4])\n",
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"plt.xlabel(\"$x_1$\")\n",
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"plt.ylabel(\"$x_2$\", labelpad=10, rotation=0)\n",
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"plt.grid(True)\n",
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"\n",
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"plt.subplot(122)\n",
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"plt.scatter(t, X[:, 1], c=t, cmap=darker_hot)\n",
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"plt.scatter(t, X_swiss[:, 1], c=t, cmap=darker_hot)\n",
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"plt.axis([4, 14.8, axes[2], axes[3]])\n",
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"plt.xlabel(\"$z_1$\")\n",
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"plt.grid(True)\n",
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@ -364,9 +368,9 @@
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"x3s = np.linspace(axes[4], axes[5], 10)\n",
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"x2, x3 = np.meshgrid(x2s, x3s)\n",
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"\n",
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"positive_class = X[:, 0] > 5\n",
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"X_pos = X[positive_class]\n",
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"X_neg = X[~positive_class]\n",
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"positive_class = X_swiss[:, 0] > 5\n",
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"X_pos = X_swiss[positive_class]\n",
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"X_neg = X_swiss[~positive_class]\n",
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"\n",
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"fig = plt.figure(figsize=(6, 5))\n",
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"ax = plt.subplot(1, 1, 1, projection='3d')\n",
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@ -380,8 +384,8 @@
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"\n",
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"fig = plt.figure(figsize=(5, 4))\n",
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"ax = plt.subplot(1, 1, 1)\n",
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"ax.plot(t[positive_class], X[positive_class, 1], \"gs\")\n",
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"ax.plot(t[~positive_class], X[~positive_class, 1], \"y^\")\n",
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"ax.plot(t[positive_class], X_swiss[positive_class, 1], \"gs\")\n",
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"ax.plot(t[~positive_class], X_swiss[~positive_class, 1], \"y^\")\n",
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"ax.axis([4, 15, axes[2], axes[3]])\n",
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"ax.set_xlabel(\"$z_1$\")\n",
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"ax.set_ylabel(\"$z_2$\", rotation=0, labelpad=8)\n",
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@ -389,9 +393,9 @@
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"save_fig(\"manifold_decision_boundary_plot2\")\n",
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"plt.show()\n",
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"\n",
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"positive_class = 2 * (t[:] - 4) > X[:, 1]\n",
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"X_pos = X[positive_class]\n",
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"X_neg = X[~positive_class]\n",
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"positive_class = 2 * (t[:] - 4) > X_swiss[:, 1]\n",
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"X_pos = X_swiss[positive_class]\n",
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"X_neg = X_swiss[~positive_class]\n",
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"\n",
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"fig = plt.figure(figsize=(6, 5))\n",
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"ax = plt.subplot(1, 1, 1, projection='3d')\n",
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@ -412,8 +416,8 @@
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"\n",
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"fig = plt.figure(figsize=(5, 4))\n",
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"ax = plt.subplot(1, 1, 1)\n",
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"ax.plot(t[positive_class], X[positive_class, 1], \"gs\")\n",
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"ax.plot(t[~positive_class], X[~positive_class, 1], \"y^\")\n",
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"ax.plot(t[positive_class], X_swiss[positive_class, 1], \"gs\")\n",
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"ax.plot(t[~positive_class], X_swiss[~positive_class, 1], \"y^\")\n",
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"ax.plot([4, 15], [0, 22], \"b-\", linewidth=2)\n",
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"ax.axis([4, 15, axes[2], axes[3]])\n",
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"ax.set_xlabel(\"$z_1$\")\n",
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@ -436,23 +440,25 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"# not in the book\n",
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"\n",
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"angle = np.pi / 5\n",
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"stretch = 5\n",
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"m = 200\n",
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"\n",
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"np.random.seed(3)\n",
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"X = np.random.randn(m, 2) / 10\n",
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"X = X @ np.array([[stretch, 0], [0, 1]]) # stretch\n",
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"X = X @ [[np.cos(angle), np.sin(angle)],\n",
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" [np.sin(angle), np.cos(angle)]] # rotate\n",
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"X_line = np.random.randn(m, 2) / 10\n",
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"X_line = X_line @ np.array([[stretch, 0], [0, 1]]) # stretch\n",
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"X_line = X_line @ [[np.cos(angle), np.sin(angle)],\n",
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" [np.sin(angle), np.cos(angle)]] # rotate\n",
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"\n",
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"u1 = np.array([np.cos(angle), np.sin(angle)])\n",
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"u2 = np.array([np.cos(angle - 2 * np.pi / 6), np.sin(angle - 2 * np.pi / 6)])\n",
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"u3 = np.array([np.cos(angle - np.pi / 2), np.sin(angle - np.pi / 2)])\n",
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"\n",
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"X_proj1 = X @ u1.reshape(-1, 1)\n",
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"X_proj2 = X @ u2.reshape(-1, 1)\n",
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"X_proj3 = X @ u3.reshape(-1, 1)\n",
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"X_proj1 = X_line @ u1.reshape(-1, 1)\n",
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"X_proj2 = X_line @ u2.reshape(-1, 1)\n",
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"X_proj3 = X_line @ u3.reshape(-1, 1)\n",
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"\n",
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"plt.figure(figsize=(8, 4))\n",
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"plt.subplot2grid((3, 2), (0, 0), rowspan=3)\n",
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@ -462,7 +468,7 @@
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" linewidth=2)\n",
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"plt.plot([-1.4, 1.4], [-1.4 * u3[1] / u3[0], 1.4 * u3[1] / u3[0]], \"k:\",\n",
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" linewidth=2)\n",
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"plt.plot(X[:, 0], X[:, 1], \"ro\", alpha=0.5)\n",
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"plt.plot(X_line[:, 0], X_line[:, 1], \"ro\", alpha=0.5)\n",
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"plt.arrow(0, 0, u1[0], u1[1], head_width=0.1, linewidth=4, alpha=0.9,\n",
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" length_includes_head=True, head_length=0.1, fc=\"b\", ec=\"b\", zorder=10)\n",
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"plt.arrow(0, 0, u3[0], u3[1], head_width=0.1, linewidth=1, alpha=0.9,\n",
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@ -632,14 +638,14 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The first dimension explains about 68% of the variance, while the second explains about 28%."
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"The first dimension explains about 76% of the variance, while the second explains about 15%."
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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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"By projecting down to 2D, we lost about 4% of the variance:"
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"By projecting down to 2D, we lost about 9% of the variance:"
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]
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},
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{
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@ -673,7 +679,7 @@
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"pca = PCA()\n",
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"pca.fit(X_train)\n",
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"cumsum = np.cumsum(pca.explained_variance_ratio_)\n",
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"d = np.argmax(cumsum >= 0.95) + 1 # d == 154"
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"d = np.argmax(cumsum >= 0.95) + 1 # d equals 154"
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]
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},
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{
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"execution_count": 23,
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"metadata": {},
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"outputs": [],
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"source": [
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"X_reduced_pca = X_reduced # not in the book (saved for comparison below)"
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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": 24,
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"metadata": {},
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"outputs": [],
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"source": [
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"pca.n_components_"
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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": 46,
|
||||
"execution_count": 45,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1111,7 +1108,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 47,
|
||||
"execution_count": 46,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1123,7 +1120,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 48,
|
||||
"execution_count": 47,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1135,7 +1132,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 49,
|
||||
"execution_count": 48,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1155,7 +1152,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 50,
|
||||
"execution_count": 49,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1186,7 +1183,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 51,
|
||||
"execution_count": 50,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1198,7 +1195,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 52,
|
||||
"execution_count": 51,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1271,7 +1268,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 53,
|
||||
"execution_count": 52,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1291,7 +1288,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 54,
|
||||
"execution_count": 53,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1300,7 +1297,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 55,
|
||||
"execution_count": 54,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1309,7 +1306,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 56,
|
||||
"execution_count": 55,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1328,7 +1325,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 57,
|
||||
"execution_count": 56,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1347,7 +1344,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 58,
|
||||
"execution_count": 57,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1371,7 +1368,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 59,
|
||||
"execution_count": 58,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1397,7 +1394,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 60,
|
||||
"execution_count": 59,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1409,7 +1406,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 61,
|
||||
"execution_count": 60,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1426,7 +1423,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 62,
|
||||
"execution_count": 61,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1443,7 +1440,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 63,
|
||||
"execution_count": 62,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1488,7 +1485,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 64,
|
||||
"execution_count": 63,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1504,7 +1501,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 65,
|
||||
"execution_count": 64,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1524,7 +1521,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 66,
|
||||
"execution_count": 65,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1552,7 +1549,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 67,
|
||||
"execution_count": 66,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1574,7 +1571,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 68,
|
||||
"execution_count": 67,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1589,7 +1586,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 69,
|
||||
"execution_count": 68,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1625,7 +1622,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 70,
|
||||
"execution_count": 69,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1672,7 +1669,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 71,
|
||||
"execution_count": 70,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1688,7 +1685,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 72,
|
||||
"execution_count": 71,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1704,7 +1701,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 73,
|
||||
"execution_count": 72,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1734,11 +1731,12 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 74,
|
||||
"execution_count": 73,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"%time X_pca_reduced = PCA(n_components=2).fit_transform(X_sample)\n",
|
||||
"pca = PCA(n_components=2, random_state=42)\n",
|
||||
"%time X_pca_reduced = pca.fit_transform(X_sample)\n",
|
||||
"plot_digits(X_pca_reduced, y_sample)\n",
|
||||
"plt.show()"
|
||||
]
|
||||
|
|
@ -1752,7 +1750,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 75,
|
||||
"execution_count": 74,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1771,7 +1769,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 76,
|
||||
"execution_count": 75,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1780,7 +1778,7 @@
|
|||
"\n",
|
||||
"%time X_pca_lle_reduced = pca_lle.fit_transform(X_sample)\n",
|
||||
"plot_digits(X_pca_lle_reduced, y_sample)\n",
|
||||
"plt.show()tight_layout="
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
|
|
@ -1801,12 +1799,12 @@
|
|||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"**Warning**: the following cell will take about 10-15 minutes to run, depending on your hardware:"
|
||||
"**Warning**: the following cell will take about 10 minutes to run, depending on your hardware:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 77,
|
||||
"execution_count": 76,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1826,12 +1824,12 @@
|
|||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"**Warning**: the following cell will take about 10-15 minutes to run, depending on your hardware:"
|
||||
"**Warning**: the following cell will take about 10 minutes to run, depending on your hardware:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 78,
|
||||
"execution_count": 77,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -1859,7 +1857,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 79,
|
||||
"execution_count": 78,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
|
|||
Loading…
Reference in New Issue