{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Support Vector Machines\n", "\n", "### Verschiedene Methoden um Daten zu simulieren\n", "Eine Sammlung von verschieden Methoden um Daten zu simulieren: \n", "* Methode um Daten mit zwei Klassen zu simulieren, welche linear trennbar sind (seperable=True/False gibt an ob die Daten perfekt trennbar sind oder teilweise überlappen): \n", "generate_linear_data(n_samples=1000,seperable=True,scaled=True) \n", " \n", "* Methode um Daten mit zwei Klassen zu simulieren, welche nicht linear trennbar sind: \n", "generate_non_linear_data(n_samples=1000) \n", " \n", "* Transformiert Daten von $\\mathbb{R}^2$ in $\\mathbb{R}^3$: \n", "transform_data_2_3d(X,y) " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/vnd.plotly.v1+html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "from mpl_toolkits.mplot3d import Axes3D\n", "import pylab as pl\n", "import matplotlib\n", "matplotlib.rcParams.update({'font.size': 22})\n", "\n", "from plotly.offline import init_notebook_mode, iplot\n", "from plotly.graph_objs import *\n", "import plotly.tools as tls\n", "init_notebook_mode(connected=True)\n", "\n", "import scipy as sp\n", "from sklearn.datasets import make_circles, make_blobs\n", "from sklearn.preprocessing import StandardScaler\n", "from sklearn.svm import SVC\n", "\n", "'''\n", "Generate Linear Seperable Data for SVM Demo\n", "'''\n", "def generate_linear_data(n_samples=800,seperable=True,scaled=True):\n", " if seperable:\n", " X1, Y1 = make_blobs(n_samples=n_samples,n_features=2,centers=2,shuffle=True,\n", " center_box=[-4,4],random_state=42)\n", " else:\n", " X1, Y1 = make_blobs(n_samples=n_samples,n_features=2,centers=2,shuffle=True,\n", " center_box=[-3,3],random_state=42)\n", " \n", " if scaled:\n", " scaler = StandardScaler()\n", " X1 = scaler.fit_transform(X1)\n", " \n", " return X1, Y1\n", "\n", "'''\n", "Generate Non Linear Seperable Data for SVM Demo\n", "'''\n", "def generate_non_linear_data(n_samples=800):\n", " X1, Y1 = make_circles(n_samples=n_samples, noise=0.05, factor=0.4,random_state=42)\n", " return X1, Y1\n", "\n", "'''\n", "Map 2D data into 3D Feature Space\n", "'''\n", "def transform_data_2_3d(X,y):\n", " #transform data using the following transformation: [x1, x2, x1**2.0 + x2**2.0]\n", " poly_kernel = lambda x1,x2: sp.array([x1,x2,x1**2.0+x2**2.0])\n", " newX = sp.array([poly_kernel(x1,x2) for x1,x2 in zip(sp.ravel(X[:,0]),sp.ravel(X[:,1]))])\n", " return newX, y" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Methode um Daten zu visualisieren\n", "Diese Methode plotted die simulierten Daten, entweder statisch oder interactive (interactive=True/False). Falls, ein SVM Model mit übergeben wird, wird zusätzlich die Entscheidungsfunktion und die Margin geplottet. Falls, die Daten in $\\mathbb{R}^3$ sind werden diese in 3D geplottet." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def plot_data(X,y,svm=None,interactive=True):\n", " if X.shape[1]==2:\n", " fig = pl.figure(figsize=(11,6))\n", " if svm:\n", " w = svm.coef_[0]\n", " b = svm.intercept_[0]\n", "\n", " x0 = sp.linspace(-1,1,50)\n", " decision_boundary = -w[0]/float(w[1]) * x0 - b/float(w[1])\n", "\n", " margin = 1/float(w[1])\n", " margin_up = decision_boundary + margin\n", " margin_down = decision_boundary - margin\n", "\n", " svs = svm.support_vectors_\n", " pl.plot(svs[:, 0], svs[:, 1],\"o\",markersize=17, \n", " markeredgecolor=\"#8e44ad\",\n", " markeredgewidth=2,\n", " markerfacecolor=\"#ffffff\",alpha=0.5)\n", " pl.plot(x0, decision_boundary, \"k-\", linewidth=2,label=\"Entscheidungsfunktion\")\n", " pl.plot(x0, margin_up, \"k--\", color=\"#16a085\", linewidth=2,label=\"Margin\")\n", " pl.plot(x0, margin_down, \"k--\", color=\"#16a085\", linewidth=2)\n", " \n", " pl.plot(X[y==0,0],X[y==0,1],\"o\",color=\"#e67e22\",label=\"Negative Class\",markersize=10)\n", " pl.plot(X[y==1,0],X[y==1,1],\"s\",color=\"#2980b9\",label=\"Positive Class\",markersize=10)\n", " pl.xlabel(\"x1\")\n", " pl.ylabel(\"x2\")\n", " \n", " if interactive:\n", " plotly_fig = tls.mpl_to_plotly(fig)\n", " iplot(plotly_fig, show_link=False)\n", " else:\n", " ax = pl.gca()\n", " ax.spines['top'].set_visible(False)\n", " ax.spines['right'].set_visible(False)\n", " ax.spines['left'].set_visible(True)\n", " ax.spines['bottom'].set_visible(True)\n", "\n", " #turn off all ticks\n", " ax.yaxis.set_ticks_position('none')\n", " ax.xaxis.set_ticks_position('none')\n", " pl.legend(ncol=4,loc=\"upper right\",frameon=False)\n", " pl.ylim([-2,3])\n", " pl.show()\n", " \n", " else: #3D plots\n", " if interactive:\n", " negClass = Scatter3d(x = newX[y==0,0], y = newX[y==0,1], z = newX[y==0,2],\n", " mode = 'markers',name=\"Negative Class\",\n", " marker=dict(opacity=0.5,size=3))\n", " posClass = Scatter3d(x = newX[y==1,0], y = newX[y==1,1], z = newX[y==1,2],\n", " mode = 'markers',name=\"Positive Class\",\n", " marker=dict(opacity=0.5,size=3))\n", " if svm!=None: \n", " tmp = sp.linspace(-1,1,51)\n", " x0,y0 = sp.meshgrid(tmp,tmp)\n", "\n", " decision_boundary = lambda x,y: (-svm.intercept_[0]-svm.coef_[0][0]*x-svm.coef_[0][1]) / svm.coef_[0][2]\n", "\n", " db_pl = Surface(x=x0,y=y0,z=decision_boundary(x0,y0),\n", " opacity=0.8,\n", " autocolorscale=False,\n", " surfacecolor=\"#8e44ad\",\n", " showscale=False,\n", " cauto=False,\n", " showlegend=False,)\n", "\n", " plotData = [posClass,negClass,db_pl]\n", " \n", " else:\n", " plotData = [posClass,negClass]\n", " \n", " layout = dict(title=None,\n", " font=dict(family='Optima', size=15, color='#7f7f7f'),\n", " margin=Margin(l=0,r=0,b=0,t=50,),\n", " )\n", "\n", " fig = dict(data=plotData,layout=layout)\n", " iplot(fig,show_link=False)\n", " else:\n", " fig = pl.figure(figsize=(11,6))\n", " ax = fig.add_subplot(111, projection='3d')\n", "\n", " ax.scatter(X[y==1,0],X[y==1,1],X[y==1,2], c='#2980b9', marker='s')\n", " ax.scatter(X[y==0,0],X[y==0,1],X[y==0,2], c='#e67e22', marker='o')\n", "\n", " if svm!=None: \n", " tmp = sp.linspace(-1,1,51)\n", " x0,y0 = sp.meshgrid(tmp,tmp)\n", "\n", " decision_boundary = lambda x,y: (-svm.intercept_[0]-svm.coef_[0][0]*x-svm.coef_[0][1]) / svm.coef_[0][2]\n", "\n", " ax.plot_surface(x0,y0,decision_boundary(x0,y0), color='#8e44ad',alpha=0.6,shade=False)\n", "\n", " pl.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Beispiel 1\n", "In dem ersten Beispiel simulieren wir linear trennbare Daten, plotten diese und wenden eine lineare SVM auf die Daten an. In diesem Beispiel betrachten wir die Hard Margin SVM, hierzu wird der Parameter C auf einen hohen Wert gesetzt.\n", "\n", "### Simuliere Daten" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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Ro6xHjx7Wtm3buuf62LFjgx5AEuo1YP369TZ48GArKiqqezO2bw1uuOTWzOxH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\n56b/6Zi4c0pbciO5BZAywp0q1vbsKfr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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "X,y = generate_linear_data(n_samples=100,seperable=True)\n", "plot_data(X,y,interactive=False) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Wende Hard-Margin SVM auf Daten an\n", "Beachte, ein hoher C-Wert führt zur Hard-Margin SVM!" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "data": [ { "marker": { "color": "#FFFFFF", "line": { "color": "#8E44AD", "width": 2 }, "opacity": 0.5, "size": 17, "symbol": "dot" }, "mode": "markers", "name": "_line0", "type": "scatter", "x": [ 0.09262402675366854, 0.7365703225667646, -0.21695235241603567 ], "xaxis": "x1", "y": [ 0.5388597658899593, 0.8052448153226546, -0.9286124578693161 ], "yaxis": "y1" }, { "line": { "color": "rgba (0, 0, 0, 1)", "dash": "solid", "width": 2 }, "mode": "lines", "name": "Entscheidungsfunktion", "type": "scatter", "x": [ -1, 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} } }, "text/html": [ "
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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "X,y = generate_non_linear_data(n_samples=1000)\n", "plot_data(X,y,interactive=True)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Transformiere Daten von $\\mathbb{R}^2$ in $\\mathbb{R}^3$" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "data": [ { "marker": { "opacity": 0.5, "size": 3 }, "mode": "markers", "name": "Positive Class", "type": "scatter3d", "x": [ 0.3560542215884903, -0.33935792244172513, -0.4330128346483529, -0.25882387604312374, -0.2227416412671963, -0.011018961921621006, 0.3649485750469348, -0.09580791773827983, -0.39852893753885654, -0.0676222463275808, 0.34337556333294394, 0.33719392291729755, 0.039804035793637504, 0.08476522693124644, -0.44042536737455973, 0.28279536459963855, 0.3528264631610884, 0.4581744820041619, 0.1117574206867209, 0.36270788879041577, -0.03282154287044477, -0.2912022989030473, 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