{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ToDo-List / Bemerkungen:\n",
    "\n",
    "- Task 1.0: (c) Die vektorweise Funktion muss noch geschrieben werden. Sie ist schon als def subseq_v2(a) in einer Zelle angelegt.\n",
    "\n",
    "- Task 1.2: I. Wer Spaß und Zeit hat kann das gerne mit der QR-Zerlegung implementieren.\n",
    "II. Es sollte ein countorplot benutzt werden -> matplotlib.pyplot.contour verstehen\n",
    "\n",
    "- Task 1.5: ACHTUNG: Bei DataFrame.loc[a:b] wird im Gegensatz zu numpy die obere Schranke miteinbezogen. Kann man matplotlib beibringen die Spalten [0:49],[100:149] zusammen zu berarbeiten? \n",
    "\n",
    "Generell: Ist es sinnvoll plt.plot zu benutzen oder doch besser plt.scatterplot?\n",
    "\n",
    "Bisher unbearbeitet: Task 1.6, 1.7 (apply gradient missing), 1.8, 1.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) Create a jupyter-notebook in which you create an array z consisting of 10000 random numbers drawn from {0, 1, 2}."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "z = np.array(np.random.randint(0, 3, 100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement two versions of a function which counts the number of appearances of the subsequence (2, 0, 1) in z. The first version should work with a loop that accesses the array z elementwise and makes elementwise comparisons."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def subseq_v1(a):\n",
    "    k = 0\n",
    "    for i in range(a.size-2):\n",
    "        if (a[i]==2 and a[i+1]==0 and a[i+2]==1):\n",
    "            k=k+1\n",
    "            i=i+3\n",
    "    return k        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "40.3 µs ± 937 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit subseq_v1(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second version should be a vectorized one (Hint: The numpy function logical_and might help you), which operates on (almost) the whole array z. Compare the runtime of the two versions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "tags": [
     "todo"
    ]
   },
   "outputs": [],
   "source": [
    "def subseq_v2(a):\n",
    "    x = np.logical_and(a == 2, np.logical_and(np.append(a[1:] == 0, [False]), np.append(a[2:] == 1, [False, False])))\n",
    "    return np.count_nonzero(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "14.7 µs ± 512 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit subseq_v2(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create n = 200 data points in the following way:\n",
    "\n",
    "(a) Draw ten random i.i.d. samples from the two-variate normal distribution $\\mathcal{N} ((\\frac{3}{2} \\, 0)^⊤,\\mathbf{I})$ and store them in a $\\mathtt{numpy}$ array $a$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = np.random.multivariate_normal([3/2, 0], np.eye(2), 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw another ten samples according to $\\mathcal{N} ((0 \\, \\frac{3}{2} )^⊤, \\mathbf{I})$  and store them in another $\\mathtt{numpy}$ array $b$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "b = np.random.multivariate_normal([0, 3/2], np.eye(2), 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use MatPlotLib to make a scatter plot (i.e. plot the points in a 2D coordinate system) of the elements in $a$ and the elements in $b$ using different colors for the two arrays."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x116e60ef0>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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SuijJ/VXV26rdUKN6VtWzwLsGrF8GPtq8/u/A3x9mO5KkdnmHryR1kOEvSR1k+EtSBxn+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\nGf6S1EH/H47daKvCveONAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10b15d048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(a[:,0], a[:,1], c='blue')\n",
    "plt.scatter(b[:,0], b[:,1], c='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Pick 100 equidistributed indices $i_1, . . . , i_{100}$ from ${1, 2, . . . , 10}$ and set the j-th data point $\\mathbf{x}_j$ to\n",
    "\n",
    "$\\mathbf{x}_j := a[i_j]+\\epsilon_j$ for all $j=1,...,100$ with $\\epsilon_j \\sim \\mathcal{N}((0 \\, 0)^⊤,\\frac{1}{4}\\mathbf{I}).$\n",
    "\n",
    "Proceed analogously for $j = 101, . . . , 200$ by substituting $a$ by $b$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def random_data(a,b,n):\n",
    "    i = np.random.randint(1, 11, n)\n",
    "    j = np.random.randint(1, 11, n)\n",
    "    epsilon_1 = np.random.multivariate_normal([0,0], np.eye(2)*(1/4), n)\n",
    "    epsilon_2 = np.random.multivariate_normal([0,0], np.eye(2)*(1/4), n)\n",
    "    x_1 = np.array(a[i-1]+epsilon_1)\n",
    "    x_2 = np.array(b[j-1]+epsilon_2)\n",
    "    return np.concatenate((x_1,x_2),axis=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = random_data(a,b,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Make a scatter plot for the data points $\\mathbf{x}_j$ with $j = 1, . . . , 200$ with different colors for the first 100 points and the second 100 points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PathCollection at 0x116f7b470>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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VfTWA1wN4Xybf2U8BvFFVfxnArQBuF5HXB7ZpUJIQdlW9X1VfOPj1WwBuDGmP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\nmUFhJ4SQzKCwE0JIZlDYCSEkM/4fj3c2ACxFeIsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116ee0240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x[0:100, 0], x[0:100, 1], c='blue')\n",
    "plt.scatter(x[100:200, 0], x[100:200, 1], c='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(c) The first $j = 1,...,100$ data points get the label $y_j = 0$, the next\n",
    "$j = 101,...,200$ ones get $y_j = 1.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = np.concatenate((np.zeros(100),np.ones(100)),axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement a linear least squares algorithm. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# computes the alpha-vector in the LLS for input data x and labels y\n",
    "def lls_solve(x,y):\n",
    "    X = np.c_[np.ones(x.shape[0]), x] #create modified input data\n",
    "    A = (X.T)@X\n",
    "    b = (X.T).dot(y)\n",
    "    alpha = np.linalg.solve(A,b)\n",
    "    return alpha"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply it to the data from task 1.1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.56098065 -0.15655837  0.20604369]\n"
     ]
    }
   ],
   "source": [
    "alpha = lls_solve(x,y)\n",
    "print(alpha)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the scattered input data as in step (b) of task 1.1 together with the separating hyperplane, i.e. the contour line given by\n",
    "\n",
    "$\\alpha_0 + \\alpha_1x_1 + \\alpha_2x_2 = \\frac{1}{2},$\n",
    "\n",
    "where $x_1$ and $x_2$ denote the coordinates in $\\mathbb{R}^2$ (not to be confused with the data $\\mathbf{x}_i$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def PlotContourLine(func, minx, maxx, miny, maxy, value=0):\n",
    "    #This plots the contourline func(x) = value\n",
    "    \n",
    "    samplenum = 100\n",
    "    #minx = -4\n",
    "    #maxx = 5\n",
    "    #miny = -3\n",
    "    #maxy = 4\n",
    "    xrange = np.arange(minx, maxx, (maxx-minx)/samplenum)\n",
    "    yrange = np.arange(miny, maxy, (maxy-miny)/samplenum)\n",
    "    \n",
    "    #This generates a two-dimensional mesh\n",
    "    X, Y = np.meshgrid(xrange,yrange)\n",
    "    \n",
    "    argsForf = np.array([X.flatten(),Y.flatten()]).T\n",
    "    Z = func(argsForf)\n",
    "    Z = np.reshape(Z,X.shape)\n",
    "    \n",
    "    plt.xlim(minx, maxx)\n",
    "    plt.ylim(miny, maxy)\n",
    "    plt.xlabel(r'$x_1$')\n",
    "    plt.ylabel(r'$x_2$')\n",
    "    #plt.contour(X, Y, Z, alpha=0.5,levels=[value],linestyles='dashed',linewidths=3)\n",
    "    Z = np.where(Z > value, 1, -1)\n",
    "    plt.contourf(X, Y, Z, alpha=0.2,colors=('red', 'blue'))\n",
    "    \n",
    "def LLS_classifier_alpha(x):\n",
    "    x_mod = np.c_[np.ones(x.shape[0]), x]\n",
    "    return x_mod@alpha    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x117451be0>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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nHZISqK+H9XdZtrrYmNEkSuETCrPrSgUIp+Utm+2DXRq7qXUOJ82b7tmoNnhw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\nVMXzDn0JAyvX8uyTcMa1o3k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x9zF44iWnAXwd+EhpLfKX+DDGLAeWpyU+rqIEndxBZEdE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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x116ea2f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x[0:100, 0], x[0:100, 1], c='blue', label='Class 0')\n",
    "plt.scatter(x[100:200, 0], x[100:200, 1], c='red', label='Class 1')\n",
    "PlotContourLine(LLS_classifier_alpha,-4, 5, -3, 4, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 = 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "# Task 1.3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Build the so-called confusion matrix for the data and the hyperplane from task 1.2,\n",
    "i.e. a matrix $C$, with entries\n",
    "\n",
    "$C_{ij}$ = # {Points classified as i, where the real label is j}.\n",
    "\n",
    "In our case this is a 2×2 matrix with $i,j \\in \\{0,1\\}$ since $|\\, \\Gamma\\, | = 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Classifier of the input data x depending on alpha\n",
    "def LLS_classifier(alpha, x):\n",
    "    x_mod = np.c_[np.ones(x.shape[0]), x]\n",
    "    return x_mod@alpha"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Confusion matrix of the input data x,\n",
    "# the classification depending on alpha\n",
    "# and the bounds of the class 0 (labeled entries)\n",
    "def confusion_matrix(alpha,x,a,b):\n",
    "    H = np.array(LLS_classifier(alpha,x))\n",
    "    C_11 = H[a:b] < 0.5 #entries of class 0, classified as class 0\n",
    "    C_12 = H[a:b] > 0.5 #entries of class 0, classified as class 1\n",
    "    C_21 = H[b:] < 0.5 #entries of class 1, classified as class 0\n",
    "    C_22 = H[b:] > 0.5 #entries of class 1, classified as class 1\n",
    "    C_31 = H[:a] < 0.5 #entries of class 1, classified as class 0\n",
    "    C_32 = H[:a] > 0.5 #entries of class 1, classified as class 1\n",
    "    return np.matrix([[np.sum(C_11),np.sum(C_12)],[np.sum(C_21)+np.sum(C_31), np.sum(C_22)+np.sum(C_32)]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[85 15]\n",
      " [ 9 91]]\n"
     ]
    }
   ],
   "source": [
    "C = confusion_matrix(alpha,x,0,100)\n",
    "print(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the accuracy $\\frac{\\text{trace(C)}}{n}$ ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def acc(C,x): return np.trace(C)/(x.shape[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.88"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc(C,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create 10 000 test points for each of the two classes in the same way as you created the training data in step (b) of task 1.1. Evaluate the LLS classifier, which was built on the training, on the test data and compute the confusion matrix and the accuracy of the test data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.84225000000000005"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_new = np.random.multivariate_normal([3/2, 0], np.eye(2), 10)\n",
    "b_new = np.random.multivariate_normal([0, 3/2], np.eye(2), 10)\n",
    "x_new = random_data(a_new,b_new,10000)\n",
    "y_new = np.concatenate((np.zeros(10000),np.ones(10000)),axis=0)\n",
    "C = confusion_matrix(alpha,x_new,0,10000)\n",
    "acc(C,x_new)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Compare your results to the ones from task 1.3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The accuracy is lower as in the training data set (what could have been expectetd). Anyway is the accurancy quiet high."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(a) Read in the Iris data set and use the data label $\\textbf{y}_i=0$ for the Iris-setosa instances and $\\mathbf{y}_i=1$ for the Iris-versicolor and Iris-virginica classes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "url = 'https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data'\n",
    "irisDataFrame = pd.read_csv(url, header=None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = np.concatenate((np.zeros(50),np.ones(100)),axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a.1. Run the LLS algorithm by using only the first two dimensions of $\\Omega$ in the input data, i.e. we only look at the first two features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "alpha = lls_solve(irisDataFrame.loc[:,0:1],y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the scattered data and the separating hyperplane as in task 1.2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x117611f28>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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l0zn3GvAAcEbe9pudc63OudbmnP8xZfCivsNUWGHuguW3T6n9y59bf0jhs6ty\n5taHeTP8qkTW1Yd/c8utPDl+fHA/IFzRsqjyzsX6F/WHMOwHPso8e1L+p/NR8aBvZs2ZK3zMbARw\nGvBspdsVT9CYXNDdp6IWdGcqvz6ee174tTR+c+tzvwYIMzodtIjppLbCz/ltz/KrSDn9KP/jBfUj\nm1/ethVw/Tnk668rvH3VKv99Vq3yH0Q9otm/f0FVNqNeMBV0PL/nTmrzP9+4+hDjArE40jsTgNsy\nef064IfOudqsblSDgsbkbryx8D5+i68Go5Q7UxXq46zjyxswHjA4W87c+jCj02Fu7fXQw8H98KtI\n6bdf0PFeftl/VVz2+Pnbsx8Kv5V0995beHA16HyzuepCVTazoiqXWsrvMf+5oJWDYfoRpg8xLhDT\n4qwUC7PIKkkCC5/NaAqeahmFoEVM4P9cqAVOIdvyfc5PyL5HvaArTjXc9zADuSrDkGIJSC+Glpu+\nea15Rt9Xy4wmWubPrHzAh8HlbsvNIYdty+85vzx72L4P5sNU7RubRP27KqbK56ugn2IJSC+Wrb1j\nJPdf91j/4Oz8ChU+K0VQ3n7KlMLPTZkSnDP3E/TLCmrLr4/TpkXb95Pawn2YwrwXUQt6b6PuXwLO\nV+mdlAsqd5w0HSs3JOOm4llB87ErMTfd75cVZr1A0I1IwvY9mx8v58OUlBub+L23Ufcv4uOFSe8o\n6EsiFSx81gU7knBT8azAPHvQ/1d+z1egoFnRvpT7+qj7nvB8euTF56I9XmIXZ4mUyrfwWVczLTOa\nklX4LKgwWair5ZCDKWFuehJ0I5I4+5704m5R9y8B56ucviRG/tz6pnPe1/c1b2GZg7NxDJYtWQIN\nwwZuaxjmbS82N93veEH8zinMPPjzzg3f97jm1SfBEDxfXelL1fXNre9iYGmEsIIKXkU+YJH/p3rm\nsd98+yuv8oqTFfKd7/j3L+icgtrKys9XA9xzT+G+Hz8Lfnw3uJw6GFbnbQ+zniFI1MeL2hA8X+X0\nJVb5hc82PPwK2zu7gAjn1sc1OBi2nTA3B4lzQBGSMbgqRSmnL4lWuPBZZzRX97miLuJV7XYq0VbC\ni4JJ5SjoS8UVKo3QbxLzop6JU4nBskJT+oq1E+V82KjbKna8JA+u1roqz5NW0JeKuuuWLrZ3vgg0\n09I6I57ZN0uWDMx/w+AGy/zy6Wed5aU7CrUTlIM/otm7uXc+v6Jlxc4pzBhGsfcoyvdP+sU63lSY\ngr5URPbq3kU1OFuOqAfL/ApyPfQwLL3af1GPXxGvBp//7fy2FzunoLb8zjnhRcGGrKiLu4WggVwZ\ntOzc+lyRD85WU5gFNUkpxpaFr+ubAAAMlklEQVSEBU7SLwGLs0qep29mHzCzb5nZrMzjRWX3UIac\n3Ln123fQ90VTU3lz68PMq4+rcFWYm2/EeSONsMeL8mbqxVS7qFpSJKDKYTnpnUuBhcDVZnY4MKsy\nXZJaEOnc+jB5zjhzoye1DbzxeNaUKf598NvnpLZDb5oOg1/wU+7xgm7qHuZm6kESkMdOjKjHm0Io\nOb1jZjc75xZlfr4eONU5NyfqDim9k3ze4GyE6Zswc9DjLNTl11ZQ0TII7l/UMzjKPV5Qkbaob6ae\nlKJqSRHh777S8/T/M/uDc+5KM/tcWb2TmlSo8Fl2cHbeORFVuYxyznicc+T9bjxeSr8XLIj2Krfc\n4wXd1D1qcf6uakHUv/syFQ36ZvZ14K+dcz/J3e6c+0bFeiVVlzu3vuKFz8LMq4+zcJVfW0FFyyDZ\nc92DirRFLQFFxqRfKQO5rwM/NbORAGZ2upkVudGn1LK7bunisVvX09nVPPjCZ6UIKlwWtE9chauW\nLIH6vOuj+gavvk2Cb4AdKKioWtSS/l6kTNErfefc1Wb2ceBBM+sG9gBXFtlNalBV59b7FS7zE2fh\nqnVrobdn4LbsY795+llJneteSpG2qCSgyJj0KzqQa2anAlfjTTCeAJztnPufSnVIA7nxiKXwWamS\nPtAX56CnSBkqNZC7FLjGOfeQmR0LLDezLzjnfhW6p1JVHSs30LVuO53M6NuWLXwW2eBsOZI+0Bfn\noKdIhZWS3nl/zs9PmdkC4MfAeyvZMYle7tz67G0H+x1a+Cy2ulBBA31JuIlvnIOecUrCeyuxK7v2\njnNuayblIzVkQOGzEmbfrFoF116bs55mm/cYKhAX/BasnNSWjEU9s2fDEwXSOLOD7k2bcFowlVqh\nbpfonNsXdUekMjpWbuD+6x7DdXaWVRohqC5U5BYs8AZEx08AzPu+9GqvoFlsnQjw8svlba8Fsf6C\nJUlUZXMIKVz4zFHq1X2u2NPshRasXHNNzJ3wkfQxhzCG4jlJSRT0h4j+9A0D7kxFU1Oowdnx4wv/\n/x/repqkLOpJSj+uvy7cFMswN4CRIUtBv8ZVam79kiUDc/pQhfU0CShOlZh+hC2QFuYGMDKkKejX\nsHIHZ8uRiPU0iehEQvpx9wr/7UFBP8wNYGRI001UakShwmfZqZdVmVsv8WoNmCm0pj1gP91gZSir\ndJVNqYLcufW5qfrOrmZa5s+M/qbikkxh1woody95FPQTLD9909TWv5hqHujqPk3OO7fwTVmKFUhL\nwniEJIqCfgJ1rNxQmbr1UrvCFkhLwniEJIpy+lWWqMJnIlJTlNOvMXfd0sW4HY9Bc0vfthfWOEZP\nCze3XkSkmIoHfTObAnwPGA8cBG52zv3fSrebZLlz6zubZtCS89zRZ8dZw74GqCiYSKTiuNLvAS53\nzv3WzEYD7WZ2n3PumRjaTpxKzq0fclQUTCRyFQ/6zrmtwNbMz7vNrAOYBKQq6GtwNoSgomAK+iKh\nxJrTN7MjgXcDj+VtXwQsApg6BOYPBxY+09z60qkomEjkYgv6ZvYmvJuv/JVzblfuc865m4GbwZu9\nE1efKsGv8FnLjCamt+nqvixaWCQSuViCvpkNwwv4tzvn7o6jzbgpfVMBWlgkErk4Zu8Y8G2gwzn3\n1Uq3F7dsKkfpmwrQwiKRyMVxpd8GfBJ4yszWZrb9nXNuZQxtRy6/8FnXuu28sKlZc+srpdDNVUQk\ntDhm7zwEWKXbqbRs+qaTGbQ092/f3uU4+uyZmlsvIjVBK3JL4A3OZm87CEz0Cp9Nn44GZ0Wkpijo\nB9DgrIgMNQr6GYULn2lwVkSGFgV9/AufHd2qufUiMrSkOujnpm9U+ExE0iCVQV9z60UkrVIX9DtW\nbtDcehFJrSEd9DtWbuDpLeMGbHNrXmDHmKM0t15EUmnIBv3+ufVdAwuftc5mngZnRSSlhkTQzy+N\n8IrSNyIiBdV00M/edrBr4MU8nV3NSt+IiBRQs0E//7aDTW2T+p6bB7q6FxEpoOaCvkojiIiEl+ig\nr9IIIiLRSmzQ70vf5CTrd2/q0uCsiMggJC7o79/Zzf3XPdaXvmk5vj9X33K8SiOIiAxG4oL+7r31\ndHZ5g7PzLjqt2t0RERlSEhf0G4bBvIUzlb4REamAump3IN/wI8Yo4IuIVEjigr6IiFSOgr6ISIoo\n6IuIpIiCvohIiijoi4ikiIK+iEiKKOiLiKSIgr6ISIoo6IuIpIiCvohIiijoi4ikiIK+iEiKKOiL\niKSIgr6ISIoo6IuIpEjFg76ZfcfMtpvZ+kq3JSIiweK40v8ucEYM7YiISBEVD/rOud8Af6x0OyIi\nUpxy+iIiKZKIoG9mi8xsjZmt6eraUe3uiIgMWYkI+s65m51zrc651qam5mp3R0RkyEpE0BcRkXjE\nMWXzTuAR4O1mttnM/rLSbYqISGENlW7AOXdBpdsQEZHSKL0jIpIiCvoiIimioC8ikiIK+iIiKaKg\nLyKSIgr6IiIpoqAvIpIiCvoiIimioC8ikiIK+iIiKaKgLyKSIgr6IiIpoqAvIpIiCvoiIimioC8i\nkiIK+iIiKaKgLyKSIgr6IiIpoqAvIpIiCvoiIimioC8ikiIK+iIiKaKgLyKSIgr6IiIpoqAvIpIi\nCvoiIimioC8ikiIK+iIiKaKgLyKSIgr6IiIpoqAvIpIiCvoiIimioC8ikiIK+iIiKaKgLyKSIrEE\nfTM7w8z+x8yeN7Mr42hTREQOVfGgb2b1wI3AAuCdwAVm9s5KtysiIoeK40p/LvC8c+5F59wbwA+A\nP42hXRERydMQQxuTgJdzHm8G5uW+wMwWAYsyD7tbW219DP2qBUcAr1a7Ewmh96Kf3ot+ei/6vb2U\nF8UR9K3ANjfggXM3AzcDmNka51xrDP1KPL0X/fRe9NN70U/vRT8zW1PK6+JI72wGpuQ8ngxsiaFd\nERHJE0fQfwJ4m5lNN7PDgD8HfhpDuyIikqfi6R3nXI+ZXQb8AqgHvuOcezpgl5sr3acaovein96L\nfnov+um96FfSe2HOueKvEhGRIUErckVEUkRBX0QkRRIT9M2s0cweN7N1Zva0mf1DtftUbWZWb2a/\nM7N7q92XajKzjWb2lJmtLXVa2lBlZuPM7C4ze9bMOszsxGr3qRrM7O2Zz0P2a5eZ/VW1+1UtZvbX\nmbi53szuNLNG39cmJadvZgaMcs69bmbDgIeAzzvnHq1y16rGzL4AtAJjnHNnVbs/1WJmG4FW51zq\nF+GY2W3AaufcLZnZcCOdc69Vu1/VlCn18gowzzm3qdr9iZuZTcKLl+90zu0zsx8CK51z3y30+sRc\n6TvP65mHwzJfyfgXqQrMbDLwIeCWavdFksHMxgCnAN8GcM69kfaAn3Eq8EIaA36OBmCEmTUAIwlY\nC5WYoA996Yy1wHbgPufcY9XuUxV9HbgCOFjtjiSAA35pZu2Zkh1pdRSwA7g1k/a7xcxGVbtTCfDn\nwJ3V7kS1OOdeAf4FeAnYCux0zv3S7/WJCvrOuV7n3Cy8VbtzzWxmtftUDWZ2FrDdOdde7b4kRJtz\n7gS8Sq1LzOyUaneoShqAE4CbnHPvBvYAqS5VnklxnQ38qNp9qRYza8IrYjkdmAiMMrNP+L0+UUE/\nK/Mn6wPAGVXuSrW0AWdnctk/AN5vZt+vbpeqxzm3JfN9O7ACr3JrGm0GNuf8BXwX3j8CabYA+K1z\n7vfV7kgVnQZscM7tcM4dAO4G3uv34sQEfTNrNrNxmZ9H4J3Is9XtVXU4565yzk12zh2J96frr5xz\nvv9yD2VmNsrMRmd/Bk4HUlmF1Tm3DXjZzLLVFE8Fnqlil5LgAlKc2sl4CXiPmY3MTIg5Fejwe3Ec\nVTZLNQG4LTMSXwf80DmX6qmKAsBbgBXeZ5kG4A7n3M+r26Wq+hxweyat8SKwsMr9qRozGwl8ALik\n2n2pJufcY2Z2F/BboAf4HQElGRIzZVNERCovMekdERGpPAV9EZEUUdAXEUkRBX0RkRRR0BcRSREF\nfRGRFFHQFwlgZr82sw9kfv6imf2/avdJZDCStDhLJIn+HvhHM2sB3o1X50WkZmlxlkgRZvYg8CZg\nvnNut5kdBSwFxjrnPlrd3omUR+kdkQBmdixeiZBu59xuAOfci865v6xuz0TCUdAX8WFmE4Db8crW\n7jGzD1a5SyKDpqAvUkCmmNfdwOXOuQ7gn4BlVe2USASU0xcpk5m9GbgWr8LjLc6566rcJZGSKeiL\niKSI0jsiIimioC8ikiIK+iIiKaKgLyKSIgr6IiIpoqAvIpIiCvoiIimioC8ikiIK+iIiKfL/ARaY\nYWq6Qn/zAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117525c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(irisDataFrame.loc[0:49, 0], irisDataFrame.loc[0:49, 1], c='blue', label='Class 0')\n",
    "plt.scatter(irisDataFrame.loc[50:149, 0], irisDataFrame.loc[50:149, 1], c='red', label='Class 1')\n",
    "PlotContourLine(LLS_classifier_alpha,3,8,0,5, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 = 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a.2. Now run the LLS algorithm by using all four features/dimensions of the input data. Compute the confusion matrix and the accuracy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "alpha = lls_solve(irisDataFrame.loc[:,0:3],y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 50   0]\n",
      " [  0 100]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = confusion_matrix(alpha,irisDataFrame.loc[:,0:3],0,50)\n",
    "print(C)\n",
    "acc(C,irisDataFrame)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(b) Finally, run the same two steps as in (a), but now try to classify Iris-versicolor instances (label $\\mathbf{y}_i = 0$) against both Iris-setosa and Iris-virginica (label $\\mathbf{y}_i=1$)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = np.concatenate((np.ones(50),np.zeros(50),np.ones(50)),axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b.1. Run the LLS algorithm by using only the first two dimensions of $\\Omega$ in the input data, i.e. we only look at the first two features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "alpha = lls_solve(irisDataFrame.loc[:,0:1],y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the scattered data and the separating hyperplane as in task 1.2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x1188275f8>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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s3LiBlpaTOeGESXzhC5/1nVp5xYplfOADY1m8+D/58pe/QGPjkZG3IXDbNLWySAFd19z3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\nMuT/AL2PydojQCLoAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1176baf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(irisDataFrame.loc[50:99, 0], irisDataFrame.loc[50:99, 1], c='blue', label='Class 0')\n",
    "plt.scatter(irisDataFrame.loc[0:49, 0], irisDataFrame.loc[0:49, 1], c='red', label='Class 1')\n",
    "plt.scatter(irisDataFrame.loc[100:149, 0], irisDataFrame.loc[100:149, 1], c='red', label='_nolegend_')\n",
    "PlotContourLine(LLS_classifier_alpha,3,8,0,5, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 + \\alpha_3 \\cdot x_3 +\\alpha_4 \\cdot x_4= 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "b.2. Now run the LLS algorithm by using all four features/dimensions of the input data. Compute the confusion matrix and the accuracy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "alpha = lls_solve(irisDataFrame.loc[:,0:3],y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.73333333333333328"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = confusion_matrix(alpha,irisDataFrame.loc[:,0:3],50,100)\n",
    "acc(C,irisDataFrame)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " What do you observe?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We observe that in part b.1, the hyperplane is not seperating the two classes at all. In part b.2 is the accuracy, considering our result in b.1, quiet high. This means that only the data in the last two columns (features three and four) allow us to distinguish between Iris-versicolor and the other species. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Converting our IrisdataFrame.loc wich is of type DataFrame to an numpy array\n",
    "def DFtA(x): #DFtA=DataFrametoArray\n",
    "    xy = x.reset_index().values\n",
    "    return xy[:,1:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "data1 = DFtA(irisDataFrame.loc[:,0:1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#defining J\n",
    "def J(alpha,data,y):\n",
    "    return (1/y.size)*(((data@alpha)-y).T)@((data@alpha)-y)\n",
    "\n",
    "#defining gradient for this specific function\n",
    "def gradJ2(alpha,data,y):\n",
    "    z = data@alpha - y\n",
    "    return (2/data[:,0].size)*(data.T@z)\n",
    "\n",
    "#defining the gradient method\n",
    "def Learning_2(n,data,y,nu,stepMax,epsilon,color,legende):\n",
    "        alpha = np.random.random(n) #initialize alpha_0 as 0\n",
    "        step_2 = 0\n",
    "        pl=np.array([step_2,J(alpha,data,y)])\n",
    "        pl.shape=(1,2)\n",
    "        while (np.linalg.norm(gradJ2(alpha,data,y))>epsilon and step_2<stepMax):\n",
    "            alpha = alpha - nu*gradJ2(alpha,data,y)\n",
    "            step_2 += 1\n",
    "            pl_hilf=np.array([step_2,J(alpha,data,y)])\n",
    "            pl_hilf.shape=(1,2)\n",
    "            pl = np.append(pl,pl_hilf,axis=0)\n",
    "        print(np.linalg.norm(gradJ2(alpha,data,y)))\n",
    "        plt.figure()\n",
    "        plt.plot(pl[:, 0], pl[:, 1], color, label=legende)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "inf\n",
      "7.01431211178e+91\n",
      "0.30496858235\n",
      "0.849628130641\n",
      "2.50702135437\n",
      "92.2248396906\n",
      "51.4669302717\n",
      "81.5866246116\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x118cd3a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11755b1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11751d4a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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6VLVmMjxwzjm0ptN88IUXeFO3rBORkCnQp2FpLseD55zDwWKR923cyBsDA2GXJCIJpkCf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118e1c080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x118d75a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x118d25470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11777f358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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VrDMwa4AEX/96eqDt4IPTE87Dh6feg5l9WFFjCJI6S1oErATuj4gFdXYfAcyJiLcaaLud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\nIZiZWcYJwczMAPh/bmonw1A4ep0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x118e81198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#running the algorithm\n",
    "nu = np.array([10**(-n) for n in range(8)]) #10^-n\n",
    "color = ['b-','g-','r-','c-','m-','y-','k-','b-']\n",
    "legende = ['nu=1','nu=10^-1','nu=10^-2','nu=10^-3','nu=10^-4','nu=10^-5','nu=10^-6','nu=10^-7']\n",
    "y=np.append(np.zeros(50), np.ones(100))\n",
    "stepMax = 100\n",
    "epsilon = 10**(-5)\n",
    "data=data1[:,:2] #getting the Data\n",
    "n = data[1,:].size+1\n",
    "data= np.c_[np.ones(150), data]\n",
    "for i in range(nu.size):\n",
    "    Learning_2(n,data,y,nu[i],stepMax,epsilon,color[i],legende[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.7"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Normalize the data from task 1.5 (a.1.). To this end, calculate the mean $\\mu_j$ and the standard deviation $\\sigma_j$ for each feature $j$ (i.e. each coordinate direction $j$ of the data set) and set the $j$-th component of the $i$-th data point to\n",
    "\n",
    "$[\\mathbf{x}_i]_j := \\frac{[\\mathbf{x}_i]_j-\\mu_j}{\\sigma_j}.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.   5.1  3.5]\n",
      " [ 1.   4.9  3. ]\n",
      " [ 1.   4.7  3.2]\n",
      " [ 1.   4.6  3.1]\n",
      " [ 1.   5.   3.6]\n",
      " [ 1.   5.4  3.9]\n",
      " [ 1.   4.6  3.4]\n",
      " [ 1.   5.   3.4]\n",
      " [ 1.   4.4  2.9]\n",
      " [ 1.   4.9  3.1]\n",
      " [ 1.   5.4  3.7]\n",
      " [ 1.   4.8  3.4]\n",
      " [ 1.   4.8  3. ]\n",
      " [ 1.   4.3  3. ]\n",
      " [ 1.   5.8  4. ]\n",
      " [ 1.   5.7  4.4]\n",
      " [ 1.   5.4  3.9]\n",
      " [ 1.   5.1  3.5]\n",
      " [ 1.   5.7  3.8]\n",
      " [ 1.   5.1  3.8]\n",
      " [ 1.   5.4  3.4]\n",
      " [ 1.   5.1  3.7]\n",
      " [ 1.   4.6  3.6]\n",
      " [ 1.   5.1  3.3]\n",
      " [ 1.   4.8  3.4]\n",
      " [ 1.   5.   3. ]\n",
      " [ 1.   5.   3.4]\n",
      " [ 1.   5.2  3.5]\n",
      " [ 1.   5.2  3.4]\n",
      " [ 1.   4.7  3.2]\n",
      " [ 1.   4.8  3.1]\n",
      " [ 1.   5.4  3.4]\n",
      " [ 1.   5.2  4.1]\n",
      " [ 1.   5.5  4.2]\n",
      " [ 1.   4.9  3.1]\n",
      " [ 1.   5.   3.2]\n",
      " [ 1.   5.5  3.5]\n",
      " [ 1.   4.9  3.1]\n",
      " [ 1.   4.4  3. ]\n",
      " [ 1.   5.1  3.4]\n",
      " [ 1.   5.   3.5]\n",
      " [ 1.   4.5  2.3]\n",
      " [ 1.   4.4  3.2]\n",
      " [ 1.   5.   3.5]\n",
      " [ 1.   5.1  3.8]\n",
      " [ 1.   4.8  3. ]\n",
      " [ 1.   5.1  3.8]\n",
      " [ 1.   4.6  3.2]\n",
      " [ 1.   5.3  3.7]\n",
      " [ 1.   5.   3.3]\n",
      " [ 1.   7.   3.2]\n",
      " [ 1.   6.4  3.2]\n",
      " [ 1.   6.9  3.1]\n",
      " [ 1.   5.5  2.3]\n",
      " [ 1.   6.5  2.8]\n",
      " [ 1.   5.7  2.8]\n",
      " [ 1.   6.3  3.3]\n",
      " [ 1.   4.9  2.4]\n",
      " [ 1.   6.6  2.9]\n",
      " [ 1.   5.2  2.7]\n",
      " [ 1.   5.   2. ]\n",
      " [ 1.   5.9  3. ]\n",
      " [ 1.   6.   2.2]\n",
      " [ 1.   6.1  2.9]\n",
      " [ 1.   5.6  2.9]\n",
      " [ 1.   6.7  3.1]\n",
      " [ 1.   5.6  3. ]\n",
      " [ 1.   5.8  2.7]\n",
      " [ 1.   6.2  2.2]\n",
      " [ 1.   5.6  2.5]\n",
      " [ 1.   5.9  3.2]\n",
      " [ 1.   6.1  2.8]\n",
      " [ 1.   6.3  2.5]\n",
      " [ 1.   6.1  2.8]\n",
      " [ 1.   6.4  2.9]\n",
      " [ 1.   6.6  3. ]\n",
      " [ 1.   6.8  2.8]\n",
      " [ 1.   6.7  3. ]\n",
      " [ 1.   6.   2.9]\n",
      " [ 1.   5.7  2.6]\n",
      " [ 1.   5.5  2.4]\n",
      " [ 1.   5.5  2.4]\n",
      " [ 1.   5.8  2.7]\n",
      " [ 1.   6.   2.7]\n",
      " [ 1.   5.4  3. ]\n",
      " [ 1.   6.   3.4]\n",
      " [ 1.   6.7  3.1]\n",
      " [ 1.   6.3  2.3]\n",
      " [ 1.   5.6  3. ]\n",
      " [ 1.   5.5  2.5]\n",
      " [ 1.   5.5  2.6]\n",
      " [ 1.   6.1  3. ]\n",
      " [ 1.   5.8  2.6]\n",
      " [ 1.   5.   2.3]\n",
      " [ 1.   5.6  2.7]\n",
      " [ 1.   5.7  3. ]\n",
      " [ 1.   5.7  2.9]\n",
      " [ 1.   6.2  2.9]\n",
      " [ 1.   5.1  2.5]\n",
      " [ 1.   5.7  2.8]\n",
      " [ 1.   6.3  3.3]\n",
      " [ 1.   5.8  2.7]\n",
      " [ 1.   7.1  3. ]\n",
      " [ 1.   6.3  2.9]\n",
      " [ 1.   6.5  3. ]\n",
      " [ 1.   7.6  3. ]\n",
      " [ 1.   4.9  2.5]\n",
      " [ 1.   7.3  2.9]\n",
      " [ 1.   6.7  2.5]\n",
      " [ 1.   7.2  3.6]\n",
      " [ 1.   6.5  3.2]\n",
      " [ 1.   6.4  2.7]\n",
      " [ 1.   6.8  3. ]\n",
      " [ 1.   5.7  2.5]\n",
      " [ 1.   5.8  2.8]\n",
      " [ 1.   6.4  3.2]\n",
      " [ 1.   6.5  3. ]\n",
      " [ 1.   7.7  3.8]\n",
      " [ 1.   7.7  2.6]\n",
      " [ 1.   6.   2.2]\n",
      " [ 1.   6.9  3.2]\n",
      " [ 1.   5.6  2.8]\n",
      " [ 1.   7.7  2.8]\n",
      " [ 1.   6.3  2.7]\n",
      " [ 1.   6.7  3.3]\n",
      " [ 1.   7.2  3.2]\n",
      " [ 1.   6.2  2.8]\n",
      " [ 1.   6.1  3. ]\n",
      " [ 1.   6.4  2.8]\n",
      " [ 1.   7.2  3. ]\n",
      " [ 1.   7.4  2.8]\n",
      " [ 1.   7.9  3.8]\n",
      " [ 1.   6.4  2.8]\n",
      " [ 1.   6.3  2.8]\n",
      " [ 1.   6.1  2.6]\n",
      " [ 1.   7.7  3. ]\n",
      " [ 1.   6.3  3.4]\n",
      " [ 1.   6.4  3.1]\n",
      " [ 1.   6.   3. ]\n",
      " [ 1.   6.9  3.1]\n",
      " [ 1.   6.7  3.1]\n",
      " [ 1.   6.9  3.1]\n",
      " [ 1.   5.8  2.7]\n",
      " [ 1.   6.8  3.2]\n",
      " [ 1.   6.7  3.3]\n",
      " [ 1.   6.7  3. ]\n",
      " [ 1.   6.3  2.5]\n",
      " [ 1.   6.5  3. ]\n",
      " [ 1.   6.2  3.4]\n",
      " [ 1.   5.9  3. ]]\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "data = data1\n",
    "data= np.c_[np.ones(150), data]\n",
    "\n",
    "def mean(x): #defining the mean\n",
    "    return np.sum(x)/x.size\n",
    "\n",
    "def var(x): #defining variance\n",
    "    return (x-mean(x)).T@(x-mean(x))/x.size\n",
    "print(data)\n",
    "\n",
    "def normalize(dataUN):\n",
    "    dataN1 = dataUN\n",
    "    dataN2 = dataUN\n",
    "    for i in range(dataUN[0,:].size):\n",
    "   \n",
    "        if(var(dataUN[:,i])==0):\n",
    "            print(var(dataUN[:,i]))\n",
    "            dataN1[:,i]=np.zeros(dataUN[:,i].size)\n",
    "        else : dataN1[:,i] = (dataUN[:,i]-mean(dataUN[:,i]))/var(dataUN[:,i])\n",
    "    return dataN1\n",
    "\n",
    "data2 = normalize(data)\n",
    "\n",
    "#Anmerkung: Mit \n",
    "# irisDataNorm = (irisDataFrame.loc[:,0:3]-irisDataFrame.mean(axis=0))/irisDataFrame.std(axis=0)\n",
    "# geht das bestimmt besser ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now run the gradient descent LLS algorithm on the normalized data. Again, choose  ν\n",
    "  as the largest value such that convergence is achieved. Compare the first 100 iteration steps by plotting the value of  J\n",
    "  vs. the iteration number for both the normalized and the unnormalized case. What do you observe?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.37954692062e+99\n",
      "8.84369081972e-06\n",
      "0.0553349249904\n",
      "3.44630677466\n",
      "3.48820081141\n",
      "9.29712050692\n",
      "9.12278370487\n",
      "6.47803592405\n"
     ]
    },
    {
     "data": {
      "image/png": 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6HTdbMoc/8i5JzYxlqUSStEmGW5KaMdyS1IzhlqRmDLckNWO4JakZwy1Jzfw/bddyrL3z\nsZ4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1193096d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11933bba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11937ba90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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P5+dz5vDHlhY+sHEjdTqaVSRiqdzHmU9OnszzCxbwvwcOsHzDBir37/c6kogMA5X7OHRx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\nEFAKBwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11934c278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1193f22e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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ra4/HyU5MpS8iMkiZmefh99cwffrtbN36EwKBXNranvc61nGp9EVETkFiYjpz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\npS8iEkdU+iIicUSlLyISR1T6IiJx5P8A0Je7Horo5IkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119429f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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1vvvuO+rXr8+DDz7IgQMHvI5WqlQIIiJBrVq1Iicnh169evH444+TmJjIO++8\n43WsUqNCEBHJ55xzzmHWrFmsXbuWqKgomjZtSq9evdi/f7/X0UqcCkFE5Diuu+46tm3bxqBBg5g7\ndy4+n4/ly5d7HatEqRBERApw1llnMXHiRDZt2sT5559P69at6dixI/v27fM6WolQIYiIFMLv95Oe\nns6YMWNYtmwZcXFxLFy4MOLGX6gQRESKoEKFCjz00ENkZmZyxRVX0LlzZ1q1asU//vEPr6MVGxWC\niMhJ8Pl8vP/++0ydOpW1a9cSHx/PrFmzOHr0qNfRTpsKQUTkJJUrV44BAwaQnZ1N3bp16dOnD02a\nNOHTTz/1OtppUSGIiJyiSy+9lNWrVzNv3jy2b99O7dq1mThxIkeOHPE62ilRIYiInAYzo0ePHgQC\nAVq2bMngwYOpW7cuW7du9TraSVMhiIgUg4suuoi0tDSWLFnCl19+id/vZ/jw4fz0009eRysyFYKI\nSDExM9q1a0cgEKBz586MGzeOlJQUPvjgA6+jFYkKQUSkmJ133nk8++yzrFq1ioMHD9KwYUP69+/P\njz/+6HW0E1IhiIiUkGbNmpGdnU2/fv2YMWMGCQkJrFq1yutYBVIhiIiUoNjYWKZNm8a6deuIiYmh\nRYsWdO/enW+//dbraP9PkQvBzMqZWaaZrTjBmnZm5szMH7zdycy25vs6ambJv3nMcjPLPvVdEBEJ\nfQ0aNGDr1q0MGzaMBQsW4PP5SEtL8zrWfziZM4QBQG5BG82sItAf2HTsPufcQudcsnMuGegC7HTO\nbc33mDZAaF9UExEpJjExMTz66KOkp6dTtWpV2rVrR9u2bdmzZ4/X0YAiFoKZVQNuAeacYNkYYCJQ\n0HusOgKL8j1nLPAAMLZISUVEIkRycjKbN29mwoQJvP766/h8Pp555hnPh+UV9QxhKvBn4LjDOsws\nBajunCvwchLQgXyFQF6BPA6c8HfUmVkvM0s3s/RIHTkrImVPdHQ0gwcPZvv27SQmJnL33XfTrFkz\n/va3v3mWqdBCMLNWwNfOuYwCtkcBU4CBJ3iOusAB51x28HYycLlz7uXCvr9z7innnN85569cuXJh\ny0VEwsqVV17J2rVrmTlzJhs3biQhIYFp06bxyy+/lHqWopwhNABSzWwnsBhoamYL8m2vCCQAa4Nr\n6gHLj72wHHQn/3l2UB+4Orj+feBKM1t7ivsgIhLWoqKi6NOnDzk5OVx33XUMGDCARo0aEQgESjdH\nYQucc0Odc9WcczXJ+8G+xjnXOd/2/c65C5xzNYNrNgKpzrl0+PUMoj15ZXLsMbOcc1WD6xsCnzjn\nri++3RIRCT+XXHIJr7/+Os/nJPNGAAAExUlEQVQ//zyffPIJKSkpjB07lkOHDpXK9z/lzyGY2Wgz\nSy3C0sbALufcjlP9XiIiZYWZ0blzZwKBAG3atGHEiBH4/X52795d8t/b61e1T4bf73fp6elexxAR\nKTXLly/n2WefZcmSJZQrV+6UnsPMMpxz/kLXqRBERCJbUQtBoytERARQIYiISJAKQUREABWCiIgE\nqRBERARQIYiISJAKQUREABWCiIgEhdUH08xsH/DFKT78AuCfxRgnHGify4ayts9lbX/h9Pe5hnOu\n0HHRYVUIp8PM0ovySb1Ion0uG8raPpe1/YXS22ddMhIREUCFICIiQWWpEJ7yOoAHtM9lQ1nb57K2\nv1BK+1xmXkMQEZETK0tnCCIicgIRXwhm1sLMPjazz8xsiNd5SoKZVTezd8ws18xyzGxA8P7zzOxN\nM/s0+N9KXmctbmZWzswyzWxF8PalZrYpuM8vmlkFrzMWJzM718yWmtlHweNdP9KPs5ndH/x7nW1m\ni8wsJtKOs5nNM7OvzSw7333HPa6WZ1rwZ9p2M7uquHJEdCGYWTlgJtAS8AEdzcznbaoScQQY6JyL\nA+oBfYP7OQR42zl3BfB28HakGQDk5rv9GDAluM/fAX/0JFXJeQJ4wzn3ByCJvH2P2ONsZhcD/QG/\ncy4BKEfe73aPtOP8LNDiN/cVdFxbAlcEv3oBs4orREQXAlAH+Mw5t8M5dwhYDLT2OFOxc87tcc59\nGPzzD+T9kLiYvH2dH1w2H7jNm4Qlw8yqAbcAc4K3DWgKLA0uiah9NrPfkfc7yucCOOcOOef+RYQf\nZyAaONPMooGzgD1E2HF2zr0HfPubuws6rq2B51yejcC5ZnZRceSI9EK4GPhHvtu7gvdFLDOrCaQA\nm4Aqzrk9kFcawO+9S1YipgJ/Bo4Gb58P/Ms5dyR4O9KO92XAPuCZ4GWyOWZ2NhF8nJ1zXwKTgL+T\nVwT7gQwi+zgfU9BxLbGfa5FeCHac+yL2bVVmFgukAfc55773Ok9JMrNWwNfOuYz8dx9naSQd72jg\nKmCWcy4F+DcRdHnoeILXzVsDlwJVgbPJu2TyW5F0nAtTYn/PI70QdgHV892uBuz2KEuJMrPy5JXB\nQufcsuDdXx07lQz+92uv8pWABkCqme0k71JgU/LOGM4NXlqAyDveu4BdzrlNwdtLySuISD7ONwJ/\nc87tc84dBpYB1xLZx/mYgo5rif1ci/RC2AJcEXxHQgXyXoxa7nGmYhe8dj4XyHXOTc63aTnQLfjn\nbsCrpZ2tpDjnhjrnqjnnapJ3XNc45zoB7wDtgssibZ/3Av8ws1rBu24AAkTwcSbvUlE9Mzsr+Pf8\n2D5H7HHOp6DjuhzoGny3UT1g/7FLS6cr4j+YZmY3k/cvx3LAPOfcox5HKnZm1hBYB2Txf9fTh5H3\nOsJLwCXk/Y/V3jn32xeuwp6ZXQ886JxrZWaXkXfGcB6QCXR2zv3sZb7iZGbJ5L2IXgHYAfQg7x92\nEXuczWwU0IG8d9NlAj3Ju2YeMcfZzBYB15M31fQr4GHgFY5zXIPFOIO8dyUdAHo459KLJUekF4KI\niBRNpF8yEhGRIlIhiIgIoEIQEZEgFYKIiAAqBBERCVIhiIgIoEIQEZEgFYKIiADwv/xQsBD2okms\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11946ae10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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fH3dlFaOgEBGpZHvuCfffH57k3rgROnWCSy6B776Lu7LyUVCIiFSRX/0qNBkc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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x119407438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(nu.size):\n",
    "    Learning_2(n,data,y,nu[i],stepMax,epsilon,color[i],legende[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement the k-nearest neighbor algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.2360679774997898"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from scipy.spatial import distance\n",
    "u = np.array([1,2])\n",
    "v = np.array([0,0])\n",
    "distance.euclidean(u,v) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([163, 158,  81,  80,  56,  58, 132,   0, 103, 148,  39, 176, 171,\n",
       "       184,  98])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Gives n x k array of the k nearest training data points for every test data point x1,...,xn\n",
    "def kNearest_array(k, x_training, x_test):\n",
    "    dist_mat = distance.cdist(x_test, x_training, metric = 'euclidean')\n",
    "    output = np.zeros((len(x_test), k))\n",
    "    for j in range(len(x_test)):\n",
    "        output[j] = (np.argpartition(dist_mat[j], k)[:k]) # Partitions array to take the k smallest entries without fully sorting\n",
    "    return output.astype(int)\n",
    "kNearest_array(15, x, x)[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def kNearest_classify(k, x_training, y_training, x_test):\n",
    "    nearNeigh_y = y_training[kNearest_array(k, x_training, x_test)] #Takes the labels of the k nearest points\n",
    "    nearNeigh_val = (nearNeigh_y / k).sum(axis = 1) #Computes the sum for the algorithm\n",
    "    nearNeigh_class = (nearNeigh_val > 0.5) #Separates by labels\n",
    "    return nearNeigh_class.astype(int)\n",
    "print(kNearest_classify(5, x, y, x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#This function is used later to generate the contour plots, it's functionally analogous to kNearest_classify, just different output\n",
    "def kN_for_contour(k, x_training, y_training, x_test):\n",
    "    nearNeigh_y = y_training[kNearest_array(k, x_training, x_test)]\n",
    "    nearNeigh_val = (nearNeigh_y / k).sum(axis = 1)\n",
    "    return nearNeigh_val"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# use the same data as test- and training data\n",
    "def confusionmatrix_nearest_neighbor(x_training, y_training, y_test):\n",
    "    class0_is = (y_training == 0)\n",
    "    class1_is = (y_training == 1)\n",
    "    class0_assigned = (y_test == 0)\n",
    "    class1_assigned = (y_test == 1)\n",
    "    c00 = np.count_nonzero(np.logical_and(class0_is, class0_assigned))\n",
    "    c01 = np.count_nonzero(np.logical_and(class1_is, class0_assigned))\n",
    "    c10 = np.count_nonzero(np.logical_and(class0_is, class1_assigned))\n",
    "    c11 = np.count_nonzero(np.logical_and(class1_is, class1_assigned))\n",
    "    return np.array([[c00, c01], [c10, c11]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# x and y are the data from task 1.1\n",
    "kNearest_classify(15,x,y,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Here, the confusion matrices are computed for k = 1, 15, 30 to give the accuracy of each\n",
    "\n",
    "y_test_1 = kNearest_classify(1, x, y, x)\n",
    "y_test_15 = kNearest_classify(15, x, y, x)\n",
    "y_test_30 = kNearest_classify(30, x, y, x)\n",
    "con_nn_1 = confusionmatrix_nearest_neighbor(x, y,y_test_1)\n",
    "con_nn_15 = confusionmatrix_nearest_neighbor(x, y,y_test_15)\n",
    "con_nn_30 = confusionmatrix_nearest_neighbor(x, y,y_test_30)\n",
    "acc_nn_1 = acc(con_nn_1,x)\n",
    "acc_nn_15 = acc(con_nn_15,x)\n",
    "acc_nn_30 = acc(con_nn_30,x)\n",
    "print(con_nn_1, con_nn_15, con_nn_30, acc_nn_1, acc_nn_15, acc_nn_30)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# x and y are from task 1.1\n",
    "def f_1(z):\n",
    "    return kN_for_contour(1, x, y, z)\n",
    "\n",
    "def f_15(z):\n",
    "    return kN_for_contour(15, x, y, z)\n",
    "\n",
    "def f_30(z):\n",
    "    return kN_for_contour(30, x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "#plot for k = 1\n",
    "plt.scatter(x[:99,0], x[:99,1],c='orange', label = 'class 0')\n",
    "plt.scatter(x[100:199,0], x[100:199,1],c='blue', label = 'class 1')\n",
    "PlotContourLine(f_1,-4,4,-4,4, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 = 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot for k = 15\n",
    "plt.scatter(x[:99,0], x[:99,1],c='orange', label = 'class 0')\n",
    "plt.scatter(x[100:199,0], x[100:199,1],c='blue', label = 'class 1')\n",
    "PlotContourLine(f_15,-4,4,-4,4, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 = 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot for k = 30\n",
    "plt.scatter(x[:99,0], x[:99,1],c='orange', label = 'class 0')\n",
    "plt.scatter(x[100:199,0], x[100:199,1],c='blue', label = 'class 1')\n",
    "PlotContourLine(f_30,-4,4,-4,4, value = 0.5)\n",
    "plt.title(r'Hyperplane H: $\\alpha_0 + \\alpha_1 \\cdot x_1 + \\alpha_2 \\cdot x_2 = 0.5 $')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Task 1.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "accuracies = np.zeros(200)\n",
    "for k in range(1, 200):\n",
    "    y_assigned = kNearest_classify(k, x, y, x) # gives assigned classes for the test data array x (from task 1.1)\n",
    "    con= confusionmatrix_nearest_neighbor(x, y,y_assigned)\n",
    "    accuracies[k - 1] = acc(con,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(accuracies)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the data from task 1.4 were x_new and y_new:\n",
    "accuracies_new = np.zeros(200)\n",
    "for k in range(1, 200):\n",
    "    y_assigned_new = kNearest_classify(k, x_new, y_new, x_new) # gives assigned classes for the test data array x (from task 1.1)\n",
    "    con_new = confusionmatrix_nearest_neighbor(x_new, y_new,y_assigned_new)\n",
    "    accuracies_new[k-1] = acc(con_new,x_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "plt.plot(accuracies)\n",
    "plt.plot(accuracies_new)"
   ]
  }
 ],
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