dm_lab_2_fm.ipynb

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      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
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      "file_extension": ".py",
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      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.5"
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    "colab": {
      "name": "dm_lab-2_fm.ipynb",
      "provenance": [],
      "include_colab_link": true
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/foxtrotmike/CS909/blob/master/dm_lab_2_fm.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "H5YY8bbnZI9T",
        "colab_type": "text"
      },
      "source": [
        "# Exercise-2\n",
        "## by Fayyaz Minhas\n",
        "## Understanding how intuition fails in high dimensions\n",
        "We have seen how a nearest neighbor classifier works: by labeling a test examples based on the label of its nearest training example. One of the questions we are interested in is \"Why do we need multiple different types of classifiers if we have nearest neighbor classifier?\". In order to answer that question, we need to understand how distance, especially Euclidean distance which is typically used in the nearest neighbor classifier, works and how it is affected by the \"Curse of Dimensionality\". To do that, we are going to be doing a small computational experiemnt in the lab session: we will generate random data in a number of dimensions and try to explore how the number of data points in a fixed number of dimensions affects the notion of \"nearest\". To do that, let's do a thought experiment first: Assume you had an infinite number of points in $d$ dimensions and were interested in finding the closest and farthest points from an arbitrary point, say origin. Since the Euclidean distance between two points $p$ and $q$ is given by $d(p,q)=||p-q||=\\sqrt{(p_1-q_1)^2+(p_2-q_2)^2+...+(p_d-q_d)^2}$ and because we, in our thought experiment, have an infinite amount of points, therefore the closest point to the origin will be at a distance 0 from the origin whereas the farthest point will be $(1,1,1...,1)$ and its distance from origin will be $\\sqrt{d}$. Thus the expected difference between the distance of the farthest and closest points to the origin will be $\\sqrt(d)$. The code below generates a number $n$ of points in different number of dimensions $d$ from 1 to 10 and plots the actual difference between the closest and farthest points from the origin (henceforth called contrast!) together with the expected contrast. Please:\n",
        "\n",
        "1. Try to understand how the code works\n",
        "2. Change the number of points $n$ from 2, 4, 8, 16, 32, 64 and so on and see how the actual contrast differs from the expected contrast.\n",
        "3. Note down your observations about this and try to figure out what's going on\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "hbKLFojHZI9U",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "outputId": "51aecc53-7c46-4330-945d-1c35e9529d58"
      },
      "source": [
        "from numpy.random import rand\n",
        "import matplotlib.pyplot as plt #importing plotting module\n",
        "import numpy as np\n",
        "\n",
        "n = int(4) # to do: change this number and re-run and think about what's going on\n",
        "p = 2 \n",
        "D = np.arange(1,10,1)\n",
        "va = []\n",
        "for d in D:\n",
        "    Xp = rand(n,d)\n",
        "    Xnorm = np.linalg.norm(Xp,axis = 1, ord = p)\n",
        "    M = np.max(Xnorm)\n",
        "    m = np.min(Xnorm)\n",
        "    va.append((M,m,M-m))\n",
        "    \n",
        "plt.plot(D,va,'o-');plt.plot(D,D**(1.0/p));plt.grid()\n",
        "plt.legend(['Max','Min','Difference','Expected'])\n",
        "plt.xlabel('dimnesions')\n",
        "plt.ylabel('Value')\n",
        "plt.title('n = '+str(n)+' p = '+str(p))\n",
        "plt.show()"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
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euLGto1OUcsVqiUBKGQ1EZy/rhBAngBpA3kTwMDBbaiOb7RFC+Akhqmd/VlFu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\ngg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "U0Bk08qUZI9X",
        "colab_type": "text"
      },
      "source": [
        "You should be able to clearly see that, unless the number of points is very large, in higher dimensions, the actual contrast does not change as fast as the expected contrast. You should also see that with very high number of dimensions, the actual contrast actually flat lines. What this means is that the farthest point from the origin is actually as far away as the closest point! Thus, the notion of nearest point disappears in higher dimensions unless you have a very large number of points. This is a known effect of the curse of dimensionality which limits the applicability of nearest neighbor and other distance or similarity based prediction models. We need better models to esacape the curse of dimensionality in real life where we do not have access to infinite number of points and the number of features can be very large!"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "U1YPXeG3ZI9X",
        "colab_type": "text"
      },
      "source": [
        "# Understanding gradient Descent\n",
        "\n",
        "We discussed how gradient descent works for solving optimization problems: By taking steps in a direction opposite to the gradient. Below is a very simple implementation for this method. Based on this, can you:\n",
        "1. Try to understand how Gradient Descent Works\n",
        "2. Can you find the  minima of the following function: $f(x)=2x^2-x$. You will need to update the implementation of the function and its derivative in the code for this.\n",
        "3. What happens when you use: $f(x)=sin(3x)$? Is this function convex?\n",
        "4. What do you think, are the limitations of gradient descent for convex functions?\n",
        "5. What happens when you change the learning rate parameter (step size)?\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "6vNkYFhYZI9Y",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 295
        },
        "outputId": "786f8f59-1e93-4fc1-de83-a5f3c1f19441"
      },
      "source": [
        "import numpy as np\n",
        "\n",
        "def gd(f,df,x0=0.0,lr = 0.01,eps=1e-4,nmax=1000, history = True):\n",
        "    \"\"\"\n",
        "    Implementation of a gradient descent solver.\n",
        "        f: function, f(x) returns value of f(x) for a given x\n",
        "        df: gradient function df(x) returns the gradient at x\n",
        "        x0: initial position [Default 0.0]\n",
        "        lr: learning rate [0.001]\n",
        "        eps: min step size threshold [1e-4]\n",
        "        nmax: maximum number of iters [1000]\n",
        "        history: whether to store history of x or not [True]\n",
        "    Returns:\n",
        "        x: argmin_x f(x)\n",
        "        converged: True if the final step size is less than eps else false\n",
        "        H: history\n",
        "    \"\"\"\n",
        "    H = []\n",
        "    x = x0\n",
        "    if history:\n",
        "        H = [[x,f(x)]]\n",
        "    for i in range(nmax):\n",
        "        dx = -lr*df(x) #gradient step\n",
        "        if np.linalg.norm(dx)<eps: # if the step taken is too small, we have converged\n",
        "            break\n",
        "        if history:\n",
        "            H.append([x+dx,f(x+dx)])\n",
        "        x = x+dx #gradient update\n",
        "    converged = np.linalg.norm(dx)<eps        \n",
        "    return x,converged,np.array(H)\n",
        "    \n",
        "if __name__=='__main__':\n",
        "    import matplotlib.pyplot as plt\n",
        "    def f(x):\n",
        "        y = x**2\n",
        "        return y\n",
        "    def df(x):\n",
        "        dy = 2*x\n",
        "        return dy\n",
        "    \n",
        "\n",
        "    z = np.linspace(-3,3,100)\n",
        "    #select random initial point in the range\n",
        "    x0 = np.min(z)+(np.max(z)-np.min(z))*np.random.rand()\n",
        "    \n",
        "    x,c,H = gd(f,df,x0=x0,lr = 0.01,eps=1e-4,nmax=1000, history = True) \n",
        "    \n",
        "    plt.plot(z,f(z)); plt.plot(z,df(z));\n",
        "    plt.legend(['f(x)','df(x)'])\n",
        "    plt.xlabel('x');plt.ylabel('value')\n",
        "    s = 'Convergence in '+str(len(H))+' steps'\n",
        "    if not c:\n",
        "        s = 'No '+s\n",
        "    plt.title(s)\n",
        "    plt.plot(H[0,0],H[0,1],'ko',markersize=10)\n",
        "    plt.plot(H[:,0],H[:,1],'r.-')\n",
        "    plt.plot(H[-1,0],H[-1,1],'k*',markersize=10)    \n",
        "    plt.grid(); plt.show()"
      ],
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Gl5s0UhQk+XQdkL+PBwtHtWTc9zt467f9xJ67xsS+9XF3kR5CIg8sFtj3E0S8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\nAAA0JQAATQkAQFOfuP4MfgAAAARJREFUmzfe950k/ZkAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "BqbYOdxOZI9a",
        "colab_type": "text"
      },
      "source": [
        "## Working with Ordinary Least Squares\n",
        "\n",
        "We have discussed how OLS works: Given some training data it finds a weight vector and bias by minimizing the sum of squares error between target and predicted output of each example. In this lab session, we explore how OLS and nearest neighbor classifier can be used to solve a practical problem. But before that, we test it on a toy problem. Below is the code for plotting (you can safely ignore it!). Please see the cell after the plotting function for the toy example in which we use Sk-learn OLS [(help availabel here)](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html)."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "MKSO-xbNZI9a",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        "from numpy.random import randn #importing randn\n",
        "import numpy as np #importing numpy\n",
        "import matplotlib.pyplot as plt #importing plotting module\n",
        "import itertools\n",
        "import warnings\n",
        "\n",
        "def plotit(X,Y=None,clf=None,  conts = None, ccolors = ('b','k','r'), colors = ('c','y'), markers = ('s','o'), hold = False, transform = None,extent = None,**kwargs):\n",
        "    \"\"\"\n",
        "    A function for showing data scatter plot and classification boundary\n",
        "    of a classifier for 2D data\n",
        "        X: nxd  matrix of data points\n",
        "        Y: (optional) n vector of class labels\n",
        "        clf: (optional) classification/discriminant function handle\n",
        "        conts: (optional) contours (if None, contours are drawn for each class boundary)\n",
        "        ccolors: (optional) colors for contours   \n",
        "        colors: (optional) colors for each class (sorted wrt class id)\n",
        "            can be 'scaled' or 'random' or a list/tuple of color ids\n",
        "        markers: (optional) markers for each class (sorted wrt class id)\n",
        "        hold: Whether to hold the plot or not for overlay (default: False).\n",
        "        transform: (optional) a function handle for transforming data before passing to clf\n",
        "        kwargs: any keyword arguments to be passed to clf (if any)        \n",
        "    \"\"\"\n",
        "    if clf is not None and X.shape[1]!=2:\n",
        "        warnings.warn(\"Data Dimensionality is not 2. Unable to plot.\")\n",
        "        return\n",
        "    if markers is None:\n",
        "        markers = ('.',)\n",
        "    eps=1e-6\n",
        "    d0,d1 = (0,1)\n",
        "    if extent is None:\n",
        "        minx, maxx = np.min(X[:,d0])-0.5, np.max(X[:,d0])+0.5\n",
        "        miny, maxy = np.min(X[:,d1])-0.5, np.max(X[:,d1])+0.5\n",
        "        extent = [minx,maxx,miny,maxy]\n",
        "    else:\n",
        "        [minx,maxx,miny,maxy] = extent\n",
        "    if Y is not None:\n",
        "        classes = sorted(set(Y))\n",
        "        if conts is None or len(conts)<2:\n",
        "            #conts = list(classes)\n",
        "            vmin,vmax = classes[0]-eps,classes[-1]+eps\n",
        "        else:            \n",
        "            vmin,vmax= np.min(conts)-eps,np.max(conts)+eps\n",
        "        \n",
        "    else:\n",
        "        vmin,vmax=-2-eps,2+eps\n",
        "        if conts is None or len(conts)<2:            \n",
        "            conts = sorted([-1+eps,0,1-eps])\n",
        "        else:\n",
        "            vmin,vmax= np.min(conts)-eps,np.max(conts)+eps\n",
        "        \n",
        "    if clf is not None:\n",
        "        npts = 150\n",
        "        x = np.linspace(minx,maxx,npts)\n",
        "        y = np.linspace(miny,maxy,npts)\n",
        "        t = np.array(list(itertools.product(x,y)))\n",
        "        if transform is not None:\n",
        "            t = transform(t)\n",
        "        z = clf(t,**kwargs)\n",
        "        \n",
        "        z = np.reshape(z,(npts,npts)).T        \n",
        "        \n",
        "        \n",
        "        plt.contour(x,y,z,conts,linewidths = [2],colors=ccolors,extent=extent, label='f(x)=0')\n",
        "        #plt.imshow(np.flipud(z), extent = extent, cmap=plt.cm.Purples, vmin = -2, vmax = +2); plt.colorbar()\n",
        "        plt.pcolormesh(x, y, z,cmap=plt.cm.Purples,vmin=vmin,vmax=vmax);plt.colorbar()\n",
        "        plt.axis(extent)\n",
        "    \n",
        "    if Y is not None:        \n",
        "        for i,y in enumerate(classes):\n",
        "            \n",
        "            if colors is None or colors=='scaled':\n",
        "                cc = np.array([[i,i,i]])/float(len(classes))\n",
        "            elif colors =='random':\n",
        "                cc = np.array([[np.random.rand(),np.random.rand(),np.random.rand()]])\n",
        "            else:\n",
        "                cc = colors[i%len(colors)]\n",
        "            \n",
        "            mm = markers[i%len(markers)]\n",
        "            plt.scatter(X[Y==y,d0],X[Y==y,d1], marker = mm,c = cc, s = 50)     \n",
        "         \n",
        "    else:\n",
        "        plt.scatter(X[:,d0],X[:,d1],marker = markers[0], c = 'k', s = 5)\n",
        "    plt.xlabel('$x_1$')\n",
        "    plt.ylabel('$x_2$')   \n",
        "    if not hold:\n",
        "        plt.grid()        \n",
        "        plt.show()\n",
        "    return extent\n",
        "\n"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ezWHU1ccZI9c",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 318
        },
        "outputId": "2c8efec7-bdf9-4b01-d717-ec544062e616"
      },
      "source": [
        "import numpy as np\n",
        "X = np.array([[-1,0],[0,1],[4,4],[2,3]])\n",
        "y =np.array([-1,-1,+1,+1])\n",
        "from sklearn.linear_model import LinearRegression as OLS\n",
        "clf = OLS()\n",
        "clf = clf.fit(X, y)\n",
        "plotit(X,y,clf=clf.predict,conts=[0.0])\n",
        "print(\"Weights: \",clf.coef_)"
      ],
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:63: UserWarning: The following kwargs were not used by contour: 'label'\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "display_data",
          "data": {
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+MP5zMSGaQv4wAOI4OWWh4ZcI/GFU4a0D8/jGty7mhReWpEy/Lr30DJvaj/Kh\nD50mkaAof5jcNMf0Zxq1P4xr3BW+Gakkf5hCYZga8IepJibwRs0yrn1AVlnGcwT2ARl9lbrIGrLb\nNO8iaQT2AQCjowl27lrM01taOPwL9+7UyeQYN914jLs39bJy5XlPvLN2t4akN4YJuvuc7jNq+wAR\n7/Z3ZdoHpNZEc2Lpk28fUG3i+r5M4I1pw4kT9XRva2HHzmbOnnVNv+bNG2LTpne4/ZZzzJtnpl/T\nEonvKqsJ/BRm/Nl5WApkbqilXH+YdJ0ismgK+MPkG085/jCq8M67c9jS1coLLy5ibMw9d9GFZ+lo\nP8otN59i1twjjAwtxd1tmp6Bl+IPk5OpU0F/GBD37khl+sNIgTBMLfjDVJO4vj0T+ClGYfH2QzXj\n1ykcNsn8EhjPPiDYjggkGsIFvmA7wVz3osQyfxgmkRCGh4Xde5t4eksL7/48bfp145rj3LWxlw9e\ndj51Ewp/0dJtt0B8PbCQ65eHjTF4XAn7ABFIhnwplGofUGgnatwFPYi75BDP92sCb8SK06fr2P5M\nC1u3N3P6tGv6NXv2MOvW9tK2oZeFC4c9UbdQjJEmpvpuAl/rROIPE3Z96Aw+MwxTrD8MkDLwCoZo\nCtr9hvjDQGYYJjVT9j+DAv4whw7PprOrhef3NDE66patWH6O9rYebr/tBPX1mmH6lZkRI6n/Oibq\nD5O3TlgK5AR2l2Z4v2SNb6L+MNnn4jqLLYa4vncT+BokZR8ggXh2GfYBMH7sPPP6kDh9gZCK/5zw\nBL7OG3OY02NO2KaAaBa6/V0iIYyOwk9fWsTTnS0ceDtt+nXD9SfZ2NHD1Vef87bIZ4Zasu0DRkYI\nWBUEBL4E+4BCYZio7QMQ93Mo3z4gnqJWMkIq6yhumMAbU46zZ5Ps3NVM19YWTpxwTb9mzBhh7W2u\n93pr6wiAhWGMorAYvFFR8vnDiEAy+/Z3E/CHyW47p68y/WFS7SQTIGN5F1mdZMimn6xZcVgYxp+5\nvn90Ft3bWtm1q4mhYXfG3dJ8nva2Hu5Ye5zGxjFv52jmbHw8f5jRUaG+PvM/pVL9YbIXWyvqD+OV\nx9ofpsrE9WMxgZ8EirUPEEdSt5Irxz4guyz7OSzbJadOAfuAYDviDOemSWZkuISHMpyQEE0i4TA2\nBq/uX8CWzlb2vz4v9Rlec/UpNrb3sGpVP47jj9mZkH2ADIjn3z5O7LyAfUBQ0IPvuRL2AYi3oSvG\n9gHVJq6fkwm8UZOcP5/gud3NdHa30tfXCLimX7ff2kdbWw8rlg8BaWE1jHKIqb6bwFeLCfnDSOD2\nd2X4w+Q75z+X6w+TbieByBYC5iEAACAASURBVAjJ7NvfhcyKwxZQAfqOzaR7WwvP7FzC4KDbTlPT\nIG3rj7LujmPMnatu/bxhmNL9YUSgPtsPvkR/mMLeL8XUwf2si/CHEe89xN0fpmpINLfsq0WKFngR\naQc+DnxVVfeJyAOqurmI61YA3waaAQU2q+pXJjrgqUIU/jDiQLIxcwNOUNSL3XxUsK/QL4HS/GH8\n9t2wAdT7Ap8liPk2/ajCm2/N5+nOVl55dX7K9Ovyy07T0d7Dh1afToVhEoncDUluO4HwS4n+MO5G\np6wvihL9YQrF0KP2h0EEJxCiMVEvD/cLM6K2RDYBXwESwNdV9ZGs8w24engDcAz4FVV9xzv3OeBT\nwCjw71X1qXLHU8oM/pPA7wB/LCILgVVFXjcC/KGqviAic4C9IvKTKG4oa0xtBgcd9uxdQufWVt5/\nP236dfNNfbS3HeXClQOAhWGMyhPFDF5EEsBXgXbgELBbRJ7I0rpPASdU9WIRuRf4c+BXROQK4F7g\nSmAp8LSIXKqqo+WMqRSBP6OqJ4EHReQR4EPFXKSqR4Aj3uszIrIfWAbEUuDHn50X7w/j3v7OzynP\nbW9COe6hdcrzh/GfU7PhrM8gO0Rz7HgD27a3sP2ZZs6dc9/f/PlDnvd6HwsWjHnvK3PzUaX8YUTC\n8uBLy03Pt0haEX8YMmf9RvlE9FGuAQ6o6kG3TXkcuIdMrbsH+FPv9Q+AvxH3B3kP8LiqDgI/E5ED\nXnvPljOgUgT+R/4LVX1IRP73UjsTkZXAdcBzWeUPAA8ALJi3iPv+5LJSmy6Khcsbue/RCrQd+O1I\nvcz6lztdIWS24P9bnvr/HARh7uIkG/9dS2idjCayy0quk+7XLVPvxZh3PJY6mb7efwqOGerqR1m+\n8kzGORFQhf37F/LjH6/k+edbGEt5r5/gw3e/w003v08yqe41qXGE9OEVjrlDQ72hjox6fQ3lfubZ\nn3fwPYvA8PAAhw+/kvWZSW47IS+yy3LrZNTKOVeo7ZzzXlH/2X52PLM9t5Eapr+/n+07tk32MMIJ\n+5vMT5OI7Akcbw6EqpcB7wXOHQJuzLo+VUdVR0TkFLDIK9+Vde2yYgeVj3EFXkS+Avy+qv4wWK6q\nf11KRyIyG/gHr63TWW1tBjYDzJGl+vcPvl5K00Vz36OXUYm2C8/O/Vj8+HWyZ94dDzTz9GO9WefS\ns/ZSYvBhu0rT7eT5b6NIAzD/2EkIS5ef5uiR+d45YXhEePHFxXR2t/Leodle+Rg3rTlG+4ajXHLJ\neQBGBpsZGfTj7JkzcAmLr0doAPbeey+zcuU1Ge2WagCWkx5ZQQOw7Tu2cftta5lK1PyYi5/B96nq\n6gqOJFKKmcGfAZ4QkV9R1XMishH4gqreWmwnIlKHK+7fUdV/nOBYa4aq+cMEFlnL8YcJllXCHyZV\nxxOquqTDqVN1bH+mlW3bmznT73qvz5kzzJ3retlwZy9NTaPeeJKp691np2AYphL+MMEQzUT9YcKE\nPbuO+cPUJm7IK5J1nsPAisDxcq8srM4hEUkC83AXW4u5tmTGFXhV/WMRuQ/YKiJDQD/wULEdePGl\nbwD7VfUvJzzSSSblD8MExXsC/jDBGHw5/jDBskr4w/h1EwnhwNvzeeKfL2H33kUp068PfOAcG9uP\ncsvNJ5kx0xfC/LtN86Y5VsgfRiTwBTZBf5igoAfrmD/M1CCij303cImIXIgrzvcC92XVeQK4Hze2\n/jGgU1VVRJ4A/l5E/hJ3kfUS4PlyB1RMiGYD8FvAWaAV+KSqvlFCH7cCvw68LCL7vLL/qKpPljrY\napLPPsB9nXWuQvYB7k7WPLe/K8U+YJy+yrEP8Lf6v7hvEVu6Wnj74Bz3sxDlQ6tPsLG9hyuucE2/\ngjnqYfYB/nO2sFfDPiAZCM34zyXZB4TlrZMt8FnnTNRrg4jy4L2Y+meAp3DTJB9T1VdF5GFgj6o+\ngTvZ/R/eIupx3C8BvHrfx12QHQF+t9wMGiguRPN54E9UdYeIXA18T0T+QFU7i+lAVXdQSoTLmDKc\nOZNk565Wtm5v5uRJ1/Rr1qxh1t7ex4b1PbQ0m+mXUfsIkc3g8SauT2aVfSHwegD4t3mu/TLw5WhG\n4lJMiGZ94PXLInIXbjz9ligHMtkU6w8TLKu0P4wIaZ/ysDpF+sOE1inDH+bQ4Zl0bW3lueebGB52\n6yxtPc/Gjh7a2l/HYYk35qw0xywPmOC54OYkJ2cGHh47d9v0/7MIWy9Iz9iD7znbH0Ykc3HVfy7F\nHyY4YwfMH2aKEdefSclWBap6xAvbTHkmZB9AQDgqbB8Q9FbPDsOUYh/gji08pFGMfYAfs/7pSwvY\n0tXK62/MTX2G117jeq9fe00/yTqHuoYxxkby7zYtZB/g910odl4J+wARIZn1xVeqfUBY3rr/bDtN\nax8T+ACqej7qgRi1yblzCXY930L31hb6jrmmXw0No6y9rY+2DUdZutQ1/bIwjDFlEWL7JTztzMZE\npCx/mOyyzLoR+8OIkEgtsk7cHwbyZ5Tk84d5/2gj3VuX8uxzi1OmX4sXD9DR1sPa248xZ45v+pUO\nw/i7b+tCbn/nPxfyh/GfsxdJK+0PI7g3sYbAf3Ul+sNkz/LjKhhxREj/bOPGtBB4X/gQIVEfdou6\n4u0DgnUqbR/g3t/UzSEvxz4gWFYoJ1xEeG3/fDq7W3n5lfmpz++Ky09z16Zerlt1irp6BxBSpl/B\nTJukw+hoyLb/QPijkH2Af+zkfOEEzlXCPiAQg5+wfUBWDN6YWsT1xzYtBN4ozMCgw67nltC9rSVl\n+lVXN8bNNx2jo+0oF1wwkBJAw4gdEaVJ1iKxFfiw2bk7I06WZR+QXeY+p2ftpYRoCtsHuFk0yYas\nOhOwD3Bf54Zh+o41sG17K9ufWcL58+6vwoIFg7Rt6GXdHb3Mn5cOw+Rmr4QvoA4MCPX+5qyQGXgh\n+4BUOyGLq357lbAPEMI2LJVmH2BMbeL6Y4yNwBdjHyACifpEefYBZAo6VM4+QJwRkjPqst5PrliO\nZx/gvk6L5JtvzaWzu5V9P12Q8l6/5OJ+7trUw+obTlLf4Idh0uI9noujH34ZHExnpIRtnCpkH5B+\nX7kZLf5xRewDJP2ZmX3A9CSuP9PYCLxRmKEhhz17m+ja1sqhQ7MASCTGuHHNMTrae7jownMWhjGm\nJUJ8F8WntMCX6g/jb/3PrluKP0xmXyELsSEz9ux2ivaHkdHULfsm6g9z+kwD3Vub2b6jmf6z7n8D\nc+cMs359L+vv7GHRwjFvPMkQIy9/1l7YH8b/LBMJ9zZyuXd0yqyTeS6zjgTyzycyg5+IP4wExmP+\nMNMQIbYxmikl8OX6w+CHaCYaO8+3CSojlTJb4MPEuzh/GHEkLZYl+sMc/Nlsurpb2fviwpTp1wUX\nnGVTRw833XiCxkb/vSfK9ofxx+d48Wz/hh9hu00L+cP476FQnL0i/jDBmLyJ+rQkrj/vKSXwRmFG\nRlzTr66tS3nnXdf0y3GU1TccY2NHD5dc3J+Kj5s9kGGkiam+17bAR+0PIyIkGxNl+cMEyyrtDyNC\nYDacfwH17Ll6tu9oZuv2Fk6dqgdg1swR1t3RS3t7L02Lhr3ZdDJnkTQKf5jgmB0H6lIhmuxrJHTG\nDpkz8uzNQpX2h5HA52pMR8Q2OlUTEQkV73L9YcQRkvW5WSzFbj7KG5+vkD+MI5IWy5D4+uFfzOLp\nzhae393EyIh77bJl5+ho7+HWm48zc6b/JRQWhskffinVH8Z/L+l7smaOueAu0xLj7BXxh4nn37ZR\nJCJk/j7EiJoTeDeV0ckU3YjsA0Rw8+ALCfx49gEZZRW2DwiZwYPw05cWsqWrlTffmuu9L2XVtSe5\na1MvV115hmSdAzhFpTdGYR+Qaifh3dGpLrOdUmLobgyejDKzDzAqjcXgjUnl7LkEz+xsZuu2Fo4d\nd02/GhtHuf22Xja299LcPGhpjoYxQWKq77Un8MHdpkWlJZbgD5OTJlliiCZsxp5TJ0J/GEeEvmOz\n6Oxq4dnnFjM05PbfvGSA9vYe1t1xnJkz0mmOORuEEk7+mXsg7l+uP4zfjttvwHq3oMlXZgw92743\nWGb+MEalievvSM0JvGsIlqiIfQAi44ZoxrMPCO0rQvsAd5jCa/sX0LX5El56aXHqo7nqylNs7Ohl\n1bWnqav3Uw6LjK/nLOQG4+u5Y4Ti7QPAj4+7IZqw29+l6xB6rmAYxuwDjAoS59+Z2hP4aczAgMOz\nu5bQta2Vnp4ZANTXj3LLzcfY2N7DsmVm+mUYlaAa+i4iC4HvASuBd4CPq+qJrDqrgL8F5gKjwJdV\n9XveuW8CdwCnvOq/qar7KEDNCbxrsJXwZsP5Fzz957AZO4T7w6S8aEJm4EWlQIbM2P0+IHM2XIo/\nzLHjjXRtbWHnziWcH3B/JIsWDXL33Qe57eYhZs8eDYRh0tcW6w/j9h8eWonCH8Z/Ts3gs0I0wZ2k\nhfxh/Dr5PGDMH8aoDOn03QrzELBFVR8RkYe8489m1TkH/IaqviUiS4G9IvKUqp70zv+Rqv6g2A5r\nT+C9OHkl7ANS8f1y7AMCZRO1D0gkBFV48615PN3Zwksvp02/PnjpGTZt7GXNmtPMmHmEkeGlhIdh\nircPyDeOnDoTtA/wnx3H3cma+pILqVPIPsD/GWWHYcw+wKg0Vfp1ugdY573+FtBNlsCr6puB178Q\nkR5gMXCSCVBzAh93XNOvxWzpbuUXv3C915PJMW5a00dHRw8rLzifETs3DKOyBBfvi6BJRPYEjjer\n6uYir21W1SPe6/eB5sLjkjVAPfB2oPjLIvIFYAvwkKoOFmqj5gRexL1NXSX8YUSEZEOyLH8YCA9z\n+H3l84c5dbqRrdta2P7MEs56pl/z5g2x4c4eNmw4xry5I1677ntP7WSt925/V4Y/TMZYK+APk1En\nZJE1GEYp5A/jl4tknbNZu1FhSvjV6lPV1fnbkaeBlpBTnw8eqKqKiBZopxX4H8D9qjrmFX8O94uh\nHtiMO/t/uNBga0/gHajzMl2itg/AEZIzglk0pdsHQG4oI198XRXeeXcOT3e28sKLCxkbc89ddOFZ\nNnb0cMvNp0gmNTfN0dt8NDYmJJNZt7+bgH2A22Z4pkwU9gH+sxsjz8wGyngObGIKsw/wfkQm6EbV\nkYi2M6tqW94+RI6KSKuqHvEEvCdPvbnAj4DPq+quQNv+7H9QRP478OB446k5gY8Dw8PC7j1NbOlu\n5ec/nw1AIjHGmg+5pl8fvPS8V2ZhGMOoCaozl3gCuB94xHv+Yc4wROqBfwK+nb2YGvhyEOCXgVfG\n67D2BN7Pgy8yRFOKP4w4QqIhPA++WH8Yt054iKa/v4Htz7SwdfsSTp92Tb9mzx7mzjt6aWvrZeGC\nYc8aIHNWni9/fWhIqM8K0UzEH8a9LjsMk7k4ml0GmTP48esQuP1d+vN26+AiEroxKfhsdgJG1ZGq\n/d49AnxfRD4FvAt8HEBEVgO/raqf9srWAotE5De96/x0yO+IyGJ3xOwDfnu8DmtO4CWwGSlqfxiR\nsYyNThPyhwk5d+jQbJ7uamX3nkUp7/UVy13Tr9tvO0F9fWYYplh/mOHhtDNjOf4wwesr4Q/jH4vg\n/rFkhWiCom6CbtQaQnVuuq2qx4ANIeV7gE97r/8O+Ls8168vtc+aE/ipwugo7PvpIrZ0t3LwYNr0\n6/rrTtDedpQrr+j34tEWhjGMWieuyz01J/DiBDY6FZNFU4I/jDhDbhZNGf4w/f1Jdu5qpmtrCydO\nNAAwY8YId9zeR3t7D60tI279ZDLj+on4w4hIypmxPH+Y8fPXM8+RU6eQP4z72abrJFIhGrLqpnPc\nbQHVqCXi+vtYewIfEqKJyh9GZJi6+sSE/GHef38mnd2t7HquiaFht+2WlgE62o5yx9oTNDaOeemN\nxcXXg2X5/GFEoL4u01t9Iv4wGWUV8Ifxy/0smtzb32VeYxi1Rlx/NWtO4GuJsTF49bUFbOlq5Y03\n56fKr77qJG0bjnLtNWdwHAvDGMZUJjgpiRu1J/COl0VTAX8YEfdmFOP5w5w/n2DH80vo7G6lr8/1\nXm9oGOW2W4/R3tbDiuVD3niSqev9a6P0hxER7+Yd4+S4FxN+yaoTpT+Mf417TnIWWQ2j1onrr2rN\nCbw4QnJGXUX8YdyYdiJDSP26iYRwtKeR7q0t7Ni5hMFB9wujqWmQ9g093LnuOHPnuRvPHCdfGCZa\nfxgRyQnRTMQfZrw65frDpOtIrK1XjfgS1UanWqP2BF4ksMgarX2AG9N2MmLeqvDGm/PZ0tXCy6/M\nT5l+XX7ZGTZ2HOVDq70wTCC+HhT0zL6jtQ/w/+MAyrMPCDln9gGGkSauv8YVF3gReQz4CNCjqldV\nur9iGRx02L23ma7uFo68nzb9uuXmPto3HOXCC10PnyrZiAJwy/G3OMdY6vjRkQHu+sWrqeNZ4vDS\niiurNh7DmC7EdaJSjRn8N4G/Ab5dTGXX0reuIv4wIsLp0zPYur2V7c8s4dw59+3Pnz9E+4Ze7lzX\nx4IFrsDmpDkmcm9OHbU/TFDcwzirYxlGXsX6w7ivSX0GGc/mD2NMc9yw4mSPojJUXOBVdZuIrCz6\nAoGEF6KJwj4gkXBDMm8dmMO3vnMRzz/fkgrDXPxL/XS0H+WmG0/R0Ohfnx2GyR9+ido+oBiCC6vF\n2gdArnibfYBh+Ehsf8drLgYfJcPDwvO7F9PZ3crP35sFgGv6dZz2tqNccvE5r8zSHA1jOhPX/0xr\nQuBF5AHgAYBFixZz5XWjIKMIw14F/ylzgTDfuePHG/jXf72Anzz9AU6fdnebzp07yMaN79DRfoiF\niwb9ftNNSEjbgQ7GxtwHwMiIV2mQnGuyf1HSh+n28/Xx6PD5jNLlo6M82t+fUbb/9b3Zbz3HCS8j\nIyDP723mtfnGXDr9/f1s37Ft4g1MAlNtzFNtvDAFxhxPfa8NgffuiLIZYNmyD+rBAzNLsg/w6/zs\nndl0dreye/dCRsfcOhd84BybNvZw800nmLvgfYYHW9FRfyu9J7ol2AcExxO1fcCmd36a8bk82t/P\ng7NnZ5T1Xbk6dW0p9gGFwjBR7jTdvmMbt9+2tux2qslUG/NUGy/U+JjFZvA1y+io8MKLC+naupSD\nP5sDuKZfN1x/nI62o1x++dnUTaANwzCyEWyRdcKIyHdxbzTbJCKHgC+q6jfy13f9V8bzhzlzJskz\nz7awdVszJ066YZiZM0e4Y20fbRt6aGlOm35l7mRNlOUP444jc1YelT9MMSRCsmjG84fJdy5YbhjT\nGSemfwfVyKL5RCn1/e352fYB7rNw6NBMOre6pl8jI2750qXn2djRy223HmeWu5aaYfrlCzri3pO1\nHPuA0DoR2QfMdhz6x/KnSs52nLQ3fkn2ASbohlGIuP5dTIkQzdgYvPTyQrq2tvLmW/NS5ddcfZKN\nHT1cdeXpQBhm6oZj3v7gdUB6lv/ivuc4cc0aIL6/gIZRC8T1z6vmBN63E0gkHAYHk+zYuYTurS30\nHXNNvxobRrn99j7aN/SwbNlw6k5J2SGVMH+YkRHfi2bi/jCZ5zLrRO4PI5IxY/f7gNL8YQzDyE/Q\nriNu1KDAC8eOzaJ7Wys7dy1OmX4tXjzAxvZe1t7exxx3LZVEMhEq6BDuDzM66t7jtBx/mGCdSvvD\nSLCPkDi7+cMYRjRU409GRBYC3wNWAu8AH1fVEyH1RoGXvcOfq+pHvfILgceBRcBe4NdVdahQnzUn\n8L29DXzxS9elji+/7BQbO45y7TWnUsZbUzkMYxhG7VGlOdFDwBZVfUREHvKOPxtS77yqrgop/3Pg\nr1T1cRH5b8CngL8t1GHNCXwiodTVuaZfmzb2smL5gDebTubkqpfqDyMDeAu4xfvDBGftYTN2yAyx\nROoPI+kxmj+MYVSOKlkV3IObUQjwLaCbcIHPQdw/8vXAfYHr/5SpJvALFozwuc/+lPnz1BP0sDDM\nxPxhfH/1UvxhCsbOQwQ+Sn8YIf2FY/4whlEZqngPg2ZVPeK9fh9ozlOvUUT2ACPAI6r6z7hhmZOq\nOuLVOQQsG6/DmhP4ZEKZM3sUC8MYhlGDNHni67PZ24kPgIg8DbSEXPf54IGqqohonj4uUNXDInIR\n0CkiLwOnJjLYmhN4Eaj3ctXHu/1dIiREkz0DD1oe+DfQCLMTCJuxZz8XynpJ1fEzW7JDKoHjoJVA\nRp2AnUAqiybrfRmGET0lzOD7VHV1vpOq2lagj6Mi0qqqR0SkFejJ08Zh7/mgiHQD1wH/AMwXkaQ3\ni18OHB5vsDUo8JLabRq1P0z6ln3F+8MUFnhy6kTpDyOYsBtGNajSMtYTwP3AI97zD3PHIQuAc6o6\nKCJNwK3AX3gz/i7gY7iZNKHXZ1N7Au8Idd4MPmr7AH8GX4p9QGYdQs8FRT1S+wDTdsOIE48A3xeR\nTwHvAh8HEJHVwG+r6qeBy4GvicgYbpz6EVV9zbv+s8DjIvJnwItAXssXn5oTeMMwjGpTDS8aVT0G\nbAgp3wN82nu9E7g6z/UHgTWl9FlzAi/ibUaqgD9MKvxTwu5SP3wSNsuXkDrmD2MYU5CY/gnWoMD7\njo/R2weIZO5anZB9AJmC7vfhlpt9gGFMNbx8hlhScwJvGIZRXSTzLmgxouYEXkSoq09UxB9GBJLe\nTbj9c9l1xvOHgdyNSeYPYxhTnJj+edagwENd0qmIfQC4bZZlH5BVFjw2+wDDmJrYDT8MwzDiSjz1\nvfYE3s2iSVTEH0YEkkknpE7x/jDuOTLrmj+MYUxp4vqXW4MC74pwJewD3Bh8+G7Tou0DMEE3jDgh\nxDekao5ehmEYMaUGZ/BeiKYC/jDiLbKW4w8TLDMMIwZYHnz1SMXJK+APg2RvWEoLfdH+MIZhxI64\n/n3XnMAbhmFUm5jqe+0JvIikNiNF7Q8D4CQc84cxDCOA7WStKu4Gpuj9YUQ8zxrzhzEMI0hM//xr\nTuCDuepR2weA+6Vhu00Nw/Bx0yQnexSVoeYE3jAMo9rEVN9rT+BF0n4x0fvDkLp9n2EYBuClScZT\n4mtP4PEdH80+wDCM6hBTfbedrIZhGKm7foz3KKsLWSgiPxGRt7znBSF17hSRfYHHgIj8snfumyLy\ns8C5VeP1WXsCL2kbYD+Txkl4D//YcXA8L/j0Ob/MSYd4bPZuGEYRSJGPMnkI2KKqlwBbvOMMVLVL\nVVep6ipgPXAO+NdAlT/yz6vqvvE6rMEQTTrTxewDDMOoNFXMorkHWOe9/hbQDXy2QP2PAT9W1XMT\n7bD2ZvCGYRhVxk+5Hu9RJs2qesR7/T7QPE79e4HvZpV9WUReEpG/EpGG8TqsuRk8kt5tav4whmHU\nGE0isidwvFlVN/sHIvI00BJy3eeDB6qqIqL5OhGRVuBq4KlA8edwvxjqgc24s/+HCw225gRecHeb\nRmEfsODZZzgzOpo6frS/nzt3bEsdz0kkOHHzrWWP2TCMKUxp66d9qro630lVbcvbjchREWlV1SOe\ngPcU6OfjwD+p6nCgbX/2Pygi/x14cLzBViVEIyKbROQNETkgIjkLC5UiKO4TOW8YxnShKsusTwD3\ne6/vB35YoO4nyArPeF8KiDvT/WXglfE6rLjAi0gC+CpwF3AF8AkRuSL/BaQzZ7xMmIjiX4ZhGKFU\nIUsS4BGgXUTeAtq8Y0RktYh8PT0WWQmsALZmXf8dEXkZeBloAv5svA6rEaJZAxxQ1YMAIvI47mry\na2GVBbHdpoZhxA5VPQZsCCnfA3w6cPwOsCyk3vpS+6yGwC8D3gscHwJuDFYQkQeABwCam5vZHoiT\nl8Oj/f0Zx8tHR3PKouqrEvT399f0+MKwMVeeqTZeqO0xC/GNENTEIqu3Cr0ZYPUNq/X229ZG0u6d\nWb9Qj/b38+Ds2RllIxH1VQm279hGVJ9FtbAxV56pNl6YmmOOA9UQ+MO48SSf5V6ZYRhGTRDTCXxV\nsmh2A5eIyIUiUo+bvP9EFfo1DMOY1lR8Bq+qIyLyGdyE/QTwmKq+Wul+wc1zL5QKOSeRqMYwDMOo\nZWJ8x4+qxOBV9UngyWr0FSR7E9P2HdtqOuZuGMbkEE95r5FFVsMwjMkkphN4E3jDMIy4KrztKDIM\nw4gpNoM3DGPaE8/5uwm8YRhGbBXeBN4wjGmNmyUZT4U3gTcMY9oTT3k3gTcMY7oT0R21axETeMMw\njJgqvAm8YRjTnnjKuwm8YRhGbBXeBN4wjGlOfG/4YTtZDcMwYooJvGEY055q3HRbRP6tiLwqImMi\nsrpAvU0i8oaIHBCRhwLlF4rIc17597z7axTEBN4wDKM6vAL8z0Dem9OKSAL4KnAXcAXwCRG5wjv9\n58BfqerFwAngU+N1aAJvGMa0xt/JWsyjHFR1v6q+MU61NcABVT2oqkPA48A94na+HviBV+9bwC+P\n12fNLbLufWFvX7Iu8W6Fmm8C+irUdiWYauMFG3M1mGrjhcqN+YJyG9j7wt6nknWJpiKrN4rInsDx\nZlXdXO4YAiwD3gscHwJuBBYBJ1V1JFC+bLzGak7gVXVxpdoWkT2qmjf2VWtMtfGCjbkaTLXxQm2P\nWVU3RdWWiDwNtISc+ryq/jCqfoql5gTeMAxjqqKqbWU2cRhYEThe7pUdA+aLSNKbxfvlBbEYvGEY\nRu2wG7jEy5ipB+4FnlBVBbqAj3n17gfG/Y9gugl8lLGyajDVxgs25mow1cYLU3PMkSIi/0ZEDgE3\nAz8Skae88qUi8iSANzv/DPAUsB/4vqq+6jXxWeAPROQAbkz+G+P26X4xGIZhGHFjus3gDcMwpg0m\n8IZhGDFlWgl8sVuFa4F825VrFRF5TER6ROSVyR5LMYjIChHpEpHXvN+J35vsMY2HiDSKyPMi8lNv\nzP/nZI+pGEQkISIvisi/TPZYphvTSuApYqtwLTDOduVa5ZtAZPnEVWAE+ENVvQK4CfjdKfAZDwLr\nVfVaYBWwSURumuQx9NYlPAAAAoFJREFUFcPv4S4YGlVmWgl8kVuFa4HQ7cqTPKaCqOo24Phkj6NY\nVPWIqr7gvT6DK0Dj7gycTNSl3zus8x41nSUhIsuBDwNfn+yxTEemlcBPIcK2K9e0+ExlRGQlcB3w\n3OSOZHy8cMc+oAf4iarW+pj/C/B/AGOTPZDpSOwEXkSeFpFXQh41PQM2JgcRmQ38A/D7qnp6sscz\nHqo6qqqrcHcyrhGRqyZ7TPkQkY8APaq6d7LHMl2JnVVBBFuFa4F825WNCBGROlxx/46q/uNkj6cU\nVPWkiHThrnvU6sL2rcBHReRuoBGYKyJ/p6q/NsnjmjbEbgYfE0K3K0/ymGKFZ7/6DWC/qv7lZI+n\nGERksYjM917PANqB1yd3VPlR1c+p6nJVXYn7O9xp4l5dppXA59sqXGuMs125JhGR7wLPAh8UkUMi\nMu7NCCaZW4FfB9aLyD7vcfdkD2ocWoEuEXkJdxLwE1W11EMjL2ZVYBiGEVOm1QzeMAxjOmECbxiG\nEVNM4A3DMGKKCbxhGEZMMYE3DMOIKSbwhmEYMcUE3qhJPCvfdu/1n4nIX0/2mAxjqhE7qwIjNnwR\neFhEluAagX10ksdjGFMO2+hk1CwishWYDaxT1TMichHweWCeqn6s8NWGYViIxqhJRORq3K35Q55f\nO54/fq1bIBhGzWACb9QcItIKfAf3Jif9IjKV7hRlGDWDCbxRU4jITOAfcW+ntx/4Em483jCMErEY\nvDFlEJFFwJdxbXK/rqr/aZKHZBg1jQm8YRhGTLEQjWEYRkwxgTcMw4gpJvCGYRgxxQTeMAwjppjA\nG4ZhxBQTeMMwjJhiAm8YhhFTTOANwzBiigm8YRhGTPn/AZSYXxmz9olcAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "Weights:  [-0.57142857  1.28571429]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "AVgWNvSHZI9d",
        "colab_type": "text"
      },
      "source": [
        "1. What are your observations about the line that is drawn by this predictor?\n",
        "2. What happens to the line if you change the data?\n",
        "3. Is this line optimal? \n",
        "4. Can you predict the score and class label for a new test point, say [2,0]? Is this point classified correctly?\n",
        "5. You can use clf.coef_ to get the weight vector learned by the OLS method. What do these weights and their signs tell you about your data? Try changing the data and observe how that impacts the weight vector.\n",
        "6. What happens when your data is not linearly separable?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RpKV63D0ZI9e",
        "colab_type": "text"
      },
      "source": [
        "## Trying out Perceptron!\n",
        "\n",
        "Let's try a single Perceptron on the same toy problem. More help on using Perceptron in sk-learn is available [here](https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.Perceptron.html). Here are some questions for you:\n",
        "\n",
        "1. What are your observations about the line that is drawn by this predictor?\n",
        "2. What happens to the line if you change the data?\n",
        "3. Is this line optimal? \n",
        "4. Can you predict the score and class label for a new test point, say [2,0]? Is this point classified correctly?\n",
        "5. You can use clf.coef_ to get the weight vector learned by the OLS method. What do these weights and their signs tell you about your data? Try changing the data and observe how that impacts the weight vector.\n",
        "6. What happens when your data is not linearly separable?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "b1eqqLKMZI9e",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 318
        },
        "outputId": "bf4a0d74-33b7-48ca-f015-a3a15aebf1d0"
      },
      "source": [
        "from sklearn.linear_model import Perceptron\n",
        "clf = Perceptron()\n",
        "clf.fit(X,y)\n",
        "plotit(X,y,clf=clf.decision_function,conts=[0.0])\n",
        "print(\"Weights: \",clf.coef_)"
      ],
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:63: UserWarning: The following kwargs were not used by contour: 'label'\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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nI6IfuBO4SknjS4C70/NuB66u1KaiTl+qP1GSXgT+sU7VzwVeqlPd9TDV4gXH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\n",
            "text/plain": [
              "<Figure size 432x288 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "Weights:  [[2. 0.]]\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "97krRNHtZI9g",
        "colab_type": "text"
      },
      "source": [
        "## Classifying digits\n",
        "Below is a code for classifying between MNIST digits (7 vs 9) using nearest neighbor and OLS classifiers. Please try to understand the code and answer the following questions."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DdYgrb4lZI9g",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "outputId": "1cf32631-68c6-4e11-e8ac-f2566fe0c389"
      },
      "source": [
        "import numpy as np\n",
        "from sklearn.datasets import load_digits\n",
        "import random\n",
        "from sklearn.neighbors import KNeighborsClassifier\n",
        "from sklearn.linear_model import LinearRegression as OLS\n",
        "import matplotlib.pyplot as plt\n",
        "digN,digP = 7,9\n",
        "X,y = load_digits(n_class=10, return_X_y=True)\n",
        "idx = (y==digP) + (y==digN) #Just focus on 7 and 9\n",
        "X,y = X[idx,:], y[idx]\n",
        "y[y==digN]=-1.0 #set (negative class)\n",
        "y[y==digP]=+1.0 #set (positive class)\n",
        "idx = list(range(len(y)))\n",
        "random.shuffle(idx) # Just shuffle the data\n",
        "X,y = X[idx,:], y[idx]\n",
        "plt.gray();plt.matshow(X[0].reshape((8,-1))) # show an example\n",
        "\n",
        "Ntr = 15 #select number of training examples\n",
        "Xtr,Ytr = X[:Ntr], y[:Ntr]\n",
        "Xtt,Ytt = X[Ntr:], y[Ntr:]\n",
        "\n",
        "clf = KNeighborsClassifier(n_neighbors=1)\n",
        "clf = clf.fit(Xtr, Ytr)\n",
        "Z = clf.predict(Xtt)\n",
        "print('Accuracy of kNN:',np.mean(Z==Ytt))\n",
        "plt.gray();plt.matshow(Xtt[0].reshape((8,-1))); plt.title('Predicted:'+str(Z[0]))\n",
        "error_idx = np.nonzero(Ytt!=np.sign(Z))[0][0] # show one error\n",
        "plt.gray();plt.matshow(Xtt[error_idx].reshape((8,-1))); plt.title('Misclassified: Predicted as '+str(Z[error_idx])) # show a misclassification\n",
        "\n",
        "\n",
        "clf = OLS()\n",
        "clf = clf.fit(Xtr, Ytr)\n",
        "Z = clf.predict(Xtt)\n",
        "print('Accuracy of OLS:',np.mean(np.sign(Z)==Ytt))\n",
        "plt.gray();plt.matshow(Xtt[0].reshape((8,-1))); plt.title('Predicted:'+str(Z[0]))\n",
        "error_idx = np.nonzero(Ytt!=np.sign(Z))[0][0] # show one error\n",
        "plt.gray();plt.matshow(Xtt[error_idx].reshape((8,-1))); plt.title('Misclassified: Predicted as '+str(Z[error_idx])) #show a misclassification\n"
      ],
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Accuracy of kNN: 0.9476744186046512\n",
            "Accuracy of OLS: 0.9534883720930233\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Text(0.5, 1.05, 'Misclassified: Predicted as -0.5603215968934636')"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 6
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 0 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPoAAAECCAYAAADXWsr9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nHzsUmIOq1q+tAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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mvatJ0LcCWyPi8fr+PVTBN7MJYtSgR8QLwBZJJ9YPnQNsLNoqM2tV06PunwFuq4+4Pwd8\nqlyTzKxtjYIeEesBv/c2m6A8M84sAQfdLAEH3SwBB90sAQfdLAEH3SwBB90sAQfdLIGmM+NskAUL\nFnRab8WKFZ3Ws8nHPbpZAg66WQIOulkCDrpZAg66WQIOulkCDrpZAg66WQIOulkCDrpZAqMGXdKJ\nktYP+npV0tIuGmdm7Rh1rntE/BiYDyBpKrANuL9wu8ysRWMdup8DPBsRz5dojJmVMdaz1y4G7hju\nCUlLgCUH3CIza13jHr1evOEi4O7hnvfaa2a9ayxD9/OBdRHxYqnGmFkZYwn6JbzLsN3MelujoNfr\noZ8L3Fe2OWZWQtO1114HjizcFjMrxDPjzBJw0M0ScNDNEnDQzRJw0M0ScNDNEnDQzRJw0M0SUES0\nv1NpJ7A/p7IeBbzUcnN6oZbruV5X9eZGxOyhDxYJ+v6StKars9+6rOV6rjfe9Tx0N0vAQTdLoNeC\nfsMkreV6rjeu9XrqPbqZldFrPbqZFeCgmyXgoJsl4KCbJeCgmyXw/7C9n2oaAkVhAAAAAElFTkSu\nQmCC\n",
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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Z1aPKUfcZkqbl6T2B44D7mm6YmdWnylH3/YBLJPWRPhi+GRHXNdssM6tTlaPu/wkc0kJb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\n4KCbFeD/AfSWCP81x1s3AAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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0swI46GYFaOxY9w1pYGCg03qzZ8/utF7X5+6aM2dOp/XOOOOMTut1vT77gbfoZgVw0M0K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\nB92sAA66WQEcdLMC/D+WMxutA0XT+AAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<Figure size 288x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "aPWp7xpDZI9i",
        "colab_type": "text"
      },
      "source": [
        "1. What is the relationship between accuracy over test examples and training set size for both classifiers? How can you design an experiment to do this?\n",
        "2. Which classifier performs better and why?\n",
        "3. Here, we are using each pixel as a feature. Can you extract better features that allow you to do better classification? \n",
        "4. You can plot the weight vector of the OLS as an image by using plt.matshow(clf.coef_.reshape(8,-1). What weights are not important? Why? What are the most important weights? why?"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "B4mgVXMNZI9i",
        "colab_type": "text"
      },
      "source": [
        "## Using Perceptron for Digit Classification\n",
        "\n",
        "Use the codes provided above and compare the predictive performance of a perceptron for this problem."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "gPi_uN1OZI9j",
        "colab_type": "code",
        "colab": {}
      },
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}