linear_regression_playground.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Linear Regression Playground"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Goal: explore the following via intentional practice:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*1.0 Modelling Approaches:*\n",
    "* Linear regression from scratch using Numpy (single variable + Gradient Descent) [**DONE**] [Viz GD + Error Surface]\n",
    "* Linear regression from scratch using Numpy (Multiple variables + Gradient Descent) **[DONE]** \n",
    "* Linear regression from scratch using Numpy (Multiple variables + Polynomials + Gradient Descent) **[DONE]** \n",
    "* Linear regression using the Normal equation (Chesky, QR) [**DONE**]\n",
    "* Linear regression using SKlearn [**DONE**]\n",
    "* Repeat all of the above for (multi-variable, polynomial, learning rates, batch,\n",
    "\n",
    "*2.0 Playground:*\n",
    "* Explore the effect of different learning rates [**DONE** PLAY AROUND IN COURSERA WORKBOOK]\n",
    "* Explore the effect of not scaling features [**DONE** PLAY AROUND IN COURSERA WORKBOOK]\n",
    "* Explore the effect of outliers on model performance\n",
    "* Explore the use of batch, stochastic and min-batch (run times, convergence, error rates) (Numpy + scikit learn) **[DONE]** \n",
    "* Explore the effect of regularization on the model\n",
    "* Memory, runtime and complexity -> Computational linear algebra (increase number of examples)\n",
    "\n",
    "*3.0 Package Library Explorer:*\n",
    "* Do working example with Scikit learn, Tensorflow, Pytorch, Keras\n",
    "* Read and understand their documentation + how modules/classes are packaged\n",
    "* Playaround with features explored above (learning rate, algo, regularization etc.)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1.0 Modelling Approaches (Linear)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1 Prepare Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#import packages\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'retina'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#create artificial dataset for experimentation\n",
    "\n",
    "X = 2* np.random.rand(100,1) #100x1 shape...random numbers from normal distribution\n",
    "\n",
    "y= 4 + 3*(X) + np.random.rand(100,1)\n",
    "\n",
    "#Add one column to x for incercept\n",
    "X_ones = np.c_[np.ones((100,1)), X]\n",
    "m=len(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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ePRzbdGtf6AZXPZlZ1zVX7cyfD8uWpbP1pUvh6adh3brJ0Z20vz8ltkZso92qdCpx1ZOZ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\nFGZmVsiJwszMCjlRmJlZIScKMzMr9P8B1iQGemWSadYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"blue\")\n",
    "plt.scatter(X, y, c=colors,s=15)\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.2 Normal Equations (1 Variable)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy.linalg import inv\n",
    "\n",
    "theta = np.linalg.inv(X_ones.T.dot(X_ones)).dot(X_ones.T).dot(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[4.49121059],\n",
       "       [3.05120183]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Not a bad result....we were hoping for Theta0=4 and Theta1=3. The noise made it impossible to get it bang on"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Now lets make preductions with theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_test=np.array([[0.5],[2],[1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Add one column to x for incercept\n",
    "X_test_ones = np.c_[np.ones((3,1)), X_test]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1. , 0.5],\n",
       "       [1. , 2. ],\n",
       "       [1. , 1. ]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_test_ones"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_predict=X_test_ones.dot(theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6.01681151],\n",
       "       [10.59361426],\n",
       "       [ 7.54241242]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_predict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Lets plot the models predictions in a new color"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bv/IGItJfDOimpLRJ9I4+5i0embgvDNuQ/qI/Xg4LLwW6m3vSmoaUJxCRgWpQ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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"red\")\n",
    "plt.scatter(X_test, y_predict, c=\"red\",s=50)\n",
    "plt.scatter(X, y, c=\"blue\",s=15)\n",
    "plt.plot(X, np.dot(X_ones, theta), label='Linear Reg Normal Eqn', color='green')\n",
    "\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2.5,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#what is the mean square error using the NEqn (good to compare to others)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.04376849]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dt=np.dot(X_ones,theta)-y\n",
    "cost=np.dot(dt.T,dt)/2/m\n",
    "cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Try and calculate the R^2 error based on Sklearn implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.97121345]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u=np.dot(dt.T,dt)\n",
    "y_mean=np.mean(y)\n",
    "dz=(y-y_mean)\n",
    "v=np.dot(dz.T,dz)\n",
    "r_squared=(1-u/v)\n",
    "r_squared\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 Sci-Kit Learn Implementation (1 Variable)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LinearRegression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "reg = LinearRegression()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "460 µs ± 59 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit reg.fit(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.05120183]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.49121059])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.intercept_ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6.01681151],\n",
       "       [10.59361426],\n",
       "       [ 7.54241242]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.predict(X_test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "predicted_line=reg.predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9712134527445793"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.score(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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JUMzN+Bs716WLiT5w2qsTj1MoGGSaOBGWLg39GpYuDa8hd05CKqeczVXvAVYCZ5rZNjO7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\nkRgFBhERifn/0imGt3f+4fkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"red\")\n",
    "plt.scatter(X_test, y_predict, c=\"red\",s=50)\n",
    "plt.scatter(X, y, c=\"blue\",s=15)\n",
    "plt.plot(X, np.dot(X_ones, theta), label='Linear Reg Normal Eqn', color='green')\n",
    "plt.plot(X, predicted_line , label='Linear Reg SkLearn', color='cyan')\n",
    "\n",
    "\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2.5,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.3 Sci-Kit Learn Implementation (1 Variable - Stochastic GD)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.linear_model import SGDRegressor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "582 µs ± 35.9 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "sgd_reg=SGDRegressor(max_iter=1000, tol=1e-3, penalty=None)\n",
    "%timeit sgd_reg.fit(X, y.ravel())\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([4.11996711])"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg.intercept_\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([3.39725234])"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 5.81859328, 10.91447179,  7.51721945])"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg.predict(X_test)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "predicted_line=sgd_reg.predict(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.9575394062080488"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sgd_reg.score(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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QTI/3H1piR/vRE1U8XVxsDmezpspl2fLlcO215ctLA2DAIEmSVOeGG+paFn7B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\nrkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 277,
       "width": 390
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"red\")\n",
    "plt.scatter(X_test, y_predict, c=\"red\",s=50)\n",
    "plt.scatter(X, y, c=\"blue\",s=15)\n",
    "plt.plot(X, np.dot(X_ones, theta), label='Linear Reg Normal Eqn', color='green')\n",
    "plt.plot(X, predicted_line , label='Linear Reg SkLearn', color='cyan')\n",
    "\n",
    "\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2.5,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.4 Gradient Descent Implementation (Manual Via Numpy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0],\n",
       "       [0]])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_initial=np.array([[0], [0]])\n",
    "theta_initial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def computeCost(X, y, theta):\n",
    "    dt=np.dot(X,theta)-y\n",
    "    cost=np.dot(dt.T,dt)/2/m\n",
    "    return cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[27.70061813]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "computeCost(X_ones, y, theta_initial)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def gradientDescent(X, y, theta, alpha, num_iters):\n",
    "    #Cost function history\n",
    "    J_history=np.zeros(num_iters)\n",
    "    \n",
    "    #start loop\n",
    "    theta_new=theta.copy()\n",
    "    \n",
    "    for ii in range(num_iters):\n",
    "        #update theta\n",
    "        theta_new -= alpha*np.dot(X.T, (np.dot(X,theta_new)-y))/m\n",
    "        print(theta_new)\n",
    "        #Store cost function results\n",
    "        J_history[ii]=computeCost(X,y,theta_new)\n",
    "        print(J_history[ii])\n",
    "    return theta_new, J_history"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.        , 1.96749131],\n",
       "       [1.        , 1.46772954],\n",
       "       [1.        , 0.26091792],\n",
       "       [1.        , 0.54235331],\n",
       "       [1.        , 0.16730865]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_ones[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0],\n",
       "       [0]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta_initial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 1)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.shape(theta_initial)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The memory_profiler extension is already loaded. To reload it, use:\n",
      "  %reload_ext memory_profiler\n"
     ]
    }
   ],
   "source": [
    "%load_ext memory_profiler"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
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      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "[[4.49121059]\n",
      " [3.05120183]]\n",
      "0.04376849120345776\n",
      "peak memory: 104.11 MiB, increment: 0.05 MiB\n"
     ]
    }
   ],
   "source": [
    "%memit theta_new, J_history = gradientDescent(X_ones, y, theta, 0.01, 1500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6.01681151],\n",
       "       [10.59361426],\n",
       "       [ 7.54241242]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_predict=X_test_ones.dot(theta_new)\n",
    "y_predict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.97121345]])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dt=np.dot(X_ones,theta_new)-y\n",
    "cost=np.dot(dt.T,dt)/2/m\n",
    "\n",
    "u=np.dot(dt.T,dt)\n",
    "y_mean=np.mean(y)\n",
    "dz=(y-y_mean)\n",
    "v=np.dot(dz.T,dz)\n",
    "r_squared=(1-u/v)\n",
    "r_squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#Vizualize the Cost Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.04376849, 0.04376849, 0.04376849, ..., 0.04376849, 0.04376849,\n",
       "       0.04376849])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J_history"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0,0.5,'$J(\\\\theta)$')"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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NGEgkSZIkNWMgkSRJktSMgUSSJElSM9MikETEVhHx4Yi4MSJWRsSyiDg1IjYZZz2b1nLL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\n/wf10b0hDV31sAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 263,
       "width": 402
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "# Plot cost function vs. iteration\n",
    "plt.plot(J_history)\n",
    "plt.xlabel(r'Iteration number')\n",
    "plt.ylabel(r'$J(\\theta)$')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Not bad....let's look at our results\n",
    "\n",
    "**Parameters:**\n",
    "* Normal Eqn: theta0 (4.6), theta1 (2.88)\n",
    "* Sklearn:    theta0 (4.5), theta1 (2.99)\n",
    "* Grad Desc:  theta0 (4.6), theta1 (2.89)\n",
    "\n",
    "**Predictions:**\n",
    "* Normal Eqn: (6.08, 10.4, 7.5)\n",
    "* Sklearn:  (6.03, 10.5, 7.5)  \n",
    "* Grad Desc: (6.08, 10.4, 7.5)\n",
    "\n",
    "**R^2 Errors:**\n",
    "* Normal Eqn: (0.966) -> Cost(0.044)\n",
    "* Sklearn:  (0.967)  \n",
    "* Grad Desc: (0.967) -> Cost(0.0428)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2.0 Modelling Approaches (Polynomial)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 563,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "m=200\n",
    "X = 6 * np.random.rand(m,1) - 3\n",
    "y= 0.5 * X**2 + X + 2 + np.random.randn(m,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 564,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.31697701],\n",
       "       [-1.69177574],\n",
       "       [ 2.32985145]])"
      ]
     },
     "execution_count": 564,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 565,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.90400183],\n",
       "       [1.50789165],\n",
       "       [6.98329867]])"
      ]
     },
     "execution_count": 565,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 566,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x_squared_feature=np.square(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 567,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.10047443],\n",
       "       [2.86210514],\n",
       "       [5.42820777]])"
      ]
     },
     "execution_count": 567,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_squared_feature[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 568,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.        , -0.31697701,  0.10047443],\n",
       "       [ 1.        , -1.69177574,  2.86210514],\n",
       "       [ 1.        ,  2.32985145,  5.42820777]])"
      ]
     },
     "execution_count": 568,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Add one column to x for incercept\n",
    "X_squares = np.c_[X,x_squared_feature]\n",
    "X_ones = np.c_[np.ones((200,1)), X_squares]\n",
    "\n",
    "\n",
    "X_ones[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 569,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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DgZlZzTkQmM2CpC9LeqKKK3RafTgQmM3O75PMlzArLQcCswwkvULS36f3IViU\n3oPg9Ij4KrC76PKZzcbIziw2GyURcY+kzwMfBY4GboiIyixLYPXmQGCW3X8lWQPrGeB9BZfFbGCc\nGjLL7nnAYmAJcFTBZTEbGAcCs+w2AFcANwL/o+CymA2MU0NmGUh6B3AgIv4yXXF0k6TzgI8ApwGL\n0xVk3xkRXymyrGa98uqjZmY159SQmVnNORCYmdWcA4GZWc05EJiZ1ZwDgZlZzTkQmJnVnAOBmVnN\n/X+/oq8O8u4XZAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"blue\")\n",
    "plt.scatter(X, y, c=colors,s=15)\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,6,0,10])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 Normal Equations (Polynomial -> 1 Variable)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " "
     ]
    }
   ],
   "source": [
    "from numpy.linalg import inv\n",
    "\n",
    "%prun theta = np.linalg.inv(X_ones.T.dot(X_ones)).dot(X_ones.T).dot(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 571,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2.03445191],\n",
       "       [1.00372013],\n",
       "       [0.49238608]])"
      ]
     },
     "execution_count": 571,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 572,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_test=np.array([[0.5,1.0],[2,3.0],[1,2.0]])\n",
    "X_test_X=np.array([[0.5],[2],[1]])\n",
    "\n",
    "#Add one column to x for incercept\n",
    "X_test_ones = np.c_[np.ones((3,1)), X_test]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 573,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.02869806],\n",
       "       [5.51905041],\n",
       "       [4.0229442 ]])"
      ]
     },
     "execution_count": 573,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_predict=X_test_ones.dot(theta)\n",
    "y_predict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 574,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_plot=X_ones[:,1].reshape(m,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 575,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.31697701],\n",
       "       [-1.69177574],\n",
       "       [ 2.32985145],\n",
       "       [-0.58200224],\n",
       "       [ 1.77433612],\n",
       "       [ 1.29631061],\n",
       "       [ 1.49089583],\n",
       "       [ 0.74723881],\n",
       "       [-1.00189344],\n",
       "       [-2.44803816],\n",
       "       [ 1.70375476],\n",
       "       [ 2.0586729 ],\n",
       "       [ 1.40616418],\n",
       "       [ 0.07232019],\n",
       "       [-2.42014573],\n",
       "       [-1.55249985],\n",
       "       [-2.71653905],\n",
       "       [ 2.98350545],\n",
       "       [ 0.48272679],\n",
       "       [-0.10586655],\n",
       "       [-0.09572834],\n",
       "       [ 0.09154972],\n",
       "       [-1.46724533],\n",
       "       [-2.46297487],\n",
       "       [ 1.77383243],\n",
       "       [-1.4883623 ],\n",
       "       [-1.58019779],\n",
       "       [-0.38470022],\n",
       "       [-2.70779745],\n",
       "       [-1.15157409],\n",
       "       [-0.3103182 ],\n",
       "       [ 2.62680724],\n",
       "       [ 0.42998495],\n",
       "       [ 2.34737278],\n",
       "       [ 2.63591483],\n",
       "       [ 0.91199673],\n",
       "       [ 0.60405871],\n",
       "       [ 0.48226409],\n",
       "       [-1.55029439],\n",
       "       [-1.18731756],\n",
       "       [ 1.2113166 ],\n",
       "       [-2.4374396 ],\n",
       "       [ 1.63960087],\n",
       "       [ 1.28664483],\n",
       "       [-2.30068449],\n",
       "       [-2.75562608],\n",
       "       [ 1.39187816],\n",
       "       [ 0.89018167],\n",
       "       [-1.71234784],\n",
       "       [ 2.90743642],\n",
       "       [-1.56521437],\n",
       "       [-1.89881992],\n",
       "       [ 0.263598  ],\n",
       "       [-0.14360132],\n",
       "       [ 0.19709711],\n",
       "       [-2.69908967],\n",
       "       [-1.10581479],\n",
       "       [ 2.02226439],\n",
       "       [-2.71498185],\n",
       "       [-1.49008597],\n",
       "       [ 1.68831531],\n",
       "       [-1.46882271],\n",
       "       [-1.79817153],\n",
       "       [ 1.60418156],\n",
       "       [ 2.86554373],\n",
       "       [ 0.00670033],\n",
       "       [-0.12098494],\n",
       "       [-0.20229705],\n",
       "       [-2.80746115],\n",
       "       [-1.43990339],\n",
       "       [-1.21771604],\n",
       "       [-2.99734372],\n",
       "       [-1.58047862],\n",
       "       [-1.13577288],\n",
       "       [ 0.06598934],\n",
       "       [ 0.48662724],\n",
       "       [ 0.57209858],\n",
       "       [-2.06492101],\n",
       "       [-1.70757648],\n",
       "       [-0.92002515],\n",
       "       [ 1.6912358 ],\n",
       "       [-1.38124001],\n",
       "       [-0.49566588],\n",
       "       [-2.60762925],\n",
       "       [ 2.82750388],\n",
       "       [ 0.22348797],\n",
       "       [ 1.62730847],\n",
       "       [ 2.28442337],\n",
       "       [ 0.62626862],\n",
       "       [-0.82394645],\n",
       "       [ 0.81145151],\n",
       "       [ 2.95890971],\n",
       "       [-2.78418458],\n",
       "       [-0.84151516],\n",
       "       [-1.85860201],\n",
       "       [-2.32922561],\n",
       "       [ 1.79849211],\n",
       "       [ 0.03293515],\n",
       "       [ 2.94809001],\n",
       "       [-2.13148054],\n",
       "       [ 1.93370684],\n",
       "       [ 0.58599126],\n",
       "       [-2.73371021],\n",
       "       [-1.42185266],\n",
       "       [-0.50898831],\n",
       "       [ 0.23294786],\n",
       "       [-1.30019177],\n",
       "       [ 0.97965503],\n",
       "       [-0.3440629 ],\n",
       "       [ 2.21974223],\n",
       "       [-1.26252111],\n",
       "       [-1.86735136],\n",
       "       [-0.41689866],\n",
       "       [-0.77155817],\n",
       "       [-2.1136044 ],\n",
       "       [ 0.31486165],\n",
       "       [-1.37794813],\n",
       "       [-2.78815807],\n",
       "       [ 0.8437994 ],\n",
       "       [-0.42243935],\n",
       "       [ 1.37720818],\n",
       "       [ 1.7880941 ],\n",
       "       [ 2.50412715],\n",
       "       [ 2.60135265],\n",
       "       [ 2.83153308],\n",
       "       [ 0.4230441 ],\n",
       "       [ 0.9808771 ],\n",
       "       [ 0.99841078],\n",
       "       [-0.26122788],\n",
       "       [ 2.34997573],\n",
       "       [ 2.56666035],\n",
       "       [ 0.71743909],\n",
       "       [ 2.50872361],\n",
       "       [-2.32656458],\n",
       "       [ 0.38031626],\n",
       "       [ 2.08336699],\n",
       "       [ 0.68601007],\n",
       "       [ 1.89717696],\n",
       "       [-0.57095655],\n",
       "       [-1.96534694],\n",
       "       [ 1.23858422],\n",
       "       [ 2.21620015],\n",
       "       [-2.63087727],\n",
       "       [ 0.89094719],\n",
       "       [ 0.5252651 ],\n",
       "       [-2.02485665],\n",
       "       [ 2.51892681],\n",
       "       [ 2.76927678],\n",
       "       [-0.03223941],\n",
       "       [ 2.75345144],\n",
       "       [ 2.52852018],\n",
       "       [-2.73266965],\n",
       "       [-2.55658549],\n",
       "       [-1.37474963],\n",
       "       [ 0.05932167],\n",
       "       [ 0.06057612],\n",
       "       [-2.35946265],\n",
       "       [-0.47478941],\n",
       "       [-0.92238026],\n",
       "       [-0.17766394],\n",
       "       [ 0.7736991 ],\n",
       "       [-0.10842226],\n",
       "       [-0.03907375],\n",
       "       [-0.66545863],\n",
       "       [-1.14934302],\n",
       "       [-1.34551975],\n",
       "       [ 0.40199247],\n",
       "       [-1.3808432 ],\n",
       "       [ 2.23009953],\n",
       "       [-1.36194893],\n",
       "       [-2.28583488],\n",
       "       [ 1.82475925],\n",
       "       [-2.792254  ],\n",
       "       [ 2.6308782 ],\n",
       "       [-1.02944769],\n",
       "       [ 0.37832071],\n",
       "       [-1.91717024],\n",
       "       [ 1.04799891],\n",
       "       [ 0.8305796 ],\n",
       "       [-0.80338197],\n",
       "       [ 2.39367801],\n",
       "       [-2.67788598],\n",
       "       [ 1.49725892],\n",
       "       [ 0.39488694],\n",
       "       [-1.85101335],\n",
       "       [-0.91851531],\n",
       "       [ 2.84454614],\n",
       "       [ 0.73103613],\n",
       "       [ 1.1150225 ],\n",
       "       [-1.95540706],\n",
       "       [-2.26527266],\n",
       "       [ 2.04005718],\n",
       "       [-2.27612494],\n",
       "       [-1.93635018],\n",
       "       [-0.98970432],\n",
       "       [-2.02747403],\n",
       "       [-2.98403573],\n",
       "       [ 2.37429957],\n",
       "       [ 2.30673828],\n",
       "       [ 0.54678389]])"
      ]
     },
     "execution_count": 575,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 578,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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s9LBfQEREqqDaGVoXz1tMZ1tnRdsoGRjc/X8Dc4GvA5cA68zsU2Z2WEV7FhGRoh3Mg2sH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\nRK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"red\")\n",
    "plt.scatter(X_test_X, y_predict, c=\"red\",s=50)\n",
    "plt.scatter(X, y, c=\"blue\",s=15)\n",
    "plt.scatter(X_plot, np.dot(X_ones, theta), label='Linear Reg Normal Eqn', color='green')\n",
    "\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2.5,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 579,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.54995058]])"
      ]
     },
     "execution_count": 579,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dt=np.dot(X_ones,theta)-y\n",
    "cost=np.dot(dt.T,dt)/2/m\n",
    "cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 580,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.81320433]])"
      ]
     },
     "execution_count": 580,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u=np.dot(dt.T,dt)\n",
    "y_mean=np.mean(y)\n",
    "dz=(y-y_mean)\n",
    "v=np.dot(dz.T,dz)\n",
    "r_squared=(1-u/v)\n",
    "r_squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 581,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Now we'll move onto to looking how we do this polynomial craic with Sci-kit learn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3 Sklearn (Polynomial -> 1 Variable)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 582,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "#first use in built Sci-Kit learn transformation to add polynomial equivilants of all X features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 583,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.preprocessing import PolynomialFeatures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 584,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "poly_features = PolynomialFeatures(degree=2, include_bias=False)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 585,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_poly = poly_features.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 587,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.31697701,  0.10047443],\n",
       "       [-1.69177574,  2.86210514],\n",
       "       [ 2.32985145,  5.42820777]])"
      ]
     },
     "execution_count": 587,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X_poly[:3]\n",
    "#Great, this looks the exact same as my handmade example above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 588,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "reg = LinearRegression()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 595,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 595,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.fit(X_poly, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 596,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.00372013, 0.49238608]])"
      ]
     },
     "execution_count": 596,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.coef_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 597,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2.03445191])"
      ]
     },
     "execution_count": 597,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.intercept_ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 608,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[3.02869806],\n",
       "       [5.51905041],\n",
       "       [4.0229442 ]])"
      ]
     },
     "execution_count": 608,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_predict=reg.predict(X_test)\n",
    "y_predict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 600,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "predicted_line=reg.predict(X_poly)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 601,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8132043340398349"
      ]
     },
     "execution_count": 601,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reg.score(X_poly, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 609,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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jtK49uANGXprSOkoGBnf/A+Ac4CvA9cATZvYnZvbyKW1ZREQmrWBet6uPrsMDmI/RdXiA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\nYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#plot the data\n",
    "colors = (\"red\")\n",
    "plt.scatter(X_test_X, y_predict, c=\"red\",s=50)\n",
    "plt.scatter(X, y, c=\"blue\",s=15)\n",
    "plt.scatter(X_plot, np.dot(X_ones, theta), label='Linear Reg Normal Eqn', color='green')\n",
    "plt.scatter(X, predicted_line , label='Linear Reg SkLearn', color='cyan')\n",
    "\n",
    "\n",
    "\n",
    "plt.title('2-D Scatter Plot Random Data')\n",
    "plt.xlabel('x1')\n",
    "plt.ylabel('y')\n",
    "plt.axis([0,2.5,0,15])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}