ML - Polynomial Linear Regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Polynomial reggression is concerned with only one variable and different order of that variable.\n",
    "\n",
    "$ y = \\beta_{0} + \\beta_{1}X_{1} + \\beta_{2}X_{1}^2 + ... + \\beta_{n}X_{1}^n$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "# %matplotlib notebook - this line makes the plot interactive Good for zooming.\n",
    "from sklearn.linear_model import LinearRegression\n",
    "from sklearn.preprocessing import PolynomialFeatures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dataset = pd.read_csv(\"Position_Salaries.csv\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Position</th>\n",
       "      <th>Level</th>\n",
       "      <th>Salary</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>Business Analyst</td>\n",
       "      <td>1</td>\n",
       "      <td>45000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>Junior Consultant</td>\n",
       "      <td>2</td>\n",
       "      <td>50000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>Senior Consultant</td>\n",
       "      <td>3</td>\n",
       "      <td>60000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>Manager</td>\n",
       "      <td>4</td>\n",
       "      <td>80000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>Country Manager</td>\n",
       "      <td>5</td>\n",
       "      <td>110000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>Region Manager</td>\n",
       "      <td>6</td>\n",
       "      <td>150000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>Partner</td>\n",
       "      <td>7</td>\n",
       "      <td>200000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>Senior Partner</td>\n",
       "      <td>8</td>\n",
       "      <td>300000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>C-level</td>\n",
       "      <td>9</td>\n",
       "      <td>500000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>CEO</td>\n",
       "      <td>10</td>\n",
       "      <td>1000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "            Position  Level   Salary\n",
       "0   Business Analyst      1    45000\n",
       "1  Junior Consultant      2    50000\n",
       "2  Senior Consultant      3    60000\n",
       "3            Manager      4    80000\n",
       "4    Country Manager      5   110000\n",
       "5     Region Manager      6   150000\n",
       "6            Partner      7   200000\n",
       "7     Senior Partner      8   300000\n",
       "8            C-level      9   500000\n",
       "9                CEO     10  1000000"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dataset"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9BlFaMfdQRNza0+sR8cUTXpWZmdWMPqfm0gq1T1egFjMzq0H9nZr7R0nfApZSXAkBgIh4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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1120ed438>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(dataset[\"Level\"], dataset[\"Salary\"])\n",
    "plt.title(\"Salary vs. Level\")\n",
    "plt.xlabel(\"Level\")\n",
    "plt.ylabel(\"Salary\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The above graph will help guide us towards PLR."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X = dataset.iloc[:, 1:2].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y = dataset.iloc[:, 2].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1],\n",
       "       [ 2],\n",
       "       [ 3],\n",
       "       [ 4],\n",
       "       [ 5],\n",
       "       [ 6],\n",
       "       [ 7],\n",
       "       [ 8],\n",
       "       [ 9],\n",
       "       [10]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  45000,   50000,   60000,   80000,  110000,  150000,  200000,\n",
       "        300000,  500000, 1000000])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We dont need the position column as it is equivalent to the level column (it's like we already encoded the data).\n",
    "\n",
    "Note, we always want our X to be represented as a matrix, hence we put [:,1:2].values. This is equivalent to [:, 1].values but ensures it remains in matrix form instead of vector form.\n",
    "\n",
    "Furthermore, since we have such a small amount of data, it makes now sense to split it into a test and training set. There is also no need for feature scaling here!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "poly_reg = PolynomialFeatures(degree=5) # the defautlt is to features, that is, order/degree = 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Later on, I will get into how to set the appropriate degree for models. For this example, a degree of 5 was more appropriate.\n",
    "\n",
    "Ofcourse, we want to be cautious aout overfitting here. However, in this instance, since we have 'cooked' data, we know that it follows a perfect polynomial relationship and it seems that a degree of 5/6 fits it very well.\n",
    "\n",
    "Later on, I wil discuss how to avoid overfitting."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "X_poly = poly_reg.fit_transform(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>0</th>\n",
       "      <th>1</th>\n",
       "      <th>2</th>\n",
       "      <th>3</th>\n",
       "      <th>4</th>\n",
       "      <th>5</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1.0</td>\n",
       "      <td>2.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>16.0</td>\n",
       "      <td>32.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>27.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>243.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>16.0</td>\n",
       "      <td>64.0</td>\n",
       "      <td>256.0</td>\n",
       "      <td>1024.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>25.0</td>\n",
       "      <td>125.0</td>\n",
       "      <td>625.0</td>\n",
       "      <td>3125.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1.0</td>\n",
       "      <td>6.0</td>\n",
       "      <td>36.0</td>\n",
       "      <td>216.0</td>\n",
       "      <td>1296.0</td>\n",
       "      <td>7776.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>1.0</td>\n",
       "      <td>7.0</td>\n",
       "      <td>49.0</td>\n",
       "      <td>343.0</td>\n",
       "      <td>2401.0</td>\n",
       "      <td>16807.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>1.0</td>\n",
       "      <td>8.0</td>\n",
       "      <td>64.0</td>\n",
       "      <td>512.0</td>\n",
       "      <td>4096.0</td>\n",
       "      <td>32768.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>1.0</td>\n",
       "      <td>9.0</td>\n",
       "      <td>81.0</td>\n",
       "      <td>729.0</td>\n",
       "      <td>6561.0</td>\n",
       "      <td>59049.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>1.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>100.0</td>\n",
       "      <td>1000.0</td>\n",
       "      <td>10000.0</td>\n",
       "      <td>100000.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     0     1      2       3        4         5\n",
       "0  1.0   1.0    1.0     1.0      1.0       1.0\n",
       "1  1.0   2.0    4.0     8.0     16.0      32.0\n",
       "2  1.0   3.0    9.0    27.0     81.0     243.0\n",
       "3  1.0   4.0   16.0    64.0    256.0    1024.0\n",
       "4  1.0   5.0   25.0   125.0    625.0    3125.0\n",
       "5  1.0   6.0   36.0   216.0   1296.0    7776.0\n",
       "6  1.0   7.0   49.0   343.0   2401.0   16807.0\n",
       "7  1.0   8.0   64.0   512.0   4096.0   32768.0\n",
       "8  1.0   9.0   81.0   729.0   6561.0   59049.0\n",
       "9  1.0  10.0  100.0  1000.0  10000.0  100000.0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pd.DataFrame(X_poly) #poly reg class automatically adds the column of ones in order to include the constant b0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg = LinearRegression()\n",
    "lin_reg.fit(X_poly, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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RSTpV0h+AXwA/Az4FPAA8WMD4zMwqWrmX5WAHrhEBjwHXR8Qzee0TJR3T9GGZmVlEkoiO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\nIjMzy9T/B9Q58so0EQu8AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1146f5d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#visualize results.\n",
    "\n",
    "plt.scatter (X, y, color = 'red')\n",
    "plt.plot(X, lin_reg.predict(poly_reg.fit_transform(X)), color = 'blue')\n",
    "plt.title(\"PLR\")\n",
    "plt.xlabel(\"Position Level\")\n",
    "plt.ylabel(\"Salary\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, lets actually predict the salary using our model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 174878.07765151])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lin_reg.predict(poly_reg.fit_transform(6.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A prediction of ~175K with degree 6. Note, this answer will change depending on the degree.\n",
    "\n",
    "This concludes PLR."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda root]",
   "language": "python",
   "name": "conda-root-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}