tuf22191 · May 4, 2018 03:55
diff --git a/Linear Regression 5 3 2018 b/Linear Regression 5 3 2018
 {
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "#from Collections import Counter \n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need to get some data first, let's do an example of number of billionaires a function of time: \n",
    "\n",
    "data was taken from : https://www.forbes.com/special-report/2012/billionaires-25th-anniversary-timeline.html \n",
    "\n",
    "in our model we are assuming that the number of billionaires is completely determined with time , in future tutorials we might add additional features to our algorithms (like global market indicators) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "26\n",
      "26\n"
     ]
    }
   ],
   "source": [
    "time = np.arange(1987, 2013)\n",
    "num_billionaires = np.array([140,191,198,269,273,288,310,349,365,447,486,454,\n",
    "                             465,470,538,497,476,587,691,793,946,1125,793,1011,1210,\n",
    "                            1226])\n",
    "print(len(time))\n",
    "print(len(num_billionaires))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the data to see what it looks like (should be an increasing trend):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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xIjK42zlXSnnNO1kFvKE9ffZ67twBjASKgKUi8qmIPCUiUUCSMSYPwH4PsvOn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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaea3358>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dataframe = pd.DataFrame({'year':time, 'amt':num_billionaires})\n",
    "plt.plot(dataframe['year'], dataframe['amt'])\n",
    "plt.title('Amt of Billionaires as a Function of Year')\n",
    "plt.show() #page 25\n",
    "#talk about income disparity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Great! Let's try to fit a linear regression model. Let's code one up first. Recall that the formula for Linear Regression (also called L2 norm minimization or Least Squares Regression): \n",
    "\n",
    "$$\n",
    " \\hat{w} = (X^T X)^{-1}X^T \\hat{t}\\\n",
    "$$\n",
    "\n",
    "Where X is an augmented data matrix (we will construct this by ourselves). The $\\hat{t}$ is the vector of the output variables (vector because we stack all the output results of each of training data points). For us the output variable is the number of billlionaires (which is a function/output of time <- which is the input)\n",
    "\n",
    "So lets assume for our billionaires data, we will have a straight line so we will assume that the formula for amount of billionaires with respect to time is:   \n",
    "\n",
    "$$ \n",
    "   f(\\hat{w}) = w_0 + w_1t\\      \\tag 1\\\\\n",
    "$$\n",
    "\n",
    "  where $\\hat{w}$ is a vector comprised of $[ w_0 , w_1 ]$ . We will now populate the $X$ matrix and the $X^T$ matrix, and the $\\hat{t}$ vector from our training data (the time and num_billionaires). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "X_mat = np.ones(len(time))\n",
    "X_mat = np.vstack([X_mat, time]).T\n",
    "#print(X_mat)   #looks exactly the way we need it for our algorithm \n",
    "X_mat_transpose = np.array(X_mat.T)\n",
    "#print(X_mat_transpose)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "output_t_vector = np.array(num_billionaires) #using np.array creates a copy of the vector\n",
    "output_t_vector = output_t_vector.reshape((len(output_t_vector), 1)) #must reshape for easier data manipulation\n",
    "#print(output_t_vector) # this the t vector from equation 1 , we now have everything for linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(26L, 2L)\n",
      "(2L, 26L)\n",
      "(26L, 1L)\n"
     ]
    }
   ],
   "source": [
    "#double check the shapes \n",
    "print(X_mat.shape)\n",
    "print(X_mat_transpose.shape)\n",
    "print(output_t_vector.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -7.77259703e+04   3.91535043e+01]\n"
     ]
    }
   ],
   "source": [
    "#and now for the algorithm\n",
    "def train_least_squares(x_mat, x_mat_transposed, t_vect):\n",
    "    #just follow equation 1 above 1 step at a time\n",
    "    #multiply x_mat by its transpose\n",
    "    w_vector = np.matmul(x_mat_transposed,x_mat) #https://docs.scipy.org/doc/numpy-1.14.0/reference/generated/numpy.matmul.html\n",
    "    #take inverse\n",
    "    w_vector = np.linalg.inv(w_vector)\n",
    "    #multiply by x_mat's transpose\n",
    "    w_vector = np.matmul(w_vector, x_mat_transposed)\n",
    "    #multiply by T vector \n",
    "    w_vector = np.matmul(w_vector, t_vect)\n",
    "    w_vector = w_vector.flatten() # converts the shape of w_vector from matrix 1-d array just a nuance to make things easier to read off for us\n",
    "    return w_vector # return the coefficients w_0 and w_1'\n",
    "\n",
    "w_vect_trained = train_least_squares(X_mat, X_mat_transpose, output_t_vector)\n",
    "print(w_vect_trained)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9FABaWVM79IZm4jRU7pyjQ0w7usR90cn37hzLQ98ezfe/OqSJtRWRSKIhoFbW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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb50c8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# so let's plot the line along with the scatter plot \n",
    "plt.scatter(dataframe['year'], dataframe['amt'])\n",
    "\n",
    "line_points = [(w_vect_trained[0]+w_vect_trained[1]*year) for year in dataframe['year']]\n",
    "\n",
    "plt.plot(dataframe['year'], line_points)\n",
    "plt.show() #p25"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Beautiful, the line seems to fit our model pretty accurately! However, there looks like we could add a sinusoidal to the data. In a future tutorial, we will perhaps look at ways to do a little bit more complex regression fitting but using this same framework. Furthermore, we can test our error because the data goes up to 2012 and so we can add 6 more years to see how good our predictions might be! We can then talk about overfitting/regularization/etc! \n",
    "\n",
    "But for now friends, take care!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.14"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
 }