{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear Regression\n",
    "\n",
    "*By James Peret – 30/07/2017*\n",
    "\n",
    "> Linear regression is a prediction method that is more than 200 years old.\n",
    "\n",
    "In statistics, [linear regression](https://en.wikipedia.org/wiki/Linear_regression) is an approach for modeling the relationship between a scalar dependent variable y and one or more explanatory variables (or independent variables) denoted X. The case of one explanatory variable is called simple linear regression.\n",
    "\n",
    "[Simple linear regression](https://en.wikipedia.org/wiki/Simple_linear_regression) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables. The adjective simple refers to the fact that the outcome variable is related to a single predictor.\n",
    "\n",
    "### Libraries\n",
    "\n",
    "We will be using [pandas](https://pandas.pydata.org) for handling data structures and [matplotlib](https://matplotlib.org/) for plotting graphs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# imports\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# this allows images to appear directly in the notebook\n",
    "from IPython.display import Image\n",
    "from IPython.core.display import HTML \n",
    "\n",
    "# this allows plots to appear directly in the notebook\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Data\n",
    "\n",
    "In this notebook we will be using a dataset of housing prices from the city of São Paulo, Brazil. The data was scraped from the web using this [notebook](../data/housing-prices-sao-paulo.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>meters</th>\n",
       "      <th>price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>100.0</td>\n",
       "      <td>1150.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>102.0</td>\n",
       "      <td>760.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>63.0</td>\n",
       "      <td>519.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>30.0</td>\n",
       "      <td>278.2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>55.0</td>\n",
       "      <td>400.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   meters   price\n",
       "0   100.0  1150.0\n",
       "1   102.0   760.0\n",
       "2    63.0   519.0\n",
       "3    30.0   278.2\n",
       "4    55.0   400.0"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "filename = '../data/housing-prices-sao-paulo.txt'\n",
    "data = pd.read_csv(filename, sep=\",\")\n",
    "data.columns = [\"meters\", \"price\"]\n",
    "data = data[data.price < 10000]\n",
    "data = data[data.meters > 25]\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1069, 2)"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# print the shape of the DataFrame\n",
    "data.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will plot this data using *panda's plot function* and see how it looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10e7064a8>"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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emNkXwym7XzSz9WY2N0xfkY2Zlc3sc2Z2TQb7LzKzfzGzJ8zsNDMb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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10da14b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "data.plot(x=\"meters\",  y=\"price\", s=2, kind='scatter',  style=\"o\")\n",
    "plt.xlabel(\"Size (m²)\")\n",
    "plt.ylabel(\"Price (1000 of R$)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple Linear Regression with sklearn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  438.17463796   846.37629325  1254.57794855  1662.77960384  2070.98125914]\n"
     ]
    }
   ],
   "source": [
    "from sklearn import linear_model\n",
    "X = data[[0]]\n",
    "y = data.price\n",
    "z = pd.DataFrame([50, 100, 150, 200, 250], columns=['size_in_meters'])\n",
    "lm = linear_model.LinearRegression()\n",
    "model = lm.fit(X,y)\n",
    "predictions = lm.predict(z)\n",
    "print(predictions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1076849b0>"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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h6x9tokFLZW5M6opnV1UDcIeqDgXOB2519y2fAaxQ1TJghXsMTkqTMvfnFuBh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\njBxQHHZRX8DtbvAqys9t14d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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d736748>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = data.plot(x=\"meters\",  y=\"price\", s=2, kind='scatter', style=\"o\")\n",
    "p.plot(X, model.predict(X), color='red')\n",
    "plt.xlabel(\"Size (m²)\")\n",
    "plt.ylabel(\"Price (1000 of R$)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References\n",
    "\n",
    "- [DAT4/notebooks/08_linear_regression.ipynb](https://github.com/justmarkham/DAT4/blob/master/notebooks/08_linear_regression.ipynb) - Introduction to Linear Regression\n",
    "- [Simple and Multiple Linear Regression in Python](https://medium.com/towards-data-science/simple-and-multiple-linear-regression-in-python-c928425168f9) - Quick introduction to linear regression in Python\n",
    "- [Basic linear regressions in Python](http://jmduke.com/posts/basic-linear-regressions-in-python/)\n",
    "- [How To Implement Simple Linear Regression From Scratch With Python](http://machinelearningmastery.com/implement-simple-linear-regression-scratch-python/)\n",
    "- [Linear Regression In scikit-learn](https://chrisalbon.com/machine-learning/linear_regression_scikit-learn.html)\n",
    "- [Example of Regression Analysis](http://facweb.cs.depaul.edu/mobasher/classes/CSC478/Notes/IPython%20Notebook%20-%20Regression.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}