unit01_simp_lin_reg/demo01_auto_mpg.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simple Linear Regression for Automobile mpg Data\n",
    "\n",
    "In this demo, you will see how to:\n",
    "* Load data from a text fileausing the `pandas` package\n",
    "* Create a scatter plot of data\n",
    "* Handle missing data\n",
    "* Fit a simple linear model\n",
    "* Plot the linear fit with the test data\n",
    "* Use a nonlinear transformation for an improved fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Loading the Data\n",
    "\n",
    "The python [`pandas`](http://pandas.pydata.org/) library is a powerful package for data analysis.  In this course, we will use a small portion of its features -- just reading and writing data from files.  After reading the data, we will convert it to `numpy` for all numerical processing including running machine learning algorithms.\n",
    "\n",
    "We begin by loading the packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data for this demo comes from a survey of cars to determine the relation of mpg to engine characteristics.  The data can be found in the UCI library:\n",
    "https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Try 1:  Loading the Data Incorrectly\n",
    "\n",
    "The pandas has very good methods for loading data from ASCII tables. In this case, we want to read the data in the file:\n",
    "https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data\n",
    "\n",
    "Since the file is a CSV file (comma-separated-values), we can try to use the `read_csv` command:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This creates a pandas *dataframe*. We can see the first six lines of the dataframe with `head` command:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "ename": "CertificateError",
     "evalue": "hostname 'archive.ics.uci.edu' doesn't match 'casl3-cluster1-trusted.net.nyu.edu'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mCertificateError\u001b[0m                          Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-963a27b56c1c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mdf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mpd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mread_csv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/pandas/io/parsers.py\u001b[0m in \u001b[0;36mparser_f\u001b[0;34m(filepath_or_buffer, sep, delimiter, header, names, index_col, usecols, squeeze, prefix, mangle_dupe_cols, dtype, engine, converters, true_values, false_values, skipinitialspace, skiprows, nrows, na_values, keep_default_na, na_filter, verbose, skip_blank_lines, parse_dates, infer_datetime_format, keep_date_col, date_parser, dayfirst, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, escapechar, comment, encoding, dialect, tupleize_cols, error_bad_lines, warn_bad_lines, skipfooter, skip_footer, doublequote, delim_whitespace, as_recarray, compact_ints, use_unsigned, low_memory, buffer_lines, memory_map, float_precision)\u001b[0m\n\u001b[1;32m    653\u001b[0m                     skip_blank_lines=skip_blank_lines)\n\u001b[1;32m    654\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 655\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0m_read\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfilepath_or_buffer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkwds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    656\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    657\u001b[0m     \u001b[0mparser_f\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__name__\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/pandas/io/parsers.py\u001b[0m in \u001b[0;36m_read\u001b[0;34m(filepath_or_buffer, kwds)\u001b[0m\n\u001b[1;32m    390\u001b[0m     \u001b[0mcompression\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_infer_compression\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfilepath_or_buffer\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcompression\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    391\u001b[0m     filepath_or_buffer, _, compression = get_filepath_or_buffer(\n\u001b[0;32m--> 392\u001b[0;31m         filepath_or_buffer, encoding, compression)\n\u001b[0m\u001b[1;32m    393\u001b[0m     \u001b[0mkwds\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'compression'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcompression\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    394\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/pandas/io/common.py\u001b[0m in \u001b[0;36mget_filepath_or_buffer\u001b[0;34m(filepath_or_buffer, encoding, compression)\u001b[0m\n\u001b[1;32m    184\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0m_is_url\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfilepath_or_buffer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    185\u001b[0m         \u001b[0murl\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfilepath_or_buffer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 186\u001b[0;31m         \u001b[0mreq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_urlopen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0murl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    187\u001b[0m         \u001b[0mcontent_encoding\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mreq\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mheaders\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Content-Encoding'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    188\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mcontent_encoding\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m'gzip'\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36murlopen\u001b[0;34m(url, data, timeout, cafile, capath, cadefault, context)\u001b[0m\n\u001b[1;32m    221\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    222\u001b[0m         \u001b[0mopener\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_opener\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 223\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mopener\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mopen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0murl\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtimeout\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    224\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    225\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0minstall_opener\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mopener\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36mopen\u001b[0;34m(self, fullurl, data, timeout)\u001b[0m\n\u001b[1;32m    524\u001b[0m             \u001b[0mreq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmeth\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    525\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 526\u001b[0;31m         \u001b[0mresponse\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_open\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mreq\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    527\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    528\u001b[0m         \u001b[0;31m# post-process response\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36m_open\u001b[0;34m(self, req, data)\u001b[0m\n\u001b[1;32m    542\u001b[0m         \u001b[0mprotocol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mreq\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    543\u001b[0m         result = self._call_chain(self.handle_open, protocol, protocol +\n\u001b[0;32m--> 544\u001b[0;31m                                   '_open', req)\n\u001b[0m\u001b[1;32m    545\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    546\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36m_call_chain\u001b[0;34m(self, chain, kind, meth_name, *args)\u001b[0m\n\u001b[1;32m    502\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mhandler\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mhandlers\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    503\u001b[0m             \u001b[0mfunc\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mhandler\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmeth_name\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 504\u001b[0;31m             \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    505\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mresult\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    506\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0mresult\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36mhttps_open\u001b[0;34m(self, req)\u001b[0m\n\u001b[1;32m   1359\u001b[0m         \u001b[0;32mdef\u001b[0m \u001b[0mhttps_open\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mreq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1360\u001b[0m             return self.do_open(http.client.HTTPSConnection, req,\n\u001b[0;32m-> 1361\u001b[0;31m                 context=self._context, check_hostname=self._check_hostname)\n\u001b[0m\u001b[1;32m   1362\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1363\u001b[0m         \u001b[0mhttps_request\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAbstractHTTPHandler\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdo_request_\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/urllib/request.py\u001b[0m in \u001b[0;36mdo_open\u001b[0;34m(self, http_class, req, **http_conn_args)\u001b[0m\n\u001b[1;32m   1316\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1317\u001b[0m                 h.request(req.get_method(), req.selector, req.data, headers,\n\u001b[0;32m-> 1318\u001b[0;31m                           encode_chunked=req.has_header('Transfer-encoding'))\n\u001b[0m\u001b[1;32m   1319\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mOSError\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0merr\u001b[0m\u001b[0;34m:\u001b[0m \u001b[0;31m# timeout error\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1320\u001b[0m                 \u001b[0;32mraise\u001b[0m \u001b[0mURLError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36mrequest\u001b[0;34m(self, method, url, body, headers, encode_chunked)\u001b[0m\n\u001b[1;32m   1237\u001b[0m                 encode_chunked=False):\n\u001b[1;32m   1238\u001b[0m         \u001b[0;34m\"\"\"Send a complete request to the server.\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1239\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_send_request\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmethod\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0murl\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbody\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mheaders\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mencode_chunked\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1240\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1241\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0m_send_request\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0murl\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mbody\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mheaders\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mencode_chunked\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36m_send_request\u001b[0;34m(self, method, url, body, headers, encode_chunked)\u001b[0m\n\u001b[1;32m   1283\u001b[0m             \u001b[0;31m# default charset of iso-8859-1.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1284\u001b[0m             \u001b[0mbody\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_encode\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbody\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'body'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1285\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mendheaders\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbody\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mencode_chunked\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mencode_chunked\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1286\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1287\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mgetresponse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36mendheaders\u001b[0;34m(self, message_body, encode_chunked)\u001b[0m\n\u001b[1;32m   1232\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1233\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mCannotSendHeader\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1234\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_send_output\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmessage_body\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mencode_chunked\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mencode_chunked\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1235\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1236\u001b[0m     def request(self, method, url, body=None, headers={}, *,\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36m_send_output\u001b[0;34m(self, message_body, encode_chunked)\u001b[0m\n\u001b[1;32m   1024\u001b[0m         \u001b[0mmsg\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34mb\"\\r\\n\"\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mjoin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_buffer\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1025\u001b[0m         \u001b[0;32mdel\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_buffer\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1026\u001b[0;31m         \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmsg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1027\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1028\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mmessage_body\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36msend\u001b[0;34m(self, data)\u001b[0m\n\u001b[1;32m    962\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msock\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    963\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mauto_open\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 964\u001b[0;31m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mconnect\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    965\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    966\u001b[0m                 \u001b[0;32mraise\u001b[0m \u001b[0mNotConnected\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/http/client.py\u001b[0m in \u001b[0;36mconnect\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m   1398\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1399\u001b[0m             self.sock = self._context.wrap_socket(self.sock,\n\u001b[0;32m-> 1400\u001b[0;31m                                                   server_hostname=server_hostname)\n\u001b[0m\u001b[1;32m   1401\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_context\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcheck_hostname\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_check_hostname\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1402\u001b[0m                 \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/ssl.py\u001b[0m in \u001b[0;36mwrap_socket\u001b[0;34m(self, sock, server_side, do_handshake_on_connect, suppress_ragged_eofs, server_hostname, session)\u001b[0m\n\u001b[1;32m    405\u001b[0m                          \u001b[0msuppress_ragged_eofs\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msuppress_ragged_eofs\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    406\u001b[0m                          \u001b[0mserver_hostname\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mserver_hostname\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 407\u001b[0;31m                          _context=self, _session=session)\n\u001b[0m\u001b[1;32m    408\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    409\u001b[0m     def wrap_bio(self, incoming, outgoing, server_side=False,\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/ssl.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, sock, keyfile, certfile, server_side, cert_reqs, ssl_version, ca_certs, do_handshake_on_connect, family, type, proto, fileno, suppress_ragged_eofs, npn_protocols, ciphers, server_hostname, _context, _session)\u001b[0m\n\u001b[1;32m    812\u001b[0m                         \u001b[0;31m# non-blocking\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    813\u001b[0m                         \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"do_handshake_on_connect should not be specified for non-blocking sockets\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 814\u001b[0;31m                     \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdo_handshake\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    815\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    816\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mOSError\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/ssl.py\u001b[0m in \u001b[0;36mdo_handshake\u001b[0;34m(self, block)\u001b[0m\n\u001b[1;32m   1066\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mtimeout\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m0.0\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mblock\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1067\u001b[0m                 \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msettimeout\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1068\u001b[0;31m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_sslobj\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mdo_handshake\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1069\u001b[0m         \u001b[0;32mfinally\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1070\u001b[0m             \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msettimeout\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtimeout\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/ssl.py\u001b[0m in \u001b[0;36mdo_handshake\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    692\u001b[0m                 raise ValueError(\"check_hostname needs server_hostname \"\n\u001b[1;32m    693\u001b[0m                                  \"argument\")\n\u001b[0;32m--> 694\u001b[0;31m             \u001b[0mmatch_hostname\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetpeercert\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mserver_hostname\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    695\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    696\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0munwrap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/ssl.py\u001b[0m in \u001b[0;36mmatch_hostname\u001b[0;34m(cert, hostname)\u001b[0m\n\u001b[1;32m    329\u001b[0m         raise CertificateError(\"hostname %r \"\n\u001b[1;32m    330\u001b[0m             \u001b[0;34m\"doesn't match %r\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 331\u001b[0;31m             % (hostname, dnsnames[0]))\n\u001b[0m\u001b[1;32m    332\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    333\u001b[0m         raise CertificateError(\"no appropriate commonName or \"\n",
      "\u001b[0;31mCertificateError\u001b[0m: hostname 'archive.ics.uci.edu' doesn't match 'casl3-cluster1-trusted.net.nyu.edu'"
     ]
    }
   ],
   "source": [
    "df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/auto-mpg/auto-mpg.data')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There were three errors:\n",
    "* All the data appeared in one column.  That is, the columns were not \"delimited\" correctly\n",
    "* The first line got mistook as a header\n",
    "* The columns are missing their header names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>2</th>\n",
       "      <td>16.0   8   304.0      150.0      3433.      12...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>17.0   8   302.0      140.0      3449.      10...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>15.0   8   429.0      198.0      4341.      10...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>14.0   8   454.0      220.0      4354.       9...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  18.0   8   307.0      130.0      3504.      12.0   70  1\\t\"chevrolet chevelle malibu\"\n",
       "0  15.0   8   350.0      165.0      3693.      11...                                   \n",
       "1  18.0   8   318.0      150.0      3436.      11...                                   \n",
       "2  16.0   8   304.0      150.0      3433.      12...                                   \n",
       "3  17.0   8   302.0      140.0      3449.      10...                                   \n",
       "4  15.0   8   429.0      198.0      4341.      10...                                   \n",
       "5  14.0   8   454.0      220.0      4354.       9...                                   "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Try 2: Fixing the Errors in the loading\n",
    "\n",
    "The problems above are common.  Often it takes a few times to load the data correctly.  That is why it is \n",
    "good to look at the first few elements of the dataframe before proceeding.\n",
    "After some googling you can find out that you need to specify some other options to the `read_csv` command.\n",
    "First, you need to supply the names of the columns.  In this case, I have supplied them manually based on the\n",
    "description in the UCI website:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "names = ['mpg', 'cylinders','displacement', 'horsepower', \n",
    "         'weight', 'acceleration', 'model year', 'origin', 'car name']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/'+\n",
    "                 'auto-mpg/auto-mpg.data',\n",
    "                 header=None,delim_whitespace=True,names=names,na_values='?')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you re-run `head` command now, you can see the loading was correct. You can see the column names, index, and values:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating the Data\n",
    "We can get the `shape` of the data, which indicates the number of samples and number of attributes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(398, 9)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also see the three components of the `dataframe` object.  The dataframe is stored in a table (similar to a SQL table if you know databases).  In this case, there is one row for each car and the attributes of the car are stored in the columns.  The command `df.columns` returns the names of the columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mpg</th>\n",
       "      <th>cylinders</th>\n",
       "      <th>displacement</th>\n",
       "      <th>horsepower</th>\n",
       "      <th>weight</th>\n",
       "      <th>acceleration</th>\n",
       "      <th>model year</th>\n",
       "      <th>origin</th>\n",
       "      <th>car name</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>18.0</td>\n",
       "      <td>8</td>\n",
       "      <td>307.0</td>\n",
       "      <td>130.0</td>\n",
       "      <td>3504.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>chevrolet chevelle malibu</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>15.0</td>\n",
       "      <td>8</td>\n",
       "      <td>350.0</td>\n",
       "      <td>165.0</td>\n",
       "      <td>3693.0</td>\n",
       "      <td>11.5</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>buick skylark 320</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>18.0</td>\n",
       "      <td>8</td>\n",
       "      <td>318.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>3436.0</td>\n",
       "      <td>11.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>plymouth satellite</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>16.0</td>\n",
       "      <td>8</td>\n",
       "      <td>304.0</td>\n",
       "      <td>150.0</td>\n",
       "      <td>3433.0</td>\n",
       "      <td>12.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>amc rebel sst</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>17.0</td>\n",
       "      <td>8</td>\n",
       "      <td>302.0</td>\n",
       "      <td>140.0</td>\n",
       "      <td>3449.0</td>\n",
       "      <td>10.5</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>ford torino</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>15.0</td>\n",
       "      <td>8</td>\n",
       "      <td>429.0</td>\n",
       "      <td>198.0</td>\n",
       "      <td>4341.0</td>\n",
       "      <td>10.0</td>\n",
       "      <td>70</td>\n",
       "      <td>1</td>\n",
       "      <td>ford galaxie 500</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    mpg  cylinders  displacement  horsepower  weight  acceleration  \\\n",
       "0  18.0          8         307.0       130.0  3504.0          12.0   \n",
       "1  15.0          8         350.0       165.0  3693.0          11.5   \n",
       "2  18.0          8         318.0       150.0  3436.0          11.0   \n",
       "3  16.0          8         304.0       150.0  3433.0          12.0   \n",
       "4  17.0          8         302.0       140.0  3449.0          10.5   \n",
       "5  15.0          8         429.0       198.0  4341.0          10.0   \n",
       "\n",
       "   model year  origin                   car name  \n",
       "0          70       1  chevrolet chevelle malibu  \n",
       "1          70       1          buick skylark 320  \n",
       "2          70       1         plymouth satellite  \n",
       "3          70       1              amc rebel sst  \n",
       "4          70       1                ford torino  \n",
       "5          70       1           ford galaxie 500  "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.head(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['mpg', 'cylinders', 'displacement', 'horsepower', 'weight',\n",
       "       'acceleration', 'model year', 'origin', 'car name'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The field `df.index` returns the indices of the rows.  In this case, they are just enumerated 0,1,..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "RangeIndex(start=0, stop=398, step=1)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.index"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, `df.values` is a 2D array with values of the attributes for each car.  Note that the data can be *heterogeneous*:  Some entries are integers, some are floating point values and some are strings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[18.0, 8, 307.0, ..., 70, 1, 'chevrolet chevelle malibu'],\n",
       "       [15.0, 8, 350.0, ..., 70, 1, 'buick skylark 320'],\n",
       "       [18.0, 8, 318.0, ..., 70, 1, 'plymouth satellite'],\n",
       "       ..., \n",
       "       [32.0, 4, 135.0, ..., 82, 1, 'dodge rampage'],\n",
       "       [28.0, 4, 120.0, ..., 82, 1, 'ford ranger'],\n",
       "       [31.0, 4, 119.0, ..., 82, 1, 'chevy s-10']], dtype=object)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `df.columns` attribute is not a python list, but a `pandas`-specific data structure called an `Index`.  To convert to a list, use the `tolist()` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['mpg',\n",
       " 'cylinders',\n",
       " 'displacement',\n",
       " 'horsepower',\n",
       " 'weight',\n",
       " 'acceleration',\n",
       " 'model year',\n",
       " 'origin',\n",
       " 'car name']"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns.tolist()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can select subsets of the attributes with indexing.  For example, this selects one attribute, which returns what is called a pandas `Series`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    8\n",
       "1    8\n",
       "2    8\n",
       "3    8\n",
       "4    8\n",
       "5    8\n",
       "Name: cylinders, dtype: int64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df['cylinders']\n",
    "df2.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also select a list of column names which returns another dataframe.  Note the use of the double brackets `[[ ... ]]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style>\n",
       "    .dataframe thead tr:only-child th {\n",
       "        text-align: right;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: left;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>cylinders</th>\n",
       "      <th>horsepower</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>8</td>\n",
       "      <td>130.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>8</td>\n",
       "      <td>165.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8</td>\n",
       "      <td>150.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>8</td>\n",
       "      <td>150.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>8</td>\n",
       "      <td>140.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>8</td>\n",
       "      <td>198.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   cylinders  horsepower\n",
       "0          8       130.0\n",
       "1          8       165.0\n",
       "2          8       150.0\n",
       "3          8       150.0\n",
       "4          8       140.0\n",
       "5          8       198.0"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2 = df[['cylinders','horsepower']]\n",
    "df2.head(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Plotting the Data\n",
    "We load the `matplotlib` module to plot the data.  This module has excellent plotting routines that are very similar to those in MATLAB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we need to convert the dataframes to numpy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "xstr = 'displacement'\n",
    "x = np.array(df[xstr])\n",
    "y = np.array(df['mpg'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can create a scatter plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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fzJclAw6NFFhz5x7AwjyTwlpVNi5N7SNIAj8b8dS2ykXiqnH4RrlBL3vVqaHj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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d80ff60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulating Numpy arrays\n",
    "\n",
    "Once the data is converted to a numpy array, we can perform many useful simple calculations.  For example, we can compute the sample mean:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean displacement = 193.4, mean mpg=  23.5\n"
     ]
    }
   ],
   "source": [
    "mx = np.mean(x)\n",
    "my = np.mean(y)\n",
    "print('Mean {0:s} = {1:5.1f}, mean mpg= {2:5.1f}'.format(xstr, mx, my))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fraction of cars with > 25 mpg:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we can repeat the `read_csv` command with the correct options. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.39698492462311558"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(y > 25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample mean displacement for the cars that have mpg > 25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "110.08227848101266"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = (y>25)\n",
    "np.mean(x*I)/np.mean(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also do the previous command with [boolean indexing](https://docs.scipy.org/doc/numpy-1.13.0/user/basics.indexing.html)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "110.08227848101266"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(x[I])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Missing Data and NaN Values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, try a different field, horsepower"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xstr = 'horsepower'\n",
    "x = np.array(df[xstr])\n",
    "y = np.array(df['mpg'])\n",
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When you get the mean, it gives `nan` which means not a number.  The reason is that there was missing data in the orginal file and the `load_csv` function put `nan` values in the places where the data was missing.  This is very common.  To remove the rows with the missing data, we can use the `dropna` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(392, 2)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df1 = df[['mpg','horsepower']]\n",
    "df2 = df1.dropna()\n",
    "df2.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "We can see that some of the rows have been dropped.  Specifically, the number of samples went from 396 to 392.  We can now compute the mean using the reduced dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "104.46938775510205"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array(df2['horsepower'])\n",
    "y = np.array(df2['mpg'])\n",
    "np.mean(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And, we can plot the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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B8UqyuiUo4kzcSdu5m6Xh/VhUboYRh6Y2DS27KOsbTinAjQvPDXUURzWJ735gV1m5BGGE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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115cd52b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x,y,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing and Plotting a Linear Fit\n",
    "We can now try to fit a linear model, $\\hat{y} = \\beta_0 + \\beta_1 x$.\n",
    "From class, the formulae are:\n",
    "$$\\beta_1 = s_{yx}/s_{xx}, \\quad \\beta_0 = \\bar{y} - \\beta_1\\bar{x},$$\n",
    "where $\\bar{x}$ and $\\bar{y}$ are the sample means and $s_{yx}$ and $s_{xx}$ are the cross- and auto-covariances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "xbar     = 104.47,       ybar=  23.45\n",
      "sqrt(sxx)=  38.44,  sqrt(syy)=   7.80\n",
      "beta0=  39.94, beta1=  -0.16\n"
     ]
    }
   ],
   "source": [
    "xm = np.mean(x)\n",
    "ym = np.mean(y)\n",
    "syy = np.mean((y-ym)**2)\n",
    "syx = np.mean((y-ym)*(x-xm))\n",
    "sxx = np.mean((x-xm)**2)\n",
    "beta1 = syx/sxx\n",
    "beta0 = ym - beta1*xm\n",
    "\n",
    "print(\"xbar     ={0:7.2f},       ybar={1:7.2f}\".format(xm,ym))\n",
    "print(\"sqrt(sxx)={0:7.2f},  sqrt(syy)={1:7.2f}\".format(np.sqrt(sxx),np.sqrt(syy)))\n",
    "print(\"beta0={0:7.2f}, beta1={1:7.2f}\".format(beta0,beta1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create a plot of the regression line on top of the scatter plot.  The vector `xplt` are the x-coordinates of the two endpoints of the line.  They are chosen so that the line fits nicely in the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AbZmGwpDrF/2iY536RTuq3PjifAGv9sT+uuyqqe+orRUohB9Ne27bM5IHOZ2r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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115d48470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Points on the regression line\n",
    "xplt = np.array([20,250])          \n",
    "yplt = beta1*xplt + beta0\n",
    "\n",
    "plt.plot(x,y,'o')                    # Plot the data points\n",
    "plt.plot(xplt,yplt,'-',linewidth=3)  # Plot the regression line\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The (normalized) residual sum of squares is RSS=$(1/n)\\sum_i (y_i-\\hat{y}_i)^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSS =   23.94\n"
     ]
    }
   ],
   "source": [
    "yhat=beta0+beta1*x\n",
    "RSS = np.mean((y-yhat)**2)\n",
    "print(\"RSS = {0:7.2f}\".format(RSS))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us see whether this is the same as the analytically derived minimal RSS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSSmin =   23.94\n"
     ]
    }
   ],
   "source": [
    "rxy=syx/np.sqrt(sxx)/np.sqrt(syy)\n",
    "RSSmin=(1-rxy*rxy)*syy\n",
    "print(\"RSSmin = {0:7.2f}\".format(RSSmin))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Nonlinear Transformation\n",
    "\n",
    "We see that the linear regression captures the general trend of the relation between `y=mpg` and `x=horsepower`.  However, the trend does not really appear linear - instead it has an inverse type relation.   So, a natural idea is to use a *nonlinear transformation*:\n",
    "* Transform the data `z=1/y` \n",
    "* Fit `z` vs. `x` with a linear model:  $\\hat{z}=\\beta_0 + \\beta_1x$.\n",
    "* Invert the nonlinear relation for a model for `y`:  $\\hat{y} = \\hat{z}=1/(\\beta_0 + \\beta_1x)$.\n",
    "\n",
    "We begin then by computing `z` and plotting a scatter plot of `z` vs. `x`.  Note that `z` represented gallons per mile (1/mpg)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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18/jGBTO46bwZTJ0Y3j5fGVtY/ANxC29XOKKzYyieJAw4bFJxkl7ojOmolY6I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\n8c5ZAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115d48c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = 1/y\n",
    "plt.plot(x,z,'o')\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('1/mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see a clear linear relation between `z` (1/mpg) and `x` (horsepower). We can fit a linear model,\n",
    "$z = \\beta_0 + \\beta_1 x$.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Use linear regression to fit `z` vs. `x`\n",
    "xm = np.mean(x)\n",
    "zm = np.mean(z)\n",
    "sxz = np.mean((z-zm)*(x-xm))\n",
    "sxx = np.mean((x-xm)**2)\n",
    "beta1_inv = sxz/sxx\n",
    "beta0_inv = zm - beta1_inv*xm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create a plot of the regression line on top of the scatter plot.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EbH0aaA2nXg+njIZAw1hJUcBHdU3jrDlFAV/cRX1O9SgpDrCnti7plBn5msNI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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115eb0f98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "z = 1/y\n",
    "xplt_inv = np.linspace(20,250,100)\n",
    "zplt_inv = beta1_inv*xplt_inv + beta0_inv\n",
    "plt.plot(x,z,'o')\n",
    "plt.plot(xplt_inv,zplt_inv,'-',linewidth=3)\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('1/mpg')\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we compute the estimate in the original domain:  $\\hat{y}=1/\\zhat{z}$.  We plot the data, original linear fit and the linear fit with inversion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11619fd30>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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2Zn9XUK9nESqjqqIxoxRBVdBooPM1MO3/4IHDMOF/oWWZyNn0/fDtw/DPRFhx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\nAhGRhPs7+1glDIwPFvAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1160fdf28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "yplt_inv = 1/zplt_inv \n",
    "plt.plot(x,y,'o')\n",
    "plt.plot(xplt,yplt,'-',linewidth=3)\n",
    "plt.plot(xplt_inv,yplt_inv,'-',linewidth=3)\n",
    "plt.xlabel(xstr)\n",
    "plt.ylabel('mpg')\n",
    "plt.grid(True)\n",
    "plt.legend(['data', 'linear', 'linear+inversion'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can conclude by comparing the RSS using the linear fit and the linear fit+inversion.  We see that we get a slightly reduced error using the nonlinear transformation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RSS =   23.94 (linear)\n",
      "RSS =   20.66 (linear+inversion)\n"
     ]
    }
   ],
   "source": [
    "zhat_inv = beta0_inv + beta1_inv*x\n",
    "yhat_inv = 1/zhat_inv\n",
    "RSS_inv = np.mean((yhat_inv-y)**2)\n",
    "print(\"RSS = {0:7.2f} (linear)\".format(RSS))\n",
    "print(\"RSS = {0:7.2f} (linear+inversion)\".format(RSS_inv))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "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": 1
}