League Analysis - Final.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <u>Dorset & Wilts 2 North</u>\n",
    "\n",
    "**Sam Jones - @SJBiomech**\n",
    "\n",
    "In this notebook I will be looking at the Dorset & Wilts 2 North league table from the past years. I will aim to briefly investigate the data and determine the factors which appear have the greatest influence on league position.\n",
    "\n",
    "Data were taken from Melksham's website since they appear to have stayed in this league each year. <br>Data access: https://www.melkshamrfc.co.uk/teams/22086/league-table?season=67487"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plots displayed in the notebook and also stored in the notebook document.\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pandas import *\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load in and Clean Data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First load in the collated league data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "league = read_excel('League Results.xlsx')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Have a look at the first few rows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Pos</th>\n",
       "      <th>Team</th>\n",
       "      <th>Season</th>\n",
       "      <th>Pl</th>\n",
       "      <th>HW</th>\n",
       "      <th>HD</th>\n",
       "      <th>HL</th>\n",
       "      <th>AW</th>\n",
       "      <th>AD</th>\n",
       "      <th>AL</th>\n",
       "      <th>...</th>\n",
       "      <th>HF</th>\n",
       "      <th>AF</th>\n",
       "      <th>F</th>\n",
       "      <th>A</th>\n",
       "      <th>Diff</th>\n",
       "      <th>TB</th>\n",
       "      <th>LB</th>\n",
       "      <th>Adj</th>\n",
       "      <th>Rand</th>\n",
       "      <th>Pts</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>Westbury</td>\n",
       "      <td>19/20</td>\n",
       "      <td>15</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>9</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>...</td>\n",
       "      <td>171</td>\n",
       "      <td>343</td>\n",
       "      <td>514</td>\n",
       "      <td>174</td>\n",
       "      <td>340</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>56</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>Swindon II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>...</td>\n",
       "      <td>283</td>\n",
       "      <td>157</td>\n",
       "      <td>440</td>\n",
       "      <td>247</td>\n",
       "      <td>193</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>Corsham II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>...</td>\n",
       "      <td>341</td>\n",
       "      <td>301</td>\n",
       "      <td>642</td>\n",
       "      <td>302</td>\n",
       "      <td>340</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>Devizes II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>...</td>\n",
       "      <td>339</td>\n",
       "      <td>181</td>\n",
       "      <td>520</td>\n",
       "      <td>295</td>\n",
       "      <td>225</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>43</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>Pewsey Vale</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>7</td>\n",
       "      <td>...</td>\n",
       "      <td>359</td>\n",
       "      <td>127</td>\n",
       "      <td>486</td>\n",
       "      <td>288</td>\n",
       "      <td>198</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>36</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 23 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   Pos         Team Season  Pl  HW  HD  HL  AW  AD  AL  ...   HF   AF    F  \\\n",
       "0    1     Westbury  19/20  15   5   0   0   9   0   1  ...  171  343  514   \n",
       "1    2   Swindon II  19/20  18   8   0   1   4   0   5  ...  283  157  440   \n",
       "2    3   Corsham II  19/20  17   7   0   2   5   0   3  ...  341  301  642   \n",
       "3    4   Devizes II  19/20  17   7   0   2   3   1   4  ...  339  181  520   \n",
       "4    5  Pewsey Vale  19/20  17   8   0   1   1   0   7  ...  359  127  486   \n",
       "\n",
       "     A  Diff   TB   LB  Adj  Rand  Pts  \n",
       "0  174   340  0.0  0.0  0.0   NaN   56  \n",
       "1  247   193  0.0  0.0  0.0   NaN   50  \n",
       "2  302   340  0.0  0.0  0.0   NaN   48  \n",
       "3  295   225  0.0  0.0  0.0   NaN   43  \n",
       "4  288   198  0.0  0.0  0.0   NaN   36  \n",
       "\n",
       "[5 rows x 23 columns]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "league.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are a few columns we do not need and so first we will drop them and inspect the first few rows of our new data frame. Since I put this dataset together myself, I also know that there are no missing values and so no further cleaning is necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Pos</th>\n",
       "      <th>Team</th>\n",
       "      <th>Season</th>\n",
       "      <th>Pl</th>\n",
       "      <th>HW</th>\n",
       "      <th>HD</th>\n",
       "      <th>HL</th>\n",
       "      <th>AW</th>\n",
       "      <th>AD</th>\n",
       "      <th>AL</th>\n",
       "      <th>W</th>\n",
       "      <th>D</th>\n",
       "      <th>L</th>\n",
       "      <th>HF</th>\n",
       "      <th>AF</th>\n",
       "      <th>F</th>\n",
       "      <th>A</th>\n",
       "      <th>Diff</th>\n",
       "      <th>Pts</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>Westbury</td>\n",
       "      <td>19/20</td>\n",
       "      <td>15</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>9</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>14</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>171</td>\n",
       "      <td>343</td>\n",
       "      <td>514</td>\n",
       "      <td>174</td>\n",
       "      <td>340</td>\n",
       "      <td>56</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>Swindon II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>12</td>\n",
       "      <td>0</td>\n",
       "      <td>6</td>\n",
       "      <td>283</td>\n",
       "      <td>157</td>\n",
       "      <td>440</td>\n",
       "      <td>247</td>\n",
       "      <td>193</td>\n",
       "      <td>50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>Corsham II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>12</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>341</td>\n",
       "      <td>301</td>\n",
       "      <td>642</td>\n",
       "      <td>302</td>\n",
       "      <td>340</td>\n",
       "      <td>48</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>Devizes II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>10</td>\n",
       "      <td>1</td>\n",
       "      <td>6</td>\n",
       "      <td>339</td>\n",
       "      <td>181</td>\n",
       "      <td>520</td>\n",
       "      <td>295</td>\n",
       "      <td>225</td>\n",
       "      <td>43</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>Pewsey Vale</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>7</td>\n",
       "      <td>9</td>\n",
       "      <td>0</td>\n",
       "      <td>8</td>\n",
       "      <td>359</td>\n",
       "      <td>127</td>\n",
       "      <td>486</td>\n",
       "      <td>288</td>\n",
       "      <td>198</td>\n",
       "      <td>36</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   Pos         Team Season  Pl  HW  HD  HL  AW  AD  AL   W  D  L   HF   AF  \\\n",
       "0    1     Westbury  19/20  15   5   0   0   9   0   1  14  0  1  171  343   \n",
       "1    2   Swindon II  19/20  18   8   0   1   4   0   5  12  0  6  283  157   \n",
       "2    3   Corsham II  19/20  17   7   0   2   5   0   3  12  0  5  341  301   \n",
       "3    4   Devizes II  19/20  17   7   0   2   3   1   4  10  1  6  339  181   \n",
       "4    5  Pewsey Vale  19/20  17   8   0   1   1   0   7   9  0  8  359  127   \n",
       "\n",
       "     F    A  Diff  Pts  \n",
       "0  514  174   340   56  \n",
       "1  440  247   193   50  \n",
       "2  642  302   340   48  \n",
       "3  520  295   225   43  \n",
       "4  486  288   198   36  "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "league.drop(columns = ['TB', 'LB', 'Adj', 'Rand']).head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculate Relative Variables\n",
    "\n",
    "Even from these first few rows we can see that the teams have not all played an equal amount of matches. Therefore, it may be a good idea to calculate variables per game. This way we can make fair, direct comparisons and even scale up to a full 22 matches if deemed necessary or of interest.\n",
    "\n",
    "First, calculate the variables of interest per game (i.e. divide variable by no. of matches played):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "league['Rel F']    = league['F']    / league['Pl']\n",
    "league['Rel A']    = league['A']    / league['Pl']\n",
    "league['Rel Diff'] = league['Diff'] / league['Pl']\n",
    "league['Rel Pts']  = league['Pts']  / league['Pl']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check new columns by inspection:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Pos</th>\n",
       "      <th>Team</th>\n",
       "      <th>Season</th>\n",
       "      <th>Pl</th>\n",
       "      <th>HW</th>\n",
       "      <th>HD</th>\n",
       "      <th>HL</th>\n",
       "      <th>AW</th>\n",
       "      <th>AD</th>\n",
       "      <th>AL</th>\n",
       "      <th>...</th>\n",
       "      <th>Diff</th>\n",
       "      <th>TB</th>\n",
       "      <th>LB</th>\n",
       "      <th>Adj</th>\n",
       "      <th>Rand</th>\n",
       "      <th>Pts</th>\n",
       "      <th>Rel F</th>\n",
       "      <th>Rel A</th>\n",
       "      <th>Rel Diff</th>\n",
       "      <th>Rel Pts</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1</td>\n",
       "      <td>Westbury</td>\n",
       "      <td>19/20</td>\n",
       "      <td>15</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>9</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>...</td>\n",
       "      <td>340</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>56</td>\n",
       "      <td>34.266667</td>\n",
       "      <td>11.600000</td>\n",
       "      <td>22.666667</td>\n",
       "      <td>3.733333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2</td>\n",
       "      <td>Swindon II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>18</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>0</td>\n",
       "      <td>5</td>\n",
       "      <td>...</td>\n",
       "      <td>193</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>50</td>\n",
       "      <td>24.444444</td>\n",
       "      <td>13.722222</td>\n",
       "      <td>10.722222</td>\n",
       "      <td>2.777778</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3</td>\n",
       "      <td>Corsham II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>5</td>\n",
       "      <td>0</td>\n",
       "      <td>3</td>\n",
       "      <td>...</td>\n",
       "      <td>340</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>48</td>\n",
       "      <td>37.764706</td>\n",
       "      <td>17.764706</td>\n",
       "      <td>20.000000</td>\n",
       "      <td>2.823529</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4</td>\n",
       "      <td>Devizes II</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>7</td>\n",
       "      <td>0</td>\n",
       "      <td>2</td>\n",
       "      <td>3</td>\n",
       "      <td>1</td>\n",
       "      <td>4</td>\n",
       "      <td>...</td>\n",
       "      <td>225</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>43</td>\n",
       "      <td>30.588235</td>\n",
       "      <td>17.352941</td>\n",
       "      <td>13.235294</td>\n",
       "      <td>2.529412</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5</td>\n",
       "      <td>Pewsey Vale</td>\n",
       "      <td>19/20</td>\n",
       "      <td>17</td>\n",
       "      <td>8</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>7</td>\n",
       "      <td>...</td>\n",
       "      <td>198</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>36</td>\n",
       "      <td>28.588235</td>\n",
       "      <td>16.941176</td>\n",
       "      <td>11.647059</td>\n",
       "      <td>2.117647</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 27 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "   Pos         Team Season  Pl  HW  HD  HL  AW  AD  AL  ...  Diff   TB   LB  \\\n",
       "0    1     Westbury  19/20  15   5   0   0   9   0   1  ...   340  0.0  0.0   \n",
       "1    2   Swindon II  19/20  18   8   0   1   4   0   5  ...   193  0.0  0.0   \n",
       "2    3   Corsham II  19/20  17   7   0   2   5   0   3  ...   340  0.0  0.0   \n",
       "3    4   Devizes II  19/20  17   7   0   2   3   1   4  ...   225  0.0  0.0   \n",
       "4    5  Pewsey Vale  19/20  17   8   0   1   1   0   7  ...   198  0.0  0.0   \n",
       "\n",
       "   Adj  Rand  Pts      Rel F      Rel A   Rel Diff   Rel Pts  \n",
       "0  0.0   NaN   56  34.266667  11.600000  22.666667  3.733333  \n",
       "1  0.0   NaN   50  24.444444  13.722222  10.722222  2.777778  \n",
       "2  0.0   NaN   48  37.764706  17.764706  20.000000  2.823529  \n",
       "3  0.0   NaN   43  30.588235  17.352941  13.235294  2.529412  \n",
       "4  0.0   NaN   36  28.588235  16.941176  11.647059  2.117647  \n",
       "\n",
       "[5 rows x 27 columns]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "league.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Perform Linear Regression Analyses to Model the Relationship Between Variables of Interest\n",
    "\n",
    "In this section we will do as the title suggests and perfrom simple linear regression analyses. This will help us to identify which variables are most important to finish higher in the league table."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First import the Statsmodel module to perform the analyses:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first linear regression will look at the influence of each teams' average points difference per match against their finishing position in the league.\n",
    "\n",
    "Let's start by plotting these variables against each other."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Final League Position')"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(league['Rel Diff'], league['Pos'], 'rx') # A look at how average points difference per match influences final league position\n",
    "plt.gca().invert_yaxis()\n",
    "plt.yticks(np.arange(min(league['Pos']), max(league['Pos'])+1, 1.0))\n",
    "plt.xlabel(\"Average Points Difference Per Match\", fontweight='bold')\n",
    "plt.ylabel(\"Final League Position\", fontweight='bold')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, there appears to be a linear relationship of some degree where a greater average points difference per match leads to finishing higher up the table.\n",
    "\n",
    "Now run the linear regression between these variables.\n",
    "\n",
    "A simple linear regression like this one will have the structure:\n",
    "\n",
    "**$Y = C + M*X$**\n",
    "\n",
    "Where:\n",
    "\n",
    "- $Y$ = Dependent variable (output/outcome/prediction/estimation)\n",
    "- $C$ = Constant (Y-Intercept)\n",
    "- $M$ = Slope of the regression line (the effect that X has on Y)\n",
    "- $X$ = Independent variable (input variable used in the prediction of Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                    Pos   R-squared:                       0.815\n",
      "Model:                            OLS   Adj. R-squared:                  0.812\n",
      "Method:                 Least Squares   F-statistic:                     256.2\n",
      "Date:                Fri, 07 Aug 2020   Prob (F-statistic):           6.03e-23\n",
      "Time:                        15:09:23   Log-Likelihood:                -108.78\n",
      "No. Observations:                  60   AIC:                             221.6\n",
      "Df Residuals:                      58   BIC:                             225.8\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          6.4385      0.195     33.056      0.000       6.049       6.828\n",
      "Rel Diff      -0.1699      0.011    -16.007      0.000      -0.191      -0.149\n",
      "==============================================================================\n",
      "Omnibus:                        0.475   Durbin-Watson:                   1.447\n",
      "Prob(Omnibus):                  0.789   Jarque-Bera (JB):                0.303\n",
      "Skew:                           0.174   Prob(JB):                        0.859\n",
      "Kurtosis:                       2.970   Cond. No.                         18.4\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "X = league['Rel Diff'] # Model input variable\n",
    "Y = league['Pos'] # Model output variable\n",
    "\n",
    "X = sm.add_constant(X) # adding a constant\n",
    "\n",
    "model = sm.OLS(Y, X).fit() # Fit using ordinary least squares\n",
    "predictions = model.predict(X) \n",
    "\n",
    "model_summary = model.summary()\n",
    "print(model_summary)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives the linear equation:\n",
    "\n",
    "$Y = 6.4385 + (-0.1699 * X)$\n",
    "\n",
    "Where:\n",
    "\n",
    "- $Y$ = League Position\n",
    "- $X$ = Points Difference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's visualise this in a graph:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(-40, 3, '$R^2$ = 0.815')"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = sm.graphics.abline_plot(model_results=model) # Plot regression line\n",
    "fig.set_size_inches(6, 5)\n",
    "ax = fig.axes[0]\n",
    "ax.scatter(league['Rel Diff'], league['Pos'], c='r', marker='x') # Scatter plot\n",
    "ax.invert_yaxis()\n",
    "plt.xlabel(\"Average Points Difference Per Match\", fontweight='bold')\n",
    "plt.ylabel(\"Final League Position\", fontweight='bold')\n",
    "plt.xlim(-50, 40)    \n",
    "plt.ylim(13, 0)\n",
    "plt.yticks(np.arange(min(league['Pos']), max(league['Pos'])+1, 1.0))\n",
    "plt.text(-40,3, '$R^2$ = {:.3f}'.format(model.rsquared), fontsize=12) # Add R-squared to graph with 3 decimal places"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Based on the results the linear regression equation appears to fit the data well. It can be said that the model explains $81.5$% of the variation in the dependent variable (final league position), which is interpreted from the R-squared value ($R^2 = 0.815$). The independent variable (average points difference per match) is also a significant ($p < 0.001$) predictor of final league position in this model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiple Linear Regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's run run a multiple linear regression and look at the influence of average points scored and average points conceded per match against final league position.\n",
    "\n",
    "A multiple linear regression like this one will have the structure:\n",
    "\n",
    "**$Y = C + B_1*X_1 + B_2*X_2$**\n",
    "\n",
    "Where:\n",
    "\n",
    "- $Y$ = Dependent variable (output/outcome/prediction/estimation)\n",
    "- $C$ = Constant (Y-Intercept)\n",
    "- $B_n$ = Independent variable coefficients (the effect that X has on Y)\n",
    "- $X_n$ = Independent variables (input variable used in the prediction of Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                    Pos   R-squared:                       0.822\n",
      "Model:                            OLS   Adj. R-squared:                  0.816\n",
      "Method:                 Least Squares   F-statistic:                     131.9\n",
      "Date:                Fri, 07 Aug 2020   Prob (F-statistic):           4.11e-22\n",
      "Time:                        15:09:24   Log-Likelihood:                -107.64\n",
      "No. Observations:                  60   AIC:                             221.3\n",
      "Df Residuals:                      57   BIC:                             227.6\n",
      "Df Model:                           2                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "const          4.4040      1.380      3.191      0.002       1.640       7.168\n",
      "Rel F         -0.1236      0.033     -3.768      0.000      -0.189      -0.058\n",
      "Rel A          0.2116      0.030      7.073      0.000       0.152       0.271\n",
      "==============================================================================\n",
      "Omnibus:                        3.580   Durbin-Watson:                   1.405\n",
      "Prob(Omnibus):                  0.167   Jarque-Bera (JB):                2.761\n",
      "Skew:                           0.504   Prob(JB):                        0.251\n",
      "Kurtosis:                       3.298   Cond. No.                         237.\n",
      "==============================================================================\n",
      "\n",
      "Warnings:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n"
     ]
    }
   ],
   "source": [
    "X2 = league[['Rel F', 'Rel A']] # Model input variable\n",
    "Y2 = league['Pos'] # Model output variable\n",
    "\n",
    "X2 = sm.add_constant(X2) # adding a constant\n",
    "\n",
    "model2 = sm.OLS(Y2, X2).fit() # Fit using ordinary least squares\n",
    "predictions = model2.predict(X2) \n",
    "\n",
    "model2_summary = model2.summary()\n",
    "print(model2_summary)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This gives the linear equation:\n",
    "\n",
    "$Y = 4.4040 + (-0.1236 * X_1) + (0.2116 * X_2)$\n",
    "\n",
    "Where:\n",
    "\n",
    "- $Y$ = League Position\n",
    "- $X_1$ = Points Scored\n",
    "- $X_2$ = Points Conceded"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, let's visualise this. It's not as simple to plot a multiple regression so we will just plot two scatter plots - one for each of the independent variables. Unfortunately this does mean that we can't plot the regression equation on the figures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Final League Position')"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 1080x432 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "fig.set_size_inches(15, 6)\n",
    "\n",
    "plt.subplot(1, 2, 1)\n",
    "plt.plot(league['Rel F'], league['Pos'], 'rx')\n",
    "plt.gca().invert_yaxis()\n",
    "plt.yticks(np.arange(min(league['Pos']), max(league['Pos'])+1, 1.0))\n",
    "plt.xlabel(\"Average Points Scored Per Match\", fontweight='bold')\n",
    "plt.ylabel(\"Final League Position\", fontweight='bold')\n",
    "\n",
    "plt.subplot(1, 2, 2)\n",
    "plt.plot(league['Rel A'], league['Pos'], 'rx')\n",
    "plt.gca().invert_yaxis()\n",
    "plt.yticks(np.arange(min(league['Pos']), max(league['Pos'])+1, 1.0))\n",
    "plt.xlabel(\"Average Points Conceded Per Match\", fontweight='bold')\n",
    "plt.ylabel(\"Final League Position\", fontweight='bold')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results of this multiple linear regression indicate that splitting points difference into points scored and points conceded explains slightly more of the variation ($82.2$%; $R^2 = 0.822$) in the final league position than using points difference alone ($81.5$%; $R^2 = 0.815$). Both of the independent variables are also significant ($p < 0.001$) predictors of final league position, with the average number of points conceded per match having a greater influence than points scored. This can be observed from the coefficient values (Points scored coeff. = $-0.1236$; Points conceded coeff. =  $0.2116$) and also from the figures above - where, based on visual inspection, there appears to be a slightly stronger correlation between points conceded and final league position than points scored and final league position."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Summary\n",
    "\n",
    "This brief piece of analysis looks at the table from one of my local leagues over the past few seasons. It demonstrates how a team's final position in the league can be modelled using linear regression and variables such as points scored and conceded per match, and also overall points difference per match. \n",
    "\n",
    "Obviously there are many points of interest that could be singled out for discussion. One I find interesting is how points conceded is weighted nearly twice as much as points scored (in the multiple linear regression), indicating that good defence would be more important than attack in achieveing a higher league position. Another that stood out to me was the constants given by the models. The simple linear regression using just points difference resulted in a constant of 6.4385. This suggests that if a single team achieved a points difference of zero in every match (i.e. draw every match) then they would finish 6th in the league (6.4385 rounded to the nearest integer). Whereas, using the multiple linear regression, and separating points difference into two separate independent variables of points scored and points conceded, resulted in a constant of 4.4040. This value suggests that if a team draws every match then they will finish 4th in the league. Though the latter would reqiure the team not to score or concede, i.e. a score of 0-0, and other factors such as bonus points are not considered.\n",
    "\n",
    "Whilst there are clearly more influenicing factors at play that have not been accounted for and collecting more seasons worth of data would improve the accuracy of the models, the aim of this work was simply to practise performing linear regression analyses in Python. $\\square$"
   ]
  }
 ],
 "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}