simples regressão linear em Julia

simple_linear_regression-julia.ipynb

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   "cell_type": "code",
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   "source": [
    "using DataFrames, CSV, Query"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = CSV.read(\"co2_por_nacao.csv\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>Nation</th><th>Year</th><th>Total CO2 emissions from fossil-fuels and cement production (thousand metric tons of C)</th><th>Emissions from solid fuel consumption</th><th>Emissions from liquid fuel consumption</th><th>Emissions from gas fuel consumption</th><th>Emissions from cement production</th><th>Emissions from gas flaring</th><th>Per capita CO2 emissions (metric tons of carbon)</th><th>Emissions from bunker fuels (not included in the totals)</th></tr></thead><tbody><tr><th>1</th><td>AFGHANISTAN</td><td>1949</td><td>4</td><td>4</td><td>0</td><td>0</td><td>0</td><td>missing</td><td>missing</td><td>0</td></tr><tr><th>2</th><td>AFGHANISTAN</td><td>1950</td><td>23</td><td>6</td><td>18</td><td>0</td><td>0</td><td>0</td><td>0.0</td><td>0</td></tr><tr><th>3</th><td>AFGHANISTAN</td><td>1951</td><td>25</td><td>7</td><td>18</td><td>0</td><td>0</td><td>0</td><td>0.0</td><td>0</td></tr><tr><th>4</th><td>AFGHANISTAN</td><td>1952</td><td>25</td><td>9</td><td>17</td><td>0</td><td>0</td><td>0</td><td>0.0</td><td>0</td></tr><tr><th>5</th><td>AFGHANISTAN</td><td>1953</td><td>29</td><td>10</td><td>18</td><td>0</td><td>0</td><td>0</td><td>0.0</td><td>0</td></tr><tr><th>6</th><td>AFGHANISTAN</td><td>1954</td><td>29</td><td>12</td><td>18</td><td>0</td><td>0</td><td>0</td><td>0.0</td><td>0</td></tr></tbody></table>"
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       "6×10 DataFrame. Omitted printing of 8 columns\n",
       "│ Row │ Nation      │ Year │\n",
       "├─────┼─────────────┼──────┤\n",
       "│ 1   │ AFGHANISTAN │ 1949 │\n",
       "│ 2   │ AFGHANISTAN │ 1950 │\n",
       "│ 3   │ AFGHANISTAN │ 1951 │\n",
       "│ 4   │ AFGHANISTAN │ 1952 │\n",
       "│ 5   │ AFGHANISTAN │ 1953 │\n",
       "│ 6   │ AFGHANISTAN │ 1954 │"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "head(df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "col = Symbol(\"Total CO2 emissions from fossil-fuels and cement production (thousand metric tons of C)\")\n",
    "nation_str = \"JAPAN\";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "nation_df = @from i in df begin\n",
    "    @where i.Nation == nation_str\n",
    "    @select i\n",
    "    @collect DataFrame\n",
    "end\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nunique</th><th>nmissing</th><th>eltype</th></tr></thead><tbody><tr><th>1</th><td>Nation</td><td></td><td>JAPAN</td><td></td><td>JAPAN</td><td>1</td><td>0</td><td>CategoricalString{UInt32}</td></tr><tr><th>2</th><td>Year</td><td>1981.5</td><td>1950</td><td>1981.5</td><td>2013</td><td></td><td>0</td><td>Int64</td></tr><tr><th>3</th><td>Total CO2 emissions from fossil-fuels and cement production (thousand metric tons of C)</td><td>2.22702e5</td><td>27991</td><td>249744.0</td><td>345244</td><td></td><td>0</td><td>Int64</td></tr><tr><th>4</th><td>Emissions from solid fuel consumption</td><td>70233.7</td><td>26127</td><td>64468.5</td><td>125349</td><td></td><td>0</td><td>Int64</td></tr><tr><th>5</th><td>Emissions from liquid fuel consumption</td><td>1.24351e5</td><td>1220</td><td>1.53707e5</td><td>189585</td><td></td><td>0</td><td>Int64</td></tr><tr><th>6</th><td>Emissions from gas fuel consumption</td><td>20150.4</td><td>37</td><td>13419.5</td><td>66389</td><td></td><td>0</td><td>Int64</td></tr><tr><th>7</th><td>Emissions from cement production</td><td>7961.31</td><td>607</td><td>9275.0</td><td>12851</td><td></td><td>0</td><td>Int64</td></tr><tr><th>8</th><td>Emissions from gas flaring</td><td>5.6875</td><td>0</td><td>0.0</td><td>30</td><td></td><td>0</td><td>Int64</td></tr><tr><th>9</th><td>Per capita CO2 emissions (metric tons of carbon)</td><td>1.89266</td><td>0.34</td><td>2.195</td><td>2.73</td><td></td><td>0</td><td>Float64</td></tr><tr><th>10</th><td>Emissions from bunker fuels (not included in the totals)</td><td>8173.27</td><td>429</td><td>8651.5</td><td>18070</td><td></td><td>0</td><td>Int64</td></tr></tbody></table>"
      ],
      "text/plain": [
       "10×8 DataFrame. Omitted printing of 8 columns\n",
       "│ Row │ ├─────┼\n",
       "│ 1   │ │ 2   │ │ 3   │ │ 4   │ │ 5   │ │ 6   │ │ 7   │ │ 8   │ │ 9   │ │ 10  │ "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "describe(nation_df)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# calculando os coeficientes para regressão linear:\n",
    "\n",
    "$$ \\beta = \\frac{\\sum^n_{n=1} x_iy_i - \\frac{1}{n} \\sum^n_{i=1} x_i \\sum^n_{j=1} y_i}{\\sum^n_{i=1} (x^2_i) - \\frac{1}{n}(\\sum^n_{i=1} x_i)^2}  $$\n",
    "\n",
    "$$ \\alpha = \\frac{\\sum^n_{n=1} y_i}{n} - \\beta \\frac{\\sum^n_{i=1} x_i}{n} $$\n",
    "\n",
    "$$ f(x) =  \\alpha + \\beta x$$\n",
    "\n",
    "*fonte: https://pt.wikipedia.org/wiki/Regress%C3%A3o_linear_simples*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = nation_df[:Year], nation_df[col]\n",
    "n = length(x)\n",
    ";"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5492.159088827839"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "β = (sum(x.*y) -1/n * sum(x) * sum(y) ) / (sum(x.^2) -1/n * sum(x)^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0660010843887363e7"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "α = sum(y)/n - β * sum(x)/n"
   ]
  }
 ],
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