Taller1.ipynb

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    "## Universidad Nacional De Colombia\n",
    "## Introduction to Optimization\n",
    "### First Excercices\n",
    "\n",
    "### Presented by Daniel Felipe Quinones Ordonez"
   ]
  },
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   "source": [
    "# First Point"
   ]
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    "#### Related with the nature of the data, which conditions should meet matrices  A  and  b  to be solvable from the least squares perspective.\n",
    "\n",
    "#### Answer:\n",
    "When we have a data set what we are looking is that there is no linear dependency among the points, or observations in our case. Also the \"well\" behavior of the data will imply a good condition number of the resultant matrix. That is \n",
    "\n",
    "$k = ||A||*||A||^{-1}$\n",
    "\n",
    "Based on that we see that the data matrix must have inverse, that could give us additional conditions over the data, that depending of our resolution algorithm, approach, constraints in the problem, could give us a better approximation to the best solution possible. Some examples ot those conditions are if the matrix is a Vandermonde mayrtix, the inverse is granted for obsevation values, all differentes among themselves.[1]\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "### References:\n",
    "\n",
    "[1] Rescued from: Polynomial regression, Wikipedia. https://en.wikipedia.org/wiki/Polynomial_regression"
   ]
  },
  {
   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "# Second Point"
   ]
  },
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   "id": "e78f745f",
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   "source": [
    "Importing the data of bitocins values of a whole month"
   ]
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  {
   "cell_type": "code",
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    "using CSV\n",
    "using DataFrames\n",
    "\n",
    "bitcoinData = CSV.read(\"C:\\\\Users\\\\danqu\\\\Documents\\\\UNAL\\\\CC\\\\introOpti\\\\bitcoin.csv\", DataFrame);\n",
    "bitcoinData.Price .= replace.(bitcoinData.Price, \",\" => \"\");\n",
    "bitcoinData.PriceFloat = parse.(Float64, bitcoinData.Price);"
   ]
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   "id": "7dedccb0",
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       " 37224.6\n",
       "     ⋮\n",
       " 40000.1\n",
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     "execution_count": 130,
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   "source": [
    "bMarch = bitcoinData.PriceFloat"
   ]
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  {
   "cell_type": "code",
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   "id": "f9774efa",
   "metadata": {},
   "outputs": [],
   "source": [
    "bitcoinData;"
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  {
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   "id": "46970d33",
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   "source": [
    "using Plots\n",
    "\n",
    "plot(bMarch);"
   ]
  },
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   "id": "5d0bd743",
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       "29×4 Matrix{Float64}:\n",
       " 1.0   1.0    1.0      1.0\n",
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       " 1.0  10.0  100.0   1000.0\n",
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       " 1.0  12.0  144.0   1728.0\n",
       " 1.0  13.0  169.0   2197.0\n",
       " ⋮                 \n",
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       " 1.0  20.0  400.0   8000.0\n",
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       " 1.0  22.0  484.0  10648.0\n",
       " 1.0  23.0  529.0  12167.0\n",
       " 1.0  24.0  576.0  13824.0\n",
       " 1.0  25.0  625.0  15625.0\n",
       " 1.0  26.0  676.0  17576.0\n",
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       " 1.0  28.0  784.0  21952.0\n",
       " 1.0  29.0  841.0  24389.0"
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    "M = size(bMarch, 1);\n",
    "Am = Array((1:M));\n",
    "Amarch = [ones(M) Am Am.^2];\n",
    "Amarch3 = [ones(M) Am Am.^2 Am.^3];\n",
    "Amarch3"
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  {
   "cell_type": "code",
   "execution_count": 158,
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       "4-element Vector{Float64}:\n",
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   "source": [
    "using LinearAlgebra\n",
    "\n",
    "qrAmarch3 = Amarch3\\bMarch\n",
    "qrAmarch = Amarch\\bMarch\n"
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    "estimator = qrAmarch[3]*Amarch[:,3] + qrAmarch[2]*Amarch[:,2] + qrAmarch[1]*Amarch[:,1];\n",
    "estimator3 = qrAmarch3[4]*Amarch3[:,4] + qrAmarch3[3]*Amarch3[:,3] + qrAmarch3[2]*Amarch3[:,2] + qrAmarch3[1]*Amarch3[:,1];\n",
    "#prediction = x[5]*A[:,5] + x[4]*A[:,4] +  x[3]*A[:,3] +  x[2]*A[:,2] + x[1]*ones(N,1) #Setting the equation of the estimator"
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     "execution_count": 162,
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   ],
   "source": [
    "plot(Amarch[:,2], [bMarch, estimator, estimator3], labels = [\"Bitcoin Price\" \"Comodities Estimator\"], lw = [2 3])\n",
    "xlabel!(\"Days\")\n",
    "ylabel!(\"Price(USD)\")"
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  {
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   "id": "f78e99f3",
   "metadata": {},
   "source": [
    "# Third Point"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "782f46e7",
   "metadata": {},
   "source": [
    "Third point: we going to evaluate the relation among somo comodities and the bitcoin price. The comodities to take into account are: gold, silver, gas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "2328d612",
   "metadata": {},
   "outputs": [],
   "source": [
    "using CSV\n",
    "using DataFrames\n",
    "\n",
    "comoditiesData = CSV.read(\"C:\\\\Users\\\\danqu\\\\Documents\\\\UNAL\\\\CC\\\\introOpti\\\\comoditiesPrices.csv\", DataFrame);\n",
    "comoditiesData = dropmissing(comoditiesData); #removes the missing data\n",
    "comoditiesData.BitcoinPrice .= replace.(comoditiesData.BitcoinPrice, \",\" => \"\");\n",
    "comoditiesData.BitcoinPrice64 = parse.(Float64, comoditiesData.BitcoinPrice);\n",
    "comoditiesData = select!(comoditiesData, Not(:Date)); #removes the date column\n",
    "comoditiesData = select!(comoditiesData, Not(:BitcoinPrice)); #removes the BitcoinPrice column for be a string column\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "a22251e7",
   "metadata": {},
   "outputs": [
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       "<div class=\"data-frame\"><p>258 rows × 4 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>GoldPrice</th><th>SilverPrice</th><th>OilPrice</th><th>BitcoinPrice64</th></tr><tr><th></th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>1725.1</td><td>25.227</td><td>57.76</td><td>54452.5</td></tr><tr><th>2</th><td>1733.2</td><td>25.231</td><td>61.18</td><td>52325.4</td></tr><tr><th>3</th><td>1725.1</td><td>25.047</td><td>58.56</td><td>51322.3</td></tr><tr><th>4</th><td>1732.78</td><td>25.114</td><td>60.97</td><td>55036.1</td></tr><tr><th>5</th><td>1713.16</td><td>24.771</td><td>61.56</td><td>57616.2</td></tr><tr><th>6</th><td>1685.16</td><td>24.137</td><td>60.55</td><td>58771.3</td></tr><tr><th>7</th><td>1715.24</td><td>24.532</td><td>59.16</td><td>58763.7</td></tr><tr><th>8</th><td>1728.4</td><td>24.948</td><td>61.45</td><td>58718.3</td></tr><tr><th>9</th><td>1728.8</td><td>24.78</td><td>58.65</td><td>58993.4</td></tr><tr><th>10</th><td>1743.0</td><td>25.227</td><td>59.33</td><td>57996.3</td></tr><tr><th>11</th><td>1741.6</td><td>25.247</td><td>59.77</td><td>55948.7</td></tr><tr><th>12</th><td>1758.2</td><td>25.585</td><td>59.6</td><td>58077.4</td></tr><tr><th>13</th><td>1744.8</td><td>25.325</td><td>59.32</td><td>58118.7</td></tr><tr><th>14</th><td>1732.7</td><td>24.867</td><td>59.7</td><td>59863.8</td></tr><tr><th>15</th><td>1747.6</td><td>25.426</td><td>60.18</td><td>63540.9</td></tr><tr><th>16</th><td>1736.3</td><td>25.524</td><td>63.15</td><td>62980.4</td></tr><tr><th>17</th><td>1766.8</td><td>25.964</td><td>63.47</td><td>63216.0</td></tr><tr><th>18</th><td>1780.2</td><td>26.105</td><td>63.154</td><td>61379.7</td></tr><tr><th>19</th><td>1770.6</td><td>25.837</td><td>63.41</td><td>55646.1</td></tr><tr><th>20</th><td>1778.4</td><td>25.84</td><td>62.624</td><td>56483.2</td></tr><tr><th>21</th><td>1793.1</td><td>26.57</td><td>61.35</td><td>53820.2</td></tr><tr><th>22</th><td>1782.0</td><td>26.18</td><td>61.43</td><td>51729.5</td></tr><tr><th>23</th><td>1777.8</td><td>26.075</td><td>62.14</td><td>51143.6</td></tr><tr><th>24</th><td>1780.1</td><td>26.209</td><td>61.91</td><td>54020.5</td></tr><tr><th>25</th><td>1778.8</td><td>26.418</td><td>62.94</td><td>55036.5</td></tr><tr><th>26</th><td>1773.9</td><td>26.0982</td><td>63.86</td><td>54841.4</td></tr><tr><th>27</th><td>1768.3</td><td>26.0722</td><td>65.01</td><td>53560.8</td></tr><tr><th>28</th><td>1767.7</td><td>25.869</td><td>63.58</td><td>57720.3</td></tr><tr><th>29</th><td>1791.8</td><td>26.96</td><td>64.49</td><td>57169.8</td></tr><tr><th>30</th><td>1776.0</td><td>26.558</td><td>65.69</td><td>53741.5</td></tr><tr><th>&vellip;</th><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td><td>&vellip;</td></tr></tbody></table></div>"
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       "\t29 & 1791.8 & 26.96 & 64.49 & 57169.8 \\\\\n",
       "\t30 & 1776.0 & 26.558 & 65.69 & 53741.5 \\\\\n",
       "\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
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       "  ⋮  │     ⋮           ⋮          ⋮            ⋮\n",
       " 249 │   2000.4        26.256   108.7           39416.3\n",
       " 250 │   1985.0        26.16    106.02          38730.2\n",
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       " 258 │   1937.3        25.189   109.768         42912.0\n",
       "\u001b[36m                                        237 rows omitted\u001b[0m"
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      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "#x = 0:size(comoditiesData.SilverPrice);\n",
    "plot(comoditiesData.SilverPrice)\n",
    "#plot!(x, comoditiesData.SilverPrice)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "id": "ddcacd3b",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  814.819,1098.3 822.718,1073.66 830.617,1032.34 838.515,1002.39 846.414,1009.59 854.312,970.162 862.211,987.98 870.11,972.816 878.008,915.57 885.907,900.406 \n",
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       "  1051.78,1101.33 1059.68,1057.36 1067.57,1064.56 1075.47,1077.45 1083.37,1082.76 1091.27,1079.35 1099.17,995.183 1107.07,986.464 1114.97,1052.81 1122.86,1036.89 \n",
       "  1130.76,945.141 1138.66,973.195 1146.56,950.448 1154.46,958.03 1162.36,974.711 1170.26,890.928 1178.16,890.928 1186.05,1024.37 1193.95,1043.71 1201.85,1018.69 \n",
       "  1209.75,1048.64 1217.65,1039.54 1225.55,991.392 1233.45,1038.4 1241.34,1182.46 1249.24,1202.18 1257.14,1155.17 1265.04,1100.95 1272.94,1098.68 1280.84,1208.62 \n",
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       "  1367.72,1171.85 1375.62,1178.29 1383.52,1164.64 1391.42,1030.44 1399.32,1018.31 1407.21,1130.52 1415.11,1140.38 1423.01,1122.18 1430.91,1067.59 1438.81,1078.97 \n",
       "  1446.71,1024.37 1454.61,984.568 1462.5,1035.37 1470.4,1014.9 1478.3,1000.49 1486.2,1071.38 1494.1,1026.27 1502,1050.53 1509.9,1147.21 1517.79,1034.99 \n",
       "  1525.69,946.657 1533.59,904.197 1541.49,893.582 1549.39,827.238 1557.29,768.097 1565.19,750.658 1573.09,757.861 1580.98,805.25 1588.88,744.213 1596.78,777.575 \n",
       "  1604.68,814.727 1612.58,986.464 1620.48,1071.76 1628.38,1067.9 1636.27,1061.38 1644.17,1070.85 1652.07,1101.64 1659.97,1069.87 1667.87,1151.75 1675.77,1071.38 \n",
       "  1683.67,1088.06 1691.56,1068.35 1699.46,1065.32 1707.36,1098.68 1715.26,1067.97 1723.16,1054.7 1731.06,1115.36 1738.96,1144.93 1746.85,1017.17 1754.75,991.771 \n",
       "  1762.65,1030.82 1770.55,1053.19 1778.45,1002.01 1786.35,965.992 1794.25,965.992 1802.14,965.992 1810.04,976.986 1817.94,969.025 1825.84,988.359 1833.74,956.893 \n",
       "  1841.64,901.922 1849.54,1009.97 1857.43,954.998 1865.33,915.191 1873.23,1051.29 1881.13,1020.2 1889.03,1014.9 1896.93,940.212 1904.83,906.851 1912.72,929.218 \n",
       "  1920.62,947.795 1928.52,963.338 1936.42,846.573 1944.32,848.847 1952.22,889.791 1960.12,852.259 1968.02,811.315 1975.91,896.008 1983.81,1033.62 1991.71,1063.73 \n",
       "  1999.61,1025.06 2007.51,1004.66 2015.41,971.299 2023.31,994.804 2031.2,980.777 2039.1,927.702 2047,904.576 2054.9,871.594 2062.8,868.561 2070.7,850.743 \n",
       "  2078.6,747.246 2086.49,797.288 2094.39,739.285 2102.29,623.657 2110.19,631.997 2118.09,631.997 2125.99,603.185 2133.89,591.812 2141.78,531.533 2149.68,678.248 \n",
       "  2157.58,628.585 2165.48,465.189 2173.38,546.698 2181.28,495.139 2189.18,378.753 2197.07,267.674 2204.97,87.9763 2212.87,296.865 2220.77,250.614 2228.67,308.997 \n",
       "  2236.57,400.741 2244.47,518.644 2252.36,596.361 2260.26,467.464 2268.16,520.16 2276.06,519.402 2283.96,549.731 2291.86,489.831 \n",
       "  \"/>\n",
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       "  2013.43,147.058 2156.88,147.058 \n",
       "  \"/>\n",
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      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(comoditiesData.GoldPrice)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "id": "37e48014",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[54452.5, 52325.4, 51322.3, 55036.1, 57616.2, 58771.3, 58763.7, 58718.3, 58993.4, 57996.3, 55948.7, 58077.4, 58118.7, 59863.8, 63540.9, 62980.4, 63216.0, 61379.7, 55646.1, 56483.2, 53820.2, 51729.5, 51143.6, 54020.5, 55036.5, 54841.4, 53560.8, 57720.3, 57169.8, 53741.5, 57441.3, 56405.4, 57337.2, 55848.9, 56695.7, 49384.2, 49704.6, 49839.8, 43541.3, 42897.3, 36720.5, 40717.2, 37297.4, 38750.6, 38378.3, 39249.2, 38417.3, 35662.5, 36687.6, 37555.8, 39187.3, 36851.3, 33578.0, 33382.9, 37332.2, 36649.4, 37314.6, 40529.4, 40156.1, 38336.0, 38052.0, 35749.4, 31692.0, 32496.4, 33674.3, 34665.8, 31594.0, 34475.9, 35834.7, 35026.9, 33543.6, 33813.4, 34225.6, 33867.8, 32866.3, 33797.4, 33113.0, 32728.1, 32820.7, 31840.5, 31394.0, 30837.2, 29793.8, 32131.4, 32298.9, 33603.3, 37276.6, 39452.0, 40003.2, 40001.4, 42203.4, 39168.4, 38130.3, 39736.9, 40867.2, 42795.4, 46284.3, 45593.8, 45564.3, 44403.4, 47809.1, 45996.3, 44691.6, 44723.8, 46755.9, 49324.0, 49539.7, 47714.7, 48994.5, 46831.6, 49064.3, 46992.7, 47130.4, 48819.4, 49274.3, 49999.0, 52672.1, 46779.6, 46061.4, 46385.6, 44842.8, 44949.5, 47077.5, 48130.6, 47748.0, 47282.8, 42870.6, 40651.3, 43551.6, 44869.2, 42819.9, 42172.6, 41022.3, 41536.8, 43823.3, 48146.0, 49227.3, 51469.3, 55323.2, 53783.9, 53914.7, 57477.3, 56015.9, 57380.1, 57345.8, 61672.5, 62056.3, 64278.5, 65979.1, 62210.2, 60690.3, 63067.0, 60310.8, 58438.1, 60582.5, 62242.8, 60915.3, 63221.7, 62918.5, 61412.6, 60974.3, 67527.9, 66904.4, 64932.6, 64806.7, 64134.5, 63597.9, 60089.1, 60367.8, 56955.3, 58083.4, 56308.8, 57573.2, 57171.7, 53730.9, 57794.8, 56882.9, 57210.3, 56496.0, 53631.5, 50562.1, 50595.2, 50473.9, 47596.6, 47137.2, 46703.6, 48366.5, 48871.5, 47628.2, 46172.2, 46895.2, 48902.0, 48599.7, 50801.0, 50786.1, 50779.9, 50678.2, 47545.2, 46461.7, 47123.3, 46219.5, 46430.2, 45837.3, 43425.9, 43097.5, 41546.7, 41832.2, 42733.2, 43901.0, 42560.0, 43073.3, 42364.6, 41677.8, 40715.9, 36475.5, 36686.3, 36950.6, 36800.4, 37164.3, 37745.1, 38498.6, 38709.7, 36890.2, 37287.8, 41567.6, 43833.9, 44038.2, 44388.9, 43521.0, 42388.4, 42550.3, 44544.4, 43883.6, 40552.8, 40000.1, 37017.7, 38248.2, 37224.6, 38339.2, 39209.6, 43188.2, 44420.3, 43912.8, 42463.0, 39142.7, 38024.8, 38744.8, 41929.0, 39416.3, 38730.2, 39671.1, 39285.7, 41118.7, 40914.8, 41767.8, 41022.0, 42373.0, 42912.0]\n"
     ]
    }
   ],
   "source": [
    "N = size(comoditiesData.BitcoinPrice64, 1);\n",
    "A = Array((1:N));\n",
    "A = [ones(N) A comoditiesData.GoldPrice comoditiesData.SilverPrice comoditiesData.OilPrice]; #Setting the matrix of the model\n",
    "b = Array(comoditiesData.BitcoinPrice64); #vector with the current prices of bitcoin. (That's what we want to estimate)\n",
    "println(b);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "id": "e5c3673c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "258×5 Matrix{Float64}:\n",
       " 1.0    1.0  1725.1   25.227   57.76\n",
       " 1.0    2.0  1733.2   25.231   61.18\n",
       " 1.0    3.0  1725.1   25.047   58.56\n",
       " 1.0    4.0  1732.78  25.114   60.97\n",
       " 1.0    5.0  1713.16  24.771   61.56\n",
       " 1.0    6.0  1685.16  24.137   60.55\n",
       " 1.0    7.0  1715.24  24.532   59.16\n",
       " 1.0    8.0  1728.4   24.948   61.45\n",
       " 1.0    9.0  1728.8   24.78    58.65\n",
       " 1.0   10.0  1743.0   25.227   59.33\n",
       " 1.0   11.0  1741.6   25.247   59.77\n",
       " 1.0   12.0  1758.2   25.585   59.6\n",
       " 1.0   13.0  1744.8   25.325   59.32\n",
       " ⋮                            \n",
       " 1.0  247.0  2043.3   26.895  119.4\n",
       " 1.0  248.0  1988.2   25.816  123.7\n",
       " 1.0  249.0  2000.4   26.256  108.7\n",
       " 1.0  250.0  1985.0   26.16   106.02\n",
       " 1.0  251.0  1960.8   25.298  109.33\n",
       " 1.0  252.0  1929.7   25.158  103.01\n",
       " 1.0  253.0  1909.2   24.71    96.44\n",
       " 1.0  254.0  1943.2   25.616   95.04\n",
       " 1.0  255.0  1929.3   25.087  102.714\n",
       " 1.0  256.0  1929.5   25.313  104.056\n",
       " 1.0  257.0  1921.5   24.904  110.83\n",
       " 1.0  258.0  1937.3   25.189  109.768"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "id": "2e71a0a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 149356.75648746028\n",
       "    -13.5361390622061\n",
       "    -46.107082527854864\n",
       "   -833.1726678203818\n",
       "     43.67697314031704"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "#Using the QR descomposition implemented by Julia\n",
    "qrA = qr(A);\n",
    "x = qrA \\ b;\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "id": "e031a08f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "258×1 Matrix{Float64}:\n",
       " 51308.22735707559\n",
       " 51067.26640700636\n",
       " 51466.067737671096\n",
       " 51147.868141319166\n",
       " 52350.500600668645\n",
       " 54112.080500912794\n",
       " 52321.8291229586\n",
       " 51454.94421650788\n",
       " 51440.64272783547\n",
       " 50429.65817609744\n",
       " 50483.22636739956\n",
       " 49415.27521121781\n",
       " 50223.96931918287\n",
       "     ⋮\n",
       " 34609.580101854226\n",
       " 38223.34850315837\n",
       " 36625.5553863106\n",
       " 37284.998606272085\n",
       " 39250.01948513958\n",
       " 40511.01931594171\n",
       " 41528.98201035216\n",
       " 39131.802865901205\n",
       " 40535.08060713195\n",
       " 40382.64052659107\n",
       " 41374.59648494274\n",
       " 40348.7292861366"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "prediction = x[5]*A[:,5] + x[4]*A[:,4] +  x[3]*A[:,3] +  x[2]*A[:,2] + x[1]*ones(N,1) #Setting the equation of the estimator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "id": "b0ab5e64",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "data": [
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         "colorbar": {
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         "legendgroup": "Bitcoin Price",
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      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Plotting the results. Comparing the current bitcoin's prices with the values given by the comodities estimator\n",
    "plot(A[:,2], [b, prediction], labels = [\"Bitcoin Price\" \"Comodities Estimator\"], lw = [2 3])\n",
    "xlabel!(\"Days\")\n",
    "ylabel!(\"Price(USD)\")\n"
   ]
  }
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