ornstein uhlenbeck plus noise julia.ipynb

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      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
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      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
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      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
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      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
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      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Project.toml`\n",
      " \u001b[90m [c7f686f2]\u001b[39m\u001b[92m + MCMCChains v3.0.6\u001b[39m\n",
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.4/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "import Pkg\n",
    "Pkg.add(\"DifferentialEquations\")\n",
    "Pkg.add(\"Turing\")\n",
    "Pkg.add(\"Distributions\")\n",
    "Pkg.add(\"StatsPlots\")\n",
    "Pkg.add(\"Plots\")\n",
    "Pkg.add(\"MCMCChains\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
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     "text": [
      "┌ Info: Precompiling Plots [91a5bcdd-55d7-5caf-9e0b-520d859cae80]\n",
      "└ @ Base loading.jl:1260\n"
     ]
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       "  975.392,730.294 977.476,711.061 979.56,666.47 981.643,740.692 983.727,743.877 985.811,776.241 987.894,616.151 989.978,666.061 992.062,716.092 994.145,790.731 \n",
       "  996.229,740.087 998.313,707.775 1000.4,558.578 1002.48,444.812 1004.56,539.48 1006.65,619.159 1008.73,548.085 1010.81,411.928 1012.9,442.487 1014.98,412.097 \n",
       "  1017.07,511.656 1019.15,522.797 1021.23,499.485 1023.32,525.601 1025.4,517.431 1027.48,442.494 1029.57,472.017 1031.65,654.564 1033.73,568.383 1035.82,534.669 \n",
       "  1037.9,576.318 1039.99,443.654 1042.07,512.749 1044.15,435.457 1046.24,394.98 1048.32,484.404 1050.4,469.609 1052.49,598.234 1054.57,652.988 1056.66,715.354 \n",
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       "  1100.41,1330.75 1102.5,1241.39 1104.58,1127.11 1106.66,931.253 1108.75,896.591 1110.83,851.43 1112.91,907.784 1115,934.201 1117.08,995.793 1119.16,939.62 \n",
       "  1121.25,820.806 1123.33,732.56 1125.42,664.961 1127.5,564.105 1129.58,617.049 1131.67,570.626 1133.75,754.867 1135.83,875.015 1137.92,704.944 1140,626.272 \n",
       "  1142.09,643.425 1144.17,584.608 1146.25,565.697 1148.34,452.069 1150.42,523.423 1152.5,508.601 1154.59,740.555 1156.67,702.393 1158.75,735.465 1160.84,760.525 \n",
       "  1162.92,853.248 1165.01,1009.41 1167.09,932.926 1169.17,903.059 1171.26,835.654 1173.34,691.125 1175.42,741.007 1177.51,763.855 1179.59,936.426 1181.67,882.395 \n",
       "  1183.76,818.513 1185.84,631.359 1187.93,682.295 1190.01,722.483 1192.09,800.804 1194.18,867.361 1196.26,877.523 1198.34,886.286 1200.43,979.469 1202.51,1114.28 \n",
       "  1204.6,1040.29 1206.68,893.432 1208.76,846.304 1210.85,897.207 1212.93,748.917 1215.01,718.388 1217.1,775.613 1219.18,693.336 1221.26,903.579 1223.35,719.981 \n",
       "  1225.43,876.361 1227.52,841.396 1229.6,681.917 1231.68,497.955 1233.77,576.48 1235.85,571.447 1237.93,531.035 1240.02,411.098 1242.1,458.585 1244.18,473.325 \n",
       "  1246.27,473.34 1248.35,481.046 1250.44,584.752 1252.52,653.686 1254.6,602.076 1256.69,710 1258.77,892.927 1260.85,926.429 1262.94,856.394 1265.02,871.413 \n",
       "  1267.11,790.649 1269.19,802.111 1271.27,920.01 1273.36,977.593 1275.44,864.049 1277.52,903.341 1279.61,949.282 1281.69,843.239 1283.77,760.742 1285.86,743.233 \n",
       "  1287.94,702.243 1290.03,654.487 1292.11,697.922 1294.19,762.29 1296.28,914.677 1298.36,1008.39 1300.44,1004.37 1302.53,893.786 1304.61,1092.74 1306.69,914.302 \n",
       "  1308.78,827.126 1310.86,971.921 1312.95,931.001 1315.03,1017.05 1317.11,864.966 1319.2,996.229 1321.28,901.204 1323.36,802.238 1325.45,719.092 1327.53,749.703 \n",
       "  1329.61,807.278 1331.7,967.575 1333.78,935.589 1335.87,868.917 1337.95,989.353 1340.03,981.696 1342.12,912.946 1344.2,926.761 1346.28,939.71 1348.37,853.059 \n",
       "  1350.45,845.96 1352.54,749.783 1354.62,886.996 1356.7,810.72 1358.79,893.623 1360.87,675.418 1362.95,717.335 1365.04,587.528 1367.12,446.95 1369.2,458.046 \n",
       "  1371.29,512.419 1373.37,710.708 1375.46,675.349 1377.54,749.876 1379.62,767.433 1381.71,803.285 1383.79,796.308 1385.87,849.973 1387.96,928.402 1390.04,768.583 \n",
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       "  1454.63,1151.32 1456.72,1050.02 1458.8,1062.56 1460.89,1012.22 1462.97,1059.76 1465.05,1014.92 1467.14,990.427 1469.22,929.197 1471.3,850.243 1473.39,941.361 \n",
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       "  1496.31,1266.83 1498.39,1128.26 1500.48,1089.26 1502.56,931.563 1504.64,966.806 1506.73,998.244 1508.81,955.697 1510.89,995.317 1512.98,1015.55 1515.06,1038.26 \n",
       "  1517.14,1008.29 1519.23,1146.59 1521.31,1050.93 1523.4,1082.96 1525.48,1025.36 1527.56,897.862 1529.65,898.451 1531.73,906.683 1533.81,1029.38 1535.9,937.699 \n",
       "  1537.98,785.048 1540.06,709.464 1542.15,792.992 1544.23,780.216 1546.32,593.162 1548.4,583.519 1550.48,657.946 1552.57,615.788 1554.65,633.996 1556.73,626.85 \n",
       "  1558.82,654.684 1560.9,674.287 1562.99,638.284 1565.07,553.631 1567.15,602.808 1569.24,352.411 1571.32,476.844 1573.4,478.731 1575.49,434.956 1577.57,466.408 \n",
       "  1579.65,455.701 1581.74,432.602 1583.82,578.445 1585.91,626.011 1587.99,665.267 1590.07,661.446 1592.16,600.201 1594.24,550.009 1596.32,388.513 1598.41,393.035 \n",
       "  1600.49,548.104 1602.57,543.646 1604.66,517.445 1606.74,675.006 1608.83,798.14 1610.91,1027.04 1612.99,944.54 1615.08,894.931 1617.16,916.797 1619.24,863.745 \n",
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       "  1683.84,557.632 1685.92,559.544 1688,619.083 1690.09,787.624 1692.17,928.208 1694.26,786.228 1696.34,817.938 1698.42,904.32 1700.51,793.217 1702.59,716.527 \n",
       "  1704.67,663.337 1706.76,389.587 1708.84,339.391 1710.93,430.862 1713.01,394.711 1715.09,399.546 1717.18,403.363 1719.26,461.169 1721.34,613.866 1723.43,559.846 \n",
       "  1725.51,696.02 1727.59,868.677 1729.68,755.109 1731.76,597.208 1733.85,559.927 1735.93,781.756 1738.01,864.374 1740.1,843.812 1742.18,901.652 1744.26,848.659 \n",
       "  1746.35,806.364 1748.43,724.151 1750.51,669.16 1752.6,809.785 1754.68,750.058 1756.77,813.019 1758.85,799.691 1760.93,851.974 1763.02,998.194 1765.1,1067.88 \n",
       "  1767.18,1027.5 1769.27,1084.34 1771.35,892.878 1773.44,796.003 1775.52,919.214 1777.6,920.758 1779.69,915.776 1781.77,808.416 1783.85,793.804 1785.94,709.831 \n",
       "  1788.02,790.834 1790.1,754.148 1792.19,702.836 1794.27,605.63 1796.36,447.169 1798.44,315.927 1800.52,252.046 1802.61,230.21 1804.69,276.168 1806.77,186.812 \n",
       "  1808.86,255.264 1810.94,191.909 1813.02,95.5052 1815.11,116.109 1817.19,216.478 1819.28,275.868 1821.36,286.998 1823.44,88.0053 1825.53,203.328 1827.61,212.395 \n",
       "  1829.69,270.189 1831.78,358.249 1833.86,598.324 1835.94,473.219 1838.03,464.907 1840.11,358.354 1842.2,451.85 1844.28,507.577 1846.36,645.33 1848.45,683.162 \n",
       "  1850.53,695.427 1852.61,712.201 1854.7,755.236 1856.78,778.782 1858.87,800.221 1860.95,820.453 1863.03,717.36 1865.12,723.799 1867.2,711.444 1869.28,720.513 \n",
       "  1871.37,703.054 1873.45,771.443 1875.53,783.845 1877.62,618.938 1879.7,719.996 1881.79,645.045 1883.87,544.466 1885.95,656.432 1888.04,722.624 1890.12,646.496 \n",
       "  1892.2,645.624 1894.29,418.829 1896.37,346.686 1898.45,370.037 1900.54,454.509 1902.62,480.615 1904.71,471.905 1906.79,504.294 1908.87,664.03 1910.96,627.714 \n",
       "  1913.04,634.875 1915.12,630.042 1917.21,661.418 1919.29,630.552 1921.38,684.969 1923.46,500.235 1925.54,441.136 1927.63,465.386 1929.71,612.398 1931.79,592.983 \n",
       "  1933.88,640.017 1935.96,636.97 1938.04,695.093 1940.13,717.18 1942.21,607.805 1944.3,728.326 1946.38,713.753 1948.46,672.733 1950.55,711.536 1952.63,866.439 \n",
       "  1954.71,753.82 1956.8,582.594 1958.88,562.801 1960.96,734.002 1963.05,743.93 1965.13,945.767 1967.22,923.843 1969.3,1091.69 1971.38,1043.39 1973.47,1005.27 \n",
       "  1975.55,1089.62 1977.63,1231.25 1979.72,1348.29 1981.8,1316.24 1983.89,1178.61 1985.97,1113.97 1988.05,970.479 1990.14,813.629 1992.22,845.951 1994.3,796.281 \n",
       "  1996.39,770.684 1998.47,658.346 2000.55,590.493 2002.64,499.018 2004.72,515.318 2006.81,564.657 2008.89,586.815 2010.97,578.512 2013.06,647.854 2015.14,663.659 \n",
       "  2017.22,775.96 2019.31,948.078 2021.39,1076.84 2023.47,1151.52 2025.56,1115.66 2027.64,1259.52 2029.73,1094.05 2031.81,1077.67 2033.89,1112.69 2035.98,1158.39 \n",
       "  2038.06,1171.32 2040.14,1056.5 2042.23,1167.73 2044.31,1176.19 2046.39,1111.12 2048.48,1033.57 2050.56,1077.65 2052.65,981.405 2054.73,1025.64 2056.81,1034.96 \n",
       "  2058.9,985.108 2060.98,1042.09 2063.06,1120.46 2065.15,986.968 2067.23,1092.3 2069.32,932.922 2071.4,802.087 2073.48,767.114 2075.57,784.542 2077.65,906.173 \n",
       "  2079.73,706.08 2081.82,790.832 2083.9,781.576 2085.98,757.607 2088.07,692.85 2090.15,695.078 2092.24,561.598 2094.32,462.896 2096.4,563.803 2098.49,571.638 \n",
       "  2100.57,745.892 2102.65,671.732 2104.74,623.929 2106.82,632.256 2108.9,799.002 2110.99,729.044 2113.07,697.203 2115.16,785.811 2117.24,752.425 2119.32,684.794 \n",
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       "  2142.24,605.38 2144.33,619.453 2146.41,575.008 2148.49,589.382 2150.58,675.855 2152.66,781.34 2154.75,735.325 2156.83,647.313 2158.91,716.924 2161,859.587 \n",
       "  2163.08,788.522 2165.16,743.074 2167.25,726.126 2169.33,890.207 2171.41,883.769 2173.5,820.482 2175.58,818.863 2177.67,1041.23 2179.75,1050 2181.83,1010.26 \n",
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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DifferentialEquations\n",
    "using Plots\n",
    "\n",
    "μ = 0.0\n",
    "σ = sqrt(2)\n",
    "Θ = 1.0\n",
    "W = OrnsteinUhlenbeckProcess(Θ,μ,σ,0.0,1.0)\n",
    "prob = NoiseProblem(W,(0.0,100.0))\n",
    "sol = solve(prob;dt=0.1)\n",
    "plot(sol.u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
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       "  1350.45,824.39 1352.54,740.175 1354.62,934.654 1356.7,707.81 1358.79,951.724 1360.87,760.78 1362.95,781.724 1365.04,629.149 1367.12,494.812 1369.2,483.997 \n",
       "  1371.29,520.916 1373.37,670.419 1375.46,655.444 1377.54,807.805 1379.62,699.2 1381.71,851.565 1383.79,814.969 1385.87,809.157 1387.96,915.036 1390.04,674.206 \n",
       "  1392.12,883.777 1394.21,778.85 1396.29,761.47 1398.38,824.864 1400.46,711.869 1402.54,678.305 1404.63,709.119 1406.71,708.532 1408.79,524.451 1410.88,529.754 \n",
       "  1412.96,681.733 1415.05,710.987 1417.13,788.726 1419.21,794.231 1421.3,804.543 1423.38,870.134 1425.46,761.395 1427.55,874.469 1429.63,763.281 1431.71,735.07 \n",
       "  1433.8,604.217 1435.88,563.985 1437.97,633.02 1440.05,724.008 1442.13,776.889 1444.22,824.283 1446.3,902.135 1448.38,1009.84 1450.47,1136.14 1452.55,1226.4 \n",
       "  1454.63,1192.78 1456.72,1071.2 1458.8,1074.23 1460.89,1024.78 1462.97,1115.05 1465.05,979.402 1467.14,992.736 1469.22,948.555 1471.3,856.141 1473.39,940.099 \n",
       "  1475.47,780.789 1477.55,605.402 1479.64,532.631 1481.72,568.447 1483.81,575.066 1485.89,664.843 1487.97,896.087 1490.06,953.055 1492.14,1153.82 1494.22,1120.63 \n",
       "  1496.31,1300.19 1498.39,1100.19 1500.48,1072.94 1502.56,998.115 1504.64,1039.95 1506.73,937.942 1508.81,917.081 1510.89,1027.48 1512.98,1055.24 1515.06,998.541 \n",
       "  1517.14,962.451 1519.23,1132.44 1521.31,996.736 1523.4,983.695 1525.48,1113.09 1527.56,922.847 1529.65,945.827 1531.73,805.518 1533.81,1044.61 1535.9,909.587 \n",
       "  1537.98,774.466 1540.06,753.404 1542.15,694.484 1544.23,781.911 1546.32,603.99 1548.4,599.254 1550.48,664.624 1552.57,660.265 1554.65,584.3 1556.73,660.632 \n",
       "  1558.82,636.053 1560.9,618.944 1562.99,638.519 1565.07,591.269 1567.15,649.235 1569.24,411.79 1571.32,538.402 1573.4,478.761 1575.49,443.397 1577.57,458.472 \n",
       "  1579.65,424.796 1581.74,389.185 1583.82,628.368 1585.91,623.812 1587.99,631.692 1590.07,651.155 1592.16,627.185 1594.24,478.331 1596.32,375.357 1598.41,395.505 \n",
       "  1600.49,541.388 1602.57,614.776 1604.66,487.083 1606.74,672.856 1608.83,814.82 1610.91,1056.32 1612.99,906.068 1615.08,860.16 1617.16,879.295 1619.24,904.895 \n",
       "  1621.33,823.299 1623.41,871.089 1625.5,929.565 1627.58,1017.16 1629.66,891.843 1631.75,879.242 1633.83,706.42 1635.91,743.7 1638,624.49 1640.08,760.942 \n",
       "  1642.16,969.789 1644.25,919.715 1646.33,882.06 1648.42,854.286 1650.5,778.16 1652.58,778.332 1654.67,548.417 1656.75,538.046 1658.83,430.745 1660.92,556.143 \n",
       "  1663,748.928 1665.08,766.608 1667.17,665.573 1669.25,644.592 1671.34,709.259 1673.42,751.955 1675.5,742.635 1677.59,685.115 1679.67,691.389 1681.75,620.614 \n",
       "  1683.84,517.748 1685.92,547.816 1688,669.924 1690.09,831.518 1692.17,991.553 1694.26,759.228 1696.34,815.918 1698.42,969.487 1700.51,763.048 1702.59,683.022 \n",
       "  1704.67,652.084 1706.76,513.986 1708.84,303.869 1710.93,466.374 1713.01,379.622 1715.09,416.605 1717.18,413.884 1719.26,509.106 1721.34,607.472 1723.43,613.163 \n",
       "  1725.51,729.378 1727.59,873.039 1729.68,718.101 1731.76,596.877 1733.85,588.707 1735.93,701.807 1738.01,879.788 1740.1,878.116 1742.18,911.547 1744.26,853.47 \n",
       "  1746.35,822.279 1748.43,748.687 1750.51,643.728 1752.6,795.726 1754.68,739.065 1756.77,887.657 1758.85,853.888 1760.93,880.115 1763.02,1006.55 1765.1,998.089 \n",
       "  1767.18,1037.36 1769.27,1042.29 1771.35,919.619 1773.44,823.987 1775.52,942.281 1777.6,987.578 1779.69,903.077 1781.77,841.75 1783.85,753.396 1785.94,719.839 \n",
       "  1788.02,757.983 1790.1,810.305 1792.19,721.474 1794.27,629.671 1796.36,401.397 1798.44,411.061 1800.52,298.528 1802.61,225.576 1804.69,352.19 1806.77,201.372 \n",
       "  1808.86,311.94 1810.94,217.714 1813.02,108.377 1815.11,88.0053 1817.19,171.498 1819.28,308.706 1821.36,369.918 1823.44,172.326 1825.53,265.839 1827.61,252.105 \n",
       "  1829.69,266.454 1831.78,375.96 1833.86,572.205 1835.94,435.617 1838.03,488.127 1840.11,451.414 1842.2,506.208 1844.28,491.522 1846.36,698.794 1848.45,642.737 \n",
       "  1850.53,784.157 1852.61,804.669 1854.7,777.365 1856.78,775.787 1858.87,817.029 1860.95,754.909 1863.03,689.607 1865.12,763.105 1867.2,672.812 1869.28,728.388 \n",
       "  1871.37,632.036 1873.45,710.9 1875.53,672.016 1877.62,609.651 1879.7,645.185 1881.79,608.684 1883.87,483.581 1885.95,694.524 1888.04,781.146 1890.12,681.501 \n",
       "  1892.2,693.783 1894.29,446.805 1896.37,390.121 1898.45,412.533 1900.54,485.245 1902.62,463.662 1904.71,540.288 1906.79,526.262 1908.87,649.327 1910.96,627.865 \n",
       "  1913.04,704.764 1915.12,631.338 1917.21,713.013 1919.29,651.408 1921.38,629.581 1923.46,553.426 1925.54,474.671 1927.63,554.73 1929.71,606.072 1931.79,608.291 \n",
       "  1933.88,628.773 1935.96,620.635 1938.04,631.582 1940.13,690.835 1942.21,628.074 1944.3,744.739 1946.38,721.292 1948.46,626.172 1950.55,712.62 1952.63,987.476 \n",
       "  1954.71,761.908 1956.8,546.585 1958.88,565.485 1960.96,736.268 1963.05,780.663 1965.13,966.572 1967.22,984.122 1969.3,1042.26 1971.38,1057.83 1973.47,881.647 \n",
       "  1975.55,1096.63 1977.63,1191.1 1979.72,1304.95 1981.8,1357.24 1983.89,1146.8 1985.97,1069.77 1988.05,932.689 1990.14,811.895 1992.22,847.021 1994.3,793.711 \n",
       "  1996.39,787.637 1998.47,689.336 2000.55,578.453 2002.64,492.898 2004.72,578.874 2006.81,598.909 2008.89,587.515 2010.97,527.539 2013.06,669.161 2015.14,664.858 \n",
       "  2017.22,722.717 2019.31,937.729 2021.39,1142.19 2023.47,1086.75 2025.56,1132.9 2027.64,1286.03 2029.73,1050.7 2031.81,1052.3 2033.89,1111.43 2035.98,1079.9 \n",
       "  2038.06,1203.68 2040.14,1043.65 2042.23,1162.43 2044.31,1219.75 2046.39,1074.6 2048.48,1069.84 2050.56,1080.45 2052.65,1018.41 2054.73,934.927 2056.81,1001.75 \n",
       "  2058.9,992.983 2060.98,1053.03 2063.06,1072.44 2065.15,1009.28 2067.23,1028.85 2069.32,937.893 2071.4,813.175 2073.48,766.897 2075.57,890.485 2077.65,963.205 \n",
       "  2079.73,761.744 2081.82,820.816 2083.9,810.313 2085.98,705.651 2088.07,595.767 2090.15,708.05 2092.24,454.089 2094.32,482.251 2096.4,608.046 2098.49,590.727 \n",
       "  2100.57,730.435 2102.65,706.931 2104.74,662.023 2106.82,682.379 2108.9,886.572 2110.99,750.537 2113.07,660.834 2115.16,782.285 2117.24,707.767 2119.32,615.423 \n",
       "  2121.41,774.867 2123.49,837.504 2125.57,723.042 2127.66,668.521 2129.74,670.942 2131.83,682.75 2133.91,559.977 2135.99,614.988 2138.08,751.996 2140.16,644.098 \n",
       "  2142.24,631.859 2144.33,643.632 2146.41,597.602 2148.49,607.722 2150.58,645.211 2152.66,762.922 2154.75,794.95 2156.83,666.212 2158.91,710.978 2161,887.615 \n",
       "  2163.08,813.136 2165.16,747.046 2167.25,750.277 2169.33,889.361 2171.41,895.183 2173.5,900.035 2175.58,777.389 2177.67,1073.21 2179.75,1081.55 2181.83,954.535 \n",
       "  2183.92,902.679 2186,954.446 2188.08,783.016 2190.17,773.892 2192.25,802.11 2194.34,673.722 2196.42,786.5 2198.5,654.362 2200.59,650.975 2202.67,772.22 \n",
       "  2204.75,770.382 2206.84,877.323 2208.92,949.209 2211,829.374 2213.09,771.576 2215.17,509.533 2217.26,611.414 2219.34,569.546 2221.42,389.134 2223.51,402.759 \n",
       "  2225.59,333.794 2227.67,416.512 2229.76,416.048 2231.84,466.363 2233.92,402.148 2236.01,374.863 2238.09,584.402 2240.18,549.617 2242.26,803.682 2244.34,763.194 \n",
       "  2246.43,816.385 2248.51,684.635 2250.59,806.168 2252.68,875.657 2254.76,869.807 2256.84,834.834 2258.93,903.872 2261.01,774.006 2263.1,844.135 2265.18,1028.61 \n",
       "  2267.26,913.334 2269.35,959.657 2271.43,999.102 2273.51,749.867 2275.6,841.9 2277.68,846.474 2279.77,914.751 2281.85,784.434 2283.93,682.759 2286.02,728.547 \n",
       "  2288.1,701.667 2290.18,662.59 \n",
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       "</svg>\n"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Distributions, Random\n",
    "ou_data = sol.u\n",
    "noise = rand(Normal(0,0.2), length(ou_data))\n",
    "plot(ou_data.+noise)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling Turing [fce5fe82-541a-59a6-adf8-730c64b5f9a0]\n",
      "└ @ Base loading.jl:1260\n",
      "WARNING: could not import MCMCChains.AbstractChains into Turing\n",
      "ERROR: LoadError: LoadError: UndefVarError: AbstractChains not defined\n",
      "Stacktrace:\n",
      " [1] include(::Module, ::String) at ./Base.jl:377\n",
      " [2] include(::String) at /Users/hstrey/.julia/packages/Turing/xFzfF/src/Turing.jl:1\n",
      " [3] top-level scope at /Users/hstrey/.julia/packages/Turing/xFzfF/src/Turing.jl:124\n",
      " [4] include(::Module, ::String) at ./Base.jl:377\n",
      " [5] top-level scope at none:2\n",
      " [6] eval at ./boot.jl:331 [inlined]\n",
      " [7] eval(::Expr) at ./client.jl:449\n",
      " [8] top-level scope at ./none:3\n",
      "in expression starting at /Users/hstrey/.julia/packages/Turing/xFzfF/src/utilities/Utilities.jl:6\n",
      "in expression starting at /Users/hstrey/.julia/packages/Turing/xFzfF/src/Turing.jl:124\n"
     ]
    },
    {
     "ename": "ErrorException",
     "evalue": "Failed to precompile Turing [fce5fe82-541a-59a6-adf8-730c64b5f9a0] to /Users/hstrey/.julia/compiled/v1.4/Turing/gm4QC_8bdJ6.ji.",
     "output_type": "error",
     "traceback": [
      "Failed to precompile Turing [fce5fe82-541a-59a6-adf8-730c64b5f9a0] to /Users/hstrey/.julia/compiled/v1.4/Turing/gm4QC_8bdJ6.ji.",
      "",
      "Stacktrace:",
      " [1] error(::String) at ./error.jl:33",
      " [2] compilecache(::Base.PkgId, ::String) at ./loading.jl:1272",
      " [3] _require(::Base.PkgId) at ./loading.jl:1029",
      " [4] require(::Base.PkgId) at ./loading.jl:927",
      " [5] require(::Module, ::Symbol) at ./loading.jl:922",
      " [6] top-level scope at In[16]:1"
     ]
    }
   ],
   "source": [
    "using Turing\n",
    "using Distributions\n",
    "using LinearAlgebra\n",
    "using MCMCChains\n",
    "\n",
    "# Ornstein-Uhlenbeck process with added Gaussian noise\n",
    "@model oupn(rn,T,delta_t,::Type{R}=Vector{Float64}) where {R} = begin\n",
    "    ampl ~ Uniform(0.0,5.0)\n",
    "    tau ~ Uniform(0.0,5.0)\n",
    "    noise_ampl ~ Uniform(0.0,1)\n",
    "    \n",
    "    b = exp(-delta_t/tau)\n",
    "    r = R(undef, T)\n",
    "    \n",
    "    r[1] ~ Normal(0,sqrt(ampl))\n",
    "    \n",
    "    for i=2:T\n",
    "        r[i] ~ Normal(r[i-1]*b,sqrt(ampl*(1-b^2)))\n",
    "    end\n",
    "    rn ~ MvNormal(r,sqrt(ampl))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling StatsPlots [f3b207a7-027a-5e70-b257-86293d7955fd]\n",
      "└ @ Base loading.jl:1260\n",
      "ERROR: ERROR: ERROR: ERROR: LoadError: LoadError: LoadError: LoadError: ERROR: LoadError: LoadError: InterruptException:\n",
      "Stacktrace:InterruptException:\n",
      "Stacktrace:InterruptException:\n",
      "Stacktrace:InterruptException:\n",
      "Stacktrace:\n",
      " [1] \n",
      " [1] \n",
      " [1] poptaskrefpoptaskrefgrow_to!\n",
      " [1] (poptaskref((::(::::Base.InvasiveLinkedListSynchronized{Task}) at ./task.jl:702\n",
      " [2] wait() at ./task.jl:709\n",
      " [3] wait(::Base.GenericCondition{Base.Threads.SpinLock}) at ./condition.jl:106\n",
      " [4] wait(::Base.Process) at ./process.jl:622\n",
      " [5] success at ./process.jl:483 [inlined]\n",
      " [6] compilecache(::Base.PkgId, ::String) at ./loading.jl:1262\n",
      " [7] _require(::Base.PkgId) at ./loading.jl:1029\n",
      " [8] require(::Base.PkgId) at ./loading.jl:927\n",
      " [9] require(::Module, ::Symbol) at ./loading.jl:922\n",
      " [10] include(::Module, ::String) at ./Base.jl:377\n",
      " [11] top-level scope at none:2\n",
      " [12] eval at ./boot.jl:331 [inlined]\n",
      " [13] eval(::Expr) at ./client.jl:449\n",
      " [14] top-level scope at ./none:3\n",
      "in expression starting at /Users/hstrey/.julia/packages/FFMPEG/vocw2/src/FFMPEG.jl:3\n",
      "::Dict{Pkg.BinaryPlatforms.Linux,String}, ::Base.Generator{Array{Pkg.BinaryPlatforms.Platform,1},Opus_jll.var\"#5#6\"}, ::Int64) at ./dict.jl:159\n",
      " [2] grow_to!(::Dict{Any,Any}, ::Base.Generator{Array{Pkg.BinaryPlatforms.Platform,1},Opus_jll.var\"#5#6\"}) at ./dict.jl:144\n",
      " [3] dict_with_eltype at ./abstractdict.jl:549 [inlined]\n",
      " [4] Dict(::Base.Generator{Array{Pkg.BinaryPlatforms.Platform,1},Opus_jll.var\"#5#6\"}) at ./dict.jl:128\n",
      " [5] top-level scope at /Users/hstrey/.julia/packages/Opus_jll/aQt9C/src/Opus_jll.jl:21\n",
      " [6] include(::Module, ::String) at ./Base.jl:377\n",
      " [7] top-level scope at none:2\n",
      " [8] eval at ./boot.jl:331 [inlined]\n",
      " [9] eval(::Expr) at ./client.jl:449\n",
      " [10] top-level scope at ./none:3\n",
      "in expression starting at /Users/hstrey/.julia/packages/Opus_jll/aQt9C/src/Opus_jll.jl:21\n",
      "Base.InvasiveLinkedListSynchronized{Task}) at ./task.jl:702\n",
      " [2] wait() at ./task.jl:709\n",
      " [3] wait(::Base.GenericCondition{Base.Threads.SpinLock}) at ./condition.jl:106\n",
      " [4] wait(::Base.Process) at ./process.jl:622\n",
      " [5] success at ./process.jl:483 [inlined]\n",
      " [6] compilecache(::Base.PkgId, ::String) at ./loading.jl:1262\n",
      " [7] _require(::Base.PkgId) at ./loading.jl:1029\n",
      " [8] require(::Base.PkgId) at ./loading.jl:927\n",
      " [9] require(::Module, ::Symbol) at ./loading.jl:922\n",
      " [10] include(::Module, ::String) at ./Base.jl:377\n",
      " [11] include(::String) at /Users/hstrey/.julia/packages/FFMPEG_jll/1XNMx/src/FFMPEG_jll.jl:1\n",
      " [12] top-level scope at /Users/hstrey/.julia/packages/FFMPEG_jll/1XNMx/src/FFMPEG_jll.jl:47\n",
      " [13] include(::Module, ::String) at ./Base.jl:377\n",
      " [14] top-level scope at none:2\n",
      " [15] eval at ./boot.jl:331 [inlined]\n",
      " [16] eval(::Expr) at ./client.jl:449\n",
      " [17] top-level scope at ./none:3\n",
      "in expression starting at /Users/hstrey/.julia/packages/FFMPEG_jll/1XNMx/src/wrappers/x86_64-apple-darwin14.jl:17\n",
      "in expression starting at /Users/hstrey/.julia/packages/FFMPEG_jll/1XNMx/src/FFMPEG_jll.jl:43\n",
      "Base.InvasiveLinkedListSynchronized{Task}) at ./task.jl:702\n",
      " [2] wait() at ./task.jl:709\n",
      " [3] wait(::Base.GenericCondition{Base.Threads.SpinLock}) at ./condition.jl:106\n",
      " [4] wait(::Base.Process) at ./process.jl:622\n",
      " [5] success at ./process.jl:483 [inlined]\n",
      " [6] compilecache(::Base.PkgId, ::String) at ./loading.jl:1262\n",
      " [7] _require(::Base.PkgId) at ./loading.jl:1029\n",
      " [8] require(::Base.PkgId) at ./loading.jl:927\n",
      " [9] require(::Module, ::Symbol) at ./loading.jl:922\n",
      " [10] include(::Module, ::String) at ./Base.jl:377\n",
      " [11] top-level scope at none:2\n",
      " [12] eval at ./boot.jl:331 [inlined]\n",
      " [13] eval(::Expr) at ./client.jl:449\n",
      " [14] top-level scope at ./none:3\n",
      "in expression starting at /Users/hstrey/.julia/packages/StatsPlots/1ip2r/src/StatsPlots.jl:7\n"
     ]
    },
    {
     "ename": "ErrorException",
     "evalue": "Failed to precompile StatsPlots [f3b207a7-027a-5e70-b257-86293d7955fd] to /Users/hstrey/.julia/compiled/v1.4/StatsPlots/SiylL_8bdJ6.ji.",
     "output_type": "error",
     "traceback": [
      "Failed to precompile StatsPlots [f3b207a7-027a-5e70-b257-86293d7955fd] to /Users/hstrey/.julia/compiled/v1.4/StatsPlots/SiylL_8bdJ6.ji.",
      "",
      "Stacktrace:",
      " [1] error(::String) at ./error.jl:33",
      " [2] compilecache(::Base.PkgId, ::String) at ./loading.jl:1272",
      " [3] _require(::Base.PkgId) at ./loading.jl:1029",
      " [4] require(::Base.PkgId) at ./loading.jl:927",
      " [5] require(::Module, ::Symbol) at ./loading.jl:922",
      " [6] top-level scope at In[5]:1"
     ]
    }
   ],
   "source": [
    "using StatsPlots\n",
    "\n",
    "chn = sample(oupn(ou_data.+noise,length(ou_data),0.1), NUTS(0.65), 2000)\n",
    "\n",
    "# Summarise results (currently requires the master branch from MCMCChains)\n",
    "describe(chn)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: plot not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: plot not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[6]:1"
     ]
    }
   ],
   "source": [
    "# Plot and save results\n",
    "p = plot(chn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: chn not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: chn not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[7]:1"
     ]
    }
   ],
   "source": [
    "ampl_data = chn[:ampl].value.data\n",
    "print(mean(ampl_data[:,1,1]))\n",
    "plot(ampl_data[:,1,1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: chn not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: chn not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[8]:1"
     ]
    }
   ],
   "source": [
    "r_data = chn[:r].value.data\n",
    "r_mean=mean(r_data,dims=1)[1,:,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "InterruptException:\n",
      "Stacktrace:"
     ]
    },
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: plot not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: plot not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[9]:1"
     ]
    }
   ],
   "source": [
    "plot(r_mean)\n",
    "plot!(sol.u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: plot not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: plot not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[10]:1"
     ]
    }
   ],
   "source": [
    "plot(noise)\n",
    "plot!(r_mean.-sol.u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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