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SIR.ipynb
{
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"cell_type": "markdown",
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"# Parameter inference for a simple SIR model with the package `Bridge`\n",
"\n",
"## Introduction\n",
"\n",
"Mathieu Besançon made a nice blog post on Chris' `DifferentialEquations` ecosystem,\n",
"https://mbesancon.github.io/post/2017-12-14-diffeq-julia/ . \n",
"\n",
"One of the examples is a simple model tracking individuals in a population with three states $S$, $I$, $R$. \n",
"\n",
"Let $u = (u_1, u_2, u_2)$ be the vector of individuals in these states.\n",
"\n",
"* At any moment an individual of state $I$ can \"convince\" an individual of state $S$ to join the fraction $I$\n",
"$$\n",
"S,I \\mapsto 2I\n",
"$$\n",
"This transition happens proportional to the number of possible pairs $I, S$ which can match up, so proportional\n",
"to $u_1 u_2$ \n",
"with some proportionality constant $\\alpha$.\n",
"\n",
"* At any moment an individual of state $I$ can attain state $R$\n",
"$$\n",
"I \\mapsto R\n",
"$$\n",
"This reaction happens proportional to the number of individuals in state $I$, so proportional to $u_2$\n",
"with proportionality constant $\\beta$.\n",
"\n",
"In the SIR interpretation of the model, where $I$, $S$ and $R$ stand for\n",
"(S)usceptible, (I)nfected, and (R)ecovered individuals, the $I,S \\mapsto 2I$ is understood as infection and $I \\mapsto R$ as recovery (with subsequent immunity) of the individual. \n",
"\n",
"Intepreting these transitions as continuous processes, the population can be modeled by an\n",
"ordinary differential equation\n",
"\n",
"$$\n",
"\\frac{d}{dt} u(t) = \\begin{bmatrix} -\\alpha u_1(t)u_2(t)\\\\ \\alpha u_1(t)u_2(t) - \\beta u_2(t)\\\\ \\beta u_2(t)\\end{bmatrix}\n",
"$$\n",
"\n",
"In the following I show how to estimate the model parameters $\\alpha$ and $\\beta$ from observations of a noisy version of the model.\n",
"\n",
"\n",
"I am using my package `Bridge` and `StaticArray`'s and some more."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"] add Bridge Plots LaTeXStrings"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"using Bridge\n",
"using Plots\n",
"using LaTeXStrings\n",
"using StaticArrays\n",
"using LinearAlgebra\n",
"using Statistics"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Stochastic model\n",
"\n",
"\n",
"\n",
"The dynamics of the population is given by a stochastic differential equation\n",
"\n",
"$$\n",
" d X_t = b(t, X_t)\\,dt + \\sigma(t, X_t)\\, d W_t\\qquad (\\star)\n",
"$$\n",
"\n",
"with drift vector $b$ and dispersion coefficient $\\sigma$\n",
"\n",
"$$\n",
" b(t, u) = \\begin{bmatrix} -\\alpha u_1u_2\\\\ \\alpha u_1u_2 - \\beta u_2\\\\ \\beta u_2\\end{bmatrix}\n",
"\\qquad\n",
"\\sigma(t, u)= \\begin{bmatrix}\n",
"\\sigma_1 u_1 u_2 & 0\\\\\n",
"-\\sigma_1u_1u_2& -\\sigma_2 u_2\\\\\n",
"0& \\sigma_2 u_2\n",
"\\end{bmatrix}.\n",
"$$\n",
"\n",
"with $\\alpha = 0.8, \\beta = 3.0, \\sigma_1 = 0.07, \\sigma_2 = 0.4$.\n",
"\n",
"\n",
"\n",
"Because $\\sigma$ is a $3x2$ matrix, there is a 2 dimensional noise process $W$ acting on all three components.\n",
"\n",
"This equation models a *closed system*, because of the entries of $b$ and the columns of $\\sigma$ add up to $0$. The population $u_1 + u_2 + u_3$ stays constant. See Mathieu's explanations.\n",
"\n",
"\n",
"In Julia, using `Bridge` and `StaticArray`s, this SDE model reads as follows."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"σ (generic function with 24 methods)"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import Bridge: b, σ, a\n",
"\n",
"struct CIR <: ContinuousTimeProcess{SVector{3,Float64}}\n",
" α::Float64\n",
" β::Float64\n",
" σ1::Float64\n",
" σ2::Float64\n",
"end\n",
"Ptrue = P = CIR(0.8, 3.0, 0.07, 0.4)\n",
"\n",
"b(t, u, P::CIR) = @SVector [-P.α*u[1]*u[2], P.α*u[1]*u[2] - P.β*u[2], P.β*u[2]]\n",
"σ(t, u, P::CIR) = @SMatrix Float64[\n",
" P.σ1*u[1]*u[2] 0.0 \n",
" -P.σ1*u[1]*u[2] -P.σ2*u[2]\n",
" 0.0 P.σ2*u[2]\n",
" ]\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Solving the system\n",
"\n",
"The interpretation of eq. $(\\star)$ is that the population\n",
"follows an ordinary differential equation as\n",
"$ d u_t = b(t, u_t)\\, dt$ above, \n",
"but with superimposed Gaussian noise, such that\n",
"\n",
"$$\n",
" X_{t+\\Delta} \\approx X_{t} + b(t, X_t) d\\Delta + \\sigma(t, X_t) \\sqrt{\\Delta} Z,\n",
"$$\n",
"\n",
"where $Z$ is a *two*-dimensional standard normal vector. $\\sqrt{\\Delta}$ is the scaling of the noise typical for stochastic differential equations. This approximation is the so called Euler-Maruyama scheme implemented in `Bridge`.\n",
"\n",
"The following lines simulate and plot a 2d Brownian motion (Wiener process) $W$ with `Bridge`."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
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" 714.427,913.33 716.45,893.63 718.473,881.091 720.495,881.16 722.518,912.296 724.54,933.999 726.563,923.892 728.586,929.275 730.608,954.857 732.631,976.362 \n",
" 734.654,989.599 736.676,1010.71 738.699,984.457 740.722,983.041 742.744,1040.07 744.767,1055.99 746.789,1086.6 748.812,1090.93 750.835,1066.92 752.857,1063.03 \n",
" 754.88,1022.96 756.903,1033.61 758.925,1048.25 760.948,1008.51 762.97,997.87 764.993,980.743 767.016,1017.62 769.038,1014.08 771.061,1005.9 773.084,991.193 \n",
" 775.106,972.172 777.129,962.05 779.151,977.457 781.174,916.105 783.197,896.337 785.219,940.17 787.242,912.765 789.265,901.224 791.287,912.335 793.31,890.763 \n",
" 795.332,894.244 797.355,877.731 799.378,884.428 801.4,867.189 803.423,848.844 805.446,836.086 807.468,869.456 809.491,834.05 811.513,824.141 813.536,833.236 \n",
" 815.559,845.53 817.581,879.321 819.604,838.201 821.627,830.152 823.649,839.619 825.672,814.235 827.695,776.909 829.717,810.001 831.74,806.649 833.762,792.865 \n",
" 835.785,815.257 837.808,846.55 839.83,793.735 841.853,766.036 843.876,803.547 845.898,805.378 847.921,777.794 849.943,768.277 851.966,756.43 853.989,746.945 \n",
" 856.011,757.106 858.034,783.084 860.057,767.402 862.079,778.398 864.102,788.201 866.124,799.365 868.147,807.591 870.17,776.356 872.192,720.287 874.215,698.962 \n",
" 876.238,701.555 878.26,725.693 880.283,718 882.305,685.564 884.328,668.444 886.351,653.536 888.373,625.07 890.396,626.673 892.419,635.622 894.441,591.483 \n",
" 896.464,571.29 898.486,551.54 900.509,502.291 902.532,551.476 904.554,546.488 906.577,570.05 908.6,558.853 910.622,544.226 912.645,543.64 914.668,574.148 \n",
" 916.69,584.459 918.713,571.487 920.735,560.4 922.758,539.946 924.781,515.34 926.803,515.428 928.826,524.102 930.849,626.615 932.871,644.477 934.894,625.858 \n",
" 936.916,594.124 938.939,611.778 940.962,583.284 942.984,599.051 945.007,576.23 947.03,580.964 949.052,603.548 951.075,594.419 953.097,553.397 955.12,598.582 \n",
" 957.143,584.556 959.165,584.037 961.188,594.22 963.211,559.348 965.233,532.256 967.256,542.072 969.278,514.038 971.301,546.353 973.324,510.394 975.346,482.359 \n",
" 977.369,472.878 979.392,490.076 981.414,506.164 983.437,539.972 985.459,517.609 987.482,507.386 989.505,508.059 991.527,517.905 993.55,536.904 995.573,555.313 \n",
" 997.595,521.097 999.618,538.591 1001.64,579.372 1003.66,574.54 1005.69,605.518 1007.71,603.441 1009.73,613.628 1011.75,628.847 1013.78,657.849 1015.8,702.425 \n",
" 1017.82,739.502 1019.84,710.355 1021.87,698.715 1023.89,662.246 1025.91,621.14 1027.93,632.952 1029.96,647.777 1031.98,650.318 1034,658.368 1036.03,666.329 \n",
" 1038.05,636.471 1040.07,638.288 1042.09,646.877 1044.12,665.3 1046.14,661.342 1048.16,634.349 1050.18,639.642 1052.21,617.447 1054.23,615.498 1056.25,654.245 \n",
" 1058.27,650.917 1060.3,645.41 1062.32,662.116 1064.34,631.785 1066.36,641.832 1068.39,647.343 1070.41,644.209 1072.43,680.838 1074.46,672.318 1076.48,664.678 \n",
" 1078.5,630.401 1080.52,584.527 1082.55,551.519 1084.57,555.225 1086.59,520.9 1088.61,530.617 1090.64,493.195 1092.66,505.928 1094.68,487.134 1096.7,459.761 \n",
" 1098.73,492.874 1100.75,448.225 1102.77,444.197 1104.79,415.849 1106.82,380.325 1108.84,351.802 1110.86,327.617 1112.89,338.748 1114.91,310.194 1116.93,315.068 \n",
" 1118.95,309.644 1120.98,316.836 1123,307.641 1125.02,317.872 1127.04,303.724 1129.07,284.411 1131.09,261.834 1133.11,275.415 1135.13,298.647 1137.16,286.451 \n",
" 1139.18,281.318 1141.2,291.032 1143.22,293.982 1145.25,274.854 1147.27,261.045 1149.29,261.071 1151.31,296.444 1153.34,284.951 1155.36,263.344 1157.38,276.75 \n",
" 1159.41,286.45 1161.43,267.24 1163.45,213.577 1165.47,185.309 1167.5,169.94 1169.52,167.645 1171.54,146.608 1173.56,185.446 1175.59,149.676 1177.61,193.131 \n",
" 1179.63,209.238 1181.65,218.252 1183.68,193.567 1185.7,198.893 1187.72,215.683 1189.74,211.612 1191.77,218.862 1193.79,205.609 1195.81,216.516 1197.84,228.011 \n",
" 1199.86,233.931 1201.88,248.996 1203.9,215.244 1205.93,188.671 1207.95,202.354 1209.97,222.801 1211.99,231.262 1214.02,213.874 1216.04,219.645 1218.06,221.523 \n",
" 1220.08,223.12 1222.11,254.524 1224.13,241.983 1226.15,246.879 1228.17,208.777 1230.2,189.338 1232.22,194.595 1234.24,236.402 1236.27,286.83 1238.29,300.038 \n",
" 1240.31,287.867 1242.33,318.019 1244.36,342.935 1246.38,325.162 1248.4,314.219 1250.42,326.763 1252.45,353.136 1254.47,392.241 1256.49,396.307 1258.51,411.068 \n",
" 1260.54,423.146 1262.56,435.976 1264.58,471.377 1266.6,438.918 1268.63,402.81 1270.65,362.469 1272.67,391.191 1274.7,311.009 1276.72,316.181 1278.74,330.795 \n",
" 1280.76,333.43 1282.79,325.779 1284.81,325.232 1286.83,325.43 1288.85,320.399 1290.88,344.456 1292.9,329.489 1294.92,304.202 1296.94,333.942 1298.97,314.165 \n",
" 1300.99,290.304 1303.01,278.276 1305.03,276.588 1307.06,297.518 1309.08,341.665 1311.1,328.131 1313.13,321.068 1315.15,300.454 1317.17,287.631 1319.19,307.641 \n",
" 1321.22,314.473 1323.24,322.92 1325.26,296.584 1327.28,304.797 1329.31,281.094 1331.33,295.658 1333.35,296.773 1335.37,265.122 1337.4,272.825 1339.42,299.002 \n",
" 1341.44,286.351 1343.46,300.148 1345.49,320.32 1347.51,298.592 1349.53,277.946 1351.56,289.994 1353.58,272.1 1355.6,305.41 1357.62,292.17 1359.65,274.591 \n",
" 1361.67,222.141 1363.69,187.102 1365.71,176.226 1367.74,163.227 1369.76,156.789 1371.78,151.815 1373.8,163.158 1375.83,163.996 1377.85,176.971 1379.87,163.681 \n",
" 1381.89,169.31 1383.92,193.999 1385.94,193.393 1387.96,187.799 1389.99,129.994 1392.01,156.732 1394.03,155.145 1396.05,187.259 1398.08,150.703 1400.1,144.946 \n",
" 1402.12,150.229 1404.14,161.772 1406.17,177.526 1408.19,167.442 1410.21,170.989 1412.23,160.629 1414.26,131.512 1416.28,103.227 1418.3,86.9346 1420.32,86.2547 \n",
" 1422.35,130.143 1424.37,131.66 1426.39,171.527 1428.41,130.66 1430.44,130.33 1432.46,180.329 1434.48,161.657 1436.51,205.216 1438.53,222.627 1440.55,211.535 \n",
" 1442.57,217.845 1444.6,219.31 1446.62,207.926 1448.64,209.643 1450.66,210.738 1452.69,211.1 1454.71,190.123 1456.73,183.734 1458.75,168.827 1460.78,179.065 \n",
" 1462.8,198.782 1464.82,218.778 1466.84,215.189 1468.87,244.391 1470.89,257.649 1472.91,290.545 1474.94,309.445 1476.96,271.105 1478.98,269.597 1481,252.512 \n",
" 1483.03,220.671 1485.05,244.692 1487.07,288.113 1489.09,283.12 1491.12,249.727 1493.14,279.678 1495.16,282.218 1497.18,265.225 1499.21,265.776 1501.23,241.847 \n",
" 1503.25,225.377 1505.27,259.356 1507.3,270.899 1509.32,245.242 1511.34,241.258 1513.37,253.319 1515.39,278.071 1517.41,291.841 1519.43,259.331 1521.46,224.57 \n",
" 1523.48,206.425 1525.5,187.503 1527.52,204.551 1529.55,196.997 1531.57,212.412 1533.59,206.622 1535.61,196.856 1537.64,177.899 1539.66,228.041 1541.68,228.987 \n",
" 1543.7,201.059 1545.73,257.563 1547.75,270.831 1549.77,262.858 1551.8,240.272 1553.82,232.765 1555.84,252.445 1557.86,235.804 1559.89,239.045 1561.91,298.174 \n",
" 1563.93,302.097 1565.95,282.308 1567.98,236.562 1570,244.912 1572.02,260.018 1574.04,257.755 1576.07,228.685 1578.09,238.741 1580.11,245.586 1582.13,249.819 \n",
" 1584.16,266.237 1586.18,261.253 1588.2,220.507 1590.23,202.42 1592.25,194.354 1594.27,212.001 1596.29,227.54 1598.32,230.81 1600.34,205.64 1602.36,254.678 \n",
" 1604.38,257.418 1606.41,271.27 1608.43,257.244 1610.45,197.125 1612.47,197.993 1614.5,213.215 1616.52,177.904 1618.54,198.549 1620.56,189.162 1622.59,188.298 \n",
" 1624.61,192.675 1626.63,211.64 1628.66,238.461 1630.68,260.352 1632.7,282.7 1634.72,296.218 1636.75,298.944 1638.77,291.778 1640.79,286.063 1642.81,302.602 \n",
" 1644.84,333.21 1646.86,358.836 1648.88,393.88 1650.9,385.581 1652.93,378.961 1654.95,386.033 1656.97,416.836 1658.99,394.678 1661.02,412.517 1663.04,410.405 \n",
" 1665.06,423.205 1667.09,428.113 1669.11,441.751 1671.13,417.09 1673.15,451.757 1675.18,461.706 1677.2,424.151 1679.22,387.261 1681.24,380.844 1683.27,398.05 \n",
" 1685.29,374.671 1687.31,386.269 1689.33,353.593 1691.36,381.19 1693.38,399.877 1695.4,378.543 1697.42,375.378 1699.45,370.705 1701.47,396.37 1703.49,394.549 \n",
" 1705.51,382.985 1707.54,394.028 1709.56,377.723 1711.58,398.366 1713.61,396.533 1715.63,369.604 1717.65,391.37 1719.67,371.52 1721.7,357.349 1723.72,366.433 \n",
" 1725.74,377.301 1727.76,369.448 1729.79,374.71 1731.81,373.982 1733.83,393.825 1735.85,402.798 1737.88,396.072 1739.9,386.448 1741.92,389.591 1743.94,428.581 \n",
" 1745.97,473.281 1747.99,492.69 1750.01,522.354 1752.04,524.862 1754.06,529.259 1756.08,571.611 1758.1,544.86 1760.13,549.468 1762.15,529.664 1764.17,533.424 \n",
" 1766.19,530.067 1768.22,516.493 1770.24,554.911 1772.26,569.261 1774.28,521.676 1776.31,516.672 1778.33,517.156 1780.35,535.311 1782.37,614.408 1784.4,609.839 \n",
" 1786.42,605.51 1788.44,594.537 1790.47,564.615 1792.49,537.196 1794.51,536.765 1796.53,526.183 1798.56,533.809 1800.58,517.918 1802.6,517.789 1804.62,510.515 \n",
" 1806.65,522.994 1808.67,502.23 1810.69,513.607 1812.71,493.769 1814.74,507.645 1816.76,480.363 1818.78,444.306 1820.8,427.621 1822.83,462.254 1824.85,468.71 \n",
" 1826.87,449.948 1828.9,415.304 1830.92,431.7 1832.94,405.093 1834.96,397.841 1836.99,363.673 1839.01,371.499 1841.03,390.416 1843.05,443.123 1845.08,448.412 \n",
" 1847.1,444.886 1849.12,469.079 1851.14,452.917 1853.17,461.367 1855.19,472.268 1857.21,451.566 1859.23,457.36 1861.26,451.938 1863.28,492.047 1865.3,500.184 \n",
" 1867.33,493.399 1869.35,508.062 1871.37,509.277 1873.39,519.487 1875.42,489.206 1877.44,527.574 1879.46,481.947 1881.48,502.135 1883.51,521.869 1885.53,525.042 \n",
" 1887.55,519.949 1889.57,525.947 1891.6,511.794 1893.62,488.845 1895.64,485.014 1897.66,514.544 1899.69,489.434 1901.71,529.995 1903.73,562.581 1905.76,543.866 \n",
" 1907.78,552.241 1909.8,546.742 1911.82,513.374 1913.85,483.704 1915.87,471.803 1917.89,497.448 1919.91,502.674 1921.94,512.391 1923.96,528.672 1925.98,524.23 \n",
" 1928,525.464 1930.03,540.895 1932.05,534.422 1934.07,538.613 1936.09,573.225 1938.12,568.056 1940.14,564.044 1942.16,561.391 1944.19,578.22 1946.21,570.336 \n",
" 1948.23,590.937 1950.25,601.896 1952.28,625.161 1954.3,612.296 1956.32,647.545 1958.34,626.036 1960.37,643.157 1962.39,648.218 1964.41,614.616 1966.43,596.277 \n",
" 1968.46,601.75 1970.48,611.727 1972.5,666.377 1974.52,687.688 1976.55,719.888 1978.57,748.996 1980.59,702.243 1982.62,695.231 1984.64,676.586 1986.66,687.994 \n",
" 1988.68,730.743 1990.71,690.523 1992.73,671.75 1994.75,713.593 1996.77,686.395 1998.8,669.184 2000.82,660.194 2002.84,604.746 2004.86,612.19 2006.89,625.538 \n",
" 2008.91,643.785 2010.93,650.863 2012.95,666.112 2014.98,689.578 2017,703.271 2019.02,689.37 2021.04,684.942 2023.07,679.662 2025.09,653.452 2027.11,627.096 \n",
" 2029.14,651.088 2031.16,667.861 2033.18,660.079 2035.2,621.883 2037.23,624.931 2039.25,629.796 2041.27,665.195 2043.29,623.383 2045.32,649.234 2047.34,653.716 \n",
" 2049.36,651.789 2051.38,677.541 2053.41,673.638 2055.43,693.816 2057.45,697.584 2059.47,676.596 2061.5,688.847 2063.52,657.02 2065.54,678.097 2067.57,667.066 \n",
" 2069.59,668.022 2071.61,682.036 2073.63,642.856 2075.66,626.006 2077.68,632.148 2079.7,651.708 2081.72,637.884 2083.75,663.378 2085.77,674.295 2087.79,671.427 \n",
" 2089.81,675.157 2091.84,683.786 2093.86,665.449 2095.88,693.523 2097.9,687.301 2099.93,674.02 2101.95,677.137 2103.97,647.549 2106,648.067 2108.02,631.46 \n",
" 2110.04,650.772 2112.06,689.66 2114.09,710.705 2116.11,704.058 2118.13,705.616 2120.15,645.328 2122.18,633.663 2124.2,617.348 2126.22,589.868 2128.24,585.451 \n",
" 2130.27,594.019 2132.29,639.672 2134.31,630.489 2136.33,580.792 2138.36,573.943 2140.38,551.539 2142.4,577.887 2144.43,570.443 2146.45,543.524 2148.47,575.507 \n",
" 2150.49,597.398 2152.52,579.311 2154.54,587.147 2156.56,557.909 2158.58,583.97 2160.61,588.524 2162.63,580.958 2164.65,548.692 2166.67,548.544 2168.7,493.724 \n",
" 2170.72,437.44 2172.74,452.664 2174.76,438.957 2176.79,439.819 2178.81,434.601 2180.83,447.186 2182.86,421.68 2184.88,412.544 2186.9,384.96 2188.92,382.723 \n",
" 2190.95,376.92 2192.97,362.518 2194.99,359.154 2197.01,378.263 2199.04,408.654 2201.06,396.229 2203.08,425.716 2205.1,400.329 2207.13,444.474 2209.15,414.401 \n",
" 2211.17,383.457 2213.19,383.606 2215.22,382.245 2217.24,406.396 2219.26,387.256 2221.29,396.574 2223.31,416.297 2225.33,431.256 2227.35,463.144 2229.38,433.981 \n",
" 2231.4,441.031 2233.42,434.447 2235.44,439.607 2237.47,412.204 2239.49,396.466 2241.51,404.847 2243.53,419.598 2245.56,412.211 2247.58,417.982 2249.6,426.95 \n",
" 2251.62,421.48 2253.65,429.273 2255.67,452.983 2257.69,454.678 2259.72,455.789 2261.74,491.901 2263.76,488.974 2265.78,483.403 2267.81,477.953 2269.83,469.415 \n",
" 2271.85,472.336 2273.87,470.206 2275.9,454.381 2277.92,443.662 2279.94,415.611 2281.96,448.108 2283.99,484.579 2286.01,459.951 2288.03,503.564 2290.05,479.228 \n",
" 2292.08,444.008 \n",
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" 289.676,816.713 291.698,844.371 293.721,831.78 295.743,802.678 297.766,792.532 299.789,835.46 301.811,832.731 303.834,856.118 305.857,852.564 307.879,837.724 \n",
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" 390.807,849.184 392.83,856.868 394.852,879.671 396.875,855.841 398.897,865.339 400.92,895.26 402.943,955.581 404.965,987.809 406.988,1010.69 409.011,992.198 \n",
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" 431.259,892.335 433.282,874.855 435.305,912.421 437.327,913.811 439.35,941.292 441.373,945.623 443.395,948.118 445.418,958.09 447.44,975.845 449.463,972.913 \n",
" 451.486,970.004 453.508,928.298 455.531,952.264 457.554,978.782 459.576,990.101 461.599,963.571 463.622,973.759 465.644,950.198 467.667,942.077 469.689,957.029 \n",
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" 512.165,835.413 514.187,820.11 516.21,805.931 518.232,823.925 520.255,810.509 522.278,821.574 524.3,824.955 526.323,806.962 528.346,793.059 530.368,821.76 \n",
" 532.391,846.483 534.413,866.771 536.436,901.565 538.459,922.639 540.481,907.271 542.504,887.522 544.527,870.989 546.549,842.997 548.572,774.387 550.595,774.902 \n",
" 552.617,773.293 554.64,804.831 556.662,811.896 558.685,774.756 560.708,783.969 562.73,798.889 564.753,783.043 566.776,755.437 568.798,718.332 570.821,744.095 \n",
" 572.843,744.291 574.866,738.57 576.889,765.621 578.911,753.452 580.934,783.367 582.957,770.58 584.979,723.783 587.002,694.812 589.024,667.629 591.047,661.472 \n",
" 593.07,647.552 595.092,689.895 597.115,674.49 599.138,692.098 601.16,685.095 603.183,679.454 605.205,701.641 607.228,727.754 609.251,713.732 611.273,705.107 \n",
" 613.296,715.615 615.319,690.743 617.341,714.523 619.364,728.509 621.386,728.156 623.409,724.483 625.432,739.463 627.454,728.953 629.477,707.477 631.5,702.273 \n",
" 633.522,720.045 635.545,746.074 637.568,729.284 639.59,751.241 641.613,737.233 643.635,723.168 645.658,710.237 647.681,693.179 649.703,650.411 651.726,676.447 \n",
" 653.749,629.378 655.771,629.648 657.794,587.91 659.816,547.502 661.839,543.797 663.862,508.429 665.884,534.78 667.907,531 669.93,495.632 671.952,449.834 \n",
" 673.975,445.982 675.997,469.019 678.02,519.293 680.043,513.975 682.065,540.203 684.088,574.234 686.111,536.896 688.133,532.479 690.156,555.898 692.178,535.521 \n",
" 694.201,523.657 696.224,578.427 698.246,559.115 700.269,566.189 702.292,599.612 704.314,618.795 706.337,599.191 708.359,587.415 710.382,625.33 712.405,648.267 \n",
" 714.427,678.605 716.45,696.955 718.473,708.228 720.495,728.447 722.518,749.578 724.54,759.426 726.563,760.961 728.586,782.437 730.608,784.742 732.631,810.498 \n",
" 734.654,791.921 736.676,830.395 738.699,872.7 740.722,859.82 742.744,856.902 744.767,801.51 746.789,827.318 748.812,872.402 750.835,868.852 752.857,887.777 \n",
" 754.88,896.539 756.903,886.83 758.925,904.627 760.948,949.913 762.97,940.175 764.993,947.382 767.016,993.404 769.038,1006.73 771.061,960.95 773.084,979.478 \n",
" 775.106,961.521 777.129,919.49 779.151,924.988 781.174,953.79 783.197,931.849 785.219,933.434 787.242,937.806 789.265,938.373 791.287,923.369 793.31,923.574 \n",
" 795.332,949.731 797.355,936.381 799.378,877.15 801.4,900.039 803.423,912.126 805.446,904.477 807.468,931.406 809.491,907.536 811.513,896.617 813.536,888.354 \n",
" 815.559,874.812 817.581,911.974 819.604,900.587 821.627,931.206 823.649,952.516 825.672,965.211 827.695,999.799 829.717,986.252 831.74,969.115 833.762,949.734 \n",
" 835.785,948.202 837.808,1026.22 839.83,1042.98 841.853,1046.63 843.876,1029.54 845.898,1047.33 847.921,1079.24 849.943,1080.85 851.966,1066.9 853.989,1075.89 \n",
" 856.011,1079.98 858.034,1065.34 860.057,1072.05 862.079,1075.98 864.102,1070.79 866.124,1078.67 868.147,1094.45 870.17,1065.07 872.192,1057.16 874.215,1038.5 \n",
" 876.238,1029.75 878.26,1010.55 880.283,1019.26 882.305,1031.98 884.328,1027.05 886.351,1003.65 888.373,1002.18 890.396,1036.53 892.419,1041.48 894.441,1052.72 \n",
" 896.464,1046.38 898.486,1038.15 900.509,1065.65 902.532,1061.52 904.554,1058.4 906.577,1069.44 908.6,1067.62 910.622,1048.97 912.645,1038.65 914.668,1043.36 \n",
" 916.69,1077.15 918.713,1048.18 920.735,1063.31 922.758,1055.81 924.781,1074.22 926.803,1086.11 928.826,1106.19 930.849,1055.84 932.871,1075.31 934.894,1132.8 \n",
" 936.916,1149.26 938.939,1153.03 940.962,1158.45 942.984,1195.38 945.007,1171.39 947.03,1167.91 949.052,1156.69 951.075,1085.02 953.097,1095.45 955.12,1092.57 \n",
" 957.143,1097.3 959.165,1097.81 961.188,1087.81 963.211,1055.96 965.233,1066.91 967.256,1065.17 969.278,1052.78 971.301,1080.02 973.324,1099.7 975.346,1053.43 \n",
" 977.369,1005.08 979.392,1032.76 981.414,1005.25 983.437,992.453 985.459,1004.58 987.482,1016.67 989.505,1032.25 991.527,982.235 993.55,992.913 995.573,976.956 \n",
" 997.595,939.045 999.618,967.661 1001.64,946.851 1003.66,944.373 1005.69,935.122 1007.71,912.268 1009.73,892.682 1011.75,869.801 1013.78,858.933 1015.8,865.942 \n",
" 1017.82,923.145 1019.84,942.129 1021.87,913.395 1023.89,918.681 1025.91,916.851 1027.93,931.015 1029.96,917.574 1031.98,904.001 1034,880.189 1036.03,833.722 \n",
" 1038.05,772.688 1040.07,786.607 1042.09,775.692 1044.12,815.705 1046.14,800.99 1048.16,794.996 1050.18,802.286 1052.21,801.006 1054.23,744.699 1056.25,765.547 \n",
" 1058.27,778.066 1060.3,749.304 1062.32,771.839 1064.34,748.569 1066.36,746.22 1068.39,777.329 1070.41,788.94 1072.43,775.547 1074.46,794.944 1076.48,788.427 \n",
" 1078.5,780.423 1080.52,780.029 1082.55,726.014 1084.57,735.862 1086.59,742.295 1088.61,744.468 1090.64,764.954 1092.66,741.307 1094.68,739.557 1096.7,757.658 \n",
" 1098.73,740.966 1100.75,723.894 1102.77,687.814 1104.79,684.306 1106.82,683.877 1108.84,658.204 1110.86,625.262 1112.89,650.276 1114.91,646.236 1116.93,647.897 \n",
" 1118.95,645.794 1120.98,711.248 1123,750.406 1125.02,762.535 1127.04,738.597 1129.07,777.354 1131.09,780.941 1133.11,775.89 1135.13,762.852 1137.16,799.009 \n",
" 1139.18,800.255 1141.2,803.908 1143.22,793.922 1145.25,766.448 1147.27,729.225 1149.29,723.704 1151.31,731.712 1153.34,743.052 1155.36,733.138 1157.38,747.889 \n",
" 1159.41,762.334 1161.43,746.395 1163.45,702.824 1165.47,683.3 1167.5,683.677 1169.52,717.317 1171.54,710.565 1173.56,739.381 1175.59,730.944 1177.61,700.826 \n",
" 1179.63,733.698 1181.65,762.036 1183.68,732.698 1185.7,700.634 1187.72,692.414 1189.74,716.868 1191.77,682.482 1193.79,661.708 1195.81,685.908 1197.84,677.969 \n",
" 1199.86,667.814 1201.88,695.979 1203.9,685.84 1205.93,715.646 1207.95,700.059 1209.97,721.808 1211.99,729.68 1214.02,729.76 1216.04,706.038 1218.06,678.02 \n",
" 1220.08,691.411 1222.11,704.466 1224.13,705.728 1226.15,693.412 1228.17,674.965 1230.2,675.58 1232.22,697.665 1234.24,688.672 1236.27,659.642 1238.29,656.997 \n",
" 1240.31,666.947 1242.33,689.672 1244.36,710.51 1246.38,695.876 1248.4,684.015 1250.42,657.564 1252.45,672.517 1254.47,634.787 1256.49,631.395 1258.51,618.008 \n",
" 1260.54,589.803 1262.56,623.186 1264.58,613.598 1266.6,661.641 1268.63,679.164 1270.65,703.801 1272.67,718.28 1274.7,719.391 1276.72,744.196 1278.74,737.712 \n",
" 1280.76,758.488 1282.79,778.125 1284.81,757.603 1286.83,765.474 1288.85,764.788 1290.88,728.116 1292.9,720.457 1294.92,713.546 1296.94,703.674 1298.97,674.551 \n",
" 1300.99,678.353 1303.01,658.305 1305.03,639.777 1307.06,640.439 1309.08,643.195 1311.1,654.804 1313.13,660.81 1315.15,665.254 1317.17,678.645 1319.19,672.919 \n",
" 1321.22,688.193 1323.24,664.702 1325.26,673.524 1327.28,641.934 1329.31,642.594 1331.33,664.424 1333.35,699.414 1335.37,698.716 1337.4,653.534 1339.42,615.648 \n",
" 1341.44,630.668 1343.46,655.414 1345.49,652.248 1347.51,684.288 1349.53,701.552 1351.56,737.187 1353.58,743.538 1355.6,765.723 1357.62,782.753 1359.65,749.638 \n",
" 1361.67,716.897 1363.69,733.798 1365.71,715.76 1367.74,696.606 1369.76,694.735 1371.78,713.329 1373.8,717.31 1375.83,719.974 1377.85,757.033 1379.87,728.201 \n",
" 1381.89,736.532 1383.92,772.148 1385.94,793.428 1387.96,751.376 1389.99,768.234 1392.01,728.981 1394.03,749.089 1396.05,719.105 1398.08,688.471 1400.1,707.918 \n",
" 1402.12,687.181 1404.14,657.904 1406.17,663.176 1408.19,640.81 1410.21,657.893 1412.23,665.193 1414.26,670.079 1416.28,676.254 1418.3,674.134 1420.32,705.453 \n",
" 1422.35,753.926 1424.37,739.479 1426.39,744.445 1428.41,746.953 1430.44,730.451 1432.46,712.983 1434.48,697.053 1436.51,678.39 1438.53,675.507 1440.55,660.751 \n",
" 1442.57,647.149 1444.6,627.827 1446.62,661.797 1448.64,670.041 1450.66,684.562 1452.69,659.09 1454.71,680.93 1456.73,705.141 1458.75,740.797 1460.78,716.413 \n",
" 1462.8,754.134 1464.82,753.379 1466.84,767.441 1468.87,782.048 1470.89,773.978 1472.91,749.703 1474.94,737.474 1476.96,760.097 1478.98,751.385 1481,762.195 \n",
" 1483.03,779.899 1485.05,756.43 1487.07,767.396 1489.09,764.223 1491.12,794.895 1493.14,789.778 1495.16,809.084 1497.18,829.845 1499.21,796.787 1501.23,839.314 \n",
" 1503.25,845.896 1505.27,864.908 1507.3,860.517 1509.32,856.462 1511.34,848.105 1513.37,888.356 1515.39,885.805 1517.41,894.68 1519.43,918.742 1521.46,910.012 \n",
" 1523.48,919.467 1525.5,955.057 1527.52,981.135 1529.55,986.049 1531.57,1000.56 1533.59,984.437 1535.61,1008.91 1537.64,996.521 1539.66,980.202 1541.68,981.536 \n",
" 1543.7,968.284 1545.73,958.93 1547.75,951.986 1549.77,984.424 1551.8,1005.19 1553.82,977.394 1555.84,971.246 1557.86,945.672 1559.89,960.407 1561.91,974.112 \n",
" 1563.93,986.321 1565.95,955.187 1567.98,935.497 1570,951.425 1572.02,926.633 1574.04,881.884 1576.07,896.468 1578.09,896.023 1580.11,895.582 1582.13,916.628 \n",
" 1584.16,908.519 1586.18,874.45 1588.2,864.791 1590.23,835.297 1592.25,821.063 1594.27,801.506 1596.29,806.665 1598.32,786.616 1600.34,806.666 1602.36,799.417 \n",
" 1604.38,804.373 1606.41,771.847 1608.43,749.429 1610.45,765.348 1612.47,773.39 1614.5,791.005 1616.52,787.719 1618.54,753.67 1620.56,769.845 1622.59,750.776 \n",
" 1624.61,726.889 1626.63,668.567 1628.66,671.705 1630.68,660.637 1632.7,702.929 1634.72,732.045 1636.75,767.061 1638.77,770.577 1640.79,759.804 1642.81,803.404 \n",
" 1644.84,812.07 1646.86,810.558 1648.88,805.983 1650.9,839.432 1652.93,850.109 1654.95,861.814 1656.97,829.14 1658.99,854.568 1661.02,837.366 1663.04,842.935 \n",
" 1665.06,815.392 1667.09,853.448 1669.11,835.06 1671.13,855.902 1673.15,909.341 1675.18,936.489 1677.2,943.459 1679.22,918.62 1681.24,909.649 1683.27,924.237 \n",
" 1685.29,933.926 1687.31,903.833 1689.33,910.952 1691.36,896.79 1693.38,877.133 1695.4,882.113 1697.42,915.78 1699.45,897.528 1701.47,890.052 1703.49,859.98 \n",
" 1705.51,831.073 1707.54,834.559 1709.56,852.424 1711.58,853.752 1713.61,828.161 1715.63,831.867 1717.65,812.071 1719.67,837.255 1721.7,845.982 1723.72,861.657 \n",
" 1725.74,889.214 1727.76,886.593 1729.79,890.376 1731.81,845.177 1733.83,835.637 1735.85,866.589 1737.88,867.027 1739.9,826.224 1741.92,812.788 1743.94,757.73 \n",
" 1745.97,728.149 1747.99,707.352 1750.01,681.338 1752.04,660.931 1754.06,628.963 1756.08,637.305 1758.1,611.025 1760.13,629.222 1762.15,629.952 1764.17,659.463 \n",
" 1766.19,649.066 1768.22,637.546 1770.24,644.897 1772.26,644.63 1774.28,627.728 1776.31,673.776 1778.33,695.917 1780.35,663.505 1782.37,647.456 1784.4,646.979 \n",
" 1786.42,648.302 1788.44,635.402 1790.47,638.718 1792.49,645.581 1794.51,658.799 1796.53,674.457 1798.56,646.779 1800.58,647.339 1802.6,637.959 1804.62,636.439 \n",
" 1806.65,604.231 1808.67,578.249 1810.69,560.216 1812.71,573.843 1814.74,601.897 1816.76,564.537 1818.78,552.579 1820.8,581.06 1822.83,581.92 1824.85,574.981 \n",
" 1826.87,564.264 1828.9,541.922 1830.92,565.888 1832.94,564.34 1834.96,592.994 1836.99,574.055 1839.01,565.563 1841.03,578.838 1843.05,610.509 1845.08,594.025 \n",
" 1847.1,614.463 1849.12,581.765 1851.14,575.665 1853.17,595.24 1855.19,616.758 1857.21,614.026 1859.23,644.249 1861.26,655.09 1863.28,609.547 1865.3,607.36 \n",
" 1867.33,600.154 1869.35,620.408 1871.37,660.738 1873.39,643.015 1875.42,664.348 1877.44,707.465 1879.46,683.752 1881.48,653.095 1883.51,677.747 1885.53,685.42 \n",
" 1887.55,685.243 1889.57,666.691 1891.6,665.252 1893.62,639.217 1895.64,603.526 1897.66,592.31 1899.69,603.555 1901.71,625.168 1903.73,621.14 1905.76,628.042 \n",
" 1907.78,641.984 1909.8,637.778 1911.82,590.437 1913.85,591.239 1915.87,587.683 1917.89,584.199 1919.91,591.196 1921.94,614.864 1923.96,590.213 1925.98,639.909 \n",
" 1928,645.371 1930.03,607.174 1932.05,615.181 1934.07,625.387 1936.09,643.603 1938.12,657.432 1940.14,687.349 1942.16,690.097 1944.19,674.326 1946.21,675.364 \n",
" 1948.23,638.996 1950.25,678.881 1952.28,685.348 1954.3,635.496 1956.32,610.995 1958.34,612.959 1960.37,598.653 1962.39,615.375 1964.41,650.797 1966.43,664.414 \n",
" 1968.46,612.2 1970.48,638.12 1972.5,600.275 1974.52,609.31 1976.55,645.425 1978.57,638.842 1980.59,669.7 1982.62,668.704 1984.64,652.188 1986.66,670.86 \n",
" 1988.68,663.375 1990.71,691.116 1992.73,662.793 1994.75,633.664 1996.77,609.587 1998.8,612.494 2000.82,580.301 2002.84,577.526 2004.86,614.359 2006.89,585.192 \n",
" 2008.91,541.731 2010.93,577.348 2012.95,573.642 2014.98,599.719 2017,541.286 2019.02,555.284 2021.04,585.672 2023.07,612.132 2025.09,651.21 2027.11,702.93 \n",
" 2029.14,726.503 2031.16,716.609 2033.18,748.813 2035.2,749.531 2037.23,803.682 2039.25,847.425 2041.27,865.737 2043.29,847.891 2045.32,828.91 2047.34,834.352 \n",
" 2049.36,880.751 2051.38,913.446 2053.41,901.495 2055.43,871.176 2057.45,845.188 2059.47,838.539 2061.5,771.512 2063.52,732.759 2065.54,748.027 2067.57,721.957 \n",
" 2069.59,733.83 2071.61,768.653 2073.63,803.16 2075.66,828.978 2077.68,839.187 2079.7,803.124 2081.72,790.594 2083.75,740.034 2085.77,720.437 2087.79,752.373 \n",
" 2089.81,734.117 2091.84,717.886 2093.86,759.251 2095.88,779.876 2097.9,786.777 2099.93,811.237 2101.95,831.539 2103.97,786.636 2106,813.079 2108.02,788.022 \n",
" 2110.04,802.038 2112.06,779.309 2114.09,797.873 2116.11,820.417 2118.13,850.482 2120.15,839.879 2122.18,882.878 2124.2,926.463 2126.22,953.858 2128.24,943.802 \n",
" 2130.27,955.778 2132.29,948.956 2134.31,939.565 2136.33,928.35 2138.36,917.598 2140.38,966.234 2142.4,1000.69 2144.43,1044.25 2146.45,1014.81 2148.47,1048.72 \n",
" 2150.49,1056 2152.52,1102.13 2154.54,1062.69 2156.56,1056.2 2158.58,1057.44 2160.61,1047.39 2162.63,1072.44 2164.65,1077.29 2166.67,1091.63 2168.7,1097.53 \n",
" 2170.72,1122.44 2172.74,1125.75 2174.76,1105.03 2176.79,1106.27 2178.81,1146.67 2180.83,1168.22 2182.86,1231 2184.88,1217.93 2186.9,1242.44 2188.92,1244.93 \n",
" 2190.95,1240.77 2192.97,1266.14 2194.99,1280.58 2197.01,1258.19 2199.04,1307.38 2201.06,1320.1 2203.08,1289.6 2205.1,1276.61 2207.13,1310.41 2209.15,1336.35 \n",
" 2211.17,1348.09 2213.19,1349.74 2215.22,1338.34 2217.24,1310.56 2219.26,1340.43 2221.29,1386.61 2223.31,1384.44 2225.33,1335.83 2227.35,1355.69 2229.38,1332.09 \n",
" 2231.4,1319.92 2233.42,1326.09 2235.44,1318.78 2237.47,1315.09 2239.49,1323.98 2241.51,1286.55 2243.53,1264.99 2245.56,1248.98 2247.58,1247.3 2249.6,1238.48 \n",
" 2251.62,1230.47 2253.65,1213.15 2255.67,1210.53 2257.69,1194.12 2259.72,1208.39 2261.74,1225.11 2263.76,1247.58 2265.78,1259.8 2267.81,1241.86 2269.83,1253.91 \n",
" 2271.85,1255.89 2273.87,1241.84 2275.9,1247.56 2277.92,1253.56 2279.94,1266.02 2281.96,1304.23 2283.99,1270.68 2286.01,1269.63 2288.03,1270.91 2290.05,1276.78 \n",
" 2292.08,1241.02 \n",
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},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u₀ = @SVector [49.0, 1.0, 0.0]\n",
"tt = 0.0:0.001:1.0\n",
"\n",
"W = sample(tt, Wiener{SVector{2,Float64}}())\n",
"\n",
"plot(W.tt, first.(W.yy), label=L\"W_1(t)\")\n",
"plot!(W.tt, last.(W.yy), label=L\"W_2(t)\")\n",
"\n",
"xlabel!(\"t\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Solve the SDE starting in $u_0$ on the grid $t_i = 0.001i$ using the Euler-Maruyama approximation."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"\n",
"X = solve(Bridge.EulerMaruyama(), u₀, W, P);"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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" 1184.03,1372.8 1186.11,1372.83 1188.2,1372.83 1190.29,1372.84 1192.37,1372.82 1194.46,1372.81 1196.55,1372.82 1198.63,1372.86 1200.72,1372.89 1202.81,1372.91 \n",
" 1204.9,1372.9 1206.98,1372.93 1209.07,1372.94 1211.16,1372.94 1213.24,1372.94 1215.33,1372.95 1217.42,1372.97 1219.51,1372.99 1221.59,1373 1223.68,1373.01 \n",
" 1225.77,1373.01 1227.85,1373.02 1229.94,1373.04 1232.03,1373.03 1234.11,1373.02 1236.2,1373.01 1238.29,1373.02 1240.38,1372.99 1242.46,1373 1244.55,1373.01 \n",
" 1246.64,1373.02 1248.72,1373.02 1250.81,1373.02 1252.9,1373.02 1254.98,1373.03 1257.07,1373.04 1259.16,1373.04 1261.25,1373.03 1263.33,1373.04 1265.42,1373.04 \n",
" 1267.51,1373.03 1269.59,1373.03 1271.68,1373.04 1273.77,1373.05 1275.85,1373.06 1277.94,1373.06 1280.03,1373.06 1282.12,1373.06 1284.2,1373.06 1286.29,1373.07 \n",
" 1288.38,1373.07 1290.46,1373.08 1292.55,1373.07 1294.64,1373.08 1296.72,1373.07 1298.81,1373.08 1300.9,1373.08 1302.99,1373.08 1305.07,1373.08 1307.16,1373.09 \n",
" 1309.25,1373.09 1311.33,1373.1 1313.42,1373.11 1315.51,1373.1 1317.59,1373.1 1319.68,1373.1 1321.77,1373.1 1323.86,1373.11 1325.94,1373.11 1328.03,1373.11 \n",
" 1330.12,1373.1 1332.2,1373.09 1334.29,1373.09 1336.38,1373.09 1338.47,1373.09 1340.55,1373.09 1342.64,1373.1 1344.73,1373.1 1346.81,1373.1 1348.9,1373.1 \n",
" 1350.99,1373.11 1353.07,1373.12 1355.16,1373.12 1357.25,1373.12 1359.34,1373.11 1361.42,1373.12 1363.51,1373.12 1365.6,1373.13 1367.68,1373.12 1369.77,1373.12 \n",
" 1371.86,1373.13 1373.94,1373.13 1376.03,1373.14 1378.12,1373.13 1380.21,1373.14 1382.29,1373.14 1384.38,1373.13 1386.47,1373.13 1388.55,1373.13 1390.64,1373.13 \n",
" 1392.73,1373.14 1394.81,1373.14 1396.9,1373.15 1398.99,1373.15 1401.08,1373.15 1403.16,1373.16 1405.25,1373.16 1407.34,1373.17 1409.42,1373.17 1411.51,1373.17 \n",
" 1413.6,1373.17 1415.68,1373.18 1417.77,1373.18 1419.86,1373.18 1421.95,1373.18 1424.03,1373.18 1426.12,1373.18 1428.21,1373.18 1430.29,1373.18 1432.38,1373.18 \n",
" 1434.47,1373.19 1436.55,1373.19 1438.64,1373.19 1440.73,1373.2 1442.82,1373.2 1444.9,1373.21 1446.99,1373.21 1449.08,1373.21 1451.16,1373.21 1453.25,1373.21 \n",
" 1455.34,1373.2 1457.43,1373.21 1459.51,1373.21 1461.6,1373.21 1463.69,1373.21 1465.77,1373.22 1467.86,1373.22 1469.95,1373.22 1472.03,1373.22 1474.12,1373.22 \n",
" 1476.21,1373.21 1478.3,1373.22 1480.38,1373.22 1482.47,1373.22 1484.56,1373.22 1486.64,1373.22 1488.73,1373.23 1490.82,1373.23 1492.9,1373.23 1494.99,1373.22 \n",
" 1497.08,1373.22 1499.17,1373.22 1501.25,1373.23 1503.34,1373.23 1505.43,1373.23 1507.51,1373.23 1509.6,1373.23 1511.69,1373.23 1513.77,1373.23 1515.86,1373.24 \n",
" 1517.95,1373.23 1520.04,1373.24 1522.12,1373.24 1524.21,1373.24 1526.3,1373.24 1528.38,1373.24 1530.47,1373.24 1532.56,1373.24 1534.64,1373.24 1536.73,1373.25 \n",
" 1538.82,1373.25 1540.91,1373.25 1542.99,1373.25 1545.08,1373.25 1547.17,1373.25 1549.25,1373.25 1551.34,1373.25 1553.43,1373.25 1555.52,1373.25 1557.6,1373.25 \n",
" 1559.69,1373.26 1561.78,1373.26 1563.86,1373.25 1565.95,1373.25 1568.04,1373.25 1570.12,1373.26 1572.21,1373.26 1574.3,1373.26 1576.39,1373.26 1578.47,1373.26 \n",
" 1580.56,1373.26 1582.65,1373.26 1584.73,1373.26 1586.82,1373.26 1588.91,1373.26 1590.99,1373.26 1593.08,1373.26 1595.17,1373.26 1597.26,1373.26 1599.34,1373.26 \n",
" 1601.43,1373.26 1603.52,1373.27 1605.6,1373.27 1607.69,1373.27 1609.78,1373.27 1611.86,1373.27 1613.95,1373.27 1616.04,1373.27 1618.13,1373.27 1620.21,1373.28 \n",
" 1622.3,1373.28 1624.39,1373.28 1626.47,1373.28 1628.56,1373.28 1630.65,1373.28 1632.73,1373.28 1634.82,1373.28 1636.91,1373.28 1639,1373.29 1641.08,1373.29 \n",
" 1643.17,1373.29 1645.26,1373.29 1647.34,1373.29 1649.43,1373.29 1651.52,1373.29 1653.6,1373.29 1655.69,1373.29 1657.78,1373.29 1659.87,1373.29 1661.95,1373.29 \n",
" 1664.04,1373.29 1666.13,1373.29 1668.21,1373.29 1670.3,1373.29 1672.39,1373.29 1674.48,1373.29 1676.56,1373.29 1678.65,1373.29 1680.74,1373.29 1682.82,1373.29 \n",
" 1684.91,1373.29 1687,1373.29 1689.08,1373.29 1691.17,1373.29 1693.26,1373.29 1695.35,1373.29 1697.43,1373.3 1699.52,1373.29 1701.61,1373.29 1703.69,1373.3 \n",
" 1705.78,1373.3 1707.87,1373.3 1709.95,1373.3 1712.04,1373.3 1714.13,1373.3 1716.22,1373.3 1718.3,1373.3 1720.39,1373.3 1722.48,1373.3 1724.56,1373.3 \n",
" 1726.65,1373.3 1728.74,1373.3 1730.82,1373.3 1732.91,1373.3 1735,1373.3 1737.09,1373.31 1739.17,1373.31 1741.26,1373.31 1743.35,1373.31 1745.43,1373.31 \n",
" 1747.52,1373.31 1749.61,1373.31 1751.69,1373.31 1753.78,1373.31 1755.87,1373.31 1757.96,1373.31 1760.04,1373.31 1762.13,1373.31 1764.22,1373.31 1766.3,1373.31 \n",
" 1768.39,1373.31 1770.48,1373.31 1772.56,1373.31 1774.65,1373.31 1776.74,1373.31 1778.83,1373.31 1780.91,1373.31 1783,1373.31 1785.09,1373.31 1787.17,1373.31 \n",
" 1789.26,1373.31 1791.35,1373.31 1793.44,1373.31 1795.52,1373.31 1797.61,1373.31 1799.7,1373.31 1801.78,1373.31 1803.87,1373.31 1805.96,1373.31 1808.04,1373.31 \n",
" 1810.13,1373.31 1812.22,1373.31 1814.31,1373.31 1816.39,1373.31 1818.48,1373.31 1820.57,1373.31 1822.65,1373.31 1824.74,1373.31 1826.83,1373.31 1828.91,1373.31 \n",
" 1831,1373.31 1833.09,1373.31 1835.18,1373.31 1837.26,1373.31 1839.35,1373.31 1841.44,1373.31 1843.52,1373.31 1845.61,1373.31 1847.7,1373.31 1849.78,1373.31 \n",
" 1851.87,1373.31 1853.96,1373.31 1856.05,1373.32 1858.13,1373.32 1860.22,1373.32 1862.31,1373.32 1864.39,1373.32 1866.48,1373.32 1868.57,1373.32 1870.65,1373.32 \n",
" 1872.74,1373.32 1874.83,1373.32 1876.92,1373.32 1879,1373.32 1881.09,1373.32 1883.18,1373.32 1885.26,1373.32 1887.35,1373.32 1889.44,1373.32 1891.52,1373.32 \n",
" 1893.61,1373.32 1895.7,1373.32 1897.79,1373.32 1899.87,1373.32 1901.96,1373.32 1904.05,1373.32 1906.13,1373.32 1908.22,1373.32 1910.31,1373.32 1912.4,1373.32 \n",
" 1914.48,1373.32 1916.57,1373.32 1918.66,1373.32 1920.74,1373.32 1922.83,1373.32 1924.92,1373.32 1927,1373.32 1929.09,1373.32 1931.18,1373.32 1933.27,1373.32 \n",
" 1935.35,1373.32 1937.44,1373.32 1939.53,1373.32 1941.61,1373.32 1943.7,1373.32 1945.79,1373.32 1947.87,1373.32 1949.96,1373.32 1952.05,1373.32 1954.14,1373.32 \n",
" 1956.22,1373.32 1958.31,1373.32 1960.4,1373.32 1962.48,1373.32 1964.57,1373.32 1966.66,1373.32 1968.74,1373.32 1970.83,1373.32 1972.92,1373.32 1975.01,1373.32 \n",
" 1977.09,1373.32 1979.18,1373.32 1981.27,1373.32 1983.35,1373.32 1985.44,1373.32 1987.53,1373.32 1989.61,1373.32 1991.7,1373.32 1993.79,1373.32 1995.88,1373.32 \n",
" 1997.96,1373.32 2000.05,1373.33 2002.14,1373.33 2004.22,1373.33 2006.31,1373.33 2008.4,1373.33 2010.48,1373.33 2012.57,1373.33 2014.66,1373.33 2016.75,1373.33 \n",
" 2018.83,1373.33 2020.92,1373.33 2023.01,1373.33 2025.09,1373.33 2027.18,1373.33 2029.27,1373.33 2031.36,1373.33 2033.44,1373.33 2035.53,1373.33 2037.62,1373.33 \n",
" 2039.7,1373.33 2041.79,1373.33 2043.88,1373.33 2045.96,1373.33 2048.05,1373.33 2050.14,1373.33 2052.23,1373.33 2054.31,1373.33 2056.4,1373.33 2058.49,1373.33 \n",
" 2060.57,1373.33 2062.66,1373.33 2064.75,1373.33 2066.83,1373.33 2068.92,1373.33 2071.01,1373.33 2073.1,1373.33 2075.18,1373.33 2077.27,1373.33 2079.36,1373.33 \n",
" 2081.44,1373.33 2083.53,1373.33 2085.62,1373.33 2087.7,1373.33 2089.79,1373.33 2091.88,1373.33 2093.97,1373.33 2096.05,1373.33 2098.14,1373.33 2100.23,1373.33 \n",
" 2102.31,1373.33 2104.4,1373.33 2106.49,1373.33 2108.57,1373.33 2110.66,1373.33 2112.75,1373.33 2114.84,1373.33 2116.92,1373.33 2119.01,1373.33 2121.1,1373.33 \n",
" 2123.18,1373.33 2125.27,1373.33 2127.36,1373.33 2129.44,1373.33 2131.53,1373.33 2133.62,1373.33 2135.71,1373.33 2137.79,1373.33 2139.88,1373.33 2141.97,1373.33 \n",
" 2144.05,1373.33 2146.14,1373.33 2148.23,1373.33 2150.32,1373.33 2152.4,1373.33 2154.49,1373.33 2156.58,1373.33 2158.66,1373.33 2160.75,1373.33 2162.84,1373.33 \n",
" 2164.92,1373.33 2167.01,1373.33 2169.1,1373.33 2171.19,1373.33 2173.27,1373.33 2175.36,1373.33 2177.45,1373.33 2179.53,1373.33 2181.62,1373.33 2183.71,1373.33 \n",
" 2185.79,1373.33 2187.88,1373.33 2189.97,1373.33 2192.06,1373.33 2194.14,1373.33 2196.23,1373.33 2198.32,1373.33 2200.4,1373.33 2202.49,1373.33 2204.58,1373.33 \n",
" 2206.66,1373.33 2208.75,1373.33 2210.84,1373.33 2212.93,1373.33 2215.01,1373.33 2217.1,1373.33 2219.19,1373.33 2221.27,1373.33 2223.36,1373.33 2225.45,1373.33 \n",
" 2227.53,1373.33 2229.62,1373.33 2231.71,1373.33 2233.8,1373.33 2235.88,1373.33 2237.97,1373.33 2240.06,1373.33 2242.14,1373.33 2244.23,1373.33 2246.32,1373.33 \n",
" 2248.41,1373.33 2250.49,1373.33 2252.58,1373.33 2254.67,1373.33 2256.75,1373.33 2258.84,1373.33 2260.93,1373.33 2263.01,1373.33 2265.1,1373.33 2267.19,1373.33 \n",
" 2269.28,1373.33 2271.36,1373.33 2273.45,1373.33 2275.54,1373.33 2277.62,1373.33 2279.71,1373.33 2281.8,1373.33 2283.88,1373.33 2285.97,1373.33 2288.06,1373.33 \n",
" 2290.15,1373.33 \n",
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" 223.998,1343.62 226.085,1340.1 228.172,1336.66 230.259,1333.86 232.346,1321.89 234.433,1313.1 236.52,1302.69 238.607,1303.32 240.694,1305.23 242.781,1310.2 \n",
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" 599.661,558.298 601.748,558.439 603.835,580.748 605.922,595.565 608.009,595.024 610.096,613.877 612.183,604.669 614.27,611.734 616.357,627.928 618.444,640.699 \n",
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" 766.622,608.39 768.709,594.858 770.796,602.452 772.883,592.322 774.97,585.54 777.057,582.954 779.144,571.501 781.231,578.714 783.318,588.167 785.405,598.631 \n",
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" 850.103,630.719 852.19,636.324 854.277,628.465 856.364,631.361 858.451,634.755 860.538,632.228 862.625,635.195 864.712,644.737 866.799,650.779 868.886,650.667 \n",
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" 1017.06,866.7 1019.15,875.889 1021.24,871.441 1023.33,879.128 1025.41,881.195 1027.5,874.589 1029.59,873.028 1031.67,877.997 1033.76,874.423 1035.85,877.631 \n",
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" 1058.8,906.017 1060.89,911.668 1062.98,921.818 1065.07,924.033 1067.15,925.519 1069.24,932.945 1071.33,941.929 1073.41,937.512 1075.5,939.774 1077.59,940.676 \n",
" 1079.67,942.449 1081.76,928.867 1083.85,921.035 1085.94,919.476 1088.02,926.567 1090.11,918.803 1092.2,919.328 1094.28,921.865 1096.37,926.275 1098.46,919.108 \n",
" 1100.55,920.168 1102.63,920.631 1104.72,924.355 1106.81,932.211 1108.89,942.188 1110.98,944.724 1113.07,944.149 1115.15,942.877 1117.24,946.43 1119.33,944.369 \n",
" 1121.42,942.372 1123.5,947.294 1125.59,958.398 1127.68,963.931 1129.76,965.086 1131.85,959.071 1133.94,961.802 1136.02,956.734 1138.11,959.866 1140.2,967.604 \n",
" 1142.29,961.771 1144.37,956.845 1146.46,964.546 1148.55,972.661 1150.63,975.572 1152.72,971.639 1154.81,980.103 1156.89,985.589 1158.98,981.793 1161.07,984.587 \n",
" 1163.16,987.82 1165.24,983.239 1167.33,986.512 1169.42,981.616 1171.5,985.987 1173.59,982.689 1175.68,982.228 1177.76,983.391 1179.85,989.421 1181.94,996.23 \n",
" 1184.03,994.694 1186.11,993.198 1188.2,994.088 1190.29,997.676 1192.37,1002.47 1194.46,1003.47 1196.55,1000.27 1198.63,1003.12 1200.72,1009.85 1202.81,1011.44 \n",
" 1204.9,1010.63 1206.98,1007.35 1209.07,1004.42 1211.16,1008.37 1213.24,1011.75 1215.33,1017.86 1217.42,1016.11 1219.51,1024.26 1221.59,1025.92 1223.68,1029.4 \n",
" 1225.77,1035.53 1227.85,1030.6 1229.94,1033.34 1232.03,1025.77 1234.11,1023.61 1236.2,1020.14 1238.29,1018.49 1240.38,1019.38 1242.46,1015.81 1244.55,1018.09 \n",
" 1246.64,1015.27 1248.72,1012.64 1250.81,1017.61 1252.9,1017.2 1254.98,1018.4 1257.07,1026.3 1259.16,1028.75 1261.25,1031.04 1263.33,1033.83 1265.42,1040.06 \n",
" 1267.51,1040.4 1269.59,1044.92 1271.68,1049.1 1273.77,1049.95 1275.85,1050.43 1277.94,1049.43 1280.03,1049.38 1282.12,1049.6 1284.2,1048.29 1286.29,1050.23 \n",
" 1288.38,1048.6 1290.46,1053.58 1292.55,1053.06 1294.64,1059.35 1296.72,1060.18 1298.81,1057.52 1300.9,1052.64 1302.99,1053.73 1305.07,1062.29 1307.16,1069.41 \n",
" 1309.25,1067.92 1311.33,1064.85 1313.42,1066.29 1315.51,1062.03 1317.59,1060.14 1319.68,1055.19 1321.77,1055.09 1323.86,1052.31 1325.94,1050.4 1328.03,1057 \n",
" 1330.12,1063.41 1332.2,1061.59 1334.29,1065.49 1336.38,1069.52 1338.47,1070.73 1340.55,1068.67 1342.64,1068.94 1344.73,1069.43 1346.81,1064.4 1348.9,1070.02 \n",
" 1350.99,1069.6 1353.07,1064.8 1355.16,1062.27 1357.25,1070.09 1359.34,1068.32 1361.42,1075.53 1363.51,1073.27 1365.6,1078.9 1367.68,1084.54 1369.77,1082.44 \n",
" 1371.86,1086.49 1373.94,1091.77 1376.03,1091.83 1378.12,1095.99 1380.21,1094.32 1382.29,1094.09 1384.38,1094.21 1386.47,1094.14 1388.55,1095.29 1390.64,1091.54 \n",
" 1392.73,1085.18 1394.81,1088.24 1396.9,1088.34 1398.99,1088.82 1401.08,1092.15 1403.16,1095.56 1405.25,1098.73 1407.34,1102.24 1409.42,1103.46 1411.51,1106.37 \n",
" 1413.6,1109.08 1415.68,1112.56 1417.77,1108.68 1419.86,1108.32 1421.95,1107.09 1424.03,1111.46 1426.12,1109.23 1428.21,1106.66 1430.29,1102.45 1432.38,1106.74 \n",
" 1434.47,1102.24 1436.55,1103.16 1438.64,1101.96 1440.73,1100.69 1442.82,1102.66 1444.9,1106.93 1446.99,1109.44 1449.08,1107.09 1451.16,1109.11 1453.25,1108.4 \n",
" 1455.34,1106.73 1457.43,1110.82 1459.51,1110.08 1461.6,1111.31 1463.69,1107.87 1465.77,1109.37 1467.86,1107.48 1469.95,1105.37 1472.03,1110.84 1474.12,1105.75 \n",
" 1476.21,1105.63 1478.3,1103.74 1480.38,1105.17 1482.47,1106.55 1484.56,1108.53 1486.64,1103.71 1488.73,1104.87 1490.82,1104.42 1492.9,1101.82 1494.99,1103.89 \n",
" 1497.08,1103.36 1499.17,1099.11 1501.25,1096.16 1503.34,1096.27 1505.43,1094.99 1507.51,1098.18 1509.6,1095.46 1511.69,1098.11 1513.77,1101.29 1515.86,1101.92 \n",
" 1517.95,1104.63 1520.04,1106.75 1522.12,1108.52 1524.21,1104.8 1526.3,1102.67 1528.38,1107.44 1530.47,1109.1 1532.56,1113.45 1534.64,1112.21 1536.73,1111.1 \n",
" 1538.82,1110.2 1540.91,1115.31 1542.99,1118.76 1545.08,1117.39 1547.17,1121.49 1549.25,1128.18 1551.34,1127.04 1553.43,1127.83 1555.52,1128.63 1557.6,1126.65 \n",
" 1559.69,1128.44 1561.78,1133.57 1563.86,1135.51 1565.95,1139.92 1568.04,1142.37 1570.12,1145.44 1572.21,1145.5 1574.3,1148.59 1576.39,1146.89 1578.47,1148.43 \n",
" 1580.56,1148.52 1582.65,1153.04 1584.73,1156.3 1586.82,1155.14 1588.91,1154.87 1590.99,1153.49 1593.08,1154.54 1595.17,1159.11 1597.26,1157.93 1599.34,1160.74 \n",
" 1601.43,1164.05 1603.52,1171.11 1605.6,1171.38 1607.69,1173.16 1609.78,1169.3 1611.86,1166.78 1613.95,1163.59 1616.04,1163.83 1618.13,1165.65 1620.21,1161.5 \n",
" 1622.3,1161.17 1624.39,1161.97 1626.47,1163.11 1628.56,1160.04 1630.65,1159.48 1632.73,1158.8 1634.82,1163.14 1636.91,1160.95 1639,1163.51 1641.08,1163.53 \n",
" 1643.17,1167.2 1645.26,1163.68 1647.34,1166.34 1649.43,1164.69 1651.52,1159.44 1653.6,1157.03 1655.69,1156.88 1657.78,1160.37 1659.87,1162.01 1661.95,1161.02 \n",
" 1664.04,1160.57 1666.13,1164.58 1668.21,1164.43 1670.3,1166.61 1672.39,1169.37 1674.48,1169.45 1676.56,1166.44 1678.65,1169.05 1680.74,1170.47 1682.82,1174.29 \n",
" 1684.91,1177.92 1687,1178.14 1689.08,1176.89 1691.17,1177.34 1693.26,1180.57 1695.35,1180.78 1697.43,1183.36 1699.52,1181.41 1701.61,1181.11 1703.69,1180.09 \n",
" 1705.78,1177.87 1707.87,1178.73 1709.95,1178.92 1712.04,1184.13 1714.13,1185.65 1716.22,1183.15 1718.3,1183.68 1720.39,1188.32 1722.48,1190.19 1724.56,1196.05 \n",
" 1726.65,1199.34 1728.74,1201.77 1730.82,1204.63 1732.91,1206.95 1735,1210.25 1737.09,1210.02 1739.17,1212.77 1741.26,1211.71 1743.35,1212.14 1745.43,1210.11 \n",
" 1747.52,1211.5 1749.61,1212.97 1751.69,1212.82 1753.78,1213.33 1755.87,1215.23 1757.96,1211.87 1760.04,1210.47 1762.13,1213.74 1764.22,1215.57 1766.3,1216.08 \n",
" 1768.39,1216.44 1770.48,1217.98 1772.56,1218.18 1774.65,1218.08 1776.74,1217.47 1778.83,1216.65 1780.91,1219.4 1783,1219.82 1785.09,1221.04 1787.17,1221.62 \n",
" 1789.26,1224.64 1791.35,1227.13 1793.44,1228.95 1795.52,1228.35 1797.61,1226.64 1799.7,1229.97 1801.78,1231.3 1803.87,1229.6 1805.96,1229.96 1808.04,1230.92 \n",
" 1810.13,1232.15 1812.22,1234.23 1814.31,1232.89 1816.39,1233.43 1818.48,1231.74 1820.57,1233.58 1822.65,1234.62 1824.74,1234.07 1826.83,1232.16 1828.91,1233.81 \n",
" 1831,1232.73 1833.09,1235.57 1835.18,1236.42 1837.26,1235.42 1839.35,1234.27 1841.44,1234.89 1843.52,1233.1 1845.61,1232.72 1847.7,1236.52 1849.78,1237.08 \n",
" 1851.87,1238.01 1853.96,1236.97 1856.05,1234.49 1858.13,1236.2 1860.22,1235.07 1862.31,1232.34 1864.39,1234.53 1866.48,1237.18 1868.57,1235.82 1870.65,1235.68 \n",
" 1872.74,1236.11 1874.83,1237.86 1876.92,1238.37 1879,1240.62 1881.09,1243.52 1883.18,1244.67 1885.26,1244.3 1887.35,1243.22 1889.44,1243.88 1891.52,1243.8 \n",
" 1893.61,1243.24 1895.7,1243.91 1897.79,1247.53 1899.87,1247.85 1901.96,1248.47 1904.05,1249.07 1906.13,1248.98 1908.22,1247.81 1910.31,1249.81 1912.4,1246.95 \n",
" 1914.48,1246.97 1916.57,1249.89 1918.66,1249.74 1920.74,1249.44 1922.83,1248.63 1924.92,1248.09 1927,1246.49 1929.09,1246.69 1931.18,1248.12 1933.27,1248.43 \n",
" 1935.35,1251.2 1937.44,1249 1939.53,1248.95 1941.61,1252.59 1943.7,1254.51 1945.79,1254.74 1947.87,1255.99 1949.96,1255.31 1952.05,1253.46 1954.14,1252.96 \n",
" 1956.22,1256.63 1958.31,1255.39 1960.4,1258.09 1962.48,1257.89 1964.57,1256.04 1966.66,1256.8 1968.74,1255.25 1970.83,1255.67 1972.92,1257.05 1975.01,1256.25 \n",
" 1977.09,1257.06 1979.18,1255.71 1981.27,1257.82 1983.35,1259.94 1985.44,1261.72 1987.53,1261.88 1989.61,1264.11 1991.7,1264.59 1993.79,1262.81 1995.88,1264.84 \n",
" 1997.96,1267.65 2000.05,1265.98 2002.14,1266.51 2004.22,1265.37 2006.31,1269.01 2008.4,1268.56 2010.48,1267.19 2012.57,1266.03 2014.66,1264.15 2016.75,1261.5 \n",
" 2018.83,1260.45 2020.92,1261.37 2023.01,1259.81 2025.09,1260.11 2027.18,1257.22 2029.27,1254.89 2031.36,1254.11 2033.44,1255.58 2035.53,1257.11 2037.62,1257.13 \n",
" 2039.7,1254.64 2041.79,1252.95 2043.88,1254.07 2045.96,1256.33 2048.05,1258.28 2050.14,1259.03 2052.23,1263.41 2054.31,1265.98 2056.4,1265.44 2058.49,1267.25 \n",
" 2060.57,1266.9 2062.66,1265.27 2064.75,1263.63 2066.83,1262.47 2068.92,1262.2 2071.01,1264.65 2073.1,1265.69 2075.18,1268.88 2077.27,1270.27 2079.36,1268.85 \n",
" 2081.44,1270.17 2083.53,1271.36 2085.62,1269.44 2087.7,1268.62 2089.79,1268.56 2091.88,1267.52 2093.97,1266.71 2096.05,1269.55 2098.14,1268.42 2100.23,1270.12 \n",
" 2102.31,1269.66 2104.4,1271.21 2106.49,1270.52 2108.57,1269.61 2110.66,1268.28 2112.75,1269.18 2114.84,1267.13 2116.92,1265.02 2119.01,1263.78 2121.1,1264.69 \n",
" 2123.18,1264.33 2125.27,1265.05 2127.36,1265.91 2129.44,1266.86 2131.53,1267.79 2133.62,1265.4 2135.71,1263.76 2137.79,1261.58 2139.88,1263.65 2141.97,1262.02 \n",
" 2144.05,1261.92 2146.14,1259.55 2148.23,1262.26 2150.32,1262.97 2152.4,1263.23 2154.49,1264.14 2156.58,1263.03 2158.66,1263.08 2160.75,1262.58 2162.84,1262.56 \n",
" 2164.92,1261.44 2167.01,1261.58 2169.1,1263.14 2171.19,1263.4 2173.27,1261.39 2175.36,1260.45 2177.45,1257.06 2179.53,1258.21 2181.62,1257.07 2183.71,1257.26 \n",
" 2185.79,1257.87 2187.88,1256.67 2189.97,1256.13 2192.06,1257.87 2194.14,1255.22 2196.23,1254.78 2198.32,1257.04 2200.4,1258.19 2202.49,1256.48 2204.58,1255.24 \n",
" 2206.66,1254.86 2208.75,1255.12 2210.84,1256.18 2212.93,1258.24 2215.01,1256.78 2217.1,1254.29 2219.19,1254.79 2221.27,1258.18 2223.36,1257.32 2225.45,1259.11 \n",
" 2227.53,1260.18 2229.62,1260.16 2231.71,1260.93 2233.8,1261.49 2235.88,1261.3 2237.97,1263.84 2240.06,1265.42 2242.14,1266.65 2244.23,1267.06 2246.32,1267.88 \n",
" 2248.41,1268.64 2250.49,1269.91 2252.58,1270.36 2254.67,1271.56 2256.75,1271.1 2258.84,1270.5 2260.93,1269.6 2263.01,1269.24 2265.1,1270.54 2267.19,1270.19 \n",
" 2269.28,1270.39 2271.36,1271.46 2273.45,1271.46 2275.54,1271.45 2277.62,1271.08 2279.71,1269.33 2281.8,1271.48 2283.88,1271.84 2285.97,1272.08 2288.06,1272.07 \n",
" 2290.15,1274.28 \n",
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" 1204.9,423.156 1206.98,426.408 1209.07,429.327 1211.16,425.377 1213.24,422.002 1215.33,415.88 1217.42,417.613 1219.51,409.443 1221.59,407.773 1223.68,404.281 \n",
" 1225.77,398.14 1227.85,403.066 1229.94,400.307 1232.03,407.889 1234.11,410.054 1236.2,413.542 1238.29,415.175 1240.38,414.318 1242.46,417.88 1244.55,415.587 \n",
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" 1288.38,385.019 1290.46,380.028 1292.55,380.554 1294.64,374.265 1296.72,373.432 1298.81,376.093 1300.9,380.965 1302.99,379.885 1305.07,371.321 1307.16,364.182 \n",
" 1309.25,365.674 1311.33,368.738 1313.42,367.298 1315.51,371.558 1317.59,373.454 1319.68,378.392 1321.77,378.502 1323.86,381.265 1325.94,383.181 1328.03,376.58 \n",
" 1330.12,370.177 1332.2,372.006 1334.29,368.109 1336.38,364.08 1338.47,362.869 1340.55,364.925 1342.64,364.649 1344.73,364.163 1346.81,369.183 1348.9,363.565 \n",
" 1350.99,363.986 1353.07,368.772 1355.16,371.304 1357.25,363.481 1359.34,365.264 1361.42,358.043 1363.51,360.304 1365.6,354.665 1367.68,349.032 1369.77,351.123 \n",
" 1371.86,347.073 1373.94,341.79 1376.03,341.727 1378.12,337.567 1380.21,339.23 1382.29,339.465 1384.38,339.346 1386.47,339.416 1388.55,338.267 1390.64,342.019 \n",
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},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(X.tt, Bridge.mat(X.yy)', label=[L\"X_1\",L\"X_2\",L\"X_3\"])\n",
"xlabel!(\"t\")"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.000093 seconds (10 allocations: 464 bytes)\n"
]
}
],
"source": [
"@time solve!(Bridge.EulerMaruyama(), X, u₀, W, P);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"*Quite fast. Try to beat that or ask Chris to appreciate. ;-) *"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Parameter inference\n",
"\n",
"Assume that we observe the path shown in the picture and want to now the intensities $\\alpha$ and $\\beta$ of the transitions and also the noise parameters $\\sigma_1$, $\\sigma_2$. In the Bayesian paradigm, we assume that there are Gaussian priors on $\\alpha$ and $\\beta$.\n",
"\n",
"*A nice feature of the model is that $\\alpha$ and $\\beta$ enter as multiplicative constants into the drift $b$. Therefore the posterior of the drift parameters given $\\sigma_1$ and $\\sigma_2$ is conjugate Gaussian.*\n",
"\n",
"The following function samples from the posterior distribution of these parameters. It is written to be generally useful. If your drift is say\n",
"\n",
"```\n",
"b(t, u, ::Example) = m*u + b\n",
"```\n",
"then define\n",
"```\n",
"paramgrad(t, u, ::Example) = u\n",
"paramintercept(t, u, ::Example) = b\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"conjugate_posterior"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"paramgrad(t, u, ::CIR) = @SMatrix [-u[1]*u[2] 0.; u[1]*u[2] -u[2]; 0. u[2]]\n",
"paramintercept(t, u, ::CIR) = @SVector [0.0, 0.0, 0.0]\n",
"\"\"\"\n",
" conjugate_posterior(Y, Ξ, P)\n",
"\n",
"Sample the posterior distribution of the conjugate drift parameters from path `Y`,\n",
"prior precision matrix `Ξ` under model `P` with non-conjugate parameters in `P` fixed.\n",
"\"\"\"\n",
"function conjugate_posterior(Y, Ξ, P)\n",
" t, y = Y.tt[1], Y.yy[1]\n",
" ϕ = paramgrad(t, y, P)\n",
" mu = zero(ϕ[1, :])\n",
" G = zero(mu*mu')\n",
"\n",
" for i in 1:length(Y)-1\n",
" ϕ = paramgrad(t, y, P)\n",
" Gϕ = SMatrix{3,2,Float64}(pinv(Matrix(a(t, y, P)))*ϕ) # a is sigma*sigma'. Todo: smoothing like this is very slow\n",
" zi = ϕ'*Gϕ\n",
" t2, y2 = Y.tt[i + 1], Y.yy[i + 1]\n",
" dy = y2 - y \n",
" ds = t2 - t\n",
" mu = mu + Gϕ'*(dy - paramintercept(t, y, P)*ds)\n",
" t, y = t2, y2\n",
" G = G + zi*ds\n",
" end\n",
" WW = G + Ξ\n",
"\n",
" WL = Bridge.cholupper(Hermitian(WW))'\n",
" th° = WL'\\(randn(typeof(mu))+WL\\mu)\n",
"end "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us visualise that."
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
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"source": [
"Π = [conjugate_posterior(X, 0.1*I, P) for i in 1:1000]\n",
"scatter(first.(Π), last.(Π), color=\"blue\", markersize = 0.3, label=\"posterior samples\")\n",
"scatter!([Ptrue.α], [Ptrue.β], color=\"red\", label=\"truth\")\n",
"ylabel!(LaTeXString(\"\\\\beta\"))\n",
"xlabel!(LaTeXString(\"\\\\alpha\"))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Estimating $\\sigma_1$, $\\sigma_2$\n",
"\n",
"It remains to the other two parameters. Let's be pragmatic and just derive estimation equations assuming the drift is known.\n",
"\n",
"\n",
"From equation $(\\star)$ and \n",
"\n",
"$$\n",
"\\sigma(t, u) = \\begin{bmatrix}\n",
"\\sigma_1 u_1 u_2 & 0\\\\\n",
"-\\sigma_1u_1u_2& -\\sigma_2 u_2\\\\\n",
"0& \\sigma_2 u_2\n",
"\\end{bmatrix}\n",
"$$\n",
"\n",
"we find\n",
"\n",
"$$\n",
"\\frac{M^1_t}{X_t^1 X_t^2 \\sqrt{\\Delta t}} \\sim N(0, \\sigma_1^2)\n",
"\\quad\\frac{M^3_t}{ X_t^2 \\sqrt{\\Delta t}} \\sim N(0, \\sigma_2^2)\n",
"$$\n",
"\n",
"where $X_t^1$, $X_t^2$, etc. denote the first, second, etc. component of $X$ and\n",
"\n",
"$$\n",
" M_t = X_{t+\\Delta} - X_t - b(t, X_t) \\Delta t\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Ptrue.σ1 = 0.07\n",
"σ1est = 0.0691475811777478\n"
]
}
],
"source": [
"second(x) = getindex(x, 2)\n",
"M = [X.yy[i]-X.yy[i-1]-b(X.tt[i], X.yy[i], P)*(X.tt[i]-X.tt[i-1]) for i in 2:length(X)]\n",
"\n",
"σ1est = std(first.(M)./sqrt.(diff(X.tt))./first.(X.yy[1:end-1])./second.(X.yy[1:end-1]))\n",
"@show Ptrue.σ1\n",
"@show σ1est;\n"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"name": "stdout",
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"text": [
"Ptrue.σ2 = 0.4\n",
"σ2est = 0.40025568517832083\n"
]
}
],
"source": [
"σ2est = std(last.(M)./sqrt.(diff(X.tt))./second.(X.yy[1:end-1]))\n",
"@show Ptrue.σ2\n",
"@show σ2est;\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# MCMC for the parameters\n",
"\n",
"Note that to estimate $\\sigma_1$, $\\sigma_2$ one needs to know $\\alpha$ and $\\beta$. We can estimate all parameters with the help of a Markov chain Monte Carlo procedure, iterating the two estimation steps.\n",
"\n",
"One of the nice features of Julia is that direct implementations of MCMC algorithm are performant and quite easy to deal with.\n",
"\n",
"Because we can sample the drift parameters from the posterior, but only derived estimation equations for $\\sigma_1$, $\\sigma_2$ this is a sampler with one Gibbs step and one EM-type step. That means for example that Bayesian uncertainty quantification is lost. Especially the spread of the MCMC sample distribution of the parameters $\\sigma_1$ and $\\sigma_2$ parameters below is much smaller than the error, see the pictures below. This is a classical problem with SDE models."
]
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"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"iterations = 1000\n",
"P = CIR(2.1, 5.0, 0.17, 0.1)\n",
"params = zeros(4, iterations)\n",
"M = zeros(SVector{3,Float64}, length(tt)-1,)\n",
"for iter in 1:iterations\n",
" \n",
" # step 1: sample conjugate parameter α and β conditional on σ1, σ2\n",
" \n",
" α, β = conjugate_posterior(X, 0.1*I, P)\n",
" \n",
" P = CIR(α, β, P.σ1, P.σ2)\n",
" \n",
" # step 2: estimate σ1, σ2\n",
"\n",
" for i in 2:length(X)\n",
" M[i-1] = (X.yy[i]-X.yy[i-1]-b(X.tt[i], X.yy[i], P)*(X.tt[i]-X.tt[i-1])) \n",
" end\n",
"\n",
" σ1 = std(first.(M)[1:end]./sqrt.(diff(tt))./first.(X.yy[1:end-1])./second.(X.yy[1:end-1]))\n",
" σ2 = std(last.(M)./sqrt.(diff(X.tt))./second.(X.yy[1:end-1]))\n",
" \n",
" P = CIR(P.α, P.β, σ1, σ2)\n",
" \n",
" # record sample\n",
" \n",
" params[:, iter] = [α, β, σ1, σ2]\n",
"end"
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{
"cell_type": "code",
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" 1467.06,242.831 1468,932.677 1468.95,824.124 1469.89,793.525 1470.83,780.351 1471.78,775.942 1472.72,1042.94 1473.66,543.887 1474.61,1023.78 1475.55,874.271 \n",
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" 1495.36,929.379 1496.3,906.646 1497.25,650.633 1498.19,733.293 1499.13,658.417 1500.08,670.163 1501.02,534.963 1501.96,804.188 1502.91,668.176 1503.85,1086.19 \n",
" 1504.79,1132.05 1505.73,732.495 1506.68,792.827 1507.62,1147.11 1508.56,612.901 1509.51,826.653 1510.45,583.029 1511.39,1008.45 1512.34,482.636 1513.28,604.161 \n",
" 1514.22,835.497 1515.17,685.257 1516.11,685.462 1517.05,898.76 1518,720.33 1518.94,879.324 1519.88,550.455 1520.83,1074.13 1521.77,455.949 1522.71,671.174 \n",
" 1523.66,707.154 1524.6,841.163 1525.54,1121.31 1526.49,1079.05 1527.43,902.356 1528.37,947.172 1529.32,404.376 1530.26,894.779 1531.2,999.268 1532.15,815.099 \n",
" 1533.09,715.818 1534.03,931.714 1534.98,241.221 1535.92,919.173 1536.86,985.451 1537.81,975.405 1538.75,601.343 1539.69,828.371 1540.64,742.578 1541.58,718.369 \n",
" 1542.52,735.54 1543.47,698.868 1544.41,1119.08 1545.35,632.764 1546.3,966.264 1547.24,633.742 1548.18,917.622 1549.12,1225.72 1550.07,646.718 1551.01,921.968 \n",
" 1551.95,717.196 1552.9,949.223 1553.84,848.096 1554.78,739.669 1555.73,829.572 1556.67,838.436 1557.61,612.635 1558.56,447.592 1559.5,826.032 1560.44,487.22 \n",
" 1561.39,401.301 1562.33,579.646 1563.27,879.888 1564.22,917.72 1565.16,734.808 1566.1,659.562 1567.05,717.804 1567.99,595.918 1568.93,704.165 1569.88,1210.03 \n",
" 1570.82,409.638 1571.76,719.672 1572.71,830.818 1573.65,579.946 1574.59,921.471 1575.54,779.522 1576.48,741.809 1577.42,1006.71 1578.37,744.471 1579.31,804.986 \n",
" 1580.25,813.583 1581.2,946.856 1582.14,944.759 1583.08,862.841 1584.03,901.599 1584.97,728.831 1585.91,590.835 1586.86,903.506 1587.8,519.188 1588.74,681.929 \n",
" 1589.69,791.611 1590.63,726.202 1591.57,760.677 1592.51,908.338 1593.46,691.573 1594.4,807.531 1595.34,379.634 1596.29,808.958 1597.23,886.702 1598.17,961.268 \n",
" 1599.12,689.624 1600.06,612.005 1601,571.152 1601.95,636.446 1602.89,437.513 1603.83,505.874 1604.78,372.92 1605.72,641.321 1606.66,807.577 1607.61,1157.02 \n",
" 1608.55,905.065 1609.49,785.372 1610.44,780.481 1611.38,998.696 1612.32,697.492 1613.27,948.824 1614.21,599.314 1615.15,483.907 1616.1,576.7 1617.04,924.298 \n",
" 1617.98,956.066 1618.93,422.293 1619.87,598.097 1620.81,541.761 1621.76,639.043 1622.7,525.055 1623.64,1068.51 1624.59,1446.71 1625.53,663.126 1626.47,736.893 \n",
" 1627.42,732.339 1628.36,1014.39 1629.3,1266.99 1630.25,943.785 1631.19,807.763 1632.13,1182.5 1633.08,696.071 1634.02,798.683 1634.96,1008.84 1635.91,669.544 \n",
" 1636.85,669.386 1637.79,1030.01 1638.73,706.038 1639.68,911.17 1640.62,789.246 1641.56,1067.36 1642.51,595.56 1643.45,460.363 1644.39,863.43 1645.34,1080.93 \n",
" 1646.28,1337.34 1647.22,591.689 1648.17,732.985 1649.11,687.894 1650.05,761.649 1651,799.099 1651.94,831.657 1652.88,840.924 1653.83,800.827 1654.77,842.406 \n",
" 1655.71,608.848 1656.66,727.435 1657.6,614.271 1658.54,697.373 1659.49,462.855 1660.43,546.582 1661.37,580.937 1662.32,609.73 1663.26,672.97 1664.2,807.955 \n",
" 1665.15,991.66 1666.09,557.988 1667.03,295.054 1667.98,975.482 1668.92,486.21 1669.86,1004.21 1670.81,523.454 1671.75,1029.76 1672.69,795.964 1673.64,877.912 \n",
" 1674.58,673.403 1675.52,718.368 1676.47,631.83 1677.41,388.291 1678.35,785.802 1679.3,812.983 1680.24,890.392 1681.18,812.508 1682.12,603.362 1683.07,561.157 \n",
" 1684.01,890.795 1684.95,651.452 1685.9,796.772 1686.84,668.799 1687.78,892.658 1688.73,949.737 1689.67,749.722 1690.61,606.665 1691.56,850.395 1692.5,575.731 \n",
" 1693.44,527.354 1694.39,820.593 1695.33,752.572 1696.27,856.538 1697.22,832.378 1698.16,613.379 1699.1,889.438 1700.05,717.451 1700.99,1032.41 1701.93,791.847 \n",
" 1702.88,832.891 1703.82,437.914 1704.76,434.183 1705.71,1175.72 1706.65,841.979 1707.59,366.183 1708.54,772.452 1709.48,995.336 1710.42,537.5 1711.37,678.135 \n",
" 1712.31,589.52 1713.25,914.478 1714.2,932.756 1715.14,832.613 1716.08,1110.8 1717.03,531.895 1717.97,709.716 1718.91,701.2 1719.86,951.267 1720.8,882.999 \n",
" 1721.74,420.039 1722.69,577.451 1723.63,816.01 1724.57,705.794 1725.51,669.763 1726.46,939.181 1727.4,723.206 1728.34,710.343 1729.29,480.269 1730.23,664.505 \n",
" 1731.17,1040.49 1732.12,536.348 1733.06,631.424 1734,846.356 1734.95,727.702 1735.89,342.947 1736.83,601.836 1737.78,923.3 1738.72,652.698 1739.66,644.273 \n",
" 1740.61,620.2 1741.55,1078.29 1742.49,770.679 1743.44,631.93 1744.38,720.974 1745.32,626.757 1746.27,400.688 1747.21,964.123 1748.15,653.812 1749.1,836.501 \n",
" 1750.04,763.217 1750.98,563.854 1751.93,795.931 1752.87,564.111 1753.81,444.005 1754.76,1017.54 1755.7,644.97 1756.64,786.689 1757.59,716.194 1758.53,684.175 \n",
" 1759.47,809.738 1760.42,678.416 1761.36,532.269 1762.3,740.643 1763.25,236.384 1764.19,621.904 1765.13,510.851 1766.08,773.99 1767.02,840.696 1767.96,464.608 \n",
" 1768.9,851.228 1769.85,563.446 1770.79,504.738 1771.73,583.138 1772.68,832.04 1773.62,593.613 1774.56,684.286 1775.51,762.173 1776.45,1169.49 1777.39,716.438 \n",
" 1778.34,686.666 1779.28,857.502 1780.22,670.577 1781.17,663.539 1782.11,814.063 1783.05,548.027 1784,883.509 1784.94,593.68 1785.88,479.103 1786.83,806.931 \n",
" 1787.77,990.145 1788.71,656.763 1789.66,922.925 1790.6,634.393 1791.54,391.651 1792.49,748.021 1793.43,870.181 1794.37,699.195 1795.32,691.121 1796.26,471.718 \n",
" 1797.2,621.866 1798.15,868.801 1799.09,933.834 1800.03,885.72 1800.98,421.574 1801.92,487.855 1802.86,1038.69 1803.81,964.166 1804.75,702.873 1805.69,1014.23 \n",
" 1806.64,572.453 1807.58,575.404 1808.52,934.852 1809.47,548.883 1810.41,588.163 1811.35,685.726 1812.3,815.015 1813.24,477.457 1814.18,506.877 1815.12,784.989 \n",
" 1816.07,950.757 1817.01,245.702 1817.95,1097.08 1818.9,1207.87 1819.84,878.951 1820.78,729.219 1821.73,1227.66 1822.67,792.219 1823.61,639.931 1824.56,943.395 \n",
" 1825.5,977.37 1826.44,608.386 1827.39,891.166 1828.33,665.444 1829.27,644.146 1830.22,703.91 1831.16,1074.75 1832.1,825.157 1833.05,1081.58 1833.99,845.476 \n",
" 1834.93,1284.7 1835.88,1067.94 1836.82,635.989 1837.76,561.55 1838.71,955.148 1839.65,865.398 1840.59,812.312 1841.54,659.941 1842.48,975.221 1843.42,1064.48 \n",
" 1844.37,799.437 1845.31,1021.44 1846.25,491.89 1847.2,536.358 1848.14,884.573 1849.08,980.904 1850.03,954.278 1850.97,831.407 1851.91,781.062 1852.86,758.561 \n",
" 1853.8,625.302 1854.74,887.341 1855.69,1092.96 1856.63,992.087 1857.57,855.848 1858.51,1019.05 1859.46,906.482 1860.4,737.88 1861.34,568.193 1862.29,620.332 \n",
" 1863.23,569.387 1864.17,601.394 1865.12,829.322 1866.06,668.738 1867,985.861 1867.95,799.821 1868.89,802.904 1869.83,569.137 1870.78,418.895 1871.72,932.827 \n",
" 1872.66,708.607 1873.61,613.908 1874.55,745.842 1875.49,967.956 1876.44,591.407 1877.38,723.994 1878.32,1041.83 1879.27,826.515 1880.21,578.973 1881.15,539.252 \n",
" 1882.1,565.902 1883.04,648.661 1883.98,699.687 1884.93,1076.79 1885.87,715.632 1886.81,491.862 1887.76,703.076 1888.7,809.24 1889.64,732.294 1890.59,601.059 \n",
" 1891.53,575.86 1892.47,953.767 1893.42,755.881 1894.36,753.144 1895.3,560.383 1896.25,887.918 1897.19,982.224 1898.13,982.163 1899.08,874.937 1900.02,554.401 \n",
" 1900.96,880.802 1901.9,1257.17 1902.85,836.398 1903.79,1051.02 1904.73,802.761 1905.68,727.563 1906.62,755.046 1907.56,1177.58 1908.51,738.667 1909.45,595.197 \n",
" 1910.39,592.595 1911.34,541.561 1912.28,862.228 1913.22,866.087 1914.17,788.143 1915.11,858.65 1916.05,471.921 1917,906.451 1917.94,1078.39 1918.88,1206.62 \n",
" 1919.83,337.289 1920.77,1189.6 1921.71,712.656 1922.66,990.197 1923.6,459.863 1924.54,406.55 1925.49,703.76 1926.43,667.658 1927.37,820.383 1928.32,792.24 \n",
" 1929.26,672.211 1930.2,573.618 1931.15,1063.14 1932.09,981.807 1933.03,1072.33 1933.98,649.655 1934.92,822.887 1935.86,908.495 1936.81,877.678 1937.75,882.22 \n",
" 1938.69,805.711 1939.64,990.18 1940.58,1152.63 1941.52,867.133 1942.47,538.913 1943.41,888.312 1944.35,1056.36 1945.29,424.175 1946.24,927.827 1947.18,647.934 \n",
" 1948.12,882.101 1949.07,876.056 1950.01,673.69 1950.95,676.36 1951.9,930.171 1952.84,622.607 1953.78,822.848 1954.73,992.686 1955.67,820.865 1956.61,836.94 \n",
" 1957.56,741.854 1958.5,619.482 1959.44,506.184 1960.39,729.845 1961.33,878.417 1962.27,697.326 1963.22,632.926 1964.16,799.425 1965.1,892.834 1966.05,629.676 \n",
" 1966.99,689.142 1967.93,1155.81 1968.88,792.802 1969.82,717.291 1970.76,654.359 1971.71,697.044 1972.65,676.387 1973.59,787.909 1974.54,631.859 1975.48,1139.99 \n",
" 1976.42,401.791 1977.37,749.737 1978.31,849.7 1979.25,708.391 1980.2,565.351 1981.14,1189.74 1982.08,1010.93 1983.03,559.531 1983.97,534.347 1984.91,720.022 \n",
" 1985.86,819.274 1986.8,848.586 1987.74,784.262 1988.69,718.823 1989.63,848.559 1990.57,1039.95 1991.51,734.966 1992.46,913.062 1993.4,513.339 1994.34,203.986 \n",
" 1995.29,768.003 1996.23,1218.04 1997.17,615.365 1998.12,535.3 1999.06,646.481 2000,828.98 2000.95,1081.84 2001.89,807.009 2002.83,920.931 2003.78,747.327 \n",
" 2004.72,717.701 2005.66,637.64 2006.61,876.117 2007.55,726.516 2008.49,1326.18 2009.44,586.609 2010.38,576.73 2011.32,590.408 2012.27,826.31 2013.21,820.027 \n",
" 2014.15,563.586 2015.1,948.008 2016.04,775.263 2016.98,664.071 2017.93,928.517 2018.87,732.356 2019.81,498.966 2020.76,490.365 2021.7,88.0053 2022.64,714.568 \n",
" 2023.59,588.882 2024.53,1003.39 2025.47,827.641 2026.42,593.748 2027.36,983.971 2028.3,883.991 2029.25,523.273 2030.19,605.871 2031.13,860.22 2032.08,914.839 \n",
" 2033.02,278.979 2033.96,933.295 2034.9,1021.03 2035.85,449.067 2036.79,477.295 2037.73,329.694 2038.68,627.082 2039.62,1191.95 2040.56,857.311 2041.51,629.315 \n",
" 2042.45,527.49 2043.39,830.926 2044.34,1204.36 2045.28,642.634 2046.22,527.342 2047.17,789.217 2048.11,558.825 2049.05,196.326 2050,708.459 2050.94,946.886 \n",
" 2051.88,943.895 2052.83,881.858 2053.77,1007.5 2054.71,919.607 2055.66,499.057 2056.6,582.399 2057.54,796.181 2058.49,727.663 2059.43,431.73 2060.37,818.192 \n",
" 2061.32,1088.2 2062.26,486.974 2063.2,713.7 2064.15,740.018 2065.09,761.259 2066.03,331.187 2066.98,846.31 2067.92,437.482 2068.86,693.812 2069.81,893.988 \n",
" 2070.75,754.094 2071.69,328.552 2072.64,689.467 2073.58,295.735 2074.52,298.838 2075.47,919.52 2076.41,546.712 2077.35,971.32 2078.29,827.858 2079.24,763.101 \n",
" 2080.18,1221.84 2081.12,633.923 2082.07,818.84 2083.01,821.156 2083.95,1270.85 2084.9,987.097 2085.84,536.079 2086.78,882.266 2087.73,1165.69 2088.67,1089.75 \n",
" 2089.61,1153.98 2090.56,925.446 2091.5,240.005 2092.44,465.697 2093.39,1220.73 2094.33,806.413 2095.27,540.728 2096.22,809.932 2097.16,516.953 2098.1,623.549 \n",
" 2099.05,623.955 2099.99,838.181 2100.93,768.793 2101.88,465.236 2102.82,792.571 2103.76,644.462 2104.71,487.398 2105.65,1079.47 2106.59,700.188 2107.54,1115.05 \n",
" 2108.48,846.394 2109.42,993.151 2110.37,814.166 2111.31,976.196 2112.25,620.409 2113.2,916.346 2114.14,948.446 2115.08,731.958 2116.03,742.828 2116.97,958.428 \n",
" 2117.91,332.446 2118.86,1080.5 2119.8,471.113 2120.74,726.605 2121.68,610.274 2122.63,529.99 2123.57,929.895 2124.51,523.827 2125.46,969.831 2126.4,871.749 \n",
" 2127.34,777.236 2128.29,863.586 2129.23,542.565 2130.17,642.615 2131.12,1299.32 2132.06,897.674 2133,691.562 2133.95,987.865 2134.89,657.271 2135.83,746.784 \n",
" 2136.78,774.491 2137.72,992.381 2138.66,940.924 2139.61,838.593 2140.55,695.955 2141.49,655.105 2142.44,259.475 2143.38,721.674 2144.32,743.599 2145.27,504.988 \n",
" 2146.21,835.55 2147.15,753.925 2148.1,641.924 2149.04,794.338 2149.98,859.085 2150.93,940.8 2151.87,924.25 2152.81,672.08 2153.76,749.324 2154.7,621.606 \n",
" 2155.64,1041.08 2156.59,629.489 2157.53,667.31 2158.47,839.367 2159.42,365.39 2160.36,540.859 2161.3,377.843 2162.25,990.168 2163.19,802.071 2164.13,809.866 \n",
" 2165.08,626.286 2166.02,1043.12 2166.96,788.794 2167.9,855.897 2168.85,1153.23 2169.79,1084.79 2170.73,814.33 2171.68,1007.33 2172.62,651.314 2173.56,768.402 \n",
" 2174.51,401.65 2175.45,818.01 2176.39,850.324 2177.34,762.021 2178.28,1140.18 2179.22,808.501 2180.17,689.855 2181.11,793.577 2182.05,725.936 2183,615.049 \n",
" 2183.94,901.46 2184.88,697.009 2185.83,876.615 2186.77,788.043 2187.71,502.225 2188.66,579.038 2189.6,1039.6 2190.54,348.327 2191.49,547.186 2192.43,976.528 \n",
" 2193.37,832.298 2194.32,1016.31 2195.26,938.77 2196.2,804.843 2197.15,932.754 2198.09,861.007 2199.03,758.387 2199.98,811.88 2200.92,877.446 2201.86,663.562 \n",
" 2202.81,686.08 2203.75,533.781 2204.69,742.525 2205.64,467.101 2206.58,813.99 2207.52,1255.87 2208.47,1054.46 2209.41,958.137 2210.35,419.724 2211.29,1032.62 \n",
" 2212.24,910.254 2213.18,895.532 2214.12,713.844 2215.07,904.462 2216.01,750.589 2216.95,886.858 2217.9,708.576 2218.84,969.753 2219.78,425.833 2220.73,627.48 \n",
" 2221.67,795.448 2222.61,1022.67 2223.56,597.795 2224.5,672.899 2225.44,705.203 2226.39,727.6 2227.33,1130.6 2228.27,1024.83 2229.22,1182.32 2230.16,511.006 \n",
" 2231.1,563.62 2232.05,1144.71 2232.99,858.495 2233.93,710.437 2234.88,510.395 2235.82,873.165 2236.76,931.996 2237.71,680.235 2238.65,537.573 2239.59,1018.61 \n",
" 2240.54,1005.49 2241.48,507.112 2242.42,632.35 2243.37,839.858 2244.31,878.142 2245.25,872.019 2246.2,897.752 2247.14,604.107 2248.08,1030.47 2249.03,677.7 \n",
" 2249.97,1016.17 2250.91,851.383 2251.86,888.285 2252.8,948.244 2253.74,1019.11 2254.68,1092.17 2255.63,915.612 2256.57,612.18 2257.51,727.186 2258.46,825.038 \n",
" 2259.4,521.139 2260.34,766.961 2261.29,413.552 2262.23,976.995 2263.17,835.587 2264.12,659.64 2265.06,841.345 2266,782.561 2266.95,481.073 2267.89,588.5 \n",
" 2268.83,589.937 2269.78,621.859 2270.72,758.396 2271.66,590.928 2272.61,869.457 2273.55,1247.09 2274.49,711.969 2275.44,785.73 2276.38,912.286 2277.32,616.053 \n",
" 2278.27,770.93 2279.21,675.494 2280.15,644.214 2281.1,559.285 2282.04,563.13 2282.98,452.067 2283.93,1257.35 2284.87,924.163 2285.81,408.657 2286.76,822.758 \n",
" 2287.7,778.49 2288.64,785.549 2289.59,890.028 2290.53,870.434 2291.47,715.124 2292.42,905.431 2293.36,663.959 2294.3,652.657 2295.25,406.207 2296.19,806.734 \n",
" 2297.13,541.393 2298.08,1165.77 2299.02,675.867 2299.96,697.421 2300.9,746.366 2301.85,871.419 2302.79,406.854 2303.73,792.602 2304.68,862.469 2305.62,1089.4 \n",
" 2306.56,681.704 2307.51,724.286 2308.45,644.588 2309.39,499.439 2310.34,623.739 2311.28,755.345 2312.22,540.38 2313.17,862.697 2314.11,611.044 2315.05,668.324 \n",
" 2316,821.428 2316.94,558.08 2317.88,485.345 2318.83,495.023 2319.77,716.024 2320.71,1027.5 2321.66,837.647 2322.6,742.403 2323.54,941.431 2324.49,732.004 \n",
" \n",
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" 1382.17,614.501 1383.11,614.501 1384.05,614.501 1385,614.501 1385.94,614.501 1386.88,614.501 1387.83,614.501 1388.77,614.501 1389.71,614.501 1390.66,614.501 \n",
" 1391.6,614.501 1392.54,614.501 1393.49,614.501 1394.43,614.501 1395.37,614.501 1396.32,614.501 1397.26,614.501 1398.2,614.501 1399.15,614.501 1400.09,614.501 \n",
" 1401.03,614.501 1401.98,614.501 1402.92,614.501 1403.86,614.501 1404.81,614.501 1405.75,614.501 1406.69,614.501 1407.64,614.501 1408.58,614.501 1409.52,614.501 \n",
" 1410.47,614.501 1411.41,614.501 1412.35,614.501 1413.3,614.501 1414.24,614.501 1415.18,614.501 1416.12,614.501 1417.07,614.501 1418.01,614.501 1418.95,614.501 \n",
" 1419.9,614.501 1420.84,614.501 1421.78,614.501 1422.73,614.501 1423.67,614.501 1424.61,614.501 1425.56,614.501 1426.5,614.501 1427.44,614.501 1428.39,614.501 \n",
" 1429.33,614.501 1430.27,614.501 1431.22,614.501 1432.16,614.501 1433.1,614.501 1434.05,614.501 1434.99,614.501 1435.93,614.501 1436.88,614.501 1437.82,614.501 \n",
" 1438.76,614.501 1439.71,614.501 1440.65,614.501 1441.59,614.501 1442.54,614.501 1443.48,614.501 1444.42,614.501 1445.37,614.501 1446.31,614.501 1447.25,614.501 \n",
" 1448.2,614.501 1449.14,614.501 1450.08,614.501 1451.03,614.501 1451.97,614.501 1452.91,614.501 1453.86,614.501 1454.8,614.501 1455.74,614.501 1456.69,614.501 \n",
" 1457.63,614.501 1458.57,614.501 1459.52,614.501 1460.46,614.501 1461.4,614.501 1462.34,614.501 1463.29,614.501 1464.23,614.501 1465.17,614.501 1466.12,614.501 \n",
" 1467.06,614.501 1468,614.501 1468.95,614.501 1469.89,614.501 1470.83,614.501 1471.78,614.501 1472.72,614.501 1473.66,614.501 1474.61,614.501 1475.55,614.501 \n",
" 1476.49,614.501 1477.44,614.501 1478.38,614.501 1479.32,614.501 1480.27,614.501 1481.21,614.501 1482.15,614.501 1483.1,614.501 1484.04,614.501 1484.98,614.501 \n",
" 1485.93,614.501 1486.87,614.501 1487.81,614.501 1488.76,614.501 1489.7,614.501 1490.64,614.501 1491.59,614.501 1492.53,614.501 1493.47,614.501 1494.42,614.501 \n",
" 1495.36,614.501 1496.3,614.501 1497.25,614.501 1498.19,614.501 1499.13,614.501 1500.08,614.501 1501.02,614.501 1501.96,614.501 1502.91,614.501 1503.85,614.501 \n",
" 1504.79,614.501 1505.73,614.501 1506.68,614.501 1507.62,614.501 1508.56,614.501 1509.51,614.501 1510.45,614.501 1511.39,614.501 1512.34,614.501 1513.28,614.501 \n",
" 1514.22,614.501 1515.17,614.501 1516.11,614.501 1517.05,614.501 1518,614.501 1518.94,614.501 1519.88,614.501 1520.83,614.501 1521.77,614.501 1522.71,614.501 \n",
" 1523.66,614.501 1524.6,614.501 1525.54,614.501 1526.49,614.501 1527.43,614.501 1528.37,614.501 1529.32,614.501 1530.26,614.501 1531.2,614.501 1532.15,614.501 \n",
" 1533.09,614.501 1534.03,614.501 1534.98,614.501 1535.92,614.501 1536.86,614.501 1537.81,614.501 1538.75,614.501 1539.69,614.501 1540.64,614.501 1541.58,614.501 \n",
" 1542.52,614.501 1543.47,614.501 1544.41,614.501 1545.35,614.501 1546.3,614.501 1547.24,614.501 1548.18,614.501 1549.12,614.501 1550.07,614.501 1551.01,614.501 \n",
" 1551.95,614.501 1552.9,614.501 1553.84,614.501 1554.78,614.501 1555.73,614.501 1556.67,614.501 1557.61,614.501 1558.56,614.501 1559.5,614.501 1560.44,614.501 \n",
" 1561.39,614.501 1562.33,614.501 1563.27,614.501 1564.22,614.501 1565.16,614.501 1566.1,614.501 1567.05,614.501 1567.99,614.501 1568.93,614.501 1569.88,614.501 \n",
" 1570.82,614.501 1571.76,614.501 1572.71,614.501 1573.65,614.501 1574.59,614.501 1575.54,614.501 1576.48,614.501 1577.42,614.501 1578.37,614.501 1579.31,614.501 \n",
" 1580.25,614.501 1581.2,614.501 1582.14,614.501 1583.08,614.501 1584.03,614.501 1584.97,614.501 1585.91,614.501 1586.86,614.501 1587.8,614.501 1588.74,614.501 \n",
" 1589.69,614.501 1590.63,614.501 1591.57,614.501 1592.51,614.501 1593.46,614.501 1594.4,614.501 1595.34,614.501 1596.29,614.501 1597.23,614.501 1598.17,614.501 \n",
" 1599.12,614.501 1600.06,614.501 1601,614.501 1601.95,614.501 1602.89,614.501 1603.83,614.501 1604.78,614.501 1605.72,614.501 1606.66,614.501 1607.61,614.501 \n",
" 1608.55,614.501 1609.49,614.501 1610.44,614.501 1611.38,614.501 1612.32,614.501 1613.27,614.501 1614.21,614.501 1615.15,614.501 1616.1,614.501 1617.04,614.501 \n",
" 1617.98,614.501 1618.93,614.501 1619.87,614.501 1620.81,614.501 1621.76,614.501 1622.7,614.501 1623.64,614.501 1624.59,614.501 1625.53,614.501 1626.47,614.501 \n",
" 1627.42,614.501 1628.36,614.501 1629.3,614.501 1630.25,614.501 1631.19,614.501 1632.13,614.501 1633.08,614.501 1634.02,614.501 1634.96,614.501 1635.91,614.501 \n",
" 1636.85,614.501 1637.79,614.501 1638.73,614.501 1639.68,614.501 1640.62,614.501 1641.56,614.501 1642.51,614.501 1643.45,614.501 1644.39,614.501 1645.34,614.501 \n",
" 1646.28,614.501 1647.22,614.501 1648.17,614.501 1649.11,614.501 1650.05,614.501 1651,614.501 1651.94,614.501 1652.88,614.501 1653.83,614.501 1654.77,614.501 \n",
" 1655.71,614.501 1656.66,614.501 1657.6,614.501 1658.54,614.501 1659.49,614.501 1660.43,614.501 1661.37,614.501 1662.32,614.501 1663.26,614.501 1664.2,614.501 \n",
" 1665.15,614.501 1666.09,614.501 1667.03,614.501 1667.98,614.501 1668.92,614.501 1669.86,614.501 1670.81,614.501 1671.75,614.501 1672.69,614.501 1673.64,614.501 \n",
" 1674.58,614.501 1675.52,614.501 1676.47,614.501 1677.41,614.501 1678.35,614.501 1679.3,614.501 1680.24,614.501 1681.18,614.501 1682.12,614.501 1683.07,614.501 \n",
" 1684.01,614.501 1684.95,614.501 1685.9,614.501 1686.84,614.501 1687.78,614.501 1688.73,614.501 1689.67,614.501 1690.61,614.501 1691.56,614.501 1692.5,614.501 \n",
" 1693.44,614.501 1694.39,614.501 1695.33,614.501 1696.27,614.501 1697.22,614.501 1698.16,614.501 1699.1,614.501 1700.05,614.501 1700.99,614.501 1701.93,614.501 \n",
" 1702.88,614.501 1703.82,614.501 1704.76,614.501 1705.71,614.501 1706.65,614.501 1707.59,614.501 1708.54,614.501 1709.48,614.501 1710.42,614.501 1711.37,614.501 \n",
" 1712.31,614.501 1713.25,614.501 1714.2,614.501 1715.14,614.501 1716.08,614.501 1717.03,614.501 1717.97,614.501 1718.91,614.501 1719.86,614.501 1720.8,614.501 \n",
" 1721.74,614.501 1722.69,614.501 1723.63,614.501 1724.57,614.501 1725.51,614.501 1726.46,614.501 1727.4,614.501 1728.34,614.501 1729.29,614.501 1730.23,614.501 \n",
" 1731.17,614.501 1732.12,614.501 1733.06,614.501 1734,614.501 1734.95,614.501 1735.89,614.501 1736.83,614.501 1737.78,614.501 1738.72,614.501 1739.66,614.501 \n",
" 1740.61,614.501 1741.55,614.501 1742.49,614.501 1743.44,614.501 1744.38,614.501 1745.32,614.501 1746.27,614.501 1747.21,614.501 1748.15,614.501 1749.1,614.501 \n",
" 1750.04,614.501 1750.98,614.501 1751.93,614.501 1752.87,614.501 1753.81,614.501 1754.76,614.501 1755.7,614.501 1756.64,614.501 1757.59,614.501 1758.53,614.501 \n",
" 1759.47,614.501 1760.42,614.501 1761.36,614.501 1762.3,614.501 1763.25,614.501 1764.19,614.501 1765.13,614.501 1766.08,614.501 1767.02,614.501 1767.96,614.501 \n",
" 1768.9,614.501 1769.85,614.501 1770.79,614.501 1771.73,614.501 1772.68,614.501 1773.62,614.501 1774.56,614.501 1775.51,614.501 1776.45,614.501 1777.39,614.501 \n",
" 1778.34,614.501 1779.28,614.501 1780.22,614.501 1781.17,614.501 1782.11,614.501 1783.05,614.501 1784,614.501 1784.94,614.501 1785.88,614.501 1786.83,614.501 \n",
" 1787.77,614.501 1788.71,614.501 1789.66,614.501 1790.6,614.501 1791.54,614.501 1792.49,614.501 1793.43,614.501 1794.37,614.501 1795.32,614.501 1796.26,614.501 \n",
" 1797.2,614.501 1798.15,614.501 1799.09,614.501 1800.03,614.501 1800.98,614.501 1801.92,614.501 1802.86,614.501 1803.81,614.501 1804.75,614.501 1805.69,614.501 \n",
" 1806.64,614.501 1807.58,614.501 1808.52,614.501 1809.47,614.501 1810.41,614.501 1811.35,614.501 1812.3,614.501 1813.24,614.501 1814.18,614.501 1815.12,614.501 \n",
" 1816.07,614.501 1817.01,614.501 1817.95,614.501 1818.9,614.501 1819.84,614.501 1820.78,614.501 1821.73,614.501 1822.67,614.501 1823.61,614.501 1824.56,614.501 \n",
" 1825.5,614.501 1826.44,614.501 1827.39,614.501 1828.33,614.501 1829.27,614.501 1830.22,614.501 1831.16,614.501 1832.1,614.501 1833.05,614.501 1833.99,614.501 \n",
" 1834.93,614.501 1835.88,614.501 1836.82,614.501 1837.76,614.501 1838.71,614.501 1839.65,614.501 1840.59,614.501 1841.54,614.501 1842.48,614.501 1843.42,614.501 \n",
" 1844.37,614.501 1845.31,614.501 1846.25,614.501 1847.2,614.501 1848.14,614.501 1849.08,614.501 1850.03,614.501 1850.97,614.501 1851.91,614.501 1852.86,614.501 \n",
" 1853.8,614.501 1854.74,614.501 1855.69,614.501 1856.63,614.501 1857.57,614.501 1858.51,614.501 1859.46,614.501 1860.4,614.501 1861.34,614.501 1862.29,614.501 \n",
" 1863.23,614.501 1864.17,614.501 1865.12,614.501 1866.06,614.501 1867,614.501 1867.95,614.501 1868.89,614.501 1869.83,614.501 1870.78,614.501 1871.72,614.501 \n",
" 1872.66,614.501 1873.61,614.501 1874.55,614.501 1875.49,614.501 1876.44,614.501 1877.38,614.501 1878.32,614.501 1879.27,614.501 1880.21,614.501 1881.15,614.501 \n",
" 1882.1,614.501 1883.04,614.501 1883.98,614.501 1884.93,614.501 1885.87,614.501 1886.81,614.501 1887.76,614.501 1888.7,614.501 1889.64,614.501 1890.59,614.501 \n",
" 1891.53,614.501 1892.47,614.501 1893.42,614.501 1894.36,614.501 1895.3,614.501 1896.25,614.501 1897.19,614.501 1898.13,614.501 1899.08,614.501 1900.02,614.501 \n",
" 1900.96,614.501 1901.9,614.501 1902.85,614.501 1903.79,614.501 1904.73,614.501 1905.68,614.501 1906.62,614.501 1907.56,614.501 1908.51,614.501 1909.45,614.501 \n",
" 1910.39,614.501 1911.34,614.501 1912.28,614.501 1913.22,614.501 1914.17,614.501 1915.11,614.501 1916.05,614.501 1917,614.501 1917.94,614.501 1918.88,614.501 \n",
" 1919.83,614.501 1920.77,614.501 1921.71,614.501 1922.66,614.501 1923.6,614.501 1924.54,614.501 1925.49,614.501 1926.43,614.501 1927.37,614.501 1928.32,614.501 \n",
" 1929.26,614.501 1930.2,614.501 1931.15,614.501 1932.09,614.501 1933.03,614.501 1933.98,614.501 1934.92,614.501 1935.86,614.501 1936.81,614.501 1937.75,614.501 \n",
" 1938.69,614.501 1939.64,614.501 1940.58,614.501 1941.52,614.501 1942.47,614.501 1943.41,614.501 1944.35,614.501 1945.29,614.501 1946.24,614.501 1947.18,614.501 \n",
" 1948.12,614.501 1949.07,614.501 1950.01,614.501 1950.95,614.501 1951.9,614.501 1952.84,614.501 1953.78,614.501 1954.73,614.501 1955.67,614.501 1956.61,614.501 \n",
" 1957.56,614.501 1958.5,614.501 1959.44,614.501 1960.39,614.501 1961.33,614.501 1962.27,614.501 1963.22,614.501 1964.16,614.501 1965.1,614.501 1966.05,614.501 \n",
" 1966.99,614.501 1967.93,614.501 1968.88,614.501 1969.82,614.501 1970.76,614.501 1971.71,614.501 1972.65,614.501 1973.59,614.501 1974.54,614.501 1975.48,614.501 \n",
" 1976.42,614.501 1977.37,614.501 1978.31,614.501 1979.25,614.501 1980.2,614.501 1981.14,614.501 1982.08,614.501 1983.03,614.501 1983.97,614.501 1984.91,614.501 \n",
" 1985.86,614.501 1986.8,614.501 1987.74,614.501 1988.69,614.501 1989.63,614.501 1990.57,614.501 1991.51,614.501 1992.46,614.501 1993.4,614.501 1994.34,614.501 \n",
" 1995.29,614.501 1996.23,614.501 1997.17,614.501 1998.12,614.501 1999.06,614.501 2000,614.501 2000.95,614.501 2001.89,614.501 2002.83,614.501 2003.78,614.501 \n",
" 2004.72,614.501 2005.66,614.501 2006.61,614.501 2007.55,614.501 2008.49,614.501 2009.44,614.501 2010.38,614.501 2011.32,614.501 2012.27,614.501 2013.21,614.501 \n",
" 2014.15,614.501 2015.1,614.501 2016.04,614.501 2016.98,614.501 2017.93,614.501 2018.87,614.501 2019.81,614.501 2020.76,614.501 2021.7,614.501 2022.64,614.501 \n",
" 2023.59,614.501 2024.53,614.501 2025.47,614.501 2026.42,614.501 2027.36,614.501 2028.3,614.501 2029.25,614.501 2030.19,614.501 2031.13,614.501 2032.08,614.501 \n",
" 2033.02,614.501 2033.96,614.501 2034.9,614.501 2035.85,614.501 2036.79,614.501 2037.73,614.501 2038.68,614.501 2039.62,614.501 2040.56,614.501 2041.51,614.501 \n",
" 2042.45,614.501 2043.39,614.501 2044.34,614.501 2045.28,614.501 2046.22,614.501 2047.17,614.501 2048.11,614.501 2049.05,614.501 2050,614.501 2050.94,614.501 \n",
" 2051.88,614.501 2052.83,614.501 2053.77,614.501 2054.71,614.501 2055.66,614.501 2056.6,614.501 2057.54,614.501 2058.49,614.501 2059.43,614.501 2060.37,614.501 \n",
" 2061.32,614.501 2062.26,614.501 2063.2,614.501 2064.15,614.501 2065.09,614.501 2066.03,614.501 2066.98,614.501 2067.92,614.501 2068.86,614.501 2069.81,614.501 \n",
" 2070.75,614.501 2071.69,614.501 2072.64,614.501 2073.58,614.501 2074.52,614.501 2075.47,614.501 2076.41,614.501 2077.35,614.501 2078.29,614.501 2079.24,614.501 \n",
" 2080.18,614.501 2081.12,614.501 2082.07,614.501 2083.01,614.501 2083.95,614.501 2084.9,614.501 2085.84,614.501 2086.78,614.501 2087.73,614.501 2088.67,614.501 \n",
" 2089.61,614.501 2090.56,614.501 2091.5,614.501 2092.44,614.501 2093.39,614.501 2094.33,614.501 2095.27,614.501 2096.22,614.501 2097.16,614.501 2098.1,614.501 \n",
" 2099.05,614.501 2099.99,614.501 2100.93,614.501 2101.88,614.501 2102.82,614.501 2103.76,614.501 2104.71,614.501 2105.65,614.501 2106.59,614.501 2107.54,614.501 \n",
" 2108.48,614.501 2109.42,614.501 2110.37,614.501 2111.31,614.501 2112.25,614.501 2113.2,614.501 2114.14,614.501 2115.08,614.501 2116.03,614.501 2116.97,614.501 \n",
" 2117.91,614.501 2118.86,614.501 2119.8,614.501 2120.74,614.501 2121.68,614.501 2122.63,614.501 2123.57,614.501 2124.51,614.501 2125.46,614.501 2126.4,614.501 \n",
" 2127.34,614.501 2128.29,614.501 2129.23,614.501 2130.17,614.501 2131.12,614.501 2132.06,614.501 2133,614.501 2133.95,614.501 2134.89,614.501 2135.83,614.501 \n",
" 2136.78,614.501 2137.72,614.501 2138.66,614.501 2139.61,614.501 2140.55,614.501 2141.49,614.501 2142.44,614.501 2143.38,614.501 2144.32,614.501 2145.27,614.501 \n",
" 2146.21,614.501 2147.15,614.501 2148.1,614.501 2149.04,614.501 2149.98,614.501 2150.93,614.501 2151.87,614.501 2152.81,614.501 2153.76,614.501 2154.7,614.501 \n",
" 2155.64,614.501 2156.59,614.501 2157.53,614.501 2158.47,614.501 2159.42,614.501 2160.36,614.501 2161.3,614.501 2162.25,614.501 2163.19,614.501 2164.13,614.501 \n",
" 2165.08,614.501 2166.02,614.501 2166.96,614.501 2167.9,614.501 2168.85,614.501 2169.79,614.501 2170.73,614.501 2171.68,614.501 2172.62,614.501 2173.56,614.501 \n",
" 2174.51,614.501 2175.45,614.501 2176.39,614.501 2177.34,614.501 2178.28,614.501 2179.22,614.501 2180.17,614.501 2181.11,614.501 2182.05,614.501 2183,614.501 \n",
" 2183.94,614.501 2184.88,614.501 2185.83,614.501 2186.77,614.501 2187.71,614.501 2188.66,614.501 2189.6,614.501 2190.54,614.501 2191.49,614.501 2192.43,614.501 \n",
" 2193.37,614.501 2194.32,614.501 2195.26,614.501 2196.2,614.501 2197.15,614.501 2198.09,614.501 2199.03,614.501 2199.98,614.501 2200.92,614.501 2201.86,614.501 \n",
" 2202.81,614.501 2203.75,614.501 2204.69,614.501 2205.64,614.501 2206.58,614.501 2207.52,614.501 2208.47,614.501 2209.41,614.501 2210.35,614.501 2211.29,614.501 \n",
" 2212.24,614.501 2213.18,614.501 2214.12,614.501 2215.07,614.501 2216.01,614.501 2216.95,614.501 2217.9,614.501 2218.84,614.501 2219.78,614.501 2220.73,614.501 \n",
" 2221.67,614.501 2222.61,614.501 2223.56,614.501 2224.5,614.501 2225.44,614.501 2226.39,614.501 2227.33,614.501 2228.27,614.501 2229.22,614.501 2230.16,614.501 \n",
" 2231.1,614.501 2232.05,614.501 2232.99,614.501 2233.93,614.501 2234.88,614.501 2235.82,614.501 2236.76,614.501 2237.71,614.501 2238.65,614.501 2239.59,614.501 \n",
" 2240.54,614.501 2241.48,614.501 2242.42,614.501 2243.37,614.501 2244.31,614.501 2245.25,614.501 2246.2,614.501 2247.14,614.501 2248.08,614.501 2249.03,614.501 \n",
" 2249.97,614.501 2250.91,614.501 2251.86,614.501 2252.8,614.501 2253.74,614.501 2254.68,614.501 2255.63,614.501 2256.57,614.501 2257.51,614.501 2258.46,614.501 \n",
" 2259.4,614.501 2260.34,614.501 2261.29,614.501 2262.23,614.501 2263.17,614.501 2264.12,614.501 2265.06,614.501 2266,614.501 2266.95,614.501 2267.89,614.501 \n",
" 2268.83,614.501 2269.78,614.501 2270.72,614.501 2271.66,614.501 2272.61,614.501 2273.55,614.501 2274.49,614.501 2275.44,614.501 2276.38,614.501 2277.32,614.501 \n",
" 2278.27,614.501 2279.21,614.501 2280.15,614.501 2281.1,614.501 2282.04,614.501 2282.98,614.501 2283.93,614.501 2284.87,614.501 2285.81,614.501 2286.76,614.501 \n",
" 2287.7,614.501 2288.64,614.501 2289.59,614.501 2290.53,614.501 2291.47,614.501 2292.42,614.501 2293.36,614.501 2294.3,614.501 2295.25,614.501 2296.19,614.501 \n",
" 2297.13,614.501 2298.08,614.501 2299.02,614.501 2299.96,614.501 2300.9,614.501 2301.85,614.501 2302.79,614.501 2303.73,614.501 2304.68,614.501 2305.62,614.501 \n",
" 2306.56,614.501 2307.51,614.501 2308.45,614.501 2309.39,614.501 2310.34,614.501 2311.28,614.501 2312.22,614.501 2313.17,614.501 2314.11,614.501 2315.05,614.501 \n",
" 2316,614.501 2316.94,614.501 2317.88,614.501 2318.83,614.501 2319.77,614.501 2320.71,614.501 2321.66,614.501 2322.6,614.501 2323.54,614.501 2324.49,614.501 \n",
" \n",
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" 1391.6,765.714 1392.54,772.841 1393.49,762.49 1394.43,754.339 1395.37,755.33 1396.32,755.755 1397.26,752.811 1398.2,754.244 1399.15,745.031 1400.09,741.386 \n",
" 1401.03,738.303 1401.98,746.181 1402.92,740.446 1403.86,760.531 1404.81,763.207 1405.75,762.798 1406.69,776.085 1407.64,770.346 1408.58,770.033 1409.52,765.827 \n",
" 1410.47,768.001 1411.41,768.232 1412.35,761.302 1413.3,764.184 1414.24,756.261 1415.18,763.835 1416.12,774.882 1417.07,768.877 1418.01,754.593 1418.95,753.582 \n",
" 1419.9,754.253 1420.84,751.162 1421.78,747.494 1422.73,751.63 1423.67,753.359 1424.61,759.769 1425.56,758.993 1426.5,755.433 1427.44,749.419 1428.39,749.915 \n",
" 1429.33,745.133 1430.27,740.735 1431.22,734.38 1432.16,738.24 1433.1,742.426 1434.05,744.465 1434.99,741.301 1435.93,733.448 1436.88,731.593 1437.82,734.167 \n",
" 1438.76,739.694 1439.71,737.438 1440.65,743.582 1441.59,746.373 1442.54,747.841 1443.48,750.687 1444.42,751.773 1445.37,754.513 1446.31,761.133 1447.25,762.141 \n",
" 1448.2,762.06 1449.14,761.012 1450.08,759.92 1451.03,764.313 1451.97,763.403 1452.91,762.45 1453.86,760.926 1454.8,761.754 1455.74,763.935 1456.69,761.764 \n",
" 1457.63,759.102 1458.57,760.92 1459.52,759.626 1460.46,759.9 1461.4,761.72 1462.34,759.095 1463.29,759.596 1464.23,760.859 1465.17,760.537 1466.12,762.407 \n",
" 1467.06,756.698 1468,758.61 1468.95,759.315 1469.89,759.679 1470.83,759.896 1471.78,760.064 1472.72,762.98 1473.66,760.744 1474.61,763.401 1475.55,764.51 \n",
" 1476.49,763.582 1477.44,764.661 1478.38,759.605 1479.32,760.666 1480.27,758.212 1481.21,756.448 1482.15,756.505 1483.1,759.138 1484.04,761.674 1484.98,761.749 \n",
" 1485.93,761.324 1486.87,762.486 1487.81,763.286 1488.76,761.887 1489.7,764.242 1490.64,763.03 1491.59,765.3 1492.53,766.4 1493.47,767.733 1494.42,768.065 \n",
" 1495.36,769.398 1496.3,770.523 1497.25,769.548 1498.19,769.256 1499.13,768.369 1500.08,767.59 1501.02,765.758 1501.96,766.058 1502.91,765.299 1503.85,767.768 \n",
" 1504.79,770.548 1505.73,770.26 1506.68,770.43 1507.62,773.241 1508.56,772.053 1509.51,772.455 1510.45,771.072 1511.39,772.792 1512.34,770.705 1513.28,769.515 \n",
" 1514.22,769.983 1515.17,769.386 1516.11,768.8 1517.05,769.702 1518,769.362 1518.94,770.115 1519.88,768.62 1520.83,770.685 1521.77,768.572 1522.71,767.923 \n",
" 1523.66,767.521 1524.6,768.005 1525.54,770.314 1526.49,772.319 1527.43,773.158 1528.37,774.273 1529.32,771.917 1530.26,772.695 1531.2,774.12 1532.15,774.376 \n",
" 1533.09,774.012 1534.03,774.986 1534.98,771.711 1535.92,772.61 1536.86,773.9 1537.81,775.114 1538.75,774.074 1539.69,774.397 1540.64,774.209 1541.58,773.88 \n",
" 1542.52,773.656 1543.47,773.221 1544.41,775.22 1545.35,774.402 1546.3,775.498 1547.24,774.692 1548.18,775.5 1549.12,778.029 1550.07,777.296 1551.01,778.099 \n",
" 1551.95,777.763 1552.9,778.705 1553.84,779.084 1554.78,778.87 1555.73,779.144 1556.67,779.463 1557.61,778.571 1558.56,776.81 1559.5,777.071 1560.44,775.545 \n",
" 1561.39,773.586 1562.33,772.576 1563.27,773.132 1564.22,773.877 1565.16,773.677 1566.1,773.094 1567.05,772.814 1567.99,771.92 1568.93,771.58 1569.88,773.772 \n",
" 1570.82,771.96 1571.76,771.702 1572.71,771.993 1573.65,771.051 1574.59,771.785 1575.54,771.823 1576.48,771.678 1577.42,772.808 1578.37,772.672 1579.31,772.826 \n",
" 1580.25,773.019 1581.2,773.839 1582.14,774.642 1583.08,775.054 1584.03,775.642 1584.97,775.426 1585.91,774.575 1586.86,775.166 1587.8,773.998 1588.74,773.579 \n",
" 1589.69,773.661 1590.63,773.447 1591.57,773.39 1592.51,773.992 1593.46,773.626 1594.4,773.776 1595.34,772.039 1596.29,772.201 1597.23,772.701 1598.17,773.521 \n",
" 1599.12,773.158 1600.06,772.463 1601,771.599 1601.95,771.022 1602.89,769.603 1603.83,768.485 1604.78,766.816 1605.72,766.289 1606.66,766.462 1607.61,768.089 \n",
" 1608.55,768.657 1609.49,768.726 1610.44,768.775 1611.38,769.717 1612.32,769.422 1613.27,770.151 1614.21,769.46 1615.15,768.308 1616.1,767.539 1617.04,768.166 \n",
" 1617.98,768.915 1618.93,767.539 1619.87,766.869 1620.81,765.983 1621.76,765.485 1622.7,764.546 1623.64,765.729 1624.59,768.368 1625.53,767.962 1626.47,767.842 \n",
" 1627.42,767.706 1628.36,768.648 1629.3,770.543 1630.25,771.199 1631.19,771.337 1632.13,772.883 1633.08,772.595 1634.02,772.692 1634.96,773.57 1635.91,773.185 \n",
" 1636.85,772.802 1637.79,773.748 1638.73,773.5 1639.68,774.002 1640.62,774.057 1641.56,775.12 1642.51,774.472 1643.45,773.342 1644.39,773.665 1645.34,774.762 \n",
" 1646.28,776.764 1647.22,776.108 1648.17,775.956 1649.11,775.646 1650.05,775.596 1651,775.679 1651.94,775.874 1652.88,776.1 1653.83,776.185 1654.77,776.414 \n",
" 1655.71,775.838 1656.66,775.672 1657.6,775.121 1658.54,774.857 1659.49,773.799 1660.43,773.031 1661.37,772.385 1662.32,771.839 1663.26,771.508 1664.2,771.63 \n",
" 1665.15,772.361 1666.09,771.651 1667.03,770.078 1667.98,770.753 1668.92,769.821 1669.86,770.587 1670.81,769.782 1671.75,770.626 1672.69,770.708 1673.64,771.053 \n",
" 1674.58,770.739 1675.52,770.572 1676.47,770.128 1677.41,768.912 1678.35,768.966 1679.3,769.105 1680.24,769.488 1681.18,769.623 1682.12,769.102 1683.07,768.452 \n",
" 1684.01,768.833 1684.95,768.469 1685.9,768.556 1686.84,768.248 1687.78,768.631 1688.73,769.187 1689.67,769.127 1690.61,768.632 1691.56,768.88 1692.5,768.295 \n",
" 1693.44,767.567 1694.39,767.727 1695.33,767.681 1696.27,767.947 1697.22,768.14 1698.16,767.679 1699.1,768.04 1700.05,767.891 1700.99,768.671 1701.93,768.739 \n",
" 1702.88,768.927 1703.82,767.96 1704.76,766.986 1705.71,768.175 1706.65,768.388 1707.59,767.226 1708.54,767.241 1709.48,767.897 1710.42,767.236 1711.37,766.982 \n",
" 1712.31,766.476 1713.25,766.897 1714.2,767.367 1715.14,767.551 1716.08,768.518 1717.03,767.853 1717.97,767.69 1718.91,767.504 1719.86,768.016 1720.8,768.336 \n",
" 1721.74,767.371 1722.69,766.846 1723.63,766.982 1724.57,766.814 1725.51,766.548 1726.46,767.019 1727.4,766.9 1728.34,766.746 1729.29,765.97 1730.23,765.696 \n",
" 1731.17,766.436 1732.12,765.818 1733.06,765.458 1734,765.674 1734.95,765.573 1735.89,764.449 1736.83,764.017 1737.78,764.439 1738.72,764.144 1739.66,763.828 \n",
" 1740.61,763.451 1741.55,764.276 1742.49,764.292 1743.44,763.948 1744.38,763.836 1745.32,763.481 1746.27,762.543 1747.21,763.063 1748.15,762.782 1749.1,762.971 \n",
" 1750.04,762.972 1750.98,762.464 1751.93,762.549 1752.87,762.045 1753.81,761.24 1754.76,761.887 1755.7,761.593 1756.64,761.656 1757.59,761.542 1758.53,761.349 \n",
" 1759.47,761.469 1760.42,761.263 1761.36,760.694 1762.3,760.645 1763.25,759.35 1764.19,759.012 1765.13,758.402 1766.08,758.44 1767.02,758.641 1767.96,757.924 \n",
" 1768.9,758.151 1769.85,757.679 1770.79,757.066 1771.73,756.646 1772.68,756.828 1773.62,756.435 1774.56,756.262 1775.51,756.277 1776.45,757.263 1777.39,757.166 \n",
" 1778.34,756.998 1779.28,757.236 1780.22,757.031 1781.17,756.811 1782.11,756.946 1783.05,756.455 1784,756.753 1784.94,756.372 1785.88,755.725 1786.83,755.845 \n",
" 1787.77,756.388 1788.71,756.158 1789.66,756.543 1790.6,756.261 1791.54,755.423 1792.49,755.406 1793.43,755.669 1794.37,755.54 1795.32,755.393 1796.26,754.748 \n",
" 1797.2,754.447 1798.15,754.706 1799.09,755.11 1800.03,755.404 1800.98,754.654 1801.92,754.056 1802.86,754.693 1803.81,755.16 1804.75,755.044 1805.69,755.62 \n",
" 1806.64,755.214 1807.58,754.816 1808.52,755.213 1809.47,754.759 1810.41,754.393 1811.35,754.242 1812.3,754.375 1813.24,753.77 1814.18,753.232 1815.12,753.301 \n",
" 1816.07,753.73 1817.01,752.63 1817.95,753.374 1818.9,754.354 1819.84,754.622 1820.78,754.567 1821.73,755.58 1822.67,755.658 1823.61,755.412 1824.56,755.812 \n",
" 1825.5,756.282 1826.44,755.969 1827.39,756.255 1828.33,756.063 1829.27,755.827 1830.22,755.718 1831.16,756.387 1832.1,756.531 1833.05,757.21 1833.99,757.393 \n",
" 1834.93,758.49 1835.88,759.132 1836.82,758.877 1837.76,758.469 1838.71,758.875 1839.65,759.094 1840.59,759.203 1841.54,759 1842.48,759.442 1843.42,760.064 \n",
" 1844.37,760.145 1845.31,760.676 1846.25,760.13 1847.2,759.677 1848.14,759.93 1849.08,760.375 1850.03,760.765 1850.97,760.907 1851.91,760.948 1852.86,760.943 \n",
" 1853.8,760.672 1854.74,760.924 1855.69,761.585 1856.63,762.042 1857.57,762.228 1858.51,762.735 1859.46,763.019 1860.4,762.969 1861.34,762.587 1862.29,762.308 \n",
" 1863.23,761.93 1864.17,761.617 1865.12,761.749 1866.06,761.568 1867,762.003 1867.95,762.076 1868.89,762.155 1869.83,761.783 1870.78,761.122 1871.72,761.452 \n",
" 1872.66,761.351 1873.61,761.068 1874.55,761.039 1875.49,761.434 1876.44,761.11 1877.38,761.04 1878.32,761.573 1879.27,761.696 1880.21,761.35 1881.15,760.931 \n",
" 1882.1,760.564 1883.04,760.353 1883.98,760.24 1884.93,760.832 1885.87,760.748 1886.81,760.246 1887.76,760.14 1888.7,760.231 1889.64,760.179 1890.59,759.885 \n",
" 1891.53,759.544 1892.47,759.903 1893.42,759.895 1894.36,759.883 1895.3,759.517 1896.25,759.752 1897.19,760.159 1898.13,760.564 1899.08,760.772 1900.02,760.397 \n",
" 1900.96,760.616 1901.9,761.515 1902.85,761.651 1903.79,762.173 1904.73,762.246 1905.68,762.184 1906.62,762.171 1907.56,762.915 1908.51,762.872 1909.45,762.572 \n",
" 1910.39,762.269 1911.34,761.877 1912.28,762.055 1913.22,762.239 1914.17,762.285 1915.11,762.456 1916.05,761.943 1917,762.198 1917.94,762.753 1918.88,763.532 \n",
" 1919.83,762.785 1920.77,763.532 1921.71,763.443 1922.66,763.838 1923.6,763.309 1924.54,762.69 1925.49,762.588 1926.43,762.423 1927.37,762.524 1928.32,762.575 \n",
" 1929.26,762.419 1930.2,762.095 1931.15,762.611 1932.09,762.987 1933.03,763.515 1933.98,763.321 1934.92,763.423 1935.86,763.669 1936.81,763.863 1937.75,764.063 \n",
" 1938.69,764.134 1939.64,764.516 1940.58,765.17 1941.52,765.342 1942.47,764.961 1943.41,765.168 1944.35,765.656 1945.29,765.085 1946.24,765.357 1947.18,765.161 \n",
" 1948.12,765.356 1949.07,765.54 1950.01,765.387 1950.95,765.24 1951.9,765.512 1952.84,765.277 1953.78,765.371 1954.73,765.745 1955.67,765.836 1956.61,765.952 \n",
" 1957.56,765.913 1958.5,765.674 1959.44,765.25 1960.39,765.193 1961.33,765.377 1962.27,765.266 1963.22,765.052 1964.16,765.107 1965.1,765.314 1966.05,765.095 \n",
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],
"source": [
"p1 = plot(params[1, :], linewidth=0.2, label = \"posterior samples\")\n",
"plot!(Ptrue.α*ones(iterations), label = LaTeXString(\"true \\\\alpha\"))\n",
"plot!(Bridge.runmean(params[1, :]), label = \"running mean\")\n",
"p2 = plot(params[2, :], linewidth=0.2, label = \"posterior samples\")\n",
"plot!(Ptrue.β*ones(iterations), label = LaTeXString(\"true \\\\beta\"))\n",
"plot!(Bridge.runmean(params[2, :]), label = \"running mean\")\n",
"plot(p1,p2)\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"scrolled": true
},
"outputs": [
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" 1503.03,679.349 1503.87,914.264 1504.71,616.903 1505.56,1224.02 1506.4,830.189 1507.24,755.343 1508.08,1123.57 1508.93,618.024 1509.77,764.034 1510.61,646.524 \n",
" 1511.45,835.965 1512.29,778.177 1513.14,541.985 1513.98,862.011 1514.82,489.183 1515.66,1031.28 1516.51,1174.35 1517.35,549.192 1518.19,212.705 1519.03,716.887 \n",
" 1519.87,783.892 1520.72,627.052 1521.56,596.042 1522.4,932.074 1523.24,832.186 1524.08,1050.51 1524.93,726.082 1525.77,590.665 1526.61,462.977 1527.45,776.979 \n",
" 1528.3,508.393 1529.14,518.8 1529.98,405.914 1530.82,945.397 1531.66,970.991 1532.51,859.373 1533.35,566.606 1534.19,287.185 1535.03,626.674 1535.88,888.708 \n",
" 1536.72,1073.54 1537.56,602.406 1538.4,1126.55 1539.24,924.821 1540.09,844.518 1540.93,938.29 1541.77,826.211 1542.61,940.705 1543.45,1212.85 1544.3,834.424 \n",
" 1545.14,759.157 1545.98,689.376 1546.82,683.973 1547.67,1087.17 1548.51,698.813 1549.35,693.754 1550.19,647.781 1551.03,828.254 1551.88,936.603 1552.72,592.831 \n",
" 1553.56,548.685 1554.4,910.786 1555.25,656.211 1556.09,785.43 1556.93,917.195 1557.77,538.476 1558.61,805.379 1559.46,873.516 1560.3,734.962 1561.14,931.433 \n",
" 1561.98,243.948 1562.82,935.283 1563.67,826.875 1564.51,796.291 1565.35,783.12 1566.19,778.712 1567.04,1045.26 1567.88,546.355 1568.72,1026.15 1569.56,876.972 \n",
" 1570.4,673.567 1571.25,876.279 1572.09,245.043 1572.93,872.613 1573.77,505.4 1574.62,573.695 1575.46,765.341 1576.3,1043.24 1577.14,1037.85 1577.98,772.772 \n",
" 1578.83,717.236 1579.67,894.222 1580.51,855.589 1581.35,606.449 1582.19,1034.96 1583.04,626.343 1583.88,1030.97 1584.72,897.834 1585.56,927.561 1586.41,810.317 \n",
" 1587.25,931.992 1588.09,909.3 1588.93,653.32 1589.77,736.055 1590.62,661.114 1591.46,672.875 1592.3,537.407 1593.14,806.949 1593.99,670.886 1594.83,1088.35 \n",
" 1595.67,1134.02 1596.51,735.257 1597.35,795.592 1598.2,1149.01 1599.04,615.526 1599.88,829.402 1600.72,585.594 1601.56,1010.87 1602.41,484.918 1603.25,606.77 \n",
" 1604.09,838.239 1604.93,687.985 1605.78,688.191 1606.62,901.427 1607.46,723.086 1608.3,882.019 1609.14,552.941 1609.99,1076.33 1610.83,458.135 1611.67,673.887 \n",
" 1612.51,709.902 1613.36,843.901 1614.2,1123.33 1615.04,1081.24 1615.88,905.017 1616.72,949.749 1617.57,406.353 1618.41,897.452 1619.25,1001.72 1620.09,817.855 \n",
" 1620.93,718.571 1621.78,934.323 1622.62,242.328 1623.46,921.806 1624.3,987.936 1625.15,977.916 1625.99,603.946 1626.83,831.119 1627.67,745.344 1628.51,721.124 \n",
" 1629.36,738.304 1630.2,701.61 1631.04,1121.11 1631.88,635.424 1632.72,968.797 1633.57,636.403 1634.41,920.257 1635.25,1227.23 1636.09,649.399 1636.94,924.595 \n",
" 1637.78,719.95 1638.62,951.795 1639.46,850.827 1640.3,742.434 1641.15,832.319 1641.99,841.176 1642.83,615.26 1643.67,449.746 1644.52,828.781 1645.36,489.517 \n",
" 1646.2,403.265 1647.04,582.203 1647.88,882.582 1648.73,920.354 1649.57,737.571 1650.41,662.262 1651.25,720.558 1652.09,598.51 1652.94,706.91 1653.78,1211.63 \n",
" 1654.62,411.638 1655.46,722.428 1656.31,833.563 1657.15,582.504 1657.99,924.099 1658.83,782.291 1659.67,744.575 1660.52,1009.14 1661.36,747.237 1662.2,807.748 \n",
" 1663.04,816.34 1663.89,949.434 1664.73,947.34 1665.57,865.556 1666.41,904.261 1667.25,731.592 1668.1,593.416 1668.94,906.165 1669.78,521.587 1670.62,684.654 \n",
" 1671.46,794.377 1672.31,728.962 1673.15,763.447 1673.99,910.989 1674.83,694.308 1675.68,810.291 1676.52,381.5 1677.36,811.717 1678.2,889.387 1679.04,963.813 \n",
" 1679.89,692.357 1680.73,614.629 1681.57,573.69 1682.41,639.112 1683.26,439.628 1684.1,508.232 1684.94,374.754 1685.78,643.995 1686.62,810.337 1687.47,1158.88 \n",
" 1688.31,907.722 1689.15,788.14 1689.99,783.249 1690.83,1001.15 1691.68,700.232 1692.52,951.397 1693.36,601.913 1694.2,486.193 1695.05,579.251 1695.89,926.921 \n",
" 1696.73,958.623 1697.57,424.346 1698.41,600.694 1699.26,544.224 1700.1,641.713 1700.94,527.471 1701.78,1070.74 1702.63,1446.71 1703.47,665.83 1704.31,739.657 \n",
" 1705.15,735.101 1705.99,1016.79 1706.84,1268.26 1707.68,946.368 1708.52,810.523 1709.36,1184.24 1710.2,698.81 1711.05,801.447 1711.89,1011.26 1712.73,672.256 \n",
" 1713.57,672.097 1714.42,1032.36 1715.26,708.785 1716.1,913.816 1716.94,792.013 1717.78,1069.59 1718.63,598.152 1719.47,462.566 1720.31,866.145 1721.15,1083.11 \n",
" 1722,1338.16 1722.84,594.273 1723.68,735.747 1724.52,690.625 1725.36,764.419 1726.21,801.863 1727.05,834.402 1727.89,843.662 1728.73,803.59 1729.57,845.142 \n",
" 1730.42,611.466 1731.26,730.195 1732.1,616.9 1732.94,700.113 1733.79,465.067 1734.63,549.058 1735.47,583.497 1736.31,612.349 1737.15,675.685 1738,810.715 \n",
" 1738.84,994.128 1739.68,560.494 1740.52,296.483 1741.37,977.992 1742.21,488.504 1743.05,1006.64 1743.89,525.866 1744.73,1032.12 1745.58,798.729 1746.42,880.609 \n",
" 1747.26,676.119 1748.1,721.123 1748.94,634.489 1749.79,390.197 1750.63,788.57 1751.47,815.74 1752.31,893.071 1753.16,815.266 1754,605.969 1754.84,563.671 \n",
" 1755.68,893.473 1756.52,654.141 1757.37,799.536 1758.21,671.509 1759.05,895.334 1759.89,952.308 1760.73,752.489 1761.58,609.278 1762.42,853.123 1763.26,578.28 \n",
" 1764.1,529.776 1764.95,823.346 1765.79,755.34 1766.63,859.26 1767.47,835.122 1768.31,616.005 1769.16,892.118 1770,720.206 1770.84,1034.76 1771.68,794.613 \n",
" 1772.53,835.635 1773.37,440.031 1774.21,436.285 1775.05,1177.49 1775.89,844.715 1776.74,367.985 1777.58,775.222 1778.42,997.794 1779.26,539.952 1780.1,680.856 \n",
" 1780.95,592.099 1781.79,917.118 1782.63,935.363 1783.47,835.357 1784.32,1112.86 1785.16,534.331 1786,712.466 1786.84,703.943 1787.68,953.835 1788.53,885.689 \n",
" 1789.37,422.083 1790.21,580.003 1791.05,818.765 1791.9,708.541 1792.74,672.475 1793.58,941.775 1794.42,725.963 1795.26,713.093 1796.11,482.543 1796.95,667.211 \n",
" 1797.79,1042.82 1798.63,538.796 1799.47,634.082 1800.32,849.088 1801.16,730.462 1802,344.635 1802.84,604.44 1803.69,925.924 1804.53,655.388 1805.37,646.951 \n",
" 1806.21,622.838 1807.05,1080.48 1807.9,773.449 1808.74,634.589 1809.58,723.73 1810.42,629.407 1811.27,402.649 1812.11,966.661 1812.95,656.504 1813.79,839.242 \n",
" 1814.63,765.987 1815.48,566.374 1816.32,798.696 1817.16,566.632 1818,446.146 1818.84,1019.93 1819.69,647.649 1820.53,789.456 1821.37,718.947 1822.21,686.903 \n",
" 1823.06,812.497 1823.9,681.138 1824.74,534.706 1825.58,743.408 1826.42,237.461 1827.27,624.546 1828.11,513.225 1828.95,776.759 1829.79,843.433 1830.64,466.826 \n",
" 1831.48,853.956 1832.32,565.965 1833.16,507.093 1834,585.703 1834.85,834.784 1835.69,596.2 1836.53,687.013 1837.37,764.943 1838.21,1171.29 1839.06,719.192 \n",
" 1839.9,689.396 1840.74,860.224 1841.58,673.29 1842.43,666.244 1843.27,816.819 1844.11,550.507 1844.95,886.198 1845.79,596.268 1846.64,481.373 1847.48,809.691 \n",
" 1848.32,992.618 1849.16,659.459 1850.01,925.55 1850.85,637.056 1851.69,393.572 1852.53,750.788 1853.37,872.888 1854.22,701.936 1855.06,693.855 1855.9,473.962 \n",
" 1856.74,624.508 1857.58,871.51 1858.43,936.439 1859.27,888.406 1860.11,423.624 1860.95,490.154 1861.8,1041.02 1862.64,966.704 1863.48,705.618 1864.32,1016.63 \n",
" 1865.16,574.994 1866.01,577.951 1866.85,937.454 1867.69,551.365 1868.53,590.739 1869.38,688.455 1870.22,817.772 1871.06,479.721 1871.9,509.238 1872.74,787.756 \n",
" 1873.59,953.325 1874.43,246.837 1875.27,1099.2 1876.11,1209.47 1876.95,881.646 1877.8,731.979 1878.64,1229.16 1879.48,794.985 1880.32,642.602 1881.17,945.979 \n",
" 1882.01,979.876 1882.85,611.004 1883.69,893.845 1884.53,668.151 1885.38,646.824 1886.22,706.655 1887.06,1076.95 1887.9,827.907 1888.74,1083.76 1889.59,848.209 \n",
" 1890.43,1285.86 1891.27,1070.17 1892.11,638.654 1892.96,564.065 1893.8,957.706 1894.64,868.111 1895.48,815.07 1896.32,662.64 1897.17,977.732 1898.01,1066.72 \n",
" 1898.85,802.201 1899.69,1023.83 1900.54,494.202 1901.38,538.806 1902.22,887.261 1903.06,983.401 1903.9,956.838 1904.75,834.152 1905.59,783.831 1906.43,761.33 \n",
" 1907.27,627.95 1908.11,890.025 1908.96,1095.1 1909.8,994.555 1910.64,858.571 1911.48,1021.44 1912.33,909.136 1913.17,740.644 1914.01,570.724 1914.85,622.971 \n",
" 1915.69,571.92 1916.54,603.998 1917.38,832.069 1918.22,671.449 1919.06,988.346 1919.91,802.584 1920.75,805.666 1921.59,571.67 1922.43,420.934 1923.27,935.434 \n",
" 1924.12,711.356 1924.96,616.536 1925.8,748.609 1926.64,970.485 1927.48,593.99 1928.33,726.752 1929.17,1044.14 1930.01,829.264 1930.85,581.528 1931.7,541.708 \n",
" 1932.54,568.427 1933.38,651.346 1934.22,702.429 1935.06,1078.99 1935.91,718.385 1936.75,494.175 1937.59,705.821 1938.43,811.999 1939.28,735.056 1940.12,603.661 \n",
" 1940.96,578.409 1941.8,956.329 1942.64,758.649 1943.49,755.913 1944.33,562.894 1945.17,890.601 1946.01,984.717 1946.85,984.657 1947.7,877.638 1948.54,556.898 \n",
" 1949.38,883.495 1950.22,1258.5 1951.07,839.139 1951.91,1053.3 1952.75,805.523 1953.59,730.323 1954.43,757.814 1955.28,1179.34 1956.12,741.432 1956.96,597.787 \n",
" 1957.8,595.18 1958.65,544.023 1959.49,864.944 1960.33,868.799 1961.17,790.91 1962.01,861.37 1962.86,474.165 1963.7,909.105 1964.54,1080.58 1965.38,1208.23 \n",
" 1966.22,338.948 1967.07,1191.3 1967.91,715.407 1968.75,992.669 1969.59,462.064 1970.44,408.537 1971.28,706.505 1972.12,670.367 1972.96,823.136 1973.8,795.006 \n",
" 1974.65,674.925 1975.49,576.162 1976.33,1065.39 1977.17,984.302 1978.02,1074.55 1978.86,652.341 1979.7,825.638 1980.54,911.146 1981.38,880.375 1982.23,884.911 \n",
" 1983.07,808.472 1983.91,992.652 1984.75,1154.51 1985.59,869.843 1986.44,541.368 1987.28,890.994 1988.12,1058.63 1988.96,426.236 1989.81,930.443 1990.65,650.617 \n",
" 1991.49,884.792 1992.33,878.755 1993.17,676.406 1994.02,679.08 1994.86,932.782 1995.7,625.25 1996.54,825.599 1997.39,995.152 1998.23,823.618 1999.07,839.681 \n",
" 1999.91,744.62 2000.75,622.12 2001.6,508.543 2002.44,732.606 2003.28,881.113 2004.12,700.065 2004.96,635.586 2005.81,802.189 2006.65,895.51 2007.49,632.331 \n",
" 2008.33,691.875 2009.18,1157.67 2010.02,795.568 2010.86,720.045 2011.7,657.051 2012.54,699.783 2013.39,679.106 2014.23,790.676 2015.07,634.518 2015.91,1141.93 \n",
" 2016.75,403.757 2017.6,752.505 2018.44,852.429 2019.28,711.139 2020.12,567.875 2020.97,1191.44 2021.81,1013.34 2022.65,562.04 2023.49,536.79 2024.33,722.778 \n",
" 2025.18,822.027 2026.02,851.316 2026.86,787.03 2027.7,721.578 2028.55,851.289 2029.39,1042.27 2030.23,737.729 2031.07,915.705 2031.91,515.72 2032.76,204.85 \n",
" 2033.6,770.773 2034.44,1219.59 2035.28,617.995 2036.12,537.745 2036.97,649.162 2037.81,831.727 2038.65,1084.02 2039.49,809.769 2040.34,923.56 2041.18,750.094 \n",
" 2042.02,720.456 2042.86,640.308 2043.7,878.816 2044.55,729.275 2045.39,1327.08 2046.23,589.182 2047.07,579.28 2047.92,592.989 2048.76,829.059 2049.6,822.78 \n",
" 2050.44,566.105 2051.28,950.583 2052.13,778.032 2052.97,666.776 2053.81,931.132 2054.65,735.118 2055.49,501.302 2056.34,492.672 2057.18,88.0053 2058.02,717.321 \n",
" 2058.86,591.46 2059.71,1005.83 2060.55,830.389 2061.39,596.336 2062.23,986.461 2063.07,886.68 2063.92,525.684 2064.76,608.483 2065.6,862.938 2066.44,917.479 \n",
" 2067.29,280.316 2068.13,935.901 2068.97,1023.42 2069.81,451.228 2070.65,479.558 2071.5,331.312 2072.34,629.733 2073.18,1193.64 2074.02,860.032 2074.86,631.97 \n",
" 2075.71,529.913 2076.55,833.672 2077.39,1205.98 2078.23,645.309 2079.08,529.765 2079.92,791.984 2080.76,561.332 2081.6,197.138 2082.44,711.207 2083.29,949.463 \n",
" 2084.13,946.478 2084.97,884.55 2085.81,1009.92 2086.66,922.238 2087.5,501.393 2088.34,584.962 2089.18,798.946 2090.02,730.423 2090.87,433.822 2091.71,820.946 \n",
" 2092.55,1090.35 2093.39,489.271 2094.23,716.452 2095.08,742.783 2095.92,764.029 2096.76,332.814 2097.6,849.043 2098.45,439.597 2099.29,696.549 2100.13,896.662 \n",
" 2100.97,756.862 2101.81,330.165 2102.66,692.2 2103.5,297.168 2104.34,300.289 2105.18,922.152 2106.03,549.188 2106.87,973.841 2107.71,830.606 2108.55,765.871 \n",
" 2109.39,1223.37 2110.24,636.585 2111.08,821.594 2111.92,823.908 2112.76,1272.1 2113.6,989.578 2114.45,538.526 2115.29,884.957 2116.13,1167.5 2116.97,1091.9 \n",
" 2117.82,1155.86 2118.66,928.066 2119.5,241.105 2120.34,467.919 2121.18,1222.27 2122.03,809.174 2122.87,543.189 2123.71,812.691 2124.55,519.346 2125.39,626.194 \n",
" 2126.24,626.6 2127.08,840.921 2127.92,771.563 2128.76,467.456 2129.61,795.337 2130.45,647.14 2131.29,489.696 2132.13,1081.65 2132.97,702.93 2133.82,1117.09 \n",
" 2134.66,849.127 2135.5,995.615 2136.34,816.923 2137.19,978.705 2138.03,623.048 2138.87,918.983 2139.71,951.019 2140.55,734.72 2141.4,745.594 2142.24,960.979 \n",
" 2143.08,334.079 2143.92,1082.68 2144.76,473.354 2145.61,729.364 2146.45,612.894 2147.29,532.42 2148.13,932.507 2148.98,526.239 2149.82,972.356 2150.66,874.454 \n",
" 2151.5,780.005 2152.34,866.3 2153.19,545.03 2154.03,645.29 2154.87,1300.39 2155.71,900.342 2156.56,694.297 2157.4,990.344 2158.24,659.967 2159.08,749.551 \n",
" 2159.92,777.26 2160.77,994.848 2161.61,943.513 2162.45,841.332 2163.29,698.694 2164.13,657.799 2164.98,260.695 2165.82,724.431 2166.66,746.365 2167.5,507.344 \n",
" 2168.35,838.292 2169.19,756.693 2170.03,644.598 2170.87,797.103 2171.71,861.804 2172.56,943.39 2173.4,926.873 2174.24,674.794 2175.08,752.092 2175.93,624.247 \n",
" 2176.77,1043.4 2177.61,632.143 2178.45,670.019 2179.29,842.106 2180.14,367.189 2180.98,543.319 2181.82,379.701 2182.66,992.641 2183.5,804.833 2184.35,812.625 \n",
" 2185.19,628.936 2186.03,1045.43 2186.87,791.56 2187.72,858.62 2188.56,1155.11 2189.4,1086.95 2190.24,817.087 2191.08,1009.76 2191.93,654.002 2192.77,771.172 \n",
" 2193.61,403.615 2194.45,820.765 2195.3,853.052 2196.14,764.791 2196.98,1142.12 2197.82,811.26 2198.66,692.588 2199.51,796.343 2200.35,728.695 2201.19,617.679 \n",
" 2202.03,904.123 2202.87,699.748 2203.72,879.313 2204.56,790.81 2205.4,504.572 2206.24,581.594 2207.09,1041.93 2207.93,350.041 2208.77,549.664 2209.61,979.036 \n",
" 2210.45,835.042 2211.3,1018.71 2212.14,941.365 2212.98,807.605 2213.82,935.361 2214.67,863.724 2215.51,761.156 2216.35,814.638 2217.19,880.143 2218.03,666.266 \n",
" 2218.88,688.809 2219.72,536.222 2220.56,745.291 2221.4,469.329 2222.24,816.747 2223.09,1257.21 2223.93,1056.73 2224.77,960.689 2225.61,421.767 2226.46,1034.97 \n",
" 2227.3,912.902 2228.14,898.204 2228.98,716.596 2229.82,907.12 2230.67,753.357 2231.51,889.542 2232.35,711.324 2233.19,972.278 2234.04,427.902 2234.88,630.131 \n",
" 2235.72,798.213 2236.56,1025.05 2237.4,600.391 2238.25,675.615 2239.09,707.949 2239.93,730.36 2240.77,1132.58 2241.61,1027.21 2242.46,1184.05 2243.3,513.38 \n",
" 2244.14,566.139 2244.98,1146.62 2245.83,861.215 2246.67,713.187 2247.51,512.767 2248.35,875.868 2249.19,934.603 2250.04,682.958 2250.88,540.025 2251.72,1021.01 \n",
" 2252.56,1007.92 2253.4,509.474 2254.25,635.009 2255.09,842.596 2255.93,880.839 2256.77,874.723 2257.62,900.42 2258.46,606.715 2259.3,1032.82 2260.14,680.421 \n",
" 2260.98,1018.57 2261.83,854.11 2262.67,890.968 2263.51,950.818 2264.35,1021.5 2265.2,1094.31 2266.04,918.251 2266.88,614.804 2267.72,729.946 2268.56,827.788 \n",
" 2269.41,523.543 2270.25,769.731 2271.09,415.568 2271.93,979.502 2272.77,838.329 2273.62,662.339 2274.46,844.082 2275.3,785.329 2276.14,483.349 2276.99,591.077 \n",
" 2277.83,592.517 2278.67,624.5 2279.51,761.165 2280.35,593.51 2281.2,872.165 2282.04,1248.48 2282.88,714.719 2283.72,788.498 2284.57,914.93 2285.41,618.684 \n",
" 2286.25,773.7 2287.09,678.212 2287.93,646.892 2288.78,561.793 2289.62,565.648 2290.46,454.239 2291.3,1258.68 2292.14,926.786 2292.99,410.653 2293.83,825.509 \n",
" 2294.67,781.259 2295.51,788.317 2296.36,892.708 2297.2,873.14 2298.04,717.877 2298.88,908.087 2299.72,666.664 2300.57,655.347 2301.41,408.193 2302.25,809.494 \n",
" 2303.09,543.855 2303.94,1167.59 2304.78,678.586 2305.62,700.161 2306.46,749.133 2307.3,874.124 2308.15,408.842 2308.99,795.368 2309.83,865.185 2310.67,1091.55 \n",
" 2311.51,684.429 2312.36,727.044 2313.2,647.266 2314.04,501.777 2314.88,626.384 2315.73,758.114 2316.57,542.839 2317.41,865.413 2318.25,613.666 2319.09,671.034 \n",
" 2319.94,824.18 2320.78,560.586 2321.62,487.636 2322.46,497.347 2323.31,718.777 2324.15,1029.87 2324.99,840.387 2325.83,745.169 2326.67,944.02 2327.52,734.766 \n",
" \n",
" \"/>\n",
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" 1486.19,1243.72 1487.03,1243.72 1487.87,1243.72 1488.71,1243.72 1489.56,1243.72 1490.4,1243.72 1491.24,1243.72 1492.08,1243.72 1492.92,1243.72 1493.77,1243.72 \n",
" 1494.61,1243.72 1495.45,1243.72 1496.29,1243.72 1497.14,1243.72 1497.98,1243.72 1498.82,1243.72 1499.66,1243.72 1500.5,1243.72 1501.35,1243.72 1502.19,1243.72 \n",
" 1503.03,1243.72 1503.87,1243.72 1504.71,1243.72 1505.56,1243.72 1506.4,1243.72 1507.24,1243.72 1508.08,1243.72 1508.93,1243.72 1509.77,1243.72 1510.61,1243.72 \n",
" 1511.45,1243.72 1512.29,1243.72 1513.14,1243.72 1513.98,1243.72 1514.82,1243.72 1515.66,1243.72 1516.51,1243.72 1517.35,1243.72 1518.19,1243.72 1519.03,1243.72 \n",
" 1519.87,1243.72 1520.72,1243.72 1521.56,1243.72 1522.4,1243.72 1523.24,1243.72 1524.08,1243.72 1524.93,1243.72 1525.77,1243.72 1526.61,1243.72 1527.45,1243.72 \n",
" 1528.3,1243.72 1529.14,1243.72 1529.98,1243.72 1530.82,1243.72 1531.66,1243.72 1532.51,1243.72 1533.35,1243.72 1534.19,1243.72 1535.03,1243.72 1535.88,1243.72 \n",
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" 1545.14,1243.72 1545.98,1243.72 1546.82,1243.72 1547.67,1243.72 1548.51,1243.72 1549.35,1243.72 1550.19,1243.72 1551.03,1243.72 1551.88,1243.72 1552.72,1243.72 \n",
" 1553.56,1243.72 1554.4,1243.72 1555.25,1243.72 1556.09,1243.72 1556.93,1243.72 1557.77,1243.72 1558.61,1243.72 1559.46,1243.72 1560.3,1243.72 1561.14,1243.72 \n",
" 1561.98,1243.72 1562.82,1243.72 1563.67,1243.72 1564.51,1243.72 1565.35,1243.72 1566.19,1243.72 1567.04,1243.72 1567.88,1243.72 1568.72,1243.72 1569.56,1243.72 \n",
" 1570.4,1243.72 1571.25,1243.72 1572.09,1243.72 1572.93,1243.72 1573.77,1243.72 1574.62,1243.72 1575.46,1243.72 1576.3,1243.72 1577.14,1243.72 1577.98,1243.72 \n",
" 1578.83,1243.72 1579.67,1243.72 1580.51,1243.72 1581.35,1243.72 1582.19,1243.72 1583.04,1243.72 1583.88,1243.72 1584.72,1243.72 1585.56,1243.72 1586.41,1243.72 \n",
" 1587.25,1243.72 1588.09,1243.72 1588.93,1243.72 1589.77,1243.72 1590.62,1243.72 1591.46,1243.72 1592.3,1243.72 1593.14,1243.72 1593.99,1243.72 1594.83,1243.72 \n",
" 1595.67,1243.72 1596.51,1243.72 1597.35,1243.72 1598.2,1243.72 1599.04,1243.72 1599.88,1243.72 1600.72,1243.72 1601.56,1243.72 1602.41,1243.72 1603.25,1243.72 \n",
" 1604.09,1243.72 1604.93,1243.72 1605.78,1243.72 1606.62,1243.72 1607.46,1243.72 1608.3,1243.72 1609.14,1243.72 1609.99,1243.72 1610.83,1243.72 1611.67,1243.72 \n",
" 1612.51,1243.72 1613.36,1243.72 1614.2,1243.72 1615.04,1243.72 1615.88,1243.72 1616.72,1243.72 1617.57,1243.72 1618.41,1243.72 1619.25,1243.72 1620.09,1243.72 \n",
" 1620.93,1243.72 1621.78,1243.72 1622.62,1243.72 1623.46,1243.72 1624.3,1243.72 1625.15,1243.72 1625.99,1243.72 1626.83,1243.72 1627.67,1243.72 1628.51,1243.72 \n",
" 1629.36,1243.72 1630.2,1243.72 1631.04,1243.72 1631.88,1243.72 1632.72,1243.72 1633.57,1243.72 1634.41,1243.72 1635.25,1243.72 1636.09,1243.72 1636.94,1243.72 \n",
" 1637.78,1243.72 1638.62,1243.72 1639.46,1243.72 1640.3,1243.72 1641.15,1243.72 1641.99,1243.72 1642.83,1243.72 1643.67,1243.72 1644.52,1243.72 1645.36,1243.72 \n",
" 1646.2,1243.72 1647.04,1243.72 1647.88,1243.72 1648.73,1243.72 1649.57,1243.72 1650.41,1243.72 1651.25,1243.72 1652.09,1243.72 1652.94,1243.72 1653.78,1243.72 \n",
" 1654.62,1243.72 1655.46,1243.72 1656.31,1243.72 1657.15,1243.72 1657.99,1243.72 1658.83,1243.72 1659.67,1243.72 1660.52,1243.72 1661.36,1243.72 1662.2,1243.72 \n",
" 1663.04,1243.72 1663.89,1243.72 1664.73,1243.72 1665.57,1243.72 1666.41,1243.72 1667.25,1243.72 1668.1,1243.72 1668.94,1243.72 1669.78,1243.72 1670.62,1243.72 \n",
" 1671.46,1243.72 1672.31,1243.72 1673.15,1243.72 1673.99,1243.72 1674.83,1243.72 1675.68,1243.72 1676.52,1243.72 1677.36,1243.72 1678.2,1243.72 1679.04,1243.72 \n",
" 1679.89,1243.72 1680.73,1243.72 1681.57,1243.72 1682.41,1243.72 1683.26,1243.72 1684.1,1243.72 1684.94,1243.72 1685.78,1243.72 1686.62,1243.72 1687.47,1243.72 \n",
" 1688.31,1243.72 1689.15,1243.72 1689.99,1243.72 1690.83,1243.72 1691.68,1243.72 1692.52,1243.72 1693.36,1243.72 1694.2,1243.72 1695.05,1243.72 1695.89,1243.72 \n",
" 1696.73,1243.72 1697.57,1243.72 1698.41,1243.72 1699.26,1243.72 1700.1,1243.72 1700.94,1243.72 1701.78,1243.72 1702.63,1243.72 1703.47,1243.72 1704.31,1243.72 \n",
" 1705.15,1243.72 1705.99,1243.72 1706.84,1243.72 1707.68,1243.72 1708.52,1243.72 1709.36,1243.72 1710.2,1243.72 1711.05,1243.72 1711.89,1243.72 1712.73,1243.72 \n",
" 1713.57,1243.72 1714.42,1243.72 1715.26,1243.72 1716.1,1243.72 1716.94,1243.72 1717.78,1243.72 1718.63,1243.72 1719.47,1243.72 1720.31,1243.72 1721.15,1243.72 \n",
" 1722,1243.72 1722.84,1243.72 1723.68,1243.72 1724.52,1243.72 1725.36,1243.72 1726.21,1243.72 1727.05,1243.72 1727.89,1243.72 1728.73,1243.72 1729.57,1243.72 \n",
" 1730.42,1243.72 1731.26,1243.72 1732.1,1243.72 1732.94,1243.72 1733.79,1243.72 1734.63,1243.72 1735.47,1243.72 1736.31,1243.72 1737.15,1243.72 1738,1243.72 \n",
" 1738.84,1243.72 1739.68,1243.72 1740.52,1243.72 1741.37,1243.72 1742.21,1243.72 1743.05,1243.72 1743.89,1243.72 1744.73,1243.72 1745.58,1243.72 1746.42,1243.72 \n",
" 1747.26,1243.72 1748.1,1243.72 1748.94,1243.72 1749.79,1243.72 1750.63,1243.72 1751.47,1243.72 1752.31,1243.72 1753.16,1243.72 1754,1243.72 1754.84,1243.72 \n",
" 1755.68,1243.72 1756.52,1243.72 1757.37,1243.72 1758.21,1243.72 1759.05,1243.72 1759.89,1243.72 1760.73,1243.72 1761.58,1243.72 1762.42,1243.72 1763.26,1243.72 \n",
" 1764.1,1243.72 1764.95,1243.72 1765.79,1243.72 1766.63,1243.72 1767.47,1243.72 1768.31,1243.72 1769.16,1243.72 1770,1243.72 1770.84,1243.72 1771.68,1243.72 \n",
" 1772.53,1243.72 1773.37,1243.72 1774.21,1243.72 1775.05,1243.72 1775.89,1243.72 1776.74,1243.72 1777.58,1243.72 1778.42,1243.72 1779.26,1243.72 1780.1,1243.72 \n",
" 1780.95,1243.72 1781.79,1243.72 1782.63,1243.72 1783.47,1243.72 1784.32,1243.72 1785.16,1243.72 1786,1243.72 1786.84,1243.72 1787.68,1243.72 1788.53,1243.72 \n",
" 1789.37,1243.72 1790.21,1243.72 1791.05,1243.72 1791.9,1243.72 1792.74,1243.72 1793.58,1243.72 1794.42,1243.72 1795.26,1243.72 1796.11,1243.72 1796.95,1243.72 \n",
" 1797.79,1243.72 1798.63,1243.72 1799.47,1243.72 1800.32,1243.72 1801.16,1243.72 1802,1243.72 1802.84,1243.72 1803.69,1243.72 1804.53,1243.72 1805.37,1243.72 \n",
" 1806.21,1243.72 1807.05,1243.72 1807.9,1243.72 1808.74,1243.72 1809.58,1243.72 1810.42,1243.72 1811.27,1243.72 1812.11,1243.72 1812.95,1243.72 1813.79,1243.72 \n",
" 1814.63,1243.72 1815.48,1243.72 1816.32,1243.72 1817.16,1243.72 1818,1243.72 1818.84,1243.72 1819.69,1243.72 1820.53,1243.72 1821.37,1243.72 1822.21,1243.72 \n",
" 1823.06,1243.72 1823.9,1243.72 1824.74,1243.72 1825.58,1243.72 1826.42,1243.72 1827.27,1243.72 1828.11,1243.72 1828.95,1243.72 1829.79,1243.72 1830.64,1243.72 \n",
" 1831.48,1243.72 1832.32,1243.72 1833.16,1243.72 1834,1243.72 1834.85,1243.72 1835.69,1243.72 1836.53,1243.72 1837.37,1243.72 1838.21,1243.72 1839.06,1243.72 \n",
" 1839.9,1243.72 1840.74,1243.72 1841.58,1243.72 1842.43,1243.72 1843.27,1243.72 1844.11,1243.72 1844.95,1243.72 1845.79,1243.72 1846.64,1243.72 1847.48,1243.72 \n",
" 1848.32,1243.72 1849.16,1243.72 1850.01,1243.72 1850.85,1243.72 1851.69,1243.72 1852.53,1243.72 1853.37,1243.72 1854.22,1243.72 1855.06,1243.72 1855.9,1243.72 \n",
" 1856.74,1243.72 1857.58,1243.72 1858.43,1243.72 1859.27,1243.72 1860.11,1243.72 1860.95,1243.72 1861.8,1243.72 1862.64,1243.72 1863.48,1243.72 1864.32,1243.72 \n",
" 1865.16,1243.72 1866.01,1243.72 1866.85,1243.72 1867.69,1243.72 1868.53,1243.72 1869.38,1243.72 1870.22,1243.72 1871.06,1243.72 1871.9,1243.72 1872.74,1243.72 \n",
" 1873.59,1243.72 1874.43,1243.72 1875.27,1243.72 1876.11,1243.72 1876.95,1243.72 1877.8,1243.72 1878.64,1243.72 1879.48,1243.72 1880.32,1243.72 1881.17,1243.72 \n",
" 1882.01,1243.72 1882.85,1243.72 1883.69,1243.72 1884.53,1243.72 1885.38,1243.72 1886.22,1243.72 1887.06,1243.72 1887.9,1243.72 1888.74,1243.72 1889.59,1243.72 \n",
" 1890.43,1243.72 1891.27,1243.72 1892.11,1243.72 1892.96,1243.72 1893.8,1243.72 1894.64,1243.72 1895.48,1243.72 1896.32,1243.72 1897.17,1243.72 1898.01,1243.72 \n",
" 1898.85,1243.72 1899.69,1243.72 1900.54,1243.72 1901.38,1243.72 1902.22,1243.72 1903.06,1243.72 1903.9,1243.72 1904.75,1243.72 1905.59,1243.72 1906.43,1243.72 \n",
" 1907.27,1243.72 1908.11,1243.72 1908.96,1243.72 1909.8,1243.72 1910.64,1243.72 1911.48,1243.72 1912.33,1243.72 1913.17,1243.72 1914.01,1243.72 1914.85,1243.72 \n",
" 1915.69,1243.72 1916.54,1243.72 1917.38,1243.72 1918.22,1243.72 1919.06,1243.72 1919.91,1243.72 1920.75,1243.72 1921.59,1243.72 1922.43,1243.72 1923.27,1243.72 \n",
" 1924.12,1243.72 1924.96,1243.72 1925.8,1243.72 1926.64,1243.72 1927.48,1243.72 1928.33,1243.72 1929.17,1243.72 1930.01,1243.72 1930.85,1243.72 1931.7,1243.72 \n",
" 1932.54,1243.72 1933.38,1243.72 1934.22,1243.72 1935.06,1243.72 1935.91,1243.72 1936.75,1243.72 1937.59,1243.72 1938.43,1243.72 1939.28,1243.72 1940.12,1243.72 \n",
" 1940.96,1243.72 1941.8,1243.72 1942.64,1243.72 1943.49,1243.72 1944.33,1243.72 1945.17,1243.72 1946.01,1243.72 1946.85,1243.72 1947.7,1243.72 1948.54,1243.72 \n",
" 1949.38,1243.72 1950.22,1243.72 1951.07,1243.72 1951.91,1243.72 1952.75,1243.72 1953.59,1243.72 1954.43,1243.72 1955.28,1243.72 1956.12,1243.72 1956.96,1243.72 \n",
" 1957.8,1243.72 1958.65,1243.72 1959.49,1243.72 1960.33,1243.72 1961.17,1243.72 1962.01,1243.72 1962.86,1243.72 1963.7,1243.72 1964.54,1243.72 1965.38,1243.72 \n",
" 1966.22,1243.72 1967.07,1243.72 1967.91,1243.72 1968.75,1243.72 1969.59,1243.72 1970.44,1243.72 1971.28,1243.72 1972.12,1243.72 1972.96,1243.72 1973.8,1243.72 \n",
" 1974.65,1243.72 1975.49,1243.72 1976.33,1243.72 1977.17,1243.72 1978.02,1243.72 1978.86,1243.72 1979.7,1243.72 1980.54,1243.72 1981.38,1243.72 1982.23,1243.72 \n",
" 1983.07,1243.72 1983.91,1243.72 1984.75,1243.72 1985.59,1243.72 1986.44,1243.72 1987.28,1243.72 1988.12,1243.72 1988.96,1243.72 1989.81,1243.72 1990.65,1243.72 \n",
" 1991.49,1243.72 1992.33,1243.72 1993.17,1243.72 1994.02,1243.72 1994.86,1243.72 1995.7,1243.72 1996.54,1243.72 1997.39,1243.72 1998.23,1243.72 1999.07,1243.72 \n",
" 1999.91,1243.72 2000.75,1243.72 2001.6,1243.72 2002.44,1243.72 2003.28,1243.72 2004.12,1243.72 2004.96,1243.72 2005.81,1243.72 2006.65,1243.72 2007.49,1243.72 \n",
" 2008.33,1243.72 2009.18,1243.72 2010.02,1243.72 2010.86,1243.72 2011.7,1243.72 2012.54,1243.72 2013.39,1243.72 2014.23,1243.72 2015.07,1243.72 2015.91,1243.72 \n",
" 2016.75,1243.72 2017.6,1243.72 2018.44,1243.72 2019.28,1243.72 2020.12,1243.72 2020.97,1243.72 2021.81,1243.72 2022.65,1243.72 2023.49,1243.72 2024.33,1243.72 \n",
" 2025.18,1243.72 2026.02,1243.72 2026.86,1243.72 2027.7,1243.72 2028.55,1243.72 2029.39,1243.72 2030.23,1243.72 2031.07,1243.72 2031.91,1243.72 2032.76,1243.72 \n",
" 2033.6,1243.72 2034.44,1243.72 2035.28,1243.72 2036.12,1243.72 2036.97,1243.72 2037.81,1243.72 2038.65,1243.72 2039.49,1243.72 2040.34,1243.72 2041.18,1243.72 \n",
" 2042.02,1243.72 2042.86,1243.72 2043.7,1243.72 2044.55,1243.72 2045.39,1243.72 2046.23,1243.72 2047.07,1243.72 2047.92,1243.72 2048.76,1243.72 2049.6,1243.72 \n",
" 2050.44,1243.72 2051.28,1243.72 2052.13,1243.72 2052.97,1243.72 2053.81,1243.72 2054.65,1243.72 2055.49,1243.72 2056.34,1243.72 2057.18,1243.72 2058.02,1243.72 \n",
" 2058.86,1243.72 2059.71,1243.72 2060.55,1243.72 2061.39,1243.72 2062.23,1243.72 2063.07,1243.72 2063.92,1243.72 2064.76,1243.72 2065.6,1243.72 2066.44,1243.72 \n",
" 2067.29,1243.72 2068.13,1243.72 2068.97,1243.72 2069.81,1243.72 2070.65,1243.72 2071.5,1243.72 2072.34,1243.72 2073.18,1243.72 2074.02,1243.72 2074.86,1243.72 \n",
" 2075.71,1243.72 2076.55,1243.72 2077.39,1243.72 2078.23,1243.72 2079.08,1243.72 2079.92,1243.72 2080.76,1243.72 2081.6,1243.72 2082.44,1243.72 2083.29,1243.72 \n",
" 2084.13,1243.72 2084.97,1243.72 2085.81,1243.72 2086.66,1243.72 2087.5,1243.72 2088.34,1243.72 2089.18,1243.72 2090.02,1243.72 2090.87,1243.72 2091.71,1243.72 \n",
" 2092.55,1243.72 2093.39,1243.72 2094.23,1243.72 2095.08,1243.72 2095.92,1243.72 2096.76,1243.72 2097.6,1243.72 2098.45,1243.72 2099.29,1243.72 2100.13,1243.72 \n",
" 2100.97,1243.72 2101.81,1243.72 2102.66,1243.72 2103.5,1243.72 2104.34,1243.72 2105.18,1243.72 2106.03,1243.72 2106.87,1243.72 2107.71,1243.72 2108.55,1243.72 \n",
" 2109.39,1243.72 2110.24,1243.72 2111.08,1243.72 2111.92,1243.72 2112.76,1243.72 2113.6,1243.72 2114.45,1243.72 2115.29,1243.72 2116.13,1243.72 2116.97,1243.72 \n",
" 2117.82,1243.72 2118.66,1243.72 2119.5,1243.72 2120.34,1243.72 2121.18,1243.72 2122.03,1243.72 2122.87,1243.72 2123.71,1243.72 2124.55,1243.72 2125.39,1243.72 \n",
" 2126.24,1243.72 2127.08,1243.72 2127.92,1243.72 2128.76,1243.72 2129.61,1243.72 2130.45,1243.72 2131.29,1243.72 2132.13,1243.72 2132.97,1243.72 2133.82,1243.72 \n",
" 2134.66,1243.72 2135.5,1243.72 2136.34,1243.72 2137.19,1243.72 2138.03,1243.72 2138.87,1243.72 2139.71,1243.72 2140.55,1243.72 2141.4,1243.72 2142.24,1243.72 \n",
" 2143.08,1243.72 2143.92,1243.72 2144.76,1243.72 2145.61,1243.72 2146.45,1243.72 2147.29,1243.72 2148.13,1243.72 2148.98,1243.72 2149.82,1243.72 2150.66,1243.72 \n",
" 2151.5,1243.72 2152.34,1243.72 2153.19,1243.72 2154.03,1243.72 2154.87,1243.72 2155.71,1243.72 2156.56,1243.72 2157.4,1243.72 2158.24,1243.72 2159.08,1243.72 \n",
" 2159.92,1243.72 2160.77,1243.72 2161.61,1243.72 2162.45,1243.72 2163.29,1243.72 2164.13,1243.72 2164.98,1243.72 2165.82,1243.72 2166.66,1243.72 2167.5,1243.72 \n",
" 2168.35,1243.72 2169.19,1243.72 2170.03,1243.72 2170.87,1243.72 2171.71,1243.72 2172.56,1243.72 2173.4,1243.72 2174.24,1243.72 2175.08,1243.72 2175.93,1243.72 \n",
" 2176.77,1243.72 2177.61,1243.72 2178.45,1243.72 2179.29,1243.72 2180.14,1243.72 2180.98,1243.72 2181.82,1243.72 2182.66,1243.72 2183.5,1243.72 2184.35,1243.72 \n",
" 2185.19,1243.72 2186.03,1243.72 2186.87,1243.72 2187.72,1243.72 2188.56,1243.72 2189.4,1243.72 2190.24,1243.72 2191.08,1243.72 2191.93,1243.72 2192.77,1243.72 \n",
" 2193.61,1243.72 2194.45,1243.72 2195.3,1243.72 2196.14,1243.72 2196.98,1243.72 2197.82,1243.72 2198.66,1243.72 2199.51,1243.72 2200.35,1243.72 2201.19,1243.72 \n",
" 2202.03,1243.72 2202.87,1243.72 2203.72,1243.72 2204.56,1243.72 2205.4,1243.72 2206.24,1243.72 2207.09,1243.72 2207.93,1243.72 2208.77,1243.72 2209.61,1243.72 \n",
" 2210.45,1243.72 2211.3,1243.72 2212.14,1243.72 2212.98,1243.72 2213.82,1243.72 2214.67,1243.72 2215.51,1243.72 2216.35,1243.72 2217.19,1243.72 2218.03,1243.72 \n",
" 2218.88,1243.72 2219.72,1243.72 2220.56,1243.72 2221.4,1243.72 2222.24,1243.72 2223.09,1243.72 2223.93,1243.72 2224.77,1243.72 2225.61,1243.72 2226.46,1243.72 \n",
" 2227.3,1243.72 2228.14,1243.72 2228.98,1243.72 2229.82,1243.72 2230.67,1243.72 2231.51,1243.72 2232.35,1243.72 2233.19,1243.72 2234.04,1243.72 2234.88,1243.72 \n",
" 2235.72,1243.72 2236.56,1243.72 2237.4,1243.72 2238.25,1243.72 2239.09,1243.72 2239.93,1243.72 2240.77,1243.72 2241.61,1243.72 2242.46,1243.72 2243.3,1243.72 \n",
" 2244.14,1243.72 2244.98,1243.72 2245.83,1243.72 2246.67,1243.72 2247.51,1243.72 2248.35,1243.72 2249.19,1243.72 2250.04,1243.72 2250.88,1243.72 2251.72,1243.72 \n",
" 2252.56,1243.72 2253.4,1243.72 2254.25,1243.72 2255.09,1243.72 2255.93,1243.72 2256.77,1243.72 2257.62,1243.72 2258.46,1243.72 2259.3,1243.72 2260.14,1243.72 \n",
" 2260.98,1243.72 2261.83,1243.72 2262.67,1243.72 2263.51,1243.72 2264.35,1243.72 2265.2,1243.72 2266.04,1243.72 2266.88,1243.72 2267.72,1243.72 2268.56,1243.72 \n",
" 2269.41,1243.72 2270.25,1243.72 2271.09,1243.72 2271.93,1243.72 2272.77,1243.72 2273.62,1243.72 2274.46,1243.72 2275.3,1243.72 2276.14,1243.72 2276.99,1243.72 \n",
" 2277.83,1243.72 2278.67,1243.72 2279.51,1243.72 2280.35,1243.72 2281.2,1243.72 2282.04,1243.72 2282.88,1243.72 2283.72,1243.72 2284.57,1243.72 2285.41,1243.72 \n",
" 2286.25,1243.72 2287.09,1243.72 2287.93,1243.72 2288.78,1243.72 2289.62,1243.72 2290.46,1243.72 2291.3,1243.72 2292.14,1243.72 2292.99,1243.72 2293.83,1243.72 \n",
" 2294.67,1243.72 2295.51,1243.72 2296.36,1243.72 2297.2,1243.72 2298.04,1243.72 2298.88,1243.72 2299.72,1243.72 2300.57,1243.72 2301.41,1243.72 2302.25,1243.72 \n",
" 2303.09,1243.72 2303.94,1243.72 2304.78,1243.72 2305.62,1243.72 2306.46,1243.72 2307.3,1243.72 2308.15,1243.72 2308.99,1243.72 2309.83,1243.72 2310.67,1243.72 \n",
" 2311.51,1243.72 2312.36,1243.72 2313.2,1243.72 2314.04,1243.72 2314.88,1243.72 2315.73,1243.72 2316.57,1243.72 2317.41,1243.72 2318.25,1243.72 2319.09,1243.72 \n",
" 2319.94,1243.72 2320.78,1243.72 2321.62,1243.72 2322.46,1243.72 2323.31,1243.72 2324.15,1243.72 2324.99,1243.72 2325.83,1243.72 2326.67,1243.72 2327.52,1243.72 \n",
" \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip7703)\" style=\"stroke:#3da44d; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1486.19,638.211 1487.03,640.899 1487.87,743.564 1488.71,734.49 1489.56,732.103 1490.4,741.073 1491.24,707.072 1492.08,723.12 1492.92,770.992 1493.77,740.552 \n",
" 1494.61,768.231 1495.45,775.376 1496.29,765.035 1497.14,756.893 1497.98,757.899 1498.82,758.336 1499.66,755.403 1500.5,756.845 1501.35,747.629 1502.19,743.991 \n",
" 1503.03,740.913 1503.87,748.792 1504.71,743.058 1505.56,763.098 1506.4,765.782 1507.24,765.38 1508.08,778.646 1508.93,772.91 1509.77,772.604 1510.61,768.401 \n",
" 1511.45,770.581 1512.29,770.818 1513.14,763.884 1513.98,766.77 1514.82,758.839 1515.66,766.406 1516.51,777.432 1517.35,771.426 1518.19,757.099 1519.03,756.094 \n",
" 1519.87,756.772 1520.72,753.684 1521.56,750.017 1522.4,754.155 1523.24,755.889 1524.08,762.294 1524.93,761.523 1525.77,757.964 1526.61,751.944 1527.45,752.444 \n",
" 1528.3,747.659 1529.14,743.258 1529.98,736.893 1530.82,740.754 1531.66,744.94 1532.51,746.984 1533.35,743.819 1534.19,735.946 1535.03,734.094 1535.88,736.671 \n",
" 1536.72,742.194 1537.56,739.939 1538.4,746.076 1539.24,748.868 1540.09,750.34 1540.93,753.188 1541.77,754.278 1542.61,757.019 1543.45,763.625 1544.3,764.637 \n",
" 1545.14,764.56 1545.98,763.515 1546.82,762.426 1547.67,766.814 1548.51,765.908 1549.35,764.958 1550.19,763.436 1551.03,764.267 1551.88,766.449 1552.72,764.279 \n",
" 1553.56,761.617 1554.4,763.436 1555.25,762.144 1556.09,762.421 1556.93,764.242 1557.77,761.617 1558.61,762.12 1559.46,763.386 1560.3,763.067 1561.14,764.937 \n",
" 1561.98,759.212 1562.82,761.126 1563.67,761.833 1564.51,762.2 1565.35,762.42 1566.19,762.589 1567.04,765.504 1567.88,763.267 1568.72,765.923 1569.56,767.033 \n",
" 1570.4,766.108 1571.25,767.188 1572.09,762.119 1572.93,763.181 1573.77,760.726 1574.62,758.962 1575.46,759.021 1576.3,761.653 1577.14,764.187 1577.98,764.265 \n",
" 1578.83,763.841 1579.67,765.005 1580.51,765.807 1581.35,764.409 1582.19,766.762 1583.04,765.551 1583.88,767.82 1584.72,768.921 1585.56,770.255 1586.41,770.588 \n",
" 1587.25,771.922 1588.09,773.048 1588.93,772.075 1589.77,771.785 1590.62,770.899 1591.46,770.121 1592.3,768.289 1593.14,768.591 1593.99,767.833 1594.83,770.299 \n",
" 1595.67,773.075 1596.51,772.789 1597.35,772.96 1598.2,775.767 1599.04,774.58 1599.88,774.983 1600.72,773.601 1601.56,775.32 1602.41,773.231 1603.25,772.042 \n",
" 1604.09,772.511 1604.93,771.916 1605.78,771.33 1606.62,772.234 1607.46,771.895 1608.3,772.649 1609.14,771.155 1609.99,773.217 1610.83,771.102 1611.67,770.454 \n",
" 1612.51,770.053 1613.36,770.539 1614.2,772.844 1615.04,774.847 1615.88,775.687 1616.72,776.803 1617.57,774.443 1618.41,775.222 1619.25,776.646 1620.09,776.904 \n",
" 1620.93,776.541 1621.78,777.515 1622.62,774.232 1623.46,775.132 1624.3,776.422 1625.15,777.635 1625.99,776.595 1626.83,776.92 1627.67,776.733 1628.51,776.406 \n",
" 1629.36,776.183 1630.2,775.749 1631.04,777.746 1631.88,776.928 1632.72,778.024 1633.57,777.22 1634.41,778.028 1635.25,780.551 1636.09,779.819 1636.94,780.623 \n",
" 1637.78,780.288 1638.62,781.23 1639.46,781.61 1640.3,781.397 1641.15,781.673 1641.99,781.993 1642.83,781.101 1643.67,779.339 1644.52,779.6 1645.36,778.073 \n",
" 1646.2,776.111 1647.04,775.101 1647.88,775.658 1648.73,776.404 1649.57,776.205 1650.41,775.623 1651.25,775.344 1652.09,774.451 1652.94,774.111 1653.78,776.299 \n",
" 1654.62,774.485 1655.46,774.227 1656.31,774.519 1657.15,773.578 1657.99,774.312 1658.83,774.351 1659.67,774.207 1660.52,775.337 1661.36,775.202 1662.2,775.357 \n",
" 1663.04,775.551 1663.89,776.372 1664.73,777.174 1665.57,777.587 1666.41,778.176 1667.25,777.961 1668.1,777.11 1668.94,777.702 1669.78,776.533 1670.62,776.115 \n",
" 1671.46,776.198 1672.31,775.985 1673.15,775.929 1673.99,776.532 1674.83,776.166 1675.68,776.317 1676.52,774.578 1677.36,774.741 1678.2,775.242 1679.04,776.061 \n",
" 1679.89,775.699 1680.73,775.005 1681.57,774.141 1682.41,773.564 1683.26,772.143 1684.1,771.025 1684.94,769.352 1685.78,768.826 1686.62,768.999 1687.47,770.624 \n",
" 1688.31,771.193 1689.15,771.263 1689.99,771.312 1690.83,772.254 1691.68,771.96 1692.52,772.69 1693.36,771.998 1694.2,770.846 1695.05,770.076 1695.89,770.704 \n",
" 1696.73,771.452 1697.57,770.075 1698.41,769.405 1699.26,768.519 1700.1,768.022 1700.94,767.082 1701.78,768.264 1702.63,770.893 1703.47,770.488 1704.31,770.369 \n",
" 1705.15,770.234 1705.99,771.175 1706.84,773.065 1707.68,773.721 1708.52,773.86 1709.36,775.403 1710.2,775.116 1711.05,775.214 1711.89,776.092 1712.73,775.707 \n",
" 1713.57,775.325 1714.42,776.27 1715.26,776.023 1716.1,776.526 1716.94,776.582 1717.78,777.644 1718.63,776.996 1719.47,775.865 1720.31,776.188 1721.15,777.284 \n",
" 1722,779.28 1722.84,778.624 1723.68,778.473 1724.52,778.163 1725.36,778.115 1726.21,778.198 1727.05,778.394 1727.89,778.621 1728.73,778.707 1729.57,778.936 \n",
" 1730.42,778.361 1731.26,778.196 1732.1,777.645 1732.94,777.382 1733.79,776.323 1734.63,775.555 1735.47,774.908 1736.31,774.363 1737.15,774.033 1738,774.155 \n",
" 1738.84,774.886 1739.68,774.176 1740.52,772.6 1741.37,773.275 1742.21,772.341 1743.05,773.107 1743.89,772.302 1744.73,773.145 1745.58,773.228 1746.42,773.575 \n",
" 1747.26,773.261 1748.1,773.094 1748.94,772.651 1749.79,771.433 1750.63,771.488 1751.47,771.628 1752.31,772.011 1753.16,772.147 1754,771.626 1754.84,770.976 \n",
" 1755.68,771.358 1756.52,770.994 1757.37,771.082 1758.21,770.775 1759.05,771.158 1759.89,771.714 1760.73,771.655 1761.58,771.16 1762.42,771.409 1763.26,770.824 \n",
" 1764.1,770.095 1764.95,770.256 1765.79,770.211 1766.63,770.478 1767.47,770.671 1768.31,770.21 1769.16,770.572 1770,770.423 1770.84,771.203 1771.68,771.272 \n",
" 1772.53,771.46 1773.37,770.491 1774.21,769.517 1775.05,770.703 1775.89,770.917 1776.74,769.753 1777.58,769.769 1778.42,770.424 1779.26,769.763 1780.1,769.509 \n",
" 1780.95,769.004 1781.79,769.425 1782.63,769.895 1783.47,770.08 1784.32,771.045 1785.16,770.38 1786,770.218 1786.84,770.033 1787.68,770.545 1788.53,770.865 \n",
" 1789.37,769.899 1790.21,769.374 1791.05,769.51 1791.9,769.343 1792.74,769.077 1793.58,769.549 1794.42,769.43 1795.26,769.277 1796.11,768.5 1796.95,768.227 \n",
" 1797.79,768.967 1798.63,768.348 1799.47,767.988 1800.32,768.205 1801.16,768.104 1802,766.978 1802.84,766.547 1803.69,766.968 1804.53,766.674 1805.37,766.359 \n",
" 1806.21,765.982 1807.05,766.806 1807.9,766.823 1808.74,766.479 1809.58,766.368 1810.42,766.013 1811.27,765.074 1812.11,765.593 1812.95,765.313 1813.79,765.502 \n",
" 1814.63,765.504 1815.48,764.996 1816.32,765.081 1817.16,764.578 1818,763.772 1818.84,764.418 1819.69,764.124 1820.53,764.188 1821.37,764.075 1822.21,763.882 \n",
" 1823.06,764.003 1823.9,763.797 1824.74,763.228 1825.58,763.179 1826.42,761.881 1827.27,761.543 1828.11,760.933 1828.95,760.972 1829.79,761.173 1830.64,760.455 \n",
" 1831.48,760.683 1832.32,760.21 1833.16,759.597 1834,759.177 1834.85,759.359 1835.69,758.967 1836.53,758.795 1837.37,758.809 1838.21,759.794 1839.06,759.697 \n",
" 1839.9,759.53 1840.74,759.769 1841.58,759.564 1842.43,759.344 1843.27,759.48 1844.11,758.989 1844.95,759.287 1845.79,758.906 1846.64,758.259 1847.48,758.379 \n",
" 1848.32,758.922 1849.16,758.692 1850.01,759.077 1850.85,758.796 1851.69,757.957 1852.53,757.94 1853.37,758.203 1854.22,758.075 1855.06,757.928 1855.9,757.283 \n",
" 1856.74,756.982 1857.58,757.241 1858.43,757.646 1859.27,757.94 1860.11,757.189 1860.95,756.59 1861.8,757.226 1862.64,757.694 1863.48,757.578 1864.32,758.154 \n",
" 1865.16,757.748 1866.01,757.35 1866.85,757.747 1867.69,757.293 1868.53,756.927 1869.38,756.777 1870.22,756.91 1871.06,756.305 1871.9,755.766 1872.74,755.836 \n",
" 1873.59,756.264 1874.43,755.162 1875.27,755.905 1876.11,756.882 1876.95,757.151 1877.8,757.097 1878.64,758.107 1879.48,758.186 1880.32,757.94 1881.17,758.34 \n",
" 1882.01,758.81 1882.85,758.497 1883.69,758.783 1884.53,758.592 1885.38,758.357 1886.22,758.248 1887.06,758.916 1887.9,759.061 1888.74,759.738 1889.59,759.923 \n",
" 1890.43,761.016 1891.27,761.658 1892.11,761.403 1892.96,760.995 1893.8,761.401 1894.64,761.62 1895.48,761.73 1896.32,761.527 1897.17,761.969 1898.01,762.591 \n",
" 1898.85,762.672 1899.69,763.203 1900.54,762.657 1901.38,762.204 1902.22,762.457 1903.06,762.902 1903.9,763.292 1904.75,763.434 1905.59,763.475 1906.43,763.471 \n",
" 1907.27,763.201 1908.11,763.453 1908.96,764.113 1909.8,764.57 1910.64,764.756 1911.48,765.263 1912.33,765.547 1913.17,765.498 1914.01,765.115 1914.85,764.837 \n",
" 1915.69,764.459 1916.54,764.146 1917.38,764.278 1918.22,764.097 1919.06,764.533 1919.91,764.607 1920.75,764.686 1921.59,764.313 1922.43,763.652 1923.27,763.982 \n",
" 1924.12,763.881 1924.96,763.599 1925.8,763.57 1926.64,763.965 1927.48,763.641 1928.33,763.571 1929.17,764.104 1930.01,764.227 1930.85,763.882 1931.7,763.462 \n",
" 1932.54,763.095 1933.38,762.885 1934.22,762.772 1935.06,763.364 1935.91,763.28 1936.75,762.778 1937.59,762.672 1938.43,762.763 1939.28,762.712 1940.12,762.417 \n",
" 1940.96,762.077 1941.8,762.436 1942.64,762.429 1943.49,762.417 1944.33,762.051 1945.17,762.286 1946.01,762.693 1946.85,763.098 1947.7,763.306 1948.54,762.931 \n",
" 1949.38,763.15 1950.22,764.047 1951.07,764.183 1951.91,764.705 1952.75,764.778 1953.59,764.716 1954.43,764.704 1955.28,765.447 1956.12,765.404 1956.96,765.105 \n",
" 1957.8,764.802 1958.65,764.409 1959.49,764.588 1960.33,764.772 1961.17,764.819 1962.01,764.989 1962.86,764.476 1963.7,764.731 1964.54,765.286 1965.38,766.063 \n",
" 1966.22,765.315 1967.07,766.06 1967.91,765.972 1968.75,766.366 1969.59,765.837 1970.44,765.217 1971.28,765.115 1972.12,764.951 1972.96,765.052 1973.8,765.103 \n",
" 1974.65,764.948 1975.49,764.624 1976.33,765.14 1977.17,765.515 1978.02,766.043 1978.86,765.849 1979.7,765.951 1980.54,766.198 1981.38,766.392 1982.23,766.593 \n",
" 1983.07,766.664 1983.91,767.045 1984.75,767.699 1985.59,767.871 1986.44,767.49 1987.28,767.697 1988.12,768.185 1988.96,767.613 1989.81,767.885 1990.65,767.689 \n",
" 1991.49,767.884 1992.33,768.068 1993.17,767.916 1994.02,767.769 1994.86,768.042 1995.7,767.806 1996.54,767.901 1997.39,768.275 1998.23,768.366 1999.07,768.483 \n",
" 1999.91,768.444 2000.75,768.205 2001.6,767.781 2002.44,767.724 2003.28,767.908 2004.12,767.798 2004.96,767.584 2005.81,767.64 2006.65,767.846 2007.49,767.628 \n",
" 2008.33,767.506 2009.18,768.133 2010.02,768.177 2010.86,768.1 2011.7,767.922 2012.54,767.814 2013.39,767.672 2014.23,767.709 2015.07,767.497 2015.91,768.091 \n",
" 2016.75,767.514 2017.6,767.49 2018.44,767.624 2019.28,767.535 2020.12,767.221 2020.97,767.888 2021.81,768.273 2022.65,767.95 2023.49,767.588 2024.33,767.518 \n",
" 2025.18,767.603 2026.02,767.734 2026.86,767.764 2027.7,767.692 2028.55,767.821 2029.39,768.246 2030.23,768.199 2031.07,768.427 2031.91,768.037 2032.76,767.171 \n",
" 2033.6,767.176 2034.44,767.87 2035.28,767.641 2036.12,767.289 2036.97,767.109 2037.81,767.207 2038.65,767.69 2039.49,767.754 2040.34,767.99 2041.18,767.963 \n",
" 2042.02,767.891 2042.86,767.698 2043.7,767.866 2044.55,767.808 2045.39,768.649 2046.23,768.379 2047.07,768.096 2047.92,767.834 2048.76,767.925 2049.6,768.007 \n",
" 2050.44,767.706 2051.28,767.978 2052.13,767.993 2052.97,767.843 2053.81,768.085 2054.65,768.036 2055.49,767.642 2056.34,767.237 2057.18,766.236 2058.02,766.164 \n",
" 2058.86,765.908 2059.71,766.26 2060.55,766.354 2061.39,766.105 2062.23,766.427 2063.07,766.602 2063.92,766.251 2064.76,766.022 2065.6,766.163 2066.44,766.382 \n",
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" 2159.92,764.689 2160.77,764.976 2161.61,765.199 2162.45,765.293 2163.29,765.211 2164.13,765.077 2164.98,764.452 2165.82,764.403 2166.66,764.38 2167.5,764.063 \n",
" 2168.35,764.155 2169.19,764.145 2170.03,763.998 2170.87,764.039 2171.71,764.159 2172.56,764.379 2173.4,764.578 2174.24,764.468 2175.08,764.453 2175.93,764.282 \n",
" 2176.77,764.622 2177.61,764.46 2178.45,764.346 2179.29,764.44 2180.14,763.959 2180.98,763.691 2181.82,763.227 2182.66,763.504 2183.5,763.554 2184.35,763.613 \n",
" 2185.19,763.451 2186.03,763.79 2186.87,763.823 2187.72,763.937 2188.56,764.406 2189.4,764.791 2190.24,764.854 2191.08,765.146 2191.93,765.014 2192.77,765.021 \n",
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" 2202.03,765.162 2202.87,765.085 2203.72,765.219 2204.56,765.249 2205.4,764.944 2206.24,764.73 2207.09,765.053 2207.93,764.569 2208.77,764.319 2209.61,764.569 \n",
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" 2294.67,767.437 2295.51,767.459 2296.36,767.589 2297.2,767.698 2298.04,767.647 2298.88,767.792 2299.72,767.688 2300.57,767.571 2301.41,767.201 2302.25,767.244 \n",
" 2303.09,767.014 2303.94,767.426 2304.78,767.335 2305.62,767.266 2306.46,767.247 2307.3,767.357 2308.15,766.99 2308.99,767.019 2309.83,767.119 2310.67,767.45 \n",
" 2311.51,767.366 2312.36,767.325 2313.2,767.202 2314.04,766.933 2314.88,766.79 2315.73,766.781 2316.57,766.554 2317.41,766.654 2318.25,766.5 2319.09,766.403 \n",
" 2319.94,766.462 2320.78,766.254 2321.62,765.973 2322.46,765.703 2323.31,765.656 2324.15,765.921 2324.99,765.996 2325.83,765.975 2326.67,766.153 2327.52,766.122 \n",
" \n",
" \"/>\n",
"<path clip-path=\"url(#clip7700)\" d=\"\n",
"M1746.47 372.684 L2280.76 372.684 L2280.76 130.764 L1746.47 130.764 Z\n",
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"<polyline clip-path=\"url(#clip7700)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
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" \"/>\n",
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"</g>\n",
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]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p1 = plot(params[3, :], linewidth=0.2, label = \"EM samples\")\n",
"plot!(Ptrue.σ1*ones(iterations), label = LaTeXString(\"true \\\\sigma_1\"))\n",
"plot!(Bridge.runmean(params[3, :]), label = \"running mean\")\n",
"p2 = plot(params[4, :], linewidth=0.2, label = \"EM samples\")\n",
"plot!(Ptrue.σ2*ones(iterations), label = LaTeXString(\"true \\\\sigma_2\"))\n",
"plot!(Bridge.runmean(params[4, :]), label = \"running mean\")\n",
"plot(p1,p2)\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Reference\n",
"\n",
"* van der Meulen, S., van Zanten: Reversible jump MCMC for nonparametric drift estimation for diffusion processes. *Computational Statistics & Data Analysis 71*, 2014, ISSN 0167-9473, https://doi.org/10.1016/j.csda.2013.03.002 ."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 2020 1.1.1",
"language": "julia",
"name": "julia-2020-1.1"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.1.1"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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