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Gauss–Markov process

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C. B. Mehr and J. A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522
299: 705:{\displaystyle {\textbf {S}}_{x}(s)={\frac {2\sigma ^{2}\beta }{-s^{2}+\beta ^{2}}}={\frac {{\sqrt {2\beta }}\,\sigma }{(s+\beta )}}\cdot {\frac {{\sqrt {2\beta }}\,\sigma }{(-s+\beta )}}.} 913: 437: 1448: 332: 1272: 827: 1875: 868: 1405: 1385: 1789: 345: 801: 1706: 1716: 1390: 222:
Property (3) means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).
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If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function
2020: 2002: 1982: 1977: 1696: 1528: 1508: 1355: 1298: 1137: 1047: 1488: 2095: 2050: 2040: 1781: 1726: 1701: 1670: 1650: 1410: 1395: 1262: 38: 2090: 1930: 1855: 1660: 1420: 1330: 1220: 431: 57:. A stationary Gauss–Markov process is unique up to rescaling; such a process is also known as an 2060: 2025: 1940: 1910: 1680: 1675: 1498: 1335: 1000: 938: 877: 821: 523:{\displaystyle {\textbf {S}}_{x}(j\omega )={\frac {2\sigma ^{2}\beta }{\omega ^{2}+\beta ^{2}}}.} 307: 46: 1741: 2080: 1885: 1536: 1293: 1210: 1179: 1072: 1052: 1042: 898: 893: 807: 780: 776: 747: 65: 1746: 1483: 2100: 1987: 1870: 1240: 1215: 1164: 1092: 1015: 968: 768: 427: 50: 2065: 1965: 1950: 1711: 1645: 1323: 1267: 1250: 995: 339: 1880: 1112: 2070: 2035: 1955: 1561: 1308: 1225: 1194: 1189: 1169: 1159: 1102: 1097: 1077: 1057: 1022: 990: 973: 738: 215: 54: 2133: 1972: 1513: 1350: 1345: 1303: 1245: 1067: 983: 923: 769: 713: 302: 42: 17: 2030: 1992: 1546: 1478: 1367: 1362: 1174: 1107: 1082: 918: 1610: 2075: 1594: 1589: 1584: 1574: 1377: 1318: 1313: 1277: 1037: 928: 2085: 1625: 1569: 1453: 530:(Note that the Cauchy distribution and this spectrum differ by scale factors.) 1579: 237: 1406:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
846: 417:{\displaystyle {\textbf {R}}_{x}(\tau )=\sigma ^{2}e^{-\beta |\tau |}.} 850: 720:
There are also some trivial exceptions to all of the above.
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Autoregressive conditional heteroskedasticity (ARCH) model
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Ornstein–Uhlenbeck process § Mathematical properties
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Independent and identically distributed random variables
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Bob Schutz, Byron Tapley, George H. Born (2004-06-26).
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3D-Position Tracking and Control for All-Terrain Robots
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Autoregressive integrated moving average (ARIMA) model
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The above yields the following spectral factorization:
539: 440: 348: 310: 245: 2013: 1818: 1780: 1689: 1603: 1560: 1527: 1419: 1376: 1286: 1203: 959: 884: 294:{\displaystyle {\textbf {E}}(X^{2}(t))=\sigma ^{2}} 704: 522: 416: 326: 293: 1273:Stochastic chains with memory of variable length 737:C. E. Rasmussen & C. K. I. Williams (2006). 430:(PSD) function that has the same shape as the 862: 8: 826:: CS1 maint: multiple names: authors list ( 170:) and a strictly increasing scalar function 84:) possesses the three following properties: 1401:Autoregressive–moving-average (ARMA) model 869: 855: 847: 672: 662: 659: 632: 622: 619: 607: 594: 576: 566: 548: 542: 541: 538: 508: 495: 480: 470: 449: 443: 442: 439: 404: 396: 389: 379: 357: 351: 350: 347: 315: 309: 285: 260: 247: 246: 244: 135:) is a non-decreasing scalar function of 740:Gaussian Processes for Machine Learning 729: 236:A stationary Gauss–Markov process with 49:that satisfy the requirements for both 1707:Doob's martingale convergence theorems 819: 1459:Constant elasticity of variance (CEV) 1449:Chan–Karolyi–Longstaff–Sanders (CKLS) 7: 543: 444: 352: 248: 96:) is a non-zero scalar function of 1946:Skorokhod's representation theorem 1727:Law of large numbers (weak/strong) 25: 1916:Martingale representation theorem 746:. MIT Press. p. Appendix B. 159:)) is also a Gauss–Markov process 35:Gauss–Markov stochastic processes 1961:Stochastic differential equation 1851:Doob's optional stopping theorem 1846:Doob–Meyer decomposition theorem 124:) is also a Gauss–Markov process 1831:Convergence of random variables 1717:Fisher–Tippett–Gnedenko theorem 803:Statistical Orbit Determination 1429:Binomial options pricing model 693: 678: 650: 638: 560: 554: 464: 455: 405: 397: 369: 363: 334:has the following properties. 275: 272: 266: 253: 1: 1896:Kolmogorov continuity theorem 1732:Law of the iterated logarithm 1901:Kolmogorov extension theorem 1580:Generalized queueing network 1088:Interacting particle systems 64:Gauss–Markov processes obey 27:Not to be confused with the 1033:Continuous-time random walk 327:{\displaystyle \beta ^{-1}} 76:Every Gauss–Markov process 31:of mathematical statistics. 2156: 2041:Extreme value theory (EVT) 1841:Doob decomposition theorem 1133:Ornstein–Uhlenbeck process 904:Chinese restaurant process 229: 59:Ornstein–Uhlenbeck process 26: 2109: 1921:Optional stopping theorem 1722:Large deviation principle 1474:Heath–Jarrow–Morton (HJM) 1411:Moving-average (MA) model 1396:Autoregressive (AR) model 1221:Hidden Markov model (HMM) 1155:Schramm–Loewner evolution 1836:Doléans-Dade exponential 1666:Progressively measurable 1464:Cox–Ingersoll–Ross (CIR) 2056:Mathematical statistics 2046:Large deviations theory 1876:Infinitesimal generator 1737:Maximal ergodic theorem 1656:Piecewise-deterministic 1258:Random dynamical system 1123:Markov additive process 1891:Karhunen–Loève theorem 1826:Cameron–Martin formula 1790:Burkholder–Davis–Gundy 1185:Variance gamma process 767:Lamon, Pierre (2008). 712:which is important in 706: 524: 418: 328: 295: 2021:Actuarial mathematics 1983:Uniform integrability 1978:Stratonovich integral 1906:Lévy–Prokhorov metric 1810:Marcinkiewicz–Zygmund 1697:Central limit theorem 1299:Gaussian random field 1128:McKean–Vlasov process 1048:Dyson Brownian motion 909:Galton–Watson process 775:. Springer. pp.  707: 525: 419: 329: 296: 2096:Time series analysis 2051:Mathematical finance 1936:Reflection principle 1263:Regenerative process 1063:Fleming–Viot process 878:Stochastic processes 537: 438: 346: 308: 243: 47:stochastic processes 39:Carl Friedrich Gauss 29:Gauss–Markov theorem 18:Gauss-Markov process 2091:Stochastic analysis 1931:Quadratic variation 1926:Prokhorov's theorem 1861:Feynman–Kac formula 1331:Markov random field 979:Birth–death process 432:Cauchy distribution 2061:Probability theory 1941:Skorokhod integral 1911:Malliavin calculus 1494:Korn-Kreer-Lenssen 1378:Time series models 1341:Pitman–Yor process 702: 520: 414: 324: 291: 214:) is the standard 66:Langevin equations 51:Gaussian processes 2127: 2126: 2081:Signal processing 1800:Doob's upcrossing 1795:Doob's martingale 1759:Engelbert–Schmidt 1702:Donsker's theorem 1636:Feller-continuous 1504:Rendleman–Bartter 1294:Dirichlet process 1211:Branching process 1180:Telegraph process 1073:Geometric process 1053:Empirical process 1043:Diffusion process 899:Branching process 894:Bernoulli process 813:978-0-08-054173-0 786:978-3-540-78286-5 697: 670: 654: 630: 614: 545: 515: 446: 354: 250: 16:(Redirected from 2147: 2140:Markov processes 2101:Machine learning 1988:Usual hypotheses 1871:Girsanov theorem 1856:Dynkin's formula 1621:Continuous paths 1529:Actuarial models 1469:Garman–Kohlhagen 1439:Black–Karasinski 1434:Black–Derman–Toy 1421:Financial models 1287:Fields and other 1216:Gaussian process 1165:Sigma-martingale 969:Additive process 871: 864: 857: 848: 841: 838: 832: 831: 825: 817: 797: 791: 790: 774: 764: 758: 757: 745: 734: 716:and other areas. 714:Wiener filtering 711: 709: 708: 703: 698: 696: 676: 671: 663: 660: 655: 653: 636: 631: 623: 620: 615: 613: 612: 611: 599: 598: 585: 581: 580: 567: 553: 552: 547: 546: 529: 527: 526: 521: 516: 514: 513: 512: 500: 499: 489: 485: 484: 471: 454: 453: 448: 447: 428:spectral density 423: 421: 420: 415: 410: 409: 408: 400: 384: 383: 362: 361: 356: 355: 333: 331: 330: 325: 323: 322: 300: 298: 297: 292: 290: 289: 265: 264: 252: 251: 226:Other properties 72:Basic properties 55:Markov processes 21: 2155: 2154: 2150: 2149: 2148: 2146: 2145: 2144: 2130: 2129: 2128: 2123: 2105: 2066:Queueing theory 2009: 1951:Skorokhod space 1814: 1805:Kunita–Watanabe 1776: 1742:Sanov's theorem 1712:Ergodic theorem 1685: 1681:Time-reversible 1599: 1562:Queueing models 1556: 1552:Sparre–Anderson 1542:Cramér–Lundberg 1523: 1509:SABR volatility 1415: 1372: 1324:Boolean network 1282: 1268:Renewal process 1199: 1148:Non-homogeneous 1138:Poisson process 1028:Contact process 991:Brownian motion 961:Continuous time 955: 949:Maximal entropy 880: 875: 845: 844: 839: 835: 818: 814: 806:. p. 230. 799: 798: 794: 787: 766: 765: 761: 754: 743: 736: 735: 731: 726: 677: 661: 637: 621: 603: 590: 586: 572: 568: 540: 535: 534: 504: 491: 490: 476: 472: 441: 436: 435: 385: 375: 349: 344: 343: 340:autocorrelation 311: 306: 305: 281: 256: 241: 240: 234: 228: 74: 32: 23: 22: 15: 12: 11: 5: 2153: 2151: 2143: 2142: 2132: 2131: 2125: 2124: 2122: 2121: 2116: 2114:List of topics 2110: 2107: 2106: 2104: 2103: 2098: 2093: 2088: 2083: 2078: 2073: 2071:Renewal theory 2068: 2063: 2058: 2053: 2048: 2043: 2038: 2036:Ergodic theory 2033: 2028: 2026:Control theory 2023: 2017: 2015: 2011: 2010: 2008: 2007: 2006: 2005: 2000: 1990: 1985: 1980: 1975: 1970: 1969: 1968: 1958: 1956:Snell envelope 1953: 1948: 1943: 1938: 1933: 1928: 1923: 1918: 1913: 1908: 1903: 1898: 1893: 1888: 1883: 1878: 1873: 1868: 1863: 1858: 1853: 1848: 1843: 1838: 1833: 1828: 1822: 1820: 1816: 1815: 1813: 1812: 1807: 1802: 1797: 1792: 1786: 1784: 1778: 1777: 1775: 1774: 1755:Borel–Cantelli 1744: 1739: 1734: 1729: 1724: 1719: 1714: 1709: 1704: 1699: 1693: 1691: 1690:Limit theorems 1687: 1686: 1684: 1683: 1678: 1673: 1668: 1663: 1658: 1653: 1648: 1643: 1638: 1633: 1628: 1623: 1618: 1613: 1607: 1605: 1601: 1600: 1598: 1597: 1592: 1587: 1582: 1577: 1572: 1566: 1564: 1558: 1557: 1555: 1554: 1549: 1544: 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2012: 2004: 2001: 1999: 1996: 1995: 1994: 1991: 1989: 1986: 1984: 1981: 1979: 1976: 1974: 1973:Stopping time 1971: 1967: 1964: 1963: 1962: 1959: 1957: 1954: 1952: 1949: 1947: 1944: 1942: 1939: 1937: 1934: 1932: 1929: 1927: 1924: 1922: 1919: 1917: 1914: 1912: 1909: 1907: 1904: 1902: 1899: 1897: 1894: 1892: 1889: 1887: 1884: 1882: 1879: 1877: 1874: 1872: 1869: 1867: 1864: 1862: 1859: 1857: 1854: 1852: 1849: 1847: 1844: 1842: 1839: 1837: 1834: 1832: 1829: 1827: 1824: 1823: 1821: 1817: 1811: 1808: 1806: 1803: 1801: 1798: 1796: 1793: 1791: 1788: 1787: 1785: 1783: 1779: 1772: 1768: 1764: 1763:Hewitt–Savage 1760: 1756: 1752: 1748: 1747:Zero–one laws 1745: 1743: 1740: 1738: 1735: 1733: 1730: 1728: 1725: 1723: 1720: 1718: 1715: 1713: 1710: 1708: 1705: 1703: 1700: 1698: 1695: 1694: 1692: 1688: 1682: 1679: 1677: 1674: 1672: 1669: 1667: 1664: 1662: 1659: 1657: 1654: 1652: 1649: 1647: 1644: 1642: 1639: 1637: 1634: 1632: 1629: 1627: 1624: 1622: 1619: 1617: 1614: 1612: 1609: 1608: 1606: 1602: 1596: 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1183: 1181: 1178: 1176: 1173: 1171: 1168: 1166: 1163: 1161: 1158: 1156: 1153: 1149: 1146: 1144: 1141: 1140: 1139: 1136: 1134: 1131: 1129: 1126: 1124: 1121: 1119: 1116: 1114: 1111: 1109: 1106: 1104: 1101: 1099: 1096: 1094: 1093:Itô diffusion 1091: 1089: 1086: 1084: 1081: 1079: 1076: 1074: 1071: 1069: 1068:Gamma process 1066: 1064: 1061: 1059: 1056: 1054: 1051: 1049: 1046: 1044: 1041: 1039: 1036: 1034: 1031: 1029: 1026: 1024: 1021: 1017: 1014: 1012: 1009: 1007: 1004: 1002: 999: 997: 994: 993: 992: 989: 985: 982: 981: 980: 977: 975: 972: 970: 967: 966: 964: 962: 958: 950: 947: 945: 942: 940: 939:Self-avoiding 937: 935: 932: 931: 930: 927: 925: 924:Moran process 922: 920: 917: 915: 912: 910: 907: 905: 902: 900: 897: 895: 892: 891: 889: 887: 886:Discrete time 883: 879: 872: 867: 865: 860: 858: 853: 852: 849: 837: 834: 829: 823: 815: 809: 805: 804: 796: 793: 788: 782: 778: 773: 772: 763: 760: 755: 749: 742: 741: 733: 730: 723: 721: 715: 699: 690: 687: 684: 681: 673: 667: 664: 656: 647: 644: 641: 633: 627: 624: 616: 608: 604: 600: 595: 591: 587: 582: 577: 573: 569: 563: 557: 549: 532: 517: 509: 505: 501: 496: 492: 486: 481: 477: 473: 467: 461: 458: 450: 433: 429: 425: 411: 401: 393: 390: 386: 380: 376: 372: 366: 358: 341: 337: 336: 335: 319: 316: 312: 304: 303:time constant 286: 282: 278: 269: 261: 257: 239: 233: 225: 223: 217: 213: 209: 205: 201: 197: 193: 189: 185: 181: 177: 173: 169: 165: 161: 158: 154: 150: 146: 142: 138: 134: 130: 126: 123: 119: 115: 111: 107: 103: 99: 95: 91: 87: 86: 85: 83: 79: 71: 69: 67: 62: 60: 56: 52: 48: 44: 43:Andrey Markov 40: 37:(named after 36: 30: 19: 2031:Econometrics 1993:Wiener space 1881:Itô integral 1782:Inequalities 1671:Self-similar 1641:Gauss–Markov 1640: 1631:Exchangeable 1611:Càdlàg paths 1547:Risk process 1499:LIBOR market 1368:Random graph 1363:Random field 1175:Superprocess 1113:Lévy process 1108:Jump process 1083:Hunt process 919:Markov chain 836: 802: 795: 770: 762: 739: 732: 719: 338:Exponential 235: 221: 211: 207: 203: 199: 195: 191: 187: 183: 179: 178:) such that 175: 171: 167: 163: 156: 152: 148: 144: 140: 136: 132: 128: 121: 117: 113: 109: 105: 101: 97: 93: 89: 81: 77: 75: 63: 34: 33: 2076:Ruin theory 2014:Disciplines 1886:Itô's lemma 1661:Predictable 1336:Percolation 1319:Potts model 1314:Ising model 1278:White noise 1236:Differences 1098:Itô process 1038:Cox process 934:Loop-erased 929:Random walk 2086:Statistics 1866:Filtration 1767:Kolmogorov 1751:Blumenthal 1676:Stationary 1616:Continuous 1604:Properties 1489:Hull–White 1231:Martingale 1118:Local time 1006:Fractional 984:pure birth 753:026218253X 724:References 206:)), where 1998:Classical 1011:Geometric 1001:Excursion 822:cite book 691:β 682:− 674:σ 668:β 657:⋅ 648:β 634:σ 628:β 605:β 588:− 583:β 574:σ 506:β 493:ω 487:β 478:σ 462:ω 402:τ 394:β 391:− 377:σ 367:τ 317:− 313:β 283:σ 2134:Category 2119:Category 2003:Abstract 1537:Bühlmann 1143:Compound 426:A power 238:variance 1626:Ergodic 1514:Vašíček 1356:Poisson 1016:Meander 139:, then 100:, then 1966:Tanaka 1651:Mixing 1646:Markov 1519:Wilkie 1484:Ho–Lee 1479:Heston 1251:Super- 996:Bridge 944:Biased 810:  783:  750:  45:) are 1819:Tools 1595:M/M/c 1590:M/M/1 1585:M/G/1 1575:Fluid 1241:Local 779:–95. 744:(PDF) 1771:Lévy 1570:Bulk 1454:Chen 1246:Sub- 1204:Both 828:link 808:ISBN 781:ISBN 748:ISBN 301:and 186:) = 147:) = 108:) = 53:and 41:and 1351:Cox 127:If 88:If 2136:: 1769:, 1765:, 1761:, 1757:, 1753:, 824:}} 820:{{ 777:93 434:: 342:: 68:. 61:. 1773:) 1749:( 870:e 863:t 856:v 830:) 816:. 789:. 756:. 700:. 694:) 688:+ 685:s 679:( 665:2 651:) 645:+ 642:s 639:( 625:2 617:= 609:2 601:+ 596:2 592:s 578:2 570:2 564:= 561:) 558:s 555:( 550:x 544:S 518:. 510:2 502:+ 497:2 482:2 474:2 468:= 465:) 459:j 456:( 451:x 445:S 412:. 406:| 398:| 387:e 381:2 373:= 370:) 364:( 359:x 353:R 320:1 287:2 279:= 276:) 273:) 270:t 267:( 262:2 258:X 254:( 249:E 218:. 212:t 210:( 208:W 204:t 202:( 200:f 198:( 196:W 194:) 192:t 190:( 188:h 184:t 182:( 180:X 176:t 174:( 172:f 168:t 166:( 164:h 157:t 155:( 153:f 151:( 149:X 145:t 143:( 141:Z 137:t 133:t 131:( 129:f 122:t 120:( 118:X 116:) 114:t 112:( 110:h 106:t 104:( 102:Z 98:t 94:t 92:( 90:h 82:t 80:( 78:X 20:)

Index

Gauss-Markov process
Gauss–Markov theorem
Carl Friedrich Gauss
Andrey Markov
stochastic processes
Gaussian processes
Markov processes
Ornstein–Uhlenbeck process
Langevin equations
Wiener process
Ornstein–Uhlenbeck process § Mathematical properties
variance
time constant
autocorrelation
spectral density
Cauchy distribution
Wiener filtering
Gaussian Processes for Machine Learning
ISBN
026218253X
3D-Position Tracking and Control for All-Terrain Robots
93
ISBN
978-3-540-78286-5
Statistical Orbit Determination
ISBN
978-0-08-054173-0
cite book
link
v

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