312:
1374:
868:
1072:
702:
590:
174:
1205:
1235:
734:
487:
967:
1379:
This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean,
133:
607:
1219:
variation of the residence time. It is proportional to the natural log of a normalized residence time. Noting the exponential in
Equation (
506:
307:{\displaystyle \tau (y_{0})=\inf\{t\geq t_{0}:y(t)\in \{y_{\operatorname {avg} }-y_{\min },\ y_{\operatorname {avg} }+y_{\max }\}\},}
1369:{\displaystyle {\hat {\mu }}=\ln \left(N_{0}{\bar {\tau }}\right)={\frac {\min(y_{\min },\ y_{\max })^{2}}{2\sigma _{y}^{2}}}.}
1115:
863:{\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df}}{\int _{0}^{\infty }{\Phi _{y}(f)\,df}}}},}
1417:
can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.
400:
1625:
1426:
1615:
1620:
1446:
1575:; Kabamba, Pierre T.; Girard, Anouck R. (2014). "Safety Margins for Flight Through Stochastic Gusts".
1436:
1441:
497:
1566:. Proceedings of 26th Conference on Decision and Control. Los Angeles: IEEE. pp. 1734–1739.
1216:
888:
148:
1531:, p. 495, an alternate approach to defining the logarithmic residence time and computing
1431:
1067:{\displaystyle \tau (y_{0})=\inf\{t\geq t_{0}:y(t)\in \partial \Psi \mid y_{0}\in \Psi \}.}
318:
8:
1557:. Proceedings of 25th Conference on Decision and Control. Athens: IEEE. pp. 494–498.
49:
26:
341:
is equal to one of the critical values forming the boundary of the interval, assuming
1592:
1584:
493:
376:
1476:
1474:
496:
and a boundary far from the mean, the residence time equals the inverse of the
1609:
1510:
1471:
1597:
1572:
29:
to reach a certain boundary value, usually a boundary far from the mean.
1588:
697:{\displaystyle N(y_{\max })=N_{0}e^{-y_{\max }^{2}/2\sigma _{y}^{2}},}
1459:
585:{\displaystyle {\bar {\tau }}=N^{-1}(\min(y_{\min },\ y_{\max })),}
1498:
1092:
rather than being equal to one of two discrete values, assuming
1522:
1486:
1570:
1516:
1480:
364:
proceeds randomly from its initial value to the boundary,
1200:{\displaystyle {\bar {\tau }}(y_{0})=\operatorname {E} .}
1077:
In this case, this infimum is the smallest time at which
898:
1238:
1118:
970:
737:
610:
509:
403:
177:
321:. This is the smallest time after the initial time
1368:
1199:
1066:
862:
696:
584:
481:
306:
1561:
1552:
1528:
1504:
1492:
1465:
946:. In this case, define the first passage time of
18:Statistical parameter of random process evolution
1607:
1325:
1309:
1298:
993:
656:
622:
568:
552:
541:
290:
261:
200:
891:of the Gaussian distribution over a frequency
728:is the variance of the Gaussian distribution,
1210:
25:is the average amount of time it takes for a
1058:
996:
298:
295:
240:
203:
1577:Journal of Guidance, Control, and Dynamics
1596:
845:
800:
482:{\displaystyle {\bar {\tau }}(y_{0})=E.}
1562:Meerkov, S. M.; Runolfsson, T. (1987).
1553:Meerkov, S. M.; Runolfsson, T. (1986).
1608:
903:Suppose that instead of being scalar,
1408:In general, the normalization factor
899:Generalization to multiple dimensions
1229:of a Gaussian process is defined as
1215:The logarithmic residence time is a
601:
13:
1150:
1055:
1036:
1033:
827:
820:
782:
765:
595:where the frequency of exceedance
14:
1637:
1105:. The mean of this time is the
500:of the smaller critical value,
1331:
1301:
1281:
1245:
1191:
1175:
1162:
1156:
1144:
1131:
1125:
1027:
1021:
987:
974:
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836:
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234:
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1:
1546:
1529:Meerkov & Runolfsson 1986
1505:Meerkov & Runolfsson 1987
1493:Meerkov & Runolfsson 1986
1466:Meerkov & Runolfsson 1987
1427:Cumulative frequency analysis
32:
7:
1420:
1221:
710:
10:
1642:
1571:Richardson, Johnhenri R.;
1447:Mean time between failures
1227:logarithmic residence time
1211:Logarithmic residence time
942:and has a smooth boundary
1452:
1437:First-hitting-time model
350:is within the interval.
81:and two critical values
1442:Frequency of exceedance
957:from within the domain
498:frequency of exceedance
1583:(6). AIAA: 2026–2030.
1517:Richardson et al. 2014
1481:Richardson et al. 2014
1370:
1201:
1088:is on the boundary of
1068:
889:power spectral density
864:
698:
586:
483:
308:
1564:Output Aiming Control
1468:, pp. 1734–1735.
1371:
1202:
1069:
865:
699:
587:
484:
309:
1626:Reliability analysis
1432:Extreme value theory
1236:
1116:
968:
735:
608:
507:
401:
175:
1359:
824:
769:
688:
665:
317:where "inf" is the
132:. Define the first
52:with initial value
21:In statistics, the
1616:Extreme value data
1366:
1345:
1197:
1064:
929:. Define a domain
860:
810:
755:
694:
674:
651:
582:
479:
304:
50:stochastic process
48:is a real, scalar
1621:Survival analysis
1589:10.2514/1.G000299
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494:Gaussian process
488:
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147:from within the
146:
131:
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80:
71:
47:
1641:
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1636:
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1631:
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1606:
1605:
1573:Atkins, Ella M.
1549:
1544:
1543:
1538:
1532:
1527:
1523:
1519:, p. 2028.
1515:
1511:
1507:, p. 1734.
1503:
1499:
1491:
1487:
1483:, p. 2027.
1479:
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1423:
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377:random variable
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5:
1639:
1629:
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1623:
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1604:
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1598:2027.42/140648
1568:
1559:
1555:Aiming Control
1548:
1545:
1542:
1541:
1536:
1521:
1509:
1497:
1495:, p. 494.
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1124:
1107:residence time
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973:
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933:that contains
914:has dimension
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392:residence time
385:
379:. The mean of
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61:
34:
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27:random process
23:residence time
17:
9:
6:
4:
3:
2:
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1257:
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1242:
1232:
1231:
1230:
1228:
1224:
1223:
1218:
1217:dimensionless
1194:
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849:
846:
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748:
743:
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684:
679:
675:
671:
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24:
16:
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952:
948:
935:
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920:
909:
905:
902:
882:
877:
872:
723:
719:
708:
594:
491:
391:
382:
375:is itself a
367:
359:
355:
352:
343:
336:
332:
323:
316:
160:
153:
142:
138:
134:passage time
124:
114:
105:
98:
91:
84:
74:
65:
58:
54:
43:
39:
36:
22:
20:
15:
1610:Categories
1547:References
1101:is within
33:Definition
1347:σ
1282:¯
1279:τ
1258:
1246:^
1243:μ
1179:∣
1160:τ
1154:
1126:¯
1123:τ
1056:Ψ
1053:∈
1040:∣
1037:Ψ
1034:∂
1031:∈
1003:≥
972:τ
828:Φ
821:∞
812:∫
783:Φ
766:∞
757:∫
676:σ
649:−
531:−
517:¯
514:τ
461:∣
442:τ
411:¯
408:τ
254:−
238:∈
210:≥
179:τ
112:}, where
1421:See also
353:Because
149:interval
37:Suppose
1225:), the
887:is the
390:is the
319:infimum
72:, mean
1318:
561:
492:For a
270:
130:> 0
120:> 0
1453:Notes
931:Ψ ⊂ ℝ
927:) ∈ ℝ
918:, or
330:that
1381:min(
873:and
122:and
64:) =
1593:hdl
1585:doi
1393:max
1386:min
1326:max
1310:min
1299:min
994:inf
961:as
939:avg
657:max
623:max
599:is
569:max
553:min
542:min
291:max
278:avg
262:min
249:avg
201:inf
168:as
164:max
157:min
136:of
128:max
118:min
109:max
102:avg
95:min
88:avg
78:avg
1612::
1591:.
1581:37
1579:.
1473:^
1405:.
1395:)/
1388:,
1255:ln
1109:,
944:∂Ψ
895:.
394:,
381:τ(
366:τ(
159:,
152:(−
104:+
97:,
90:−
1601:.
1595::
1587::
1537:0
1534:N
1414:0
1411:N
1401:y
1397:σ
1390:y
1383:y
1364:.
1356:2
1351:y
1343:2
1336:2
1332:)
1322:y
1315:,
1306:y
1302:(
1293:=
1289:)
1271:0
1267:N
1262:(
1252:=
1222:1
1195:.
1192:]
1187:0
1183:y
1176:)
1171:0
1167:y
1163:(
1157:[
1151:E
1148:=
1145:)
1140:0
1136:y
1132:(
1103:Ψ
1098:0
1095:y
1090:Ψ
1086:)
1084:t
1082:(
1080:y
1062:.
1059:}
1048:0
1044:y
1028:)
1025:t
1022:(
1019:y
1016::
1011:0
1007:t
1000:t
997:{
991:=
988:)
983:0
979:y
975:(
959:Ψ
955:)
953:t
951:(
949:y
936:y
925:t
923:(
921:y
916:p
912:)
910:t
908:(
906:y
893:f
885:)
883:f
881:(
878:y
875:Φ
858:,
850:f
847:d
843:)
840:f
837:(
832:y
816:0
805:f
802:d
798:)
795:f
792:(
787:y
777:2
773:f
761:0
749:=
744:0
740:N
724:y
721:σ
713:)
711:1
709:(
692:,
685:2
680:y
672:2
668:/
662:2
653:y
645:e
639:0
635:N
631:=
628:)
619:y
615:(
612:N
597:N
580:,
577:)
574:)
565:y
558:,
549:y
545:(
539:(
534:1
527:N
523:=
477:.
474:]
469:0
465:y
458:)
453:0
449:y
445:(
439:[
436:E
433:=
430:)
425:0
421:y
417:(
388:)
386:0
383:y
373:)
371:0
368:y
362:)
360:t
358:(
356:y
347:0
344:y
339:)
337:t
335:(
333:y
327:0
324:t
302:,
299:}
296:}
287:y
283:+
274:y
267:,
258:y
245:y
241:{
235:)
232:t
229:(
226:y
223::
218:0
214:t
207:t
204:{
198:=
195:)
190:0
186:y
182:(
166:)
161:y
154:y
145:)
143:t
141:(
139:y
125:y
115:y
106:y
99:y
92:y
85:y
83:{
75:y
69:0
66:y
62:0
59:t
57:(
55:y
46:)
44:t
42:(
40:y
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