35:, is the frequency with which a random process exceeds some critical value. Typically, the critical value is far from the mean. It is usually defined in terms of the number of peaks of the random process that are outside the boundary. It has applications related to predicting extreme events, such as major
64:
is an event where the instantaneous value of the process crosses the critical value with positive slope. This article assumes the two methods of counting exceedance are equivalent and that the process has one upcrossing and one peak per exceedance. However, processes, especially continuous processes
59:
exceeds some critical value, usually a critical value far from the process' mean, per unit time. Counting exceedance of the critical value can be accomplished either by counting peaks of the process that exceed the critical value or by counting upcrossings of the critical value, where an
421:
268:
287:
785:
675:
697:. This probability can be useful to estimate whether an extreme event will occur during a specified time period, such as the lifespan of a structure or the duration of an operation.
158:
65:
with high frequency components to their power spectral densities, may have multiple upcrossings or multiple peaks in rapid succession before the process reverts to its mean.
426:
For a
Gaussian process, the approximation that the number of peaks above the critical value and the number of upcrossings of the critical value are the same is good for
511:. For many types of distributions of the underlying random process, including Gaussian processes, the number of peaks above the critical value
809:, and the probability of exceedance can be computed by simply multiplying the frequency of exceedance by the specified length of time.
1119:
1100:
1078:
1207:
1202:
842:
416:{\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df}}{\int _{0}^{\infty }{\Phi _{y}(f)\,df}}}}.}
1197:
543:
723:
597:
1212:
1027:
525:
1132:(1945). "Mathematical Analysis of Random Noise: Part III Statistical Properties of Random Noise Currents".
100:
263:{\displaystyle N(y_{\max })=N_{0}e^{-{\tfrac {1}{2}}\left({\tfrac {y_{\max }}{\sigma _{y}}}\right)^{2}}.}
521:
36:
1157:; Kabamba, Pierre T.; Girard, Anouck R. (2014). "Safety Margins for Flight Through Stochastic Gusts".
847:
524:
as the critical value becomes arbitrarily large. The interarrival times of this
Poisson process are
1005:"Understanding the "Probability of Exceedance" Forecast Graphs for Temperature and Precipitation"
56:
717:
is small, for example for the frequency of a rare event occurring in a short time period, then
1115:
1096:
1074:
852:
498:
As the random process evolves over time, the number of peaks that exceeded the critical value
1174:
1166:
1141:
508:
481:
74:
120:
is a frequency. Over time, this
Gaussian process has peaks that exceed some critical value
1129:
443:
17:
1004:
1088:
981:
837:
546:
or mean time before the very first peak, is the inverse of the frequency of exceedance
1191:
493:
281:
is the frequency of upcrossings of 0 and is related to the power spectral density as
1145:
1154:
1073:. Washington, DC: American institute of Aeronautics and Astronautics, Inc.
1179:
88:
1170:
790:
Under this assumption, the frequency of exceedance is equal to the
40:
1112:
Extremes and
Related Properties of Random Sequences and Processes
572:
grows as a
Poisson process, then the probability that at time
1095:. Vol. 1 (3rd ed.). New York: John Wiley and Sons.
967:
907:
1110:
Leadbetter, M. R.; Lindgren, Georg; Rootzén, Holger (1983).
1093:
471:
does not converge. Hoblit gives methods for approximating
449:
For power spectral densities that decay less steeply than
931:
876:
874:
872:
870:
868:
528:
with rate of decay equal to the frequency of exceedance
891:
889:
220:
203:
726:
600:
290:
161:
542:. Thus, the mean time between peaks, including the
779:
669:
415:
262:
1071:Gust Loads on Aircraft: Concepts and Applications
763:
651:
227:
173:
780:{\displaystyle p_{ex}(t)\approx N(y_{\max })t.}
670:{\displaystyle p_{ex}(t)=1-e^{-N(y_{\max })t},}
69:Frequency of exceedance for a Gaussian process
8:
824:Hydrology and loads on hydraulic structures
1159:Journal of Guidance, Control, and Dynamics
576:there has not yet been any peak exceeding
1178:
762:
731:
725:
650:
636:
605:
599:
480:in such cases with applications aimed at
398:
383:
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317:
312:
304:
295:
289:
249:
236:
226:
219:
202:
198:
188:
172:
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693:has been exceeded at least once by time
130:. Counting the number of upcrossings of
932:Leadbetter, Lindgren & Rootzén 1983
864:
792:probability of exceedance per unit time
1052:
1028:"Section 2: Probability of Exceedance"
955:
943:
919:
880:
982:"Earthquake Hazards 101 – the Basics"
7:
1034:. Texas Department of Transportation
895:
980:Earthquake Hazards Program (2016).
462:, the integral in the numerator of
1003:Climate Prediction Center (2002).
488:Time and probability of exceedance
380:
373:
335:
318:
25:
563:If the number of peaks exceeding
818:Probability of major earthquakes
768:
755:
746:
740:
656:
643:
620:
614:
395:
389:
350:
344:
178:
165:
1:
1134:Bell System Technical Journal
1114:. New York: Springer–Verlag.
843:Cumulative frequency analysis
73:Consider a scalar, zero-mean
1069:Hoblit, Frederic M. (1988).
1229:
1153:Richardson, Johnhenri R.;
1007:. National Weather Service
491:
934:, pp. 176, 238, 260.
682:probability of exceedance
526:exponentially distributed
55:is the number of times a
33:annual rate of exceedance
18:Probability of exceedance
1146:10.1002/(ISSN)1538-7305c
984:. U.S. Geological Survey
1032:Hydraulic Design Manual
684:, the probability that
141:frequency of exceedance
53:frequency of exceedance
31:, sometimes called the
29:frequency of exceedance
1165:(6). AIAA: 2026–2030.
968:Richardson et al. 2014
908:Richardson et al. 2014
827:Gust loads on aircraft
781:
671:
507:grows and is itself a
417:
264:
101:power spectral density
1026:Garcia, Rene (2015).
910:, pp. 2029–2030.
782:
672:
492:Further information:
418:
265:
1208:Stochastic processes
1203:Reliability analysis
848:Extreme value theory
724:
598:
288:
159:
946:, pp. 446–448.
922:, pp. 229–235.
821:Weather forecasting
377:
322:
1198:Extreme value data
777:
667:
591:. Its complement,
413:
363:
308:
260:
243:
212:
57:stochastic process
1213:Survival analysis
1171:10.2514/1.G000299
958:, pp. 65–66.
898:, pp. 54–55.
883:, pp. 51–54.
444:narrow band noise
408:
407:
242:
211:
16:(Redirected from
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75:Gaussian process
21:
1228:
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1219:
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1188:
1187:
1155:Atkins, Ella M.
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1109:
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1089:Feller, William
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970:, p. 2027.
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522:Poisson process
520:converges to a
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49:
23:
22:
15:
12:
11:
5:
1226:
1224:
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1205:
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1189:
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1180:2027.42/140648
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1107:
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1079:
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960:
948:
936:
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900:
885:
863:
862:
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857:
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853:Rice's formula
850:
845:
840:
838:100-year flood
833:
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814:
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631:
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611:
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581:
568:
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544:residence time
537:
516:
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24:
14:
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10:
9:
6:
4:
3:
2:
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1214:
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1206:
1204:
1201:
1199:
1196:
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1193:
1181:
1176:
1172:
1168:
1164:
1160:
1156:
1151:
1147:
1143:
1140:(1): 46–156.
1139:
1135:
1131:
1127:
1123:
1121:9781461254515
1117:
1113:
1108:
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1102:9780471257080
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996:
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945:
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933:
928:
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913:
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554:
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495:
494:Return period
487:
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459:
453:
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384:
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38:
34:
30:
19:
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1158:
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1133:
1111:
1092:
1070:
1048:
1036:. Retrieved
1031:
1021:
1009:. Retrieved
998:
986:. Retrieved
975:
963:
951:
939:
927:
915:
903:
813:Applications
805:
800:
796:
791:
789:
713:
706:
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681:
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562:
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534:
530:
513:
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457:
451:
448:
436:
428:
425:
274:
272:
152:is given by
145:
140:
132:
122:
111:
106:
94:
82:
78:
72:
61:
52:
50:
32:
28:
26:
1130:Rice, S. O.
1053:Hoblit 1988
956:Hoblit 1988
944:Feller 1968
920:Hoblit 1988
881:Hoblit 1988
37:earthquakes
1192:Categories
1080:0930403452
1063:References
1055:, Chap. 4.
62:upcrossing
47:Definition
1038:April 26,
1011:April 26,
988:April 26,
896:Rice 1945
750:≈
638:−
630:−
381:Φ
374:∞
365:∫
336:Φ
319:∞
310:∫
234:σ
200:−
1091:(1968).
832:See also
442:and for
116:, where
89:variance
680:is the
1118:
1099:
1077:
440:> 2
139:, the
128:> 0
41:floods
859:Notes
87:with
1116:ISBN
1097:ISBN
1075:ISBN
1040:2016
1013:2016
990:2016
99:and
51:The
39:and
27:The
1175:hdl
1167:doi
1142:doi
764:max
710:max
700:If
690:max
652:max
585:is
582:max
569:max
556:max
538:max
517:max
504:max
455:as
432:max
228:max
174:max
149:max
143:of
136:max
126:max
1194::
1173:.
1163:37
1161:.
1138:24
1136:.
1030:.
888:^
867:^
801:ex
794:,
560:.
484:.
460:→∞
446:.
434:/σ
43:.
1183:.
1177::
1169::
1148:.
1144::
1124:.
1105:.
1083:.
1042:.
1015:.
992:.
806:t
804:/
797:p
775:.
772:t
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760:y
756:(
753:N
747:)
744:t
741:(
736:x
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707:y
705:(
703:N
695:t
687:y
665:,
660:t
657:)
648:y
644:(
641:N
634:e
627:1
624:=
621:)
618:t
615:(
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607:e
603:p
588:e
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574:t
566:y
558:)
553:y
551:(
549:N
540:)
535:y
533:(
531:N
514:y
501:y
477:0
474:N
468:0
465:N
458:f
452:f
437:y
429:y
411:.
403:f
400:d
396:)
393:f
390:(
385:y
369:0
358:f
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351:)
348:f
345:(
340:y
330:2
326:f
314:0
302:=
297:0
293:N
278:0
275:N
258:.
251:2
246:)
238:y
224:y
217:(
209:2
206:1
196:e
190:0
186:N
182:=
179:)
170:y
166:(
163:N
146:y
133:y
123:y
118:f
114:)
112:f
110:(
107:y
104:Φ
95:y
92:σ
85:)
83:t
81:(
79:y
20:)
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