24:, 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
53:
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
48:
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
410:
257:
276:
774:
664:
686:. 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.
147:
54:
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.
415:
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
500:. For many types of distributions of the underlying random process, including Gaussian processes, the number of peaks above the critical value
798:, and the probability of exceedance can be computed by simply multiplying the frequency of exceedance by the specified length of time.
405:{\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df}}{\int _{0}^{\infty }{\Phi _{y}(f)\,df}}}}.}
1108:
1089:
712:
586:
1016:
1067:
1196:
1191:
831:
1121:(1945). "Mathematical Analysis of Random Noise: Part III Statistical Properties of Random Noise Currents".
1186:
532:
252:{\displaystyle N(y_{\max })=N_{0}e^{-{\tfrac {1}{2}}\left({\tfrac {y_{\max }}{\sigma _{y}}}\right)^{2}}.}
1201:
1146:; Kabamba, Pierre T.; Girard, Anouck R. (2014). "Safety Margins for Flight Through Stochastic Gusts".
513:
as the critical value becomes arbitrarily large. The interarrival times of this
Poisson process are
514:
994:"Understanding the "Probability of Exceedance" Forecast Graphs for Temperature and Precipitation"
706:
is small, for example for the frequency of a rare event occurring in a short time period, then
89:
487:
As the random process evolves over time, the number of peaks that exceeded the critical value
510:
25:
836:
109:
is a frequency. Over time, this
Gaussian process has peaks that exceed some critical value
8:
993:
970:
535:
or mean time before the very first peak, is the inverse of the frequency of exceedance
45:
1104:
1085:
1063:
841:
270:
is the frequency of upcrossings of 0 and is related to the power spectral density as
1163:
1155:
1130:
497:
470:
63:
1134:
1118:
914:
890:
432:
1077:
826:
1180:
482:
950:
1062:. Washington, DC: American institute of Aeronautics and Astronautics, Inc.
1168:
1143:
1159:
779:
Under this assumption, the frequency of exceedance is equal to the
77:
1101:
Extremes and
Related Properties of Random Sequences and Processes
57:
561:
grows as a
Poisson process, then the probability that at time
1141:
1084:. Vol. 1 (3rd ed.). New York: John Wiley and Sons.
956:
896:
29:
1099:
Leadbetter, M. R.; Lindgren, Georg; Rootzén, Holger (1983).
1082:
968:
460:
does not converge. Hoblit gives methods for approximating
438:
For power spectral densities that decay less steeply than
1098:
1035:
991:
920:
865:
863:
861:
859:
857:
926:
902:
517:
with rate of decay equal to the frequency of exceedance
880:
878:
938:
854:
209:
192:
715:
589:
476:
279:
150:
875:
531:. Thus, the mean time between peaks, including the
768:
658:
404:
251:
1060:Gust Loads on Aircraft: Concepts and Applications
1178:
752:
640:
216:
162:
769:{\displaystyle p_{ex}(t)\approx N(y_{\max })t.}
659:{\displaystyle p_{ex}(t)=1-e^{-N(y_{\max })t},}
58:Frequency of exceedance for a Gaussian process
813:Hydrology and loads on hydraulic structures
1148:Journal of Guidance, Control, and Dynamics
565:there has not yet been any peak exceeding
1167:
469:in such cases with applications aimed at
387:
342:
682:has been exceeded at least once by time
119:. Counting the number of upcrossings of
921:Leadbetter, Lindgren & Rootzén 1983
781:probability of exceedance per unit time
1179:
1076:
1057:
1041:
1017:"Section 2: Probability of Exceedance"
1014:
944:
932:
908:
869:
971:"Earthquake Hazards 101 – the Basics"
1117:
1023:. Texas Department of Transportation
884:
969:Earthquake Hazards Program (2016).
451:, the integral in the numerator of
13:
992:Climate Prediction Center (2002).
477:Time and probability of exceedance
369:
362:
324:
307:
14:
1213:
552:If the number of peaks exceeding
807:Probability of major earthquakes
1008:
985:
801:
962:
757:
744:
735:
729:
645:
632:
609:
603:
384:
378:
339:
333:
167:
154:
1:
1123:Bell System Technical Journal
1103:. New York: Springer–Verlag.
1051:
832:Cumulative frequency analysis
62:Consider a scalar, zero-mean
35:
1058:Hoblit, Frederic M. (1988).
7:
820:
10:
1218:
1142:Richardson, Johnhenri R.;
996:. National Weather Service
480:
923:, pp. 176, 238, 260.
671:probability of exceedance
515:exponentially distributed
44:is the number of times a
22:annual rate of exceedance
1135:10.1002/(ISSN)1538-7305c
973:. U.S. Geological Survey
847:
1021:Hydraulic Design Manual
673:, the probability that
130:frequency of exceedance
42:frequency of exceedance
20:, sometimes called the
18:frequency of exceedance
1154:(6). AIAA: 2026–2030.
957:Richardson et al. 2014
897:Richardson et al. 2014
816:Gust loads on aircraft
770:
660:
496:grows and is itself a
406:
253:
90:power spectral density
1015:Garcia, Rene (2015).
899:, pp. 2029–2030.
771:
661:
481:Further information:
407:
254:
1197:Stochastic processes
1192:Reliability analysis
837:Extreme value theory
713:
587:
277:
148:
935:, pp. 446–448.
911:, pp. 229–235.
810:Weather forecasting
366:
311:
1187:Extreme value data
766:
656:
580:. Its complement,
402:
352:
297:
249:
232:
201:
46:stochastic process
1202:Survival analysis
1160:10.2514/1.G000299
947:, pp. 65–66.
887:, pp. 54–55.
872:, pp. 51–54.
433:narrow band noise
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231:
200:
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498:counting process
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471:continuous gusts
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64:Gaussian process
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1176:
1144:Atkins, Ella M.
1111:
1092:
1078:Feller, William
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959:, p. 2027.
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511:Poisson process
509:converges to a
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1169:2027.42/140648
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844:
842:Rice's formula
839:
834:
829:
827:100-year flood
822:
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533:residence time
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9:
6:
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3:
2:
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1182:
1170:
1165:
1161:
1157:
1153:
1149:
1145:
1140:
1136:
1132:
1129:(1): 46–156.
1128:
1124:
1120:
1116:
1112:
1110:9781461254515
1106:
1102:
1097:
1093:
1091:9780471257080
1087:
1083:
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1065:
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1038:
1022:
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1011:
995:
988:
972:
965:
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953:
946:
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929:
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569:
556:
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525:
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491:
484:
483:Return period
474:
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33:
31:
27:
23:
19:
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1100:
1081:
1059:
1037:
1025:. Retrieved
1020:
1010:
998:. Retrieved
987:
975:. Retrieved
964:
952:
940:
928:
916:
904:
892:
802:Applications
794:
789:
785:
780:
778:
702:
695:
691:
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675:
670:
668:
576:
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554:
551:
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523:
519:
502:
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453:
446:
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417:
414:
263:
261:
141:is given by
134:
129:
121:
111:
100:
95:
83:
71:
67:
61:
50:
41:
39:
21:
17:
15:
1119:Rice, S. O.
1042:Hoblit 1988
945:Hoblit 1988
933:Feller 1968
909:Hoblit 1988
870:Hoblit 1988
26:earthquakes
1181:Categories
1069:0930403452
1052:References
1044:, Chap. 4.
51:upcrossing
36:Definition
1027:April 26,
1000:April 26,
977:April 26,
885:Rice 1945
739:≈
627:−
619:−
370:Φ
363:∞
354:∫
325:Φ
308:∞
299:∫
223:σ
189:−
1080:(1968).
821:See also
431:and for
105:, where
78:variance
669:is the
1107:
1088:
1066:
429:> 2
128:, the
117:> 0
30:floods
848:Notes
76:with
1105:ISBN
1086:ISBN
1064:ISBN
1029:2016
1002:2016
979:2016
88:and
40:The
28:and
16:The
1164:hdl
1156:doi
1131:doi
753:max
699:max
689:If
679:max
641:max
574:is
571:max
558:max
545:max
527:max
506:max
493:max
444:as
421:max
217:max
163:max
138:max
132:of
125:max
115:max
1183::
1162:.
1152:37
1150:.
1127:24
1125:.
1019:.
877:^
856:^
790:ex
783:,
549:.
473:.
449:→∞
435:.
423:/σ
32:.
1172:.
1166::
1158::
1137:.
1133::
1113:.
1094:.
1072:.
1031:.
1004:.
981:.
795:t
793:/
786:p
764:.
761:t
758:)
749:y
745:(
742:N
736:)
733:t
730:(
725:x
722:e
718:p
703:t
701:)
696:y
694:(
692:N
684:t
676:y
654:,
649:t
646:)
637:y
633:(
630:N
623:e
616:1
613:=
610:)
607:t
604:(
599:x
596:e
592:p
577:e
568:y
563:t
555:y
547:)
542:y
540:(
538:N
529:)
524:y
522:(
520:N
503:y
490:y
466:0
463:N
457:0
454:N
447:f
441:f
426:y
418:y
400:.
392:f
389:d
385:)
382:f
379:(
374:y
358:0
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337:f
334:(
329:y
319:2
315:f
303:0
291:=
286:0
282:N
267:0
264:N
247:.
240:2
235:)
227:y
213:y
206:(
198:2
195:1
185:e
179:0
175:N
171:=
168:)
159:y
155:(
152:N
135:y
122:y
112:y
107:f
103:)
101:f
99:(
96:y
93:Φ
84:y
81:σ
74:)
72:t
70:(
68:y
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