extval_recurrence_table
Calculates the recurrence interval (return period), cumulative and exceedence probabilities based upon a time series.
Available in version 6.4.0 and later.
Prototype
function extval_recurrence_table ( time [*] : numeric, x : numeric, dims [*] : integer, opt [1] : logical ) return_val
Arguments
timeA numeric array containing 'time' steps. Actually any unique numbering can be used. For example, if N=dimsizes(x), then time=ispan(1,N,1)
xA numeric array containing observations corresponding to each 'time' step. Currently, the only variables allowed are x(ntim), x(time,npts) and x(time,nlat,mlon).
dimsCurrently not used. Set to zero.
optA logical scalar to which optional attributes may be attached. If opt=False, the returned array is in the order input to the function. If opt=True and the attrobute rank_order is present and opt@rank_order=True, the returned table will be in order by rank.
Return value
Return seven (7) ranked table entries:
- 'time' entry
- 'x' value associated with the 'time'
- Cumulative Probability Rank
- Cumulative Probability
- Exceedance Probability Rank
- Exceedance Probability
- Recurrence Interval
Description
The annual exceedance probability (AEP) is the chance or probability of an event (eg., a rainfall or flooding event) occurring annually. AEP is usually expressed as a percentage. Bigger rainfall events occur (are exceeded) less often and will therefore have a lesser annual probability.
It should be noted that there are multiple methods for estimating probabilities of exceedance or non-exceedance of ranked data (R).
R/N ; state of California
(R-0.5)/N ; Hazen
R/(N+1) ; Weibull <==== used here
(R-0.44 )/(N+0.12) ; Gringoren (WMO, 1983)
(R-0.375)/(N+0.25) ; Sevruk and Geiger
However, the differences between the various methods are typically quite small. See
Table 1 and Table 2 of Reces (2004).
extval_recurrence_table
uses the 'Weibull' recurrence interval formulation.
From Wikipedia: It has been shown by Gumbel that the estimator R/(N+1) for the probability of an event wherei R is the rank number of the observed value in the data series and N is the total number of observations is an unbiased estimator of the cumulative probability around the mode of the distribution. Therefore, this estimator is often used as a plotting position .
References:
Raes, D (2004): Frequency Analysis of Rainfall Data ICTP (International Center for Theoretical Physics (2444-10)
See Also
Examples
Example 1
This example is from http://geog.uoregon.edu/amarcus/geog422/Handout_Recurrence_calcs.htm.
yyyy = ispan(1931, 1940, 1) ; year prc = (/1306, 1345, 1032, 1580, 1293, 1497, 1469, 1392, 1037, 1633/) ; mm table = extval_recurrence_table(yyyy, prc, 0, False) printVarSummary (table) ; [10] x [7] write_matrix (table, "2f7.0,5f9.2", False)The (edited) output is:
Cum. Cum. Exceed Exceed Recurrence
yyyy prc Prob. Prob. Prob. Prob. Interval
Rank Rank
---- ---- ---- ---- ---- ---- ----
1931 1306 4.00 0.36 7.00 0.64 1.57
1932 1345 5.00 0.45 6.00 0.55 1.83
1933 1032 1.00 0.09 10.00 0.91 1.10
1934 1580 9.00 0.82 2.00 0.18 5.50
1935 1293 3.00 0.27 8.00 0.73 1.38
1936 1497 8.00 0.73 3.00 0.27 3.67
1937 1469 7.00 0.64 4.00 0.36 2.75
1938 1392 6.00 0.55 5.00 0.45 2.20
1939 1037 2.00 0.18 9.00 0.82 1.22
1940 1633 10.00 0.91 1.00 0.09 11.00
Example 2
Same data as Example 1 but the table is returned in 'rank order'.
opt = True opt@rank_order = True TABLE = extval_recurrence_table(yyyy, prc, 0, opt) write_matrix (TABLE, "2f7.0,5f9.2", False)The (edited) output is:
Cum. Cum. Exceed Exceed Recurrence
yyyy prc Prob. Prob. Prob. Prob. Interval
Rank Rank
---- ---- ---- ---- ---- ---- ----
1933 1032 1.00 0.09 10.00 0.91 1.10
1939 1037 2.00 0.18 9.00 0.82 1.22
1935 1293 3.00 0.27 8.00 0.73 1.38
1931 1306 4.00 0.36 7.00 0.64 1.57
1932 1345 5.00 0.45 6.00 0.55 1.83
1938 1392 6.00 0.55 5.00 0.45 2.20
1937 1469 7.00 0.64 4.00 0.36 2.75
1936 1497 8.00 0.73 3.00 0.27 3.67
1934 1580 9.00 0.82 2.00 0.18 5.50
1940 1633 10.00 0.91 1.00 0.09 11.00
Example 3
Let x(time,npts) then:
opt = False ;opt = True ;opt@rank_order = True time = x&time ; any unique time table = extval_recurrence_table(time, x, 0, opt) printVarSummary(table) ; (npts, ntim, 7)Example 4
Let x(time,lat,lon) then:
opt = False ;opt = True ;opt@rank_order = True time = x&time ; any unique time table = extval_recurrence_table(time, x, 0, opt) printVarSummary(table) ; (nlat, mlon, ntim, 7)Example 5
Generically, this is the same as Example 2 but for different data. This data is used for Example 6 on the 'Extreme Value' application page.
flood = (/ 312,590,248,670,365,770,465,545,315,115,232,260,655,675 \
, 455,1020,700,570,853,395,926,99,680,121,976,916,921,191 \
, 187,377,128,582,744,710,520,672,645,655,918,512,255,1126 \
,1386,1394,600,950,731,700,1407,1284,165,1496,809 /)
nflood = dimsizes(flood) ; nflood=53
yyyy = ispan(1,nflood,1)
opt = True
opt@rank_order = True
table = extval_recurrence_table(yyyy, flood, 0, opt)
printVarSummary(table)
The (edited) output is:
Cum. Cum. Exceed Exceed Recurrence
yyyy Flood Prob. Prob. Prob. Prob. Interval
Rank Rank
---- ---- ---- ---- ---- ---- ----
22. 99. 1.00 1.85 53.00 98.15 1.02
10. 115. 2.00 3.70 52.00 96.30 1.04
24. 121. 3.00 5.56 51.00 94.44 1.06
31. 128. 4.00 7.41 50.00 92.59 1.08
51. 165. 5.00 9.26 49.00 90.74 1.10
29. 187. 6.00 11.11 48.00 88.89 1.12
28. 191. 7.00 12.96 47.00 87.04 1.15
11. 232. 8.00 14.81 46.00 85.19 1.17
3. 248. 9.00 16.67 45.00 83.33 1.20
41. 255. 10.00 18.52 44.00 81.48 1.23
12. 260. 11.00 20.37 43.00 79.63 1.26
1. 312. 12.00 22.22 42.00 77.78 1.29
9. 315. 13.00 24.07 41.00 75.93 1.32
5. 365. 14.00 25.93 40.00 74.07 1.35
30. 377. 15.00 27.78 39.00 72.22 1.38
20. 395. 16.00 29.63 38.00 70.37 1.42
15. 455. 17.00 31.48 37.00 68.52 1.46
7. 465. 18.00 33.33 36.00 66.67 1.50
40. 512. 19.00 35.19 35.00 64.81 1.54
35. 520. 20.00 37.04 34.00 62.96 1.59
8. 545. 21.00 38.89 33.00 61.11 1.64
18. 570. 22.00 40.74 32.00 59.26 1.69
32. 582. 23.00 42.59 31.00 57.41 1.74
2. 590. 24.00 44.44 30.00 55.56 1.80
45. 600. 25.00 46.30 29.00 53.70 1.86
37. 645. 26.00 48.15 28.00 51.85 1.93
13. 655. 27.00 50.00 27.00 50.00 2.00
38. 655. 28.00 51.85 26.00 48.15 2.08
4. 670. 29.00 53.70 25.00 46.30 2.16
36. 672. 30.00 55.56 24.00 44.44 2.25
14. 675. 31.00 57.41 23.00 42.59 2.35
23. 680. 32.00 59.26 22.00 40.74 2.45
17. 700. 33.00 61.11 21.00 38.89 2.57
48. 700. 34.00 62.96 20.00 37.04 2.70
34. 710. 35.00 64.81 19.00 35.19 2.84
47. 731. 36.00 66.67 18.00 33.33 3.00
33. 744. 37.00 68.52 17.00 31.48 3.18
6. 770. 38.00 70.37 16.00 29.63 3.38
53. 809. 39.00 72.22 15.00 27.78 3.60
19. 853. 40.00 74.07 14.00 25.93 3.86
26. 916. 41.00 75.93 13.00 24.07 4.15
39. 918. 42.00 77.78 12.00 22.22 4.50
27. 921. 43.00 79.63 11.00 20.37 4.91
21. 926. 44.00 81.48 10.00 18.52 5.40
46. 950. 45.00 83.33 9.00 16.67 6.00
25. 976. 46.00 85.19 8.00 14.81 6.75
16. 1020. 47.00 87.04 7.00 12.96 7.71
42. 1126. 48.00 88.89 6.00 11.11 9.00
50. 1284. 49.00 90.74 5.00 9.26 10.80
43. 1386. 50.00 92.59 4.00 7.41 13.50
44. 1394. 51.00 94.44 3.00 5.56 18.00
49. 1407. 52.00 96.30 2.00 3.70 27.00
52. 1496. 53.00 98.15 1.00 1.85 54.00