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There is a consequence of the 0-value problem treatment over dry or very dry areas (described above), as some or all of the low anomaly signals will not be impossible to occur. Please consider some of these idealised examples, with a fixed portion of the ensemble members being 0 and the rest having a simple average rank, in order to simplify the rank-mean and dominant anomaly category computation:
No-1 example: The
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1-10 climate percentiles are 0, so the average rank of the
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0 ensemble members will always be around 5.5 with the evenly distributed rank representation:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Dominant anomaly category | Comment |
|---|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 11 (the lowest possible rank for a non-zero member if 1-10 percentiles in the climatology are 0) | (0 * 5.5 + 51 * 11)/51 = 11 | Low (10-25%) | Lowest possible overall rank (and dominant anomaly category) |
| 0 | 51 | NA | 20 | (0 * 5.5 + 51 * 20)/51 = 20 | Low (10-25%) | |
| 0 | 51 | NA | 50 | (0 * 5.5 + 51 * 50)/51 = 50 | Near normal (40-60%) | |
| 0 | 51 | NA | 70 | (0 * 5.5 + 51 * 70)/51 = 70 | Bit high (60-75%) | |
| 0 | 51 | NA | 100 | (0 * 5.5 + 51 * 100)/51 = 100 | Extreme high (90%<) | |
| 11 | 40 | 5.5 | 11 | (11 * 5.5 + 40 * 11)/51 = 9.81 | Extreme low (<10%) | |
| 11 | 40 | 5.5 | 20 | (11 * 5.5 + 40 * 20)/51 = 16.87 | Low (10-25%) | |
| 11 | 40 | 5.5 | 50 | (11 * 5.5 + 40 * 50)/51 = 40.40 | Near normal (40-60%) | |
| 11 | 40 | 5.5 | 70 | (11 * 5.5 + 40 * 70)/51 = 56.08 | Near normal (40-60%) | |
| 11 | 40 | 5.5 | 100 | (11 * 5.5 + 40 * 100)/51 = 79.61 | High (75-90%) | |
| 21 | 30 | 5.5 | 11 | (21 * 5.5 + 30 * 11)/51 = 8.73 | Extreme low (<10%) | |
| 21 | 30 | 5.5 | 20 | (21 * 5.5 + 30 * 20)/51 = 14.02 | Low (10-25%) | |
| 21 | 30 | 5.5 | 50 | (21 * 5.5 + 30 * 50)/51 = 31.67 | Bit low (25-40%) | |
| 21 | 30 | 5.5 | 70 | (21 * 5.5 + 30 * 70)/51 = 43.44 | Near normal (40-60%) | |
| 21 | 30 | 5.5 | 100 | (21 * 5.5 + 30 * 50)/51 = 61.08 | Bit high (60-75%) | |
| 36 | 15 | 5.5 | 11 | (36 * 5.5 + 15 * 11)/51 = 7.11 | Extreme low (<10%) | |
| 36 | 15 | 5.5 | 20 | (36 * 5.5 + 15 * 20)/51 = 9.76 | Extreme low (<10%) | |
| 36 | 15 | 5.5 | 50 | (36 * 5.5 + 15 * 50)/51 = 18.58 | Low (10-25%) | |
| 36 | 15 | 5.5 | 70 | (36 * 5.5 + 15 * 70)/51 = 24.47 | Low (10-25%) | |
| 36 | 15 | 5.5 | 100 | (36 * 5.5 + 15 * 50)/51 = 33.29 | Bit low (25-40%) | |
| 51 | 0 | 5.5 | NA (no member to rank) | (51 * 5.5 + 0)/51 = 5.5 | Extreme low (<10%) |
No-2 example: The 1-30 climate percentiles are 0, so the average rank of the 0 ensemble members will always be around 15.5 with the evenly distributed rank representation:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Dominant anomaly category | Comment |
|---|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 31 (the lowest possible rank for a non-zero member if 1-30 percentiles in the climatology are 0) | (0 * 15.5 + 51 * 31)/51 = 31 | Bit low (25-40%) | Lowest possible rank (and dominant anomaly category) |
| 0 | 51 | NA | 50 | (0 * 15.5 + 51 * 50)/51 = 50 | Near normal (40-60%) | |
| 0 | 51 | NA | 70 | (0 * 15.5 + 51 * 70)/51 = 70 | Bit high (60-75%) | |
| 0 | 51 | NA | 100 | (0 * 15.5 + 51 * 100)/51 = 100 | Extreme high (90%<) | |
| 11 | 40 | 5.5 | 11 | (11 * 15.5 + 40 * 31)/51 = 27.65 | Bit low (25-40%) | |
| 11 | 40 | 5.5 | 50 | (11 * 15.5 + 40 * 50)/51 = 42.55 | Near normal (40-60%) | |
| 11 | 40 | 5.5 | 70 | (11 * 15.5 + 40 * 70)/51 = 58.24 | Near normal (40-60%) | |
| 11 | 40 | 5.5 | 100 | (11 * 15.5 + 40 * 100)/51 = 81.77 | High (75-90%) | |
| 21 | 30 | 5.5 | 11 | (21 * 15.5 + 30 * 31)/51 = 24.61 | Low (10-25%) | |
| 21 | 30 | 5.5 | 50 | (21 * 15.5 + 30 * 50)/51 = 35.79 | Bit low (25-40%) | |
| 21 | 30 | 5.5 | 70 | (21 * 15.5 + 30 * 70)/51 = 47.55 | Near normal (40-60%) | |
| 21 | 30 | 5.5 | 100 | (21 * 15.5 + 30 * 100)/51 = 65.20 | Bit high (60-75%) | |
| 36 | 15 | 5.5 | 11 | (36 * 15.5 + 15 * 11)/51 = 20.05 | Low (10-25%) | |
| 36 | 15 | 5.5 | 50 | (36 * 15.5 + 15 * 50)/51 = 25.64 | Bit low (25-40%) | |
| 36 | 15 | 5.5 | 70 | (36 * 15.5 + 15 * 70)/51 = 31.52 | Bit low (25-40%) | |
| 36 | 15 | 5.5 | 100 | (36 * 15.5 + 15 * 100)/51 = 40.35 | Near normal (40-60%) | |
| 51 | 0 | 5.5 | NA (no member to rank) | (51 * 15.5 + 0)/51 = 15.5 | Low (10-25%) |
- If all ensemble members are 0, then the ranks will spread evenly from 1 to 10 and the rank-mean will be around 5.5, so in the Extreme low category.
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