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In this section, examples are given with simplified rank distributions in order to demonstrate the rank-mean and rank-std computation for the expected anomaly and uncertainty category. Based on these examples, the users can have a feel on how the rank-mean and rank-std values change with the changing underlying distributions. In addition, the impact of the 0-value in the climatology and in the ensemble forecasts is also demonstrated for different severity of the 0-value problem, with the complexity of these dry cases.
In these simplified examples, a fixed portion of the climatological and/or ensemble forecast distribution is 0, while the non-zero ensemble members have just one rank for the very dry cases and 5 for the non-zero cases, with members having the same rank in each group. This way, the computation methodology can be demonstrated in a simple way that is easier to interpret.
The non-zero value examples highlight that shifting the same rank distribution 'up' or 'down' (i.e. for wetter or drier) does not change the standard deviation (and thus the uncertainty). Also, after 'narrowing' the rank distribution over the same mean value (i.e. making it cluster more), the mean does not change, but the uncertainty drops markedly. Moreover, in a similar manner, by adding extreme members (i.e. near 1 or 100 very extreme members), even if only very few members, the uncertainty will increased quite substantially.
For those forecasts, when some portion or all of the climatological percentiles are 0, it is a general rule of thumb that as the percentage of zero climate percentiles increase, it gets more and more difficult to end up with negative expected forecast anomalies. The lowest possible rank-mean values are going to happen for forecasts with all 0 values, in which case the forecast rank-mean is going to be determined by the size of the 0-value section of the climatology. For example, if the lowest 30% of the climatology is 0, maybe in southern Spain somewhere on a small river catchment, then the rank-mean of the forecast of a constant 0-value will be about 15, which will put this forecast in the expected anomaly category of 'Low'. But, with 70% of climatology being 0, say further into the Sahara, there the driest possible ensemble forecast of only 0 values is only going to be rank-mean of 35, so with expected anomaly category of 'Bit low'. Drier than this anomaly is simply physically not possible for such a climatologically dry place.
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| title | Examples with no 0-value section in the climatology with and 5 ensemble forecast rank groups only |
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Few examples are shown here, when there is no 0-value in the climatology, so all ensemble forecast members can be ranked without any issue. For simplicity, 5 groups are used in the forecast only. The table below shows the numbers and the related average ranks for the 5 groups, with the rank-mean, rank-std and expected anomaly and uncertainty categories determined from those cases. For example, in the first row, 10 ensemble members are in the first group, which will all have the rank of 40. Then 10 members will be in the 2nd group with the rank of 45, and so on. The rank-mean of this simplified forecast distribution will be very close to 50 (mean of 40-45-50-55-60 with almost the same population in each group) and the rank-std will be about 7. This puts this forecast case into the 'Normal' expected anomaly category (rank-mean between 40 and 60) and the 'Low' uncertainty category (rank-set below 10). The even distribution is represented first below, for which it is shown that by shifting the same rank distribution up or down does not change the standard deviation (and uncertainty). This is true for any variety of rank distributions. Also, after 'narrowing' the rank distribution, the mean does not change, but the uncertainty drops markedly. Moreover, in a similar manner, by adding extreme members (i.e. 1 or 100 or near that), even if only with very few members (2 in this example below), the uncertainty can be increased quite substantially. | Number of ensemble members in each group | Common rank in each group | Rank-mean | Expected forecast anomaly category | Rank-std | Forecast uncertainty category |
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| N1 | N2 | N3 | N4 | N5 | R1 | R2 | R3 | R4 | R5 |
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| 10 | 10 | 11 | 10 | 10 | 40 | 45 | 50 | 55 | 60 | 50.0 | Near normal (40-60) | 7.00 | Low uncertainty | | 10 | 10 | 11 | 10 | 10 | 30 | 40 | 50 | 60 | 70 | 50.0 | Near normal (40-60) | 14.00 | Medium uncertainty | | 10 | 10 | 11 | 10 | 10 | 10 | 30 | 50 | 70 | 90 | 50.0 | Near normal (40-60) | 28.00 | High uncertainty |
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| | 10 | 10 | 11 | 10 | 10 | 60 | 65 | 70 | 75 | 80 | 70.0 | Bit high (60-75) | 7.00 | Low uncertainty | | 10 | 10 | 11 | 10 | 10 | 50 | 60 | 70 | 80 | 90 | 70.0 | Bit high (60-75) | 14.00 | Medium uncertainty |
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| | 0 | 10 | 31 | 10 | 0 |
| 45 | 50 | 55 |
| 50.0 | Near normal (40-60) | 3.13 | Low uncertainty | | 0 | 10 | 31 | 10 | 0 |
| 40 | 50 | 60 |
| 50.0 | Near normal (40-60) | 6.26 | Low uncertainty | | 0 | 10 | 31 | 10 | 0 |
| 30 | 50 | 70 |
| 50.0 | Near normal (40-60) | 12.52 | Medium uncertainty |
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| | 2 | 10 | 27 | 10 | 2 | 1 | 45 | 50 | 55 | 100 | 50.03 | Near normal (40-60) | 14.21 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 40 | 50 | 60 | 100 | 50.03 | Near normal (40-60) | 15.21 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 30 | 50 | 70 | 100 | 50.0 | Near normal (40-60) | 18.64 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 20 | 50 | 80 | 100 | 50.0 | Near normal (40-60) | 23.34 | High uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 10 | 50 | 90 | 100 | 50.0 | Near normal (40-60) | 28.62 | High uncertainty |
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