This page describes the way the anomaly and uncertainty of the ensemble forecasts in the sub-seasonal and seasonal products are determined using the climatology as reference. This includes also how the expected forecast anomaly category (one of the 7 predefined ones) and the uncertainty category (of the 3 predefined ones of low/medium/high) of the ensemble forecasts are determined. This is a generic procedure, which is the same for both EFAS and GloFAS, as it is executed the same way for each river pixel, regardless of the resolution, and also the same for the sub-seasonal and seasonal products, as it works in the exact same way regardless of whether it is weekly mean values, as in the sub-seasonal, or monthly mean values, as in the seasonal.
The characterisation of the forecast signal in both the sub-seasonal and seasonal is based on the ensemble member's extremity in the context of the model climatological distribution, which is explained further below.
Climatological percentiles and forecast anomaly categories
From the climate sample, 99 climate percentiles are determined, which represent equally likely (1% chance) segments of the river discharge value range that occurred in the 20-year climatological sample (both sub-seasonal and seasonal is currently based on 20 years). Figure 1 shows an example generic climate distribution, either based on weekly means or monthly means, with the percentiles represented along the y-axis. Only the deciles (every 10%), the quartiles (25%, 50% and 75%), of which the middle (50%) is also called median, and few of the extreme percentiles are indicated near the minimum and maximum of the climatological range shown by black crosses. Each of these percentiles have an equivalent river discharge value along the x-axis. From one percentile to the next, the river discharge value range is divided into 100 equally likely bins (separated by the percentiles), some of which is indicated in Figure 1, such as bin1 of values below the 1st percentile, bin2 of values between the 1st and 2nd percentiles or bin 100 of river discharge values above the 99th percentiles, etc.
Figure 1. Schematic of the forecast anomaly categories, defined by the climatological distribution.
Based on the percentiles and the related 100 bins, there are seven categories defined, which will be used as anomaly categories (Table 1). These are also indicated by vertical lines separating them in Figure 1. The two most extreme categories are the bottom and top 10% of the climatological distribution (<10% as 'Extreme low' and 90%< as 'Extreme high'). Then the moderately low and high river discharge categories from 10-25% ('Low') and 75-90% ('High'). The smallest negative and positive anomalies are defined by 25-40% ('Bit low') and 60-75% ('Bit high'). Finally, the normal condition category is defined as 40-60%, so the middle 1/5th of the distribution, the area called 'Normal' in Figure 1.
In addition, for some web products the middle three categories of 'Bit low', 'Normal' and 'Bit high' are merged into one larger 'Near normal' category, covering the percentiles from 25th to 75th. This 'Near normal' category is also indicated in Figure 1 and Table 1.
| Anomaly categories | Name | Ranks | Description | |
|---|---|---|---|---|
| Cat-1 | Extreme low | 1-10 | bottom 10% of the climatological distribution | |
| Cat-2 | Low | 10-25 | 15% from the 1st decile to the 1st quartile | |
| Cat-3 | Bit low |
(25-75) | 25-40 | 15% from the 1st quartile to the 2nd quintile |
| Cat-4 | Normal | 40-60 | 20% from the 2nd to the 3rd quintile | |
| Cat-5 | Bit high | 60-75 | 15% from the 3rd quintile to the 3rd quartile | |
| Cat-6 | High | 74-90 | 15% from the 3rd quartile to the 9th decile | |
| Cat-7 | Extreme high | 90-100 | top 10% of the climatological distribution | |
Table 1: Definition and description of the 7 anomaly categories. The possible value ranges in the 'Ranks column' are inclusive at the start and exclusive at the end, so for example for Cat-1 the possible ranks are 1, 2, 3, ... and 10. For some web products, the middle three categories (Cat3, Cat-4 and Cat-5) are combined into one extended 'Near normal' category.
Extremity rank computation for ensemble members
The forecast has 51 ensemble members, again for both EFAS/GloFAS and both sub-seasonal or seasonal, regardless. The members are all checked for climatological extremity and placed in one of the 100 climate bins. This will be the anomaly or extremity level of the ensemble members, which can be called hereafter rank, as one of the values from 1 to 100. For example, 1 will mean the forecast value is below the 1st climate percentile (i.e. extremely anomalously low, less than the value that happened in the climatological period only 1% of the time), then 2 will mean the value is between the 1st and 2nd climate percentiles (i.e. slightly less extremely low), etc., and finally 100 will mean the forecast value is above the 99th climate percentile (i.e. extremely high as higher than 99% of all the considered reforecasts), representing the model climate conditions for this time of year, location and lead time.
Figure 2 shows the process of determining the ranks for each ensemble member. In this example, the lowest member gets the rank of 54 (red r54 on the graph in Figure 2) by moving vertically until crossing the climatological distribution and then moving horizontally to the y-axis to determine the two bounding percentiles and thus the right percentile bin. In this case, the lowest ensemble member value is between the 53rd and 54th percentile, which results in bin-54. Then all ensemble members, similarly, get a bin number, the 2nd lowest values with bin-60 and so on until the largest ensemble member value getting bin-97, as the river discharge value is between the 96th and 97th percentiles.
Figure 2. Schematic of the forecast extremity ranking of the 51 ensemble members and the 7 anomaly categories in the context of the climatological distribution.
The probability of the 7 anomaly categories is calculated by counting of the ensemble members in each category and then dividing by 51, the total number of members. In the example of Figure 2, there is no member in the 3 low anomaly categories, while the 'Normal' category has 2, resulting in 3.9% probability, the 'Bit high' category 13, with 27.5%, the 'High' category 17, as 33.3%, and finally the 'Extreme high' category has 18 ensemble members, with 35.3% probability. The inset table in Figure 2 shows the numbers and the probabilities, but also shows the size (in terms of probabilities) of the 7 categories. For ease of interpretation, the 7 categories are displayed here with different colours. This highlights, e.g., that the normal flow category's 3.9% probability is much lower than the climatologically expected probability of 20%, however, the three high flow categories have each much higher probabilities than the climatological reference probability, especially the extreme high category, where the forecast probability (35.3%) is more than double the corresponding climatological probability (15%). In addition, the extended 'Near normal' category would have 15 members with 31.4% probability, which is lower than the climatological probability of 50%.
Extremity rank computation for ensemble members with 0 values
The forecast extremity rank computation can be done for any value above 0 m3/s. However, it becomes undefined when the values drop to 0, as there is no way to differentiate amongst values which are the same. The hydrological simulations of EFAS and GloFAS are less reliable and more prone to any random noise when we approach 0, so everything below 0.1 m3/s will be considered as 0 for the sub-seasonal and seasonal products. This problem can also happen for non-zero identical values, but normally the simulation should not produce a lot of identical non-zero values, unless there is some specific process, like reservoir operation rule, etc., which might generate such signal. There is no indication that the non-zero constant value is an issue at all in CEMS-flood, but it is clear that the 0 value is actually a major problem, as large parts of the world has dry enough areas often combined with small enough catchments to have near zero or totally 0 river discharge values. Some further explanation of the 0-value treatment is given in the expandable section below.
For the forecast rank computation in the 0-value singularity case, a special solution was developed. All the 0 ensemble member values (all below 0.1 m3/s) get an evenly-representing rank assigned from any of the percentiles that have 0 values (i.e. below 0.1 m3/s) in the model climatology. In practice, this will mean, the 'rank-undefined' section of the ensemble forecast is going to be spread evenly across the 'rank-undefined' section of the climatology during the rank computation.
Figure 3 demonstrates the process on an idealised example, where the lowest 77 percentiles are 0 in the climatology and 23 out of 51 ensemble members are also 0 (see Figure 3a). The 23 ensemble members with 0 value then are spread across the 0-value range of the climatology from 1 to 77 (see Figure 3b). This way the ranks of the 23 members will be assigned from 1 to 77 with equal as possible spacing in between (see Figure 3c). Finally, the remaining non-zero ensemble members also get their ranks in the usual way, as described above in Figure 2. Finally, the schematic of ranks of all 51 members are provided in Figure 3d.
a) | b) |
c) | d) |
Figure 3. Schematic of the forecast extremity ranking calculation for areas with 0 river discharge values.
In the extreme case of all climate percentiles being 0, which happen over river pixels of the driest places of the world, such as the Sahara, the ensemble forecast member ranks can either be 100 for any non-zero value, regardless of the magnitude of the river discharge, or the evenly spread ranks from 1 to 100, as a representation of the totally 0 climatology. In the absolute most extreme case of all 99 climate percentiles being 0 and all 51 members being 0 in the forecast, the ranks of the forecast will be from 1 to 100 in equal representation. This means, this forecast will be a perfect representation of the climatological distribution, or with another word a forecast showing climatologically expected probabilities for all anomaly categories from Extreme low to Extreme high.
Expected forecast anomaly category computation
The ensemble forecasts have 51 members, which will be assigned an extremity rank each. Using these 51 ranks, the forecasts will be put in one of the 7 anomaly categories (as described in Table 1). This is done based on the arithmetic mean of the 51 ensemble member rank values, which all can be from 1 to 100 (rank-mean) (see Figure 4):
This rank-mean will also be a number between 1 and 100, but this time a real (not integer) number. If the anomaly is 50.5, that is exactly the normal (median) condition, i.e. no anomaly whatsoever. If the anomaly is below 50.5, then drier than normal conditions are forecast, if above 50.5, then wetter than normal. The lower/higher the anomaly value is below/above 50.5, the drier/wetter the conditions are predicted to be. The lowest/highest possible value is 1/100, if all ensemble members are 1/100 (the most extremely dry/wet). Then, based on this rank-mean, we define the expected forecast anomaly category (one of the 7 categories in Table 1) for the whole ensemble forecast, by placing the rank-mean into the right categories, as defined in Table 1 above. For example, all rank-mean values from 40.0 to 60.0, interpreted as 40.0<= <60.0, will be assigned to 'Normal', or category-4.
The ensemble forecast anomaly was not based on the most probable of the 7 anomaly categories, as that would make it prone to jumpiness. For example, in the super uncertain case of 6, 8, 7, 7, 7, 9, 7 members being in each of the 7 anomaly categories, the forecast category (the expected one) would be the 'High' category (cat-6), as that has the most members (9). However, it is likely that nearby river pixels could easily be only slightly different with 7, 9, 7, 7, 7, 7, 7 members in each category, in which case the forecast anomaly category would be the 'Low' category (cat-2), as now that has the most (again 9) members. It is worth mentioning that very uncertain cases are especially likely to happen at longer ranges. These two forecasts are only slightly different in terms of distribution, but the expected forecast anomaly categories would be almost the complete opposite of each other, making the signal look possibly very jumpy geographically. With the mean-rank definition, we avoid this and simply assign the 'Normal' category (cat-4) for both of these forecasts, as the mean of the ranks are certainly very close and both will be quite near the median.
Figure 4. Schematic of the forecast extremity ranking of the 51 ensemble members and the calculation of the expected forecast anomaly category for the whole ensemble.
Forecast uncertainty category computation
In addition to the expected forecast anomaly computation for the whole ensemble, as one of 7 predefined categories, the forecast uncertainty is also represented in some of the sub-seasonal and seasonal products, namely on the new 'Seasonal outlook - River network' and 'Seasonal outlook - Basin summary' products. The forecast uncertainty is defined by the standard deviation (std) of the ensemble member ranks, which all can be from 1 to 100 (rank-std):
If the ensemble member ranks cluster well, and the spread of the ranks is low, then the forecast uncertainty will be low and conversely the confidence will be high. One specific example is the even distribution with 51 values spread from 1 to 100 evenly, as ranks of 1, 3, 5,..., 47, 49, 50 (or 51), 52, 54,..., 96, 98 and 100. This distribution has a mean of very close to 50.5 and a standard deviation of very close to 29.0. Then another example can be the most uneven distribution of rank values of 1, 1, 1, ..., 1, 100, 100, 100,..., 100, with either 25 values of 1 and 26 values of 100 or vice versa. In this case the rank-mean is either 49.5 or 51.5 (depending on either 1 or 100 has 26 and not 25 values) and the standard deviation is in both case the same 49.5. Another specific example is when all values are the same, so there is no variability amongst the 51 ranks at all, in which case the rank-mean is the same value and the rank-std is 0.
Based on these specific examples and the rank-std ranging from 0 to 49.5, with the climatology-matching distribution's ranks-std of 29, three uncertainty categories were defined, using the easy to remember generic category split values of 10 and 20. (Table 2). These work well enough for all lead times and geographical areas and will give an indication of generally how uncertain the expected forecast anomaly is for that pixel or area and that lead time.
| Uncertainty categories | Name | Rank STD |
|---|---|---|
| Cat-1 | Low uncertainty | 0-10 |
| Cat-2 | Medium uncertainty | 10-20 |
| Cat-3 | High uncertainty | 20< |
Table 2: Uncertainty categories defined by the standard deviation of the ensemble member ranks.
Simplified forecast examples to help interpreting the expected anomaly and uncertainty category information
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 calculations. 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.
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 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, then the rank-mean of the forecast of all 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 climatological place.
So, for the option of 10% of climatology being 0, the absolute minimum possible forecast rank-mean is 5.5, while for 30% it will be 15.5 and for the totally dry climatology, where all 99 percentiles are zero, the rank-mean will be 50.5. The extent of which the rank-mean of the forecast increases depends on how many of the ensemble members will be non-zero and with which actual rank (determined by the non-zero section of the climatology). For example, one of the most extreme cases is when all 99 climatological percentiles are 0 and none of the ensemble forecast members are 0. For this super unlikely to occur event, the rank-mean of the forecast will always be 100 (and the expected forecast anomaly category 'Extreme high'), regardless of the actual ensemble member values (i.e. how much higher they are than 0). So, even if all forecast ensemble member river discharge values are very low, say from 0.12 to 0.23, the forecast rank-mean will still be 100 and the forecast anomaly category 'Extreme high'.
In these examples, again for simplicity reasons, the climatological and forecast values will only be in one of 2 categories, either 0-value or non 0-value. This way, the main impact of the 0/non-0 value issue can be demonstrated. In the tables below, the numbers and the related average ranks are given for the two groups of 0 and non-0 ensemble members, with the rank-mean, rank-std and expected anomaly and uncertainty categories determined from those cases.
There are 4 tables, with 10% / 30% / 70% and 100% of 0 values in the climatology (i.e increasingly dry climate). For example, in the 7th row of the 1st table with 10% of 0 in the climatology, 11 ensemble members are 0-value and the remaining 40 are greater than 0. The average rank for the 0 members are 5.5 (as this is given by the method of handling the 0-value issue with equal representation, explained above), while the average rank for the non-zero members is given as 11. The related rank-mean is then 9.81, making this forecast into the 'Extreme low' expected category while the rank-std is 2.26, with low uncertainty category.
These tables demonstrate the complex interaction between the dryness of the climatology and ensemble forecasts, reflected in the forecast rank-mean and rank-std values and the subsequent expected anomaly and uncertainty categories. They also demonstrate, how less likely it becomes to have negative anomalies as the climatology becomes drier and drier.
Example No-1: No zero section in climatology, 5 number/rank groups:
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 other 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.
in a Below, there are examples with simplified rank distributions and specific cases of very dry rivers (0 values) that will demonstrate how the forecast anomaly category and uncertainty category generation work.
For example, when the X% of the climatological distribution is 0, then the average rank of the 0-value ensemble members will always be X/2+0.5 with the even rank representation for the 0-value case explained above (e.g. for 10% being 0 in the climate, the average rank of the 0-value forecast ensemble members will be 5.5).
Example No-2: Lowest 10% of the climatology is 0, 2 number/rank groups:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
|---|---|---|---|---|---|---|---|
| 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) | 0 | Low uncertainty |
| 0 | 51 | NA | 20 | (0 * 5.5 + 51 * 20)/51 = 20 | Low (10-25) | 0 | Low uncertainty |
| 0 | 51 | NA | 50 | (0 * 5.5 + 51 * 50)/51 = 50 | Near normal (40-60) | 0 | Low uncertainty |
| 0 | 51 | NA | 70 | (0 * 5.5 + 51 * 70)/51 = 70 | Bit high (60-75) | 0 | Low uncertainty |
| 0 | 51 | NA | 100 | (0 * 5.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
| 11 | 40 | 5.5 | 11 | (11 * 5.5 + 40 * 11)/51 = 9.81 | Extreme low (<10) | 2.26 | Low uncertainty |
| 11 | 40 | 5.5 | 20 | (11 * 5.5 + 40 * 20)/51 = 16.87 | Low (10-25) | 5.96 | Low uncertainty |
| 11 | 40 | 5.5 | 50 | (11 * 5.5 + 40 * 50)/51 = 40.40 | Near normal (40-60) | 18.3 | Medium uncertainty |
| 11 | 40 | 5.5 | 70 | (11 * 5.5 + 40 * 70)/51 = 56.08 | Near normal (40-60) | 26.52 | High uncertainty |
| 11 | 40 | 5.5 | 100 | (11 * 5.5 + 40 * 100)/51 = 79.61 | High (75-90) | 38.86 | High uncertainty |
| 21 | 30 | 5.5 | 11 | (21 * 5.5 + 30 * 11)/51 = 8.73 | Extreme low (<10) | 2.70 | Low uncertainty |
| 21 | 30 | 5.5 | 20 | (21 * 5.5 + 30 * 20)/51 = 14.02 | Low (10-25) | 7.13 | Low uncertainty |
| 21 | 30 | 5.5 | 50 | (21 * 5.5 + 30 * 50)/51 = 31.67 | Bit low (25-40) | 21.90 | High uncertainty |
| 21 | 30 | 5.5 | 70 | (21 * 5.5 + 30 * 70)/51 = 43.44 | Near normal (40-60) | 31.74 | High uncertainty |
| 21 | 30 | 5.5 | 100 | (21 * 5.5 + 30 * 50)/51 = 61.08 | Bit high (60-75) | 46.50 | High uncertainty |
| 36 | 15 | 5.5 | 11 | (36 * 5.5 + 15 * 11)/51 = 7.11 | Extreme low (<10) | 2.50 | Low uncertainty |
| 36 | 15 | 5.5 | 20 | (36 * 5.5 + 15 * 20)/51 = 9.76 | Extreme low (<10) | 6.60 | Low uncertainty |
| 36 | 15 | 5.5 | 50 | (36 * 5.5 + 15 * 50)/51 = 18.58 | Low (10-25) | 20.27 | High uncertainty |
| 36 | 15 | 5.5 | 70 | (36 * 5.5 + 15 * 70)/51 = 24.47 | Low (10-25) | 29.38 | High uncertainty |
| 36 | 15 | 5.5 | 100 | (36 * 5.5 + 15 * 50)/51 = 33.29 | Bit low (25-40) | 43.05 | High uncertainty |
| 51 | 0 | 5.5 | NA (no member to rank) | (51 * 5.5 + 0)/51 = 5.5 | Extreme low (<10) | 0 | Low uncertainty |
Example No-3: Lowest 30% of the climatology is 0, 2 number/rank groups:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
|---|---|---|---|---|---|---|---|
| 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) | 0 | Low uncertainty |
| 0 | 51 | NA | 50 | (0 * 15.5 + 51 * 50)/51 = 50 | Near normal (40-60) | 0 | Low uncertainty |
| 0 | 51 | NA | 70 | (0 * 15.5 + 51 * 70)/51 = 70 | Bit high (60-75) | 0 | Low uncertainty |
| 0 | 51 | NA | 100 | (0 * 15.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
| 11 | 40 | 15.5 | 31 | (11 * 15.5 + 40 * 31)/51 = 27.65 | Bit low (25-40) | 6.37 | Low uncertainty |
| 11 | 40 | 15.5 | 50 | (11 * 15.5 + 40 * 50)/51 = 42.55 | Near normal (40-60) | 14.18 | Medium uncertainty |
| 11 | 40 | 15.5 | 70 | (11 * 15.5 + 40 * 70)/51 = 58.24 | Near normal (40-60) | 22.41 | High uncertainty |
| 11 | 40 | 15.5 | 100 | (11 * 15.5 + 40 * 100)/51 = 81.77 | High (75-90) | 34.75 | High uncertainty |
| 21 | 30 | 15.5 | 31 | (21 * 15.5 + 30 * 31)/51 = 24.61 | Low (10-25) | 7.62 | Low uncertainty |
| 21 | 30 | 15.5 | 50 | (21 * 15.5 + 30 * 50)/51 = 35.79 | Bit low (25-40) | 16.97 | Medium uncertainty |
| 21 | 30 | 15.5 | 70 | (21 * 15.5 + 30 * 70)/51 = 47.55 | Near normal (40-60) | 26.82 | High uncertainty |
| 21 | 30 | 15.5 | 100 | (21 * 15.5 + 30 * 100)/51 = 65.20 | Bit high (60-75) | 41.58 | High uncertainty |
| 36 | 15 | 15.5 | 31 | (36 * 15.5 + 15 * 31)/51 = 20.05 | Low (10-25) | 7.06 | Low uncertainty |
| 36 | 15 | 15.5 | 50 | (36 * 15.5 + 15 * 50)/51 = 25.64 | Bit low (25-40) | 15.71 | Medium uncertainty |
| 36 | 15 | 15.5 | 70 | (36 * 15.5 + 15 * 70)/51 = 31.52 | Bit low (25-40) | 24.83 | High uncertainty |
| 36 | 15 | 15.5 | 100 | (36 * 15.5 + 15 * 100)/51 = 40.35 | Near normal (40-60) | 38.50 | High uncertainty |
| 51 | 0 | 15.5 | NA (no member to rank) | (51 * 15.5 + 0)/51 = 15.5 | Low (10-25) | 0 | Low uncertainty |
Example No-4: Lowest 70% of the climatology is 0, 2 number/rank groups:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
|---|---|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 71 (the lowest possible rank for a non-zero member if 1-70 percentiles in the climatology are 0) | (0 * 35.5 + 51 * 71)/51 = 71 | Bit high (60-75) | 0 | Low uncertainty |
| 0 | 51 | NA | 100 | (0 * 35.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
| 11 | 40 | 35.5 | 71 | (11 * 35.5 + 40 * 71)/51 = 63.34 | Bit high (60-75) | 14.60 | Medium uncertainty |
| 11 | 40 | 35.5 | 100 | (11 * 35.5 + 40 * 100)/51 = 86.08 | High (75-90) | 26.52 | High uncertainty |
| 21 | 30 | 35.5 | 71 | (21 * 35.5 + 30 * 71)/51 = 56.38 | Near normal (40-60) | 17.47 | Medium uncertainty |
| 21 | 30 | 35.5 | 100 | (21 * 35.5 + 30 * 100)/51 = 73.44 | Bit high (60-75) | 31.74 | High uncertainty |
| 36 | 15 | 35.5 | 71 | (36 * 35.5 + 15 * 71)/51 = 45.94 | Near normal (40-60) | 16.17 | Medium uncertainty |
| 36 | 15 | 35.5 | 100 | (36 * 35.5 + 15 * 100)/51 = 54.47 | Near normal (40-60) | 29.38 | High uncertainty |
| 51 | 0 | 35.5 | NA (no member to rank) | (51 * 35.5 + 0)/51 = 35.5 | Bit low (25-40) | 0 | Low uncertainty |
Example No-5: All percentiles of the climatology (1-99) are 0, 2 number/rank groups:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
|---|---|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 100 (the lowest possible rank for a non-zero member if 1-99 percentiles in the climatology are 0) | (0 * 50.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
| 11 | 40 | 50.5 | 100 | (11 * 50.5 + 40 * 100)/51 = 89.32 | High (75-90) | 20.35 | High uncertainty |
| 21 | 30 | 35.5 | 100 | (21 * 50.5 + 30 * 100)/51 = 79.61 | High (75-90) | 24.36 | High uncertainty |
| 36 | 15 | 35.5 | 100 | (36 * 50.5 + 15 * 100)/51 = 65.05 | Bit high (60-75) | 22.55 | High uncertainty |
| 51 | 0 | 35.5 | NA (no member to rank) | (51 * 50.5 + 0)/51 = 50.5 | Near normal (40-60) | 0 | Low uncertainty |