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titleEamples with some percentage of 0-value section in the climatology

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.

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titleTable with 10% of 0-value in climatology. Click here to expand...
Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanForecast anomaly categoryRank-stdUncertainty category
051NA (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 = 11Low (10-25)0Low uncertainty
051NA20(0 * 5.5 + 51 * 20)/51 = 20Low (10-25)0Low uncertainty
051NA50(0 * 5.5 + 51 * 50)/51 = 50Near normal (40-60)0Low uncertainty
051NA70(0 * 5.5 + 51 * 70)/51 = 70Bit high (60-75)0Low uncertainty
051NA100(0 * 5.5 + 51 * 100)/51 = 100Extreme high (90<)0Low uncertainty








11405.511(11 * 5.5 + 40 * 11)/51 = 9.81Extreme low (<10)2.26Low uncertainty
11405.520(11 * 5.5 + 40 * 20)/51 = 16.87Low (10-25)5.96Low uncertainty
11405.550(11 * 5.5 + 40 * 50)/51 = 40.40Near normal (40-60)18.3Medium uncertainty
11405.570(11 * 5.5 + 40 * 70)/51 = 56.08Near normal (40-60)26.52High uncertainty
11405.5100(11 * 5.5 + 40 * 100)/51 = 79.61High (75-90)38.86High uncertainty








21305.511

(21 * 5.5 + 30 * 11)/51 = 8.73

Extreme low (<10)2.70Low uncertainty
21305.520(21 * 5.5 + 30 * 20)/51 = 14.02Low (10-25)7.13Low uncertainty
21305.550(21 * 5.5 + 30 * 50)/51 = 31.67Bit low (25-40)21.90High uncertainty
21305.570(21 * 5.5 + 30 * 70)/51 = 43.44Near normal (40-60)31.74High uncertainty
21305.5100(21 * 5.5 + 30 * 50)/51 = 61.08Bit high (60-75)46.50High uncertainty








36155.511

(36 * 5.5 + 15 * 11)/51 = 7.11

Extreme low (<10)2.50Low uncertainty
36155.520(36 * 5.5 + 15 * 20)/51 = 9.76Extreme low (<10)6.60Low uncertainty
36155.550(36 * 5.5 + 15 * 50)/51 = 18.58Low (10-25)20.27High uncertainty
36155.570(36 * 5.5 + 15 * 70)/51 = 24.47Low (10-25)29.38High uncertainty
36155.5100(36 * 5.5 + 15 * 50)/51 = 33.29Bit low (25-40)43.05High uncertainty








5105.5NA (no member to rank)

(51 * 5.5 + 0)/51 = 5.5

Extreme low (<10)0Low uncertainty
Expand
titleTable with 30% of 10% 0-value in climatology. Click here to expand...
Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanForecast anomaly categoryRank-stdUncertainty category
051NA (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 = 31Bit low (25-40)0Low uncertainty
051NA50(0 * 15.5 + 51 * 50)/51 = 50Near normal (40-60)0Low uncertainty
051NA70(0 * 15.5 + 51 * 70)/51 = 70Bit high (60-75)0Low uncertainty
051NA100(0 * 15.5 + 51 * 100)/51 = 100Extreme high (90<)0Low uncertainty








114015.531(11 * 15.5 + 40 * 31)/51 = 27.65Bit low (25-40)6.37Low uncertainty
114015.550(11 * 15.5 + 40 * 50)/51 = 42.55Near normal (40-60)14.18Medium uncertainty
114015.570(11 * 15.5 + 40 * 70)/51 = 58.24Near normal (40-60)22.41High uncertainty
114015.5100(11 * 15.5 + 40 * 100)/51 = 81.77High (75-90)34.75High uncertainty








213015.531

(21 * 15.5 + 30 * 31)/51 = 24.61

Low (10-25)7.62Low uncertainty
213015.550(21 * 15.5 + 30 * 50)/51 = 35.79Bit low (25-40)16.97Medium uncertainty
213015.570(21 * 15.5 + 30 * 70)/51 = 47.55Near normal (40-60)26.82High uncertainty
213015.5100

(21 * 15.5 + 30 * 100)/51 = 65.20

Bit high (60-75)41.58High uncertainty








361515.531

(36 * 15.5 + 15 * 31)/51 = 20.05

Low (10-25)7.06Low uncertainty
361515.550(36 * 15.5 + 15 * 50)/51 = 25.64Bit low (25-40)15.71Medium uncertainty
361515.570(36 * 15.5 + 15 * 70)/51 = 31.52Bit low (25-40)24.83High uncertainty
361515.5100(36 * 15.5 + 15 * 100)/51 = 40.35Near normal (40-60)38.50High uncertainty








51015.5NA (no member to rank)

(51 * 15.5 + 0)/51 = 15.5

Low (10-25)0Low uncertainty
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titleTable with 70% of 0-value in climatology. Click here to expand...
Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanForecast anomaly categoryRank-stdUncertainty category
051NA (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 = 71Bit high (60-75)0Low uncertainty
051NA100(0 * 35.5 + 51 * 100)/51 = 100Extreme high (90<)0Low uncertainty
114035.571(11 * 35.5 + 40 * 71)/51 = 63.34Bit high (60-75)14.60Medium uncertainty
114035.5100(11 * 35.5 + 40 * 100)/51 = 86.08High (75-90)26.52High uncertainty
213035.571

(21 * 35.5 + 30 * 71)/51 = 56.38

Near normal (40-60)17.47Medium uncertainty
213035.5100

(21 * 35.5 + 30 * 100)/51 = 73.44

Bit high (60-75)31.74High uncertainty
361535.571

(36 * 35.5 + 15 * 71)/51 = 45.94

Near normal (40-60)16.17Medium uncertainty
361535.5100(36 * 35.5 + 15 * 100)/51 = 54.47Near normal (40-60)29.38High uncertainty
51035.5NA (no member to rank)

(51 * 35.5 + 0)/51 = 35.5

Bit low (25-40)0Low uncertainty
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titleTable with 100% of 0-value in climatology. Click here to expand...
Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanForecast anomaly categoryRank-stdUncertainty category
051NA (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 = 100Extreme high (90<)0Low uncertainty
114050.5100(11 * 50.5 + 40 * 100)/51 = 89.32High (75-90)20.35High uncertainty
213035.5100

(21 * 50.5 + 30 * 100)/51 = 79.61

High (75-90)24.36High uncertainty
361535.5100(36 * 50.5 + 15 * 100)/51 = 65.05Bit high (60-75)22.55High uncertainty
51035.5NA (no member to rank)

(51 * 50.5 + 0)/51 = 50.5

Near normal (40-60)0Low uncertainty



Example No-1: No zero section in climatology, 5 number/rank groups:

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