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title	Examples with no 0-value section in the climatology and 5 ensemble forecast rank groups only

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.

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title	Table with no 0-value section in climatology. Click here to expand...

Number of ensemble members in each group					Common rank in each group					Rank-mean	Expected forecast anomaly category	Rank-std	Forecast uncertainty category
N1	N2	N3	N4	N5	R1	R2	R3	R4	R5	Rank-mean	Expected forecast anomaly category	Rank-std	Forecast uncertainty category
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

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

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

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

Panel

title	Examples 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-value 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-value 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 an example of 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 climate becomes drier and drier.

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title	Table with 10% of 0-value in climatology. Click here to expand...

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

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title	Table with 10% 0-value in climatology. Click here to expand...

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

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title	Table with 70% of 0-value in climatology. Click here to expand...

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

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title	Table with 100% of 0-value in climatology. Click here to expand...

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

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