In this paper, an analysis of adapted 20 extreme precipitation indices is calculated for a limited mountain area in southern Vietnam. The daily precipitation data from four stations in the period of more than 30 years are selected. The statistical characteristics of maximum, minimum, mean, standard deviation, skewness, and kurtoris for each index are also analysed. A variety of distributions such as Normal, Lognormal, Beta, Gamma, Exponential, Loglogistic, and Johnson is used to find the best fit probability distribution for this area on the basic of the highest score. The scores are estimated based on the ranking of statistical goodness of fit test. The goodness of fit tests is the Anderson-Darling and Shapiro-Wilks tests. The best fit distribution for each index of extreme precipitation at each station is found out. Results revealed that the Johnson distribution is the best fit distribution to the data of very heavy precipitation days greater than 50 mm. Over a limited mountain area, it is difficult to fit a probability distribution to the precipitation fraction due to extremely wet days, number of extremely wet days, and number of extremely wet days when precipitation greater than 99 percentage. The lognormal, Johnson, and Loglogistic distribution are the best choices to fit most of the extreme precipitation indices over this area.
In the field of climate research, precipitation is considered to be one of the key terms for balancing the energy budget, and one of the most challenging aspects of climate modeling, especially for convection precipitation parameterization schemes. Therefore, high quality estimations of precipitation distribution, amounts and intensity are important to fully interpret the climate regime and the effects of climate on other fields (e.g., water and agriculture) at a variety of scales from global to local. For daily precipitation, a Markow chain was firstly suggested for the representation of the sequence of wet and dry days [
Wilks [
Semenow [
Olofintoye [
Su [
In most of the hydrological models, a long series of precipitation data is significantly of great importance to assess the effects of precipitation regime on water resources, environmental and agricultural planning fields. A temporally and spatially continuous distribution of precipitation can robust the quality of computing results. However, it is quite difficult to find enough evenly spread weather variables like precipitation to cover the entire nation in general and the area of interest in particular, both in space and time resolution. The reason for this is the limitations of observation system (e.g., density of precipitation measuring network). Furthermore, insufficient terrestrial meteorological observations are considered to be the most important source of uncertainty in the different studies (e.g., [
Located on the eastern margin of the Indochinese peninsula, Vietnam’s climate is strongly dominated by a typical climate of tropical monsoon of a peninsula in the Southeast of the European-Asian continent. The geographical characteristics of Vietnam is a long coastline of 3260 kilometers, series of mountain in the direction from northwest to southeast (in the northeast mountain regions), from west to east (in the central regions), central highland, and a limited low mountain area in the Southeast Vietnam. Besides that, with a long and narrow shape, the climate of Vietnam is significantly complex from place to place and time to time. Under a global warming, change in extreme weather events (e.g., heavy precipitation) is unevenly, especially for a limited mountain area in southern Vietnam in which the weather patterns are dominated by a variety of natural conditions (e.g., oceanic air mass and elevation). Thus, it is a cruel task to analyse and define the variability in extreme precipitation characteristics as well as its distribution to get significant information for a limited area. This information will be very useful for different objects, especially in the studies in flood events, urban flooding, flash flood, or environmental planning fields.
As pointed out by the Expert Team of the World Meteorological Organization and Climate Variability and Predictability on Climate Change Detection and Indices, 11 [
A limited mountain area of southern Vietnam is selected to investigate extreme precipitation indices (
In this study, the distribution of probability included normal, lognormal, gamma, loglogistic, beta, Johnson, and exponential were taken into account to define the best fit probability distribution for extreme precipitation. The description of various probability distribution functions and density functions, ranges and the parameters involved are shown in
Nr. | Station | Station code | Coordinates | Elevation (m a.s.l) | Period | |
---|---|---|---|---|---|---|
Latitude (N) | Longitude (E) | |||||
1. | Long Thanh | 30901006 | 10˚27 | 106˚33 | 41 | 1978-2013 |
2. | Cam My | 30901004 | 10˚51 | 107˚09 | 145 | 1981-2013 |
3. | Thong Nhat | 30901013 | 10˚57 | 107˚00 | 61 | 1981-2010 |
4. | Xuan Loc | 30901016 | 10˚57’ | 107˚14’ | 190 | 1990-2010 |
Nr. | Indices | Abbrev. | Unit | Nr. | Indices | Abbrev. | Unit |
---|---|---|---|---|---|---|---|
1. | Annual total precipitation | RR | mm | 2. | No. of wet days (≥0.5 mm) | RR5 | days |
3. | Moderate precipitation days (≥10 mm) | RR10 | days | 4. | Heavy precipitation days (≥25 mm) | RR25 | days |
5. | Very heavy precipitation days (≥50 mm) | RR50 | days | 6. | Extreme heavy precipitation days (≥100 mm) | RR100 | days |
7. | Highest 1-day precipitation amount | RX1day | mm | 8. | Highest 3-day precipitation amount | RX3day | mm |
9. | Highest 5-day precipitation amount | RX5day | mm | 10. | Highest 7-day precipitation amount | RX7day | mm |
11. | Maximum no. of consecutive wet days | MCWD | days | 12. | Maximum no. of consecutive dry days | MCDD | days |
13. | Simple precipitation intensity index | SDII | mm/wd | 14. | No. of moderate wet days >75 percentage | R75p | days |
15. | Precipitation fraction due to moderate wet days | R75pTOT | % | 16. | No. of very wet days >95 percentage | R95p | days |
17. | Precipitation fraction due to very wet days | R95pTOT | % | 18. | No. of extremely wet days >99 percentage | R99p | days |
19. | Precipitation fraction due to extremely wet days | R99pTOT | % | 20. | Annual total precipitation in wet days | PRCPTOT | mm |
Distributions | Probability density functions | Ranges | Parameters |
---|---|---|---|
Normal | −∞ < x < ∞ | α: standard deviation β: mean | |
Beta | 0 ≤ x ≤ 1 α, β > 0 | α: shape parameter β: scale parameter | |
Exponential | γ ≤ x < ∞ -∞ < α < ∞ β > 0 | α: location parameter β: scale parameter | |
Gamma | γ ≤ x < ∞ −∞ < α < ∞ β > 0 γ > 0 | α: location parameter β: scale parameter γ: shape parameter | |
Loglogistic | −∞ < x < ∞ −∞ < α < ∞ β > 0 | α: location parameter β: scale parameter | |
Lognormal | γ < x < ∞ −∞ < α < ∞ −∞ < β < ∞ γ > 0 | α: location parameter β: scale parameter γ: shape parameter | |
Johnson | m − γ/2 < x < m + γ/2 −∞ < m < ∞ −∞ < β < ∞ α > 0 γ > 0 | m: location parameter α: shape parameter β: shape parameter γ: scale parameter |
The goodness of fit test refers to measuring how well do the compatibility of random sample with the theoretical probability distribution. A goodness of fit statistic tests is applied for testing the following null hypothesis:
H0: the model of extreme precipitation parameter fits the specified distribution
HA: the model of extreme precipitation parameter does not fit the specified distribution.
The Anderson-Darling and Shapiro-Wilks tests were used to identify if a sample comes from a population with a specific distribution. The chi-square test at α (0.05) level of significance for the selection of the best fit probability distribution was applied. Several studies related to these tests can be found in [
where
This test allows comparing the fit of an observed cumulative distribution function to an expected cumulative distribution function.
The Shapiro-Wilks (SW) test, suggested in [
where
ai are constants generated from the means, variances and covariances of the order statistics of a sample of size n from a normal distribution.
Statistical characteristics of extreme precipitation indices for a limited mountain area in Vietnam are shown in Supplementary (from Tables 5-8). They are mean, standard deviation, skewness, kurtoris, maximum, and minimum values. It was observed that the maximum and minimum of very heavy precipitation days and extreme heavy precipitation can be reached sixteen and thirteen days in a year at Long Thanh station, respectively. The maximum of maximum number of consecutive wet days is calculated for Long Thanh and Cam My station (49 days), followed by Xuan Loc (36 days). Meanwhile, the maximum of maximum number of consecutive dry days is calculated for Cam My station (126 days), followed by Long Thanh station (114 days). Specifically, the maximum of highest 7-day precipitation amount are calculated for Thong N hat station (508.9 mm), followed by Xuan Loc station (442.2 mm). The maximum value of coefficient of skewness was observed at the station Xuan Loc for the maximum number of consecutive wet days.
In this study, the statistic of each test were tested at α = 0.05 level of significance. Based on minimum test statistic value, the ranking of different probability distributions were marked from 1 to 7 for the Anderson-Darling and Shapiro-Wilks tests of mentioned probability distributions. In case the value of test is not significant at α level, they are marked as zero. To find the best fit distribution, the maximum score of probability distribution was totaled based on the cumulative ranking. With the highest scored obtained is selected as the best fit distribution. The value of test is additionally considered as a criterion in case the same scores are seen between the probability distributions. The p-values of statistical tests are presented in Supplementary (from Tables 9-12).
Based on the calculated total test score obtained for each index for seven probability distributions, the best selected probability distributions for each data set are presented in
Stations Indices | Thong Nhat | Long Thanh | Cam My | Xuan Loc |
---|---|---|---|---|
RR | Lognormal | Loglogistic’ | Lognormal | Johnson |
RR5 | Gamma | Johnson | Gamma | Johnson |
RR10 | Lognormal | Johnson | Johnson | Loglogistic |
RR25 | Beta | Lognormal | Lognormal | Johnson |
RR50 | Johnson | Johnson | Johnson | Johnson |
RR100 | 0 | 0 | 0 | 0 |
RX1day | Lognormal | Johnson | Lognormal | Johnson |
RX3day | Johnson | Lognormal | Loglogistic | Johnson |
RX5day | Johnson | Lognormal | Loglogistic | Johnson |
RX7day | Lognormal | Lognormal | Loglogistic | Johnson |
MCWD | Johnson | Gamma | Johnson | Johnson |
MCDD | Gamma | Lognormal | Beta | Lognormal |
SDII | Johnson | Lognormal | Johnson | Loglogistic |
R75p | Gamma | Johnson | Lognormal | Loglogistic |
R75pTOT | Johnson | Lognormal | Lognormal | Loglogistic |
R95p | Gamma | Johnson | Beta | Loglogistic |
R95pTOT | Lognormal | Johnson | Gamma | Loglogistic |
R99p | 0 | 0 | 0 | 0 |
R99pTOT | 0 | 0 | 0 | Lognormal |
PRCPTOT | Lognormal | Loglogistic | Lognormal | Johnson |
extremely wet days when precipitation greater than 99 percentage, and extreme heavy precipitation days (≥100 mm) at α level except the lognormal distribution is the best fit distribution to the precipitation fraction due to extremely wet days at Xuan Loc station.
It was shown that the best fit distributions of lognormal, Johnson, and loglositic fit to most of the extreme precipitation indices over this area. Specially, it was found out that the Johnson distribution is the best fit distribution to the data of very heavy precipitation days greater than 50 mm for a limited mountain area as presented in
The results indicated a large range of fluctuation during the period of study for the maximum number of consecutive wet days from 4 days (minimum) to 49 days (maximum) and the maximum number of consecutive dry days from 13 days (minimum) to 126 days (maximum), respectively. The number of heavy precipitation could be up to 25% of days in a year (e.g., at Xuan Loc station). The maximum of annual precipitation (nearly 2900 mm) was seen at Long Thanh station. Analysis results revealed a potential precipitation amount over this area. The highest precipitation amount of 1, 3, 5, and 7-days could significantly contribute to potential the extreme flood events due to a large recorded precipitation amount.
It was seen that the best probability distributions were different for different extreme precipitation indices. In general, the distributions of Johnson, Loglogistic, and Lognormal are the best choices for most of extreme precipitation indices for a limited mountain area. Over this area, the best probability distributions are Lognormal and Loglogistic for the highest precipitation amount of 3, 5, and 7 days, respectively. Therefore, the author gives a recommendation that it should be firstly investigated the Lognormal, Loglogistic, and Johnson distributions in the studies dealing with extreme precipitation indices for other limited mountain areas in which are normally challenging to gather data.
Thanh, N.T. (2017) Fitting a Probability Distribution to Extreme Precipitation for a Limited Mountain Area in Vietnam. Journal of Geoscience and Environment Protection, 5, 92-107. https://doi.org/10.4236/gep.2017.55007
Maximum | Minimum | Mean | Standard deviation | Skewness | Kurtoris | |
---|---|---|---|---|---|---|
RR | 2645.0 | 1677.6 | 2140.1 | 227.0 | 0.4 | 0.5 |
RR5 | 170.0 | 59.0 | 81.1 | 23.2 | 3.0 | 11.3 |
RR10 | 89.0 | 25.0 | 38.2 | 13.7 | 2.9 | 9.8 |
RR25 | 26.0 | 0.0 | 18.2 | 5.4 | −1.9 | 5.8 |
RR50 | 11.0 | 0.0 | 6.7 | 2.7 | −0.3 | 0.9 |
RR100 | 3.0 | 0.0 | 1.1 | 1.0 | 0.5 | −0.8 |
RX1day | 246.7 | 23.8 | 109.6 | 44.3 | 1.4 | 4.2 |
RX3day | 313.7 | 51.9 | 167.8 | 52.6 | 0.5 | 2.5 |
RX5day | 375.2 | 78.3 | 206.9 | 59.5 | 0.8 | 2.8 |
RX7day | 442.2 | 105.6 | 242.2 | 73.4 | 1.1 | 2.3 |
MCWD | 36.0 | 10.0 | 28.8 | 50.1 | 4.5 | 20.4 |
MCDD | 105.0 | 15.0 | 53.0 | 24.5 | 0.5 | −0.1 |
SDII | 17.7 | 8.0 | 15.0 | 2.1 | −1.9 | 5.6 |
R75p | 46.0 | 7.0 | 36.3 | 8.1 | −2.4 | 8.1 |
R75pTOT | 77.6 | 8.9 | 68.3 | 14.2 | −4.0 | 17.2 |
R95p | 11.0 | 0.0 | 7.3 | 3.0 | −0.6 | 0.0 |
R95pTOT | 48.0 | 0.0 | 31.7 | 11.0 | −1.3 | 2.4 |
R99p | 4.0 | 0.0 | 1.5 | 1.3 | 0.7 | −0.5 |
R99pTOT | 26.9 | 0.0 | 10.6 | 8.5 | 0.5 | −0.8 |
PRCPTOT | 2639.4 | 1675.7 | 2136.8 | 226.3 | 0.4 | 0.5 |
Maximum | Minimum | Mean | Standard deviation | Skewness | Kurtoris | |
---|---|---|---|---|---|---|
RR | 2870.9 | 1260.6 | 1983.2 | 311.8 | 0.3 | 1.1 |
RR5 | 113.0 | 11.0 | 63.9 | 30.3 | −0.2 | −1.2 |
RR10 | 72.0 | 18.0 | 35.3 | 10.3 | 1.4 | 3.4 |
RR25 | 29.0 | 10.0 | 18.2 | 4.5 | 0.0 | −0.1 |
RR50 | 13.0 | 0.0 | 6.5 | 2.8 | 0.2 | 0.2 |
RR100 | 3.0 | 0.0 | 0.8 | 1.0 | 0.9 | −0.1 |
RX1day | 226.5 | 39.9 | 106.7 | 40.6 | 1.4 | 2.0 |
RX3day | 307.2 | 90.4 | 160.4 | 51.5 | 1.3 | 1.5 |
RX5day | 350.2 | 118.5 | 195.0 | 57.5 | 1.3 | 1.5 |
RX7day | 432.2 | 144.7 | 226.7 | 65.7 | 1.5 | 2.4 |
MCWD | 49.0 | 4.0 | 14.8 | 9.5 | 1.7 | 3.7 |
MCDD | 114.0 | 22.0 | 61.1 | 25.0 | 0.3 | −0.7 |
SDII | 31.9 | 9.8 | 17.0 | 4.8 | 1.0 | 1.2 |
R75p | 44.0 | 17.0 | 30.6 | 6.4 | −0.1 | −0.5 |
R75pTOT | 82.7 | 33.4 | 64.0 | 12.9 | −0.9 | 0.5 |
R95p | 13.0 | 0.0 | 6.0 | 2.9 | 0.2 | −0.2 |
R95pTOT | 42.6 | 0.0 | 22.6 | 11.1 | 0.0 | −0.9 |
R99p | 5.0 | 0.0 | 1.2 | 1.2 | 1.1 | 1.7 |
R99pTOT | 26.4 | 0.0 | 6.8 | 6.8 | 0.9 | 0.5 |
PRCPTOT | 2868.0 | 1260.6 | 1980.4 | 311.7 | 0.3 | 1.0 |
Maximum | Minimum | Mean | Standard deviation | Skewness | Kurtoris | |
---|---|---|---|---|---|---|
RR | 2460.6 | 624.3 | 1646.9 | 490.3 | −0.6 | −0.3 |
RR5 | 94.0 | 34.0 | 61.8 | 14.8 | −0.2 | −0.6 |
RR10 | 49.0 | 17.0 | 34.3 | 7.8 | −0.1 | −0.6 |
RR25 | 28.0 | 0.0 | 14.9 | 7.9 | −0.6 | −0.3 |
RR50 | 10.0 | 0.0 | 3.4 | 3.0 | 0.5 | −1.0 |
RR100 | 3.0 | 0.0 | 0.8 | 0.8 | 0.9 | 0.5 |
RX1day | 169.0 | 18.1 | 84.6 | 38.3 | 0.0 | −0.3 |
RX3day | 323.8 | 35.2 | 135.9 | 62.7 | 0.7 | 1.2 |
RX5day | 347.3 | 45.4 | 166.8 | 67.4 | 0.4 | 0.6 |
RX7day | 366.7 | 60.1 | 194.4 | 72.3 | 0.0 | 0.2 |
MCWD | 49.0 | 4.0 | 12.9 | 8.4 | 2.9 | 10.5 |
MCDD | 126.0 | 31.0 | 74.8 | 28.4 | 0.1 | −1.2 |
SDII | 19.6 | 6.7 | 14.1 | 3.1 | −0.8 | 0.5 |
R75p | 50.0 | 1.0 | 28.7 | 12.7 | −0.9 | 0.0 |
R75pTOT | 73.3 | 2.8 | 56.1 | 21.0 | −1.6 | 1.2 |
R95p | 13.0 | 0.0 | 5.8 | 4.3 | 0.1 | −1.2 |
R95pTOT | 41.3 | 0.0 | 19.5 | 13.5 | −0.1 | −1.2 |
R99p | 4.0 | 0.0 | 1.2 | 1.2 | 0.8 | −0.3 |
R99pTOT | 20.2 | 0.0 | 6.2 | 5.9 | 0.7 | −0.3 |
PRCPTOT | 2460.6 | 624.3 | 1646.7 | 490.3 | −0.6 | −0.3 |
Maximum | Minimum | Mean | Standard deviation | Skewness | Kurtoris | |
---|---|---|---|---|---|---|
RR | 2394.7 | 1307.4 | 1856.0 | 256.8 | 0.1 | −0.1 |
RR5 | 85.0 | 10.0 | 60.9 | 17.3 | −1.0 | 1.1 |
RR10 | 46.0 | 22.0 | 32.5 | 5.8 | 0.2 | 0.0 |
RR25 | 25.0 | 10.0 | 17.4 | 4.6 | 0.1 | −1.1 |
RR50 | 10.0 | 1.0 | 5.5 | 2.3 | 0.0 | −0.5 |
RR100 | 3.0 | 0.0 | 0.8 | 0.9 | 0.6 | −0.8 |
RX1day | 241.7 | 53.1 | 105.8 | 41.9 | 1.6 | 2.8 |
RX3day | 429.3 | 73.8 | 153.4 | 67.7 | 2.6 | 9.0 |
RX5day | 453.4 | 88.1 | 196.9 | 72.7 | 1.8 | 4.4 |
RX7day | 508.9 | 139.2 | 232.0 | 76.2 | 2.0 | 5.5 |
MCWD | 31.0 | 5.0 | 13.8 | 5.8 | 1.2 | 1.6 |
MCDD | 111.0 | 13.0 | 65.1 | 26.4 | 0.1 | −0.8 |
SDII | 26.8 | 11.5 | 15.9 | 3.3 | 1.7 | 3.5 |
R75p | 40.0 | 17.0 | 29.7 | 5.6 | −0.3 | −0.2 |
R75pTOT | 79.5 | 42.9 | 64.2 | 7.3 | −0.6 | 1.7 |
R95p | 10.0 | 1.0 | 6.0 | 2.5 | 0.1 | −0.8 |
R95pTOT | 44.7 | 3.4 | 23.2 | 9.4 | 0.3 | 0.2 |
R99p | 5.0 | 0.0 | 1.2 | 1.3 | 1.0 | 0.6 |
R99pTOT | 22.6 | 0.0 | 6.9 | 7.4 | 0.7 | −0.9 |
PRCPTOT | 2388.7 | 1306.7 | 1852.9 | 256.1 | 0.1 | −0.1 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Indices | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test |
RR | 0.9166 | 0.9552 | 0.9304 | 0.9677 | 0.8309 | 0.937 | 0.9299 | 0.9677 | 0 | 0 | 0.9274 | 0.9199 | 0.8636 | 0.9514 |
RR5 | 0.0894 | 0.0395 | 0.9633 | 0.9797 | 0.5064 | 0.3962 | 0.9827 | 0.9941 | 0 | 0 | 0.5014 | 0.4301 | 0.9784 | 0.9961 |
RR10 | 0.8456 | 0.8404 | 0.8511 | 0.8746 | 0.3977 | 0.6385 | 0.8348 | 0.8663 | 0.0114 | 0.0104 | 0.8705 | 0.8217 | 0.6232 | 0.7856 |
RR25 | 0.2677 | 0.157 | 0.3557 | 0.19 | 0.7374 | 0.696 | 0.3904 | 0.2006 | 0.0715 | 0.0224 | 0.1748 | 0.0944 | 0.6281 | 0.6498 |
RR50 | 0.1088 | 0.2523 | 0.1081 | 0.256 | 0.0997 | 0.211 | 0.1081 | 0.2561 | 0.0041 | 0.0014 | 0.0642 | 0.16 | 0.1233 | 0.2813 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0.0028 | 0.0013 | 0.9885 | 0.9972 | 0.02 | 0.0134 | 0.8604 | 0.9483 | 0.1279 | 0.0157 | 0.3056 | 0.3251 | 0.6371 | 0.7417 |
RX3day | 0 | 0 | 0.2263 | 0.44 | 0 | 0 | 0.082 | 0.1365 | 0.0028 | 0 | 0.2054 | 0.4498 | 0.8375 | 0.9354 |
RX5day | 0.0012 | 0 | 0.4147 | 0.6191 | 0.0573 | 0.0793 | 0.2081 | 0.3465 | 0.0005 | 0 | 0.5275 | 0.7779 | 0.9983 | 0.9999 |
RX7day | 0.0008 | 0 | 0.4058 | 0.54 | 0.0038 | 0 | 0.1908 | 0.3115 | 0.0028 | 0 | 0.5701 | 0.5374 | 0.3953 | 0.2813 |
MCWD | 0.0188 | 0.0194 | 0.638 | 0.8772 | 0.0752 | 0.1315 | 0.5175 | 0.814 | 0.3068 | 0.3678 | 0.4319 | 0.6213 | 0.6392 | 0.8781 |
MCDD | 0.6032 | 0.5226 | 0.652 | 0.562 | 0.0981 | 0.1117 | 0.7079 | 0.6012 | 0.0278 | 0 | 0.4251 | 0.3278 | 0.5233 | 0.6104 |
SDII | 0.0005 | 0 | 0.451 | 0.7735 | 0.0029 | 0.0045 | 0.2393 | 0.527 | 0.0088 | 0 | 0.3861 | 0.5101 | 0.9111 | 0.9507 |
R75p | 0.5962 | 0.7413 | 0.6882 | 0.8296 | 0.5096 | 0.7832 | 0.6885 | 0.8307 | 0.0045 | 0 | 0.5495 | 0.6487 | 0.5977 | 0.8395 |
R75pTOT | 0.1431 | 0.1862 | 0.2506 | 0.3996 | 0.142 | 0.1839 | 0.2373 | 0.3817 | 0 | 0 | 0.5026 | 0.7416 | 0.7694 | 0.8328 |
R95p | 0.0375 | 0.0806 | 0.0437 | 0.0905 | 0.0276 | 0.0653 | 0.0531 | 0.1026 | 0.037 | 0.0136 | 0.0212 | 0.0484 | 0.0132 | 0.0355 |
R95pTOT | 0.8246 | 0.8216 | 0.9429 | 0.9566 | 0.8972 | 0.921 | 0.9413 | 0.9551 | 0 | 0 | 0.9211 | 0.9253 | 0.9226 | 0.9408 |
R99p | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
R99pTOT | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
PRCPTOT | 0.9207 | 0.95 | 0.9359 | 0.9674 | 0.8411 | 0.936 | 0.9357 | 0.9674 | 0 | 0 | 0.9264 | 0.9158 | 0.871 | 0.9501 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | |
RR | 0.8816 | 0.8991 | 0.9272 | 0.9761 | 0.8813 | 0.8988 | 0.9346 | 0.98 | 0 | 0 | 0.9998 | 1 | 0.9998 | 1 |
RR5 | 0.0297 | 0.0401 | 0.0521 | 0.0588 | 0.2185 | 0.3597 | 0.1743 | 0.1165 | 0.0003 | 0.0005 | 0.0218 | 0.0252 | 0.2297 | 0.3973 |
RR10 | 0.0263 | 0.0056 | 0.5318 | 0.7949 | 0.3361 | 0.4842 | 0.4345 | 0.6568 | 0.0882 | 0.0396 | 0.5551 | 0.8473 | 0.6877 | 0.9294 |
RR25 | 0.2656 | 0.3691 | 0.2682 | 0.3713 | 0.1606 | 0.2494 | 0.2682 | 0.3713 | 0.0001 | 0.0002 | 0.2611 | 0.3304 | 0.2056 | 0.3021 |
RR50 | 0.322 | 0.5833 | 0.3808 | 0.6463 | 0.3218 | 0.5831 | 0.3818 | 0.6452 | 0.0243 | 0.0029 | 0.4126 | 0.6379 | 0.4163 | 0.6714 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0.0004 | 0.001 | 0.1986 | 0.3748 | 0.0283 | 0.0813 | 0.0989 | 0.228 | 0.0003 | 0 | 0.284 | 0.4345 | 0.9416 | 0.9334 |
RX3day | 0.0002 | 0.0007 | 0.4423 | 0.6359 | 0.0018 | 0.006 | 0.2453 | 0.4929 | 0.0113 | 0.002 | 0.1105 | 0.1507 | 0.129 | 0.3481 |
RX5day | 0.0006 | 0.0006 | 0.4715 | 0.5135 | 0.0021 | 0.0076 | 0.3141 | 0.454 | 0.023 | 0.0041 | 0.2927 | 0.1568 | 0.0665 | 0.1863 |
RX7day | 0.0003 | 0.0003 | 0.6563 | 0.8078 | 0.0064 | 0.0139 | 0.4084 | 0.7206 | 0.0409 | 0.0138 | 0.383 | 0.2491 | 0.1591 | 0.4311 |
MCWD | 0.0016 | 0.0003 | 0.9106 | 0.9279 | 0.2729 | 0.0509 | 0.9814 | 0.9936 | 0.9813 | 0.9938 | 0.2533 | 0.1718 | 0.9104 | 0.9276 |
MCDD | 0.6074 | 0.3296 | 0.6083 | 0.3682 | 0.4888 | 0.6376 | 0.4658 | 0.3067 | 0.0167 | 0.0144 | 0.4539 | 0.2261 | 0.4152 | 0.7461 |
SDII | 0.0276 | 0.0256 | 0.4568 | 0.6597 | 0.0002 | 0.0001 | 0.3854 | 0.6247 | 0 | 0 | 0.3072 | 0.4141 | 0.4568 | 0.6594 |
R75p | 0.9136 | 0.9612 | 0.9204 | 0.9672 | 0.9224 | 0.991 | 0.9208 | 0.9675 | 0.0128 | 0.0019 | 0.7025 | 0.7481 | 0.9428 | 0.9948 |
R75pTOT | 0.0092 | 0.0077 | 0.4566 | 0.4541 | 0.0002 | 0.0014 | 0.3805 | 0.4225 | 0 | 0 | 0.2914 | 0.1834 | 0.0739 | 0.1742 |
R95p | 0.6131 | 0.7842 | 0.5958 | 0.8178 | 0.6135 | 0.8384 | 0.5879 | 0.8655 | 0.0747 | 0.0284 | 0.4297 | 0.5836 | 0.6156 | 0.8443 |
R95pTOT | 0.547 | 0.4121 | 0.5386 | 0.4095 | 0.5873 | 0.6555 | 0.5386 | 0.4096 | 0 | 0 | 0.3629 | 0.2275 | 0.6974 | 0.848 |
R99p | 0.0001 | 0.0002 | 0 | 0.0001 | 0.0002 | 0.0003 | 0.0001 | 0.0005 | 0.0001 | 0.0005 | 0.0001 | 0.0007 | 0.0001 | 0.0003 |
R99pTOT | 0.0021 | 0.0012 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0027 | 0 | 0.0001 | 0 |
PRCPTOT | 0.8811 | 0.8967 | 0.9289 | 0.9769 | 0.8808 | 0.8964 | 0.9355 | 0.9805 | 0 | 0 | 0.9997 | 1 | 0.9996 | 1 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | |
RR | 0.0128 | 0.0368 | 0.2437 | 0.3483 | 0.0001 | 0.0006 | 0.2301 | 0.3454 | 0 | 0 | 0.0688 | 0.1168 | 0.0654 | 0.2388 |
RR5 | 0.2419 | 0.348 | 0.3235 | 0.4108 | 0.1299 | 0.2357 | 0.3688 | 0.4257 | 0.0004 | 0.0005 | 0.1852 | 0.2409 | 0.2552 | 0.3744 |
RR10 | 0.7806 | 0.8397 | 0.8169 | 0.8676 | 0.8385 | 0.9162 | 0.8344 | 0.8732 | 0.0149 | 0.0006 | 0.5504 | 0.5825 | 0.9019 | 0.9565 |
RR25 | 0.0105 | 0.0188 | 0.2274 | 0.2293 | 0.0016 | 0.0106 | 0.2073 | 0.2211 | 0 | 0 | 0.0884 | 0.0867 | 0.0004 | 0.0042 |
RR50 | 0.0036 | 0.005 | 0.0005 | 0.0009 | 0.0463 | 0.0346 | 0.0059 | 0.0061 | 0.0059 | 0.0061 | 0.0023 | 0.0037 | 0.0529 | 0.0419 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0.294 | 0.3634 | 0.3019 | 0.3675 | 0.0014 | 0.0065 | 0.3019 | 0.3675 | 0 | 0 | 0.3173 | 0.3105 | 0.0727 | 0.2294 |
RX3day | 0.4027 | 0.2004 | 0.3663 | 0.5802 | 0.1125 | 0.3266 | 0.3109 | 0.559 | 0.0004 | 0.0003 | 0.5199 | 0.5838 | 0.2887 | 0.5335 |
RX5day | 0.146 | 0.24 | 0.1044 | 0.3155 | 0.0551 | 0.2586 | 0.1061 | 0.3171 | 0 | 0 | 0.2024 | 0.3711 | 0.1129 | 0.3218 |
RX7day | 0.0412 | 0.1206 | 0.0388 | 0.1195 | 0.0065 | 0.061 | 0.0388 | 0.1195 | 0 | 0 | 0.0641 | 0.1438 | 0.0348 | 0.1126 |
MCWD | 0 | 0 | 0.3195 | 0.4954 | 0.0008 | 0.0002 | 0.0609 | 0.0671 | 0.0615 | 0.0683 | 0.3303 | 0.4461 | 0.7727 | 0.9552 |
MCDD | 0.2149 | 0.1374 | 0.2191 | 0.139 | 0.9117 | 0.9706 | 0.2197 | 0.1394 | 0.0176 | 0.0094 | 0.1089 | 0.0676 | 0.6613 | 0.8444 |
SDII | 0.0529 | 0.0732 | 0.4907 | 0.7059 | 0.2168 | 0.4098 | 0.4431 | 0.6684 | 0 | 0 | 0.4127 | 0.5104 | 0.4917 | 0.7066 |
R75p | 0.0002 | 0.0019 | 0.0296 | 0.075 | 0.0003 | 0.0042 | 0.0174 | 0.0573 | 0 | 0 | 0.0055 | 0.0235 | 0.0106 | 0.0429 |
R75pTOT | 0 | 0 | 0.5512 | 0.4278 | 0.007 | 0.013 | 0.1043 | 0.0778 | 0 | 0 | 0 | 0.0001 | 0 | 0.0003 |
R95p | 0.0782 | 0.0245 | 0.0742 | 0.0235 | 0.1262 | 0.0599 | 0.0743 | 0.0235 | 0.0003 | 0.0005 | 0.0497 | 0.0163 | 0.0722 | 0.0466 |
R95pTOT | 0.0719 | 0.0246 | 0.0865 | 0.029 | 0 | 0.0002 | 0.124 | 0.0458 | 0 | 0 | 0.0579 | 0.0189 | 0 | 0 |
R99p | 0 | 0.0003 | 0 | 0.0002 | 0.0002 | 0.0007 | 0.0001 | 0.0006 | 0.0001 | 0.0006 | 0.0001 | 0.0005 | 0.0002 | 0.0009 |
R99pTOT | 0.003 | 0.0024 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0045 | 0 | 0 | 0 |
PRCPTOT | 0.0128 | 0.0367 | 0.248 | 0.3477 | 0.0002 | 0.0007 | 0.2285 | 0.3443 | 0 | 0 | 0.0684 | 0.1165 | 0.0611 | 0.2303 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | AD test | SK test | |
RR | 0.286 | 0.6031 | 0.3799 | 0.7337 | 0.2855 | 0.6025 | 0.3687 | 0.7303 | 0 | 0 | 0.4581 | 0.7794 | 0.467 | 0.8004 |
RR5 | 0.0001 | 0 | 0.7334 | 0.821 | 0.0158 | 0.005 | 0.2913 | 0.265 | 0.2898 | 0.2645 | 0.7564 | 0.64 | 0.7339 | 0.8222 |
RR10 | 0 | 0 | 0.2873 | 0.5293 | 0.0024 | 0.0021 | 0.0756 | 0.1411 | 0.0755 | 0.1417 | 0.5939 | 0.6952 | 0.2892 | 0.5319 |
RR25 | 0.0114 | 0.0025 | 0.4179 | 0.6239 | 0.0464 | 0.0477 | 0.2874 | 0.4177 | 0 | 0 | 0.6843 | 0.8876 | 0.846 | 0.9621 |
RR50 | 0.0364 | 0.0817 | 0.028 | 0.0832 | 0.003 | 0.018 | 0.0276 | 0.0832 | 0.0006 | 0.0002 | 0.0465 | 0.1188 | 0.0873 | 0.1588 |
RR100 | 0.0045 | 0.0066 | 0.0003 | 0.0015 | 0.0055 | 0.0077 | 0.0015 | 0.0038 | 0.0015 | 0.0038 | 0.0036 | 0.0063 | 0.0053 | 0.0078 |
RX1day | 0.0046 | 0.0084 | 0.0251 | 0.073 | 0.0046 | 0.0084 | 0.0202 | 0.0592 | 0 | 0 | 0.1222 | 0.311 | 0.9645 | 0.9378 |
RX3day | 0.1964 | 0.2109 | 0.2166 | 0.2779 | 0.1956 | 0.2098 | 0.2163 | 0.3114 | 0 | 0 | 0.442 | 0.7165 | 0.6588 | 0.8013 |
RX5day | 0.0866 | 0.0955 | 0.1567 | 0.2109 | 0.0865 | 0.0953 | 0.1459 | 0.197 | 0 | 0 | 0.4691 | 0.6431 | 0.6702 | 0.8868 |
RX7day | 0.0223 | 0.035 | 0.1441 | 0.2484 | 0.0223 | 0.035 | 0.1149 | 0.2117 | 0 | 0 | 0.3701 | 0.5907 | 0.9154 | 0.9321 |
MCWD | 0.006 | 0.0081 | 0.7354 | 0.8533 | 0.0433 | 0.0955 | 0.5358 | 0.7444 | 0.5089 | 0.7029 | 0.2863 | 0.302 | 0.736 | 0.853 |
MCDD | 0.4821 | 0.4428 | 0.7718 | 0.7628 | 0.0966 | 0.2177 | 0.7461 | 0.7507 | 0.0261 | 0.0176 | 0.7113 | 0.6262 | 0.2476 | 0.476 |
SDII | 0.0384 | 0.0042 | 0.632 | 0.7424 | 0.0007 | 0.0012 | 0.0005 | 0.0005 | 0 | 0 | 0.8774 | 0.8731 | 0.6334 | 0.7431 |
R75p | 0.0037 | 0.0003 | 0.8111 | 0.8868 | 0.0532 | 0.0203 | 0.5324 | 0.5558 | 0 | 0 | 0.9345 | 0.9361 | 0.8129 | 0.888 |
R75pTOT | 0 | 0 | 0.0169 | 0.0419 | 0 | 0.0002 | 0.0001 | 0.0013 | 0 | 0 | 0.1187 | 0.0941 | 0.0171 | 0.042 |
R95p | 0.3207 | 0.1624 | 0.2469 | 0.1881 | 0.3633 | 0.218 | 0.1193 | 0.1057 | 0.0041 | 0.0006 | 0.3798 | 0.2359 | 0.2345 | 0.1731 |
R95pTOT | 0.0121 | 0.0207 | 0.1604 | 0.4926 | 0.0001 | 0.0001 | 0.1311 | 0.4411 | 0 | 0 | 0.192 | 0.4741 | 0.1743 | 0.4411 |
R99p | 0.0072 | 0.0109 | 0.0086 | 0.0125 | 0.0019 | 0.0222 | 0.0176 | 0.0226 | 0.0175 | 0.0225 | 0.0116 | 0.0165 | 0.0227 | 0.0263 |
R99pTOT | 0.2621 | 0.1352 | 0.4239 | 0.1813 | 0.0013 | 0.0074 | 0 | 0 | 0 | 0 | 0.3508 | 0.1585 | 0 | 0.0006 |
PRCPTOT | 0.2929 | 0.606 | 0.3926 | 0.7412 | 0.2913 | 0.6044 | 0.3341 | 0.6774 | 0 | 0 | 0.4742 | 0.7878 | 0.4867 | 0.8101 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | |
---|---|---|---|---|---|---|---|
RR | 10 | 14 | 6 | 13 | 0 | 8 | 8 |
RR5 | 2 | 10 | 7 | 13 | 0 | 7 | 13 |
RR10 | 10 | 13 | 4 | 10 | 0 | 11 | 6 |
RR25 | 6 | 8 | 14 | 10 | 1 | 4 | 12 |
RR50 | 10 | 10 | 7 | 11 | 0 | 5 | 14 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0 | 14 | 0 | 12 | 3 | 8 | 10 |
RX3day | 0 | 11 | 0 | 8 | 0 | 11 | 14 |
RX5day | 0 | 9 | 5 | 7 | 0 | 12 | 14 |
RX7day | 0 | 13 | 0 | 8 | 0 | 13 | 10 |
MCWD | 0 | 12 | 4 | 10 | 6 | 8 | 14 |
MCDD | 9 | 11 | 4 | 13 | 0 | 6 | 11 |
SDII | 0 | 13 | 0 | 9 | 0 | 9 | 14 |
R75p | 7 | 11 | 6 | 13 | 0 | 5 | 12 |
R75pTOT | 6 | 10 | 4 | 8 | 0 | 12 | 14 |
R95p | 5 | 6 | 4 | 14 | 0 | 0 | 0 |
R95pTOT | 4 | 14 | 6 | 12 | 0 | 8 | 10 |
R99p | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
R99pTOT | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
PRCPTOT | 9 | 14 | 6 | 13 | 0 | 8 | 9 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | |
---|---|---|---|---|---|---|---|
RR | 0 | 14 | 0 | 12 | 0 | 9 | 9 |
RR5 | 8 | 12 | 4 | 14 | 0 | 6 | 10 |
RR10 | 6 | 8 | 12 | 10 | 0 | 4 | 14 |
RR25 | 0 | 14 | 0 | 12 | 0 | 10 | 0 |
RR50 | 0 | 0 | 0 | 0 | 0 | 0 | 7 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 11 | 13 | 0 | 13 | 0 | 12 | 8 |
RX3day | 8 | 11 | 5 | 9 | 0 | 14 | 7 |
RX5day | 8 | 7 | 5 | 9 | 0 | 14 | 11 |
RX7day | 6 | 5 | 3 | 5 | 0 | 14 | 4 |
MCWD | 0 | 11 | 0 | 6 | 8 | 11 | 14 |
MCDD | 6 | 8 | 14 | 10 | 0 | 4 | 12 |
SDII | 4 | 12 | 6 | 10 | 0 | 8 | 14 |
R75p | 0 | 7 | 0 | 6 | 0 | 0 | 0 |
R75pTOT | 0 | 14 | 0 | 12 | 0 | 0 | 0 |
R95p | 6 | 4 | 14 | 5 | 0 | 0 | 3 |
R95pTOT | 5 | 6 | 0 | 7 | 0 | 4 | 0 |
R99p | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
R99pTOT | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
PRCPTOT | 0 | 14 | 0 | 12 | 0 | 9 | 9 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | |
---|---|---|---|---|---|---|---|
RR | 8 | 10 | 6 | 12 | 0 | 14 | 14 |
RR5 | 0 | 8 | 12 | 10 | 0 | 0 | 14 |
RR10 | 0 | 10 | 6 | 8 | 2 | 12 | 14 |
RR25 | 12 | 14 | 6 | 14 | 0 | 10 | 8 |
RR50 | 8 | 8 | 6 | 8 | 0 | 10 | 14 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0 | 10 | 3 | 8 | 0 | 12 | 14 |
RX3day | 0 | 14 | 0 | 12 | 0 | 8 | 10 |
RX5day | 0 | 14 | 0 | 12 | 0 | 9 | 9 |
RX7day | 0 | 14 | 0 | 12 | 0 | 9 | 9 |
MCWD | 0 | 10 | 3 | 13 | 13 | 5 | 8 |
MCDD | 10 | 12 | 11 | 7 | 0 | 5 | 9 |
SDII | 0 | 14 | 0 | 11 | 0 | 9 | 13 |
R75p | 6 | 9 | 12 | 9 | 0 | 4 | 14 |
R75pTOT | 0 | 14 | 0 | 12 | 0 | 10 | 8 |
R95p | 8 | 8 | 11 | 10 | 1 | 4 | 13 |
R95pTOT | 10 | 7 | 12 | 8 | 0 | 4 | 14 |
R99p | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
R99pTOT | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
PRCPTOT | 7 | 9 | 5 | 11 | 0 | 14 | 13 |
Normal | Lognormal | Beta | Gamma | Exponential | Loglogistic | Johnson | |
---|---|---|---|---|---|---|---|
RR | 6 | 10 | 4 | 8 | 0 | 12 | 14 |
RR5 | 0 | 11 | 0 | 8 | 6 | 12 | 13 |
RR10 | 0 | 10 | 0 | 7 | 7 | 14 | 12 |
RR25 | 0 | 10 | 0 | 8 | 0 | 12 | 14 |
RR50 | 4 | 5 | 0 | 5 | 0 | 6 | 14 |
RR100 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
RX1day | 0 | 5 | 0 | 4 | 0 | 12 | 14 |
RX3day | 6 | 9 | 4 | 9 | 0 | 12 | 14 |
RX5day | 6 | 10 | 4 | 8 | 0 | 12 | 14 |
RX7day | 0 | 10 | 0 | 8 | 0 | 12 | 14 |
MCWD | 0 | 13 | 2 | 10 | 8 | 6 | 13 |
MCDD | 7 | 14 | 4 | 12 | 0 | 10 | 7 |
SDII | 0 | 10 | 0 | 0 | 0 | 14 | 12 |
R75p | 0 | 10 | 3 | 8 | 0 | 14 | 12 |
R75pTOT | 0 | 0 | 0 | 0 | 0 | 14 | 0 |
R95p | 8 | 9 | 12 | 4 | 0 | 14 | 7 |
R95pTOT | 0 | 12 | 0 | 8 | 0 | 13 | 11 |
R99p | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
R99pTOT | 5 | 14 | 0 | 0 | 0 | 12 | 0 |
PRCPTOT | 6 | 10 | 4 | 8 | 0 | 12 | 14 |