Statistical distribution of climate time series 2.5
2.5.1 Testing and fitting the empirical- by normal distributions
The spatial distributions of the climate exhibited in this section clearly hint of different climatic regions across the larger EST study region. Because of the hydrological impact of climate change, a similar geographical pattern is to be expected for the 1971-2006 streamflow across the area. Thus, the decreasing rainfall trend observed in the northern part of the study region in Figure 2.15 may be the reason for the negative trend of the runoff measured at stream station z4 located there, as shown earlier in Figure 2.12. On the other hand, the mentioned increased rainfall in the southern part of the basin has led also to an increase of the streamflow at station z38 during the 1971-2006 time period. These relationships between rainfall and streamflow will be further quantified during the modeling of the climate and hydrological system in Chapter 6.
Statistical distribution of climate time series
Figure 2.17. Empirical kernel density functions for the monthly maximum (left panels) and minimum (right panels) temperature during years 1971-2005 at the four temperature stations fitted with the optimal theoretical normal distributions as determined by MLE.
normal distribution, whereas those of the precipitation and the streamflow show larger deviations from a normal distribution for the lower and upper quantiles, i.e. for the extremes.
The empirical cumulative density functions (ECDF) (Shorack and Wellner 1986, van der Vaart 2000, Maechler 2012) of the various hydro-climate variable are plotted together with the fitted theoretical normal distributions in Figure 2.21. Similar to the density plots of Figure 2.17 to Figure 2.19 and the Q-Q-plot of Figure 2.20, the normal distribution cannot mimic the observed distributions of the precipitation and the streamflow.
Figure 2.18. Similar to Figure 2.17 but for the monthly precipitation observed between years 1971-2005 at the 24 stations in the study region.
Figure 2.19. Similar to Figure 2.17 but for the monthly streamflow measured during years 1971-2005 at three streamgages in the study region.
Figure 2.20. Q-Q plots of empirical- over theoretical normal cumulative distribution functions for monthly maximum and minimum temperature, rainfall and streamflow.
Figure 2.21. Cumulative empirical distributions of monthly maximum and minimum temperature, rainfall and streamflow with optimal theoretical normal distribution.
Tmax48478 (°C)
Tmin48478 (°C)
PCP 48092
(mm/day) Stream Z15
(m3/s)
Tmax48478
(°C) Tmin48478
(°C)
PCP 48092 (mm/day)
Stream Z15 (m3/s)
To better quantify the fit of the empirical distribution by a theoretical (normal or others) distribution, a so-called “goodness of fit”- test can be made. There are several statistical tests available for this purpose, some of which are only applicable to tests for a normal distribution, while others can be applied to an arbitrary theoretical distribution (Shorack and Wellner 1986, van der Vaart 2000). To the latter category belongs the Kolmogorov–Smirnov (KS) test (Birnbaum and Tingey 1951, Chakravarti et al. 1967, Marsaglia et al. 2003).
The Lilliefors test (Lilliefors 1967) is a further development of the Kolmogorov–Smirnov test for a normal distribution.
This normality condition is removed again in the Anderson–Darling test (Anderson and Darling 1952) which can be applied to other well-known theoretical distributions as well, but uses another test statistic as the KS-test. In fact, the experiments, using these two test-approaches, show that the Anderson–Darling test is better than the KS- test when ties are present in the data, as is the case for many of the recorded temperatures in the study region.
Depending on the kind of theoretical distribution fitted to the observed data and the applicability of the three named tests, all of them are applied to the observed climate data series. Thus, the Anderson–Darling test is used to test the data with several theoretical non-normal distributions, i.e., log-normal (Johnson et al. 1994), Weibull (Weibull 1961), Cauchy (Ferguson 1962), Gamma (Choi and Wette 1969) and exponential (Schmidt and Makalic 2009).
All these tests, which have been programmed in the R- environment (Venables and Ripley 2002, Spano 2012), test the null hypothesis H0 that the empirical distribution belongs to the theoretical distribution assumed. The decision to accept or to reject the null hypothesis is based on the p-value, which describes the significance of the test under the assumption that the null hypothesis is true. If the p-value is less than or equal the significance level α, which in most cases is set to α
=0.05 (5%), the test rejects the null hypothesis and the alternative hypothesis H1 should be accepted. On the other hand, if the p-value is larger than α, the null hypothesis will be accepted at that level. Thus, the larger the p-value, the higher is the confidence that the null hypothesis is true.
The results of the applications of these tests to the observed climate data are listed in Table 2.8.
From the table one may notice, that only p-values of time series tests at 6 out of a total of 89 stations are high enough to accept the null hypothesis that the empirical data distribution belongs to the theoretical distribution assumed. For example, the Lilliefors test accepts only the maximum temperature- and the solar radiation series to be compatible with the fitted normal distribution.
Table 2.8. Number of climate series which are best fitted to the alternative theoretical distribution models and the number of series that reject the null hypothesis of Lilliefors test and Anderson–Darling test at α = 0.05 (p-value > 0.05).
variable total Lilliefors test best fitting theoretical distribution Anderson–Darling test
n p-value > 0.05 Exponential Gamma Log-normal Normal Weibull p-value > 0.05
PCP 24 0 24 0
%Wet 24 0 24 0
Tmax 4 1 4 1 (log-normal)
Tmin 4 0 4 1 (Weibull)
HMD 4 0 2 2 1 (Weibull)
SLR 2 1 1 1 2 (Gama, log-norm.)
Stream 3 0 1 1 1 1 (log-normal)
Total 89 2 1 1 6 74 7 6
Table 2.9. Statistical parameters of ECDF of the climate series at station 48092 and 48478, i.e.
median, ̃ , mean ( ̅) and standard deviation (σ).
station variable unit
annual probability
seasonal probability
Dry pre-monsoon monsoon_1 monsoon_2
̃ ̅ σ ̃ ̅ σ ̃ ̅ Σ ̃ ̅ σ ̃ ̅ σ 48092 PCP mm/day 1.4 2.7 3.1 0.6 1.0 1.1 3.3 3.7 2.4 4.0 4.5 2.8 2.3 2.9 2.8 48092 %Wet day/day 0.17 0.23 0.24 0.1 0.1 0.1 0.3 0.32 0.19 0.40 0.40 0.19 0.23 0.26 0.2 48478 Tmax C 32.0 32.2 1.2 32.1 32.4 1.4 33.1 33.4 1.5 31.8 32.1 1.1 32.3 32.5 1.4 48478 Tmin C 23.0 22.8 1.6 24.0 23.8 2.3 26.8 26.8 0.8 26.0 26.0 0.8 25.4 24.9 2.2 Z15 streamflow m3/s 1.31 2.62 2.96 0.1 0.3 0.9 0.9 1.3 1.3 1.8 2.8 2.8 0.7 1.8 2.4
Table 2.10. Quantiles of the ECDF of the hydro-climate time series at percentile 5%, 25%, 50%,75% and 95%, as well as the mean and the standard deviation, based on 1971-1999 observations at 4 temperature stations, 24 precipitation stations and 3 streamgages.
predictor season percentile statistical parameters
5% 25% 50% 75% 95% mean SD
Tmax (°C)
annual 30.5 31.4 32.1 32.9 34.3 32.2 1.2
dry 30.3 31.4 32.0 32.8 34.0 32.1 1.1
premonsoon 31.6 32.3 33.0 33.8 35.0 33.1 1.1
monsoon1 31.0 31.5 31.9 32.4 33.4 32.0 0.8
monsoon2 30.1 31.0 31.6 32.3 33.3 31.6 1.0
Tmin (°C)
annual 20.6 23.3 25.0 26.0 27.0 24.5 2.0
dry 20.0 22.3 23.7 25.0 26.4 23.5 2.0
premonsoon 25.0 25.8 26.3 26.8 27.7 26.3 0.8
monsoon1 24.4 25.0 25.5 25.9 26.7 25.5 0.7
monsoon2 19.9 21.7 22.9 24.0 24.5 22.7 1.5
HMD (%)
annual 66.9 73.0 76.1 78.9 83.3 75.8 4.7
dry 67.0 71.7 74.2 76.2 78.5 73.7 3.5
premonsoon 71.3 74.1 76.2 78.3 80.9 76.3 3.0
monsoon1 73.6 76.3 78.2 80.7 84.2 78.5 3.3
monsoon2 64.7 69.4 74.2 80.6 84.4 74.7 6.7
SLR (MJ/m2)
annual 14.3 16.2 17.7 19.6 22.4 18.0 2.5
dry 16.6 18.5 20.0 21.5 22.7 19.9 2.0
premonsoon 14.6 16.8 18.7 20.7 23.3 18.8 2.8
monsoon1 13.6 15.2 16.3 17.3 18.4 16.3 1.5
monsoon2 14.1 15.7 17.1 18.1 19.7 17.0 1.7
PCP (mm/day)
annual 0.0 0.6 2.5 5.4 10.2 3.5 3.5
dry 0.0 0.1 0.5 1.4 3.3 1.0 1.2
premonsoon 0.6 2.0 3.7 6.0 9.7 4.3 2.9
monsoon1 1.4 3.1 5.0 7.5 11.9 5.6 3.4
monsoon2 0.0 0.1 1.4 5.2 10.5 3.1 3.8
%Wet (day/day)
annual 0.0 0.1 0.3 0.5 0.7 0.3 0.2
dry 0.0 0.0 0.1 0.2 0.3 0.1 0.1
premonsoon 0.1 0.2 0.4 0.5 0.7 0.4 0.2
monsoon1 0.2 0.4 0.5 0.6 0.8 0.5 0.2
monsoon2 0.0 0.0 0.2 0.5 0.7 0.3 0.3
stream flow (m3/s)
annual 0.3 0.8 1.6 3.6 9.2 2.7 2.8
dry 0.6 1.7 3.3 6.4 11.1 4.3 3.5
premonsoon 0.1 0.4 0.6 0.9 1.5 0.7 0.7
monsoon1 0.4 1.0 1.6 3.1 5.8 2.2 1.7
monsoon2 0.4 1.5 2.4 4.4 10.6 3.5 3.0
Regardless of whether the empirical cumulative density functions (ECDF) can be fitted by a normal or other theoretical distribution, the salient statistical attributes of the ECDF, such as the mean, the median and the variance or standard deviation provide some valuable identification parameters of the climate variable under question. These statistical parameters are summarized in Table 2.9, and this not only for the annual-, but also for the 4-season series of the various hydro-climate variables.
An even more detailed picture is provided by the quantiles of the ECDF at certain percentile levels (Ferro et al. 2005, Schlünzen et al. 2010). These are listed in
Table 2.10 and they serve as the reference values of the hydro-climate for 1971-2006 time period. Based on these distributions of the 20th-century – hydro-climate variables, the possible changes of the climate in the future will be assessed and further discussed in Chapter 6.