Abstract
In the present explorative study, different time-series analysis methods, such as moving average, deterministic methods (linear trend with seasonality), and non-parametric Mann–Kendall trend test, were applied to monthly precipitation data from January 1871 to December 2014, with the aim of comparing the results of these methods and detecting the signs of climate change. The data set was provided by the University of Pannonia, and it contains monthly precipitation data of 144 years of measurements (1,728 data points) from the Keszthely Meteorological Station. This data set is special because few stations in Hungary can provide such long and continuous measurements with detailed historical background. The results of the research can provide insight into the signs of climate change in the past for the region of West Balaton. Parametric methods (linear trend and t-test for slope) for analyzing time series are the simplest ones to obtain insight into the changes in a variable over time. These methods have a requirement for normal distribution of the residuals that can be a limitation for their application. Non-parametric methods are distribution-free and investigators can get a more sophisticated view of the variable tendencies in time series.
Introduction
Climate change is one of the serious problems that mankind should face in the 21st century. Even the IPCC report (2013), while itself not a scientific publication, is based on more than 9,200 scientific publications, and states that a human role in the process admits of no doubt (95% is the probability that human influence has been dominant in the present changes of climate system). One of the main conclusions of the Summary for Policymakers (IPCC 2013) is that “it is extremely likely that human influence has been the dominant cause of the observed warming since the middle of the 20th century.” It is mainly supported by Chapter 10 of AR5: “detection and attribution – from global to regional” (de Larminat 2016). Climate change will probably affect all parts of the Earth, and in Central Europe, the Carpathian Region will be influenced as well. The hydrological cycle is an element of the climate system that is expected to change and the signs of these amendments can already be detected. Precipitation strongly influences the water cycle from local to global scale. Any modification in the amount or distribution of rainfall has significant impact on water availability, and therefore on water management.
The prediction of the effects of climate change on the Carpathian Region (including Hungary) is well investigated by Judit Bartholy and colleagues. This group of researchers applied Regional Climate Models to estimate projections of future climate for the Carpathian Region. Several publications underlined that the amount of precipitation will decline in the summer half-year and there is high uncertainty for the rainfall for the winter half-year (Bartholy et al. 2004, 2005, 2007, 2008, 2015; Horányi et al. 2010; Kis et al. 2014; Pongrácz et al. 2011, 2014). Recent model predictions of Kis et al. (2017) state that the spatial distribution of precipitation is not likely to change remarkably in the future in the Carpathian Region during the period of 1961−2100, but the annual distribution of precipitation is projected to be restructured. However, the hydroclimate of the region is quite variable in space and time (Kern et al. 2016), as the shallow groundwater fluctuations are driven by the Mediterranean cyclones from the Gulf of Genoa and by local/regional climate variables (Garamhegyi et al. in press). Besides the model predictions, it is interesting to search analogies of projected climate during the history of the Earth for better understanding of the processes. Prista et al. (2015) worked out chronostratigraphic analogies for IPCC scenarios, and stated that the Pliocene (mid-Piacenzian warm period) is the best analogue for warming climate in Europe.
For the tendencies of the past, Szalai et al. (2005) stated that the annual precipitation amount decreased by 11% between 1901 and 2004, according to the analysis of the Hungarian Meteorological Service. The biggest decline could be experienced in the spring; it was 25% for the aforementioned period. Bodri (2004) suggested a slow decrease in precipitation with a noticeable increase in precipitation variability for the 20th century. While Northern and Western Europe receive more precipitation in parallel with the warming tendency, Hungary, much like the Mediterranean region, gets less rainfall. The water balance shows a deficit in that the difference between water income and outflow is increasing. Between 1901 and 2009, the highest precipitation declines over the territory of Hungary occurred in the spring, nearly 20% of them (Lakatos and Bihari 2011). Bartholy and Pongrácz (2005, 2007, 2010) examined several precipitation extreme indices and suggested that regional intensity and frequency of extreme precipitation increased in the Carpathian Basin in the second half of the last century, while the total precipitation decreased.
The aim of this study is to analyze the long-term data series of the meteorological measurements of precipitation amount at Keszthely (Western Hungary, N 46°44′, E 17°14′; Fig. 1) between 1871 and 2014 from the point of view of climate change, and to compare different statistical methods (conventional “regression on time” method and non-parametric Mann–Kendall trend test) on the results of time-series analysis based on this data set.
Lake Balaton in Europe (upper left) with its artificial channel Sió (upper right) and natural water catchment (lower panel) according to Mika et al. (2010)
Citation: Central European Geology Central European Geology 60, 3; 10.1556/24.60.2017.011
Several examples can be found in the literature for the application of the Mann–Kendall trend test, for example, Patle and Libang (2014) argued on trend analysis of annual and seasonal rainfall in the northeast region of India, and Salmi et al. (2002) analyzed the trends of atmospheric pollutants in Finland. Meteorological applications can be read in Rahman and Begum (2013) who determined trends of rainfall of the largest island in Bangladesh. Ganguly et al. (2015) investigated the tendencies of rainfall in Himachal Pradesh (northern India) between 1950 and 2005. Gavrilov et al. (2015, 2016, 2018) examined trends of air temperature by Mann–Kendall test in Vojvodina, Serbia. Salami et al. (2014) applied this non-parametric trend test for the analysis of hydrometeorological variables in Nigeria. Mapurisa and Chikodzi (2014) made an assessment of trends of monthly and seasonal rainfall sums in southeastern Zimbabwe. Karmeshu (2012) investigated the temperature and precipitation changes in the northeastern United States. Hydrological utilization is provided by Hamed (2008). Burn and Hag Elnur (2002) estimated the trends and variability of 18 hydrological variables by Mann–Kendall trend test. Hirsch et al. (1991) used the method for the investigation of stream water quality. Chaudhuri and Dutta (2014) analyzed the trends of pollutants, temperature, and humidity in India. Zarei et al. (2016) examined drought indexes in Iran applying the Mann–Kendall trend test. Gocić and Trajković (2013a) analyzed precipitation and drought data sets in Serbia using the non-parametric trend test. Several other applications of the Mann–Kendall trend test related to climate change can be found in the literature, for example, Jaagus (2006), Mohsin and Gough (2010), Chattopadhyay et al. (2012), Lacombe et al. (2013), Zhang et al. (2013), and Dogan (2016).
Data and methods
Monthly amounts of precipitation were analyzed from 1871 to 2014, initially measured in the area of the ancient Georgikon Academy of Agriculture at Keszthely, then at the meteorological station of the Hungarian Meteorological Service. The data set was provided by the Department of Meteorology and Water Management of the University of Pannonia Georgikon Faculty (Keszthely). This data set is special because few stations in Hungary have continuous measurements over more than 140 years with detailed historical background (Kocsis and Anda 2006). The meteorological station of Keszthely was among those few important stations of Hungary that began measurements for the first time in the history of the Hungarian meteorological observations. A detailed history of the meteorological measurements is given by Kocsis and Bem (2007).
Linear regression and moving average
The significance of the slope can be tested by several methods. In this study, the significance of the slope coefficient β1 was tested by t-test. The regression model must be checked for normality of residuals, constant variance, and linearity of the relationship (Helsel and Hirsh 2002). This method is often called “regression on time” and the estimation method is the ordinary least squares (OLS) estimator. During the hypothesis test, an α = 5% significance level was used by one-tailed test, as it is supposed that the precipitation amount is likely to decrease, therefore β1 is expected to be negative.
Another method for tendency detection is the moving average, and as seasonal component has a probable effect on the time series, a moving average of 12 tags that should eliminate a part of the seasonal effect, is applied, and every mean seasonal deviation was calculated for each season.
Mann–Kendall trend test
The Mann–Kendall trend test is widespread in climatological and hydrological analysis for time series; since it is simple and robust, it can cope with missing values and values below the detection limit (Gavrilov et al. 2016). This non-parametric test is commonly used to detect monotonic tendencies in series of environmental data as well (Pohlert 2017). No assumption of the normality is required (Helsel and Hirsh 2002). Hamed and Rao (1998) developed a modified Mann–Kendall test for autocorrelated data. Application of this modified method is presented, for example, by Amirataee et al. (2016). Yue et al. (2002) investigated the power of the Mann–Kendall test in hydrological series.
The Mann–Kendall trend test is based upon the work of Mann (1945) and Kendall (1975), and is closely related to Kendall’s rank correlation coefficient. The methodology is introduced following the detailed descriptions given by Gilbert (1987) and Hipel and McLeod (1994) as follows:
Modified Mann–Kendall trend test for serially dependent data (seasonal Mann–Kendall trend test)
The presence of positive autocorrelation in the data increases the chance of detecting trends when none actually exist, and vice versa (Hamed and Rao 1998). This effect of the existence of autocorrelation in data is often ignored. Hamed and Rao (1998) supposed a modified non-parametric trend test, which is suitable for autocorrelated data, and gave a detailed description of the modified Mann–Kendall trend test for autocorrelated data. In this study, this type of Mann–Kendall test was also applied.
Sen’s slope estimator
Sen’s slope estimator is widely applied in hydrological and meteorological research, for example, Marofi et al. (2012), Huang et al. (2013), Guo and Xia (2014), Talaee (2014), Zamani et al. (2016), Amirataee et al. (2016), and Liuzzo et al. (2016).
Seasonal slope estimator
Addinsoft’s XLSTAT (2017) were used for carrying out the computations.
Results
Results of “regression on time” and moving average
A total of 1,728 monthly precipitation data were analyzed. Mean monthly precipitation at Keszthely is 56 mm with a standard deviation of 37 mm. As a declining tendency in precipitation is proved for the territory of Western Hungary, a decreasing trend was supposed. Linear tendency () can be detected in one-tailed t-test (β1 < 0) at α = 5%, and an alternative hypothesis can be accepted at a p value of 3.1% (Figs 2 and 3). The slope was −0.003 mm per time step (month).
Time series of monthly precipitation amounts at Keszthely between January 1871 and December 2014 (black line indicates a linear trend)
Citation: Central European Geology Central European Geology 60, 3; 10.1556/24.60.2017.011
Regression of monthly precipitation amounts by time step
Citation: Central European Geology Central European Geology 60, 3; 10.1556/24.60.2017.011
There are multiple reasons for which these fitted values and corresponding p values are not entirely trustworthy. There is a significant correlation between the residuals (Fig. 4), which is not at all surprising, as we expect to have a yearly periodicity in the precipitation.
Autocorrelation function of the residuals until lag 180, i.e., 180 months = 15 years
Citation: Central European Geology Central European Geology 60, 3; 10.1556/24.60.2017.011
A moving average with tags of 12 sums can be used as a smoothing method that can partly eliminate the effect of the seasonality in the data series. The tendency of the 12MA (moving average) is not so clear on Fig. 5.
Monthly precipitation amounts and moving average (12MA) between 1871 and 2014
Citation: Central European Geology Central European Geology 60, 3; 10.1556/24.60.2017.011
Trend analysis can be followed by the decomposition of the time-series data to trend, average seasonality, and random component. The tendency is modified by seasonal effect that can be described by corrected mean seasonal deviation. Corrected mean seasonal deviation gives the average volume of how much the seasonality increases or decreases the value given by the main trend (Table 1). Corrected mean seasonal deviations were computed using the values of moving averages, which filter the effect of seasonality and causality.
Corrected mean seasonal deviations in each season (1871−2014)
Season | Corrected mean seasonal deviation from moving average (mm) |
---|---|
January | −23.3 |
February | −22.6 |
March | −17.0 |
April | −2.7 |
May | 13.5 |
June | 19.7 |
July | 17.7 |
August | 16.5 |
September | 4.3 |
October | 2.4 |
November | 2.2 |
December | −10.6 |
Result of non-parametric methods
A parametric method, such as “regression on time,” is a commonly used method to determine the main tendency of the time series, but the requirement for the normal distribution of residuals, namely that they should be uncorrelated, is not fulfilled. Another choice for detecting tendency is the non-parametric method of the Mann–Kendall trend test.
In this case, non-parametric methods can give more appropriate results for the trend. In case the sign of the changes is determined (one-tailed test, τ < 0), significant decreasing modification can be seen with a p value of 3.24%. Sen’s slope estimator gives a slope of −0.003 mm per month, similarly to a linear trend. As the time series contains a seasonal component, the values are not serially independent. A seasonal Mann–Kendall trend test was also applied, and the one-tailed test proved the significant negative tendency at a p value of 3.86%. Sen’s slope was −0.033 mm per time step (month) by paying attention to the effect of seasonality.
The data in the time series are autocorrelated and not serially independent. The modified Mann–Kendall trend test suggested by Hamed and Rao (1998) for autocorrelated data was used to detect the supposed declining tendency of the data set (one-tailed test, τ < 0). This method showed that no significant negative trend can be detected (p value was 50%). Therefore, if autocorrelation of the data is taken into account, no significant tendency can be statistically proven, and it can be supposed that the monthly precipitation amount did not change significantly.
Discussion
A slow decrease in precipitation, together with the noticeable increase in precipitation variability, is characteristic for the 20th century (Bodri 2004). The tendency of the annual precipitation amounts between 1960 and 2009 showed a slight decrease in Hungary and a declining trend in Western Hungary which is higher than average, whereas in the northeastern part of the country precipitation amount increased (Lakatos and Bihari 2011). Lakatos and Bihari (2011) used the conventional separation of annual and seasonal precipitation amounts for research of changes in their study. The 144-year-long continuous data set of monthly precipitations had not been analyzed previously.
Conclusions
According to the “mainstream” opinion in climatology in Hungary, a decreasing tendency is supposed in monthly precipitation amounts; both parametric and non-parametric methods prove a significant negative trend in the time series of monthly precipitation amounts at Keszthely. However, the residuals of linear regression do not follow normal distribution and there is autocorrelation between them. Therefore, the results do not fulfill the requirements of diagnostic check stage. Moving averages can be used as smoothing technique that partly filter the effect of the seasonality and should provide information about the main tendency. The non-parametric Mann–Kendall trend test can be the chosen method as well, and has the advantage that it has no strict requirements for application. When analyzing monthly precipitation amounts, the effect of seasonality leads to the serial dependence of the data. This fact must be taken into account; therefore, the seasonal Mann–Kendall trend test can be used. This method in one-tailed test resulted in a significant negative tendency of monthly precipitation between 1871 and 2014 at a p value of 3.86%. Sen’s slope estimator calculated −0.033 mm decrease in precipitation sum per time step (month) over the examined period by paying attention to seasonality. The modified Mann–Kendall trend test for autocorrelated data was also used and showed that there is no significant negative tendency in the time series. This result highlights the fact that the previously detected significant negative tendencies should be false because the methods do not consider the autocorrelation in the data. As an outlook, the time series assessed in the study should be taken into account in climate studies dealing with low-frequency signals (Sen and Kern 2016).
References
Amirataee, B. , M. Montaseri , H. Sanikhani 2016: The analysis of trend variations of reference evapotranspiration via eliminating the significance effect of all autocorrelation coefficients. – Theoretical and Applied Climatology, 126, pp. 131–139.
Bartholy, J. , R. Pongrácz 2005: Extremes of ground-based and satellite measurements in the vegetation period for the Carpathian Basin. – Physics and Chemistry of the Earth, 30, pp. 81–89.
Bartholy, J. , R. Pongrácz 2007: Regional analysis of extreme temperature and precipitation indices for the Carpathian Basin from 1946 to 2001. – Global and Planetary Change, 57, pp. 83–95.
Bartholy, J. , R. Pongrácz 2010: Analysis of precipitation conditions for the Carpathian Basin based on extreme indices in the 20th century and climate simulation for 2050 and 2100. – Physics and Chemistry of the Earth, 35, pp. 43–51.
Bartholy, J. , R. Pongrácz , I. Matyasovszky , V. Schlanger 2004: A XX. Században bekövetkezett és a XXI. századra várható éghajlati tendenciák Magyarország területére [Climate tendencies occurred in the 20th century and projected for the 21st century in Hungary]. – AGRO-21 Füzetek, 33, pp. 3–18. (in Hungarian)
Bartholy, J. , J. Mika , R. Pongrácz , V. Schlanger 2005: A globális felmelegedés éghajlati sajátosságai a Kárpát-medencében [Climatic specialties of global warming in the Carpathian Basin]. – In: Takács-Sánta, A. (Ed): Éghajlatváltozás a világban és Magyarországon [Climate change in the world and in Hungary]. Alinea Kiadó-Védegylet, Budapest, pp. 105–139. (in Hungarian)
Bartholy, J. , R. Pongrácz , G. Gelybó 2007: Regional climate change in Hungary for 2071–2100. – Applied Ecology and Environment Research, 5, pp. 1–17.
Bartholy, J. , R. Pongrácz , G. Gelybó , P. Szabó 2008: Analysis of expected climate change in the Carpathian Basin using the PRUDENCE results. – Időjárás, 112, pp. 249–264.
Bartholy, J. , R. Pongrácz , A. Kis 2015: Projected changes of extreme precipitation using multi-model approach. – Időjárás, 119, pp. 129–142.
Bodri, L. 2004: Tendencies in variability of gridded temperature and precipitation in Hungary (during the period of instrumental record). – Időjárás, 108, pp. 141–153.
Burn, D.H. , M.A. Hag Elnur 2002: Detection of hydrological trends and variability. – Journal of Hydrology, 255, pp. 107–122.
Chattopadhyay, G. , P. Chakraborthy , S. Chattopadhyay 2012: Mann-Kendall trend analysis of tropospheric ozone and its modeling using ARIMA. – Theoretical and Applied Climatology, 110, pp. 321–328.
Chaudhuri, S. , D. Dutta 2014: Mann-Kendall trend test of pollutants, temperature and humidity over an urban station of India with forecast verification using different ARIMA models. – Environmental Monitoring and Assessment, 186, pp. 4719–4742.
da Silva, R.M. , C.A.G. Santos , M. Moreira , J. Corte-Real , V.C.L. Silva , I.C. Medeiros 2015: Rainfall and river flow trends using Mann-Kendall and Sen’s slope estimator statistical tests in the Cobres River basin. – Natural Hazards, 77, pp. 1205–1221.
de Larminat, P. 2016: Earth climate identification vs. anthropic global warming attribution. – Annual Reviews in Control, 42, pp. 114–125.
Dogan, M. 2016: How does the North Atlantic Oscillation affect the water levels of the Great Lakes with regard to hydro-climatic indicators? – Theoretical and Applied Climatology, 126, pp. 597–609.
Ganguly, A. , R.R. Chaudhuri , P. Sharma 2015: Analysis of trend of the precipitation data: A case study of Kangra District, Himachal Pradesh. – International Journal of Research – Granthaalayah, 3/9, pp. 87–95.
Garamhegyi, T. , J. Kovács , R. Pongrácz , P. Tanos , I.G. Hatvani in press: Investigation of the climate-driven periodicity of shallow groundwater level fluctuation in a Central-Eastern European agricultural region. – Hydrogeology Journal.
Gavrilov, M.B. , S.B. Marković , A. Jarad , V.M. Korać 2015: The analysis of temperature trends in Vojvodina (Serbia) from 1949 to 2006. – Thermal Science, 19, pp. 339–350.
Gavrilov, M.B. , I. Tosić , S.B. Marković , M. Unkašević , P. Petrović 2016: Analysis of annual and seasonal temperature trends using the Mann-Kendall test in Vojvodina, Serbia. – Időjárás, 120, pp. 183–198.
Gavrilov, M.B. , S.B. Marković , N. Janc , M. Nikolić , A. Valjarević , B. Komac , M. Zorn , M. Punišić , N. Bačević 2018: Assessing average annual air temperature trends using Mann–Kendall test in Kosovo. – Acta Geographica Slovenica, 58/1, pp. 7–25.
Gilbert, R.O. 1987: Statistical methods for environmental pollution monitoring. – John Wiley & Sons, Inc., New York, pp. 204–240.
Gocić, M. , S. Trajković 2013a: Analysis of precipitation and drought data in Serbia over the period 1980–2010. – Journal of Hydrology, 494, pp. 32–42.
Gocić, M. , S. Trajković 2013b: Analysis of changes in meteorological variables using Mann-Kendall and Sen’s slope estimator statistical tests in Serbia. – Global and Planetary Change, 100, pp. 172–182.
Guo, L. , Z. Xia 2014: Temperature and precipitation long-term trends and variations in the Ili-Balkhash Basin. – Theoretical and Applied Climatology, 115, pp. 219–229.
Hamed, K.H. 2008: Trend detection in hydrological data: The Mann-Kendall trend test under the scaling hypothesis. – Journal of Hydrology, 349, pp. 350–363.
Hamed, K.H. , A.R. Rao 1998: A modified Mann-Kendall trend test for autocorrelated data. – Journal of Hydrology, 204, pp. 182–196.
Helsel, D.R. , R.M. Hirsh 2002: Trend analysis. – Techniques of Water-Resources Investigations of the United States Geological Survey. Book 4, Hydrologic Analysis and Interpretation, Chapter A3 Statistical Methods in Water Resources, pp. 323–355.
Hipel, K.W. , A.I. McLeod 1994: Time series modelling of water resources and environmental systems. – Elsevier, Amsterdam, pp. 864–866 and pp. 924–925.
Hirsch, R.M. , R.B. Alexander , R.A. Smith 1991: Selection of methods for the detection and estimation of trends in water quality. – Water Resources Research, 27, pp. 803–813.
Hirsch, R.M. , J.R. Slack 1984: A nonparametric trend test for seasonal data with serial dependence. – Water Resources Research, 20, pp. 727–732.
Hirsch, R.M. , J.R. Slack , R.A. Smith 1982: Techniques of trend analysis for monthly water quality data. – Water Resources Research, 18, pp. 107–121.
Huang, J. , S. Sun , J. Zhang 2013: Detection of trends in precipitation during 1960–2008 in Jiangxi province, southeast China. – Theoretical and Applied Climatology, 114, pp. 237–251.
Horányi A. , I. Krüzselyi , P. Szabó , G. Szépszó (Eds) 2010: Klímamodellezési tevékenység, eredmények (Climate modeling activity, results). – Hungarian Meteorological Service, Budapest, 18 p. (in Hungarian)
IPCC, 2013: Summary for policymakers. – In: Stocker T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex, P.M. Midgley (Eds): Climate Change 2013: The Physical Science Basis. Cambridge University Press, Cambridge, pp. 3–29.
Jaagus, J. 2006: Climatic changes in Estonia during the second half of the 20th century in relationship with changes in large-scale atmospheric circulation. – Theoretical and Applied Climatology, 83, pp. 77–88.
Karmeshu, N. 2012: Trend detection in annual temperature and precipitation using the Mann-Kendall test – A case study to assess climate change on select states in the Northeastern United States. – MSc thesis, University of Pennsylvania, Philadelphia, PA, 27 p.
Kendall, M.G. 1975: Rank correlation methods. – Charles Griffin, London, 202 p.
Kern, Z. , A. Németh , M. Horoszi-Gulyás , M. Popa , T. Lavanić , I.G. Hatvani 2016: Natural proxy records of annual temperature- and hydroclimate variability from the Carpathian-Balkan Region for the past millennium: Review and recalibration. – Quaternary International, 415, pp. 109–125.
Kis, A. , R. Pongrácz , J. Bartholy 2014: Magyarországra becsült csapadéktrendek: hibakorrekció alkalmazásának hatása [Projected tendencies of precipitation for Hungary: Effect of using error-correction]. – Légkör, 59/3, pp. 117–120. (in Hungarian)
Kis, A. , R. Pongrácz , J. Bartholy 2017: Multi-model analysis of regional dry and wet conditions for the Carpathian Region. – International Journal of Climatology, 37/13, pp. 4543–4560.
Kocsis, T. , A. Anda 2006: A keszthelyi meteorológiai megfigyelések története [History of the meteorological observations at Keszthely]. – University of Pannonia, Keszthely, 60 p. (in Hungarian)
Kocsis, T. , J. Bem 2007: History of the meteorological measurements at Keszthely, one of the eldest stations in Hungary. – In: Proceedings of the 7th Annual Meeting of the European Meteorological Society, Madrid, 1–5 October 2007.
Lacombe, G. , V. Smakhtin , C.T. Hoanh 2013: Wetting tendency in the Central Mekong Basin consistent with climate change-induced atmospheric disturbances already observed in East Asia. – Theoretical and Applied Climatology, 111, pp. 251–263.
Lakatos, M. , Z. Bihari 2011: A közelmúlt megfigyelt hőmérsékleti és csapadéktendenciái [Temperature and precipitation tendencies observed in the recent past]. – In: Bartholy J., L. Bozó, L. Haszpra (Eds): Klímaváltozás 2011 [Climate Change 2011]. Hungarian Meteorological Society, Budapest, pp. 146–169. (in Hungarian)
Lavagnini, I. , D. Badocco , P. Pastore , F. Magno 2011: Theil-Sen nonparametric regression technique on univariate calibration, inverse regression and detection limits. – Talanta, 87, pp. 180–188.
Liuzzo, L. , E. Bono , V. Sammartano , G. Freni 2016: Analysis of spatial and temporal rainfall trends in Sicily during the 1921–2012 period. – Theoretical and Applied Climatology, 126, pp. 113–129.
Mann, H.B. 1945: Nonparametric tests against trend. – Econometrica, 13, pp. 245–259.
Mapurisa, B. , D. Chikodzi 2014: An assessment of trends of monthly contributions to seasonal rainfall in South-Eastern Zimbabwe. – American Journal of Climate Change, 3, pp. 50–59.
Marofi, S. , S. Soleymani , M. Salarijazi , H. Marofi 2012: Watershed-wide trend analysis of temperature characteristics in Karun-Dez watershed, southwestern Iran. – Theoretical and Applied Climatology, 110, pp. 311–320.
Mika, J. , Gy. Varga , L. Pálfy , I. Bonta , G. Bálint 2010: Could circulation anomalies cause the strong water deficit of Lake Balaton in 2000–2003? – Physics and Chemistry of the Earth, 35, pp. 2–10.
Mohsin, T. , W.A. Gough 2010: Trend analysis of long-term temperature time series in the Greater Toronto Area (GTA). – Theoretical and Applied Climatology, 101, pp. 311–327.
Patle, G.T. , A. Libang 2014: Trend analysis of annual and seasonal rainfall to climate variability in North-East region of India. – Journal of Applied and Natural Sciences, 6, pp. 480–483.
Pohlert, T. 2017: Non-parametric trend tests and change-point detection. – https://cran.r-project.org/web/packages/trend/vignettes/trend.pdf
Pongrácz, R. , J. Bartholy , E. Miklós 2011: Analysis of projected climate change for Hungary using ENSEMBLES simulations. – Applied Ecology and Environmental Research, 9, pp. 387–398.
Pongrácz, R. , J. Bartholy , A. Kis 2014: Estimation of future precipitation conditions for Hungary with special focus on dry periods. – Idojárás, 118, pp. 305–321.
Prista, G.O. , R.J. Agostinho , M.A. Cachao 2015: Observing the past to better understand the future: A synthesis of the Neogene climate in Europe and its perspectives on present climate change. – Open Geosciences, 7, pp. 65–83.
Rahman, M.A. , M. Begum 2013: Application of nonparametric test for trend detection of rainfall in the largest island of Bangladesh. – ARPN Journal of Earth Science, 2, pp. 40–44.
Rahman, M.A. , L. Yunsheng , N. Sultana 2017: Analysis and prediction of rainfall trends over Bangladesh using Mann-Kendall, Spearman’s rho tests and ARIMA model. – Meteorology and Atmospheric Physics, 129/4, pp. 409–424.
Salami, A.W , A.A. Mohammed , Z.H. Abdulmalik , O.K. Olanlokun 2014: Trend analysis of hydro-meteorological variables using the Mann-Kendall trend test: Application to the Niger River and the Benue sub-basins in Nigeria. – International Journal of Technology, 2, pp. 100–110.
Salmi, T. , A. Maatta , P. Anttila , T. Ruoho-Airola , T. Amnell 2002: Detecting trends of annual values of atmospheric pollutants by the Mann-Kendall test and Sen’s slope estimates – The Excel template application MAKESENS. – Finnish Meteorological Institute, Helsinki, 35 p.
Sen, P.K. 1968: Estimates of the regression coefficient based on Kendall’s tau. – Journal of American Statistical Association, 63, pp. 1379–1389.
Sen, A. , Z. Kern 2016: Wavelet analysis of low-frequency variability in oak tree-ring chronologies from east Central Europe. – Open Geosciences, 8, pp. 478–483.
Shadmani, M. , S. Marofi , M. Roknian 2012: Trend analysis in reference evapotranspiration using Mann-Kendall and Spearman’s rho tests in arid regions of Iran. – Water Resource Management, 26, pp. 211–224.
Szalai, S. , Z. Bihari , M. Lakatos , T. Szentimrey 2005: Magyarország éghajlatának néhány jellemzője 1901-től napjainkig [Some characteristics of the climate in Hungary from 1901 to nowadays]. – Hungarian Meteorological Service, Budapest, 12 p. (in Hungarian)
Talaee, P.H. 2014: Iranian rainfall series analysis by means on nonparametric tests. – Theoretical and Applied Climatology, 116, pp. 597–607.
XLSTAT 2017: Addinsoft. – https://www.xlstat.com/en
Yue, S. , P. Pilon , G. Cavadias 2002: Power of the Mann-Kendall and Spearman’s rho test for detecting monotonic trends in hydrological series. – Journal of Hydrology, 259, pp. 254–271.
Zarei, A.R. , M.M. Moghimi , M.R. Mahmoudi 2016: Parametric and non-parametric trend of drought in arid and semi-arid regions using RDI Index. – Water Resource Management, 30, pp. 5479–5500.
Zamani, R. , R. Mirabbasi , S. Abdollahi , D. Jhajharia 2017: Streamflow trend analysis by considering autocorrelation structure, long-term persistence, and Hurst coefficient in a semi-arid region of Iran. – Theoretical and Applied Climatology, 129/1–2, pp. 33–45.
Zhang, Q. , M. Xiao , V.P. Singh , X. Chen 2013: Copula-based risk evaluation of droughts across the Pearl River basin, China. – Theoretical and Applied Climatology, 111, pp. 119–131.