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  • 1 Budapest Business School, Hungary
  • 2 University of Pannonia, Hungary
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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.

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.

Fig. 1
Fig. 1

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

Simple linear regression of Y on t is a method to determine the tendency (Eq. 1):

Υt=β0+β1×t+εt,
where β0 is the intercept of the trend line, β1 is the slope of the trend line, t is the number of time step, and εt is the residual.

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:

In the case of determining the presence of a monotonic trend in a time series, the null hypothesis (H0) of the Mann–Kendall test is that the data come from a population where random variables are independent and identically distributed. The alternative hypothesis (Ha) is that the data follow a monotonic trend over time. The Mann–Kendall test statistic is given as (Eq. 2):

S=k=1n1j=k+1nsgn(xjxk),
where j > k and k = 1, 2,…, n − 1, j = 2, 3,…, n and n is the number of the data.

sgn(xj − xk) is calculated as follows (Eq. 3):

sgn(xjxk)={+1ifxjxk>00ifxjxk=01ifxjxk<0.

Kendall (1975) proved that S is asymptotically normally distributed with the following parameters (mean and variance; Eq. 4):

E(S)=0,Var(S)={n(n1)(2n+5)p=1gtp(tp1)(2tp+5)}/18,
where g is the number of tied groups in the data set, tp is the number of data in pth tied group, n is the number of data in the time series.

A positive value of S means that there is an increasing trend, whereas a negative value of S means the opposite, that is, a decreasing trend with time. It was proven that over n > 10 number of data, the standard normal variate Z can be used as for hypothesis test (Eq. 5):

Z={S1[Var(S)]1/2ifS>00ifS=0S+1[Var(S)]1/2ifS<0.

During the hypothesis test, an α = 5% significance level was used in a one-tailed test, as it is supposed that precipitation amount is likely to decrease; therefore, τ is expected to be negative, and the empirical significance level was determined. S is closely related to Kendall’s rank correlation coefficient (τ; Eq. 6):

τ=SD,
where D is the possible number of data pairs from n member of the data set (Eq. 7)
D=(n2),

Modified Mann–Kendall trend test for serially dependent data (seasonal Mann–Kendall trend test)

If seasonal cycles are present in the time series, it is suggested to use a trend test that removes the effect of seasonality (Gilbert 1987). Hirsch et al. (1982) and Hirsch and Slack (1984) developed the method and introduced the seasonal Mann–Kendall test for data that are serially dependent. In the modified Mann–Kendall trend test, a series of x observations recorded over K seasons for L years (without any tied values) is expressed as the following matrix (Rahman et al. 2017; Eq. 8):

X=[x11x21xK1x12x22xK2x1Lx2LxKL].

Let xil be the datum for ith season of the lth year. Each season (in reality each month) of all of the observed years is used to compute the Mann–Kendall parameter of S. Let Si computed for i season as follows (Gilbert 1987; Eq. 9):

Si=k=1ni1l=k+1nisgn(xilxik),
where l > k and ni is the number of data (over years) for ith season and (Eq. 10):
sgn(xilxik)={1ifxilxik>00ifxilxik=01ifxilxik<0.

The computation method of VAR(Si) is given by Gilbert (1987). The S′ statistic for seasonal Kendall is computed as (Eq. 11) and VAR(S′) is as follows (Eq. 12):

S=i=1KSi,
Var(S)=i=1KVar(Si).

Finally, the Z-test statistic is computed and tested by hypothesis test (Eq. 13):

Z={S1[Var(S)]1/2ifS>00ifS=0S+1[Var(S)]1/2ifS<0.

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

After detecting the non-parametric trend, Sen’s (1968) slope estimator was applied. This is a non-parametric method that can calculate the change per time unit (direction and volume). Sen’s method uses a linear model to estimate the slope of the trend, and the variance of residuals should be constant is time (da Silva et al. 2015). First, N′ slope estimates were calculated (Q; Eq. 14):

Q=xixiii,
where xi and xi are the data values at times (or during time period) i′ and i, respectively, and where i′ > i. N′ is the number of data pairs for which i′ > i. In the case where there is only one datum in each time period (Eq. 15),
N=n(n1)2,
where n is the number of time periods (Gilbert 1987; Gocić and Trajković 2013b). The N′ values of Q are ranked from smallest to largest and the median of Q values gives the slope of the tendency. The advantage of this method is that it limits the effect of outliers on the slope (Shadmani et al. 2012), and it is robust and free from restrictive statistical constraints (Lavagnini et al. 2011).

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

The seasonal slope estimator is a generalization of Sen’s slope estimator discussed above. A description of the method is given following Gilbert (1987). The individual Ni slope is calculated first for the ith season as (Eq. 16):

Qi=xilxik1k,
where l > k, and xil is the datum for ith season of the lth year and xik is the datum form the ith season of the kth year. This computation is made for each of K season. Then, N1 + N2 + … + NK′ = N′ individual slope estimates are ranked and found their median (Gilbert 1987).

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).

Fig. 2
Fig. 2

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

Fig. 3
Fig. 3

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.

Fig. 4
Fig. 4

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.

Fig. 5
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.

Table 1

Corrected mean seasonal deviations in each season (1871−2014)

SeasonCorrected mean seasonal deviation from moving average (mm)
January−23.3
February−22.6
March−17.0
April−2.7
May13.5
June19.7
July17.7
August16.5
September4.3
October2.4
November2.2
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).

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  • 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. 803813.

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  • Hirsch, R.M., J.R. Slack 1984: A nonparametric trend test for seasonal data with serial dependence. – Water Resources Research, 20, pp. 727732.

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  • Hirsch, R.M., J.R. Slack , R.A. Smith 1982: Techniques of trend analysis for monthly water quality data. – Water Resources Research, 18, pp. 107121.

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  • 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. 237251.

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  • 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)

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  • 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. 329.

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  • 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. 7788.

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    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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. 109125.

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    • Search Google Scholar
    • Export Citation
  • 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. 117120. (in Hungarian)

    • Search Google Scholar
    • Export Citation
  • 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. 45434560.

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  • 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)

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  • 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.

    • Export Citation
  • 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. 251263.

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  • 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. 146169. (in Hungarian)

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  • 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. 180188.

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  • 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. 113129.

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  • Mann, H.B. 1945: Nonparametric tests against trend. – Econometrica, 13, pp. 245259.

  • 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. 5059.

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    • Export Citation
  • 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. 311320.

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  • 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. 210.

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  • 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. 311327.

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  • 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. 480483.

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  • Pohlert, T. 2017: Non-parametric trend tests and change-point detection. – https://cran.r-project.org/web/packages/trend/vignettes/trend.pdf

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    • Export Citation
  • Pongrácz, R., J. Bartholy , E. Miklós 2011: Analysis of projected climate change for Hungary using ENSEMBLES simulations. – Applied Ecology and Environmental Research, 9, pp. 387398.

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  • Pongrácz, R., J. Bartholy , A. Kis 2014: Estimation of future precipitation conditions for Hungary with special focus on dry periods. – Idojárás, 118, pp. 305321.

    • Search Google Scholar
    • Export Citation
  • 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. 6583.

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    • Export Citation
  • 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. 4044.

    • Search Google Scholar
    • Export Citation
  • 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. 409424.

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  • 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. 100110.

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  • 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. 131139.

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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirsch, R.M., J.R. Slack 1984: A nonparametric trend test for seasonal data with serial dependence. – Water Resources Research, 20, pp. 727732.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirsch, R.M., J.R. Slack , R.A. Smith 1982: Techniques of trend analysis for monthly water quality data. – Water Resources Research, 18, pp. 107121.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 237251.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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)

    • Search Google Scholar
    • Export Citation
  • 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. 329.

    • Search Google Scholar
    • Export Citation
  • 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. 7788.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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. 109125.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 117120. (in Hungarian)

    • Search Google Scholar
    • Export Citation
  • 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. 45434560.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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)

    • Search Google Scholar
    • Export Citation
  • 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.

    • Export Citation
  • 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. 251263.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 146169. (in Hungarian)

    • Search Google Scholar
    • Export Citation
  • 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. 180188.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 113129.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mann, H.B. 1945: Nonparametric tests against trend. – Econometrica, 13, pp. 245259.

  • 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. 5059.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 311320.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 210.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 311327.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 480483.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pohlert, T. 2017: Non-parametric trend tests and change-point detection. – https://cran.r-project.org/web/packages/trend/vignettes/trend.pdf

    • Search Google Scholar
    • Export Citation
  • Pongrácz, R., J. Bartholy , E. Miklós 2011: Analysis of projected climate change for Hungary using ENSEMBLES simulations. – Applied Ecology and Environmental Research, 9, pp. 387398.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pongrácz, R., J. Bartholy , A. Kis 2014: Estimation of future precipitation conditions for Hungary with special focus on dry periods. – Idojárás, 118, pp. 305321.

    • Search Google Scholar
    • Export Citation
  • 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. 6583.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 4044.

    • Search Google Scholar
    • Export Citation
  • 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. 409424.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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. 100110.

    • Crossref
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