The dependency structure between hydrological variables is of critical importance to hydrological modelling and forecasting. When a copula capturing that dependence is fitted to a sample, information on the uncertainty of the fit is needed for subsequent hydrological calculations and reasoning. A new method is proposed to report inferential uncertainty in a copula parameter. The method is based on confidence curves constructed with the use of a pseudo maximum likelihood estimator for the copula parameter. The method was tested on synthetic data and then used as a tool in two hydrological examples. The first examines the probability of major floods in two locations on the Rhine River and its tributaries in the same calendar year. In the second example, rainfall–runoff from a karst region in Tunisia was analysed to determine a confidence interval for the delay between precipitation and runoff.

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The representation of uncertainty in results is an important aspect of statistical techniques in hydrology and climatology. Hypothesis tests and point estimates are not well suited for this purpose. Other statistical tools, such as confidence curves, are better suited to represent uncertainty. Therefore three parametric methods to construct confidence curves for the location of a sudden change in the properties of a time series, a change point (CP), are analyzed for three distributions: log-normal, gamma, and Gumbel. Two types of change are considered: a change in the mean and a change in the standard deviation. A question that confidence curves do not answer is how likely the null hypothesis of ‘no change’ is. A possible statistic to help answer this question, denoted by Un, is introduced and analyzed. It is compared to the statistic that underlies the Pettitt test. All methods perform well in terms of coverage and confidence set size. One method is based on the profile likelihood for a CP, the other two, first defined in this article, on the pseudolikelihood for a CP. The main advantage of the pseudolikelihood over the profile likelihood lies in the much lower computational cost. The confidence curves generated by the three methods are very similar. In a limited test on time series of measurements found in the literature, the methods gave results that largely matched those reported elsewhere. Some results are also given for an order one autoregressive series with a lognormal marginal distribution.

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Analysis of existing data is the first step in preparing for human modification of natural water systems or existing water infrastructure. Time series of environmental measurements form an important part of that data. Such an analysis has as its aim the determination of the future operating conditions of the modified system. The dependence between time series is important both for normal operation and for the evaluation of risks in extreme situations. One way to study this dependence is through the use of copulas. But because the analysis is statistical in nature, its results, in this case the copula parameters, contain a certain amount of uncertainty. In this paper we demonstrate an approach that can be used to represent that uncertainty in cases where the dependence is modeled by a copula. The method is based on the confidence curve concept, that is it provides confidence sets for the parameter at all confidence levels. The use of confidence curves for copula parameters is a recent development. The confidence curve construction method uses a pseudo likelihood to avoid having to fit marginals to the data. This pseudo likelihood is then used to construct a confidence curve. The method was applied to annual maximum river discharge data for different tributaries of the Rhine to see how these are correlated. @en

The availability of water resources is declining, while the demand for water continues to increase for the agricultural sector and drinking water supply in the context of climate change hazards, such as the increasing frequency of extreme weather events (drought, floods). In fact, several studies show that droughts occur more and more frequently, with a duration varying from one month to a few years. Therefore, monitoring these droughts would help in the management of groundwater in the short and medium term and thus would enhance adaptation to climate change. For this purpose, several indices allowing the prediction and characterization of these droughts have been proposed by researchers. Among these, two indices were selected: SPEI (Standardized Precipitation Evapotranspiration Index) and SGI (Standardized Groundwater Index). SPEI makes it possible to take into account precipitation and evapotranspiration on different time scales which facilitates the analysis of drought impact on water resource demands. The same is true for SGI, which allows analysis of the water table level. The objective of this work is to calculate each of these indices for several boreholes in Tunisia to assess correlations between SPEI and SGI, to project SPEI over several horizons under climate change and to predict the evolution of the SGI thanks to the correlations. Correlation between SPEI and SGI time series was examined using three different copula families, Frank, Gumbel and Clayton. Copulas were fitted to a 2D data set consisting of the SPEI time series and the time shifted SGI time series for different lag times. The Kendall @en

Climate change is incompatible with the assumption of stationarity. This has lead to a sharp increase in the detection and study of nonstationarity in hydro-meteorological processes. Most hydro-meteorological processes are still analyzed by studying time series of observations. From the perspective of statistical characteristics, a stationary time series does not show significant changes. On the contrary, a nonstationarity time series often shows a slowly increasing/decreasing trend or a sudden change. A sudden change or a change point is a time point that a time series shows a great change in its statistical characteristics, for instance in themean or the standard deviation. For stationary cases, hydrologists have a large number of statistical tools to analyse these time series. These tools can not only help hydrologists to gain a deep insight into time series, but they can also analyse the corresponding uncertainty. For nonstationary cases, the detection of changes has drawn the majority of attention, however, the uncertainty associated with the detection has still been rarely studied. Therefore, this PhD research aims at bridging the gap between nonstationarity detection and the uncertainty of detection. To be more specific the main scope is rooted in analysing the uncertainty associated with detecting a change point in hydro-meteorological time series. When it comes to representing uncertainties, a traditional choice is using a confidence interval with a certain confidence level. In this research instead, the uncertainty is represented by confidence curves because they are capable of capturing more information by including all confidence intervals at all confidence levels and they visualize uncertainty in a curve. To verify the general applicability of a confidence curve in representing uncertainties, both a discrete parameter and a continuous parameter are considered in this research. The location of a change point is considered as a discrete parameter, and the dependence parameter in copula models will be considered as a continuous one. Additionally, in order to simplify the construction of a confidence curve, several new approaches have been presented in this research. Based on results and findings, confidence curves have been proven to be more informative and theoretically they can represent uncertainties of all types of parameter of interest. With a confidence curve, hydrologists can easily read the uncertainty of the detected change point and this would also provide decision-makers a better insight into the nonstationarity of a time series of a hydro-meteorological observations. @en

In this paper, a method based on Approximate Empirical likelihood ratio and a Deviance function combined with bootstrapping (AED-BP) is proposed to construct a confidence curve for the location of a change point. The method is compared with a method based on parametric Profile Likelihood and a Deviance function combined with Monte Carlo simulation (PLD-MC). A confidence curve provides a representation of the uncertainty in the outcome of the change point analysis. To evaluate the practical usability of confidence curves constructed by AED-BP, its properties were examined and its performance was compared to that of PLD-MC. The methods were applied to both synthetic and real data. Synthetic data were generated from three parametric distributions: Fréchet with a constant shape parameter, log-normal, and gamma distributions. The real data are the hydrometeorological data analysed in other studies. The change points found in the original publications are used as a reference in this present paper. The results show that AED-BP has a performance that is similar to PLD-MC, but has an advantage in that it is not necessary to select a distribution family for the data. The AED-BP results on the Annual Maximum Runoff series for the stations Yichang and Hankou along the Yangtze river are among the first that show a possible effect of the presence of the Three Gorges dam.@en

Several commonly-used nonparametric change-point detection methods are analysed in terms of power, ability and accuracy of the estimated change-point location. The analysis is performed with synthetic data for different sample sizes, two types of change and different magnitudes of change. The methods studied are the Pettitt method, a method based on the Cramér von Mises (CvM) two-sample test statistic and a variant of the CUSUM method. The methods differ considerably in behaviour. For all methods the spread of estimated change-point location increases significantly for points near one of the ends of the sample. Series of annual maximum runoff for four stations on the Yangtze River in China are used to examine the performance of the methods on real data. It was found that the CvM-based test gave the best results, but all three methods suffer from bias and low detection rates for change points near the ends of the series.

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When investigating the hydrological cycle one does not always have the luxury of perfect knowledge. Moreover, searching for possible causes for changes in the behaviour of hydrological variables can be like looking for a needle in a haystack. Statistical tools can help to restrict our search to certain parts of that haystack, provided we have some idea of the probability of not looking in the right place. This work considers change points in hydrological time series. Numerous papers have been published on non-stationarity in hydrological time series and several methods have been developed to solve the problems related to it, such as the detection and quantification of the change including change-point and trend. Change points in hydrological time series are often the consequence of human behaviour such as urbanization, deforestation, change on land-use and the construction of large waterrelated projects. Many techniques, both parametric and non-parametric, have been proposed to infer the position of , but most of them concentrate on detection of the presence of a change point and providing a specific value for its location on the basis of a given probability bound. In this research confidence distributions are used to find and express the uncertainty about its position. The results are compared to those of commonly used tests: a Cramér von Mises statistic-based test, Pettitt’s test, and the Lee-Heighinian test. All methods are applied to a large number of simulated data series and the results are compared. @en

Information on the causes of past change points in regional hydrological behaviour may not always be available. While change points in hydrological time series are often the consequence of human activities such as urbanization, deforestation, change on land-use and the construction of large water-related projects, not every potential cause has an effect and not every cause may have been properly recorded. It is therefore interesting to take data series where potential causes are known, apply different methods for change point detection, and analyse the results in light of the historical data. For such an analysis an estimate of the uncertainty in the location of the change point would be very helpful. In this research, confidence distribution techniques are used to find and express the uncertainty about the position of . The results are compared to those of several commonly used tests: a Cramér von Mises statistic-based test, Pettitt’s test, and the Lee-Heighinian test. All methods will be applied to four discharge series from different measurement stations on the Yangtze River in China. @en