Multivariate hydrologic design methods under nonstationary conditions and application to engineering practice Multivariate hydrologic design methods under nonstationary conditions and application to... Cong Jiang et al.
Cong Jiang1, Lihua Xiong2, Lei Yan3, Jianfan Dong4, and Chong-Yu Xu2,5 Cong Jiang et al. Cong Jiang1, Lihua Xiong2, Lei Yan3, Jianfan Dong4, and Chong-Yu Xu2,5
3College of Water Conservancy and Hydropower, Hebei University of Engineering, Handan 056002, China
4Guangxi Water Resources Management Center, Nanning 530023, China
5Department of Geosciences, University of Oslo, P.O. Box 1047 Blindern, 0316 Oslo, Norway
Correspondence: Cong Jiang
Received: 27 May 2018 – Discussion started: 12 Jul 2018 – Revised: 14 Feb 2019 – Accepted: 09 Mar 2019 – Published: 22 Mar 2019
Multivariate hydrologic design under stationary conditions is traditionally performed through the use of the design criterion of the return period, which is theoretically equal to the average inter-arrival time of flood events divided by the exceedance probability of the design flood event. Under nonstationary conditions, the exceedance probability of a given multivariate flood event varies over time. This suggests that the traditional return-period concept cannot apply to engineering practice under nonstationary conditions, since by such a definition, a given multivariate flood event would correspond to a time-varying return period. In this paper, average annual reliability (AAR) was employed as the criterion for multivariate design rather than the return period to ensure that a given multivariate flood event corresponded to a unique design level under nonstationary conditions. The multivariate hydrologic design conditioned on the given AAR was estimated from the nonstationary multivariate flood distribution constructed by a dynamic C-vine copula, allowing for time-varying marginal distributions and a time-varying dependence structure. Both the most-likely design event and confidence interval for the multivariate hydrologic design conditioned on the given AAR were identified to provide supporting information for designers. The multivariate flood series from the Xijiang River, China, were chosen as a case study. The results indicated that both the marginal distributions and dependence structure of the multivariate flood series were nonstationary due to the driving forces of urbanization and reservoir regulation. The nonstationarities of both the marginal distributions and dependence structure were found to affect the outcome of the multivariate hydrologic design.
Jiang, C., Xiong, L., Yan, L., Dong, J., and Xu, C.-Y.: Multivariate hydrologic design methods under nonstationary conditions and application to engineering practice, Hydrol. Earth Syst. Sci., 23, 1683–1704, <URL>, 2019.
A complete flood event or a flood hydrograph contains multiple feature variables, such as flood peak and flood volume, which can be associated with the safety of hydraulic structures (Salvadori et al., 2004, 2007, 2011; Xiao et al., 2009; Xiong et al., 2015; Loveridge et al., 2017; Shafaei et al., 2017). For example, the water level of a reservoir is controlled not only by flood peak flow but also by flood volume (Salvadori et al., 2011). Therefore, multivariate hydrologic design, which takes into account multiple flood characteristics as well as their dependence, provides a more rational design strategy for hydraulic structures compared to univariate hydrologic design (Zheng et al., 2013, 2014; Balistrocchi and Bacchi, 2017).
Multivariate hydrologic design under stationary conditions has been widely investigated, and the design criterion is usually quantified by the return period, similar to univariate hydrologic design. Under the definition of the average recurrence interval between flood events equaling or exceeding a given threshold (Chow, 1964), the return period of a given flood event under stationary conditions theoretically equals the average inter-arrival time between flood events divided by the exceedance probability (Salvadori et al., 2011). On the other hand, the exceedance probability of a univariate flood event is usually uniquely defined without ambiguity, whereas the exceedance probability of a multivariate flood event could have multiple definitions (Salvadori and De Michele, 2004; Salvadori et al., 2011; Vandenberghe et al., 2011). To date, at least five kinds of different exceedance probabilities for a multivariate flood event have been defined: (1) the OR case in which at least one of the flood features exceeds the prescribed threshold, (2) the AND case in which all flood features exceed the prescribed thresholds, (3) the Kendall case in which the univariate representation transformed from Kendall's distribution function exceeds the prescribed threshold, (4) the survival Kendall case in which the univariate representation transformed from survival Kendall's distribution function exceeds the prescribed threshold, and (5) the structural case in which the univariate representation transformed from a structure function exceeds the prescribed threshold (Favre et al., 2004; Salvadori and De Michele, 2004, 2010; Salvadori et al., 2007, 2013, 2015, 2016; Vandenberghe et al., 2011; Requena et al., 2013; Zheng et al., 2014).
Due to climate change as well as certain anthropogenic driving forces (Milly et al., 2008), the nonstationarities of both univariate and multivariate flood series have been widely reported (Xiong and Guo, 2004; Villarini et al., 2009; Vogel et al., 2011; López and Francés, 2013; Bender et al., 2014; Xiong et al., 2015; Blöschl et al., 2017; Kundzewicz et al., 2018). The multivariate flood distribution exhibits more complex nonstationarity behaviours than the univariate distribution, including nonstationarities of individual margins and the dependence structure between the margins (Quessy et al., 2013; Bender et al., 2014; Xiong et al., 2015; Kwon et al., 2016; Sarhadi et al., 2016; Qi and Liu, 2017; Vezzoli et al., 2017; Bracken et al., 2018; Salvadori et al., 2018). Both nonstationarities of the margins and dependence structure could impact the multivariate hydrologic design. Under nonstationary conditions, the exceedance probability p of a given flood event varies from year to year; thus, the return period, calculated as the average inter-arrival time between two successive flood events divided by p, is no longer a constant (Salas and Obeysekera, 2014; Jiang et al., 2015a; Kwon et al., 2016; Sarhadi et al., 2016; Yan et al., 2017). As a result, a given flood event would correspond to a time-varying and non-unique return period. Consequently, the traditional return-period-based method for estimating hydrologic design may no longer be applicable to engineering practice under nonstationary conditions (Salas and Obeysekera, 2014).
Although increasing attention has been focused on the hydrologic designs under nonstationary conditions in recent years, the focus has mainly been on univariate designs (Obeysekera and Salas, 2014, 2016; Read and Vogel, 2016). To overcome the limitation of the traditional return period under nonstationary conditions, the concept of the return period has been revisited. Salas and Obeysekera (2014) extended two concepts of the return period into a nonstationary framework, defined as the expected waiting time (EWT) for an exceedance to occur (Olsen et al., 1998), and the time period that results in the expected number of exceedances (ENE) equal to 1 over this period (Parey et al., 2010).
Risk and reliability are both important measurements for assessing hydrologic designs (Vogel, 1987; Read and Vogel, 2015). Besides redefinitions of the return period, some risk-based or reliability-based metrics have been introduced as the hydrologic design criteria under nonstationary conditions (Rosner et al., 2014). Rootzén and Katz (2013) proposed the concept of the design life level (DLL) to quantify hydrologic risk in a nonstationary climate during the entire design life period of hydraulic structures. Read and Vogel (2015) introduced the concept of average annual reliability (AAR) to estimate the hydrologic designs under nonstationary conditions. Liang et al. (2016) defined the equivalent reliability (ER) to estimate the design flood under nonstationary conditions by linking the DLL to the return period. Salvadori et al. (2018) associated hydrologic designs with both given life times and failure probabilities to calculate bivariate design values under nonstationarity. These design criteria assess the risk or reliability of hydraulic structures associated with the flood distribution during the entire design life period, rather than for a single year. For a given design life period, these criteria can always yield a unique risk or reliability; therefore, they are applicable to the hydrologic designs under both stationary and nonstationary conditions.
Under the multivariate framework, a given design level would correspond to an infinite number of possible hydrologic design events (Hawkes, 2008; Kew et al., 2013; Zheng et al., 2015, 2017); however, these design events are generally not equivalent because their joint probability density values (i.e. likelihood) usually differ (Salvadori et al., 2011; Volpi and Fiori, 2012; Li et al., 2017; Yin et al., 2017). In engineering practice, it should be necessary to determine a typical design event as representative of a specific design level. For example, in Chinese engineering practice, a unique design flood hydrograph corresponding to a given design level is usually required to determine the scale of hydraulic structures (Yin et al., 2017). The most-likely design event, which theoretically has the largest joint probability density (likelihood) among all possible design events (Salvadori et al., 2011), appears to be the best representative candidate. Besides the most-likely design event, it is also necessary to identify the confidence interval for an infinite possible number of design events to provide a finite design range for designers (Volpi and Fiori, 2012; Yin et al., 2017). The most-likely design event and confidence interval for the bivariate hydrologic design under stationary conditions have been identified (Salvadori et al., 2011; Volpi and Fiori, 2012; Li et al., 2017; Yin et al., 2017; Salvadori et al., 2018); however, very few studies have focused on hydrologic designs with higher dimensions under nonstationary conditions.
Therefore, the objective of the present study was to address the issue of multivariate hydrologic design applying to engineering practice under nonstationary conditions, which is achieved through the following steps. First, the nonstationary multivariate flood distribution was constructed using a dynamic canonical vine (C-vine) copula (Aas et al., 2009), which was able to capture the nonstationarities of both marginal distributions and the dependence structure. The design criterion for the multivariate flood event was then quantified according to AAR rather than the traditional return period, since a given multivariate flood event would correspond to a unique AAR under both stationary and nonstationary conditions (Read and Vogel, 2015; Yan et al., 2017). The multivariate hydrologic design for any given AAR was estimated from the nonstationary multivariate flood distribution.
The aforementioned methods for the multivariate hydrologic design under nonstationary conditions were applied to the Xijiang River, China. The four-dimensional (4-D) multivariate flood series, including the annual maximum daily discharge, annual maximum 3-day flood volume, annual maximum 7-day flood volume and annual maximum 15-day flood volume of the Xijiang River were chosen as the case study data because they constitute the variables used for deriving the design flood hydrograph for hydraulic structures. It has been found that the natural flood processes of this river have been significantly altered by urbanization and reservoir regulation (Xu et al., 2014), but these two factors have not yet been taken into account in multivariate hydrologic design.
The next section of the present paper describes the study area and data. Section 3 presents the methods developed in this paper. The results of the case study are provided in Sect. 4. Finally, the conclusion and remarks are provided in Sect. 5.
The multivariate flood series of the Xijiang River, South China (see Fig. 1), were selected as a case study to illustrate the multivariate hydrologic design methods under nonstationary conditions. The drainage area of the Xijiang River basin (XRB) is 353 120 km2, with a river length of 2214 km. The basin falls within a humid subtropical monsoon climate region, with the flood season extending from May to October; therefore, floods have always been a serious natural hazard within the basin.
Figure 1Map of the Xijiang River basin (above the Dahuangjiangkou gauge).
The calculation of design floods in China involving the derivation of flood hydrographs for hydraulic structures requires not only flood peak but also flood volumes with different durations, such as 3, 7, 15 and 30 days (Ministry of Water Resources of People's Republic of China, 1996; Xiao et al., 2009; Xiong et al., 2015; Li et al., 2017). For a large catchment such as the XRB, the duration of a flood process is usually longer than 10 days. Therefore, the annual maximum daily discharge (Q1), annual maximum 3-day flood volume (V3), annual maximum 7-day flood volume (V7) and annual maximum 15-day flood volume (V15) of the Xijiang River were defined as the multivariate flood series Q1,V3,V7,V15 for deriving design flood hydrographs. The flood data were from 1951 to 2012 and observed at the Dahuangjiangkou gauge located at the main stream of the Xijiang River, draining a total catchment area of 294 669 km2, approximately 83 % of the total area of the XRB.
Rapid urbanization over recent decades has resulted in increasing river regulation projects built in the XRB, such as artificial levees for protecting urban areas from river flooding. As a result, flood flow has become increasingly constrained to the channel rather than overflowing to the floodplain, resulting in an increase in the observed river flood flow (Xu et al., 2014). For the purpose of flood control and hydropower generation, it is hard to find a river which is not impacted by reservoirs, particularly in rapidly developing China. Reservoir regulation has become an increasingly significant factor affecting flood processes of the XRB and should be seriously considered within downstream flood risk analysis and hydrologic design, particularly after 2006, when two reservoirs with considerable flood control capacities were put into operation. These are the Longtan and Baise reservoirs, with flood control capacities of 5 × 109 m3 and 1.64×109 m3 and catchment areas of 98 500 and 9600 km2, respectively. Climate change will likely result in flood nonstationarity by altering climatic conditions of the basin. Climatic conditions dominating flood processes in the XRB, such as extreme precipitation, appear to have been stationary over the past decades (Yang et al., 2010). Therefore, the current study introduced only urbanization and reservoir regulation as the potential driving forces of flood nonstationarity and ignored the effect of climate change.
The effect of urbanization on flood processes was quantified using the urban population (Pop). Given the unavailability of urban population data at the basin scale and the fact that the vast majority of cities in the XRB are distributed in Guangxi province, we used urban population data for Guangxi province to represent those of the XRB. The annual urban population data during the observation period were obtained from the China Compendium of Statistics 1949–2008 (Department of Comprehensive Statistics of National Bureau of Statistics, 2010) and the website of the National Bureau of Statistics of China (<URL>, last access: 20 March 2019). The present study assumed the design life period for hydraulic structures to be from 2013 to 2100. The urban population over the design life period was estimated based on the predicted growth rate of China's urban population reported by He (2014). The reservoir index (RI), which depends on the catchment area and flood controlling capacities of reservoirs, was used to quantify the effects of reservoir regulation on flood processes (López and Francés, 2013). As shown in Table 1, two reservoirs with flood control functions have been completed during the observation period from 1951 to 2012, and a further two are planned for operation during the design life period. Figure 2 illustrates the evolution of the urban population and reservoir index during both the observation and design life periods.
Figure 2Evolution of the urban population and reservoir index during both the observation and design life periods.
Table 1Reservoir information for the Xijiang River basin.
The present study included the following methods: (1) the nonstationary multivariate flood distribution based on a dynamic C-vine copula, allowing for both time-varying marginal distributions and a time-varying dependence structure, and (2) estimation of the multivariate hydrologic design associated with AAR under nonstationary conditions. To correspond to the case study in this paper, the multivariate flood series consisting of the annual maximum daily discharge (Q1), annual maximum 3-day flood volume (V3), annual maximum 7-day flood volume (V7) and annual maximum 15-day flood volume (V15) were chosen to illustrate the multivariate design methods under nonstationary conditions. It is worth noting that the proposed methods can be extended to other multivariate flood series, such as those consisting of flood peak, flood volume and flood duration.
3.1 Probability distribution of the nonstationary multivariate flood series
According to Sklar's theorem (Sklar, 1959), the probability distribution of the 4-D flood series Q1,V3,V7,V15 at time t measured by years (t=1,2,…,n, and n is the length of the flood series) can be formulated through a copula C(⋅) as follows:
Fq1,t,v3,t,v7,t,v15,tθt=CF1q1,t|θ1,t,F3v3,t|θ3,t,F7v7,t|θ7,t,F15v15,t|θ15,t|θc,t(1)=Cu1,t,u3,t,u7,t,u15,t|θc,t,
where F1q1,t|θ1,t, F3v3,t|θ3,t, F7v7,t|θ7,t and F15v15,t|θ15,t denote the marginal distributions for Q1, V3, V7 and V15, respectively; u1, t, u3, t, u7, t and u15, t are the marginal probabilities of Q1, V3, V7 and V15, respectively; θ1, t, θ3, t, θ7, t and θ15, t are the corresponding distribution parameters; and θc, t stands for the copula parameter vector, which describes the strength of the dependence structure. θt=θ1,t,θ3,t,θ7,t,θ15,t,θc,t is the parameter vector of the entire multivariate distribution, including the marginal distribution parameters as well as the copula parameters.
According to the multivariate distribution of Q1,V3,V7,V15 defined by Eq. (1), the corresponding density function can be written as:
fq1,t,v3,t,v7,t,v15,tθt=cF1q1,t|θ1,t,F3v3,t|θ3,t,F7v7,t|θ7,t,F15v15,t|θ15,t|θc,t⋅f1q1,t|θ1,t⋅f3v3,t|θ3,t⋅f7v7,t|θ7,t⋅f15v15,t|θ15,t=cu1,t,u3,t,u7,t,u15,t|θc,t⋅f1q1,t|θ1,t(2)⋅f3v3,t|θ3,t⋅f7v7,t|θ7,t⋅f15v15,t|θ15,t,
where f1q1,t|θ1,t, f3v3,t|θ3,t, f7v7,t|θ7,t and f15v15,t|θ15,t are the density functions of the marginal distributions for Q1, V3, V7 and V15, respectively, and c(⋅) denotes the density function of copula C(⋅). As shown by Eq. (2), the multivariate distribution of Q1,V3,V7,V15 can be separated into two modules, including the marginal distributions, i.e. f1q1,t|θ1,t, f3v3,t|θ3,t, f7v7,t|θ7,t and f15v15,t|θ15,t, as well as the dependence structure expressed by the copula density function cu1,t,u3,t,u7,t,u15,t|θc,t. Under nonstationary conditions, both the margins and dependence structure of Q1,V3,V7,V15 can vary over time t.
3.1.1 Nonstationary marginal distributions based on the time-varying moment model
The time-varying moment model that expresses the distribution parameters or moments as functions of time or some other explanatory variable or variables have been widely employed to capture the nonstationarities of univariate flood series (Strupczewski et al., 2001; Villarini et al., 2009). In this study, the nonstationary marginal distributions of the multivariate flood series Q1,V3,V7,V15 were constructed by the time-varying moment model.
Based on cause–effect analysis, the flood processes of the XRB were found to mainly be impacted by urbanization and reservoir operation. The reservoir index RI and urban population Pop were therefore used as potential covariates for marginal distribution parameters, including the location parameter μ, scale parameter σ and shape parameter ν (if any). In this study, both linear and exponential functions were considered to build the relationships between distribution parameters and covariates (Strupczewski et al., 2001; Vogel et al., 2011; Salas and Obeysekera, 2014; Jiang et al., 2015; Sarhadi et al., 2016; Read and Vogel, 2016; Yan et al., 2017). Taking the location parameter for illustration, the candidate functions of μ were generally formulated as follows.
(3) Linear : μ t = α 0 + α 1 Pop t + α 2 RI t . Exponential : μ t = exp α 0 + α 1 Pop t + α 2 RI t .
Here α0, α1 and α2 are model parameters estimated using the maximum likelihood estimate (MLE) method (Strupczewski et al., 2001). As above, the linear expression in Eq. (3) gives an additive model which suggests that the effects of the covariates RI and Pop on μ are independent, while the exponential expression defines a multiplicative model which is able to take into account the possible interaction between the covariates RI and Pop. It is important to note that Eq. (3) defines four specific nonstationary models: the first one is the most complex nonstationary model where it is assumed that both RI and Pop are the driving factors of marginal distributions, the second and third models illustrate that the marginal nonstationarity is linked only to RI and Pop, respectively, and the final one represents the simplest and stationary model, which does not contain any covariates.
Five probability distributions widely used in flood frequency analysis, namely Pearson type III (PIII), generalized extreme value (GEV), gamma, Weibull and lognormal distributions, were employed as the candidate distributions for margins (Villarini et al., 2009; Yan et al., 2017). The goodness of fit (GoF) of the probability distributions was examined by the Kolmogorov–Smirnov (KS) test with a significance level of 0.05 (Frank and Massey, 1951). The p value of the KS test was simulated by the Monte Carlo method. The relative fitting qualities of the time-varying moment models were assessed by the corrected Akaike information criterion (AICc; Hurvich and Tsai, 1989), which is stricter than the Akaike information criterion (AIC; Akaike, 1974). The best model featured with the smallest AICc value was chosen to describe the marginal distributions from the nonstationary models as expressed by Eq. (3).
3.1.2 Nonstationary dependence structure based on the dynamic C-vine copula
After estimating the marginal distributions, the nonstationary dependence structure of Q1,V3,V7,V15 as formulated by the copula density function cu1,t,u3,t,u7,t,u15,t|θc,t was constructed. Given that most applied copula functions are for bivariate random variables, cu1,t,u3,t,u7,t,u15,t|θc,t cannot be directly expressed as a specific copula function. The pair copula method has been proven to be powerful for the construction of the distribution of multivariate random variables through the decomposition of the multivariate probability density into a series of bivariate copulas (Aas et al., 2009; Xiong et al., 2015; Shafaei et al., 2017). Therefore this study constructed the dependence structure of Q1,V3,V7,V15 using the pair copula method.
Numerous pair copula decomposition forms for a multivariate distribution are available, among which two kinds of decompositions with regular vine structures prevail in practice, namely the canonical vine (C vine) and the drawable vine (D vine; Aas et al., 2009). It is known that flood peak (e.g. Q1) is the dominant feature quantifying a flood event as well as being the key factor in hydrologic design (Ministry of Water Resources of People's Republic of China, 1996). The C vine is more suitable when there is a key variable governing multivariate dependence (Aas et al., 2009). In this case, the C vine was employed to construct the joint distribution of Q1,V3,V7,V15, with Q1 elected as the key variable. Thus, the density function cu1,t,u3,t,u7,t,u15,t|θc,t can be decomposed into six bivariate pair copulas as follows:
cu1,t,u3,t,u7,t,u15,t|θc,t=c13u1,t,u3,t|θ13,t⋅c17u1,t,u7,tθ17,t⋅c115u1,t,u15,tθ115,t⋅c37|1Fu3,t|u1,t,Fu7,t|u1,tθ37|1,t⋅c315|1Fu3,t|u1,t,Fu15,t|u1,tθ315|1,t⋅c715|13Fu7,t|u1,t,u3,t,(4)Fu15,t|u1,t,u3,t|θ715|13,t],
where θc,t=θ13,t,θ17,t,θ115,t,θ37|1,t,θ315|1,t,θ715|13,t is the parameter vector in the C-vine copula, and
Fu3,t|u1,t=∂C13u1,t,u3,t|θ13,t∂u1,t,Fu7,t|u1,t=∂C17u1,t,u7,tθ17,t∂u1,t,Fu15,t|u1,t=∂C115u1,t,u15,tθ115,t∂u1,t,Fu7,t|u1,t,u3,t=∂c37|1Fu3,t|u1,t,Fu7,t|u1,tθ37|1,t∂Fu3,t|u1,t,Fu15,t|u1,t,u3,t=(5)∂C315|1Fu3,t|u1,t,Fu15,t|u1,tθ315|1,t∂Fu3,t|u1,t.
Figure 3 shows the schematic decomposition of the 4-D C-vine copula as expressed by Eq. (4). It is evident that the hierarchical structure of the 4-D C-vine copula contains three trees and six edges. The first tree (T1) includes three bivariate pair copulas, i.e. c13⋅θ13,t, c17⋅θ17,t and c115⋅θ115,t, which directly act on the marginal probabilities and describe the bivariate dependencies between the key variable Q1 and the other three variables, i.e. V3, V7 and V15. The second tree (T2) includes two bivariate pair copulas c37|1⋅θ37|1,t and c315|1⋅θ315|1,t, which act on the conditional distribution functions with u1, t as the conditioning variable. Finally, the third tree (T3) includes only one bivariate pair copula c715|13⋅θ715|13,t acting on conditional distribution functions with both u1, t and u3, t as the conditioning variables.
Figure 3Decomposition of the four-dimensional C-vine copula.
In flood frequency analysis, the upper tail of the flood distribution deserves more attention because it allows the quantification of risks of the more serious flood events. The Gumbel–Hougaard copula, an extreme-value copula widely used in hydrology, accounts for the upper tail dependence and is well-suited to the dependence structure of a multivariate flood distribution (Salvadori et al., 2007; Zhang and Singh, 2007; Xiong et al., 2015). Consequently, the present study employed the bivariate Gumbel–Hougaard copula to construct the dynamic C-vine copula formulated by Eq. (4). The bivariate Gumbel–Hougaard copula is expressed as follows:
Cu,v=exp--lnuθc+-lnvθc1/θc,(6)θc∈[1,∞),
where u and v are the bivariate marginal probabilities and θc is the single parameter measuring the dependence strength.
Similar to the nonstationary marginal distributions, the nonstationarity of the dependence structure of Q1,V3,V7,V15 was characterized by the time variations of the copula parameters in T1, T2 and T3. Both linear and exponential functions were considered to characterize the time-varying copula parameters and generally formulated as follows.
(7) Linear : θ c , t = β 0 + β 1 Pop t + β 2 RI t . Exponential : θ c , t = 1 + exp β 0 + β 1 Pop t + β 2 RI t .
Here β0, β1 and β2 are model parameters estimated using the MLE method (Aas et al., 2009). Here, the exponential expression in Eq. (7) was written as the sum of 1 and an exponential function of the covariates so that the domain range of the copula parameter θc can be satisfied under any condition. To make it easy in parameter estimation, the model parameters for each pair copula were separately estimated. The model parameters for θ13, t, θ17, t and θ115, t in T1 were first estimated, and those for the remaining copula parameters θ37|1,t, θ315|1,t and θ715|13,t in T2 and T3 were then estimated in sequence. It is worth noting that these parameters can be also simultaneously estimated. These two methods could result in possible difference in parameter estimation.
The available GoF tests for vine copulas are very limited, with the probability integral transform (PIT) test appearing to be reliable (Aas et al., 2009). Under a null hypothesis of the multivariate flood variables Q1,V3,V7,V15 following a given C-vine copula, the PIT converts the dependent flood variables into a new set of variables that are independent and uniformly distributed on [0, 1]4. The GoF of vine copulas can be obtained through determining whether the resulting variables are independent and uniform in [0, 1]. For more details of the PIT test, readers are referred to Aas et al. (2009). The best nonstationary model for each bivariate pair copula in Eq. (4) was chosen from the nonstationary models generally expressed by Eq. (7) in terms of the AICc value.
3.2 Multivariate hydrologic design under nonstationary conditions
3.2.1 Average annual reliability for multivariate flood events
The AAR introduced by Read and Vogel (2015) was calculated using the arithmetic-average method, thereby taking into account the reliability of each year with the same weighting factor. A safer design strategy should pay more attention to worse (i.e. lower) annual reliability; however, the arithmetic-average AAR is not capable of this function. The present study employed the geometric-average method to calculate AAR, which is dominated more by the minimum than arithmetic average and is theoretically able to yield safer design values. The geometric-average AAR is also equivalent to the metrics of the DLL (Rootzén and Katz, 2013) and ER (Liang et al., 2016; Yan et al., 2017).
Denoting q1,v3,v7,v15 as a given multivariate flood event, its exceedance probability pt, which is the occurrence probability of a more dangerous multivariate event than q1,v3,v7,v15 in a specific hazard scenario, would vary from year to year under nonstationary conditions. The AAR for q1,v3,v7,v15 was calculated by the geometric-average method as follows:
(8) AAR q 1 , v 3 , v 7 , v 15 = ∏ t = T 1 T 2 1 - p t 1 T 2 - T 1 + 1 ,
where T1 and T2 stand for the beginning year and ending year of the operation of an assumed hydraulic structure, respectively, T2-T1+1 is the length of the design life period of the hydraulic structure, and 1−pt measures the annual reliability of the given multivariate flood event q1,v3,v7,v15 at time t.
3.2.2 Exceedance probabilities of multivariate flood events
The present study characterized AAR by considering three widely used definitions of the exceedance probabilities of the multivariate flood event q1,v3,v7,v15, i.e. the OR, AND and Kendall cases (Salvadori and De Michele, 2004, 2010; Favre et al., 2004; Salvadori et al., 2007, 2016; Vandenberghe et al., 2011). The OR case for q1,v3,v7,v15 defines the case under which at least one of the flood features exceeds the prescribed threshold. The exceedance probability in the OR case at time t was denoted as ptor and was calculated by the following:
ptor=PQ1,t≥q1∨V3,t≥v3∨V7,t≥v7∨V15,t≥v15(9)=1-Fq1,v3,v7,v15θt,
where “∨” stands for the OR operator and F(⋅|θt) is defined in Eq. (1).
The AND case for q1,v3,v7,v15 defines the case under which all of the flood features exceed the prescribed thresholds, and the corresponding exceedance probability ptand at time t was expressed as follows:
ptand=PQ1,t>q1∧V3,t>v3∧V7,t>v7∧V15,t>v15=∫∫∫∫ΩandfQ1,t,V3,t,V7,t,V15,tθtdQ1,t⋅dV3,t⋅dV7,t⋅dV15,tΩand:q1<Q1,t<∞,v3<V3,t<∞,v7<V7,t<∞,(10)v15<V15,t<∞,
where “∧” is the AND operator and f(⋅|θt) is defined in Eq. (2).
Under the Kendall case, the multivariate flood event q1,v3,v7,v15 was first transformed into a univariate representation via Kendall's distribution function KC(⋅) as follows:
KCρt=PCU1,t,U3,t,U7,t,U15,tθc,t≤ρt(11)=PFQ1,t,V3,t,V7,t,V15,tθt≤ρt,
where ρt=Fq1,v3,v7,v15θt is the probability level corresponding to the given flood event q1,v3,v7,v15. The exceedance probability ptken in the Kendall case at time t was expressed as follows:
(12) p t ken = 1 - K C ρ t .
For general multivariate cases, the exceedance probabilities ptor, ptand and ptken could have no analytical solutions but can be numerically estimated through the Monte Carlo method (Niederreiter, 1978; Salvadori et al., 2011, 2013). The AAR in the OR, AND and Kendall cases can be calculated by replacing the exceedance probability pt in Eq. (8) by ptor, ptand and ptken, respectively.
3.2.3 Most-likely design event and confidence interval for multivariate hydrologic design
The methods identifying both the most-likely design event, denoted by zQ1*,zV3*,zV7*,zV15*, and the confidence interval for the multivariate hydrologic design zQ1,zV3,zV7,zV15 given AAR=η are introduced below. The average annual probability density, denoted by g(⋅), of zQ1,zV3,zV7,zV15 over the entire design life period from T1 to T2, was expressed as follows:
gzQ1,zV3,zV7,zV15=(13)1T2-T1+1∑t=T1T2fzQ1,zV3,zV7,zV15θt.
The probability distribution function for AAR≤η can be written as
Φη=∫∫∫∫Ω:AARq1,v3,v7,v15≤ηgq1,v3,v7,v15(14)dq1dv3dv7dv15.
By denoting the density function of Φ(η) as ϕ(η), the probability density of zQ1,zV3,zV7,zV15 conditioned on AAR=η can be expressed as follows:
g|AAR=ηzQ1,zV3,zV7,zV15(15)=gzQ1,zV3,zV7,zV15ϕη.
The most-likely design event conditioned on AAR=η was theoretically written as
zQ1*,zV3*,zV7*,zV15*=(16)argmaxg|AAR=ηzQ1,zV3,zV7,zV15.
Unfortunately, the analytical solutions of both the most-likely design event zQ1*,zV3*,zV7*,zV15* and confidence interval are unavailable but can be approximately estimated through the Monte Carlo simulation method. First, the design events with sample size N conditioned on AAR=η were generated. These design events were then sorted in descending order of their multivariate probability densities, denoted by
zQ11,zV31,zV71,zV151,zQ12,zV32,zV72,zV152,…,(17)zQ1Nc,zV3Nc,zV7Nc,zV15Nc,…,zQ1N,zV3N,zV7N,zV15N,
where Nc=N⋅pc, and pc is the critical probability level for the confidence interval. Thus, the approximate solution for zQ1*,zV3*,zV7*,zV15* is zQ11,zV31,zV71,zV151. The lower boundary for the confidence interval was expressed as follows:
(18) z Q 1 L = min z Q 1 1 , z Q 1 2 , … , z Q 1 N c z V 3 L = min z V 3 1 , z V 3 2 , … , z V 3 N c z V 7 L = min z V 7 1 , z V 7 2 , … , z V 7 N c z V 15 L = min z V 15 1 , z V 15 2 , … , z V 15 N c .
The upper boundary for the confidence interval was estimated by the following:
(19) z Q 1 U = max z Q 1 1 , z Q 1 2 , … , z Q 1 N c z V 3 U = max z V 3 1 , z V 3 2 , … , z V 3 N c z V 7 U = max z V 7 1 , z V 7 2 , … , z V 7 N c z V 15 U = max z V 15 1 , z V 15 2 , … , z V 15 N c .
3.2.4 Derivation of design flood hydrographs
In China, the design flood hydrographs for hydraulic structures are usually derived from the design flood events set against a benchmark flood hydrograph, which is chosen from the observed flood processes (Ministry of Water Resources of People's Republic of China, 1996; Xiao et al., 2009, Yin et al., 2017). For example, suppose that a flood hydrograph consists of the features of annual maximum daily discharge, 3-day flood volume, 7-day flood volume and 15-day flood volume. The four features of the benchmark flood hydrograph are denoted by Q1B, V3B, V7B and V15B, respectively. The design flood hydrograph corresponding to the multivariate hydrologic design realization zQ1,zV3,zV7,zV15 can be derived by multiplying the benchmark flood hydrograph by different amplifiers, given as described below.
The amplifier K1 for the annual maximum daily discharge was calculated by
(20) K 1 = z Q 1 Q 1 B .
The amplifier K3−1 for the 3-day flood volume except for the annual maximum daily discharge was calculated by
(21) K 3 - 1 = z V 3 - V z Q 1 V 3 B - V Q 1 B ,
where V(⋅) is the operator transforming daily discharge into flood volume. The amplifier K7−3 for the 7-day flood volume except for the 3-day flood volume was calculated by
(22) K 7 - 3 = z V 7 - z V 3 V 7 B - V 3 B .
Finally, the amplifier K15−7 for the 15-day flood volume except for the 7-day flood volume was calculated by
(23) K 15 - 7 = z V 15 - z V 7 V 15 B - V 7 B .
4.1 Nonstationary analysis for marginal distributions
The time-varying moment model was employed to perform nonstationary analysis for each marginal distribution of the multivariate flood series Q1,V3,V7,V15 of the Xijiang River. In general, the candidate distributions for all margins passed the GoF test at the 0.05 significance level. The chosen models featured with the smallest AICc values were shown in Table 2. The results indicated that the GEV distribution provided the best fit for the annual maximum daily discharge series Q1, whereas the Gamma distribution was chosen as the theoretical distribution for the flood volume series V3, V7 and V15. All estimated model parameters were found to be statistically significant at the 0.05 level. The 95 % uncertainty intervals for the estimated parameters were calculated by the parametric bootstrap method (Kyselý, 2009). In accordance with the modelling results, it can be seen that the location parameters μ for all flood series were nonstationary, while the scale and shape parameters were stationary. Through an exponential function, the location parameters μ referring to the means of the flood series were generally positively related to the urban population Pop, whereas they were negatively related to the reservoir index RI. This finding revealed the opposite roles played by urbanization and reservoir regulation on the flood processes of the XRB. In particular, more artificial levees were required to protect urban areas from flooding by constraining the flood flow to river channels, which resulted in increasing the river channel flood flow. The reservoirs played an active role in flood control by reducing the flood discharge downstream.
Table 2Results of nonstationary analysis for the marginal distributions of Q1,V3,V7,V15.
The relationships between μ and covariates were built by the exponential function in Eq. (3). α1 and α2 are the parameters related to urban population (Pop) and reservoir index (RI), respectively. The symbols a, b and c denote that the estimated model parameters are significant at the levels of 0.1, 0.05 and 0.1, respectively. The numbers in brackets are the 95 % uncertainty interval. p_KS stands for the p value of the KS test for marginal distributions.
More specific to each margin of Q1,V3,V7,V15, the location parameters μ of the three short-duration flood series, i.e. Q1, V3 and V7, were positively linked to Pop, whereas the RI was the driving factor reducing μ for all flood series, including Q1, V3, V7 and V15. Owing to the difference in covariate selections, the short-duration flood series, including Q1, V3 and V7, displayed asynchronous nonstationary behaviours with the long-duration flood series V15 occurring in the observation period of 1951–2012. As shown in Fig. 4, Q1, V3 and V7 presented significantly increasing trends during 1951–2005, particularly since the 1980s, marking the beginning of a period of rapid urbanization in China. V15 tended to follow a stationary process during 1951–2005. After the two flood control reservoirs were put into operation in 2006, all flood series, including Q1, V3, V7 and V15, exhibited a sharp decline.
Figure 4Nonstationary marginal distributions during both the observation and design life periods.
Table 3Results of nonstationary analysis for the dependence structure of Q1,V3,V7,V15.
The relationships between copula parameters and covariates were built by the exponential function in Eq. (7). β1 and β2 are the parameters related to urban population (Pop) and reservoir index (RI), respectively. The symbols a, b and c denote that the estimated model parameters are significant at the levels of 0.1, 0.05 and 0.01, respectively. The numbers in brackets are the 95 % uncertainty interval.
Figure 5Statistical correlations between flood peak and flood volumes. Three asterisks (***) indicate the statistical correlation at the 0.01 significance level.
The predicted marginal distributions for Q1,V3,V7,V15 during the design life period from 2013 to 2100 were estimated using the time-varying moment model by replacing the observed covariates for μ with those predicted. Figure 4 also shows that the mean values of Q1, V3 and V7 during the design life period increased with the growth of the urban population, following which they decreased sharply in 2023 after a larger reservoir named Datengxia is expected to be put into operation. After 2023, with no more reservoirs planned, the predicted mean values of Q1, V3 and V7 would be expected to reach their peaks in the mid-21th century followed by a slight declining trend because of a shrinking urban population. Since V15 was only related to RI, V15 would show an abrupt decline in 2023 due to the regulation of the Datengxia reservoir. In general, the predicted nonstationary marginal distributions for Q1 and V3 during 2013–2100 were roughly approximate to the marginal distributions under the assumption of stationarity, whereas the predicted nonstationary marginal distributions for V7 and V15 exhibited smaller mean values than those of the stationary distributions.
4.2 Nonstationary dependence structure for Q1,V3,V7,V15
After estimating the nonstationary marginal distributions for Q1,V3,V7,V15, the multivariate dependence structure was constructed by the dynamic C-vine copula with Q1 elected as the key variable. Figure 5 illustrates significant correlations between the flood peak Q1 and the flood volumes (i.e. V3, V7 and V15). Table 3 shows the estimation results of the dynamic C-vine copula. The PIT test for the nonstationary dependence structure of Q1,V3,V7,V15 suggested a satisfactory fitting effect, and most estimated parameters were statistically significant at the 0.05 level. The results indicated that the copula parameter θ115 for pair (Q1, V15) was found to be nonstationary and expressed as an exponential function of both the urban population Pop and reservoir index RI, whereas other copula parameters indicated stationary dependences. It was seen that the margin of Q1 displayed asynchronous nonstationarity behaviours with V15 (see Table 2 and Fig. 4). Therefore, the dependence nonstationarity of the pair (Q1, V15) could possibly be attributed to the asynchronous marginal nonstationarities.
According to the regression function, θ115 was negatively related to Pop, whereas it was positively related to RI. In other words, growing urbanization weakened the multivariate flood dependence, whereas reservoir regulation played an opposite role, enhancing the dependence. This finding indicated that human activities, including urbanization and reservoir regulation, not only changed the statistical characteristics of the marginal distributions of Q1,V3,V7,V15 but also affected the dependence of Q1,V3,V7,V15. Figure 6 shows the time variations of θ115 during the observation period of 1951–2012 as well as during the design life period of 2013–2100. Due to reservoir regulation, θ115 presented two obvious upward change points in both 2006 and 2023. Besides this, θ115 also exhibited an obvious decreasing trend with urban population growth from 1951 to the mid-21th century, followed by a slight increasing trend due to a shrinking urban population. During the design life period, the predicted nonstationary θ115 suggests a weaker dependence for Q1,V3,V7,V15 than the dependence under the stationary assumption, since it is usually smaller than the stationary estimation.
Figure 6Nonstationary copula parameter for pair (Q1, V15) during both the observation and design life periods.
In addition, the change-point detection method based on the Cramér–von Mises statistic (Bücher et al., 2014) was employed to detect possible nonstationarities in both the marginal distributions and dependence of the multivariate flood series Q1,V3,V7,V15. Readers are referred to Bücher et al. (2014) and Kojadinovic (2017) for specific steps to implement the change-point detection. The results indicated that neither the marginal distributions nor dependence displayed change points at the 0.05 significance level (see Table 4), whereas the previous analysis suggested nonstationary margins and dependence due to the joint effects of urbanization and reservoir regulation. These aforementioned inconsistencies could be attributed to the opposite roles of urbanization and reservoir regulation on shifting of the multivariate flood distribution, with urbanization generally enlarging the mean values of the flood series and weakening their dependence and reservoir regulation, decreasing the mean values and strengthening the dependence. In other words, the nonstationarities induced by these two factors may have offset each other. As a result, the nonstationarities of Q1,V3,V7,V15 might have not been captured by the statistical method based on the Cramér–von Mises statistic. This finding highlights the significance of cause–effect analysis in judging the nonstationarities of hydrologic series (Xiong et al., 2015).
Table 4Results of change-point detection for the marginal distributions and dependence of Q1,V3,V7,V15.
Figure 7Design values of the annual maximum daily discharge for different average annual reliability (AAR) varying from 0.01 to 0.99 under three nonstationary conditions.
Figure 8Design values of the 3-day flood volume for different average annual reliability (AAR) varying from 0.01 to 0.99 under three nonstationary conditions.
Figure 10Design values of the 15-day flood volume for different average annual reliability (AAR) varying from 0.01 to 0.99 under three nonstationary conditions.
Figure 11Design flood hydrographs associated with OR, AND and Kendall probabilities.
4.3 Multivariate hydrologic design characterized by average annual reliability
The multivariate hydrologic designs, characterized by AAR associated with the OR, AND and Kendall exceedance probabilities, were estimated from the predicted nonstationary multivariate distribution for Q1,V3,V7,V15 during the design life period from 2013 to 2100. The left columns in Figs. 7–10 show the most-likely design events and the 90 % confidence intervals conditioned on the AAR varying from 0.01 to 0.99. The multivariate hydrologic design events associated with both the OR and Kendall exceedance probabilities exhibited the lower boundaries, whereas the design events associated with the AND exceedance probability exhibited the upper boundaries.
The design flood hydrographs were derived from the multivariate hydrologic designs against the benchmark flood hydrograph observed in 1988. Figure 11 shows the design flood hydrographs by setting AAR equal to 0.90, 0.95 and 0.99. For any given multivariate flood event, the corresponding OR exceedance probability was larger than that of AND, with the Kendall exceedance probability somewhere in between (Vandenberghe et al., 2011). These differences among the OR, AND and Kendall exceedance probabilities indicate the different design strategies. It must be noted that the choice of design strategy in engineering practice is usually a priori and is dependent on the specific design requirements and mechanisms of failure for hydraulic structures (Serinaldi, 2015; Salvadori et al., 2016).
We calculated the univariate hydrologic design events from the predicted marginal distributions to compare the design strategies under the multivariate framework with those under the univariate framework. Figures 7–10 show that the univariate hydrologic design events exactly constituted the lower boundaries of the multivariate hydrologic design events associated with the OR exceedance probability as well as the upper boundaries of the design events associated with the AND exceedance probability. Under a given AAR, the hydrologic designs under the univariate framework were generally smaller than the most-likely design events associated with the OR exceedance probability, whereas they were larger than those associated with the AND exceedance probability; they were most approximate to those associated with the Kendall exceedance probability. The comparisons of the flood hydrographs displayed in Fig. 11 reinforced these findings.
4.4 Impacts of multivariate nonstationarity behaviours on hydrologic design values
Section 4.1 and 4.2 show the marginal distribution and dependence structure of the multivariate flood distribution of Q1,V3,V7,V15 to be nonstationary. We estimated the multivariate hydrologic design events under an assumption of stationarity to illustrate how these nonstationarities act on the multivariate hydrologic designs, i.e. both marginal distributions and the dependence structure were treated as stationary (see the right columns in Figs. 7–10). Figure 4 suggests that both the predicted nonstationary marginal distributions for Q1 and V3 during the design life period were approximate to the stationary marginal distributions. Therefore, the nonstationary and stationary marginal distributions yielded similar design values for zQ1 and zV3 (see Figs. 7 and 8). The predicted nonstationary distributions for both V7 and V15 indicated smaller mean values compared to those of the stationary distributions (see Fig. 4); therefore, the corresponding hydrologic design values estimated from the nonstationary marginal distributions were generally smaller than those estimated from the stationary marginal distributions (see Figs. 9 and 10).
The nonstationary multivariate flood distribution during the design life period was also predicted to exhibit a weaker dependence structure than that of the stationary distribution (see Fig. 6). The dependence nonstationarity was expected to have a much subtler effect on the multivariate hydrologic design compared to the marginal nonstationarities (Xiong et al., 2015). To illustrate the individual effect of the dependence nonstationarity on the multivariate hydrologic design, an artificial nonstationary condition for the multivariate flood distribution was set so that only the marginal nonstationarities were considered, whereas the dependence structure was treated as stationary. The results of the multivariate hydrologic design events are shown in the middle columns in Figs. 7–10. In general, the dependence nonstationarity had less of an effect on the multivariate hydrologic designs compared the marginal nonstationarities; however, some visible differences in both the 90 % confidence intervals were still identified. In summary, the nonstationary and weaker dependence structure generally suggested wider confidence intervals for the multivariate hydrologic design values.
5 Conclusion and remarks
The statistical characteristics of both the marginal distributions and the dependence structure of multivariate flood variables can vary with time under nonstationary conditions. It is possible that the multivariate flood distribution estimated from the historical information will not reflect the statistical characteristics of flooding in the future. As a result, the stationary-based hydrologic design would not be able to deal with potential hydrologic risks of hydraulic structures. It is necessary for hydrologic designers to take into account the physical driving forces (such as human activates and climate change) responsible for the nonstationarities of multivariate flood variables.
The present study introduced possible methods for addressing multivariate hydrologic design for application in engineering practice under nonstationary conditions. A dynamic C-vine copula allowing for both time-varying marginal distributions and time-varying dependence structure was developed to capture the nonstationarities of a multivariate flood distribution. The multivariate hydrologic design under nonstationary conditions was estimated by specifying the design criterion by average annual reliability. The most-likely design event and confidence interval were identified as the outcome of the multivariate hydrologic design. Multivariate flood series Q1,V3,V7,V15 from the Xijiang River were chosen as a case study, with the main findings given below.
Urbanization and reservoir regulation were found to be the driving forces responsible for the nonstationarities of both the marginal distributions and dependence structure of the multivariate flood series Q1,V3,V7,V15. The growth of the urban population generally resulted in an increased mean value of the individual flood series, whereas it weakened the dependence of Q1,V3,V7,V15. The increasing reservoir index had the opposite effect on the individual flood series as well as their dependence. Under a given average annual reliability, the OR exceedance probability yielded the largest design values, followed by the Kendall and the AND exceedance probabilities. Nonstationarities in both marginal distributions and dependence structure affected the outcome of the multivariate hydrologic design. It is the marginal nonstationarities that played a dominant role in affecting the multivariate hydrologic design.
There are two remarks that can be made that are related to the practical implications of the hydrologic design methods developed in the current study that are detailed below.
The first remark relates to the length of observed flood data required for multivariate and nonstationary hydrologic designs. In theory, sufficiently long observed flood data (or other extreme-value data) are required to derive robust estimations of the distribution parameters and the correct hydrologic design values (Zheng et al., 2018). However, in reality, most data series are limited in length, thus forcing us to use what we have at hand to do research or design works without fulfilling the theoretical assumptions or requirements. Some recent studies suggested that univariate flood frequency analysis under stationary conditions usually requires flood data with a continuous period of at least 30 years (Ministry of Water Resources of People's Republic of China, 1996; Engeland et al., 2018; Kobierska et al., 2018). However, determining a definitive answer to the length of observed flood data that is required for flood frequency analysis under multivariate and/or nonstationary settings poses a challenge, since this issue has not yet been fully addressed. However, it is certain that multivariate and nonstationary hydrologic designs naturally require datasets of longer length, since they usually contain more parameters to be estimated.
The second remark related to the trade-off between reducing estimation bias and increasing model uncertainty. Nonstationary models generally improve performance in fitting observation data by reducing estimation bias (Jiang et al., 2015b), but this is usually achieved at the expense of increasing model complexity, such as adding more model parameters and introducing more nonstationary covariates, which might induce additional sources of model uncertainty (Serinaldi and Kilsby, 2015; Read and Vogel, 2016). A careful balance between the model fitting effect and the model complexity should be maintained in practice when employing multivariate and nonstationary hydrologic design by keeping in mind the following two points: (1) the multivariate and nonstationary models should remain effective but should also be kept as simple as possible to avoid overfitting, and (2) to ensure a robust relationship between the distribution parameters and the explanatory covariates, the chosen covariates should be physically related to the flood processes and supported by a well-defined cause–effect analysis.
All the data used in this study can be requested by contacting the corresponding author Cong Jiang at <EMAIL>.
Appendix A: Calculating multivariate exceedance probabilities
A1 OR exceedance probability
Since the cumulative distribution function Fq1,v3,v7,v15θt has no analytical expression, the OR exceedance probability ptor at time t is calculated by the Monte Carlo method as follows:
Calculate the marginal probabilities u1,u3,u7,u15 of q1,v3,v7,v15.
Generate m samples u1,i,u3,i,u7,i,u15,i (i=1,2,…,m) from the C-vine copula expressed by Eq. (4).
Calculate Fq1,v3,v7,v15θt=1m+1∑i=1m1u1,i≤u1,u3,i≤u3,u7,i≤u7,u15,i≤u15.
Calculate ptor=1-Fq1,v3,v7,v15θt.
A2 AND exceedance probability
The AND exceedance probability ptand at time t is calculated by the Monte Carlo method as follows:
Calculate ptand=1m+1∑i=1m1u1,i≥u1,u3,i≥u3,u7,i≥u7,u15,i≥u15.
A3 The Kendall exceedance probability
The Kendall exceedance probability ptken at time t is calculated by the Monte Carlo method as follows:
Calculate ρt=Fq1,v3,v7,v15θt (see calculation steps 2–3 for OR exceedance probability).
For j=1,2,…,m, calculate vj=1m+1∑i=1m1u1,i≤u1,j,u3,i≤u3,j,u7,i≤u7,j,u15,i≤u15,j.
Calculate KCρt=1m∑i=1m1vi≤ρt.
Calculate ptken=1-KCρt.
Appendix B: Generating the multivariate design event samples
To calculate the most-likely design event and confidence interval conditioned on AAR=η, we need to generate numerous multivariate design event samples by the Monte Carlo method. Here, we give the details of generating the design event samples as follows:
Define the total number of design event samples N and the initial number of the design event sample i=0.
Generate a random integer (denoted by tr) among (T1, T1+1, … , T2).
Generate a random sample zQ1,zV3,zV7,zV15 following the multivariate distribution FzQ1,zV3,zV7,zV15θtr with the distribution parameter vector θtr.
Calculate the annual exceedance probability for each year throughout the period from T1 to T2.
Calculate AAR during the period from T1 to T2.
If AAR-η<ε (where it is a very small value, such as 0.0001), i=i+1.
If i<N, repeat steps (2)–(6).
CJ and LX developed the main ideas. CJ and LY implemented the algorithms of the methods. CJ and JD collected the data used in the case study. CJ, LX and CYX prepared the paper.
This research was jointly financially supported by the National Natural Science Foundation of China (NSFC grants <DIGITS> and <DIGITS>), the Fundamental Research Funds for the Central Universities (grant CUG170679), the Research Council of Norway (FRINATEK Project 274310) and the “111 Project” Fund of China (B18037), all of which are greatly appreciated. The authors express their gratitude to the reviewers for their constructive comments and suggestions that have helped improve the paper.
This paper was edited by Carlo De Michele and reviewed by four anonymous referees.
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Created to celebrate the centenary of Sophia de Mello Breyner Andresen's birth, "A Menina do Mar" is now adapted for the Maritime Museum and will be presented in the Museum's auditorium. Staged by the company Caixa de Palco, this performance tells us ...
The International Museum Day, celebrated on May 18th, marks the Reopening of the Ílhavo Maritime Museum (MMI), conditioned still to the fulfillment of the rules of the General Direction of Health, framed in the State of Calamity in effect on this date.
The Ílhavo Maritime Museum marks the International Museum Day, May 18, with the reopening of its doors and with strong reasons for a visit. On the Saturday after this event, May 22nd, the Museum will open the photography exhibition "Azul Azul", by ...


An anti-GMO organization violated Washington’s public disclosure law by concealing the names of more than 7,000 donors who contributed in 2013 to an initiative that would have required food makers to labea judge ruled [April 22].
The ruling was a second victory for the state Attorney General’s Office in cases stemming from the most-expensive political campaign Washington history.
Another Thurston County judge ruled in March that the Grocery Manufacturers Association was guilty of failing to disclose the names of the food and beverage companies that contributed to I-522’s defeat.


Pa 15.12.2020 More than 1.1 million EU citizens have signed the Minority SafePack Initiative. The European Commission needs to hear their voices, their expectations204
Contributions to plenary debates Asger CHRISTENSEN.171(11) Transitional provisions for support from the EAFRD and EAGF in the years 2021 and 2022 (continuation of debate) DA. 15-12-2020 P9_CRE-PROV(2020)12-15(2-074-0000) Common agricultural policy - support for strategic plans to.
EP Plenary session - Joint debate - Water legislation News - 15/12/2020. 45. 15/12/2020. HD. 05:17. s Extract - 15/12/2020. HD. 02:12. Infoclip: Drinking water directive Stockshots - 07/06/2018. See complete package: Water quality. See also. Package.
Intervention in the European Parliament Plenary Session debate on 'The need for an immediate and humanitarian EU response to the current situation in the refugee camp in Moria' Thank you. I think I share the feelings of many of you. Watching the pictures from Moria after the fire. But also before the fire. One of the goals I've been working on, to find and present a new pact on Migration.
Speech at European Parliament plenary debate. Dear President, Honourable Members, It is an honour to be here today, in the heart of European democracy to discuss the Conference on the Future of Europe with you. As a previous member of this House, I consider the European Parliament as a close friend. Today I stand before you, honourable members, still as your friend and now also as your partner.
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The extraordinary plenary session would allow. Socialist leader Iratxe García Pérez said after Thursday morning's meeting with Barnier, We will not be rushed into a consent vote of the Brexit deal.
European Parliament plenary debate on the ECB Annual Report for 2014Strasbourg, 1 February 2016. Mr Vice-President of the European Parliament, Mr Vice-President of the CommissioWe are undoubtedly at a point in time when the cohesion of Europe is being tested. Many of the challenges.
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European Parliament Plenary Session - November I 2020. Posted by Members' Research Service ⋅ November 10, 2020 ⋅ 1 Comment. an Parliament. Written by Clare Ferguson. The first item on Parliament's agenda for the November I session,s clear, is to take stock of the results, which.
Parliament's second plenary session in October will be the first ever to be held entirely virtually, due to the ongoing pandemic. Despite meeting online, however, Members will nevertheless address a full agenda that features, among other thingsand discusss well as hearing the European.
O debate Brussels, 12 November 2020. Madame Chair, Honourable Members, Fi Mureşan and Mr Tang for your report. Nearly ten months ago, the Commission adopted the Sustainable Europe Investment Plan, which is the.
Mr President, dear Members, dear colleagues of the European Parliament, Thank you for organising a very quick debate and exchange of views on the results of the United States elections. I am glad to see that you give to this event all the importance it has and to have a loo EU PARLIAMENT PLENARY: BREXIT Conclusions after October's EU Council meeting by President MICHEL and Comm. SEFCOVIC and BARNIER - (Speeches and Debate inside) Introduced by President Sassoli, the Opening std by Michel. Closing remarks by Commissioner Hahn at the European Parliament's Plenary debate on the Multiannual Financial Framework Check against delivery! Dear President, dear Honorable Members, Believe me, I've rarely been so grateful to have a debate in this honourable house. Because today's debate means we have a deal and we.European Commission. Brussels, 16/04/2020. Description: Opening of the European Parliament plenary session. Send us your feedback ; Speech Details. Speech number: 30091. Duration: 03:43. Language: (en) English. Level: Beginner. Use.
ister to speak at a debate on her country's controversial coronavirus law, according to letters seen by POLITICO. The Parliament will on Wednesday hold a discussion involving MEPs, the European Commission and EU countries on legislation that allowse.
EU citizenship cannot be traded as a commodity, the majority of speakers who want to end the golden passports schemes currently in place in some member states, including Malta, said during a.
Debate in Parliament. 16/12/2020.. T9-0359/2020 Technical information Technical information ; Procedure reference 2020/2131(INI) Procedure type. INI - Own-initiative procedure. Procedure subtype. Initiative. Legal basis. Rules of Procedure EP 54. Other legal basis. Rules of Procedure EP 159. Stage reached in procedure. Procedure completed.
Debate in Parliament. 12/03/2014. Results of vote in Parliament. 12/03/20147-0212/2014. Summary . 03/09/2014. after 1st reading in Parliament. 10/10/2014. Debate in Council. 3336. Summary . 04/12/2014. Debate in Council. 3354. Summary . 17/12/2015. Approval in committee of the. The plenary before the debate on the P on the Multiannual Financial FrameworFebruary 2020. EPA-EFE/PATRICK SEEGER. Facebook. Twitter. Linkedin. WhatsApp. Viber. Email. Print. The European Parliament's plenary session, originally scheduled to be held next week in Strasbourg, will take. The European parliament has 766 elected members from 28KEY ITEMS OF THE. Photos of the debate: European Parliament President Martin Schulz's departure is a game changer for EU institutional dynamics.The fight for the Parliament's highest-profile job is now on, and each political group has designated their candidates. In the fray: 4 Italians, 2 Belgians and one Brit. With Europe mired in Brexit, a migration crisis, austerity measures and incoming national.
We are appalled by theaisingen. Watch . DebatFuture of Europe Plenary speech by Ska Keller. More social Europe is needed. en. Watch. Debate on the Future of Europe Ska Keller addressed today the Prime Minister of. Debate in Parliament. 10/07/2007. Results of vote in Parliament. 10/07/20076-0321/2007. Summary . 24/10/2007. Modified legislative proposal published. COM(2007)0673. Summary . 10/03/2008. Council position published. 16675/2/2007. Summary . 13/03/2008.. 06/05/2008. Vote in committee.
N 0 Honourable Members, The President of the European Council has just madeEuropean Council conclusions. I would therefore like to focus my presentation on the economic and financial issues, many o Debate in Parliament. 17. T9-0384/2020 .912(RSP) Procedure type. RSP - Resolutions on topical subjects. Procedure subtype. Urgent debate or resolution. Legal basis. Rules of Procedure EP 144 Documentation gateway. The result of the vote on the Withdrawal Agreement is see /Xinhua . The monthly plenary had already been moved to Brussels instead of Strasbourg The thinking behind the decision is, according to the spokesperson, is to strike a. he subject of our discussion today is the preparation of the Special European Council, buf. The Bundestag and the European Parliament; Brussels Office; International. International Parliamentary Assemblies; International relations; Exchange programmes; Documents. close; Parliamentary Documentation; Official records; Minutes of plenary proceedings ; Legal framework; Parliamentary Library; Catalogue of the Library; Research and Documentation Services; Parliamentary Archives; Press.
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LIVE - EU PARLIAMENT Plenary (11-13 Nov) Updated : LONG-TERM BUDGET, OWN RESOURCES And RULE Of LAW Approval - Video & Streamed . EU Parliament is debating changes required and obtained for future Long-Term MFF budget and the new so-called Own resources to give the Union the finances committed by the EU Commission. Plenary: Opening remark by David SASSOLI, EP President / Picture: European.
European Parliament president David Sassoli announced the EP's medical service staten next week takes place in Strasbourg. On.
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Renew Europe calls for full transparency on vaccine approvals, requests urgent plenary debate on EU's vaccination strategy The European Medicines Agency's (EMA) approval of a second Covid vaccine, Moderna. This is another step forward in tackling the pandemic. Renew Europe has strongly defended the reinforcement of the EMA's role in this, which is vital to combine trust in the science. The European Parliament has rejected including the October 1 referendum on the agenda for its next plenary session, to be held next week in Strasbourg. The EPP and S&D groups voted against the initiative presented by the leader of the Greens/EFA, Philippe Lamberts. The European United Left (GUE), which includes left-wing parties, such as Podemos or Syriza, and the European Conservatives and. Plenary Debate On Greece With Pm Tsipras European Parliament European Council Greece Acdp President Spoke At Parliament Today About Corruption Being Exposed And Rife But Still No Prosecutions Free Online Calendar Google Calendar Online Calendar. Debate on public discrimination and hate speech . . The debate was requested upon initiative of the LGBTI Intergroup. A resolution will be voted on the topic during the December plenary France was desperate for the European Parliament to return to Strasbourg and on Monday it finally got its wish — well, kind of. Parliament President David Sassoli gave a 30-minute address in the hemicycle of the Parliament's Strasbourg building and expressed his emotion at being back in the French city. Except there were hardly any other MEPs there and he finished his speech by.
.  The MEPs (the Mion. Debates about the European Union's democratic legitimacy put national parliaments into the spotlight. Do they enhance democratic accountability by offering visible debates and electoral choice about multilevel governance? To support such accountability, saliency of EU affairs in the plenary ought to be responsive to developments in EU governance, has to be linked to decision‐making moments. European Parliament to debate situation in Hungary on Greens/EFA request Hungary Following the mass protests in Hungary around the so-called slave law and opposition to Prime Minister Viktor Orbán's autocratic rule , the European Parliament on the request of the Greens/EFA group, has tabled a plenary debate on the situation in Hungary for the end of January in Brussels (30 th or 31 st ) Tred zone .
 Source : © European Union, 2020 - EP Debates - Wednesday, 25 November 2020 - Brussels - Provisional editio A Plenary meeting held using video-conference inent, and these are set out .
Watch a plenary sitting of the German Bundestag live f Subject to parn, you will then have an opportunity to visit the dome following the tour . Br . S0. Published: 05/10/2020. Share: O. Debates on Monday 5 October and Tuesday 6 October. Iratxe García, S&D Group President We are pleased that EU leaders showed.
European Commission Speech Brussels, 10 Mar 2020 Opening Remarks of Commissioner Kyriakides at the European Parliament Plenary Debate on COVID-1 S&D Group's Plenary Times - 18 December 2020. Published: 14/12/2020. Share: S&D Group president Iratxe García The outcome of the European Council Wednesday, 16 Decemberic. Explore European Parliament's 11,108 photos on Flickr Paul-Henri Spaak building (European Parliament) Room PHSHEM . Brussels. Belgium. Inaugural plenary session (Term of office 2020-2025), with a debate on Recovery for the future of Europe - TDavid Maria SASSOLI, Charles MICHEL, Ursula VON DER LEYEN, Apostolos TZITZIKOSTAS, Geert PAUWELS, Luca VISENTINI, and Jeremy WATES. 27/10. Anoted on the.
The European Parliament on Tuesday canceled plans to return to Strasbourg next week, after the city and its surrounding area were designated as a coronavirus red zone by French authorities. Announcing the decision,t holding the session in Strasbourg would have meant Parliament staff having to quarantine on their return to Brussels. We. Microsoft partner with the European Parliament in developing an AI-based tool thatdebates in real-time, with the possibility to learn from human corrections and edits. The initiative aims to bring the democratic process closer to citizens, by ensuring that the Parliament's debates.
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May 12, 2019| Jeff Yeomans| 0 Comment
Improving Health Care by Gamifying It
Physicians often struggle to help patients change their health behaviors. Patients may know that they need to quit smoking, lose weight, or exercise more, but summoning the will to change is hard. It’s particularly difficult for the highest-risk patients who may have life circumstances — challenges such as unemployment or homelessness — that make it harder for them to focus on the long-term. But combining behavioral economics and “gamification” — putting game elements such as points and achievement levels into non-game contexts — holds promise for driving behavior change when a doctor’s advice, and patient’s good intentions, are not enough.
Other industries have long used game elements that leverage behavioral science to drive desired customer behavior (think airline loyalty programs that award points and status for miles traveled). And indeed gamification is increasingly being incorporated into health insurance design and wellness programs. However, despite its growing use, there’s only limited evidence of its effectiveness in health care, and in particular whether existing gamification makes the best use of behavioral economic principles. Members of our group recently evaluated 50 of the most popular smartphone applications for health and fitness and found that while nearly two-thirds of the apps used game elements in their design, none incorporated several key insights from behavioral economics that could effectively influence desired actions and address predictable barriers to behavior change.
A central challenge for all health-related gamification programs is engaging participation, particularly among high-risk patients. Several design elements commonly found within gamified health and wellness programs could be made more engaging by incorporating behavioral insights. For example, most programs invite patients to join, framing their choice as an opt-in decision. But we have found that opt-out framing significantly improves participation. In a randomized trial, our group tested how to engage adults with uncontrolled diabetes in a remote-monitoring program. In the traditional, opt-in approach, only 13% signed up. But when the introductory letter framed the program as standard care, but allowed patients to opt out if they wished, enrollment rates nearly tripled to 38%. We’ve found similar results when testing ways to engage patients in a medication adherence program after a heart attack.
Another common feature of gamification programs is goal-setting. The traditional approach is to assign everyone the same goal (for example, taking 10,000 steps per day) and ask to them to strive for it immediately. However, this is probably overly ambitious for many and not sufficiently ambitious for others. We have found that it is more effective for programs to establish a baseline for each individual and then engage him or her in personalized goal-setting, with goals that gradually become more demanding and that adapt to ongoing performance. For example, in a randomized trial of patients with heart disease, we combined financial incentives of $2 per day for each day step goals were met with personalized goal-setting in a program that used wearable devices to measure activity. Half the patients were assigned to use the device’s preset 10,000 step per day goal that began immediately. The other half established a personal baseline step count which increased for the first two months and then remained steady for four months. During the 6-month trial, the patients who had been assigned the preset goal had no overall change in activity. But those with personalized step goals in00 miles more than the patients in the control group. They even remained more active than the control group for two months after the incentives stopped.
In another test of gamification, we partnered withe health of generations of families but never previously participated in an intervention study. Two hundred participants over 18 years old enrolled with their families and used either a smartphone oas measured by how many steps they took each day. We established a baseline activity level for each person, and then each selected an increased step-count goal to shoot for. Families in the control arm received daily feedback on how they were doing relative to their goal, but no other interventions. The plan undertaken by families in the gamification program, on the other hand, incorporated several behavioral economic and game principles.
First, they signed a pre-commitment pledge in which they agreed to try their best to achieve their goal — a simple technique that’s known to help people stick with goals. Each time they signed onto the platform they would be shown the contract and be reminded of their pledge.
Second, they received points that were allocated up fs-aversion —
Third, since we know people in these types of programs can occasionally fall off the wagon, we replenished participants’ points each week to give families a fresh start, which leverages the tendency to be more motivated to pursue a goal when the prompt or decision is anchored at a “landmark” time, such as the beginning of the week. (New Year’s resolutions are another example of this effect.)
Finally, we incorporated social incentives. Each day one member of a family was selected at random to represent the entire family. If that person achieved his or her goal, all family members kept their points; otherwise, everyone lost points. This design encourages collaboration, accountability, and peer support. It also leverages the principle of “anticipated regret” — fear of falling short of a goal and letting others down.
During the three-month study, family members in the gamification program walked on average nearly a mile farther each day (1,661 steps) than they had at baseline — about 1,000 steps more each day than people in the control group. What’s more, while their activity dropped somewhat after the game ended, they continued to walk more than members of the control group. We’ve tested a similar approach for weight loss and found that groups lose more weight when the team members live together, further demonstrating how social incentives can be harnessed for behavior change.
Gamification is already being used widely to encourage healthy behaviors. However, many current designs cater to “super users” who already like games and are motivated to improve their fitness. These programs are unlikely to engage people at higher risk who could benefit most from changing their behavior. Our group is using gamification to help patients with uncontrolled diabetes, heart disease, and cancer. We have found that is not hard or expensive, but instead requires attention to detail. Subtle changes to program design and communications can have an outsized impact on how patients behave. That’s why embedding behavioral insights into gamification cng.
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Dog facts: 15,000 years, and they are considered to be man’s best friend. This may makethere is more to dogs than fetching and playing dead. Explore facts that will surprise you.
The reason why dogs hesitate before going out into rain is not because they’re afraid of getting wet, it’s because rain amplifies sounds and their ears are so sensitive it can be too intensive for them.
Male puppies will sometimesin order to get to know them better.
Dogs lose only a small amount of heat through sweat glands. Most of their temperature regulation is done through panting which removes body heat by evaporation from the surface of the tongue.
If puppies are not exposed to a variety of food flavors and textures in their diets, their adult food preference can be limited. However, if they are exposed to variety of food flavors as puppies then they seek out variety in foods as adults.
If a guy is having a dog with him, he’s t
Dogs depend on their sense of smell than on their sense of taste to differentiate different types of food.
Female dogs bite double the amount of males.
Smiling at your dog may make him feisty: dogs don’t see your smile as friendly, but detect your bearing teeth as a sign of aggressher’s eyes. Dogs only do it with humans.
Dog’s urine has acids that can corrode the metal.


And so, "Agenda 21" conspiracy theories might as well be real: we have launched ourselves down a path whereby, if people do what is expected to bring them fame and acclaim, they will gradually and successively convey us toward the state described by the "Agenda 21" conspiracies. Even when they are not real…
Is someone who is 3/4 English, genetically, and 1/4 something entirely different still English?
Was this diversity designed as genocide?Maybe, or maybe not, but either way, it creates genocide through outbreeding and when it is done, the original group no longer remains. At that point, it has fewer restrictions on intermixing because the damage is already done, so what are they preserving at this point?
So you can see that the conspiracy theory here is metaphorically correct. Diversity is, or rather leads to, white genocide. No one knows if it is deliberate, or where they would find such a smoking gun…
Or even farther back, there's the Zimmerman Telegram in WWI. Perhaps the attack on Fort Sumter was triggered. It's possible that the Boston Tea Party was an arranged provocation. We know this is possible because other groups do this sort of thing all the time at protests, being as obnoxious as possible in the hope that one of them gets shot and the real riot can be begin!


Funny, lighthearted and at times temperamental, cats are simply charming. Although they prefer discretion in most situations, cats surprise their owners from time to time, dropping household items or cramming into a box. But even if some cats behaviors may be difficult to understand, each one has a logical explanation.
Many cat owners complain that their felines roam the household items: boxes of paper, plastic products, but also woolen objects. In some cases, this is a behavior that occurs if the cat is ignored or bored. In others, however, it may indicate a rare disease, called pica syndrome. Also, nutritional deficiencies can be the reason why cats chew the objects around them. If your pet has symptoms of itching, you need to take him to the vet.
Cuddly and tactical, cats are used to rubbing furniture or people around them.  By releasing the pheromones from the head area, the cat is trying to tell you that you belong to her and that she is proud that you are her owner.
It is not pleasant to wake up with a mouse or some dead birds in front of the door, but for cats, the treasure brought to the owner is a proof of affection. Even after a “No, thank you,” they will continue to offer you such “gifts.” Besides, how can a cat know that people don’t eat rodents or dead cockroaches?
 And if they do not sleep, they laze a lot to conserve their energy. Thus, they are assured that they will have sufficient strength to catch the prey. Also, these fluffy animals adapt their sleep schedule according to the times when their potential prey becomes active. Therefore, they wake up early in the morning and are active late at night.
Although they are not vegetarians, cats like to eat green plants. Once reached the stomach, the leaves of the plants stimulate the reflex of vomiting, which helps to eliminate the hemorrhage from the stomach or the remains of small bones that cannot be digested. Careful! Some room plants can be toxic to your pet, so avoid making them “available”.
Cats love to stay in boxes because they always look for a place to hide. In addition, small cats like narrow places because they feel safe inside.


Share this:ShareFacebookTwitterGoogleEmail	Super Senior Emma Watson graduates from college
May 25, 2014 | 4:03pm,” “The Perks of Being a Wallflower, “This is the End,” and the recent Biblical epic “Noah.”
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Were all available exemptions in place for a Chapter 7 client to ensure personal property is protected?
Even if a client qualifies for debt elimination under Chapter 7 bankruptcy, are there better tools for his or her circumstances by filing Chapter 13 reorganization loan, reducing a car payment with a high interest rate, etc.
Our Response: Our strategy is to thoroughly review a client's circumstances prior to filing, understand the long term financial goals and recommend the best avenue for short and long term success. Next we ensure that we put together the necessary paperwork with complete accuracy. Errors can be costly and irreparable. All of our bankruptcy petitions are generated by licensed Michigan attorneys, not petition preparers or support staff. 5.) What is your attorney and staffing ratio? Importance: Having an attorney you can get a hold of is central to a positive customer service experience. Having proper support staff to ensure timely administration of your case is essential. If a firm doesn't invest in a support team, it may mean that your case doesn't get the attention it deserves or you don't achieve the results you would like. Our Response: We have team of five attorneys and eight support staff that allBeyond our convenient evening and weekend hours, we have an office cell phone available to clients for emergencies. For the last decade, we have assigned specific attorneys to each of the judge's court dockets. Each attorney has an experienced paralegal assigned to him or her to provide customer support and case administration. This long term relationship between the attorney and paralegal means a solid base of teamwork and knowledge of the Judge's ruling and Trustee preferences. In this case, experience really does count.  The invitation is there - we welcome your questions and a visit to our office to meet with one of our attorneys. To understand their education and credentials, visit our "Michigan Bankruptcy Attorney Profiles" pagor schedule your free consultation online.