Risk Analysis, Vol. 35, No. 11, 2015

DOI: 10.1111/risa.12382

Structured Coupling of Probability Loss Distributions: Assessing Joint Flood Risk in Multiple River Basins Anna Timonina,1,∗ Stefan Hochrainer-Stigler,1 Georg Pflug,1 Brenden Jongman,2 and Rodrigo Rojas3

Losses due to natural hazard events can be extraordinarily high and difficult to cope with. Therefore, there is considerable interest to estimate the potential impact of current and future extreme events at all scales in as much detail as possible. As hazards typically spread over wider areas, risk assessment must take into account interrelations between regions. Neglecting such interdependencies can lead to a severe underestimation of potential losses, especially for extreme events. This underestimation of extreme risk can lead to the failure of riskmanagement strategies when they are most needed, namely, in times of unprecedented events. In this article, we suggest a methodology to incorporate such interdependencies in risk via the use of copulas. We demonstrate that by coupling losses, dependencies can be incorporated in risk analysis, avoiding the underestimation of risk. Based on maximum discharge data of river basins and stream networks, we present and discuss different ways to couple loss distributions of basins while explicitly incorporating tail dependencies. We distinguish between coupling methods that require river structure data for the analysis and those that do not. For the later approach we propose a minimax algorithm to choose coupled basin pairs so that the underestimation of risk is avoided and the use of river structure data is not needed. The proposed methodology is especially useful for large-scale analysis and we motivate and apply our method using the case of Romania. The approach can be easily extended to other countries and natural hazards. KEY WORDS: Floods; hierarchical and ordered copulas; interdependencies between losses; Romania; vine copulas

1. INTRODUCTION

drivers of the risk increase, and climate change, whose contribution is still unclear, but may become stronger in the future.(2–5) Consequently, the call to proactively deal with (extreme) risks is now featured prominently in publications on global assessments of risks and development. For example, the World Development Report (2014) emphasizes the need to shift from unplanned and ad hoc responses to proactive and systematic risk management. The Global Risk Report published by the World Economic Forum (2014) concludes that better efforts are needed to understand, measure, and foresee the evolution of risk interdependencies. Finally, the most recent Global Assessment Report (GAR 2013) issues a stark warning that economic losses linked

Global human and economic losses due to natural hazard events exhibit a rising trend.(1) The reasons for this are manifold, including socioeconomic growth, which is considered to be one of the main 1 International

Institute for Applied Systems Analysis, RPV; A-2361 Laxenburg, Schlossplatz, 1, Austria. 2 Institute for Environmental Studies, VU University Amsterdam, De Boelelaan 1087, 1081HV Amsterdam, The Netherlands. 3 CSIRO, Land and Water, Private Bag Nr 5, POWembley, Perth, Western Australia 6913, Australia. ∗ Address correspondence to Anna Timonina, International Institute for Applied Systems Analysis, RPV; A-2361 Laxenburg, Schlossplatz, 1, Austria; tel: +43 (0) 2236 807 471; timonina@ iiasa.ac.at.

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Structured Coupling of Probability Loss Distributions to disasters are out of control and will continue to escalate unless disaster risk management becomes a core part of business investment strategies. As recent studies have shown,(6) impacts and losses due to natural disaster events could be reduced substantially with cost-effective investments. As natural hazard events are inherently random, such events and associated losses have to be assessed and managed in a risk-based manner. Risk analysis is particularly difficult for low-probability events and many sources of uncertainty must be taken into account in assessment processes.(7–9) From a risk management perspective, such low-probability events deserve special attention as they can cause extraordinary large losses, often leading to the failure of coping mechanisms (such as using savings or taking loans) and risk spreading instruments (such as insurance). One prime example is catastrophic flood events, as they spread across many regions and have the potential to cause huge losses at once. In Europe, for example, the Odra flood in 1997 affected the Czech Republic, Germany, Poland, and Slovakia simultaneously and caused a total loss of more than 5 billion euro. The 2002 flood events were even more devastating, with around 14.4 billion euro in total losses recorded in Austria, the Czech Republic, Germany, and Slovakia.(10) The recent 2013 flood affected nine EU countries at once and caused a total loss of 12 billion euro. These extreme events and corresponding losses significantly strained disaster financing mechanisms such as the European Solidarity Fund (EUSF).(11,12) As these examples indicate, taking account of the interdependency of extreme risks is important. In Europe, current flood models only consider risk information in terms of loss distribution at a very local scale. Information at larger scales is available, but is typically developed for specific scenarios or expressed in terms of average losses only.(13–18) As a consequence, risk management approaches for extremes cannot be applied at these scales as necessary probabilistic information is not available.(19) To overcome this limitation, we suggest a copula-based methodology, described and demonstrated further via an in-depth example for flood risk assessment in Romania. Copula-based approaches have been used broadly in different fields, from technology to finance.(20–23) However, only recently they were suggested as a possible way to derive risk at larger scales. For example, in Hochrainer-Stigler et al.,(24) an event-based copula was introduced that discriminated between events that are fully independent and

2103 after a given impact (or probability level) as fully dependent over a given region. Regions were defined based on Strahler orders, which counted tributaries of basins and were used for the upscaling process. In this way, their study was able to calculate flood risk estimates for all European countries. A more refined version was introduced using a Clayton copula approach to determine the dependencies of river basins using maximum river discharge data.(11) The focus of their study was on a specific ordering scheme to upscale losses and to calculate a pan-European loss distribution. Our work builds on and extends the current discussion regarding this topic. We present a detailed analysis based on new algorithms to obtain loss distributions at larger scales using copulas. It should be noted that our methodology, while applied for flood risk, could be extended to other cases and hazards as well, given a moderate set of data requirements (i.e., data to estimate copula parameters as well as loss distributions) is met. Data employed for the analysis include monthly mean and maximum river discharges for the period of 1990–2011 for all river cells in Europe (with an upstream area larger than 1,000 km2 ). Simulated losses are derived using the LISFLOOD hydrological model and an economic damage model.(25,26) This flood loss data set has been validated in previous panEuropean studies(11) and therefore provides an ideal entry point for our analysis. The article is organized as follows. In Section 2, we start with a discussion and motivation for using copulas and introduce different types of copulas for measuring tail dependencies. The section provides ordering techniques for multidependent loss distributions and also algorithms that could perform the desired tasks. In Section 3, we apply our approach and compare the suggested ordering techniques via the Romanian case study. Finally, Section 4 draws major conclusions.

2. COUPLING DEPENDENT LOSS DISTRIBUTIONS Large-scale flood risk analysis is a sophisticated procedure that consists of several steps and should incorporate interdependencies between all considered basins, regions, and countries. The neglect of underlying interdependencies (i.e., approximation of the total risk curve via the convolution of regional risk curves) eventually leads to the underestimation of risk and to the potential failure of risk management strategies.(27–30)

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In general, in order to answer the question about the flood risk in multiple regions, it is necessary to estimate the probability loss distribution that gives information on the probability of rare events (10-year event, 50-year event, 100-year event, etc.) and the amount of loss in case of these events. In order to do this, one, first of all, must know the marginal loss distributions for each of the basins. In our analysis, marginal losses are simulated using the LISFLOOD hydrological model and an economic damage model.(25,26) Second, one needs to couple marginal losses in such a way that the large-scale probability distribution is estimated correctly and fits the multiregional data on losses. For this, it is not enough to convolute marginal distributions, as this leads to the neglect of regional interdependencies, and it is necessary to find a nonlinear dependency function (i.e., copula) that transforms the marginal probability distributions into a total probability distribution taking into account all interdependencies. Third, it is necessary to understand from the available data and river structure which groups of basins can be considered as dependent, and which as independent and in which order these groups should be coupled. Therefore, this section focuses on different types of copulas (Gumbel, Frank, and Clayton) and coupling techniques in order to answer the question which copula and which coupling method should be used for the best possible estimation of the flood risk in multiple regions. 2.1. Approach

Case 1: If two river basins and, hence, their individual loss distributions are independent, a (statistical) convolution technique can be applied to produce the joint loss distribution. This could be the case if two river basins are very far away from each other or in the presence of an orographic barrier (e.g., a mountain in between them). For example, the losses in Mures River in Romania are only weakly correlated with the Bistrita losses because of the Transylvanian Western Carpathians (Apuseni Mountains) located in between them. To be more precise, let us consider two regions with losses L1 and L2 . In case these regions and losses are independent, the density of the total loss L1 + L2 can be derived through convolution:  f1 (x − y) f2 (y)dy, (3) f (x) = f1  f2 = where f1 and f2 are marginal densities of L1 and L2 correspondingly; f is the density of the total loss L1 + L2 ; and  is a convolution operator. Case 2: If two river basins are interdependent, using the independent convolution technique is no longer recommendable. Instead, one needs to couple the individual loss distributions for these basins, taking the dependency explicitly into account. This can be achieved via a copula technique, where a copula function is estimated based on the historical data of past events.

We note first that the total loss after a flood (or any hazard) event is just the sum of the losses in the individual regions that were affected by the hazard:

Definition 1 (Copula function(31,32) ). Copulas are distribution functions on [0, 1]d , d ≥ 2, with standard uniform univariate margins.

L = L1 + L2 + · · · + LN ,

Generally speaking, when regions are located close to each other, interdependencies between river basins can be expected and, hence, copula estimation and modeling are required to study risk in the form of loss distributions at an aggregate scale. It is necessary to measure the dependency between the general number of basins N, for which a multivariate copula has to be estimated. The estimated copula should be able to incorporate different strengths of dependencies between different groups (or pairs) of basins. Multivariate copulas are important for two reasons: (1) they can model the dependency structure of a random vector separately from marginal distributions and (2) they can construct new families of multivariate distributions. In most cases (for example, in the case of the Archimedean copula family), they do

(1)

where each Li , i = 1, . . . , N represents the loss in a particular region i. However, to calculate the joint probability loss distribution F(x1 , . . . , xN ) = P{L1 ≤ x1 , . . . , LN ≤ xN }

(2)

for the entire region (i.e., all regions 1, . . . , N), for dependent risks, it is not enough to only know the marginal probability distributions Fi (x) = P{Li ≤ x} of Li , i = 1, . . . , N, as will be discussed next. Conceptually, two different cases can be identified for the calculation of the joint loss distribution F(x1 , . . . , xN ). Let us start with the pair-wise coupling of losses:

Structured Coupling of Probability Loss Distributions not allow introducing different dependency parameters for different groups (or pairs) of basins; they neglect some important characteristics of the dependency phenomena. This is an important drawback that should be overcome by our suggested approach. Let C(x1 , . . . , xN ) be a copula model, meaning that: P{L1 ≤ x1 , . . . , LN ≤ xN } = C (F1 (x1 ), . . . , FN (xN )) , (4) where Fi (x) = P{Li ≤ x} are marginal distributions. If Equation (1) holds with copula C, we write: L = ⊕C Li .

(5)

The total loss distribution of L is to be calculated by the coupling of marginal loss distributions of L1 , . . . , LN over the copula C (⊕C is the sum over copula C). To incorporate different dependencies between groups of basins in this model, one could use such concepts as vine copulas,(33,34) hierarchical Archimedean copulas (HAC),(32) or some combination of them. However, for the correct use of these approaches, one needs first a model for the pair-wise coupling of dependent losses, which is introduced next. Consider two regions with losses L1 and L2 correspondingly. In case these regions and losses are dependent, the density of the total loss L1 + L2 can be obtained by the convolution over the copula C(·): f (x) = f1 C f2  = c (F1 (x − y), F2 (y)) f1 (x − y) f2 (y)dy, (6) where f1 and f2 are the marginal densities of L1 and L2 correspondingly; f is the density of the total loss L1 + L2 ; and c(·) is the copula density. To correctly study interdependent risks (such as in our special case of flood losses) it is necessary to use a copula model that correctly identifies joint loss distributions between selected regions. To study regional interdependencies, it is not enough to calculate correlations (e.g., Pearson) between different regions and countries. Such techniques are usually limited to modeling linear dependencies between losses and therefore do not adequately address tail dependencies, the most important feature for studying extreme events. Furthermore, though regional dependency can be studied by coupling losses, not

2105 all copula functions will suit our purposes. Indeed, a copula function is needed that recognizes that the dependencies between different regions may also be different, i.e., parameters describing the dependency may vary with respect to regions. Flood events usually follow the spatial network/topology of rivers and, thus, it is necessary to include this information in the analysis of losses as well. To address these complexities, we will resort to the Archimedean copula family and different hierarchical coupling approaches. In doing so, we will heavily refer to the case study area for motivation and explanation of our methods. 2.2. Copula Selection Generally speaking, for comprehensive analysis of the interdependencies between basins, a copula type Cθi j (Fi , F j ) should be chosen so that it (i) describes flood loss behavior in a satisfactory manner. For example, one should be able to model the interdependency between basins i and j to be high only for large flood events and to be low for small events. At the same time, a chosen copula should be able to (ii) explicitly model fat tail interdependencies. For convenience in mathematical analysis, (iii) coupling of copulas of the same type should again produce a copula of the same type. We consider the following three copula types from the Archimedean family for our analysis. The Clayton copula (θ ≥ −1, θ = 0) is an asymmetric Archimedean copula, exhibiting greater dependence in the negative tail than in the positive, i.e., CθC (u, v) = max([u−θ + v −θ − 1]− θ , 0), 1

which can be simplified to CθC (u, v) = [u−θ + v −θ − 1]− θ

1

(7)

as soon as θ ∈ (0, +∞) for flood loss dependencies and θ = 0 for independent losses. The Frank copula (θ = 0) is a symmetric Archimedean copula of the following type:   1 (e−θu − 1)(e−θv − 1) .(8) CθF (u, v) = − ln 1 + θ e−θ − 1 The Gumbel copula (θ ≥ 1) is an asymmetric Archimedean copula, exhibiting greater dependence in the positive tail than in the negative, i.e.,   1  CθG(u, v) = exp − (− ln u)θ + (− ln v)θ θ . (9) Notice that the Clayton copula does not describe the flood loss behavior in a satisfactory manner, as it

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exhibits smaller dependence in the positive tail than in the negative, meaning that it assumes that large flood events are less dependent than small ones. As this is not true for flood events, it is necessary to flip the Clayton copula so that the new copula exhibits greater dependence in the positive tail. Thus we introduce the Flipped Clayton copula. The density function cθFC (u, v) for parameter θ > 0 of the Flipped Clayton copula can be written as: cθFC (u, v)(1 − u)θ+1 (1 − v)θ+1  − 1 −2 = (1 + θ ) (1 − u)−θ + (1 − v)−θ − 1 θ .

(10)

Integration of the density function gives us the distribution function CθFC (u, v) of the Flipped Clayton copula: CθFC (u, v) = u + v − 1  − 1 + (1 − u)−θ + (1 − v)−θ − 1 θ (. 11) Hence, one has three possible copula types for the flood risk analysis: Flipped Clayton copula (θ > 0), i.e., CθFC (u, v) = u + v − 1  − 1 + (1 − u)−θ + (1 − v)−θ − 1 θ . Frank copula (θ = 0), i.e.,   (e−θu − 1)(e−θv − 1) 1 . CθF (u, v) = − ln 1 + θ e−θ − 1 Gumbel copula (θ ≥ 1), i.e.,   1  CθG(u, v) = exp − (− ln u)θ + (− ln v)θ θ . Notice that one can easily extend the Flipped Clayton copula for the case of θ = 0, meaning that there is no dependency between flood losses. In order to study the dependency in flood losses using these copulas, it is necessary to generate values of losses in basin j conditional on those in basin i. Suppose that v is the flood loss in one of the basins (i.e., i) that is known and that one would like to simulate loss u in another basin j conditional on loss v. For this, one needs to generate conditional copulas that are in general determined (·) as the partial derivative of Cθ (u, v) over v, i.e., (·)

(·)

∂C (u,v)

Cθ (u|v = v0 ) = θ ∂v |v=v0 . To find a corresponding loss u, we need to inverse this equation with respect to u, generating random numbers from the conditional distribution. The Flipped Clayton and Frank copulas are directly invertible. Therefore, the

generation of the u is not a complicated numerical process for these copula families: Conditional Flipped Clayton copula − 1  θ (12) u = 1 − 1 + (1 − v)−θ (r − 1+θ − 1) θ . Conditional Frank copula   r (1 − e−θ ) 1 , u = − ln 1 + θ r (e−θv − 1) − e−θv

(13)

where r is a uniform random variable on the interval (0, 1). However, the conditional Gumbel copula is not directly invertible and, hence, has to be generated differently. Consider the partial derivative of the Gumbel copula over v: 1 ∂CθG(u, v) ∂v CθG(u, v)  1−θ (− ln v)θ−1  (− ln u)θ + (− ln v)θ θ . (14) v Let us make the following change of variables   1  w = exp − (− ln u)θ + (− ln v)θ θ , =

and notice that   1  u = exp − (− ln w)θ − (− ln v)θ θ . In this case, ∂CθG(u, v) (− ln v)θ−1 = w(− ln w)1−θ , ∂v v

(15)

which is a concave function ∀w ∈ (0, 1); θ ≥ 1. Therefore, the following algorithm (see Algorithm 1) can be used for the generation of u conditional on v for the Gumbel copula: Algorithm 1. Algorithm for the generation of u conditional on v for the Gumbel copula (1) (2) (3) (4)

v is fixed and, suppose, equals to v = v0 ; Generate r randomly from the interval (0, 1); Assign w = v0 ; Iterate

θ ln w w(− ln w) − r v0 (− ln v0 ) ln v0 wnew = w − θ − 1 − ln w

while |wnew − w| > 10−6 ; (5) Assign   1  u = exp − (− ln wnew )θ − (− ln v0 )θ θ .

Structured Coupling of Probability Loss Distributions

Fig. 1. Copula for the discharge data. Parameter θ = 4.8629.

In this article, we present the results for the Flipped Clayton copula Cθ (u, v) = CθFC (u, v) only, as it satisfies all the necessary properties of (i)–(iii) and is efficient in simulation, i.e., it allows receiving accurate results via Monte Carlo generation of a large number of points (which is more difficult for the Gumbel copula). Furthermore, the Frank copula is a symmetrical copula, which is less appropriate for flood losses. In Section 3.2, we provide a comparison of losses with the Gumbel and Frank copulas. By considering a pair of river basins on the Danube and estimating the copula for the fat tails of the discharge data, one can see via Fig. 1(a) that the Flipped Clayton copula empirically well describes the flood loss behavior (see Section 4 for more details). Fig. 1(b) shows two subregions (basins) of the Danube River Basin: river basin 1 belongs to the Middle Basin of the Danube and lies in the area

2107 of dams of the Iron Gate Gorge on the border between Romania and Serbia; river basin 2 lies in the Danube Delta, which belongs to the Lower Basin of the Danube and covers an area of about 6,750 km2 . The Flipped Clayton copula estimation gives the result of θ = 4.8629 for the dependency between these two basins, based on discharge data. In more detail, Fig. 1(a) represents the contour plots of the bounds for the Flipped Clayton copula with parameter θ = 4.8629 for the level sets {0.1, 0.2, . . . , 0.9}. From the contour plot and dense behavior of points generated from this Flipped Clayton copula (blue points (color visible in on-line version)), one can see that the event dependency of basins 1 and 2 is stronger when a flood event is large, and weaker when a flood event is small (i.e., in the upper-right and lower-left corners of Fig. 1(a) correspondingly). This satisfies necessary property (i) of the dependency and indirectly shows that the Flipped Clayton copula empirically well describes the flood loss behavior for such distant basins of the Danube. However, more details are given in Section 4, including the comparison of the Gumbel and Frank copulas. Now, consider basins i and j and suppose that the marginal loss distributions Fi and F j are known. We estimate the Flipped Clayton copula Cθi j (Fi , F j ), i.e., the parameter θi j in Equation (11) by the maximum likelihood estimation techniques. Studying the interdependency between several basins within Romania (in Section 3), we will show how different basins are dependent on each other (see Fig. 7), i.e., we will consider the θ parameters of the Flipped Clayton copula between different pairs of basins and will observe their fat tail dependencies. One of the main purposes of this article is also to provide a methodology for coupling several basins, regions, or countries at once. While a multivariate copula may seem to provide a good way to deal with multiple number of basins, multivariate copulas have a serious disadvantage as they are dependent only on a single parameter θ and do not allow for modeling different dependencies between different pairs of basins. For example, the multivariate Flipped Clayton copula Equation (16) depends only on one parameter θ : n  ui − 1 Cθ (u1 , . . . , un ) = i=1

− θ1 n  (1 − ui )−θ − 1 . (16) +

i=1

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Some alternatives (such as vine or HAC copulas) to multivariate copulas should be identified to deal with the fact that different basins may depend on each other in different ways (hence, with different parameters θ ). One proposed solution for this problem is discussed next. 2.3. Coupling Multiple Dependent Loss Distributions Consider now a set of basins I ∈ {1, 2, . . . , N}, we would like to study the interdependency between all these basins and to obtain the joint probability distribution that describes the total loss over these basins. If one estimates the Flipped Clayton copula parameters between each pair of basins, the following matrix can be obtained: ⎞ ⎛ 1 θ12 . . . θ1N ⎜ θ21 1 . . . θ2N ⎟ ⎟ ⎜ (17) =⎜ . .. . . .. ⎟ , ⎝ .. . . ⎠ . θ N1 θ N2 . . . 1 where columns and rows represent basins and every element θi, j = θi j of this matrix is the Flipped Clayton copula parameter that describes the dependency between pair of basins i and j. The parametric family Cθ (u1 , . . . , u N ), to which the Flipped Clayton copula belongs, does not allow more than a one-dimensional parameter θ . This leads us to the question how one may model the dependency between different components (i.e., basins) with N × N matrix . Moreover, matrix  does not contain any information about the structure/topology of river basins, though rivers, naturally, follow some structures. Hence, we use available information on river topology in Europe to study the dependency between several basins at once. Vine copulas are a well-known tool to obtain joint distribution for N interdependent regions.(34–36) The concept of vine copulas for N regions (or basins) requires the construction of a sequence of N − 1 trees. Each tree is constructed in an iterative manner so that (1) edges in tree j become nodes in tree j + 1 and (2) two nodes in tree j + 1 are joined by an edge if the corresponding edges in tree j share a node.(34–36) All these lead to the numerical inefficiency of joint distribution calculation for basins in Romania. In Fig. 2, one can see the Bistrita river and corresponding nine basins with connections in Romania. Every line between basins represents the existing

Fig. 2. Bistrita basin structure in Romania.

water connection between them, i.e., river branches. According to its natural behavior, every river structure can be represented as a graph or a tree structure. The Bistrita river structure in Fig. 2 can be represented as a tree with nine nodes, corresponding to Tree 1 in Fig. 3(a). To study the total dependency between these nine basins from the Bistrita river structure via vine copulas, it is necessary not only to consider the underlying connections between basins, but also the existing connections between river branches. Incorporation of these connections leads us to Tree 2 from Fig. 3(b) and to the remaining trees of Figs. 3(c), (d), (g), and (h) in an iterative manner. Every following tree takes into account all the branches of the previous tree until there is only one branch left, as is shown in Fig. 3(h). Every following tree is obtained from the previous one through steps (1)–(2). Hence, to receive joint distribution for these nine basins of the Bistrita with vine coupling, we construct eight trees, as shown in Fig. 3 (Tree 1 to Tree 8). The joint density is calculated in the following way: ⎡ f (x1 , . . . , xN ) = ⎣

N−1  N− j

⎤ ci,i+ j/i+1,...,i+ j−1 ⎦

j=1 i=1

·

N 

fk(xk) ,

k=1

where N = 9 for the Bistrita river and that leads to the necessity to construct N − 1 tree structures and to estimate (N − 1)! copulas. To avoid this difficulty, we construct an approximation that represents river structures and an estimate of joint distribution without (N − 1)! copulas.

Structured Coupling of Probability Loss Distributions

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Fig. 3. A nine-dimensional vine tree structure.

An ordering technique for θi, j components of  is needed. It should produce vector θ from matrix  in such a way that (i) interdependencies between multiple basins are well represented by this set of pair-wise dependencies and (ii) the river topology of the region is taken into account. Here, we focus on hierarchical and ordered coupling techniques for the approximation of river structures as these are the most feasible techniques to be used in a practical setting with high number of basins.

Hierarchical coupling: Consider basin pairs (1, 2) and (3, 4) and suppose that the Flipped Clayton copulas for both models are estimated (i.e., we know parameters θ1,2 and θ3,4 and, hence, copulas C1,2 and C3,4 ). If these two pairs of basins were independent, we would independently convolute losses in these basin pairs to obtain joint loss distribution for basins (1, 2, 3, 4), i.e., C(x1 , x2 , x3 , x4 ). However, the losses in basins (1, 2) could be dependent on losses in basins (3, 4) and, hence, independent convolution is no longer possible. In such case, we estimate a copula

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Timonina et al. c(x1 , x2 , x3 , x4 ) = c1,2 (x2 |x1 ) · c2,3 (x3 |x2 ) · c3,4 (x4 |x3 ). (19) An obvious possibility to approximate the tree structure and to obtain the order of the Flipped Clayton copula parameters would be to sort them with respect to a given metric (e.g., geographical distances between basins or, simply, by discharge data correlations). Some possible structuring approaches are discussed next.

Fig. 4. Hierarchical coupling.

2.4. Structuring Approaches In order to use ordered copulas, we need to calculate a vector θ = (θ1 , θ2 , . . . , θ N ) of the size 1 × N out of matrix  (i.e., matrix (17)) of the size N × N. We then sample from the multivariate Flipped Clayton copula. Receiving a vector of the size 1 × N out of a matrix of the size N × N means that some information regarding basin dependencies is neglected. As we focus on the tails of probability distributions, we especially want to avoid underestimation of extremes.

Fig. 5. Ordered coupling.

function C¯ suitable for the coupling of copulas C1,2 and C3,4 : ¯ 1,2 (x1 , x2 ), C3,4 (x3 , x4 )). C(x1 , x2 , x3 , x4 ) = C(C

(18)

In this case, C1,2 , C3,4 , and C¯ may be copulas of different types. As soon as we focus on the Flipped Clayton copulas, we can guarantee that the type of copula C¯ is the same as of copulas C1,2 and C3,4 , i.e., the Flipped Clayton copula. Hierarchical copulas (see Fig. 4) follow a tree structure that may be difficult to estimate, especially if the topology of the tree has to be estimated as well. Due to the difficulty in estimation of the tree ¯ the nestructure of rivers and of the copula C, cessity of approximation arises. One possible and convenient way of approximation is using ordered copulas. Ordered coupling: Suppose that loss L1 of basin 1 influences loss L2 of basin 2. Loss L2 of basin 2 then influences loss L3 of basin 3, and loss L3 influences L4 , and so on. We will get the tree structure shown in Fig. 5. In this case, one could estimate two-dimensional copula densities c1,2 , c2,3 , and c3,4 and combine them, in the following way, to obtain a four-dimensional copula density:

Algorithm 2. Algorithm for ordering vector θ out of matrix  1. Initially, we choose the maximal θi, j from matrix —this element refers to two basins (i and j) that are most dependent in terms of flooding. We couple them first. Our Or der = [i, j] = [ j, i]. 2. Then we choose the next element (i.e., basin) k that is suitable for both basins i and j (so that it is dependent not only on the loss situation at basin i, but also on the loss situation at basin j) (notice that k = i, k = j). We must guarantee that the dependencies between pair (i, j) and each of the basins 1, . . . , N are greater than some values and then maximize over these values. Suppose we have two vectors θi and θ j . θi is a vector that contains dependencies between basin i and all other basins 1, . . . , N (the length of it is N). θ j is a vector that contains dependencies between basin j and all other basins 1, . . . , N. Hence, if we construct a new vector θ such that each element is the minimum between the rows of θi and θ j , we can guarantee that the dependency between pair (i, j) and basins 1, . . . , N is greater than the values of θ (for example, dependency between (i, j) and k is greater than θk).

Structured Coupling of Probability Loss Distributions 3. Again, it is necessary to maximize over elements of vector θ and, hence, to get index k. We now obtain a triplet of basins (i, j, k). 4. Continue iterations until a vector of length N is obtained. Therefore, we use a minimax approach to choose N pairs of basins from matrix . We show that this method adequately represents the complexity of the tree structure of rivers in Europe and underlying interdependencies between basins. Consider matrices , R, and D:  is the matrix of Flipped Clayton copula parameters θi, j between basins i and j; R is the matrix of correlations with elements ri, j that represent correlations between basins i and j; D is the matrix of geographical distances between basins i and j. Algorithm 2 describes the minimax ordering technique for the estimation of river structure and, hence, for the estimation of the multivariate Flipped Clayton copula and joint probability distribution for the whole region. In Algorithm 2 the measure for the interdependency between basins i and j is the Flipped Clayton copula parameter θi, j that refers to high dependency if θi, j is large and low dependency if θi, j is small. It is possible to use geographical distances between basins or discharge data correlations as an initial measure for the interdependencies and for ordering of the Flipped Clayton copula parameters as well. The algorithms are not very different from the one above and are shortly discussed below. Order by correlations: If one uses correlations as a measure of risk, one may follow the similar algorithm as Algorithm 2. In this case, initially chosen ri, j is the maximal value in the whole matrix R, while the maximal correlation element refers to two basins (i and j) that are most dependent in terms of flooding. The same as in Algorithm 2 we couple them at first and receive an order Or der = [i, j] = [ j, i]. Then, we choose the next element (i.e., basin) k that is suitable for both basins i and j, by the minimax approach and continue until a complete order is obtained. As the final step, we use the received order on matrix  to receive θ vector based on matrix R. Order by geographical distances: If one would like to use geographical distance as a measure of risk interdependencies, one should change Algorithm 2 so that it adapts to the fact that a smaller distance would logically mean higher interdependency, i.e., the algorithm will be maximin one instead of minimax.

2111 Initially we choose the minimal di, j from the whole matrix D—this element refers to two basins (i and j) that are geographically closest to each other. After coupling and calculating the order Or der = [i, j] = [ j, i], we choose the next element (i.e., basin) k that is suitable for both basins i and j, i.e., using the maximin approach. Continuing iterations until a vector of length N is obtained and using the final order on matrix , we obtain θ vector based on matrix D. Next, we explain available data to be used for our case study. 2.5. Data The following data(11,13,17) are available for our analysis: Geographical position: a list of basins in Europe numbered from 1 to 1,433 with corresponding locations, given in the form of their longitude and latitude; Monthly peak discharges: monthly time series data on maximal discharges at each basin (1 to 1,433) for the period of 1990–2011 that are highly correlated to the occurrence of damaging floods;(11) River structure: for each basin, there is a list of all incoming and outgoing connections to other basins that gives us a tree structure of rivers in Europe. Losses: Simulated losses derived using the LISFLOOD hydrological model and an economic damage model.(25,26) In our analysis, we ignore the direction of river flows, assuming that their influence is symmetrical (Fig. 6). 3. CASE STUDY: ROMANIA Romania was chosen as a case study as it is a very flood-prone country and has been significantly affected by floods in the past, causing significant damages. More than 1 million ha are exposed to flooding, representing 5.8% of the Romanian lands and nearly 1 million people live in high flood risk areas.(37) During the last decade, there were many floods in the country, causing significant damages. Furthermore, the river basins in Romania follow a complex topology/network, providing an excellent case study to test our methodology. The proxy for the correlation of losses over different basins is maximum discharge data obtained from each basin for the period of 1990–2011. The necessity to study fat tail interdependencies, i.e., the dependencies between low-probability/

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Fig. 7. Multivariate Flipped Clayton copula for some basins in Romania.

interdependent even for large distances and it is not appropriate to use independent convolution to obtain joint distribution for these basins. It is necessary to couple individual loss distributions. In Fig. 6(b), hierarchical basin structures in Romania are shown, which are used as an input for the estimation of the ordered or hierarchical copulas. 3.1. Testing Different Ordering Approaches Fig. 6. River basins in Romania.

high-loss events, brings our attention to the use of a copula approach in general and, particulary, as indicated in Section 2.2, we apply the Flipped Clayton copula as a good fit for the discharge data between basins. Fig. 7 shows the pair-wise Flipped Clayton copulas for four highly dependent Danube basins in Romania. Analogous to matrix (17), Fig. 7 represents pair-wise dependencies between basins I ∈ {1, 2, 3, 4} with rows and columns corresponding to the basins. To check the appropriateness of coupling approaches for the river basins in Romania, one needs to evaluate if there are highly interdependent basins. Using data on correlations between river basins we plotted a map of the basins and connected basins that are highly dependent (Fig. 6(a)). Fig. 6(a) shows that the whole Danube bank is highly

We test the use of hierarchical and ordered coupling in the Romanian case study. In order to clarify if the minimax technique (i.e., ordered coupling, see Algorithm 2) adequately represents the flood losses and structures of river basins in Romania, we test both the hierarchical and ordered coupling techniques. First of all, we divide the basins of Romania into subgroups that consist only of those basins hydrologically connected to each other (Fig. 8). Notice that the subgroups themselves are not hydrologically connected and, moreover, they are not highly correlated according to Fig. 6(a). We apply two ordering algorithms on the grouped Romanian basins. The first θ order is obtained by the minimax approach described in Algorithm 2 and the second order is based on the river flow network in Romania. To obtain the θ order based on the river flow network in Romania, we apply Algorithm 2 to the matrix  (see Equation (17)), where we assign θi, j = 0 if basins i and j are not hydrologically connected.

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Fig. 8. Different parts (groups) of basins in Romania not connected with each other by water and their loss curves.

Table II. Loss in Mures River Region for 2015 Estimated by Structuring and Minimax Techniques on Flipped Clayton Copula (in EURO bln.) Year-events Quantiles Minimax order. (L1 ) Structure order. (L2 )

Fig. 9. Loss curves corresponding to the groups of basins from Fig. 8 for 2015. Table I. Loss in Danube Region for 2015 Estimated by Structuring and Minimax Techniques on Flipped Clayton Copula (in EURO bln.) Year-events Quantiles Minimax order. (L1 ) Structure order. (L2 )

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0.8 0.9 0.95 0.98 0.99 0.996 0.998 1.979 2.284 2.544 2.832 3.019 3.243 3.391 1.984 2.287 2.543 2.836 3.027 3.263 3.424

Using these two ordering techniques, we obtain two vectors θ1 and θ2 and calculate the loss curves for predefined quantiles. We obtain L1 and L2 , respectively. Based on the obtained θ orders, we estimate losses for both groups in Fig. 9. From Fig. 9, one can see that both loss curves are close to each other, which is also represented in Tables I and II for both groups. Hence, the minimax ordering technique based on Algorithm

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0.8 0.9 0.95 0.98 0.99 0.996 0.998 1.923 2.291 2.595 2.953 3.210 3.508 3.706 1.930 2.295 2.602 2.963 3.211 3.518 3.710

2 produces the same result as the ordering in line with the true river flow structure in Romania. The minimax approach gives a good approximation of the hydrologically connected river structures in Romania. Now, we study if the same result holds for all basins in Romania, including those that are not hydrologically connected. We obtain the estimates of the total loss distribution for Romanian basins using (i) the minimax ordering with Algorithm 2, i.e., we receive L1 , and (ii) data on river structure, i.e., we receive L2 (see Fig. 10 and Table III). At the same time, we (iii) apply independent convolution to all basins in Romania, considering them as fully independent of each other, i.e., we receive L3 . For comparison, we also (iv) estimate losses for the fully dependent case, i.e., we receive L4 by summation of losses in all basins considering them as fully dependent on each other, and we (v) apply independent convolution to groups of hydrologically connected basins in Romania (see Fig. 9), considering losses of these groups as independent of each other, i.e., we receive L5 . Notice that basins inside the groups in Fig. 9 are dependent of each other.

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Fig. 10. Comparison of different ordering techniques for Romania for 2015.

Table III. Total Losses in Romania for 2015 Estimated by Structuring and Minimax Techniques on Flipped Clayton Copula (in EURO bln.) Year-events Quantiles Minimax ord. (L1 ) Structure ord. (L2 ) Ind. conv. of basins (L3 ) Fully dependent (L4 ) Ind. conv. of groups (L5 )

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0.8 0.9 0.95 0.98 0.99 0.996 0.998 4.590 5.735 6.772 8.020 8.920 10.022 10.756 4.581 5.728 6.784 8.042 8.921 10.029 10.619 4.143 4.625 5.041 5.511 5.807 6.219 6.453 7.858 9.937 11.763 14.017 15.585 17.567 18.993 4.561 5.133 5.606 6.136 6.491 6.907 7.185

In Fig. 10 we see that the loss distribution obtained by minimax approach (i.e., Algorithm 2) is highly dissimilar from the one obtained through independent convolution both of basins and of basin groups. This difference may be explained by the fact that the chosen basin structures are dependent in reality, even though there is no hydrological connection in between. A similar result holds for the structuring approach (notice, in Fig. 10, that L2 and L3 almost coincide). This is highly relevant for low-probability/highloss extreme events, which may result in damaging of basins not hydrologically connected. In Fig. 10 and Table III one can see that losses may result in high underestimation if basins or groups of basins are considered as independent.

By coupling individual loss distributions for different river basins in Romania, one can identify how these river basins are dependent on each other. For example, by coupling individual losses of different Danube subbasins, located far from each other, we may obtain high dependency (Fig. 11). At the same time, by coupling individual losses of river basins from different rivers, located closer to each other, one may not obtain high interdependency, when it is more difficult for floods to reach some basins (e.g., no hydrological connection) (see Fig. 12). Fig. 11 focuses on two Danube basins located in the Lower Basin of the Danube on the boarder of Romania and Bulgaria. These basins, connected to each other by water through Bulgaria, are 242 km away from each other; the dependency estimated by the Flipped Clayton copula gives the result of θ > 10, indicating an extremely high level of dependency even for high-probability (low-loss) events. At the same time, Fig. 12 focuses on two basins of rivers with low correlation (see also Fig. 6): these basins belong to the neighborhood of the Crisul Alb and Somes rivers. These two basins in Fig. 12 are located very close to each other (64 km); however, they are separated by the Western Carpathians. For these basins, the Flipped Clayton copula parameter is estimated to be much lower (θ = 1.5684), i.e., the dependency is much weaker than that estimated for the Danube basins. This means that the Flipped Clayton copula approach for dependency estimation is able to take the topography of the regions into account. This also means that a geographical distance cannot be used as a measure of flood dependency between basins. Notice that Figs. 11(b) and 12(b) show the contour plots of the Flipped Clayton copula for the underlying basins: the contour lines, as well as the dense behavior of generated points, represent high dependency for low- and high-probability flood events in the Danube river basins (the upper-right and the lower-left corners of Fig. 11(b)) and lower dependency for low- and high-probability flood events in the Crisul Alb and Somes river basins (the upper-right and the lower-left corners of Fig. 12(b)). Now, using Algorithm 2 and its variations for correlation measure and geographical distance measure, we can order , R, and D according to the minimax of copula parameters, correlations, and distances correspondingly. Suppose that by doing this

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Fig. 11. The dependency is high and is greater than θ > 10, though the distance between the midpoints of the basins is large and equals d = 242 km. Selection is based on Jongman et al.(11)

Fig. 12. The dependency is low and is equal to θ = 1.5684, though the distance between the midpoints of the basins is small and equals d = 64 km. Selection is based on Jongman et al.(11)

we have obtained three possible orders for the basins, respectively, i.e., Or derθ , Or derr , and Or derd . Now, applying these orders on matrix  we obtain different possible vectors θ that should represent matrix , i.e., we get θθ , θr , and θd . By hierarchically sampling multivariate Flipped Clayton copulas with parameters θθ , θr , and θd and fitting flood loss distributions to the obtained samples, we estimate three possible joint loss distributions for the considered basins by ordered coupling. Fig. 13 and Tables IV and V show how these distributions differ from each other for Flipped Clayton and Gumbel copulas that are the most

appropriate for the analysis of extreme losses (see Fig. 14). For both copulas, ordering by distances leads to a high underestimation of flood risk, in particular, concerning low-probability events. As shown, the distance between basins is not a good measure of risk or interdependencies, since it does not take into account the topography of the regions. Correlations represent the dependencies between basins better than distances, but they do not take specific joint behavior into account, and they underestimate total losses as well. In summary, distance technique is not advisable for large-scale analysis as it does not reflect

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Fig. 13. Comparison of different interdependency measures for Romania.

Table IV. Total Losses in Romania Estimated by Minimax Approach on Flipped Clayton Copula with Different Ordering Techniques for 2015 (in EURO bln.) Year-Events

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Theta parameters 4.590 5.735 6.772 8.020 8.920 10.022 10.756 Correlations 4.406 5.211 5.902 6.696 7.250 7.925 8.374 Distances 4.166 4.665 5.079 5.564 5.900 6.311 6.598 Table V. Total Losses in Romania Estimated by Minimax Approach on Gumbel Copula with Different Ordering Techniques for 2015 (in EURO bln.) Year-Events

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Theta parameters 4.662 5.821 6.911 8.213 9.112 10.114 10.783 Correlations 4.407 5.181 5.853 6.644 7.142 7.803 8.236 Distances 4.148 4.633 5.044 5.536 5.860 6.287 6.541

Fig. 14. Comparison of different copulas for loss estimation in Romania for 2015.

Table VI. Total Losses in Romania Estimated by Minimax Approach with Different Copula Types for 2015 (in EURO bln.) Year-Events Flipped Clayton copula Gumbel copula Frank copula

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4.590 5.735 6.772 8.020 8.920 10.022 10.756 4.662 5.821 6.911 8.213 9.112 10.114 10.783 4.544 5.282 5.891 6.596 7.073 7.652 7.996

possible dependencies. Using correlations does not adequately reflect increasing tail dependencies. Independence over different basins may not hold true for large-scale events as weather patterns may happen over multiple basins at once (such as in a case of heavy precipitation). An independence assumption neglects a potential increase in extreme losses. However, the maximum discharges between basins measured with the Flipped Clayton copula served well to account for an increase in tail dependence. The ordering scheme using the minimax approach takes dependencies over different basins (irrespective of locations) into account while accounting for independent as well as highly dependent situations between respective basins. Furthermore, by comparing the estimated losses in Romania with the Gumbel (see Equation (9)) and the Frank (see Equation (8)) copula types (see Table VI and Fig. 14), we can see that the difference in results is small for Flipped Clayton and Gumbel copulas and can be explained by the random method of generation of points from the copulas. The Frank copula tends to underestimate losses and, hence, does not sufficiently explain interdependencies

Structured Coupling of Probability Loss Distributions between basins. However, this may not be the case in other settings. 4. CONCLUSION Losses from extreme natural hazard events can be extraordinarily high. For the analysis of such risks, it is important to understand the probability-loss relationship (e.g., in the form of loss distributions) at all scales, ranging from the household, the country, and the global. Recently, approaches were suggested to derive such loss distributions, taking tail dependencies explicitly into account. This article focused on flood risk, which can affect multiple regions, and investigated how copulas can be used to estimate potential losses. We introduced the Flipped Clayton copula (as well as others), which served well for this purpose. Furthermore, we suggested various structured coupling methods to derive loss distributions on higher scales. We found that ordering schemes based on distances or correlations are not appropriate on these scales. Furthermore, in cases of extremes, independence assumptions based on the network structures of river basins will not adequately reflect dependencies for large-scale events. It may be beneficial to estimate theta parameters between each basin and perform a minimax approach to account for dependencies so as not to over- or underestimate risks. The proposed approach consists of the following steps: (i) estimate flood risk (i.e., probability-loss distributions) on the basin level using available techniques (in this work marginal losses are simulated using the LISFLOOD hydrological model and an economic damage model(25,26) ); (ii) choose a copula type to upscale to the multi-regional level (in this article, we show that the Flipped Clayton copula (Equation (10)) is a good fit); (iii) estimate copula parameters for each pair of basins (i.e., receive matrix (17) of the parameters); (iv) apply minimax ordering (Algorithm 2) to choose basins that most influence total dependency; (v) apply the ordered copula technique (Equation (12)) in order to receive the final loss distribution that describes the total flood risk at the large-scale level. Our approach is beneficial as it provides information on the risk of extreme losses at higher scales, highly important for determining suitable risk management strategies. Risk information may provide guidance on appropriate size of emergency funding programs that provide financial assistance to member states and/or neighborhood countries and aim to preserve financial stability (such as European

2117 Financial Stability Fund, European Solidarity Fund, Macro-Financial Assistance, etc.). The proposed copula approach allows to estimate up to which return period the existing funds can cover the losses or how large the funds should be in order to cover losses up to a certain return period. The proposed approach can be easily extended from Romania to the panEuropean level. Furthermore, the extension to other hazards is possible as long as appropriate assumptions, data, and boundaries apply. Future research should include risk analysis for multi-risk situations as well as strategies for recovery after multiple disasters. ACKNOWLEDGMENTS This research was funded by the European Commission through the ENHANCE project under the Seventh Framework Programme of the European Union (grant agreement number 308438). We are grateful to the Joint Research Centre (JRC) for supplying data on flood losses based on its LISFLOOD hydrological model. We also thank anonymous reviewers who helped in improving the article. APPENDIX We give some more examples to illustrate each step of the proposed ordering algorithms above. Example A1 (Minimax ordering). Consider basins A, B, C, D that are connected to each other by the following structure (the number on the edge refers to θi, j ):

According to the Algorithm 2 our first selection is to couple basins A and B, as soon as θ A,B = 10 is the maximal one among others. After this, notice that θ A,C = 5 and θ A,D = 2. To guarantee the dependency with all basins, we choose the minimal one; hence θ A,D = 2. Now, we do the same with θ B,C = 1 and θ B,D = 4 and we choose θ B,C = 1 as

2118 the minimal one, too. Among the θ A,D = 2 and θ B,C = 1 we choose the maximal one, i.e., θ A,D = 2. Hence, the next entry for the ordered coupling is basin D and, therefore, we have a triplet now A, B, D. As soon as in this simple example there are only four basins, the next one is, obviously, C. Hence, the order for coupling is A, B, D, C. Example A2 (Maximin ordering). Consider basins A, B, C, D, that are connected to each other by the following structure (the number on the edge refers to θi, j ):

According to the Algorithm 2 our first selection is to couple basins C and D, as soon as dC,D = 0 is the minimal distance among others. After this, notice that dC,B = 2 and dC,A = 5. To guarantee the dependency with all basins, we choose the maximal distance; hence dA,C = 5. Now, we do the same with dB,D = 4 and dA,D = 2 and we choose dB,D = 4 as the maximal one, too. Among the dA,C = 5 and dB,D = 4 we choose the minimal one, i.e., dB,D = 4. Hence, the next entry for the ordered coupling is basin B and, therefore, we have a triplet now C, D, B. As soon as in this simple example there are only four basins, the next one is, obviously, A. Hence, the order for coupling is C, D, B, A. REFERENCES 1. Munich Re. Topics: Natural catastrophes 2012. Analysis, assessments, positions. Munich Reinsurance, Munich, 2013. ¨ 2. Bouwer LM, Crompton RP, Faust E, Hoppe P, Pielke Jr. RA. Confronting disaster losses. Science, 2007; 318: 753. 3. Bouwer LM. Projections of future extreme weather losses under changes in climate and exposure. Risk Analysis, 2013; 33:915–930. DOI: 10.1111/j.1539-6924.2012.01880.x. 4. IPCC. Summary for policymakers. Managing the risks of extreme events and disasters to advance climate change adaptation. Pp. 1–19 in Field CB, Barros V, Stocker TF, Qin D, Dokken DJ, Ebi KL, Mastrandrea MD, Mach KJ, Plattner GK, Allen SK, Tignor M, Midgley PM (eds). A Special Report of Working Groups I and II of the Intergovernmental Panel on Climate Change. Cambridge, UK, and New York, USA: Cambridge University Press, 2012.

Timonina et al. 5. Kundzewicz ZW, Kanae S, Seneviratne SI, Handmer J, Nicholls N, Peduzzi P, Mechler R, Bouwer LM, Arnell N, Mach K, Muir-Wood R, Brakenridge GR, Kron W, Benito G, Honda Y, Takahashi K, Sherstyukov B. Flood risk and climate change — Global and regional perspectives. Hydrological Sciences Journal, 2014; 59: 1–28. 6. Michel-Kerjan E, Hochrainer-Stigler S, Kunreuther H, Linnerooth-Bayer J, Mechler R, Muir-Wood R, Ranger N, Vaziri P, Young M. Catastrophe risk models for evaluating disaster risk reduction investments in developing countries. Risk Analysis, 2013; 33(6): 984–999. 7. Aerts JCJH, Lin N, Botzen W, Emanuel K, deMoel H. Lowprobability flood risk modeling for New York City. Risk Analysis, 2013; 33:772–788. DOI: 10.1111/risa.12008. 8. Iman RL, Johnson ME, Watson ChC. Sensitivity analysis for computer model projections of hurricane losses. Risk Analysis, 2005; 25: 1277–1297. 9. Iman RL, Johnson ME, Watson ChC. Uncertainty analysis for computer model projections of hurricane losses. Risk Analysis, 2005; 25: 1299–1312. 10. EEA. Mapping the impacts of recent natural disasters and technological accidents in Europe. Environmental Issue Report, 2003; 35. 11. Jongman B, Hochrainer-Stigler S, Feyen L, Aerts JCJH, Mechler R, Botzen WJ, Bouwer LM, Pflug G, Rojas R, Ward PJ. Increasing stress on disaster-risk finance due to large floods. Nature Climate Change, 2014. DOI: 10.1038/NCLIMATE2124. 12. Mechler R, Hochrainer S, Aaheim A, Kundzewicz Z, Lugeri N, Moriondo M. A risk management approach for assessing adaptation to changing flood and drought risks in Europe. Pp. 200–229 in Hulme M, Neufeldt H (eds). Making Climate Change Work for Us: European Perspectives on Adaptation and Mitigation Strategies. Cambridge: Cambridge University Press, 2009. ´ 13. Feyen L, Dankers R, Bodis K, Salamon P, Barredo JI. Fluvial flood risk in Europe in present and future climates. Climatic Change, 2012; 112(1): 47–62. 14. Barredo JI. Normalised flood losses in Europe: 1970–2006. Natural Hazards & Earth System Sciences, 2009; 9(1):97–104. DOI:10.5194/nhess-9-97-2009. 15. Hirabayashi Y, Mahendran R, Koirala S, Konoshima L, Yamazaki D, Watanabe S, Kim H, Kanae S. Global flood risk under climate change. Nature Climate Change, 2013; 3: 816– 821. 16. Jongman B, Ward PJ, Aerts JCJH. Global exposure to river and coastal flooding: Long term trends and changes. Global Environmental Change, 2012; 22: 823–835. 17. Rojas R, Feyen L, Watkiss P. Climate change and river floods in the European Union: Socio-economic consequences and the costs and benefits of adaptation. Global Environmental Change; 23(6):1737–1751. DOI:10.1016/j.gloenvcha.2013.08.006. 18. Ward PJ, Jongman B, Weiland FS, Bouwman A, van Beek R, Bierkens MFP, Ligtvoet W, Winsemius HC. Assessing flood risk at the global scale: Model setup, results, and sensitivity. Environmental Research Letters, 2013; 8: 044019. 19. Mechler R, Bouwer LM, Linnerooth-Bayer J, HochrainerStigler S, Aerts JCJH, Surminski S, Williges K. Managing unnatural disaster risk from climate extremes. Nature Climate Change, 2014; 4(4):235–237. ¨ 20. Choro´s-Tomczyk B, Hardle W, Okhrin O. Valuation of collateralized debt obligations with hierarchical Archimedean copulae. Empirical Finance, 2013; 24:42–62. ¨ 21. Choro´s-Tomczyk B, Hardle W, Overback L. Copula dynamics in CDOs. Quantitative Finance, 2013; 14(9):1573–1585. DOI: 10.1080/14697688.2013.847280. ¨ 22. Jaworski P, Durante F, Hardle W (eds). Copulae in mathematical and quantitative finance. Proceedings of the Workshop

Structured Coupling of Probability Loss Distributions

23.

24.

25.

26.

27.

28.

Held in Cracow, 10–12 July 2012. Lecture Notes in Statistics, 2013:294. DOI: 10.1007/978-3-642-35407-6. ¨ Jaworski P, Durante F, Hardle W, Rychlik T (eds). Copula theory and its applications. Proceedings of the Workshop Held in Warsaw, 25–26 September 2009. Lecture Notes in Statistics, 198:327. Hochrainer-Stigler S, Lugeri N, Radziejewski M. Up-scaling of impact dependent loss distributions: A hybrid-convolution approach. Natural Hazards, 2014; 70(2):1437–1451. DOI: 10.1007/s11069-013-0885-6. van der Knijff JM, Younis J, De Roo APJ. LISFLOOD: A GIS-based distributed model for river-basin scale water balance and flood simulation. International Journal of Geographical Information Science, 2010; 24(2): 189–212. Rojas R, Feyen L, Bianchi A, Dosio A. Assessment of future flood hazard in Europe using a large ensemble of bias-corrected regional climate simulations. Journal of Geophysical Research: Atmospheres, 2012; 117(17):2156–2202. DOI: 10.1029/2012JD017461. Beven K, Hall J. Applied Uncertainty Analysis for Flood Risk Management. UK: World Scientific Publishing, 2014. CRED. EM-DAT: International Disaster Database, Centre for Research on the Epidemiology of Disasters. Belgium: Universite´ Catholique de Louvain, 2014.

2119 29. Hora SC. A note on the aggregation of event probabilities. Risk Analysis, 2013; 33:909–914. DOI: 10.1111/j.15396924.2012.01889.x. 30. Salvadori G, Michele CD, Kottegoda NT, Rosso R. Extremes in Nature. An Approach Using Copulas. New York: Springer, 2007. 31. Nelsen R. An Introduction to Copulas. New York: Springer, 2006. 32. Durante F, Hofert M, Scherer M. Multivariate hierarchical copulas with shocks. Methodology and Computing in Applied Probability, 2010; 12(4):681–694. 33. Aas K, Czado C, Frigessi A, Bakke H. Pair-copula constructions of multiple dependence. Insurance, Mathematics and Economics, 2009; 44: 182–198. 34. Bedford T, Cooke RM. Vines—A new graphical model for dependent random variables. Annals of Statistics, 2002; 30(4): 1031–1068. 35. Kurowicka D, Cooke RM. Uncertainty Analysis with High Dimensional Dependence Modelling. Chichester: Wiley, 2006. 36. Kurowicka D, Joe H. Dependence Modeling—Handbook on Vine Copulae. Singapore: World Scientific Publishing Co., 2011. 37. Pollner J. Financial and Fiscal Instruments for Catastrophe Risk Management: Addressing the Losses from Flood Hazards in Central Europe. World Bank Studies, July 2012.