Selection of Probability Distributions in Characterizing Risk of Extreme Events

Use of probability distributions by regulatory agencies often focuses on the extreme events and scenarios that correspond to the tail of probability distributions. This paper makes the case that assessment of the tail of the distribution can and often should be performed separately from assessment of the central values. Factors to consider when developing distributions that account for tail behavior include (a) the availability of data, (b) characteristics of the tail of the distribution, and (c) the value of additional information in assessment. The integration of these elements will improve the modeling of extreme events by the tail of distributions, thereby providing policy makers with critical information on the risk of extreme events. Two examples provide insight into the theme of the paper. The first demonstrates the need for a parallel analysis that separates the extreme events from the central values. The second shows a link between the selection of the tail distribution and a decision criterion. In addition, the phenomenon of breaking records in time-series data gives insight to the information that characterizes extreme values. One methodology for treating risk of extreme events explicitly adopts the conditional expected value as a measure of risk. Theoretical results concerning this measure are given to clarify some of the concepts of the risk of extreme events.     

Flood Protection Diversification to Reduce Probabilities of Extreme Losses

Recent catastrophic losses because of floods require developing resilient approaches to flood risk protection. This article assesses how diversification of a system of coastal protections might decrease the probabilities of extreme flood losses. The study compares the performance of portfolios each consisting of four types of flood protection assets in a large region of dike rings. A parametric analysis suggests conditions in which diversifications of the types of included flood protection assets decrease extreme flood losses. Increased return periods of extreme losses are associated with portfolios where the asset types have low correlations of economic risk. The effort highlights the importance of understanding correlations across asset types in planning for large‐scale flood protection. It allows explicit integration of climate change scenarios in developing flood mitigation strategy.     

Generalized Quantification of Risk Associated with Extreme Events1

In the partitioned multiobjective risk method (PMRM) the probability axis is typically partitioned into three regimes: high-exceedance low-consequence, intermediate-exceedance intermediate-consequence, and low-exceedance high-consequence (LE/HC). For each regime, the PMRM generates a conditional expected risk-function given that the damage lies within the regime. The theme of this paper is the conditional expected-risk function for the LE/HC regime. This function, denoted by f₄(.), captures the behavior of the “extreme events” of an underlying decision-making problem. The PMRM offers two advantages: (a) it isolates LE/HC events, allowing the decision-maker(s) to focus on the impacts of catastrophies; and (b) it generates more valuable information than that obtained from the common unconditional expected-risk function. Theoretical problems may arise from uncertainty about the behavior of the tail of the risk curve describing the underlying frequency of damages. When the number of physical observations in small (e.g., in flood frequency analysis), the analyst is forced to make assumptions about the density of damages. Each succeeding distributional assumption will generate a different value of f₄(.). An added dimension of difficulty is also created by the sensitivity of f₄(.) to the choice of the boundary of the LE/HC regime. This paper has two overall objectives: (a) to present distribution-free results concerning the magnitude of f₄(.); and (b) to use those results to obtain a distribution-free estimate of the sensitivity of f₄(.) to the choice of the boundary of the LE/HC regime. The above objectives are realized by extending, and further developing, existing inequalities for continuously distributed random variables.     

Spatial Risk Analysis of Power Systems Resilience During Extreme Events

The increased frequency of extreme events in recent years highlights the emerging need for the development of methods that could contribute to the mitigation of the impact of such events on critical infrastructures, as well as boost their resilience against them. This article proposes an online spatial risk analysis capable of providing an indication of the evolving risk of power systems regions subject to extreme events. A Severity Risk Index (SRI) with the support of real‐time monitoring assesses the impact of the extreme events on the power system resilience, with application to the effect of windstorms on transmission networks. The index considers the spatial and temporal evolution of the extreme event, system operating conditions, and the degraded system performance during the event. SRI is based on probabilistic risk by condensing the probability and impact of possible failure scenarios while the event is spatially moving across a power system. Due to the large number of possible failures during an extreme event, a scenario generation and reduction algorithm is applied in order to reduce the computation time. SRI provides the operator with a probabilistic assessment that could lead to effective resilience‐based decisions for risk mitigation. The IEEE 24‐bus Reliability Test System has been used to demonstrate the effectiveness of the proposed online risk analysis, which was embedded in a sequential Monte Carlo simulation for capturing the spatiotemporal effects of extreme events and evaluating the effectiveness of the proposed method.     

Risk of Extreme Events in Multiobjective Decision Trees Part 2. Rare Events

Earlier work with decision trees identified nonseparability as an obstacle to minimizing the conditional expected value, a measure of the risk of extreme events, by the well-known method of averaging out and folding back. This second of two companion papers addresses the conditional expected value that is defined as the expected outcome assuming that a random variable is observed only in the upper 100 (1 −α) percent of potential outcomes, where α is a cumulative probability preselected by the decision maker. An approach is proposed to overcome the need to evaluate all policies in order to identify the optimal policy. The approach is based in part on approximating the conditional expected value by using statistics of extremes. An existing convenient approximation of the conditional expected value is shown to be separable into two constituent elements of risk and can thus be optimized, along with other objectives including the unconditional expected value of the outcome, in a multiobjective decision tree. An example of sequential decision making for remediation or environmental contamination is provided. The importance of the results for risk analyis beyond the minimization of conditional expected values is pointed out.     

Risk of Extreme Events in Multiobjective Decision Trees Part 1. Severe Events

Earlier work with decision trees identified nonseparability as an obstacle to minimizing the conditional expected value, a measure of the risk of extreme events, by the well-known method of averaging out and folding back. This first of two companion papers addresses the conditional expected value that is defined as the expected outcome assuming the exceedance of a threshold β, where β is preselected by the decision maker. An approach is proposed to overcome the need to evaluate all policies in order to identify the optimal policy. The approach is based on the insight that the conditional expected value is separable into two constituent elements of risk and can thus be optimized along with other objectives, including the unconditional expected value of the outcome, by using a multiobjective decision tree. An example of sequential decision making for improving highway capacity is given.     

Geographic Distributions of Extreme Weather Risk Perceptions in the United States

Weather and climate disasters pose an increasing risk to life and property in the United States. Managing this risk requires objective information about the nature of the threat and subjective information about how people perceive it. Meteorologists and climatologists have a relatively firm grasp of the historical objective risk. For example, we know which parts of the United States are most likely to experience drought, heat waves, flooding, snow or ice storms, tornadoes, and hurricanes. We know less about the geographic distribution of the perceived risks of meteorological events and trends. Do subjective perceptions align with exposure to weather risks? This question is difficult to answer because analysts have yet to develop a comprehensive and spatially consistent methodology for measuring risk perceptions across geographic areas in the United States. In this project, we propose a methodology that uses multilevel regression and poststratification to estimate extreme weather and climate risk perceptions by geographic area (i.e., region, state, forecast area, and county). Then we apply the methodology using data from three national surveys (n = 9,542). This enables us to measure, map, and compare perceptions of risk from multiple weather hazards in geographic areas across the country.     

A Survey of Approaches for Assessing and Managing the Risk of Extremes

In this paper, we review methods for assessing and managing the risk of extreme events, where extreme events are defined to be rare, severe, and outside the normal range of experience of the system in question. First, we discuss several systematic approaches for identifying possible extreme events. We then discuss some issues related to risk assessment of extreme events, including what type of output is needed (e.g., a single probability vs. a probability distribution), and alternatives to the probabilistic approach. Next, we present a number of probabilistic methods. These include: guidelines for eliciting informative probability distributions from experts; maximum entropy distributions; extreme value theory; other approaches for constructing prior distributions (such as reference or noninformative priors); the use of modeling and decomposition to estimate the probability (or distribution) of interest; and bounding methods. Finally, we briefly discuss several approaches for managing the risk of extreme events, and conclude with recommendations and directions for future research.     

Projections of Future Extreme Weather Losses Under Changes in Climate and Exposure

Many attempts are made to assess future changes in extreme weather events due to anthropogenic climate change, but few studies have estimated the potential change in economic losses from such events. Projecting losses is more complex as it requires insight into the change in the weather hazard but also into exposure and vulnerability of assets. This article discusses the issues involved as well as a framework for projecting future losses, and provides an overview of some state‐of‐the‐art projections. Estimates of changes in losses from cyclones and floods are given, and particular attention is paid to the different approaches and assumptions. All projections show increases in extreme weather losses due to climate change. Flood losses are generally projected to increase more rapidly than losses from tropical and extra‐tropical cyclones. However, for the period until the year 2040, the contribution from increasing exposure and value of capital at risk to future losses is likely to be equal or larger than the contribution from anthropogenic climate change. Given the fact that the occurrence of loss events also varies over time due to natural climate variability, the signal from anthropogenic climate change is likely to be lost among the other causes for changes in risk, at least during the period until 2040. More efforts are needed to arrive at a comprehensive approach that includes quantification of changes in hazard, exposure, and vulnerability, as well as adaptation effects.     

System of Systems Engineering and Risk Management of Extreme Events: Concepts and Case Study

The domain of risk analysis is expanded to consider strategic interactions among multiple participants in the management of extreme risk in a system of systems. These risks are fraught with complexity, ambiguity, and uncertainty, which pose challenges in how participants perceive, understand, and manage risk of extreme events. In the case of extreme events affecting a system of systems, cause‐and‐effect relationships among initiating events and losses may be difficult to ascertain due to interactions of multiple systems and participants (complexity). Moreover, selection of threats, hazards, and consequences on which to focus may be unclear or contentious to participants within multiple interacting systems (ambiguity). Finally, all types of risk, by definition, involve potential losses due to uncertain events (uncertainty). Therefore, risk analysis of extreme events affecting a system of systems should address complex, ambiguous, and uncertain aspects of extreme risk. To accomplish this, a system of systems engineering methodology for risk analysis is proposed as a general approach to address extreme risk in a system of systems. Our contribution is an integrative and adaptive systems methodology to analyze risk such that strategic interactions among multiple participants are considered. A practical application of the system of systems engineering methodology is demonstrated in part by a case study of a maritime infrastructure system of systems interface, namely, the Straits of Malacca and Singapore.     

Impact Factors and Risk Analysis of Tropical Cyclones on a Highway Network

Coastal areas typically have high social and economic development and are likely to suffer huge losses due to tropical cyclones. These cyclones have a great impact on the transportation network, but there have been a limited number of studies about tropical‐cyclone‐induced transportation network functional damages, especially in Asia. This study develops an innovative measurement and analytical tool for highway network functional damage and risk in the context of a tropical cyclone, with which we explored the critical spatial characteristics of tropical cyclones with regard to functional damage to a highway network by developing linear regression models to quantify their relationship. Furthermore, we assessed the network's functional risk and calculated the return periods under different damage levels. In our analyses, we consider the real‐world highway network of Hainan province, China. Our results illustrate that the most important spatial characteristics were location (in particular, the midlands), travel distance, landfalling status, and origin coordinates. However, the trajectory direction did not obviously affect the results. Our analyses indicate that the highway network of Hainan province may suffer from a 90% functional damage scenario every 4.28 years. These results have critical policy implications for the transport sector in reference to emergency planning and disaster reduction.     

A Hybrid Methodology for Modeling Risk of Adverse Events in Complex Health‐Care Settings

In spite of increased attention to quality and efforts to provide safe medical care, adverse events (AEs) are still frequent in clinical practice. Reports from various sources indicate that a substantial number of hospitalized patients suffer treatment‐caused injuries while in the hospital. While risk cannot be entirely eliminated from health‐care activities, an important goal is to develop effective and durable mitigation strategies to render the system “safer.” In order to do this, though, we must develop models that comprehensively and realistically characterize the risk. In the health‐care domain, this can be extremely challenging due to the wide variability in the way that health‐care processes and interventions are executed and also due to the dynamic nature of risk in this particular domain. In this study, we have developed a generic methodology for evaluating dynamic changes in AE risk in acute care hospitals as a function of organizational and nonorganizational factors, using a combination of modeling formalisms. First, a system dynamics (SD) framework is used to demonstrate how organizational‐level and policy‐level contributions to risk evolve over time, and how policies and decisions may affect the general system‐level contribution to AE risk. It also captures the feedback of organizational factors and decisions over time and the nonlinearities in these feedback effects. SD is a popular approach to understanding the behavior of complex social and economic systems. It is a simulation‐based, differential equation modeling tool that is widely used in situations where the formal model is complex and an analytical solution is very difficult to obtain. Second, a Bayesian belief network (BBN) framework is used to represent patient‐level factors and also physician‐level decisions and factors in the management of an individual patient, which contribute to the risk of hospital‐acquired AE. BBNs are networks of probabilities that can capture probabilistic relations between variables and contain historical information about their relationship, and are powerful tools for modeling causes and effects in many domains. The model is intended to support hospital decisions with regard to staffing, length of stay, and investments in safety, which evolve dynamically over time. The methodology has been applied in modeling the two types of common AEs: pressure ulcers and vascular‐catheter‐associated infection, and the models have been validated with eight years of clinical data and use of expert opinion.     

Spatially Representing Vulnerability to Extreme Rain Events Using Midwestern Farmers’ Objective and Perceived Attributes of Adaptive Capacity

Potential climate‐change‐related impacts to agriculture in the upper Midwest pose serious economic and ecological risks to the U.S. and the global economy. On a local level, farmers are at the forefront of responding to the impacts of climate change. Hence, it is important to understand how farmers and their farm operations may be more or less vulnerable to changes in the climate. A vulnerability index is a tool commonly used by researchers and practitioners to represent the geographical distribution of vulnerability in response to global change. Most vulnerability assessments measure objective adaptive capacity using secondary data collected by governmental agencies. However, other scholarship on human behavior has noted that sociocultural and cognitive factors, such as risk perceptions and perceived capacity, are consequential for modulating people's actual vulnerability. Thus, traditional assessments can potentially overlook people's subjective perceptions of changes in climate and extreme weather events and the extent to which people feel prepared to take necessary steps to cope with and respond to the negative effects of climate change. This article addresses this knowledge gap by: (1) incorporating perceived adaptive capacity into a vulnerability assessment; (2) using spatial smoothing to aggregate individual‐level vulnerabilities to the county level; and (3) evaluating the relationships among different dimensions of adaptive capacity to examine whether perceived capacity should be integrated into vulnerability assessments. The result suggests that vulnerability assessments that rely only on objective measures might miss important sociocognitive dimensions of capacity. Vulnerability indices and maps presented in this article can inform engagement strategies for improving environmental sustainability in the region.     

A Method for Computing the Fraction of Attributable Risk Related to Climate Damages

The recent decision of the U.S. Supreme Court on the regulation of CO₂ emissions from new motor vehicles^{( 1 )} shows the need for a robust methodology to evaluate the fraction of attributable risk from such emissions. The methodology must enable decisionmakers to reach practically relevant conclusions on the basis of expert assessments the decisionmakers see as an expression of research in progress, rather than as knowledge consolidated beyond any reasonable doubt.^{( 2,3,4 )} This article presents such a methodology and demonstrates its use for the Alpine heat wave of 2003. In a Bayesian setting, different expert assessments on temperature trends and volatility can be formalized as probability distributions, with initial weights (priors) attached to them. By Bayesian learning, these weights can be adjusted in the light of data. The fraction of heat wave risk attributable to anthropogenic climate change can then be computed from the posterior distribution. We show that very different priors consistently lead to the result that anthropogenic climate change has contributed more than 90% to the probability of the Alpine summer heat wave in 2003. The present method can be extended to a wide range of applications where conclusions must be drawn from divergent assessments under uncertainty.     

A Probabilistic Framework for Risk Analysis of Widespread Flood Events: A Proof‐of‐Concept Study

This article presents a flood risk analysis model that considers the spatially heterogeneous nature of flood events. The basic concept of this approach is to generate a large sample of flood events that can be regarded as temporal extrapolation of flood events. These are combined with cumulative flood impact indicators, such as building damages, to finally derive time series of damages for risk estimation. Therefore, a multivariate modeling procedure that is able to take into account the spatial characteristics of flooding, the regionalization method top‐kriging, and three different impact indicators are combined in a model chain. Eventually, the expected annual flood impact (e.g., expected annual damages) and the flood impact associated with a low probability of occurrence are determined for a study area. The risk model has the potential to augment the understanding of flood risk in a region and thereby contribute to enhanced risk management of, for example, risk analysts and policymakers or insurance companies. The modeling framework was successfully applied in a proof‐of‐concept exercise in Vorarlberg (Austria). The results of the case study show that risk analysis has to be based on spatially heterogeneous flood events in order to estimate flood risk adequately.     

Robust Climate Policies Under Uncertainty: A Comparison of Robust Decision Making and Info‐Gap Methods

This study compares two widely used approaches for robustness analysis of decision problems: the info‐gap method originally developed by Ben‐Haim and the robust decision making (RDM) approach originally developed by Lempert, Popper, and Bankes. The study uses each approach to evaluate alternative paths for climate‐altering greenhouse gas emissions given the potential for nonlinear threshold responses in the climate system, significant uncertainty about such a threshold response and a variety of other key parameters, as well as the ability to learn about any threshold responses over time. Info‐gap and RDM share many similarities. Both represent uncertainty as sets of multiple plausible futures, and both seek to identify robust strategies whose performance is insensitive to uncertainties. Yet they also exhibit important differences, as they arrange their analyses in different orders, treat losses and gains in different ways, and take different approaches to imprecise probabilistic information. The study finds that the two approaches reach similar but not identical policy recommendations and that their differing attributes raise important questions about their appropriate roles in decision support applications. The comparison not only improves understanding of these specific methods, it also suggests some broader insights into robustness approaches and a framework for comparing them.     

Managing the Risk of Uncertain Threshold Responses: Comparison of Robust, Optimum, and Precautionary Approaches

Many commentators have suggested the need for new decision analysis approaches to better manage systems with deeply uncertain, poorly characterized risks. Most notably, policy challenges such as abrupt climate change involve potential nonlinear or threshold responses where both the triggering level and subsequent system response are poorly understood. This study uses a simple computer simulation model to compare several alternative frameworks for decision making under uncertainty -- optimal expected utility, the precautionary principle, and three different approaches to robust decision making -- for addressing the challenge of adding pollution to a lake without triggering unwanted and potentially irreversible eutrophication. The three robust decision approaches -- trading some optimal performance for less sensitivity to assumptions, satisficing over a wide range of futures, and keeping options open -- are found to identify similar strategies as the most robust choice. This study also suggests that these robust decision approaches offer a quantitative, decision analytic framework that captures the spirit of the precautionary principle while addressing some of its shortcomings. Finally, this study finds that robust strategies may be preferable to optimum strategies when the uncertainty is sufficiently deep and the set of alternative policy options is sufficiently rich.     

Roy’s Safety‐First Portfolio Principle in Financial Risk Management of Disastrous Events

Roy pioneers the concept and practice of risk management of disastrous events via his safety‐first principle for portfolio selection. More specifically, his safety‐first principle advocates an optimal portfolio strategy generated from minimizing the disaster probability, while subject to the budget constraint and the mean constraint that the expected final wealth is not less than a preselected disaster level. This article studies the dynamic safety‐first principle in continuous time and its application in asset and liability management. We reveal that the distortion resulting from dropping the mean constraint, as a common practice to approximate the original Roy’s setting, either leads to a trivial case or changes the problem nature completely to a target‐reaching problem, which produces a highly leveraged trading strategy. Recognizing the ill‐posed nature of the corresponding Lagrangian method when retaining the mean constraint, we invoke a wisdom observed from a limited funding‐level regulation of pension funds and modify the original safety‐first formulation accordingly by imposing an upper bound on the funding level. This model revision enables us to solve completely the safety‐first asset‐liability problem by a martingale approach and to derive an optimal policy that follows faithfully the spirit of the safety‐first principle and demonstrates a prominent nature of fighting for the best and preventing disaster from happening.     

Socioeconomic Vulnerability and Adaptation to Environmental Risk: A Case Study of Climate Change and Flooding in Bangladesh

In this article we investigate the complex relationship between environmental risk, poverty, and vulnerability in a case study carried out in one of the poorest and most flood-prone countries in the world, focusing on household and community vulnerability and adaptive coping mechanisms. Based upon the steadily growing amount of literature in this field we develop and test our own analytical model. In a large-scale household survey carried out in southeast Bangladesh, we ask almost 700 floodplain residents living without any flood protection along the River Meghna about their flood risk exposure, flood problems, flood damage, and coping mechanisms. Novel in our study is the explicit testing of the effectiveness of adaptive coping strategies to reduce flood damage costs. We show that, households with lower income and less access to productive natural assets face higher exposure to risk of flooding. Disparity in income and asset distribution at community level furthermore tends to be higher at higher risk exposure levels, implying that individually vulnerable households are also collectively more vulnerable. Regarding the identification of coping mechanisms to deal with flood events, we look at both the ex ante household level preparedness for flood events and the ex post availability of community-level support and disaster relief. We find somewhat paradoxically that the people that face the highest risk of flooding are the least well prepared, both in terms of household-level ex ante preparedness and community-level ex post flood relief.     

Current Constraints and Future Directions for Risk Adapted Land-Use Planning Practices in the High-Density Asian Setting of Ho Chi Minh City

The need to integrate adaptation efforts into land-use planning policies has been only recently recognized in Ho Chi Minh City. The city's latest planning guidance addresses both flooding resilience and mitigation of urban heat. This paper outlines the development contexts and the current barriers for adapted land-use planning within the city. The key challenge for land-use planning is communicating the important functions and services of open and natural urban spaces and effectively guiding the mainly individual developer-driven development. As the realization of non-structural adaptation measures is in strong contrast to the current market-driven private and short-term developer interests, the main development trajectories are questioned and synergies identified.