Uncertainty is inherent in the quantification of risk, and risk in relation to an extreme event in the natural environment is no different. In this blog post, I will consider why now might be a good time to re-consider how we quantify the risk from environmental extreme events.

Extreme events refer to episodes which are extremely rare but have potentially damaging or even catastrophic consequences. Examples of such extreme events include fluvial and coastal flooding, heat waves and air pollution episodes. These types of event are notoriously difficult to predict due to both their relative rarity and the, often complex, chains of pre-cursor events which generate the extreme event itself.

Extreme value analysis is a class of statistical models which can be used to describe the stochastic behaviour of the extreme events in a series of observations (Coles, 2001; Gilleland, 2013). In the context of prediction of extreme values, the advantages of an extreme value model over, say, a regression model are that (i) only the most relevant information from the full set of data is needed and (ii) the information found in the largest observations is not diluted by the (vast majority of) observations which are consistent with the average, or expected, behaviour. The subsequent model fits can be used to extrapolate far beyond the observation period, estimating events that might occur only once every one or two centuries.

These models are well-established in some areas of environmental science, for example hydrology (Stedinger, 1983; Hosking et al, 1985; Katz et al, 2002) and climate change (Parey et al, 2007; Brown et al, 2008; Cooley, 2009; Burke et al, 2010). In other areas, such as glaciology and space weather (Thomson et al, 2011; Rogers et al, 2020), their potential is still being uncovered; indeed, the deployment and development of extreme value models for novel application areas is one of the objectives of DSNE.

Risk of extreme events is most often summarised using return levels. A return period of 100 years is associated with a 100-year return level, which is defined to be the value of the process that is exceeded, on average, once in every 100 years. An alternative way to interpret the return level is to consider the annual exceedance probability, that is, the probability of exceeding the level in a given year. In the case of the 100-year return level, the AEP would be 1/100, or 0.01.

Return levels are used by the insurance industry when setting premiums on property insurance, engineers when constructing flood defences and designing transport infrastructure (e.g. bridges), housing developers and urban planners, and both local and national government for contingency planning. The widespread and diverse use of return levels means that it is vital to have a robust methodology with which to estimate them.

Currently, return levels are predicted directly from the statistical extreme value models described above. Often a frequency curve or return level plot will be produced to show how return levels increase with increasing return period (equivalently with a decrease in the AEP). As a measure of risk, the return level is not perfect:

1. It is very easy to misinterpret: the 100-year return level is not the value that is exceeded once every 100 years, it is the level exceeded on average once every 100 years. It could be exceeded two or more years in a row, or even twice in one year.

2. The return level is based on a model that assumes that the behaviour of extreme events is homogenous over time, i.e. that the frequency and magnitude of these events are not changing over time, and that events do not occur in clusters.

3. It does not directly reflect the uncertainty that follows from calibrating a statistical model using a data set which represents a short time period relative to the time horizon for which predictions of risk are required.

So, what can, and should, statisticians, data scientists and their colleagues in science, engineering and risk management do to both improve risk prediction and deliver more user-friendly measures of risk?

Of critical importance is the development of realistic statistical models that incorporate changes in the behaviour of the physical process(es) under investigation. This is especially important in the context of climate-change since it is plausible that not only the model assumptions, but also the data, are not representative of future periods. Some efforts have been made in this direction already (e.g. Eastoe and Tawn, 2009; Cooley, 2013; Cheng et al, 2014; Mentaschi et al, 2016) but there is no consensus on which (if any) of the many different approaches is `best’.

It is not just models that can go awry... development of guidelines to check how representative historical data are of current and future behaviour will help to protect against predictions that are based on samples that do not reflect the current state of the climate and/or environment.

And, if things were not already complex enough, it isn’t just data and models that can cause confusion. Popular risk metrics may also no longer be fit for purpose. As science moves further towards a “changing world” scenario, we need measures of risk that capture the extra uncertainty generated by inter-year variability, decadal (or longer) cycles and/or long-term trends. This leads us to a possible conflict between easy-to-interpret single-value metrics, such as the return level, and more plausible but harder-to-interpret conditional metrics. Conditional metrics express risk as a function of time and/or underlying driving processes (e.g. wave height as a function of wind speed and direction). Whilst such conditional metrics might be more scientifically reasonable, they are harder to interpret. Given a different return level for each combination of wind speed and wind direction, the engineer is then faced with the additional challenge of deciding against which of these values they should design their offshore structure to be protected.

This brings us, perhaps most importantly, to communication. If nothing else, it is vital that those who have experience with extreme value data sets and models engage in a dialogue with scientists, policy makes and the general public about how and why risk can change over time, why it is necessary for our estimates of risk to reflect this change and how these metrics can be interpreted.

Brown, S.J., Caesar, J. and Ferro, C.A., 2008. Global changes in extreme daily temperature since 1950. Journal of Geophysical Research: Atmospheres, 113(D5).

Burke, E.J., Perry, R.H. and Brown, S.J., 2010. An extreme value analysis of UK drought and projections of change in the future. Journal of Hydrology, 388(1-2), pp.131-143.

Cooley, D., 2009. Extreme value analysis and the study of climate change. Climatic change, 97(1), pp.77-83.

Cooley, D., 2013. Return periods and return levels under climate change. In Extremes in a changing climate (pp. 97-114). Springer, Dordrecht.

Cheng, L., AghaKouchak, A., Gilleland, E. and Katz, R.W., 2014. Non-stationary extreme value analysis in a changing climate. Climatic change, 127(2), pp.353-369.

Eastoe, E.F. and Tawn, J.A., 2009. Modelling non‐stationary extremes with application to surface level ozone. Journal of the Royal Statistical Society: Series C (Applied Statistics), 58(1), pp.25-45.

Mentaschi, L., Vousdoukas, M., Voukouvalas, E., Sartini, L., Feyen, L., Besio, G. and Alfieri, L., 2016. The transformed-stationary approach: a generic and simplified methodology for non-stationary extreme value analysis. Hydrology and Earth System Sciences, 20(9), pp.3527-3547.

Gilleland, E., Ribatet, M. and Stephenson, A.G., 2013. A software review for extreme value analysis. Extremes, 16(1), pp.103-119.

Hosking, J.R.M., Wallis, J.R. and Wood, E.F., 1985. An appraisal of the regional flood frequency procedure in the UK Flood Studies Report. Hydrological Sciences Journal, 30(1), pp.85-109

Katz, R.W., Parlange, M.B. and Naveau, P., 2002. Statistics of extremes in hydrology. Advances in water resources, 25(8-12), pp.1287-1304.

Parey, S., Malek, F., Laurent, C. and Dacunha-Castelle, D., 2007. Trends and climate evolution: Statistical approach for very high temperatures in France. Climatic Change, 81(3), pp.331-352.

Rogers, N.C., Wild, J.A., Eastoe, E.F., Gjerloev, J.W. and Thomson, A.W., 2020. A global climatological model of extreme geomagnetic field fluctuations. Journal of Space Weather and Space Climate, 10, p.5.

Stedinger, J.R., 1983. Estimating a regional flood frequency distribution. Water Resources Research, 19(2), pp.503-510.

Thomson, A.W., Dawson, E.B. and Reay, S.J., 2011. Quantifying extreme behavior in geomagnetic activity. Space Weather, 9(10).

Related Blogs

• 

  Annual meeting of the Society for Decision Making under Deep Uncertainty, 2022

• 

  The Cryosphere 2022 Conference

• 

  DSNE Reseachers receive best poster award at CDE Conference 2022

Disclaimer

The opinions expressed by our bloggers and those providing comments are personal, and may not necessarily reflect the opinions of Lancaster University. Responsibility for the accuracy of any of the information contained within blog posts belongs to the blogger.