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Aspen Opinion

Hurricanes and Randomness

May 27, 2015

With no major hurricane making U.S. landfall in the last nine years, what's the likelihood of such a break occurring by chance?

Lack of landfalls

The U.S. has experienced a "hurricane drought" for the last nine years. No major hurricane (Saffir-Simpson category 3 or higher) has made landfall over the U.S. since hurricane Wilma in 2005 and no hurricane of category 1 or higher has made landfall over Florida. This break in landfalling major hurricanes is the longest on record, as documented in the National Hurricane Center's HURDAT database1,2 (Figure 1). Despite this U.S. landfalling dearth, hurricane activity in the wider Atlantic basin has not been unusually low (except for 2013 - Figure 2), nor has the Caribbean been spared from landfalls.

Figure 1: Waiting time between major hurricanes with wind speeds of Saffir-Simpson category 3 or higher over the U.S. The final bar represents the time between the last event (Wilma in 2005) and 1st June 2015. Note that this plot only includes hurricanes that made landfall at intensity category 3 or higher and excludes major hurricanes that had a reduced intensity before landfall.

Source: Aspen Re Research and Development, HURDAT

Figure 2: Annual number of major hurricanes in the Atlantic basin 1900-2014.

Source: Aspen Re Research and Development, HURDAT

Chance: the Individual and the Many

The question naturally arises as to what is the likelihood of such a break occurring purely by chance. The issue can be viewed by characterizing the arrival times of major hurricanes over the U.S. as a random process. Randomness is an unusual property, as it implies that while individual events are by definition unpredictable, the outcome or distribution of many events is known. A series of coin tosses is the simplest example, where individual tosses are unpredictable, but with the expectation that after many tosses half would be heads.

Homogenous Process

The field of statistics provides a rich variety of tools to address such randomness that defines the world of natural hazards, with the Poisson distribution and Generalized Pareto distribution commonly adopted. Occurrence of such events can be characterized as a homogeneous Poisson process, where the arrival of storms are independent events but with a fixed average rate of arrival. The independence criterion means that the arrival time of an event has no relationship to any other events. With 55 storms in 115 years, it is the same as picking 55 random dates from the 115-year period. Independence also means that, despite having already waited nine years for a major hurricane, the chance of observing one this year is the same as any other year and independent of the waiting time.

A Poisson process can be analysed in several ways: by the number of events per time interval (e.g. years); by the times of the individual events; or by the waiting times between the events. It derives its name from the fact that the probability distribution of the numbers of events per time interval is a Poisson distribution, a distribution formulated in the mid-19th Century by the pioneering French mathematician of the same name.3 The Poisson distribution is defined only by a single parameter λ, which is the average rate of arrival of events (Equation 1). There were 55 landfalling major hurricanes over the 115-year period 1900-2014, giving a value of λ of 0.48 events per year, or an expected return period of 2.1 years between events (Figure 3).

equation 1

Equation 1: For a Poisson process, the probability of observing k hurricanes in a year is defined by the Poisson distribution. The single parameter λ is the average number of hurricanes per year.

Figure 3: Probability of having various numbers of major hurricanes per year with intensity at least category 3 over the U.S. The observations match what is expected from the Poisson distribution, which is also shown along with its 95% confidence interval.

Source: Aspen Re Research and Development, HURDAT

Exponential Distribution

Just as the Poisson process can reveal the probability of having a certain number of events per year, it can also inform the waiting time between events. Again this is just a function of the single parameter λ. In this case, waiting times are contaminated by the fact that historical landfalling major hurricanes have only occurred between the months of June and October. A better match to the statistical distribution that characterises Poisson process waiting times (called the exponential distribution) is obtained when all months outside of the hurricane months are removed. In other words, the waiting time from, for example, 31 October 2010 to 1 June 2011 is (treated as) only one day. From this statistical modelling of the observed waiting times using the exponential distribution, we can get the return period of a nine year wait directly. A nine year wait is 1377 days including only the hurricane months, which has a return period of about 86 years using the Poisson process model.

Generalized Pareto Distribution

This analysis of waiting times used the Poisson process model to extrapolate far into the tail, but unfortunately this is not the most appropriate tool for analysing such extremes. Extreme value statistics, in particular the Generalized Pareto distribution (GPD), is often used by natural hazard specialists to construct a better statistical model. The distribution of the extreme arrival times should converge with the GPD distribution if one assumes that hurricane waiting times arise from a single distribution.

We applied GPD theory to calculate the return period of waiting times for landfalling major hurricanes. For this analysis we included an additional fictitious event on 1 June 2015, since the recent long break is important to consider when modelling the extremes. In other words, we are answering the question "given that we have seen a break of nine years, what is its return period?" The resulting plot of the return period of waiting times between events (Figure 4) shows that the return period of a nine-year waiting time is 282 years, but with large uncertainty ranging from 82 years to greater than 10,000 years.

Figure 4

Figure 4: Return period of waiting times for landfalling hurricanes with category 3 or higher wind speeds over the U.S. historical events (black points) also include a fictitious event on June 1st 2015 (horizontal dashed line and longest waiting time of the black points). A Generalised Pareto distribution fit to the waiting times (solid line) shows the expected return period of a nine-year break, along with its 95% confidence interval (shading).

Source: Aspen Re Research and Development, HURDAT

Model Complexity

Statistical modelling of historical events is the simplest method in the analysis of natural hazards and should be the benchmark for any probabilistic models. Increasing complexity of models might bring additional benefits in understanding, but will also likely involve making further assumptions about the underlying physical processes. There is evidence to suggest that hurricane counts vary on inter-annual to inter-decadal time scales and it might be prudent, therefore, to increase the sophistication of the model by allowing the rate of arrival of storms λ to vary over time. Such a framework is provided by the inhomogeneous Poisson process models.

Alternatively, a more physically-based statistical model might employ sea surface temperature (SST) variability, one of the key drivers of hurricane variability, to inform the model. Such a model has been utilized in a recent study4 to determine a return period of the recent hurricane drought, by using observed SSTs since 1950 to create many realisations of synthetic hurricane tracks. This model resulted in a return period of a nine-year break in major hurricane landfalls of 177 years; less than the 282 year estimate in our analysis above but well within the range of our estimated uncertainty.

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References

  1. Jarvinen, B.R., Neumann, C.J. & Davis, M.A.S., 1984. A Tropical Cyclone Data Tape for the North Atlantic Basin, 1886:1983: Contents, limitations and uses. NOAA Technical Memorandum NWS NHC 22., Coral Gables, U.S.
  2. Landsea, C.W. et al., 2004. The Atlantic hurricane database re-analysis project: Documentation for the 1851-1910 alterations and additions to the HURDAT database. In R. J. Murnane & K. Liu, eds Hurricanes and Typhoons: Past, Present, and Future. New York: Columbia University Press, pp. 177–221.
  3. Poisson, S., 1837. Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés, Paris, France: Bachelier.
  4. Hall, T. & Hereid, K., 2015. The frequency and duration of U.S. hurricane droughts. Geophysical Research Letters, 42, pp.1–4.

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