Non Stationarity of Hurricane Number Time Series

As a method for predicting future hurricane numbers, the long baseline method is only appropriate if the landfalling hurricane number record is stationary, or at least close to stationary. A large number of studies have looked at the stationary of hurricane number records. With respect to hurricane numbers in the basin, it has been clearly shown that the record is not stationary. For instance, Elsner et al. (2001) detected the presence of statistically significant change-points in the cat 3-5 basin hurricane number series in the years 1942/43, 1964/65 and 1994/95. Using a different methodology the same authors repeated their analysis (Elsner et al., 2004) and showed the presence of statistically significant change-points in the years 1905/ 06, 1942/43 and 1994/95. In our own change-point analysis, which uses a different methodology yet again, we detected the existence of statistically significant change-points in the cat 1-5 basin hurricane number series in the years 1931/32, 1947/48, 1969/70 and 1994/95 (Jewson and Penzer, 2006). In fact, some of these change-points can be seen rather clearly by eye in the data itself, especially those in the cat 3-5 series at the end of the 1960s and in the mid 1990s: see Fig. 2. By contrast, for landfalling numbers, it is much harder to detect change-points. Elsner et al. (2004) couldn't find any, and neither could we (Jewson and Penzer, 2006). In a different kind of analysis, we did find statistically significant evidence of autocorrelation in the landfall time-series (see Khare and Jewson, 2005a, b), although the signal was rather weak.

Why is it that the basin hurricane number time-series shows such significant change-points, while the landfalling time-series does not? There are two obvious limiting-case explanations: (a) the interannual probability1 of hurricanes making landfall is constant from year to year. In this case, the landfalling number time-series would be expected to inherit properties of the basin hurricane number time-series, such as change-points. The apparent lack of change points in the landfalling

1 By 'interannual probability of landfall' we mean the probability of landfall, estimated a year before the beginning of the hurricane season. Such an estimate is unconditional with respect to ENSO state, since ENSO is not predictable that far in advance.

a Time (years)

b Time (years)

Fig. 2 Examples of Change-Points in Hurricane Numbers: (a) Levels between RMS change-points for category 1-5 hurricanes. (b) Levels between Eisner's 2004 change-points for category 3-5 hurricanes. The vertical dark grey bars indicate the number of storms in the Atlantic basin for each year and the overlaid light gray bars indicate the number of those storms which hit the U.S. coastline b Time (years)

Fig. 2 Examples of Change-Points in Hurricane Numbers: (a) Levels between RMS change-points for category 1-5 hurricanes. (b) Levels between Eisner's 2004 change-points for category 3-5 hurricanes. The vertical dark grey bars indicate the number of storms in the Atlantic basin for each year and the overlaid light gray bars indicate the number of those storms which hit the U.S. coastline time series must then be explained by the hypothesis that they are obscured by noise for statistical reasons, or (b) the interannual probability of hurricanes making landfall is not constant from year to year, and varies in such a way as to cancel the effect of the basin changes, meaning that the landfall numbers are actually stationary. Reality may be in-between these two limiting cases.

Given the importance of this question for the prediction of future numbers of landfalling hurricanes, we have investigated this in some detail. In Nzerem et al. (2006) we asked the question: if the basin series contains real change-points of the same size as the observed 1994/1995 change-point, what is the probability of detecting that change-point in the basin data and in the landfall data? Given an assumption of constant probability of landfall for each storm, it turns out that the change-point would usually be detectable in the basin data, but would usually not be detectable in the landfall data. The explanation is purely statistical: in going from basin to landfall, the number of storms reduces by a factor of four. This reduces the signal-to-noise ratio by a factor of two, and makes it twice as hard to detect any change-point. The observed change-points are of a size that this reduction in signal-to-noise is just enough to hide them in the variability of the landfall data. This analysis shows that explanation (a) above is plausible, and that, based on this result, we should not necessarily be surprised that we can't detect change-points in the landfall series, and the fact that we cannot should not lead us to conclude that they are not there. Approaching the same question from a different perspective we looked at the proportion of basin hurricanes that make landfall during active and inactive historical periods (Bellone et al., 2007). The results are very clear. The proportion of storms making landfall prior to 1948 is different to the proportion making landfall after 1948. This is most likely explained by poor observations of basin storms before that time. However, after 1948, during the period of reliable observations, the proportion of storms making landfall cannot be shown to vary (i.e. we cannot reject the null hypothesis that the proportion is constant). Another way to interpret this result is that if the proportion of storms making landfall does vary (and it very likely does vary at some level) then the size of those variations are too small to detect, and would be too small to estimate effectively and use in predictions. In Bellone et al. it is also shown that this result holds even without consideration of the change-point locations. Finally, in Hall and Jewson (2008), we studied the historical behaviour of hurricanes in years with warm and cold tropical North Atlantic sea-surface temperatures (SSTs). We find that although the number of hurricanes, their genesis sites and the characteristics of their propagation all depend on SST in statistically significant ways, the overall proportion of Atlantic storms making landfall along the U.S. coast does not.

Based on these studies, we conclude that it is reasonable to build hurricane prediction models that use the assumption that the probability of storms making landfall over the next five years is well predicted using the proportion that have made landfall since 1950. We will call this the 'constant landfall probability' model (CLP), although the model does not, strictly speaking, assume landfall probabilities are constant, but just that the proportion varies sufficiently little that we cannot properly estimate the variations and we are better off modelling it as constant. The estimate of the proportion based on this assumption is shown in Fig. 3. Note that the long-baseline model is essentially based on the opposite set of assumptions: that the proportion of storms making landfall varies in such a way as to cancel out the non-stationarity in the basin so that the mean number making landfall is approximately constant. We include predictions from both sets of assumptions in our model set, for completeness, although we believe that the data slightly favour the CLP model, as discussed above. The CLP model has various interesting implications, as we discuss below.

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