• Elsner, J. B., and A. B. Kara, 1999: Hurricanes of the North Atlantic:Climate and Society. Oxford University Press, 512 pp.

  • Fernandez-Partagas, J., and H. F. Diaz, 1996: Atlantic hurricanes in the second half of the nineteenth century. Bull. Amer. Meteor. Soc.,77, 2899–2906.

  • Landsea, C. W., R. A. Pielke, A. M. Mestas-Nunez, and J. A. Knaff, 1999: Atlantic basin hurricanes: Indices of climatic changes. Climatic Change,42, 89–129.

  • McCullagh, P., and J. A. Nelder, 1989: Generalized Linear Models. Chapman and Hall, 511 pp.

  • Neumann, C. J., B. R. Jarvinen, C. J. McAdie, and J. D. Elms, 1993: Tropical Cyclones of the North Atlantic Ocean, 1871–1992. National Climatic Data Center and the National Hurricane Center, 193 pp.

  • Silvey, S. D., 1975: Statistical Inference. Chapman and Hall, 220 pp.

  • Solow, A. R., 1989: Reconstructing a partially observed record of tropical cyclone counts. J. Climate,2, 1253–1257.

  • View in gallery

    Time series of total observed (solid) and landfalling (dashed) hurricane counts, 1930–98. The former time series is incomplete over the period 1930–45

  • View in gallery

    Estimated trend in mean annual basinwide hurricane counts, 1930–98. Also shown is a reconstruction of the observed counts for 1930–45 and the actual observed counts for 1946–98

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Testing for a Trend in a Partially Incomplete Hurricane Record

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  • 1 Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
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Abstract

The record of annual counts of basinwide North Atlantic hurricanes is incomplete prior to 1946. This has restricted efforts to identify a long-term trend in hurricane activity to the postwar period. In contrast, the complete record of U.S. landfalling hurricanes extends back to 1930 or earlier. Under the assumption that the proportion of basinwide hurricanes that make landfall is constant over time, it is possible to use the record of landfalling hurricanes to extend a test for trend in basinwide hurricane activity beyond the postwar period. This note describes and illustrates a method for doing this. The results suggest that there has been a significant reduction in basinwide hurricane activity over the period 1930–98.

Corresponding author address: Andrew R. Solow, Woods Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: asolow@whoi.edu

Abstract

The record of annual counts of basinwide North Atlantic hurricanes is incomplete prior to 1946. This has restricted efforts to identify a long-term trend in hurricane activity to the postwar period. In contrast, the complete record of U.S. landfalling hurricanes extends back to 1930 or earlier. Under the assumption that the proportion of basinwide hurricanes that make landfall is constant over time, it is possible to use the record of landfalling hurricanes to extend a test for trend in basinwide hurricane activity beyond the postwar period. This note describes and illustrates a method for doing this. The results suggest that there has been a significant reduction in basinwide hurricane activity over the period 1930–98.

Corresponding author address: Andrew R. Solow, Woods Hole Oceanographic Institution, Woods Hole, MA 02543.

Email: asolow@whoi.edu

1. Introduction

The analysis of the historical record of basinwide North Atlantic hurricane activity is limited by the unreliability of the record prior to the advent of regular aircraft reconnaissance in 1946 (Neumann et al. 1993; Elsner and Kara 1999). For example, in testing for a trend in the basinwide frequency of North Atlantic hurricanes, Landsea et al. (1999) confined the analysis to the postwar period. In contrast to the record of basinwide hurricane activity, by virtue of their conspicuousness, the record of landfalling hurricanes in the United States is believed to be complete back to 1930 or beyond. Here, a landfalling hurricane is defined as any tropical storm making at least one landfall on the United States at category 1 or higher. As shown below, the proportion of basinwide hurricanes that made landfall has remained relatively stable since 1946. Under the assumption that this stability extends back beyond 1946, this raises the possibility of using the earlier record of landfalling hurricanes to extend the test for trend in basinwide hurricane activity as well. The basic idea is that, in conjunction with the complete record of landfalling hurricanes, knowledge of this proportion provides information about the number of basinwide hurricanes in the incomplete part of the record. In this note, this idea is formalized to test for a trend in the number of basinwide North Atlantic hurricanes over the period 1930–98. The same basic idea was used informally by Fernandez-Partagas and Diaz (1996) to estimate overall North Atlantic hurricane activity for the period 1851–1900. In contrast to results for the postwar period, the analysis presented here suggests that there has been a decline in activity over the 1930–98 period.

2. Model and method

Consider the observation period t = 1, 2, . . . , n, and let the random variable Yt be the true basinwide number of hurricanes in year t. We assume that Yt has a Poisson distribution with mean:
μtμ0μ1t
where μ0 and μ1 are unknown. Interest centers on testing the null hypothesis H0: μ1 = 0 that there is no trend in the mean number of basinwide hurricanes against the general alternative hypothesis H1: μ1 ≠ 0 that there is a trend of this form.

Let the random variable Xt be the number of landfalling hurricanes in year t. We will assume that the conditional distribution of Xt given Yt = yt is binomial with parameters yt and p, where the unknown probability p that a hurricane makes landfall is assumed to be constant over the observation period. This assumption is critical to the method described here and some evidence to support it is given in the following section.

Finally, let the random variable Zt be the observed number of nonlandfalling hurricanes in year t. We will assume that the conditional distribution of Zt given Yt = yt and Xt = xt is also binomial with parameters ytxt and
i1520-0442-13-20-3696-e2
where q is unknown. Under this model, the probability that a nonlandfalling hurricane is observed is q in the first m years of the record, while in the last nm years of the record, this probability rises to 1. Thus, Xt + ZtYt for t = 1, 2, . . . , m, but Xt + Zt = Yt for t = m + 1, m + 2, . . . , n.
It is straightforward to show that, under the model outlined above, the joint probability mass function of Xt and Zt is
i1520-0442-13-20-3696-e3
The product of the first three terms in this summation represents the conditional binomial probability that Zt = z given Xt = x and Yt = y. The product of the next three terms represents the conditional binomial probability that Xt = x given Yt = y. The product of the final three terms represents the Poisson probability that Yt = y. The lower bound of the summation reflects the fact that, under the model considered here, the total number of hurricanes cannot be less than the observed number (i.e., YtXt + Zt). Note that, for t > m, (3) reduces to
i1520-0442-13-20-3696-e4
The likelihood function for the full set of observations is given by
i1520-0442-13-20-3696-e5
where xt and zt are the observed values of Xt and Zt, respectively. The maximum likelihood (ML) estimates μ̂0, μ̂1, p̂, and of the unknown parameters μ0, μ1, p, and q are found by maximizing the likelihood or its logarithm. This maximization must be done numerically. A natural statistic for testing H0 is the likelihood ratio (LR) statistic:
logL(μ̂0, μ̂1, p̂, q̂) − logL[μ̂0(0), 0, (0), (0)]},
where generally μ̂0(μ), (μ), and (μ) are the ML estimates of μ0, p, and q under the restricted model with μ1 = μ. Under H0, the LR statistic Λ has an approximate chi-squared distribution with 1 degree of freedom (Silvey 1975).
If H0 is rejected, then it may be of interest to construct a confidence interval for the trend parameter μ1. An approximate 1 − α confidence interval for μ1 is given by the set of values of μ satisfying
2[logL(μ̂0, μ̂1, p̂, q̂) − logL(μ̂0(μ), μ, (μ), (μ)] < χ21(α),
where χ21(α) is the upper α-quantile of the chi-squared distribution with 1 degree of freedom. Confidence intervals for the other parameters can be constructed in the same way.

3. Results

Figure 1 shows the time series of the total number Xt + Zt of recorded hurricanes and the number Xt of landfalling hurricanes for the n = 69-yr period 1930–98. We will treat the former time series as incomplete over the m = 16-yr period 1930–45 and as complete thereafter. Over the period 1930–45, a total of 76 hurricanes were recorded, for an annual average of 4.75. Of these, 51 were landfalling, for an annual average of 3.19. The proportion of hurricanes in the record that were landfalling over this period was 0.67. In contrast, over the period 1946–98, a total of 310 hurricanes were recorded, for an annual average of 5.85. Of these, 116 were landfalling, for an annual average of 2.19. The proportion of hurricanes that were landfalling over this period was 0.37.

Before applying the test described in the previous section, for the purposes of comparison, we tested H0 using the last nm = 53 yr of the record of basinwide hurricane counts. The approximate significance level was 0.57, so H0 could not be rejected by conventional notions of significance. This is consistent with earlier analyses (Landsea et al. 1999; Elsner and Kara 1999).Let Ŷt be the fitted value of Yt for t = m + 1, m + 2, . . . , n. As a check of the Poisson assumption, we calculated the Poisson deviance:
i1520-0442-13-20-3696-e8
for the fitted model. Under the Poisson assumption, DP has an approximate chi-squared distribution with nm − 2 degrees of freedom (McCullagh and Nelder 1989). In this case, the value of DP was 47.3 with 51 degrees of freedom for an approximate significance level of 0.65, so that the Poisson assumption is reasonable.

Turning to the methods described above, the ML estimates under the full model were μ̂0 = 2.00, μ̂1 = −0.005, = 0.39, = 0.36, while the ML estimates under the restricted model were μ̂0(0) = 1.80, (0) = 0.40, (0) = 0.43. The value of the LR statistic Λ was 2.90 with an approximate significance level of 0.09, which is at least marginally significant. The approximate 0.90 confidence interval for μ1 constructed as outlined in the previous section is (−0.009, −0.001).

In Fig. 2, the estimate of μ(t) under the full model is shown along with a rough reconstruction of basinwide hurricane counts given by
i1520-0442-13-20-3696-e9
The estimate of μ(t) declines from around 7.4 hurricanes in 1930 to around 5.4 hurricanes in 1998. It is important to emphasize that the goal of the analysis presented here is to test for a trend in annual mean hurricane activity and not to reconstruct the incomplete part of the hurricane record. The reconstructed values are shown in Fig. 2 solely as a graphical aid. A rough standard error for t for tm is [(tXt)(1 − )/]1/2. The average of this standard error for the reconstructed counts in Fig. 2 is around 2.6.
As noted, a key assumption underlying the methods described in this note is that the probability p that a hurricane makes landfall is constant over the entire observation period. Due to the incompleteness of the early part of the record, it is not possible to test this assumption. However, it is possible to test it over the period for which the records are complete. Under the assumption that the landfall probability p is constant and conditional on the observed values of Ym+1, Ym+2, . . . , Yn, the binomial deviance
i1520-0442-13-20-3696-e10
has an approximate chi-squared distribution with nm − 1 degrees of freedom, where t = cYt, t = (1 − c)Yt, and
i1520-0442-13-20-3696-e11
is the ML estimate of p based on the complete part of the record (McCullagh and Nelder 1989). For the data used here, c = 0.37 and DB = 43.33 with approximate significance level 0.80, suggesting that the assumption of constant p is reasonable for the period 1946–98.

4. Discussion

In considering the results of the previous section, it is worth revisiting the underlying logic of the method described in section 2. To begin with, we assumed that the landfall probability p is constant. This assumption appears to be reasonable for the period 1946–98. It then follows from the fact that the annual counts of landfalling hurricanes are greater on average during 1930–45 than during 1946–98 that the true annual counts of basinwide hurricanes were also greater on average during the earlier period. We have stressed the importance to this result of the assumption that p is constant. A second assumption is that the probability q of observing a nonlandfalling hurricane is also constant during this earlier period. It would be possible to relax this assumption. For example, in analyzing a partially incomplete record of tropical storms in the Australian region, Solow (1989) assumed that the sighting probability increased monotonically to 1 according to a simple parametric model. As there is no clear trend in the ratio Xt/(Xt + Zt) between 1930 and 1945, as would be expected if there were a trend in q, this additional complication does not seem justified here. However, such an extension would probably be necessary to extend this analysis back beyond 1930.

Acknowledgments

The helpful comments of two anonymous reviewers are acknowledged with gratitude. Partial support for this work was provided by the Woods Hole Oceanographic Institution as a Mellon Award to Laura Moore.

REFERENCES

  • Elsner, J. B., and A. B. Kara, 1999: Hurricanes of the North Atlantic:Climate and Society. Oxford University Press, 512 pp.

  • Fernandez-Partagas, J., and H. F. Diaz, 1996: Atlantic hurricanes in the second half of the nineteenth century. Bull. Amer. Meteor. Soc.,77, 2899–2906.

  • Landsea, C. W., R. A. Pielke, A. M. Mestas-Nunez, and J. A. Knaff, 1999: Atlantic basin hurricanes: Indices of climatic changes. Climatic Change,42, 89–129.

  • McCullagh, P., and J. A. Nelder, 1989: Generalized Linear Models. Chapman and Hall, 511 pp.

  • Neumann, C. J., B. R. Jarvinen, C. J. McAdie, and J. D. Elms, 1993: Tropical Cyclones of the North Atlantic Ocean, 1871–1992. National Climatic Data Center and the National Hurricane Center, 193 pp.

  • Silvey, S. D., 1975: Statistical Inference. Chapman and Hall, 220 pp.

  • Solow, A. R., 1989: Reconstructing a partially observed record of tropical cyclone counts. J. Climate,2, 1253–1257.

Fig. 1.
Fig. 1.

Time series of total observed (solid) and landfalling (dashed) hurricane counts, 1930–98. The former time series is incomplete over the period 1930–45

Citation: Journal of Climate 13, 20; 10.1175/1520-0442(2000)013<3696:TFATIA>2.0.CO;2

Fig. 2.
Fig. 2.

Estimated trend in mean annual basinwide hurricane counts, 1930–98. Also shown is a reconstruction of the observed counts for 1930–45 and the actual observed counts for 1946–98

Citation: Journal of Climate 13, 20; 10.1175/1520-0442(2000)013<3696:TFATIA>2.0.CO;2

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