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.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

Moore, L.

Solow, A.R.

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3696 VOLUME 13J O U R N A L O F C L I M A T Eq 2000 American Meteorological SocietyNOTES AND CORRESPONDENCETesting for a Trend in a Partially Incomplete Hurricane RecordANDREW R. SOLOW AND LAURA MOOREWoods Hole Oceanographic Institution, Woods Hole, Massachusetts12 October 1999 and 26 April 2000ABSTRACTThe record of annual counts of basinwide North Atlantic hurricanes is incomplete prior to 1946. This hasrestricted efforts to identify a long-term trend in hurricane activity to the postwar period. In contrast, the completerecord of U.S. landfalling hurricanes extends back to 1930 or earlier. Under the assumption that the proportionof basinwide hurricanes that make landfall is constant over time, it is possible to use the record of landfallinghurricanes to extend a test for trend in basinwide hurricane activity beyond the postwar period. This note describesand illustrates a method for doing this. The results suggest that there has been a significant reduction in basinwidehurricane activity over the period 1930–98.1. IntroductionThe analysis of the historical record of basinwideNorth Atlantic hurricane activity is limited by the un-reliability of the record prior to the advent of regularaircraft reconnaissance in 1946 (Neumann et al. 1993;Elsner and Kara 1999). For example, in testing for atrend in the basinwide frequency of North Atlantic hur-ricanes, Landsea et al. (1999) confined the analysis tothe postwar period. In contrast to the record of basin-wide hurricane activity, by virtue of their conspicuous-ness, the record of landfalling hurricanes in the UnitedStates is believed to be complete back to 1930 or be-yond. Here, a landfalling hurricane is defined as anytropical storm making at least one landfall on the UnitedStates at category 1 or higher. As shown below, theproportion of basinwide hurricanes that made landfallhas remained relatively stable since 1946. Under theassumption that this stability extends back beyond 1946,this raises the possibility of using the earlier record oflandfalling hurricanes to extend the test for trend inbasinwide hurricane activity as well. The basic idea isthat, in conjunction with the complete record of land-falling hurricanes, knowledge of this proportion pro-vides information about the number of basinwide hur-ricanes in the incomplete part of the record. In this note,this idea is formalized to test for a trend in the numberof basinwide North Atlantic hurricanes over the period1930–98. The same basic idea was used informally byCorresponding author address: Andrew R. Solow, Woods HoleOceanographic Institution, Woods Hole, MA 02543.E-mail: asolow@whoi.eduFernandez-Partagas and Diaz (1996) to estimate overallNorth Atlantic hurricane activity for the period 1851–1900. In contrast to results for the postwar period, theanalysis presented here suggests that there has been adecline in activity over the 1930–98 period.2. Model and methodConsider the observation period t 5 1, 2, . . . , n, andlet the random variable Yt be the true basinwide numberof hurricanes in year t. We assume that Yt has a Poissondistribution with mean:mt 5 exp(m0 1 m1t), (1)where m0 and m1 are unknown. Interest centers on testingthe null hypothesis H0: m1 5 0 that there is no trend inthe mean number of basinwide hurricanes against thegeneral alternative hypothesis H1: m1 ± 0 that there isa trend of this form.Let the random variable Xt be the number of land-falling hurricanes in year t. We will assume that theconditional distribution of Xt given Yt 5 yt is binomialwith parameters yt and p, where the unknown probabilityp that a hurricane makes landfall is assumed to be con-stant over the observation period. This assumption iscritical to the method described here and some evidenceto support it is given in the following section.Finally, let the random variable Zt be the observednumber of nonlandfalling hurricanes in year t. We willassume that the conditional distribution of Zt given Yt5 yt and Xt 5 xt is also binomial with parameters yt 2xt and15 OCTOBER 2000 3697N O T E S A N D C O R R E S P O N D E N C EFIG. 1. Time series of total observed (solid) and landfalling(dashed) hurricane counts, 1930–98. The former time series is in-complete over the period 1930–45.q t 5 1, 2, . . . , mq 5 (2)t 51 t 5 m 1 1, m 1 2, . . . , n,where q is unknown. Under this model, the probabilitythat a nonlandfalling hurricane is observed is q in thefirst m years of the record, while in the last n 2 m yearsof the record, this probability rises to 1. Thus, Xt 1 Zt# Yt for t 5 1, 2, . . . , m, but Xt 1 Zt 5 Yt for t 5 m1 1, m 1 2, . . . , n.It is straightforward to show that, under the modeloutlined above, the joint probability mass function ofXt and Zt is` y 2 x yz y2x2z x y2x yf (x, z) 5 q (1 2 q ) p (1 2 p) mOt t t t1 2 1 2z xy5x1z3 exp(2m )/y!t (3)The product of the first three terms in this summationrepresents the conditional binomial probability that Zt5 z given Xt 5 x and Yt 5 y. The product of the nextthree terms represents the conditional binomial proba-bility that Xt 5 x given Yt 5 y. The product of the finalthree terms represents the Poisson probability that Yt 5y. The lower bound of the summation reflects the factthat, under the model considered here, the total numberof hurricanes cannot be less than the observed number(i.e., Yt $ Xt 1 Zt). Note that, for t . m, (3) reducestox 1 zx z x1zf (x, z) 5 p (1 2 p) m exp(2m )/(x 1 z)! (4)t t t1 2xThe likelihood function for the full set of observationsis given bynL(m , m , p, q) 5 f (x , z ), (5)P0 1 t t tt51where xt and zt are the observed values of Xt and Zt,respectively. The maximum likelihood (ML) estimates, , p̂, and q̂ of the unknown parameters m0, m1, p,m̂ m̂0 1and q are found by maximizing the likelihood or itslogarithm. This maximization must be done numerical-ly. A natural statistic for testing H0 is the likelihoodratio (LR) statistic:L 5 2{logL(m̂ , m̂ , p̂, q̂)0 12 logL[m̂ (0), 0, p̂(0), q̂(0)]},0 (6)where generally , p̂(m), and q̂(m) are the ML es-m̂ (m)0timates of m0, p, and q under the restricted model withm1 5 m. Under H0, the LR statistic L has an approximatechi-squared distribution with 1 degree of freedom (Sil-vey 1975).If H0 is rejected, then it may be of interest to constructa confidence interval for the trend parameter m1. Anapproximate 1 2 a confidence interval for m1 is givenby the set of values of m satisfying2[logL(m̂ , m̂ , p̂, q̂) 2 logL(m̂ (m), m, p̂(m), q̂(m)]0 1 02, x (a),1 (7)where (a) is the upper a-quantile of the chi-squared2x1distribution with 1 degree of freedom. Confidence in-tervals for the other parameters can be constructed inthe same way.3. ResultsFigure 1 shows the time series of the total number Xt1 Zt of recorded hurricanes and the number Xt of land-falling hurricanes for the n 5 69-yr period 1930–98.We will treat the former time series as incomplete overthe m 5 16-yr period 1930–45 and as complete there-after. Over the period 1930–45, a total of 76 hurricaneswere recorded, for an annual average of 4.75. Of these,51 were landfalling, for an annual average of 3.19. Theproportion of hurricanes in the record that were land-falling over this period was 0.67. In contrast, over theperiod 1946–98, a total of 310 hurricanes were recorded,for an annual average of 5.85. Of these, 116 were land-falling, for an annual average of 2.19. The proportionof hurricanes that were landfalling over this period was0.37.Before applying the test described in the previoussection, for the purposes of comparison, we tested H0using the last n 2 m 5 53 yr of the record of basinwidehurricane counts. The approximate significance levelwas 0.57, so H0 could not be rejected by conventionalnotions of significance. This is consistent with earlieranalyses (Landsea et al. 1999; Elsner and Kara 1999).Let Ŷt be the fitted value of Yt for t 5 m 1 1, m 1 2,. . . , n. As a check of the Poisson assumption, we cal-culated the Poisson deviance:nˆD 5 2 Y log(Y /Y ) (8)OP t t tt5m11for the fitted model. Under the Poisson assumption, DP3698 VOLUME 13J O U R N A L O F C L I M A T EFIG. 2. Estimated trend in mean annual basinwide hurricane counts,1930–98. Also shown is a reconstruction of the observed counts for1930–45 and the actual observed counts for 1946–98.has an approximate chi-squared distribution with n 2m 2 2 degrees of freedom (McCullagh and Nelder1989). In this case, the value of DP was 47.3 with 51degrees of freedom for an approximate significance lev-el of 0.65, so that the Poisson assumption is reasonable.Turning to the methods described above, the ML es-timates under the full model were 5 2.00, 5m̂ m̂0 120.005, p̂ 5 0.39, q̂ 5 0.36, while the ML estimatesunder the restricted model were 5 1.80, p̂(0) 5m̂ (0)00.40, q̂(0) 5 0.43. The value of the LR statistic L was2.90 with an approximate significance level of 0.09,which is at least marginally significant. The approximate0.90 confidence interval for m1 constructed as outlinedin the previous section is (20.009, 20.001).In Fig. 2, the estimate of m(t) under the full modelis shown along with a rough reconstruction of basinwidehurricane counts given byX 1 (Z /q̂) t 5 1, 2, . . . , mt tỸ 5t 5X 1 Z t 5 m 1 1, m 1 2, . . . , n.t t(9)The estimate of m(t) declines from around 7.4 hurricanesin 1930 to around 5.4 hurricanes in 1998. It is importantto emphasize that the goal of the analysis presented hereis to test for a trend in annual mean hurricane activityand not to reconstruct the incomplete part of the hur-ricane record. The reconstructed values are shown inFig. 2 solely as a graphical aid. A rough standard errorfor Ỹt for t # m is [(Ỹt 2 Xt)(1 2 q̂)/q̂]1/2. The averageof this standard error for the reconstructed counts inFig. 2 is around 2.6.As noted, a key assumption underlying the methodsdescribed in this note is that the probability p that ahurricane makes landfall is constant over the entire ob-servation period. Due to the incompleteness of the earlypart of the record, it is not possible to test this as-sumption. However, it is possible to test it over theperiod for which the records are complete. Under theassumption that the landfall probability p is constantand conditional on the observed values of Ym11, Ym12,. . . , Yn, the binomial deviancenˆ ˆD 5 2 X log(X /X ) 1 Z log(Z /Z ) (10)OB t t t t t tt5m11has an approximate chi-squared distribution with n 2m 2 1 degrees of freedom, where X̂t 5 p̂cYt, Ẑt 5 (12 p̂c)Yt, andn np̂ 5 X Y (11)O Oc t t@t5m11 t5m11is the ML estimate of p based on the complete part ofthe record (McCullagh and Nelder 1989). For the dataused here, p̂c 5 0.37 and DB 5 43.33 with approximatesignificance level 0.80, suggesting that the assumptionof constant p is reasonable for the period 1946–98.4. DiscussionIn considering the results of the previous section, itis worth revisiting the underlying logic of the methoddescribed in section 2. To begin with, we assumed thatthe landfall probability p is constant. This assumptionappears to be reasonable for the period 1946–98. It thenfollows from the fact that the annual counts of land-falling hurricanes are greater on average during 1930–45 than during 1946–98 that the true annual counts ofbasinwide hurricanes were also greater on average dur-ing the earlier period. We have stressed the importanceto this result of the assumption that p is constant. Asecond assumption is that the probability q of observinga nonlandfalling hurricane is also constant during thisearlier period. It would be possible to relax this as-sumption. For example, in analyzing a partially incom-plete record of tropical storms in the Australian region,Solow (1989) assumed that the sighting probability in-creased monotonically to 1 according to a simple para-metric model. As there is no clear trend in the ratio Xt/(Xt 1 Zt) between 1930 and 1945, as would be expectedif there were a trend in q, this additional complicationdoes not seem justified here. However, such an extensionwould probably be necessary to extend this analysisback beyond 1930.Acknowledgments. The helpful comments of twoanonymous reviewers are acknowledged with gratitude.Partial support for this work was provided by the WoodsHole Oceanographic Institution as a Mellon Award toLaura Moore.REFERENCESElsner, 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 inthe second half of the nineteenth century. Bull. Amer. Meteor.Soc., 77, 2899–2906.15 OCTOBER 2000 3699N O T E S A N D C O R R E S P O N D E N C ELandsea, 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. Na-tional 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 oftropical cyclone counts. J. Climate, 2, 1253–1257.

Testing for a trend in a partially incomplete hurricane record

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Testing for a trend in a partially incomplete hurricane record

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