1. Introduction and motivation
In the past decades, intense coastal hurricanes have represented an increasing threat to communities and economies in the United States. Direct losses, indeed, have increased in an exponential manner, and a few dramatic events, especially Hurricanes Andrew in 1992 and Katrina in 2005, have had unprecedented consequences, both in economic and social terms.
Recently, hurricane seasons have been very active, and the 2005 season remains the most active season in recorded history, with 28 tropical cyclones, 15 of them reaching the hurricane level, 7 of them being category 3 or higher, and Wilma being the most intense hurricane observed over the North Atlantic. According to some authors (Emanuel 2005; Webster et al. 2005), the level of hurricane activity in the last decade has been particularly high, and intense hurricanes have been exceptionally frequent. Together with the rise in social vulnerability (e.g., Pielke 2005; Pielke and Landsea 1999; Pielke et al. 2008), these events have raised concerns about the management of hurricane risks in the United States and about the drivers of hurricane intensity and frequency. In particular, while climate change is now visible in meteorological observations with a global mean temperature increase by 0.76°C (Solomon et al. 2007), the question of whether the recent increase in hurricane activity is linked to global warming has become important. Some have argued that the level of hurricane activity in the North Atlantic exhibits interdecadal oscillations that fully explain the present level of activity (Landsea et al. 1999), while others have proposed that climate change is—at least partly—responsible for the current situation (Emanuel 2005; Webster et al. 2005). The answer to this question is of primary importance for risk management, flood management, and urban planning: if global warming is responsible for the current increase in activity, then this increase is likely to be amplified in the future, and ambitious policies should urgently be implemented to limit social vulnerability to hurricanes. If the current phase of high activity is due to natural variability, then there is no reason to think that hurricane intensity will continue to grow in the future, and the policies to be implemented to manage hurricane risks do not need to be as ambitious. Given the cost of protection infrastructures and the lifetime of these infrastructures, answering these questions is already urgent (see also Hallegatte 2006, 2007).
To answer these questions, a first necessary step is to investigate in details the large-scale drivers of hurricane activity in the North Atlantic. It is well known (e.g., Gray 1984; Bove et al. 1998; Murnane et al. 2000; Elsner et al. 2001; Jagger et al. 2001) that the El Niño–Southern Oscillation (ENSO) plays an important role and influences significantly hurricane activity in the basin. Also, the North Atlantic Oscillation (NAO) is a good candidate for large-scale process that has an influence on North Atlantic hurricanes, especially concerning their tracks (Elsner et al. 2000; Jagger et al. 2001). Of course, sea surface temperature (SST) in the basin is also known to be an essential parameter that needs to be accounted for, with a strong correlation between SST in the region and hurricane Power Dissipation Index (PDI: see, e.g., Emanuel 2005; Goldenberg et al. 2001; Jagger and Elsner 2006).
The present paper investigates the link between large-scale climate parameters and hurricane characteristics within a “downscaling” framework. Its aim, indeed, is not to forecast hurricane activity from predictors available months before the hurricane season (i.e., to produce a seasonal forecast as can be found in Elsner and Jagger 2006) but to relate contemporaneous values of large-scale (e.g., ENSO) and small-scale (e.g., hurricane maximum winds) parameters.
To do so, two main methodological approaches are available. A first one relies on physical models of hurricanes, from general circulation models with high resolution (e.g., Suji et al. 2002; Chauvin et al. 2006) to regional climate models (Knutson and Tuleya 2004) to hurricane models, specifically developed to investigate this issue (Emanuel 2006; Emanuel et al. 2006).
A second approach consists of the use of statistical models. These models extract, from historical series of climate indices and hurricane characteristics, statistically significant relationships that can then be analyzed. The present paper uses this latter approach to investigate the links between climate indices and indices of hurricane activity. Such an approach is hardly new, as numerous papers have already been published in this line (e.g., Gray 1984; Bove et al. 1998; Murnane et al. 2000; Elsner et al. 2001; Jagger et al. 2001; Goldenberg et al. 2001; Binter et al. 2006; Jagger and Elsner 2006; Camargo et al. 2007b,c). The originality of the present paper, however, lies in the choice of the indices and in the statistic tools used to extract statistically significant relationships.
First, in addition to the annual number of tropical cyclones (TCs) in the North Atlantic basin, this paper focuses on the single most intense hurricane of each season. The intensity of a hurricane is measured using the PDI as defined in Emanuel (2005). This index measures the total energy dissipation over the lifetime of the hurricane. Since the hurricane PDI and its intensity at landfall are only weakly correlated, the PDI of a hurricane is a poor indicator of the socioeconomic damages it may cause. The PDI, however, is a much more meaningful measure of hurricane intensity in a basin than only landfall intensity. Compared to maximum wind speed, the PDI is also sensitive to the length of the track, which is also important information. As a consequence, we claim that the PDI is a pertinent index of hurricane activity.
Second, this paper investigates the statistical relationships using generalized linear models (GLMs; Nelder and Wedderburn 1972) and vector generalized linear models (VGLMs; Yee and Hastie 2003) as well as generalized additive models (GAMs; Hastie and Tibshirani 1990) and vector generalized additive models (VGAMs; Yee and Wild 1996). Modeling hurricane counts by means of GLMs is a well-known technique (see, e.g., Elsner and Schmertmann 1993). Relating extreme hurricane activity levels to large-scale parameters in a continuous manner is much more recent. Jagger and Elsner (2006) compare statistics for years with either above or below normal conditions for different indexes [Southern Oscillation index (SOI), NAO, etc.], and introduce global temperature as a linear covariate in the generalized pareto distribution (GPD). In a more recent work, Jagger et al. (2008) use a hierarchical Bayesian approach to introduce NAO and SOI as linear covariates of respectively log(scale) and shape parameters of a GPD fit of extreme insured losses. But previous studies always assume linearity of the responses. In the following, we show that, by means of combination of GAMs, VGAMs, GLMs, and VGLMs we can easily investigate the level of hurricane activity as a continuous function of large-scale climate indices an put into evidence more complex nonlinear relationships.
The paper is organized as follows. Section 2 describes the data that have been used. Section 3 presents the different methodologies, starting with classical results of the extreme value theory and then describing the GLM and GAM methods. The two following sections describe the statistical relationships between indices of tropical cyclone activities and large-scale climate indices. Section 4 shows results applied to the annual number of TCs in the North Atlantic basin, while section 5 investigates the intensity of the most extreme hurricanes. Section 6 concludes and proposes leads for future research.
2. Data
The analysis is carried out using two indices of hurricane activity: the annual number of TCs (Fig. 1), and the largest PDI of a season (Fig. 2).
Tropical cyclones PDIs are calculated by the authors from the integration of cubed maximum wind speed along the track, and expressed in 109 m3 s−2. The number of TCs in a season and their wind speed are provided by the Hurricane Database (HURDAT or best track), created and maintained by the National Hurricane Center (NHC). HURDAT is considered as the best database for tropical cyclones in the North Atlantic, and provides, with a 6-h sampling time, the position and intensity estimates of tropical cyclones from 1851 to 2005. In this analysis, we considered only the TCs between 1880 and 2005, because data prior to 1880 are considered less complete and accurate. Finally, our database contains 1153 hurricane and tropical storm tracks. Most importantly, there is considerable uncertainty concerning the quality of the data in the first half of this period (see Emanuel 2005; Landsea 2005). This uncertainty makes it necessary to interpret with care our results, especially when they deal with long-term trends.
Large-scale climate indexes, namely the global mean temperature anomalies (T), SST and the detrended SST in the North Atlantic (DSST), NAO, and SOI have been derived from data by the Climate Research Unit (CRU) of the University of East Anglia (Norwich, United Kingdom). The NAO and SOI indices were directly available. The NAO is calculated from the difference in sea level pressure (SLP) between Gibraltar and a station over southwest Iceland (Jones et al. 1997). The SOI is defined as the normalized sea level pressure difference between Tahiti and Darwin (Ropelewski and Jones 1987). An El Niño event corresponds to a negative SOI, while La Niña corresponds to positive SOI. The North Atlantic SST (from 0° to 70°N) was obtained from the National Oceanic and Atmospheric Administration (NOAA) Physical Science Division (PSD). For numerical stability reasons, we use in the statistical analysis the normalized anomalies of SST, which are hereafter referred to as the nSST index. The nSST is just SST minus its climatological mean divided by its standard deviation. We also introduce the nDSST index. DSSTs are the linearly detrended SSTs, and nDSSTs are the normalized DSST. Thus, nSST and nDSST samples have zero mean and unit variance, and in addition nDSST shows no linear trend on the whole period (null slope). This index is equivalent to the Atlantic multidecadal oscillation (AMO; Enfield et al. 2001). The global mean temperature was obtained from the Climatic Research Unit of East Anglia. To investigate statistical relationships with hurricane properties, these four indices have been averaged over the hurricane season of each year (from June to November), to get one value per year and per index. In the next sections, historical data on TC activity, annual number and PDI, are related to these large-scale climate indices.
Pairs plot of those indices (PDI, TC number, T, NAO, SOI, nSST) is given in Fig. 3. It shows the positive correlation between SST and global temperature T (around 0.7).
3. Methods
a. Classical results on Extreme Value Theory
b. Linear and additive models
GLMs (Nelder and Wedderburn 1972) and GAMs (Hastie and Tibshirani 1990) are used to investigate factors influencing TC numbers. Extremes of PDI are studied by means of VGLMs (Yee and Hastie 2003) and VGAMs (Yee and Wild 1996).
Link functions log(x) and log(x + 0.5) ensure σ > 0 and ξ > −0.5. The latter condition corresponds to the case where classical maximum likelihood estimators have the standard asymptotic properties (Smith 1985).
In our study, the shape parameter (ξ) is fitted as an intercept-only parameter. It is a well-known fact that data provides little information on this parameter. Letting it be too flexible would provide numerical problems (Yee and Stephenson 2007).
For practical computation, the VGAM package (R language; Yee 2006) provides an efficient and flexible implementation of vector spline smoothers as well as procedures for VGLMs estimation under R language (R Development Core Team 2005).
When considering a nonstationary GEV distribution, one has to study joint variations of nonindependent parameters. Structural trend models are difficult to formulate in those circumstances, owing to the complex way in which different factors combine to influence extremes. Moreover, it is not advisable to fit linear trend models without evidence of their suitability. In such situations, GAMs and VGAMs provide valuable flexible exploratory tools, since one does not have to formulate a model a priori. But since they provide only pointwise approximate confidence intervals, parametric GLMs or VGLMs remain necessary for full inferences. Therefore, we tackle the problems in two times, for both TC numbers and PDI extremes. We start with an exploratory phase, conducted by means of GAMs or VGAMs: those data-driven approaches suggest proper models. Final results are obtained by means of GLMs or VGLMs.
4. Predictors of annual tropical cyclone number
As a first step, we use the annual TC number as a first index of hurricane activity. This index is obviously pertinent, as the risk of landfall and damage increases with the number of TCs. There is a continuing debate concerning the quality of TC counts before the satellite era. Nyberg et al. (2007), using biological reconstruction, suggests that available databases underestimate the number of major hurricanes by 2–3 yr−1, which means that named storms could undercounted by as many as 6–20 yr−1. Landsea (2007) investigated the evolution of the ratio of storms that make landfall, interpreted as a proxy for the number of unreported storms. The observed decrease in this ratio suggests an undercount by 3.2 named storms prior to 1966. Chang and Guo (2007) uses ship tracks to assess how many TCs may have been unreported between 1900 and 1965. They suggest that available databases may underestimate TC counts by 2.1 yr−1 before 1914 and by 1 or less after 1920. Mann et al. (2007) use statistical techniques to show that an underestimation by more than one TC per year would be inconsistent with records of other climate data. Because this literature is not conclusive yet, we used the HURDAT database with no underreporting correction.
From a pure technical point of view, we have to take into account for overdispersion of our data. While in Poisson models variance and mean should be equal, on real count datasets, variance is frequently larger than mean: in our data, the average cyclone number is equal to 9.2, and the variance is equal to 16.4. The overdispersion problem in our data is solved using a quasi-likelihood method that allows variance to be inflated by a constant (Wedderburn 1974).
a. Generalized additive model
At a first stage we model the logarithm of the rate λ of the Poisson law as a smooth function of each predictor, logarithm (log) being the canonical link function for the Poisson law. Figure 4 shows the results for six explanatory variables. Even if not considered as a real predictor, introducing the year of observation as a predictor allows smoothing data according to time. It shows how the number of TCs increases with time in the HURDAT data. One should not attribute too much confidence to this result, however, because the reliability of the database from 1880 to 1950 is questionable: hurricanes before 1950 may have been missed by observation networks. The HURDAT database, therefore, may underestimate the number of TCs in the beginning of the record.
At a first stage, additive effects and confidence intervals are estimated using one predictor at a time (the single effects in Fig. 4). This preliminary analysis allows eliminate NAO as a useful predictor. This may be because NAO over the hurricane season is generally weak. Using preseason indices may lead to different conclusions, at least on a regional scale, according to Elsner and Jagger (2006). Log(λ) exhibits a quasi-linear dependency in SST, while SOI has a more complicated behavior. Lower values of nDSST seem less informative.
log(λ) exhibits a quasi-linear dependency in nSST, as seen before;
SOI effect first increases, then seems to decrease. However, it is unclear at this stage to decide whether a “broken stick model” should be fitted, since (i) the confidence intervals are rather large at endings, and (ii) these confidence intervals are only pointwise approximations.
b. Generalized linear model
This is the formulation of a “broken stick model” (with continuity constraint). The best fit is obtained for a SOI value K = 1, its log likelihood being equal to −316.16. Compared with the log likelihood of the simpler linear response in both SOI and nSST (−318.71), a standard deviance test allows reject linearity (p value=0.02). The coefficient estimates are given in Table 1.
Expectation λ of the TC number is then straightforwardly computed as a function of SOI and SST (Fig. 6).
Our model allows rather precise modeling. Bootstrap estimates of the correlation between observed and expected TC numbers reveals an average correlation of 0.6 (see Fig. 7).
Checking the variance of the number of TCs given the covariates, we find a value of 10.6, very close to the mean (9.2) of the distribution. Conditioning on covariates allowed to quasi eliminate over dispersion: we may then consider that TC count given covariates follows a Poisson distribution.
Our analysis retrieves two well-known results: the fact that the number of TCs is increasing with SST (e.g., Gray 1984; Binter et al. 2006) and the fact that La Niña years are more active than El Niño years (e.g., Gray 1984; Elsner et al. 2001; Camargo et al. 2007a). In our results, however, the effect of ENSO is more complex than in this literature. The maximum number of TCs is reached for SOI values about 1 hPa (moderate La Niña) and starts decreasing for stronger La Niña. This result is supported by the rejection of the linearity hypothesis between the logarithm of annual number of TCs and SOI. It seems, therefore, that El Niño reduces hurricane activity but that La Niña has also a damping effect when very strong. This result is consistent with the fact that, in several analyses, the difference between El Niño years and neutral years is much larger than that between La Niña years and neutral years; see Bove et al. (1998) or Chu (2004). Because of the small number of years with SOI larger than 1 hPa, however, this result should be considered with care, and a physical analysis of the mean conditions of a strong La Niña should be carried out to understand how such conditions may inhibit hurricane activity.
The fact that SST has a much larger explanatory effect than detrended SST suggests that the increase in global temperature results in more TCs in the North Atlantic basin. According to our analysis, this relationship is exponential [as it is linear in log(λ)], and an increase by 1°C results, at least, in seven additional TCs.
These results can be translated in different ways. Using a crude classification of hot and cold years (with hot years being the 1/3 hottest years and cold years being the 1/3 coldest years), our statistic analysis suggests an average of 11.5 TCs during hot years and of 6.8 during cold years.
5. Predictors of extreme hurricane intensities
For risk analysis, the number of TCs is only one part of the needed information. The strength of the hurricanes in the basin is more important, even though its measurement is also more uncertain. To look at how hurricane strength depends on large-scale parameters, and because we are mainly interested in the most powerful hurricanes, we now consider only the hurricane of largest PDI of each year. The PDI of this hurricane is an index of hurricane activity that focuses on the most powerful hurricanes and is, therefore, of interest for risk management. In this section, we relate this index to the large-scale climate indices.
a. Additive model
In the following, we consider independent annual maxima of storm tracks PDI over the Atlantic, and we fit those maxima with the GEV distribution, according to EVT. Our main question is: which factors do influence PDI maxima? This leads us to model GEV shape and scale parameters as functions of covariates. As already stated, for computational reasons, the shape (ξ) parameter remains constant throughout this study.
Additive effects f1(μ), f2(μ), f1(σ) and f2(σ) and their corresponding confidence intervals (Fig. 8) strongly suggest that
the location parameter μ exhibits a quasi-linear dependency in SOI and nSST;
nSST effect appears to be linear on log(σ), SOI has an increasing effect that levels off for positive values.
b. VGLM fit
Note that we fit a particular changepoint model (with continuity constraint) to model the dependency of log(σ) in SOI: the response is linear for SOI values lower than K, constant for SOI ≥ K. The best fit is obtained for K = −0.55 hPa. According to a deviance test, the Gumbel approximation is valid (shape parameter estimation and corresponding standard error have both a value close to 0.07, the p value of the test being equal to 0.36), which means the PDI distribution has the properties of a light tailed distribution. So in the following we set the shape parameter equal to zero. Gumbel fit estimates are given in Table 2.
The log likelihood of this model is −646.5, to be compared with the log likelihood of the simple model with a linear response in SOI and nSST for location parameter and the logarithm of scale parameter (−647.5). Since both models have the same number of parameters, the changepoint model on log(scale) is fully justified. Based on this fit, we give in Fig. 9 the location and scale parameters as functions of SOI and SST.
Maximum likelihood estimates of quantiles Qp of extreme PDI are obtained as Qp = μ − σlog[−log(p)] for 0 < p < 1 (Fig. 10).
By the delta method (see Coles 2001 for details) standard errors of Qp estimates can be computed (Fig. 11).
Like for the annual number of TCs, our results are consistent with the literature, but they exhibit more complexity. Very classically, the extreme PDI is increasing monotonously with the North Atlantic SST (Jagger and Elsner 2006). In our results, however, the scale parameter of the Gumbel distribution is not increasing monotonically with SOI but isincreasing for SOI values lower than −0.55 hPa (El Niño years) and is stable for larger values. This complication leads to a more complex dependency of extreme PDI to SOI, even though extreme PDI is still increasing with SOI, meaning that La Niña years have more extreme hurricanes than El Niño years.
The fact that the information brought by SST and detrended SST is similar suggests that no global warming signal is visible in hurricane PDI extremes. The fact that the most extreme hurricanes tend to occur when North Atlantic SST is high (see, e.g., Emanuel 2005) can be explained by natural variability only. This result is surprising considering the trend in PDI identified in Emanuel (2005). Considering the uncertain quality of the data and the low slope of the trend compared with natural variability, however, this result does not mean that global mean temperature has no influence on hurricane intensity, but only that this potential signal cannot be extracted from the most extreme past hurricanes.
c. Diagnostics
We define pseudoresiduals of PDI as PDI minus the modeled location parameter, divided by the modeled scale. Such pseudoresiduals follow a Gumbel distribution with zero location parameter and unit-scale parameter if the model is adequate. Standard diagnostics graphical checks (probability plot and quantile plot in Fig. 12) allow verify goodness of fit.
In Fig. 13 we show observed annual PDI maxima together with Q50 (median) and Q90 quantiles computed each year according to observed SOI and SST values. The model has a good agreement, and is able to detect periods of higher PDI intensity.
The QVSS may take values ranging from −∞ to 1. When equal to zero, the QVSS indicates that there is no gain compared to the reference model. Positive values indicate a gain compared to reference. QVSS equal to 1 indicates a perfect deterministic forecast. In our study, the “climatology” reference model is the Gumbel fit with no covariates, but we got similar results using the sample quantiles. Since distribution of QVSS can hardly be assessed, we rely on bootstrap resampling to assess skill relative to climatology: 50 bootstrap samples are drawn from original data, corresponding estimations and scores are then computed. QVSS box plots are given for different values of p in Fig. 14. Since QVSS values are significantly different from zero, we may say that our model is better than climatology, a result that was not guaranteed, given the variability of the extreme PDI. We may also note that the QVSS scores are quite stable relative to p; it is even slightly better for high values of p, corresponding to high-order quantiles of the extreme PDI.
This skill suggests that the relationships extracted from historical data may be able to provide some insight into future hurricane risks, provided good seasonal forecasts for the large-scale climate indices. Even though these forecasts are currently far from perfect, their current skill, combined with the statistic relationships, would already allow for a prediction of the likely maximum of the PDI over the season.
6. Conclusions
In this article, we investigated the statistic relationships between hurricane activity in the North Atlantic and large-scale climate indices, namely SOI, NAO, SST, detrended SST in the North Atlantic, and global temperature. The originality of this paper lies in the use of the PDI of the most intense hurricane of each season as an index of hurricane activity and in the use of generalized linear models (GLMs; Nelder and Wedderburn 1972), generalized additive models (GAMs; Hastie and Tibshirani 1990), and vector generalized additive models (VGAMs; Yee and Wild 1996).
Our results are consistent with the literature. We found that a higher SST in the North Atlantic causes more TCs in the basin (at least 7 additional TCs per degree Celsius) and that these TCs were more powerful. We also showed that El Niño years had less TCs than the average year, and that these TCs were less intense. The NAO is found to have no influence on TC number or intensity, suggesting that the likely influence of NAO on landfalling hurricane statistics (Jagger and Elsner 2006) could only arise from changes in the mean track. Additional analysis, for example, using track clustering methods (Camargo et al. 2007b,c), would be necessary to give a definitive answer to this question.
Interestingly, the trend in SST is found to have an influence on TC numbers but not on the intensity of the most intense hurricanes. Considering the small amount of available data and the fact that climate change was not a linear trend along the twentieth century, this latter finding does not necessarily imply that climate change has no influence on hurricane intensity, but suggests that this potential dependency, if it exists, is hidden in the noise of natural variability. It is noteworthy that the methods of the extreme value theory, because they introduce either a threshold or a maximum, discard a lot of data and are, therefore, hardly suitable to detect a small trend in mean activity.
The interest of our method is that it allows one to extract more complex relationship than classical methods, in which above-normal and below-normal years are compared for each index. Indeed, we can express the level of hurricane activity as a continuous function of the large-scale indices. Applied to SOI and the detrended SST, this method suggests that if the annual number of TCs increases with the SOI when this index is lower than 1 hPa, it does not keep increasing with SOI when this index is larger than 1 hPa. This result suggests that El Niño inhibits hurricane genesis, but that very strong La Niña does not particularly favor hurricane formation.
Concerning PDI, we find that the distribution of its annual maxima is Gumbel, suggesting its underlying distribution is light tailed. Also, the relationship between the annual maximum of hurricane PDI and the large-scale climate indices is more complex that can be inferred from classical method. In particular, the scale parameter of the Gumbel distribution is found to increase with SOI when the SOI is lower than −0.55 hPa, but is insensitive to SOI when the SOI is larger than this value. The location parameter, on the other hand, increases with SOI for all SOI values. Also, the location and shape parameters increase unambiguously with SST. The consequence of these relationships is that the extreme PDI that can be expected for a given year depends in a complex manner on SST and SOI: the sensitivity of the extreme PDI is much more sensitive to SOI when SOI is lower than −0.55 hPa than when it is larger than this value.
The relationships described in this article suggest interesting questions for physical interpretation. For instance, the correlation between vertical wind shear over the North Atlantic and SOI is well known (Arkin 1982; Gray 1984), but a more precise analysis should be able to tell if this relationship is linear, or if this correlation arises mainly from the El Niño situation, La Niña being close to the neutral situation.
The analysis presented here is only a first investigation, and there are several ways of improving this work. A first one would be to take into account explicitly measurement errors in hurricane wind speeds like in Jagger and Elsner (2006), but their approach is difficult to introduce in our method. As a test of robustness, we compared our results based on the 1944–2005 HURDAT wind speed data, and on corrected values, using the bias correction technique proposed in Emanuel (2007). We find that using only 51 yr of data leads to sampling problems, but that the bias correction does not change qualitatively the results. This test suggests that our results are robust to the measurement errors we suspect are present in the HIRDAT database.
A second one would be to include more predictors into the analysis, like those suggested in Gray et al. (1994) or Landsea et al. (1999). For instance, these authors suggest that the stratospheric quasi-biennial oscillation (QBO) has an important influence on hurricane activity. Also mentioned in these papers, and more recently analyzed by Lau and Kim (2007) and Donnelly and Woodruff (2007), the West African surface and the subsequent aerosol loading in the atmosphere could be a significant driver of hurricane intensity. Another improvement to our analysis would be to consider indices of hurricane activity more closely related to risk management, like in Elsner and Jagger (2006), who provide empirical estimates of extreme wind speed return levels on the U.S. coastline. This information is what is needed to design buildings or to implement building norms and is, therefore, of the foremost importance. Another interesting data that could be analyzed using our methodology is the historical hurricane economic (or insured) losses, like in Jagger et al. (2008). Again, the extreme hurricane losses that can be expected with a given return time is essential because it sets the total amount of capital reserves that insurance companies should have to face hurricane risks without risking bankruptcy.
Acknowledgments
We wish to thank three anonymous reviewers and Kerry Emanuel for fruitful suggestions. We wish to thank the R development core team for developing and maintaining the R language, and Thomas Yee for releasing his exhaustive VGAM package.
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Annual maximum of PDI (in 109 m3s−2).
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Pairs plot of PDI, the number of hurricanes (referred as TCN) together with potential predictors T (anomaly, in °C), NAO (hPa), SOI (normalized index), nSST (normalized anomalies).
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Fitted single effects for each year, NAO, SOI (normalized index), T, nSST, and nDSST (normalized indices) for TC numbers. The dotted lines represent twice the pointwise asymptotic standard errors of the estimated functions.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Fitted functions for nSST and SOI (normalized indices) for TC numbers. The dotted lines represent twice the pointwise asymptotic standard errors of the estimated functions.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Contour plot of expected TC number as a function of SST and SOI (normalized index).
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Bootstrap box plot of correlation estimates between observed and expected number of TCs (50 bootstrap samples).
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Fitted functions for SOI and nSST (normalized indices) for annual PDI maxima location and log(scale) parameters. The dotted lines represent twice the pointwise asymptotic standard errors of the estimated functions.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Contour plots of PDI location and scale parameters as functions of SST and SOI (normalized index), according to the fitted changepoint model.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Contour plot of extreme PDI 90th quantile as a function of SST and SOI (normalized index), according to the fitted changepoint model.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Contour plot of extreme PDI 90th quantile standard error as a function of SST and SOI (normalized index), computed by means of the delta method.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
(left) Probability and (right) quantile plots of PDI pseudoresiduals of PDI compared to Gumbel distribution with location 0 and scale 1.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Time series of observed annual maximum PDI (o), estimated median (solid line), and 90th quantile (dotted line).
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
QVSS box plots for different values of p.
Citation: Journal of Climate 22, 3; 10.1175/2008JCLI2318.1
Maximum likelihood (ML) estimates, corresponding standard error, and t value of the parameters of the broken stick model.
Maximum likelihood estimates, corresponding standard error, and t value of the parameters of the changepoint model.
