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Predictors of Tropical Cyclone Numbers and Extreme Hurricane Intensities over the

North Atlantic using Generalized Additive and Linear Mo**de**ls.

OLIVIER MESTRE

Ecole Nationale **de** la Météorologie, Toulouse, France

Université Paul Sabatier, Laboratoire **de** Statistiques et Probabilités, Toulouse, France

STEPHANE HALLEGATTE

Ecole Nationale **de** la Météorologie, Toulouse, France

**Centre** **International** **de** **Recherche** **sur** l’Environnement et le Développement, Paris, France

(submitted 30 october 2007)

Corresponding author address: Dr Olivier Mestre, Ecole Nationale **de** la

Météorologie, 31057 Toulouse Ce**de**x, France.

E-mail: olivier.mestre@meteo.fr

1

ABSTRACT

Fluctuations of the annual number of tropical cyclones over the North-Atlantic

and of the energy dissipated by the most intense hurricane of a season are

related to a variety of predictors (global temperature, SST and **de**tren**de**d SST,

NAO, SOI) using generalized additive and linear mo**de**ls. Our study

**de**monstrates that SST and SOI are predictors of interest. The SST is found to

influence positively the annual number of tropical cyclones and the intensity

of the most intense hurricanes. The use of specific additive mo**de**ls reveals

non-linearity in the responses to SOI that has to be taken into account using

change-point mo**de**ls. The long-term trend in SST is found to influence the

annual number of tropical cyclones but does not add information for the

prediction of the most intense hurricane intensity.

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1. Introduction and motivation

In the past **de**ca**de**s, intense coastal hurricanes have represented an increasing threat to

communities and economies in the United States. Direct losses, in**de**ed, have increased in an

exponential manner, and a few dramatic events, especially Andrew in 1992 and Katrina in

2005, have had unprece**de**nted consequences, both in economic and social terms.

Recently, hurricane seasons have been very active, and the 2005 season remains the most

active season in recor**de**d 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 **de**ca**de** 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, 2008), these events have raised concerns about

the management of hurricane risks in the U.S., 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 (IPCC, 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 inter-**de**cadal

oscillations that fully explain the present level of activity (Landsea et al., 1999), while others

have proposed that climate change was – 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; Hallegatte, 2007).

To answer these questions, a first necessary step is to investigate in **de**tails 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 largescale

process that has an influence on North Atlantic hurricanes, especially concerning their

tracks (Elsner et al., 2000; Jagger et al., 2001). Of course, sea **sur**face 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 In**de**x (see,

e.g., Emanuel, 2005; Gol**de**nberg 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, in**de**ed, 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

mo**de**ls of hurricanes, from General Circulation Mo**de**ls with high resolution (e.g., Suji et al.,

2002; Chauvin et al., 2006) to Regional Climate Mo**de**ls (Knutson and Tuleya, 2004) to

hurricane mo**de**ls, specifically **de**veloped to investigate this issue (Emanuel, 2006; Emanuel et

al., 2006).

3

A second approach consists in the use of statistical mo**de**ls. These mo**de**ls 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;

Gol**de**nberg et al., 2001; Binter et al., 2006; Jagger and Elsner, 2006; Camargo et al.,

2007a,b). 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 mea**sur**ed using the Power Dissipation In**de**x (PDI), as **de**fined in Emanuel

(2005). This in**de**x mea**sur**es 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 socio-economic damages it may cause. The PDI, however,

is a much more meaningful mea**sur**e 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 in**de**x of hurricane activity.

Second, this paper investigates the statistical relationships using Generalized and Vector

Generalized Linear Mo**de**ls (GLMs, Nel**de**r and Wed**de**rburn, 1972, VGLMs, Yee and Hastie,

2003) as well as Generalized and Vector Generalized Additive Mo**de**ls (GAMs, Hastie and

Tibshirani, 1990, VGAMs, Yee and Wild, 1996). Mo**de**lling hurricane counts by means of

GLMs is a well known technique (see Elsner and Schmertmann, 1993 for example). 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 in**de**xes (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 in**sur**ed

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 evi**de**nce more complex non-linear relationships.

The paper is organized as follows. Section 2 **de**scribes the data that have been used. Section

3 presents the different methodologies, starting with classical results of the extreme value

theory and then **de**scribing the GLM and GAM methods. The two following sections **de**scribe

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 conclu**de**s 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).

FIG. 1. Annual number of cyclones

(Fig. 1 about here)

4

(Fig. 2 about here)

FiG. 2. Annual maximum of PDI (in 10 9 m 3 s -2 )

Tropical cyclones PDIs are calculated by the authors from the integration of cubed

maximum wind speed along the track, and expressed in 10 9 m 3 s -2 . The number of TCs in a

season and their wind speed are provi**de**d by the Hurricane Database (HURDAT or best

track), created and maintained by the National Hurricane Center (NHC). HURDAT is

consi**de**red as the best database for tropical cyclones in the North Atlantic, and provi**de**s, with

a 6-hour sampling time, the position and intensity estimates of tropical cyclones from 1851 to

2005. In this analysis, we consi**de**red only the TCs between 1880 and 2005, because data prior

to 1880 is consi**de**red less complete and accurate. Finally, our database contains 1153

hurricane and tropical storm tracks. Most importantly, there is consi**de**rable uncertainty

concerning the quality of the data in the first half of this period (see Emanuel, 2005 and

Landsea, 2005). This uncertainty makes it necessary to interpret with care our results,

especially when they **de**al with long-term trends.

Large-scale climate in**de**xes, namely the global mean temperature anomalies (T); the Sea

Surface Temperature (SST) and the **de**tren**de**d Sea Surface Temperature in the North Atlantic

(DSST); the Northern Atlantic Oscillation in**de**x (NAO); and the Southern Oscillation In**de**x

(SOI) have been **de**rived from data by the Climate Research Unit (CRU) of the University of

East Anglia (Norwich, GB). The NAO and SOI indices were directly available. The NAO is

calculated from the difference in SLP between Gibraltar and a station over southwest Iceland

(Jones et al., 1997). The SOI is **de**fined as the normalized sea level pres**sur**e 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

70N) 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 in**de**x.

nSST is just SST minus its climatological mean divi**de**d by its standard **de**viation. We also

introduce the nDSST in**de**x. DSSTs are the linearly **de**tren**de**d 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 in**de**x is

equivalent to the Atlantic Multi**de**cadal Oscillation (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

in**de**x. 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).

(Fig. 3 about here)

FIG. 3. Pairs plot of PDI, the number of hurricanes (referred as TCN) together with potential predictors T, NAO,

SOI, nSST.

3. Methods

a. Classical results on Extreme Value Theory

In this section, we briefly present the main results of the Extreme Value Theory (EVT,

Coles, 2001; Smith, 2003 for example). EVT focuses on the statistical behaviour of M n , the

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maximum of n in**de**pen**de**nt random variables X 1 , X 2 ,…,X n with common distribution

function F(x). If exist sequences of constants {a n >0} and {b n } such that:

⎧⎪

⎛ Mn

− b ⎞ ⎫

n ⎪

P ⎨⎜

⎟ ≤ z⎬

→ G ( z)

as n→∞

⎪⎩

⎝ a

n ⎠ ⎪⎭

then G follows the generalized extreme value distribution (GEV), of form:

1

⎧

−

⎪ ⎡ ⎛ z −µ

⎞⎤

ξ ⎫

⎪

⎧ ⎡ ⎛ z −µ

⎞⎤⎫

G ( z)

= exp ⎨− ⎢1

+ ξ⎜

⎟⎥

⎬ , if ξ≠0, G ( z)

= exp ⎨−exp

⎪ ⎣ ⎝ σ

⎢−⎜

⎟⎥⎬

if ξ=0 (Gumbel)

⎠

⎩

⎦ ⎪

⎝ σ ⎠

⎭

⎩ ⎣ ⎦⎭

provi**de**d that 1+ξ(z-µ)/σ>0, -∞

distribution. Roughly speaking, a GEV VGLM allows mo**de**l the parameters of a GEV

distribution as linear functions of covariates:

µ = β (µ) o +β (µ) 1 X 1 +β (µ) 2 X 2 +…+β (µ) q X q

log(σ) = β (σ) o +β (σ) 1 X 1 +β (σ) 2 X 2 +…+β (σ) q X q

log(ξ+0.5)

(ξ)

= β o

Additive predictors of a GEV VGAM mo**de**l can be expressed as (Yee and Stephenson,

2007)

µ = β (µ) o +f (µ) 1 (X 1 )+f (µ) 2 (X 2 )+…+f (µ) q (X q )

log(σ) = β (σ) o +f (σ) 1 (X 1 )+f (σ) 2 (X 2 )+…+f (σ) q (X q )

log(ξ+0.5)

(ξ)

= β o

where the additive predictors f j

(.)

are smooth functions of the covariates X 1 ,…, X q . Here again,

the number of predictors should remain “low”, say q=2.

Link functions log(x) and log(x+0.5) en**sur**e 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 provi**de**s little information on this parameter. Letting it be too flexible

would provi**de** numerical problems (Yee and Stephenson, 2007).

For practical computation, the VGAM package (R language, Yee, 2006) provi**de**s an

efficient and flexible implementation of vector spline smoothers as well as procedures for

VGLMs estimation un**de**r R language (R Development Core Team, 2005).

When consi**de**ring a non-stationary GEV distribution, one has to study joint variations of

non-in**de**pen**de**nt parameters. Structural trend mo**de**ls 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 mo**de**ls without evi**de**nce of their

suitability. In such situations, GAMs and VGAMs provi**de** valuable flexible exploratory tools,

since one does not have to formulate a mo**de**l a priori. But since they provi**de** only pointwise

approximate confi**de**nce 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 mo**de**ls. 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 in**de**x of hurricane activity. This

in**de**x is obviously pertinent, as the risk of landfall and damage increases with the number of

TCs. There is a continuing **de**bate concerning the quality of TC counts before the satellite era.

Nyberg et al. (2007), using biological reconstruction, suggests that available databases

un**de**restimate the number of major hurricanes by 2 to 3 per year, which means that named

storms could un**de**rcounted by as many as 6 to 20 per year. 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 **de**crease in this ratio suggests an un**de**rcount 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

un**de**restimate TC counts by 2.1 per year before 1914 and by one or less after 1920. Mann et

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al. (2007) use statistical techniques to show that an un**de**restimation 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 un**de**rreporting correction.

From a pure technical point of view, we have to take into account for over-dispersion of our

data. While in Poisson mo**de**ls 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 over-dispersion problem in our data is solved using

a quasi-likelihood method that allows variance to be inflated by a constant (Wed**de**rburn,

1974).

a. Generalized additive mo**de**l

At a first stage we mo**de**l the rate λ of the Poisson law as a smooth function of each

predictor. We use the canonical link function (log). Figure 4 shows the results for six

explanatory variables. Even if not consi**de**red 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

confi**de**nce 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 un**de**restimate the number of TCs in the beginning of the

record.

At a first stage, additive effects and confi**de**nce 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 due to the fact that NAO over the hurricane season is generally

weak. Using pre-season indices may lead to different conclusions, at least on a regional scale,

according to Elsner and Jagger (2006). Log(λ) exhibits a quasi linear **de**pen**de**ncy in SST,

while SOI has a more complicated behaviour. Lower values of nDSST seem less informative.

(Fig. 4 about here)

FIG. 4. Fitted single effects for YEAR, NAO, SOI, T, nSST and nDSST for TC numbers. The dotted lines

represent twice the pointwise asymptotic standard errors of the estimated functions.

At this stage, we may keep SOI as a predictor combined with one of the three temperature

predictors: T, nSST and nDSST, since those thermal indices are redundant. A comparison of

the log-likelihood of the fits of the three mo**de**ls gives the following results: -321.1 for

SOI+T, -314.5 for SOI+nSST, and -325.6 for SOI+nDSST. The equivalent number of **de**grees

of freedom being close to 9 for the three mo**de**ls, this comparison justifies the selection of

nSST, which is by far the best predictor when combined with SOI. When SOI and nSST are

used as predictors, T and nDSST do not add any additional information and they can,

therefore, be discar**de**d. We keep SOI and nSST as predictors, and fit the two variables

generalized additive mo**de**l:

log(λ) = β o +f 1 (SOI)+f 2 (nSST)

Additive effects f 1 , f 2 and their corresponding confi**de**nce intervals (Fig. 5) suggest that:

- log(λ) exhibits a quasi-linear **de**pen**de**ncy in nSST, as seen before

- SOI effect first increases, then seems to **de**crease. However, it is unclear at this

stage to **de**ci**de** whether a “broken stick mo**de**l” should be fitted, since (i) the

8

confi**de**nce intervals are rather large at endings, and (ii) these confi**de**nce

intervals are only pointwise approximations.

(Fig. 5 about here)

FIG. 5. Fitted functions for nSST and SOI for TC numbers. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

b. Generalized Linear Mo**de**l

According to the features revealed by means of previous additive mo**de**l, the following

generalized linear mo**de**l is fitted using maximum likelihood method, which allows for more

formal testing.

log(λ) = β o +β (1) SOI SOI+β nSST nSST SOI

conditions of a strong La Niña should be carried out to un**de**rstand how such conditions may

inhibit hurricane activity.

The fact that SST has a much larger explanatory effect than **de**tren**de**d 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 7 additional TCs.

These results can be translated in different ways. For instance, using the Hall and Jewson

(2007) classification of hot and cold years (with hot years being the 1/3 hottest years and cold

years being the 1/3 col**de**st years), our statistic analysis suggests an average of 11.5 TCs

during hot years and of 6.8 during cold years. These results are consistent with the results by

Hall and Jewson, who predict, on average, 12.8 TCs during a hot year and 8.5 during a cold

year. The difference arises probably from the fact that we consi**de**r the combined effect of

SST and SOI, and not only of SST, and from our choice to use the entire 1880-2005 period.

5. Predictors of extreme hurricane intensities

For risk analysis, the number of TCs is only one part of the nee**de**d information. The

strength of the hurricanes in the basin is more important, even though its mea**sur**ement is also

more uncertain. To look at how hurricane strength **de**pends on large-scale parameters, and

because we are mainly interested in the most powerful hurricanes, we now consi**de**r only the

hurricane of largest PDI of each year. The PDI of this hurricane is an in**de**x of hurricane

activity that focuses on the most powerful hurricanes and is, therefore, of interest for risk

management. In this section, we relate this in**de**x to the large-scale climate indices.

a. Additive mo**de**l

In the following, we consi**de**r in**de**pen**de**nt 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 mo**de**l GEV shape and

scale parameters as functions of covariates. As already stated, for computational reasons, the

shape (ξ) parameter remains constant throughout this study.

The following potential predictors are consi**de**red: Global temperature (T); normalized

anomalies of Sea Surface Temperature, with or without the long-term linear trend (nSST and

nDSST); Northern Atlantic Oscillation (NAO); and Southern Oscillation In**de**x (SOI).

Similarly to TC numbers, a preliminary analysis (not shown here) allows eliminate NAO (no

effect) and T (redundant with nSST), but here information provi**de**d by nSST and nDSST is

equivalent (log-likelihood: -642.2 when including nSST, -642.3 when including nDSST). In

the following, we only keep nSST, in or**de**r to remain coherent with the previous study on TC

numbers.

µ = β (µ) o +f (µ) 1 (SOI)+f (µ) 2 (nSST)

log(σ) = β (σ) o +f (σ) 1 (SOI)+f (σ) 2 (nSST)

(ξ)

log(ξ+0.5) = β o

Additive effects f 1 (µ) , f 2 (µ) , f 1

(σ)

and f 2 (σ) and their corresponding confi**de**nce intervals (Fig.

8) strongly suggest that:

- the location parameter µ exhibits a quasi-linear **de**pen**de**ncy in SOI and nSST

- nSST effect appears to be linear on log(σ), SOI has an increasing effect that

levels off for positive values.

10

(Fig. 8 about here)

FIG. 8: fitted functions for SOI and nSST for annual PDI maxima. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

b. VGLM fit

According to the preliminary analysis, the following vector generalized linear mo**de**l is

fitted using standard maximum likelihood method.

µ = β (µ) o +β (µ) SOI SOI+β (µ) nSST nSST

log(σ) = β (σ) o +β (σ) SOI SOI+β (σ) nSST nSST SOI

values lower than -0.55hPa (El Niño years) and is stable for larger values. This complication

leads to a more complex **de**pen**de**ncy 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 **de**tren**de**d 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 for instance, Emanuel, 2005)

can be explained by natural variability only. This result is **sur**prising consi**de**ring the trend in

PDI i**de**ntified in Emanuel (2005). Consi**de**ring 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 **de**fine pseudo-residuals of PDI as PDI minus the mo**de**lled location parameter, divi**de**d

by the mo**de**lled scale. Such pseudo-residuals follow a Gumbel distribution with zero location

parameter and unit scale parameter if the mo**de**l is a**de**quate. Standard diagnostics graphical

checks (probability plot and quantile plot in Fig. 12) allow verify goodness-of-fit.

(Fig. 12 about here)

FIG. 12. Probability (left) and quantile (right) plots of PDI pseudo-residuals of PDI compared to Gumbel

distribution with location 0 and scale 1.

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 mo**de**l has a

good agreement, and is able to **de**tect periods of higher PDI intensity.

(Fig. 13 about here)

FIG. 13. Time series of observed annual maximum PDI (o), estimated median (solid line) and 90 th quantile

(dotted line).

Since we investigate here quantiles of extreme value mo**de**ls rather than classical regression

mo**de**ls, the skill in our mo**de**l is investigated by means of the Quantile Verification Skill

Score (QVSS) proposed by Frie**de**richs and Hense (2008). QVSS is based on “proper scoring

rules” for quantiles (see Gneiting and Raftery, 2007 for **de**tails). Let PDI t be the observed PDI

at time t, and Qp t the mo**de**lled quantile at time t. Then for each probability p, the Quantile

Verification score QV(p) **de**fines a proper scoring rule:

QV p = c PDI − Qp

∑

( ) ( )

t

p t t

where c p is the “check function” **de**fined as: c p (u)=p⋅u if u≥0, and c p (u)=(p-1)⋅u if u

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

boxplots are given for different values of p in figure 14. Since QVSS values are significantly

different from zero, we may say that our mo**de**l 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 or**de**r quantiles of the extreme PDI.

(Fig. 14 about here)

FIG. 14. QVSS boxplots for different values of p

This skill suggests that the relationships extracted from historical data may be able to

provi**de** some insight into future hurricane risks, provi**de**d 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. Conclusion

In this article, we investigated the statistic relationships between hurricane activity in the

North Atlantic and large-scale climate indices, namely SOI, NAO, SST, **de**tren**de**d 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 in**de**x of hurricane activity and in the use of

Generalized Linear Mo**de**ls (GLMs, Nel**de**r and Wed**de**rburn, 1972), Generalized and Vector

Generalized Additive Mo**de**ls (GAMs, Hastie and Tibshirani, 1990, 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 **de**gree 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, e.g. using track clustering methods (Camargo et al., 2007a,b), would be

necessary to give a **de**finitive 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. Consi**de**ring 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 **de**pen**de**ncy, if it exists, is hid**de**n 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 **de**tect 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

in**de**x. In**de**ed, we can express the level of hurricane activity as a continuous function of the

large-scale indices. Applied to SOI and the **de**tren**de**d SST, this method suggests that if the

annual number of TCs increases with the SOI when this in**de**x is lower than 1hPa, it does not

keep increasing with SOI when this in**de**x is larger than 1hPa. This result suggests that El

Niño inhibits hurricane genesis, but that very strong La Niña does not particularly favour

hurricane formation.

13

Concerning PDI, we find that the distribution of its annual maxima is Gumbel, suggesting

its un**de**rlying 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 -0.55hPa, 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 **de**pends 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 -0.55hPa than when it is larger than

this value.

The relationships **de**scribed 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 mea**sur**ement 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 years 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 mea**sur**ement errors we suspect are present in the

HIRDAT data base.

A second one would be to inclu**de** 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-biennal 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 **sur**face and the subsequent aerosol

loading in the atmosphere could be a significant driver of hurricane intensity. Another

improvement to our analysis would be to consi**de**r indices of hurricane activity more closely

related to risk management, like in Elsner and Jagger (2006), who provi**de** empirical estimates

of extreme wind speed return levels on the U.S. coastline. This information is what is nee**de**d

to **de**sign buildings or to implement building norms and is, therefore, of the foremost

importance. Another interesting data that could be analysed using our methodology is the

historical hurricane economic (or in**sur**ed) 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 in**sur**ance 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 **de**velopment core team for **de**veloping and

maintaining the R language, and Thomas Yee for releasing his exhaustive VGAM package.

14

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19

Figure captions

FIG. 1. Annual number of cyclones

FiG. 2. Annual maximum of PDI (in 10 9 m 3 s -2 )

FIG. 3. Pairs plot of PDI, the number of hurricanes (referred as TCN) together with potential predictors T, NAO,

SOI, nSST.

FIG. 4. Fitted single effects for YEAR, NAO, SOI, T, nSST and nDSST for TC numbers. The dotted lines

represent twice the pointwise asymptotic standard errors of the estimated functions.

FIG. 5. Fitted functions for nSST and SOI for TC numbers. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

FIG 6. Contour plot of expected TC number as a function of SST and Southern Oscillation In**de**x.

FIG. 7. Boostrap boxplot of correlation estimates between observed and expected number of TCs (50 bootstrap

samples).

FIG. 8. Fitted functions for SOI and nSST for annual PDI maxima. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

FIG. 9. Contour plots of PDI location and scale parameters as functions of SST and Southern Oscillation In**de**x,

according to the fitted change-point mo**de**l.

FIG. 10. Contour plot of extreme PDI 90 th quantile as a function of SST and Southern Oscillation In**de**x,

according to the fitted change-point mo**de**l.

FIG. 11: Contour plot of extreme PDI 90 th quantile standard error computed by means of the **de**lta method.

FIG. 12. Probability (left) and quantile (right) plots of PDI pseudo-residuals of PDI compared to Gumbel

distribution with location 0 and scale 1.

FIG. 13. Time series of observed annual maximum PDI (o), estimated median (solid line) and 90 th quantile

(dotted line).

FIG. 14. QVSS boxplots for different values of p

20

Table legends

TABLE 1. Maximum likelihood estimates, corresponding standard error and t value of the parameters of the

broken stick mo**de**l.

TABLE 2. Maximum likelihood estimates, corresponding standard error and t value of the parameters of the

change-point mo**de**l.

21

CYCLONE NUMBER

+

ANNUAL COUNT

0 5 10 15 20 25

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FIG. 1. Annual number of cyclones

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1880 1900 1920 1940 1960 1980 2000

YEAR

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PDI ANNUAL MAXIMUM

PDI

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23

0 10 20

−1.5 0.0 1.5

−2 0 1 2

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+

+ + +

+ +

+ + +

+

+ +

+

+ +

+ +

++

++

+

++ + +

+

++ +

+ + +

+

+

+

+ +

+

+

++

+ +

+ + + +

+ +

+ + +

+

+

+

+

++

+

+

+

+ +

+ +

++ +

+

+

++

+ +

+ +

+

+

++

++

+

+

+

+

+

+

+

+

+ +

+ ++

+ ++ +

+ + + +

+ + ++ +

+ +

+

+ + +

+ +

+

+

++

+

+

+ +

+

+ +

+ NAO +

++ +

++

+

+

+

++

+ ++

++ + +

+

+

+ + +

+

+

+ ++ ++ +

+ + +

+

++

+

+ +

+

+ +

++

++

+

+

+

+ +

+

+

+

+

+ +

+ ++

+

+ +

+

++

+

+ + +

+

+

++

+

+ + + + +

+

+

+

+ + +

+ +

+

+

+

++ +

+

+

+

+

++

+

+ +

+

+ + ++ ++ ++

+

+

+ +

+

+

+ + + +

+ +

+ +

++ +

+

+ +

+

+ + +

+

+

+ +

+

+

+

+

++

+ +

+ +

+

+ ++

+ + +

+ + + +

+

++

+ +

+ + + + +

+ +

+ +

SOI

+

+

+ +

+

+

+ +

+

+

+

+ +

+

+ +

++ + +

+

++

+

+

+

+

+

+

+

+

+ + +

+

+

++ + +

+ +

+

+

−2 0 1 2

−2 0 1 2

+

+ + +

+ ++ ++

+

++ + + +

+

+

+

++

+ +

+

+

+

+

+

+ + +

+

+

+ +

+

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+

+ +

+

+ +

+

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+

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+

+ +

+

+

+

+

+

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+

+

+

++

+ ++ + +

+ +

+

+

+

+

+

+ +

+ + + +

+ +

+

+ + ++

++

+

+ +

++ ++

+++ +

+

+ ++ +

++

+ +

++ + +

+

+ + +

++ ++ +

+

+ + +

+ +

+

+++

+

+

+

++

++

+ +

+ +

+ + + +

+ + +

+ +

+

+

+

++ + +

+ + +

+

+ +

+

+

+

++

+

++

+

+

+

+

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+

+

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++

+

+

+

+

+

+ +

+

+ + +

+ +

+

+

+

+

+ +

+ + nSST

0 100 200

−6 −2 2 6

−2 0 1 2

FIG. 3. Pairs plot of PDI, the number of hurricanes (referred as TCN) together with potential predictors T, NAO,

SOI, nSST.

24

ADDITIVE EFFECT

−0.4 −0.2 0.0 0.2 0.4 0.6 0.8

ADDITIVE EFFECT

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4

1880 1900 1920 1940 1960 1980 2000

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

−2 −1 0 1 2

YEAR

NAO

SOI

ADDITIVE EFFECT

−0.5 0.0 0.5 1.0

ADDITIVE EFFECT

−1.0 −0.5 0.0 0.5 1.0

ADDITIVE EFFECT

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

ADDITIVE EFFECT

−6 −4 −2 0 2 4 6 8

−2 −1 0 1 2

−2 −1 0 1 2

T

FIG. 4. Fitted single effects for YEAR, NAO, SOI, T, nSST and nDSST for TC numbers. The dotted lines

represent twice the pointwise asymptotic standard errors of the estimated functions.

nSST

nDSST

25

ADDITIVE EFFECT

−0.5 0.0 0.5 1.0

ADDITIVE EFFECT

−0.8 −0.6 −0.4 −0.2 0.0 0.2

−2 −1 0 1 2

nSST

−2 −1 0 1 2

SOI

FIG. 5. Fitted functions for nSST and SOI for TC numbers. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

26

EXPECTED TC NUMBER

SST (°C)

21.5 22.0 22.5 23.0 23.5

−2 −1 0 1 2

SOI

FIG 6. Contour plot of expected TC number as a function of SST and Southern Oscillation In**de**x.

27

BOXPLOT OF BOOTSTRAP CORRELATION ESTIMATES

0.45 0.50 0.55 0.60 0.65 0.70

CORRELATION COEFFICIENT: OBSERVED vs EXPECTED TC NUMBER

FIG. 7. Boostrap boxplot of correlation estimates between observed and expected number of TCs (50 bootstrap

samples).

28

LOCATION

LOCATION

ADDITIVE EFFECT

−40 −20 0 20 40 60

−2 −1 0 1 2

SOI

LOG(SCALE)

ADDITIVE EFFECT

−1.5 −1.0 −0.5 0.0 0.5 1.0

ADDITIVE EFFECT

−50 0 50 100

−2 −1 0 1 2

nSST

LOG(SCALE)

ADDITIVE EFFECT

−1.5 −0.5 0.0 0.5 1.0

−2 −1 0 1 2

SOI

−2 −1 0 1 2

nSST

FIG. 8. Fitted functions for SOI and nSST for annual PDI maxima. The dotted lines represent twice the pointwise

asymptotic standard errors of the estimated functions.

29

PDI location parameter

PDI scale parameter

SST (°C)

21.5 22.0 22.5 23.0 23.5

SST (°C)

21.5 22.0 22.5 23.0 23.5

−2 −1 0 1 2

−2 −1 0 1 2

SOI

SOI

FIG. 9. Contour plots of PDI location and scale parameters as functions of SST and Southern Oscillation In**de**x,

according to the fitted change-point mo**de**l.

30

Extreme PDI Q90 quantile

SST (°C)

21.5 22.0 22.5 23.0 23.5

−2 −1 0 1 2

SOI

FIG. 10. Contour plot of extreme PDI 90 th quantile as a function of SST and Southern Oscillation In**de**x,

according to the fitted change-point mo**de**l.

31

Q90 standard error

SST (°C)

21.5 22.0 22.5 23.0 23.5

−2 −1 0 1 2

SOI

FIG. 11: Contour plot of extreme PDI 90 th quantile standard error computed by means of the **de**lta method.

32

PROBABILITY PLOT

QUANTILE PLOT

EMPIRICAL

0.0 0.2 0.4 0.6 0.8 1.0

EMPIRICAL

−1 0 1 2 3 4

0.0 0.2 0.4 0.6 0.8 1.0

−1 0 1 2 3 4 5

MODEL

MODEL

FIG. 12. Probability (left) and quantile (right) plots of PDI pseudo-residuals of PDI compared to Gumbel

distribution with location 0 and scale 1.

33

OBSERVED vs MODELLED PDI (MEDIAN,Q90)

PDI

0 50 100 150 200 250

GUMBEL DISTR. Q90

GUMBEL DISTR. MEDIAN

1880 1900 1920 1940 1960 1980 2000

YEAR

FIG. 13. Time series of observed annual maximum PDI (o), estimated median (solid line) and 90 th quantile

(dotted line).

34

BOOTSTRAP QVSS BOXPLOTS

QVSS(p)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

FIG. 14. QVSS boxplots for different values of p

p

35

TABLE 1. Maximum likelihood estimates, corresponding standard error and t value of the parameters of the

broken stick mo**de**l.

Coefficient ML Estimate Standard Error t value

β o 2.226 0.033 67.06

(1)

β SOI 0.166 0.041 4.03

(2)

β SOI -0.567 0.251 -2.09

β nSST 0.227 0.03 7.56

36

TABLE 2. Maximum likelihood estimates, corresponding standard error and t value of the parameters of the

change-point mo**de**l.

Coefficient ML Estimates Standard Error t value

(µ)

β o 66.62 3.63 18.34

(σ)

β o 3.82 0.14 26.81

(µ)

β SOI 14.95 3.09 4.83

(σ)

β SOI 0.34 0.17 1.99

(µ)

β nSST 14.13 3.42 4.12

(σ)

β nSST 0.22 0.07 3.15

37