Introduction
Much concern has been expressed, especially within the insurance and reinsurance industry, about increases in total economic damage associated with extreme weather or climate events (e.g., floods or hurricanes) in recent decades. With concomitant increases in societal vulnerability to extremes (e.g., because of development in flood plains), questions remain about the extent to which these trends are attributable to changes in climate (Changnon et al. 2000; Kunkel et al. 1999). Total economic damage fluctuates both because of variations in the frequency of extreme events and in the damage from individual events. Thus, it would be helpful in analyzing these damages to make use of a statistical approach that explicitly takes into account these different sources of variation.
A case in point is the economic damage caused by North Atlantic hurricanes making landfall in the continental United States. In an attempt to control for changes in vulnerability, this damage dataset has been adjusted for inflation, wealth, and population at risk by Pielke and Landsea (1998). Once these adjustments are made, a marked increasing trend in annual total damage disappears. Also of interest is the El Niño signal, found not only in the frequency of occurrence of such hurricanes (Gray 1984), but in the adjusted total damage as well (Pielke and Landsea 1999). In the present paper, this dataset will be reanalyzed by making use of a formal stochastic model.
A compound Poisson process is proposed as a stochastic model for the total economic damage associated with hurricanes (e.g., storms that cause damage exceeding a certain threshold). This model consists of two components, one governing the occurrence of events and another specifying the damages associated with individual events. In this way, damage totals are represented as a “random sum,” with variations in total damage being decomposed into two sources, one attributable to variations in the frequency of events and another to variations in the damage from individual events. Although the representation of extreme event damage as a random sum is relatively novel in the climate literature, it has received much attention elsewhere, even being referred to as the “bread and butter of insurance mathematics” (Embrechts et al. 1997). In particular, the statistical approach employed in the present paper is quite close to that in a study of economic damage from extreme wind storms in Sweden by Rootzén and Tajvidi (1997).
The compound Poisson process for total economic damage is described in section 2, with the event occurrence being modeled as a Poisson process and the event damage as a lognormal distribution. Then this model is applied to the hurricane damage data (section 3). Various extensions of the model are considered in section 4, including whether there are any trends in the parameters of the Poisson process for event occurrence and of the lognormal distribution for event damage and what are the effects of the El Niño phenomenon on these parameters. Section 5 consists of a discussion.
Random sum model
In this section, the compound Poisson process is formally defined, including the event occurrence and event damage components. Then the probabilistic properties of a random sum are reviewed, focusing on the decomposition of its variance. Last, techniques for the statistical estimation of the parameters of a compound Poisson process are treated.
Event occurrence
It is assumed that the occurrence of events follows a homogeneous Poisson process N(t), t ≥ 0, denoting the number of events occurring within the time interval [0, t]. This process has a single parameter λ > 0, specifying the instantaneous rate of event occurrence. A Poisson process is intended to represent events that occur at “random,” as it arises from certain axioms thought to correspond to randomness (e.g., Ross 1970, chapter 2). In more general terms, a stochastic model for a series of events is termed a point process (e.g., Guttorp 1995, chapter 5).
An example of a climate application of the Poisson process is the work of Keim and Cruise (1998), who modeled extreme events such as U.S. “nor'easters” and heavy rainfall. The Poisson distribution has been fitted to the frequency of hurricanes by Bove et al. (1998), Elsner and Bossak (2001), and Elsner et al. (1999), including distributions conditional on the state of El Niño. Solow (1995a,b) employed a more general nonparametric approach to the stochastic modeling of El Niño, allowing for point processes that are not necessarily Poisson (also see Solow and Moore 2000).
Event damage
Let Xk > 0 denote the monetary damage associated with the kth event, k = 1, 2, … , and assume that the Xks are independent and identically distributed. Let the common cumulative distribution function be denoted by F(x) = Pr{Xk ≤ x}, with mean μX and variance
Total damage
Parameter estimation
Application to hurricane damage
In this section, the results of fitting the compound Poisson process to a set of hurricane damage data (Pielke and Landsea 1998) are described. For more detailed background about hurricanes and their societal impact, the reader is referred to Pielke and Pielke (1997).
Dataset
Pielke and Landsea (1998) produced a set of damage estimates for individual North Atlantic tropical cyclones (primarily hurricanes) making landfall in the United States (data available at: http://sciencepolicy.colorado.edu/pielke/hp_roger/hurr_norm/data.html). Ranging over the time period 1925–95 (i.e., m = 71 yr), the damage data have been adjusted for inflation, wealth, and population at risk and expressed in 1995 U.S. $ billion. One potential problem with this dataset is that a disproportionate number of years with no storms were reported early in the record, suggesting a bias. To minimize any effects of the recording process and adjustment method and consistent with a focus on events with substantial economic impact, any individual storms whose adjusted damage fell below a threshold of $0.01 billion have been omitted in the present study (15 storms were eliminated, mostly late in the record).
Figure 1 shows the annual time series of total hurricane damage for this adjusted dataset, with no trend evident; instead a high degree of volatility from year to year predominates. Table 1 provides descriptive statistics for the hurricane damage data, with a total of n = 129 damaging storms occurring over the 71-yr time period.
Model fitting
Event occurrence
Figure 2 shows the annual time series of the number of damaging hurricanes, again with no obvious trend. The observed mean rate of hurricanes is slightly less than two per year [an estimate for the rate parameter λ of the Poisson process of 1.817, see (8)], and the sample variance of 1.752 is only slightly smaller (see Table 1). Recall from section 2a that equality of the mean and variance is a fundamental property of the Poisson distribution [see (2)].
Event damage
Because the storms whose damage fell below a threshold of c = $0.01 billion have been removed from the data, the logarithmic transformation is applied to the excess in damage over this threshold [i.e., Yk = ln(Xk − c) instead of (3)]. Figure 3 shows the time series of the logarithm of the adjusted damage from individual hurricanes. Although any pattern is somewhat difficult to interpret because of the varying number of storms from year to year, no trend is evident. For the untransformed damages, the sample mean is much larger than the sample median ($2.697 vs $0.349 billion; see Table 1), indicating that the distribution is highly positively skewed. A quantile–quantile (Q–Q) plot (Fig. 4) of the log-transformed damages versus the normal distribution indicates an acceptable fit, except perhaps in the extreme upper tail.
It remains to check the assumption that, conditional on the number of hurricanes within a year, the individual damages are independent and identically distributed. The damage for the first and second storms within a year is compared (35 yr with 2 or more hurricanes). The sample means of the log-transformed damage are −1.483 and −1.227, respectively (P value = 0.638 for two sample t test of equality of means), or a lack of evidence of nonidentical distributions. The sample correlation coefficient between the log-transformed damage for the first and second storms is −0.036, or a lack of evidence of dependent distributions. The damage for the second and third storms within a year could be compared as well, but the sample size is small (18 yr with three or more hurricanes). If necessary, it would be possible to relax the assumption of independence and identical distribution while still retaining the random sum representation [e.g., as done for total precipitation in Katz and Parlange (1998)].
Total damage
The adjusted annual total damage from hurricanes has a skewed distribution, with the sample mean being much larger than the sample median ($4.901 vs $1.031 billion; see Table 1). But because of the effect of summing up the damage from an, albeit random, number of storms, the degree of skewness (d = 0.617) is somewhat less than that for individual storms. To more fully characterize the sources of variation in total damage, the random sum representation can be utilized. The most general approach to decomposing the variance of total damage is taken, imposing as few conditions as possible. Specifically, the assumptions that the number of storms be Poisson-distributed and that the damages from individual storms be independent and identically distributed (made in section 2) are both relaxed.
Extensions of model
The purpose of this section is to demonstrate ways in which the compound Poisson process for hurricane damage can be generalized. These extensions include trends in the event occurrence and damage processes and the incorporation of covariates such as El Niño.
Trends
Event occurrence
Event damage
El Niño effects
As another extension, the dependence of hurricane damages on the El Niño phenomenon is explored. A connection between the frequency of hurricanes and El Niño has long been known (Gray 1984). More recently, Pielke and Landsea (1999) established a relationship between total hurricane damage and El Niño events. It remains to determine the extent to which this relationship is attributable to the connection between El Niño and the frequency of hurricanes or to any connection between El Niño and the damage associated with individual storms (i.e., through a dependence of the intensity of hurricanes on El Niño).
The classification of Trenberth (1997), as adapted by Pielke and Landsea (1999), is based on sea surface temperatures in the so-called Niño 3.4 region of the Pacific. Each year is classified into one of three possible states of El Niño (j = −1, 0, 1):
La Niña event (j = −1),
neutral event (j = 0), and
El Niño event (j = 1).
Event occurrence
The Poisson rate parameter, say λj, j = −1, 0, 1, of the occurrence of damaging hurricanes now possibly depends on the state of El Niño. Table 4a provides descriptive statistics classified according to the El Niño state, with the estimated rate being nearly one damaging hurricane per year higher during La Niña than El Niño events (a result consistent with many earlier analyses). The fit of five candidate models for the Poisson rate parameter, ranging from no dependence (i.e., λ−1 = λ0 = λ1) to complete dependence (i.e., three different rates) and including those with constraints imposed to reduce the number of parameters estimated, is compared in Table 4b. Here the “deviance” statistic listed in the table for each candidate model is a goodness-of-fit measure, essentially −2 times the maximized log likelihood function. The Bayesian information criterion (BIC), listed in the table for each model, involves penalizing the deviance for the number of parameters, with smaller values being preferable (Katz 1981; Schwarz 1978; Venables and Ripley 1999, chapters 6–7).
Two of the models involving dependence on the state of El Niño have lower BIC values than the case of no dependence; one varies the rate parameter between El Niño and non-El Niño years (i.e., λ−1 = λ0; this model has minimum BIC; see Table 4b), the other constrains the effect to be proportionately the same magnitude for both El Niño and La Niña events (i.e., λ−1/λ0 = λ0/λ1). Comparing the no-effect model with the one constraining λ−1/λ0 = λ0/λ1 (although this model does not attain minimum BIC, it is more physically appealing by allowing both La Niña and El Niño events to have an effect), a likelihood ratio test also indicates statistical significance (P value = 0.029) despite the relatively small decrease in the value of BIC. For both La Niña and El Niño events, the sample variance is quite a bit less than the mean (Table 4a), suggesting that the Poisson may not necessarily be an ideal approximation for these conditional distributions. Bove et al. (1998) also noted a departure from the Poisson distribution for hurricane counts during El Niño events.
Event damage
The mean of the log-transformed adjusted damage from individual hurricanes, say μY(j), j = −1, 0, 1, now possibly depends on the state of El Niño. Although the variance of the log-transformed damage
The fit of five candidate models for the mean of log-transformed damage, ranging from no dependence, [i.e., μY(−1) = μY(0) = μY(1)] to complete dependence (i.e., three different means) and including those with constraints imposed, is compared in Table 5c. Here the mean-square error (MSE) listed in the table for each candidate model is a goodness-of-fit measure, which the BIC penalizes for the number of parameters estimated. All three constrained models have lower BIC values than the case of no dependence, with the best one constraining the effects on the mean to be the same magnitude for both El Niño and La Niña events [i.e., μY(−1) − μY(0) = μY(0) − μY(1)]. Comparing the no-effect model with the one with minimum BIC, a partial F test also indicates statistical significance (P value = 0.006). This result is consistent with the finding of Landsea et al. (1999) that hurricane intensity is higher during La Niña than El Niño.
Total damage
The sample median of adjusted annual total damage from hurricanes ranges from $0.149 to $3.072 billion for El Niño and La Niña events, respectively (Table 6). Using the first equation in (7), the variance of total damage can be decomposed into that attributable to the El Niño phenomenon and that due to other sources (i.e., conditioning on the state of El Niño instead of the number of events). For the statistics listed in Table 6, about 3%–4% of the variance in total damage is attributable to El Niño. In other words, although the El Niño phenomenon is a significant source of variation, much of the variation in total damage remains to be explained.
The relationship (6) expresses the mean of annual total damage in terms of the parameters of both storm occurrence and damage. It can be applied to determine how much of the effect of El Niño on total damage is attributable to the individual effect of El Niño on storm occurrence or on storm damage. If only the rate of storm occurrence is varied using its sample values listed in Table 4a, then the mean of annual total damage would range from $3.468 billion in El Niño to $5.885 billion in La Niña (a difference of $2.417 billion). If only the mean of individual storm damage is varied using its sample values listed in Table 5a, then the mean of annual total damage would range from $2.833 billion in El Niño to $4.573 billion in La Niña (a difference of $1.740 billion). So each component makes a roughly comparable contribution to the effect of the El Niño phenomenon on the mean of total damage.
Discussion
A stochastic modeling approach has been advocated to treat more explicitly different sources of variation in total economic damage from hurricanes. The model has been applied in a reanalysis of a set of adjusted damage data from hurricanes (Pielke and Landsea 1998). The conclusions reached are consistent with those of Pielke and Landsea (1998) concerning the lack of any trends, as well as with other trend analyses of damages from extreme weather or climate events (e.g., Changnon et al. 2000; Kunkel et al. 1999). They also are consistent with those of Pielke and Landsea (1999) concerning the existence of a relationship with the El Niño phenomenon (cf. Landsea et al. 1999).
By enabling the variations in total damage to be attributed to either variations in event occurrence or in event damage, the present modeling approach has an inherent advantage over previous analyses. For instance, it has been shown that the connection between annual total hurricane damage and the state of El Niño is due as much to the effects of El Niño on damage from individual hurricanes as to its effects on the frequency of occurrence of hurricanes. In this regard, it would be straightforward to incorporate other covariates, including continuous variables instead of discrete ones (e.g., a continuous index of the Southern Oscillation instead of the discrete El Niño state). Possible covariates for damage from hurricanes in the North Atlantic were discussed in Landsea et al. (1999).
Although the lognormal distribution has been fitted to the damage from individual hurricanes in the present paper, there is some evidence that the damage distribution has a heavier right-hand tail (Katz 2002). Rootzén and Tajvidi (1997) also found that the damage from windstorms has a distribution whose upper tail is heavier than the lognormal. The question of how extremely high damages are distributed is naturally of particular interest to the insurance and reinsurance industry.
The stochastic approach presented is general enough that it could be readily applied to other extreme weather or climate phenomena besides hurricanes. For example, economic damage from floods raises similar issues to those for hurricanes, in that there are marked increasing trends in damage unadjusted for shifts in societal vulnerability. Despite apparent increasing trends in extreme high precipitation events (e.g., Karl and Knight 1998), it is not necessarily the case that any of these trends in flood damage are actually attributable to climate change (Pielke and Downton 2000).
Acknowledgments
Research was partially supported by NSF Grant DMS-9815344 to the NCAR Geophysical Statistics Project. I thank Roger Pielke, Jr. for suggesting this topic and for providing valuable comments. I also gratefully acknowledge the referees for their constructive suggestions.
REFERENCES
Bove, M. C., J. B. Elsner, C. W. Landsea, X. Niu, and J. J. O'Brien. 1998. Effect of El Niño on U.S. landfalling hurricanes, revisited. Bull. Amer. Meteor. Soc. 79:2477–2482.
Changnon, S. A., R. A. Pielke Jr., D. Changnon, R. T. Sylves, and R. Pulwarty. 2000. Human factors explain increased losses from weather and climate extremes. Bull. Amer. Meteor. Soc. 81:437–442.
Elsner, J. B. and B. H. Bossak. 2001. Bayesian analysis of U.S. hurricane climate. J. Climate 14:4341–4350.
Elsner, J. B., A. B. Kara, and M. A. Owens. 1999. Fluctuations in North Atlantic hurricane frequency. J. Climate 12:427–437.
Embrechts, P., C. Klüppelberg, and T. Mikosch. 1997. Modelling Extremal Events for Insurance and Finance. Springer, 660 pp.
Feller, W. 1968. An Introduction to Probability Theory and Its Applications. Vol. 1, 3d ed. John Wiley and Sons, 509 pp.
Gray, W. M. 1984. Atlantic seasonal hurricane frequency. Part I: El Niño and 30 mb quasi-biennial oscillation influences. Mon. Wea. Rev. 112:1649–1668.
Guttorp, P. 1995. Stochastic Modeling of Scientific Data. Chapman and Hall, 372 pp.
Hinkley, D. 1977. On quick choice of power transformation. Appl. Stat. 26:67–69.
Hogg, R. V. and S. A. Klugman. 1984. Loss Distributions. John Wiley and Sons, 235 pp.
Johnson, N. L. and S. Kotz. 1970. Continuous Univariate Distributions—1. Houghton Mifflin, 300 pp.
Johnson, N. L. and A. W. Kemp. 1992. Univariate Discrete Distributions. 2d ed. John Wiley and Sons, 565 pp.
Karl, T. R. and R. W. Knight. 1998. Secular trends of precipitation amount, frequency, and intensity in the United States. Bull. Amer. Meteor. Soc. 79:231–241.
Katz, R. W. 1981. On some criteria for estimating the order of a Markov chain. Technometrics 23:243–249.
Katz, R. W. . 2002. Techniques for estimating uncertainty in climate change scenarios and impact studies. Climate Res. 20:167–185.
Katz, R. W. and M. B. Parlange. 1998. Overdispersion phenomenon in stochastic modeling of precipitation. J. Climate 11:591–601.
Keim, B. D. and J. F. Cruise. 1998. A technique to measure trends in the frequency of discrete random events. J. Climate 11:848–855.
Kunkel, K. E., R. A. Pielke Jr., and S. A. Changnon. 1999. Temporal fluctuations in weather and climate extremes that cause economic and human health impacts: A review. Bull. Amer. Meteor. Soc. 80:1077–1098.
Landsea, C. W., R. A. Pielke Jr., A. M. Mestas-Nuñez, and J. A. Knaff. 1999. Atlantic basin hurricanes: Indices of climatic changes. Climatic Change 42:89–129.
Pielke, R. A. Jr, and R. A. Pielke Sr.. 1997. Hurricanes: Their Nature and Impacts on Society. John Wiley and Sons, 279 pp.
Pielke, R. A. Jr, and C. W. Landsea. 1998. Normalized hurricane damages in the United States, 1925–95. Wea. Forecasting 13:621–631.
Pielke, R. A. Jr, . 1999. La Niña, El Niño, and Atlantic hurricane damages in the United States. Bull. Amer. Meteor. Soc. 80:2027–2033.
Pielke, R. A. Jr, and M. W. Downton. 2000. Precipitation and damaging floods: Trends in the United States, 1932–97. J. Climate 13:3625–3637.
Rootzén, H. and N. Tajvidi. 1997. Extreme value statistics and windstorm losses: A case study. Scand. Actuarial J. 1997:70–94.
Ross, S. M. 1970. Applied Probability Models with Optimization Applications. Holden-Day, 198 pp.
Schwarz, G. 1978. Estimating the dimension of a model. Ann. Stat. 6:461–464.
Solow, A. R. 1995a. An exploratory analysis of a record of El Niño events, 1800–1987. J. Amer. Stat. Assoc. 90:72–77.
Solow, A. R. . 1995b. Testing for change in the frequency of El Niño events. J. Climate 8:2563–2566.
Solow, A. R. and L. Moore. 2000. Testing for a trend in a partially incomplete hurricane record. J. Climate 13:3696–3699.
Stuart, A. and J. K. Ord. 1991. Kendall's Advanced Theory of Statistics. Vol. 2, 5th ed. Oxford University Press, 605–1323.
Trenberth, K. E. 1997. The definition of El Niño. Bull. Amer. Meteor. Soc. 78:2771–2777.
Venables, W. N. and B. D. Ripley. 1999. Modern Applied Statistics with S-PLUS. 3d ed. Springer, 501 pp.
Annual time series of adjusted total North Atlantic hurricane damage (1995 U.S. $ billion), 1925–95
Citation: Journal of Applied Meteorology 41, 7; 10.1175/1520-0450(2002)041<0754:SMOHD>2.0.CO;2
Annual time series of number of damaging North Atlantic hurricanes, 1925–95
Citation: Journal of Applied Meteorology 41, 7; 10.1175/1520-0450(2002)041<0754:SMOHD>2.0.CO;2
Time series of logarithm (ln) of adjusted damage from individual North Atlantic hurricanes, 1925–95
Citation: Journal of Applied Meteorology 41, 7; 10.1175/1520-0450(2002)041<0754:SMOHD>2.0.CO;2
Q–Q plot for logarithm (ln) of adjusted damage from individual North Atlantic hurricanes vs normal distribution
Citation: Journal of Applied Meteorology 41, 7; 10.1175/1520-0450(2002)041<0754:SMOHD>2.0.CO;2
Descriptive statistics for compound Poisson model of adjusted damage (1995 U.S. $ billion) from North Atlantic hurricanes, 1925–95
Decomposition of variance of adjusted annual total damage from North Atlantic hurricanes ($ billion)
Fit and test of trends (yr−1) in parameters of compound Poisson model for adjusted North Atlantic hurricane damage
Fit and identification of conditional Poisson model for North Atlantic hurricane occurrence given El Niño state
Fit and identification of conditional lognormal distribution for adjusted damage from individual North Atlantic hurricanes given El Niño state
Descriptive statistics for conditional distribution of adjusted annual total damage ($ billion) from North Atlantic hurricanes given El Niño state
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
