Exploring possibly increasing trend of hurricane activity by a SiZer approach

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1 Exploring possibly increasing trend of hurricane activity by a SiZer approach Jesper Rydén U.U.D.M. Report 2006:32 ISSN Department of Mathematics Uppsala University

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3 Exploring possibly increasing trend of hurricane activity by a SiZer approach Jesper Rydén Department of Mathematics Uppsala University Abstract Statistical studies of hurricane activity in the North Atlantic basin are of current interest, not the least after the year of 2005 when many records were broken. An exploratory analysis of possibly increasing trend of the yearly number of hurricanes can be carried out by employing a graphic device, the SiZer map, for visualization of features with respect to both location and scale. This is a smoothing method based on scale-space ideas. Use of this method provides a useful complement to other statistical techniques for detection of trends or shifts in time series. Key words: Kernel estimation; Local regression; Smoothing; Poisson regression; Climate change 1 Introduction Discussions on climate change are intense. Broadly speaking, one aspect is whether certain events, for instance storms or hurricanes, have been more common recently; another if such events (possibly in addition) have become more severe and violent. A difficult problem related to the later issue (not discussed in this paper) is the relation to climate change: are changes to occur statistically speaking, as a part of the inherent variability of nature, or are these an effect from human activity. Focusing on hurricanes in the Atlantic, the season 2005 resulted in many broken records: 27 named storms, 14 hurricanes, 7 major hurricanes, 4 hurricanes of Category 5. Moreover, the hurricane Katrina implied very large and severe losses. Researchers agree that a change in intensity has occurred in recent years, see for instance Webster et al. (2005). When investigating a possible trend over time for the yearly number of hurricanes, analyses are often based on regression models where the number of hurricanes is supposed to be a random variable dependent on covariates like North Atlantic 1

4 oscillation index (Elsner, Jagger and Niu 2000). Landsea et al. (1999) found that only weak linear trends can be ascribed to hurricane activity and that multidecal variability is more characteristic of the region. To test for a trend in a time series could be a difficult statistical problem, in particular if residuals are correlated; cf. Woodward et al. (1997). In this paper, we take a pragmatic approach and focus on the question of increasing trend of the yearly number of hurricanes by modelling only this quantity of interest, hence no covariates involved. The purpose of the paper is to point out what can be deduced by a technique with origin in scale-space theory. Such methods have been used in many applications in image analys. Statisticians working with environmental data should be aware of this approach as a useful data-analytical tool. We will demonstrate the use of the exploratory graphical method known as the SiZer (Significance of Zero crossings of the derivative) map, which can be used to investigate jumps in time series (Chaudhuri and Marron 1999). Using statistical terminology, SiZer is a smoothing device where several bandwidths are studied simultaneously. As a benchmark, we compare with what a simpler, conventional method like Poisson regression implies; in that case, a statistical hypothesis test only tell whether to reject or not the hypothesis of a constant trend. The advantage of the SiZer approach is the possibility to investigate several time scales at the same time, presented in one single image. An application to climatology published recently was given by Weckström et al. (2006). The paper is organised as follows: Next, in Section 2, the data set of hurricanes is briefly presented. In Section 3, the SiZer methodology is reviewed and applied to hurricane data. A comparison is made with a standard approach based on Poisson regression. An index measuring the strength of hurricanes (ACE index) can be introduced; a further study of the time series of such observations is performed in Section 4. Finally, in Section 5, some conclusions are given. 2 Data set The region of the Atlantic basin has been studied in this paper. Data are courtesy of Tropical Prediction Center and are presented at a homepage of Unisys Weather, In the sequel hurricanes from are studied. Seasons offically run from 1 June to 30 November. Hurricane activity has been tabulated back to 1871, but normally the portion dating from about 1944 is considered reliable. Use of aircraft monitoring was then introduced. Before the advent of satellites and aircraft monitoring it was difficult to detect storms that did not affect land or ships. This should be kept in mind when analysing older data: the activity in 2

5 some seasons before the middle of the 20th century might be under-estimated. Hurricanes can be divided into 5 categories according to the Saffir Simpson scale which is a 1-5 rating based on the present intensity of the hurricane. The scale is used to give an estimate of the potential property damage and flooding expected along the coast from a hurricane landfall. Wind speed is the determining factor in the scale. Major hurricanes are usually defined as those in Category 3 or higher. For reference, the definitions of categories are given in Table 1. Table 1. Saffir Simpson scale defining categories of hurricanes. Category Wind speed (km/h) > Analysis by SiZer 3.1 Background The SiZer technique can be considered a smoothing method. A central question to be posed when facing data is which features are significant, as opposed to sampling artifacts. We here give a brief review; see the original papers by Chaudhuri and Marron (1999), (2000) for full details. Consider first the simple case of a random sample x 1,..., x n from a smooth density f(x). The basic idea of kernel estimation is then to estimate f by ˆf h (x) = 1 n n K h (x x i ) i=1 where h is a smoothing parameter called bandwidth and K h results from a kernel function K, K h ( ) = (1/h)K( /h). A common choice of K is a Gaussian kernel. The estimate ˆf h (x) is wiggly when h is small and very smoothed when h is large. Consider now a random sample (x 1, y 1 ),..., (x 2, y 2 ). Local linear regression is used, that is, a line y = ax + b is fitted to data using weighted least squares in a moving window, the width of which is controlled by a bandwidth h. At each location x, x [0, 1], estimates â h (x) and ˆb h (x) are found by minimizing n [ { ( i y i a n x i=1 ) }] 2 ( ) i + b K h n x where again K h ( ) = (1/h)K( /h) is a kernel function K governing the shape of the local window around x. SiZer is based on the slope estimate â h (x) and its sign. As shown 3

6 by Kim and Marron (2006), the locally weighted estimate of the standard deviation of â, ˆσ a (x), say, is given by ˆσ a (x) = { n i=1 [ ( )] i 2 ( ) } 1/2 i y i â K h n n x and significance is determined by comparing â h (x)/ˆσ a (x) with an appropriate quantile q of the Gaussian distribution. Further discussion of choices of the quantile is given in Chaudhuri and Marron 1999, Section Visualization: hurricane data In Figure 1 the hurricane data set is presented. The time series of total number of yearly hurricanes is investigated for the time period In the top panel a family of smoothed curves are shown; the original observations are shown as dots. In the bottom panel, the derivatives are visualized with respect to both location and scale (bandwidth). Here, a light shade means the smooth is significantly decreasing, a dark shade significantly increasing, and a medium grey shade is used for no significant slope. Moreover, the dotted curves indicate effective window width for each bandwidth as intervals represent ±2h. From the bottom panel we note a significantly increasing trend from around 1995 (significance level 0.05). The darker grey region at the bottom indicates there were not enough data in the smoothing window. The advantage of the SiZer map is obvious from the figure: the problem from analysis with kernel methods of choosing a suitable bandwidth is less pronounced since a collection of kernel smooths are presented simultaneously. The solid horizontal bar in the SiZer plot shows a good data-driven bandwidth as suggested by Ruppert et al. (1995). We observe that the region of significant increase occurs for bandwidths larger than that value but nevertheless the result is telling since a bandwidth or window of order 10 years is of interest in this application. In Figure 2 we study the total number of yearly major hurricanes (categories 3-5). We observe a significant increasing trend from around 1990, while a decreasing trend is found for the period (significance level 0.05). 3.3 A Poisson regression model Assume first that we study the total yearly number of hurricanes. Clearly, we model a discrete response variable, Y say, assumed to have a Poisson distribution. Suppose that the mean of Y is µ. In its most common form, the Poisson regression model is a 4

7 Overlay, families SiZer map log 10 (h) Figure 1: SiZer analysis of total number of yearly hurricanes. smooths. Bottom: SiZer map. Top: Overlay of family of special case of a generalized linear model with relates the mean to a linear function of covariates. In our case, we formulate the model µ i = t i exp(β 0 + β 1 x i ), i = 1,..., n (1) where n is the number of years considered, t i is the exposure time for each observation (one year, hence t i = 1, all i), and x i is an explanatory variable which is year. This explanatory variable is introduced since the purpose is to test for a possible trend over time: is the number of hurricanes increasing over the time period studied? The statistical test for testing if β 1 = 0 is made by using the deviance. Parameter estimates are returned as ML estimates and l being the log-likelihood function, the deviance is given by D = 2(l(β 0, β 1 ) l(β 0, β 1 )) This is used as follows: If D > χ 2 1 (α), reject the hypothesis. Comparison with table values χ 2 1 (0.05) = 3.84, χ2 1 (0.01) = 6.63 was made for various partitions of the data set: total number of (major) hurricanes , , The only rejection of the hypothesis β 1 = 0 occurred for major hurricanes (D = 14.32). This is in agreement with the findings from the SiZer maps earlier. However, note that the increase for the total number of yearly hurricanes was not considered significant for any time period using the Poisson regression. 5

8 8 Overlay, families SiZer map log 10 (h) Figure 2: SiZer analysis of total number of yearly major hurricanes. Top: Overlay of family of smooths. Bottom: SiZer map. To justify the use of Poisson regression, the possible presence of over-dispersion should be checked. A simple confidence interval for the ratio V[Y ]/E[Y ] can be constructed following Rychlik and Rydén (2006), Section 7.3 (see also Brown and Zhao 2002). We find the 0.95 confidence interval [0.64, 1.20] and hence the hypothesis of Poisson distribution is not rejected since 1 is within the interval. (We content ourselves with this simpler test, cf. Spinelli et al. (2002) for a discussion of further tests of this type.) 4 Analysis of ACE Hitherto in this paper the yearly number of hurricanes has been analysed. In this subsection the SiZer technique is used to study the intensity of hurricanes. The National Oceanic and Atmospheric Administration (NOAA) uses an Accumulated Cyclone Energy (ACE) index as a measure of total seasonal activity. This refers to the collective intensity and duration of named storms and hurricanes in the North Atlantic during a given season. The ACE index is defined as the sum of the squares of the maximum sustained surface wind speed measured every six hours for all named systems while they are at least tropical storm strength. In Figure 3, a SiZer plot is given for the time series of ACE. The bottom panel 6

9 is used to study the smoothing at different scales and we note a significant increase, starting in the end of the 1980s. For the earlier part of the data set, , on the other hand a significant decrease in ACE index is observed (significance level 0.05). This reminds about the features found in Figure Overlay, families SiZer map log 10 (h) Figure 3: SiZer analysis of ACE. Top: Overlay of family of smooths. Bottom: SiZer map. 5 Conclusions Detection of change point and/or trends are important and common problems in statistical analysis of environmental data. An exploratory analysis by the SiZer method provides a useful complement to other statistical techniques and deserves to be more well-known to statisticans in the field. The debate on increasing trend of hurricane activity has now and then been intense in the research community. The result provided by this tool, originating from kernel-estimation methodology and scale-space theory, renders a comprehensible visualization of data and in addition a statistical test for significance of slope. For applications of the type presented in this paper, by first applying the SiZer analysis a hint of interesting time periods can be obtained and further statistical tests in those regions can then be performed by other methods, for example techniques based on Poisson regression. 7

10 6 Acknowledgement I am grateful for the publicly available software for SiZer analysis in the form of Matlab routines written by Prof. James S. Marron, Department of Statistics, University of North Carolina, Chapel Hill. These were used to a large extent for the analysis carried out in this paper. References Brown, L.D. and Zhao, L.H. (2002). A test of the Poisson distribution. Sankhya, 64, Chaudhuri, P., Marron, J.S. (1999). SiZer for exploration of structures in curves. Journal of the American Statistical Association, 94, Chaudhuri, P. and Marron, J.S. (2000). Scale space view of curve estimation. Annals of Statistics, 28, Elsner, J.B., Jagger, T., and Niu, X.-F. (2000). Changes in the rates of North Atlantic major hurricane activity during the 20th century. Geophysical Research Letters, 27, Kim, C.S. and Marron, J.S. (2006). SiZer for jump detection. Nonparametric Statistics, 18, Landsea, C.W., Pielke Jr, R.A., Mestas-Nunez, A.M., and Knaff, J.A. (1999). Atlantic basin hurricanes: indices of climatic changes. Climatic Change, 42, Ruppert, D., Sheather, M.J., and Wand, M.P. (1995). An effective bandwidth selector for least squares regression. Journal of the American Statistical Association 90, Rychlik, I. and Rydén, J. (2006). Probability and Risk Analysis. An Introduction for Engineers. Springer Verlag. Spinelli, J.J., Lockhart, R.A., and Stephens, M.A. (2002). Tests for the response distribution in a Poisson regression model. Journal of statistical planning and inference, 108, Webster, P.J., Holland, G.J., Curry, J.A., and Chang, H.-R. (2005). Changes in tropical cyclone number, duration, and intensity in a warming environment. Science, 309, Weckström, J., Korhola, A., Erästö, P., and Holmström, L. (2006). Temperature patterns over the past eight centuries in Northern Ferroscandia inferred from sedimentary diatoms. Quaternary Research, 66,

11 Woodward, W.A., Bottone, S., Gray, H.L. (1997). Improved tests for trend in time series data. Journal of Agricultural, Biological, and Environmental Statistics, 2,