Cluster Models

A cluster is a group of the same or similar events close together. Clusters arise in hurricane origin locations, tracks, and landfalls. In this chapter, we look at how to analyze and model clusters. We divide the chapter into methods for time, space, and feature clustering. Of the three feature clustering is best known to climatologists. We begin by showing you how to detect and model time clusters. Consecutive hurricanes originating in the same area often take similar paths. This grouping, or clustering, increases the potential for multiple landfalls above what you expect from random events. A statistical model for landfall probability captures clustering through covariates like the North Atlantic Oscillation (NAO), which relates a steering mechanism (position and strength of the subtropical high pressure) to coastal hurricane activity. But there could be additional time correlation not related to the covariates. A model that does not account for this extra variation will underestimate the potential for multiple hits in a season. Following Jagger and Elsner (2006), you consider three coastal regions including the Gulf Coast, Florida, and the East Coast (Fig. 6.2). Regions are large enough to capture enough hurricanes, but not too large as to include many non-coastal strikes. Here you use hourly position and intensity data described in Chapter 6. For each hurricane, you note its wind speed maximum within each region. If the maximum wind exceeds 33 m s−1, then you count it as a hurricane for the region. A tropical cyclone that affects more than one region at hurricane intensity is counted in each region. Because of this, the sum of the regional counts is larger than the total count. Begin by loading annual.RData. These data were assembled in Chapter 6. Subset the data for years starting with 1866. . . . > load("annual.RData") > dat = subset(annual, Year >= 1866) . . . The covariate Southern Oscillation Index (SOI) data begins in 1866 . Next, extract all hurricane counts for the Gulf coast, Florida, and East coast regions. . . . > cts = dat[, c("G.1", "F.1", "E.1")] . . .

Data Sets

Hurricane data originate from careful analysis of past storms by operational meteorologists. The data include estimates of the hurricane position and intensity at 6-hourly intervals. Information related to landfall time, local wind speeds, damages, and deaths, as well as cyclone size, are included. The data are archived by season. Some effort is needed to make the data useful for hurricane climate studies. In this chapter, we describe the data sets used throughout this book. We show you a work flow that includes importing, interpolating, smoothing, and adding attributes. We also show you how to create subsets of the data. Code in this chapter is more complicated and it can take longer to run. You can skip this material on first reading and continue with model building in Chapter 7. You can return here when you have an updated version of the data that includes the most recent years. Most statistical models in this book use the best-track data. Here we describe these data and provide original source material. We also explain how to smooth and interpolate them. Interpolations are needed for regional hurricane analyses. The best-track data set contains the 6-hourly center locations and intensities of all known tropical cyclones across the North Atlantic basin, including the Gulf of Mexico and Caribbean Sea. The data set is called HURDAT for HURricane DATa. It is maintained by the U.S. National Oceanic and Atmospheric Administration (NOAA) at the National Hurricane Center (NHC). Center locations are given in geographic coordinates (in tenths of degrees) and the intensities, representing the one-minute near-surface (∼10 m) wind speeds, are given in knots (1 kt = .5144 m s−1) and the minimum central pressures are given in millibars (1 mb = 1 hPa). The data are provided in 6-hourly intervals starting at 00 UTC (Universal Time Coordinate). The version of HURDAT file used here contains cyclones over the period 1851 through 2010 inclusive. Information on the history and origin of these data is found in Jarvinen et al (1984). The file has a logical structure that makes it easy to read with a FORTRAN program. Each cyclone contains a header record, a series of data records, and a trailer record.

Graphs and Maps

Graphs and maps help you reason with data. They also help you communicate results. A good graph gives you the most information in the shortest time, with the least ink in the smallest space (Tufte, 1997). In this chapter, we show you how to make graphs and maps using R. A good strategy is to follow along with an open session, typing (or copying) the code as you read. Before you begin make sure you have the following data sets available in your workspace. Do this by typing . . . > SOI = read.table("SOI.txt", header=TRUE) > NAO = read.table("NAO.txt", header=TRUE) > SST = read.table("SST.txt", header=TRUE) > A = read.table("ATL.txt", header=TRUE) > US = read.table("H.txt", header=TRUE) . . . Not all the code is shown but all is available on our Web site. It is easy to make a graph. Here we provide guidance to help you make informative graphs. It is a tutorial on how to create publishable figures from your data. In R you have several choices. With the standard (base) graphics environment, you can produce a variety of plots with fine details. Most of the figures in this book use the standard graphics environment. The grid graphics environment is even more flexible. It allows you to design complex layouts with nested graphs where scaling is maintained upon resizing. The lattice and ggplot2 packages use grid graphics to create more specialized graphing functions and methods. The spplot function for example is plot method built with grid graphics that you will use to create maps. The ggplot2 package is an implementation of the grammar of graphics combining advantages from the standard and lattice graphic environments. It is worth the effort to learn. We begin with the standard graphics environment. A box plot is a graph of the five-number summary. The summary function applied to data produces the sample mean along with five other statistics including the minimum, the first quartile value, the median, the third quartile value, and the maximum. The box plot graphs these numbers. This is done using the boxplot function.

Spatial Models

In Chapter 7, annual counts were used to create rate models, and in Chapter 8, lifetime maximum winds were used to create intensity models. In this chapter, we show you how to use cyclone track data together with climate field data to create spatial models. Spatial models make use of location information in data. Geographic coordinates locate the hurricane’s center on the surface of the earth and wind speed provides an attribute. Spatial models make use of location separate from attributes. Given a common spatial framework, these models can accommodate climate data including indices (e.g., North Atlantic Oscillation) and fields (e.g., sea-surface temperature). Here, we show you how to create a spatial framework for combining hurricane data with climate data. The method tessellates the basin with hexagons and populates them with local cyclone and climate information (Elsner et al., 2012). In Chapter 5, you learned how to create a spatial data frame using functions from the sp package (Bivand et al., 2008). Let us review. Here you are interested in wind speeds along the entire track for all tropical storms and hurricanes during the 2005 North Atlantic season. You begin by creating a data frame from the best.use.RData file, where you subset on year and wind speed and convert the speed to meters per second. . . . > load("best.use.RData") > W.df = subset(best.use, Yr==2005 & WmaxS >= 34 + & Type=="*") > W.df$WmaxS = W.df$WmaxS * .5144 . . . The asterisk for Type indicates a tropical cyclone as opposed to a tropical wave or extratropical cyclone. The number of rows in your data frame is the total number of cyclone hours (3,010), and you save this by typing . . . > ch = nrow(W.df) . . . Next, assign the lon and lat columns as spatial coordinates using the coordinates function (sp). Finally, make a copy of your data frame, keeping only the spatial coordinates and the wind speed columns.

Intensity Models

Strong hurricanes, such as Camille in 1969, Andrew in 1992, and Katrina in 2005, cause catastrophic damage. It is important to have an estimate of when the next big one will occur. You also want to know what influences the strongest hurricanes and whether they are getting stronger as the earth warms. This chapter shows you how to model hurricane intensity. The data are basinwide lifetime highest intensities for individual tropical cyclones over the North Atlantic and county-level hurricane wind intervals. We begin by considering trends using the method of quantile regression and then examine extreme-value models for estimating return periods. We also look at modeling cyclone winds when the values are given by category, and use Miami-Dade County as an example. Here you consider cyclones above tropical storm intensity (≥ 17 m s−1) during the period 1967–2010, inclusive. The period is long enough to see changes but not too long that it includes intensity estimates before satellite observations. We use “intensity” and “strength” synonymously to mean the fastest wind inside the cyclone. Consider the set of events defined by the location and wind speed at which a tropical cyclone first reaches its lifetime maximum intensity (see Chapter 5). The data are in the file LMI.txt. Import and list the values in 10 columns of the first 6 rows of the data frame by typing . . . > LMI.df = read.table("LMI.txt", header=TRUE) > round(head(LMI.df)[c(1, 5:9, 12, 16)], 1). . . The data set is described in Chapter 6. Here your interest is the smoothed intensity estimate at the time of lifetime maximum (WmaxS). First, convert the wind speeds from the operational units of knots to the SI units of meter per second. . . . > LMI.df$WmaxS = LMI.df$WmaxS * .5144 . . . Next, determine the quartiles (0.25 and 0.75 quantiles) of the wind speed distribution. The quartiles divide the cumulative distribution function (CDF) into three equal-sized subsets. . . . > quantile(LMI.df$WmaxS, c(.25, .75)) 25% 75% 25.5 46.0 . . . You find that 25 percent of the cyclones have a lifetime maximum wind speed less than 26 m s−1 and 75 percent have a maximum wind speed less than 46ms−1, so that 50 percent of all cyclones have a maximum wind speed between 26 and 46 m s−1 (interquartile range–IQR).

Classical Statistics

All hurricanes are different. Statistics helps you characterize hurricanes from the typical to the extreme. In this chapter, we provide an introduction to classical (or frequentist) statistics. To get the most out of it, we again encourage you to open an R session and type in the code as you read along. Descriptive statistics are used to summarize your data. The mean and the variance are good examples. So is correlation. Data can be a set of weather records or output from a global climate model. Descriptive statistics provide answers to questions like does Jamaica experience more hurricanes than Puerto Rico? In Chapter 2, you learned some functions for summarizing your data, let us review. Recall that the data set H.txt is a list of hurricane counts by year making landfall in the United States (excluding Hawaii). To input the data and save them as a data object, type . . . > H = read. Table ("H.txt", header=TRUE) . . . Make sure the data file is located in your working directory. To check your working directory, type getwd (). Sometimes all you need are a few summary statistics from your data. You can obtain the mean and variance by typing . . . > mean (H$All); var (H$All) [1] 1.69375 [1] 2.10059 . . . Recall that the semicolon acts as a return so you can place multiple functions on the same text line. The sample mean is a measure of the central tendency and the sample variance is a measure of the spread. These are called the first-and second-moment statistics. Like all statistics, they are random variables. A random variable can be thought of as a quantity whose value is not fixed; it changes depending on the values in your sample. If you consider the number of hurricanes over a different sample of years, the sample mean will almost certainly be different. Same with the variance. The sample mean provides an estimate of the population mean (the mean over all past and future years).

R Tutorial

This chapter is a tutorial on using R. To get the most out of it, you should open an R session and type the commands into the console as you read the text. You should be able to use copy-and-paste if you have access to an electronic version of the book. All code is available on the book’s Web site. Science requires transparency and reproducibility. The R language for statistical modeling makes this easy. Developing, maintaining, and documenting your R code is simple. R contains numerous functions for organizing, graphing, and modeling your data. Directions for obtaining R, accompanying packages, and other sources of documentation are available at http://www.r-project.org/. Anyone serious about applying statistics to climate data should learn R. The book is self-contained. It presents R code and data (or links to data) that can be copied to reproduce the graphs and tables. This reproducibility provides you with an enhanced learning opportunity. Here we present a tutorial to help you get started. This can be skipped if you already know how to work with R. R is the ‘lingua franca’ of data analysis and statistical computing. It helps you perform a variety of computing tasks by giving you access to commands. This is similar to other programming languages such as Python and C++. R is particularly useful to researchers because it contains a number of built-in functions for organizing data, performing calculations, and creating graphics. R is an open-source statistical environment modeled after S. The S language was developed in the late 1980s at AT&T labs. The R project was started by Robert Gentleman and Ross Ihaka of the Statistics Department of the University of Auckland in 1995. It now has a large audience. It is currently maintained by the R core-development team, an international group of volunteer developers. To get to the R project Web site, open a browser and, in the search window, type the keywords “R project” or directly link to the Web page using http://www.r-project.org/. Directions for obtaining the software, accompanying packages, and other sources of documentation are provided at the site.

Bayesian Statistics

Classical statistics involves ways to test hypotheses and estimate confidence intervals. Bayesian statistics involves methods to calculate probabilities associated with your hypotheses. The result is a posterior distribution that combines information from your data with prior beliefs. The term “Bayesian” comes from Bayes’s theorem—a single formula for obtaining the posterior distribution. It is the cornerstone of modern statistical methods. In this chapter we introduce Bayesian statistics. We begin by considering the problem of learning about the population proportion (percentage) of all hurricanes that make landfall. We then consider the problem of estimating how many days it takes your manuscript to get accepted for publication. Again, we encourage you to follow along by typing the code into your R console. Models that have skill at forecasting hurricane counts are more relevant to society if they include an estimate of the proportion that make landfall. Before examining the data, you hold a belief about the value of this proportion. You model your belief in terms of a prior distribution. Then after examining some data, you update your belief about the proportion by computing the posterior distribution (Albert, 2009). This setting allows you to predict the likely outcomes of a new sample taken from the population, for example, the proportion of landfalls for next year. The use of the pronoun “you” focuses attention on the Bayesian viewpoint that probabilities do not exist but are how much you personally believe that something is true. Said another way, probabilities are subjective and based on all the relevant information available to you. Here you think of a population consisting of past and future hurricanes in the North Atlantic. Then let π represent the proportion of this population that hit the United States at hurricane intensity. The value of π is unknown. You are interested in learning about what the value of π could be. Bayesian statistics requires that you represent your belief about the uncertainty in this percentage with a probability distribution. The distribution reflects your subjective prior opinion about plausible values of π. Before you examine a sample of hurricanes, you think about what the value of π might be.

Hurricanes, Climate, and Statistics

This book is about hurricanes, climate, and statistics. These topics may not seem related. Hurricanes are violent winds and flooding rains, climate is about weather conditions from the past, and statistics is about numbers. But what if you wanted to estimate the probability of winds exceeding 60 ms−1 in Florida next year. The answer involves all three, hurricanes (fastest winds), climate (weather of the past), and statistics (probability). This book teaches you how to answer these questions in a rigorous and scientific way. We begin here with a short description of the topics and a few notes on what this book is about. A hurricane is an area of low air pressure over the warm tropical ocean. The low pressure creates showers and thunderstorms that start the winds rotating. The rotation helps to develop new thunderstorms. A tropical storm forms when the rotating winds exceed 17 ms−1 and a hurricane when they exceed 33 ms−1. Once formed, the winds continue to blow despite friction by an in-up-and-out circulation that imports heat at high temperature from the ocean and exports heat at lower temperature in the upper troposphere (near 16 km), which is similar to the way a steam engine converts thermal energy to mechanical motion. In short, a hurricane is powered by moisture and heat. Strong winds are a hurricane’s defining characteristic. Wind is caused by the change in air pressure between two locations. In the center of a hurricane, the air pressure, which is the weight of a column of air from the surface to the top of the atmosphere, is quite low compared with the air pressure outside the hurricane. This difference causes the air to move from the outside inward toward the center. By a combination of friction as the air rubs on the ocean below and the spin of the earth as it rotates on its axis, the air does not move directly inward but rather spirals in a counter clockwise direction toward the region of lowest pressure.

Impact Models

In this chapter, we show some broader applications of our models and methods. We focus on impact models. Hurricanes are capable of generating large financial losses. We begin with a model that estimates extreme losses conditional on climate covariates. We then describe a method for quantifying the relative change in potential losses over the decades. Financial losses from hurricanes are to some extent directly related to fluctuations in climate. Environmental factors influence the frequency and intensity of hurricanes at the coast as detailed throughout this book (see for example Chapters 7 and 8). So, it is not surprising that these same environmental signals appear in estimates of losses. Here loss is the economic damage associated with a hurricane’s direct impact. A normalization procedure adjusts the loss estimate from a past hurricane to what it would be if the same cyclone struck in a recent year by accounting for inflation and changes in wealth and population over the intervening time, plus a factor to account for changes in the number of housing units exceeding population growth. The method produces loss estimates that can be compared over time (Pielke et al. 2008). Here you focus on losses exceeding one billion ($ U.S.) that have been adjusted to 2005. The loss data are available in Losses.txt in JAGS format (see Chapter 9). Input the data by typing . . . > source("Losses.txt"). . . The log-transformed loss amounts are in the column labeled ‘y’. The annual number of loss events are in the column labeled ‘L’. The data cover the period 1900–2007. More details about these data are given in Jagger et al. (2011). You begin by plotting a time series of the number of losses and a histogram of total loss per event. . . . > layout(matrix(c(1, 2), 1, 2, byrow=TRUE), + widths=c(3/5, 2/5)) > plot(1900:2007, L, type="h", xlab="Year", + ylab="Number of Loss Events") > grid() > mtext("a", side=3, line=1, adj=0, cex=1.1) > hist(y, xlab="Loss Amount ($ log)", + ylab="Frequency", main="") > mtext("b", side=3, line=1, adj=0, cex=1.1) . . .