Statistical modeling of hot spells and heat waves

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Statistical modeling of hot spells and heat waves 3

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Eva M. Furrer*1, Richard W. Katz 1, Marcus D. Walter 2 and Reinhard 8

Furrer 3 9

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1 Institute for Mathematics Applied to Geosciences, 12

National Center for Atmospheric Research, Boulder, CO 80307 13

2 Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY 14

14850 15

3 Institut für Mathematik, Universität Zürich, CH-8057 Zürich 16

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* Email: [email protected]

Statistical Modeling of Hot Spells and Heat Waves 1

Abstract: 20

Although hot spells and heat waves are considered extreme meteorological phenomena, 21

the statistical theory of extreme values has only rarely, if ever, been applied. To address 22

this shortcoming, we extend the point process approach to extreme value analysis to 23

model the frequency, duration, and intensity of hot spells. The annual frequency of hot 24

spells is modeled by a Poisson distribution, their length by a geometric distribution. To 25

account for the temporal dependence of daily maximum temperatures within a hot spell, 26

the excesses over a high threshold are modeled by a conditional generalized Pareto 27

distribution, whose scale parameter depends on the excess on the previous day. Requiring 28

only univariate extreme value theory, our proposed approach is simple enough to be 29

readily generalized to incorporate trends in hot spell characteristics. Through a heat wave 30

simulator, the statistical modeling of hots spells can be extended to apply to more full-31

fledged heat waves, which are difficult to model directly. 32

Our statistical model for hot spells is fitted to time series of daily maximum 33

temperature during the summer heat wave season at Phoenix, AZ, USA, Fort Collins, CO, 34

USA, and Paris, France. Trends in the frequency, duration, and intensity of hot spells are 35

fitted as well. The heat wave simulator is used to convert any such trends into the 36

corresponding changes in the characteristics of heat waves. By being based at least in part 37

on extreme value theory, our proposed approach is demonstrated to be both more realistic 38

and more flexible than techniques heretofore applied to model hot spells and heat waves. 39

40

Key words: Climate change; Clustering of extremes; Generalized Pareto 41

distribution; Point process approach; Heat wave simulator. 42

43


1. Introduction 44

45

Heat waves are meteorological events that have received much attention in recent years, 46

given the mortality associated with them (Gosling et al., 2009) and given the specter of 47

trends in their frequency, duration, and severity as part of global climate change (Meehl 48

and Tebaldi, 2004). In particular, the high mortality associated with the 2003 European 49

heat wave generated much concern about whether climate change is playing a role (Schär 50

et al., 2004). Other recent heat waves of note include the 1995 event in Chicago, IL, USA 51

(Karl and Knight, 1997). Because of their rarity and because of their severity, such events 52

are naturally viewed as “extreme”. But statistical methods based on extreme value theory 53

(e.g., Coles, 2001) have only rarely, if ever, been applied to this type of meteorological 54

event in realistic climate applications. Even the statistical analysis of projections of future 55

changes in heat wave characteristics, on the basis of climate change experiments using 56

numerical models of the climate system, has generally avoided any use of the statistics of 57

extremes (Koffi and Koffi, 2008, Tebaldi et al., 2006). 58

Yet there is a long tradition of using statistical methods based on extreme value 59

theory in the analysis of simple extreme meteorological events, most commonly in the 60

form of the highest daily precipitation amount over a year or the highest temperature over 61

the summer season (Gumbel, 1958). While such analyses typically assume stationarity 62

(i.e., an unchanging climate), they are starting to be extended to the case of temporal 63

trends (e.g., Katz et al., 2002). The so-called point process approach is a parsimonious 64

way to model possibly non-stationary extremes, jointly modeling the occurrence of an 65

event (e.g., an exceedance of a high threshold) and its severity (e.g., an excess over a high 66

threshold) (Coles, 2001, Smith, 1989). This approach has recently been applied to detect 67

trends in high temperature extremes (Brown et al., 2008). Other meteorological 68

applications using the point process model are included in Furrer and Katz (2008) and 69


Katz et al. (2002). In such analyses, it is common to “decluster” the data and model only 70

cluster maxima to account for temporal dependence. In the present application, these 71

clusters constitute hot spells whose characteristics need to be modeled as well (especially 72

hot spell length and temporal dependence of excesses within a hot spell) rather than 73

discarded. 74

Hot spells and, to an even greater extent, heat waves have a complex temporal 75

structure that makes the application of extreme value theory less than routine. Although 76

some analyses have made at least limited use of the theory, the attempts to date have 77

tended to be rather ad hoc, among other things tied to somewhat arbitrary definitions of 78

hot spells or heat waves (Abaurrea et al., 2007, Katsoulis and Hatzianastassiou, 2005, 79

Khaliq et al., 2005, 2007). Part of the problem relates to the difficulty in defining a heat 80

wave, involving a choice of threshold, a minimal duration, and possibly other variables 81

besides daily maximum temperature (Robinson, 2001, Meze-Hausken, 2008). As will be 82

seen, an approach focused on hot spells, which are simply defined as consecutive days 83

with maximum temperature over a certain threshold, with the statistical modeling based 84

at least in part on extreme value theory, results in sufficient flexibility to be applicable to 85

a wide variety of more complicated definitions of a heat wave. 86

In the statistical modeling of hot spells, it is essential that the temporal dependence 87

of extreme high daily maximum temperature be realistically modeled (Kysely, 2002, 88

Mearns et al., 1984). In the statistics literature, models based on bivariate extreme value 89

theory have been proposed to account for the persistence of temperature at high (or low) 90

levels (Coles et al., 1994). In the present paper, we propose a simpler, but closely related 91

approach that only makes use of the more familiar univariate extreme value theory and 92

readily available software. All calculations in this work have been done with the free 93

software environment for statistical computing and graphics R, using the packages ismev 94

and extRemes, see http://www.R-project.org and R Development Core Team (2009). One 95


advantage of the proposed approach is being parsimonious enough to be readily extended 96

to detect trends in the statistical characteristics of hot spells and related heat waves. 97

In section 2, the statistical modeling of extreme temperature events with simple 98

structure is provided as background, emphasizing the point process approach. Summer 99

time series of daily maximum temperature at three locations, Phoenix, AZ, USA, Fort 100

Collins, CO, USA, and Paris, France are analyzed. This approach is extended to hot spells 101

in section 3, modeling hot spell length with a geometric distribution, and modeling the 102

excess on a given day within a hot spell with a conditional generalized Pareto (GP) 103

distribution whose scale parameter depends on the excess on the previous day. This 104

statistical model for hot spells is further extended to allow for trends in the frequency, 105

duration, and individual excesses of hot spells. A “heat wave simulator” is introduced in 106

section 4 to demonstrate how characteristics of more full-fledged heat waves can be 107

obtained from the underlying statistical model for hot spells. Finally, a brief discussion is 108

provided in section 5, emphasizing further extensions of the statistical modeling of hot 109

spells to make the treatment of heat waves more realistic. 110

111

2. Statistical Model for Simple Extreme Temperature Events 112

113

The appropriate statistical tools to analyze simple extreme temperature events, such as 114

excesses over high thresholds, are provided by the methods of extreme value theory. 115

Well-known in the atmospheric science and hydrology literature are two approaches: (i) 116

the modeling of block maxima (e.g., annual or seasonal maxima or, equivalently, minima) 117

using the generalized extreme value (GEV) distribution; and (ii) the peaks-over-threshold 118

(POT) modeling of threshold excesses using the generalized GP distribution. Here, we 119

advocate a third approach, closely related to the first two, which models the occurrence of 120


exceedances of a high threshold and the corresponding excesses jointly using a two-121

dimensional Poisson process. 122

123

2.1. Point Process Approach 124

The core result of extreme value theory implies that the distribution of the (appropriately 125

normalized) maximum Mn = max{X1,…,Xn} of an independent and identically distributed 126

(iid) sample X1,…,Xn from a distribution F converges to the GEV distribution. Consistent 127

with this result, the distribution of the excesses over a high threshold u is approximated 128

by a GP distribution under mild conditions on F. In the context of this paper, the block 129

maximum Mn corresponds to an annual or seasonal maximum temperature, whereas the 130

excesses over u correspond to daily maximum temperatures exceeding the threshold u. 131

The cumulative distribution function of the GEV is given by 132

1/

( ; , , ) exp 1 , 1 0,x xF x uξμ μξ σ ξ σ

σ σ

−⎧ ⎫− −⎪ ⎪⎡ ⎤= − + + >⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭ (1) 133

and that of the GP by 134

1/

( ; , , ) 1 1 , , 1 0.uu u

x xF x u x uξ

μ μξ σ ξ ξσ σ

−⎡ ⎤− −

= − + > + >⎢ ⎥⎣ ⎦

(2) 135

Here ξ denotes the shape parameter, where positive ξ implies a heavy tail, negative ξ a 136

bounded tail and the limiting case of 0ξ → an exponential tail (i.e., the Gumbel 137

distribution for block maxima and the exponential distribution for threshold excesses); 138

, 0uσ σ > denote the scale parameters and μ−∞ < < ∞ the location parameter. The scale 139

parameters of the GEV and the GP distributions are related through ( ).u uσ σ ξ μ= + − 140

We anticipate obtaining negative shape parameters, i.e., a bounded tail, for temperature as 141

indicated, for example, in Brown and Katz (1995). 142


For most practical situations, for example if the iX represent the daily maximum 143

temperature during the summer at a specific location, the independence assumption is 144

obviously not realistic. One possible way to deal with this problem is to decluster 145

excesses over the threshold u , by identifying independent clusters using an empirical rule 146

(e.g., after r consecutive observations below u a new clusters starts). Only one value per 147

cluster is kept, e.g., the first excess of the cluster or the maximum excess of the cluster, 148

reducing the sample size for further analysis. See Chapter 5 of Coles (2001) for a 149

discussion of the need to decluster and the modeling of extremes of dependent series in 150

general. A more general view on declustering schemes is provided by Ferro and Segers 151

(2003). 152

The point process approach, mentioned in the introduction, combines the modeling 153

of the occurrence of exceedances of a high threshold and their corresponding excesses in 154

one model. It uses the fact that the count of threshold exceedances within a certain time 155

window can, under the same conditions as above for the GEV to arise, be approximated 156

by a Poisson distribution with rate λ depending on the parameters , ,μ σ ξ of the limiting 157

GEV distribution of the corresponding block maximum. Chapter 7 of Coles (2001) 158

introduces in an accessible way how the Poisson process approximation is obtained and 159

summarizes mathematical and statistical details of this approach, especially the relation to 160

the well-known POT. Maximizing the likelihood of the Poisson process directly yields 161

the GEV parameters , ,μ σ ξ , and therefore indirectly the corresponding GP parameters 162

,uσ ξ . Furthermore, the Poisson rate of the number of clusters per season can be 163

expressed as [ ] 1/1 ( ) /u ξλ ξ μ σ −= + − . The point process approach has several advantages 164

over the block maxima and the POT approaches: (i) it uses considerably more data about 165

extremes than a block maximum approach resulting in more reliable results; (ii) it can be 166

formulated in terms of the GEV parameters, which are invariant to the choice of threshold, 167


allowing non-stationarities such as trends to be easily and naturally introduced through 168

covariate effects in the parameters; and (iii) it includes the threshold excess rate in the 169

inference, which is modeled separately in a POT approach. Note that parameter 170

estimation via maximum likelihood requires specialized, but straightforward numerical 171

techniques in the non-stationary case. 172

In order to fit a point process model, it is necessary to select an appropriate threshold. 173

A common approach is to fit the model using a set of candidate thresholds, and to 174

consider only values of u for which the resulting parameter estimates are approximately 175

stable. In the case of a point process model, it is also theoretically possible to vary the 176

threshold in time, but this can lead to numerical instabilities in the maximization. In the 177

case of heat waves, we will be concentrating on the summer season, so there is no need to 178

consider time-varying thresholds. 179

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2.2. Data 181

All considered models have been tested using series of daily maximum temperature at 182

three different stations, Sky Harbor International Airport in Phoenix, AZ, USA, Fort 183

Collins, CO, USA, and Parc Montsouris in Paris, France. The Phoenix data were obtained 184

from the National Climatic Data Center and span the period from 1934 to 2007, where 185

the years 1935–1937, 1939, 1945 1947 are missing and are completely left out of the 186

analysis. Note that Phoenix has experienced a heat island effect over this period, with 187

markedly increasing daily minimum temperature but less pronounced increase in daily 188

maximum temperature, see Balling et al. (1990). The Fort Collins data were obtained 189

from the Colorado Climate Center at Colorado State University, and span the period from 190

1900 to 1999 with no missing values. The Paris data were obtained from the European 191

Climate Assessment and Dataset, see Klein Tank (2002), and span the period from 1900 192

to 2008. 193


For each station we consider a summer period from June 16 to September 15 (T = 92 194

days) susceptible to the occurrence of hot spells and heat waves. Exploratory data 195

analysis confirms that daily maximum temperature does not have a marked cycle within 196

this period at these locations, so we do not model seasonality of temperature. 197

Nevertheless the use of this specific period is a convenient oversimplification since, on 198

the one hand, the heat wave season is certainly longer in Phoenix than Fort Collins or 199

Paris and, on the other hand, the length of the season itself may be subject to change. For 200

a first application of the proposed method, the simplification seems adequate but may 201

need to be relaxed in a more realistic situation. In the summer period considered, there 202

are fewer than ten additional missing values for Phoenix and only two for Paris. We set 203

the value of the daily maximum temperature on these dates to the minimum observed 204

value over the entire record period, so that they have no influence on the extremal 205

analysis. The data from the US were provided in heavily discretized form, rounded to the 206

nearest degree Fahrenheit, and we subsequently converted them to centigrade. The data 207

from France were provided rounded to the nearest one tenth of a degree centigrade. 208

Figure 1 shows the time series of annual maximum temperature at the three sites. 209

Extreme temperature events are the focus of this paper so data quality is of special 210

importance. Moreover, since detecting possible trends in hot spells is one goal, we need 211

to assume homogeneity of the data series to justify fitting the proposed model. Klein 212

Tank et al. (2002) mentioned that it is not untypical for climatic time series to be subject 213

to certain inhomogeneities as, for example, changes in station location, instrumentation 214

etc. So one should be aware that any of the detected trends could be artifacts of these 215

inhomogeneities, rather than reflecting real climate change. 216

217

2.3. Point Process Model Fit 218


The discretization of the temperature data from the US results in some numerical 219

difficulties in the fitting of the point process model, being more than normally sensitive to 220

the exact choice of the threshold. Cooley et al. (2007) ran simulations to show that using 221

thresholds in middle of the discretization interval provides numerically stable estimations, 222

which is the approach we take here. Another possibility would be to artificially add noise 223

to the observations to break the ties that cause the numerical difficulties, see Einmahl and 224

Magnus (2008). The conversion from Fahrenheit to centigrade leads to seemingly 225

arbitrary thresholds, which are simply explained as mid points between distinct data 226

values. Note that the data from Paris are much less discretized and have been used here, 227

at least in part, to ensure that the obtained results are not effects of the discretization. 228

For the traditional point process analysis, we use thresholds of 40.8◦C (i.e., 105.5◦F) 229

for Phoenix, 30.8◦C (i.e., 87.5◦F) for Fort Collins and 27◦C for Paris, which have been 230

chosen following the approach described in section 2.1 Clusters are separated by a single 231

value below the threshold, i.e., 1r = , retaining the cluster maximum excesses to be 232

treated as independent observations. Note that using the above thresholds and 1r = only 233

serves statistical purposes in the modeling of clusters of high temperature, more societally 234

meaningful thresholds and more meteorologically meaningful values for r will be used 235

when considering heat waves derived from hot spells, see section 4.2. 236

Maximum likelihood estimates of the GEV parameters at all stations, as well as the 237

above-mentioned thresholds, are given in Table 1. The estimates of the GP parameter uσ 238

and the Poisson parameter λ are derived from these values of the GEV parameters as 239

indicated in section 2.1. Table 1 includes standard errors for all parameter estimates. As 240

anticipated we obtain negative shape parameter estimates, i.e., a bounded tail, at all three 241

locations. Recall that the shape parameter is identical in both parameterizations, the GEV 242

and the Poisson-GP, of the point process approach. We test the Poisson hypothesis for the 243

number of clusters per season with a Poisson dispersion test (Rice, 1995), based on the 244


approximate 2χ distribution of the ratio of 1n − times the variance divided by the mean, 245

p-values for all stations are also given in Table 1. The hypothesis is decidedly supported 246

by the data for all three stations, with Phoenix being the strongest case. The Poisson 247

dispersion test is based on the assumption of stationarity; therefore a rejection of the null 248

hypothesis could be attributable to a Poisson distribution with a trend, rather than the lack 249

of a Poisson distribution per se. Trends will be addressed in section 3.3. 250

The top row of Figure 2 shows histograms of the number of clusters per summer 251

along with the estimated Poisson probability function for all three sites. Although 252

deviations from the estimated probability function are apparent, the general shape of the 253

histograms do not strongly contradict the Poisson assumption. In the case of Phoenix, it 254

seems that large numbers of clusters are observed more frequently than the Poisson 255

model would predict. Quantile–quantile ( )Q Q− plots for the cluster maximum excess 256

under the GP hypothesis (as derived from the point process model fit) are shown in the 257

bottom row of Figure 2, and do not indicate any major departures from the assumed point 258

process model. 259

260

3. Statistical Model for Hot Spells 261

262

In the previous section we fitted a point process model to the cluster maximum 263

temperatures, defining a high temperature cluster as consecutive days with maximum 264

temperature above the threshold u, e.g. 40.8◦C for Phoenix, where a new cluster of high 265

temperatures starts if the temperature drops below u for at least one day ( 1r = ). From a 266

more applied viewpoint, we call these clusters of high temperatures “hot spells”, see for 267

example Figure 3 for an illustration of 9 hot spells in a season of 92 days. Again, the 268

choice of threshold and of 1r = is based on statistical considerations, societal and 269


meteorological considerations will be important when applying the fitted model in the 270

analysis of heat waves. Note that the point process fit of the previous section provides a 271

Poisson model for the number of hot spells as well as a GP model for the hot spell 272

maximum excess, but it does not provide a complete description of the entire hot spell. 273

To be complete we need to additionally provide a model for the spell length, as well as a 274

model for the dependence of excesses within a spell. 275

276

3.1. Description of Hot Spell Model 277

Here, we propose to model the spell length through a geometric distribution, a simple 278

enough model to allow the easy introduction of trends through a generalized linear model 279

(GLM) framework. In addition, we propose to start from a simple GP model for the first 280

excess of a spell and to assume conditional GP distributions for the remaining excesses, 281

where the conditioning is on the excess of the previous day and the conditional 282

relationship is assumed constant over the length of the spell. Modeling the temporal 283

dependence of the excesses within a hot spell through a Markov process, making use of 284

bivariate extreme value theory, is an asymptotically correct approach, see Coles et al. 285

(1994). We use instead only univariate extreme value theory, through conditioning in 286

order to provide a simple approach that is easily applicable in practice while still making 287

to a certain extent use of the theoretical advantages of extreme value models. For both 288

types of GP model, simple and conditional, trends can be easily introduced through 289

covariate effects in the parameters. 290

Given its memoryless property, the geometric distribution is the simplest plausible 291

model for spell length. It was used by Smith et al. (1997) to model the cluster length of 292

low minimum daily temperatures, although they found some evidence that a distribution 293

with a heavier tail might be needed. The probability mass function of the geometric 294

distribution is 295


1( ) (1 )kP k θ θ−= − , k =1, 2,… 296

with the reciprocal of the parameter θ being the mean. Parameter estimation is done 297

using the method of moments (which is in this case equivalent to maximum likelihood). 298

Under a wide range of conditions, the parameter θ corresponds to the so-called extremal 299

index, which measures the tendency of the underlying process to cluster at extreme levels, 300

see Chapter 5 of Coles (2001) for a brief discussion of this index. 301

Note that this model is specific to the threshold u in the sense that, if the fitted 302

model is used at a higher threshold, the number of exceedances will no longer be 303

geometric. Asymptotically correct extreme value models are not subject to this limitation. 304

We circumvent this issue by simulating hot spells, i.e., using the original threshold, and 305

obtaining results on heat waves. 306

We model the excess on the first day of a hot spell with a GP distribution with 307

parameters uσ and ξ , which we derive from a point process model fit to data retaining 308

only the first excess per hot spell. The remaining excesses of the same spell are modeled 309

conditionally on the excess of the preceding day. More precisely, the conditional excess 310

lE on day l given the value of the excess 1lE v− = on day 1l − follows a GP distribution 311

with scale parameter ,2 ,2 ( )u u vσ σ= depending on v and constant shape parameter 2ξ . 312

Note that assuming a constant dependence structure throughout each hot spell 313

reduces the number of parameters involved and increases the amount of data available to 314

estimate each of them considerably, namely to all consecutive pairs of excesses. 315

Obviously, it is possible to extend this simple and parsimonious approach by allowing the 316

parameters of the conditional GP distribution to vary depending on which day within the 317

spell is modelled. For the station data considered here, we encountered numerical 318

problems while fitting such models, more precisely the estimated shape parameters being 319

in some cases smaller than or very close to −0.5, a theoretical bound below which the 320


maximum likelihood estimator is not valid (page 55 of Coles, 2001). The same type of 321

numerical instabilities is observed for the considered data when fitting bivariate extreme 322

value models similar to those considered in Coles et al. (1994) and Smith et al. (1997). 323

The form of the scale parameter function ,2 ( )u vσ remains to be chosen. Classical 324

functional forms are the exponential ,2 ( ) exp( )u v a b vσ = + ⋅ and linear ,2 ( )u v a b vσ = + ⋅ . 325

The exponential function is more regularly used in statistics since it ensures positivity and 326

the classical bivariate extreme value models use a corresponding function. A linear 327

functional form has the appeal of leading to a simpler model and later on to a simpler 328

simulation procedure and in practice the positivity constraint is rarely violated within the 329

range of the data. 330

The choice of the functional form has consequences for the theoretical properties of 331

the conditional GP distribution. Since we have assumed that the conditional distribution 332

of the l th excess given the value of the 1l − th excess is a GP distribution, the conditional 333

mean (expectation) of the second excess 2E given the first 1E v= , for example, is given 334

by 335

,22 1

2

( ),

1u v

E E E vσ

ξ⎡ ⎤⏐ = =⎣ ⎦ −

336

i.e., it is again a linear function of the value of the first excess, if the linear form for 337

,2 ( )u vσ is chosen. We close the description of the model by giving the formula for 338

conditional quantiles, which will be used later on: 339

340

2,212 ,2

2

( ), , ( ) (1 ) 1u

u

vF p v p ξσ

ξ σξ

−− ⎡ ⎤⎡ ⎤ = − −⎣ ⎦ ⎣ ⎦ 341


where 0<p<1. If ,2 ( )u vσ is a linear function, then the conditional quantiles are obviously 342

linear functions of v with steeper slopes for higher p resulting, for example, in a more 343

rapidly increasing inter-quartile range than the median. 344

One of the drawbacks of the proposed approach is that the unconditional distribution 345

of any given excess within a spell is not necessarily exactly a GP distribution, although it 346

should be a close approximation. Even though the GP is the asymptotically correct model, 347

it is in practice, i.e., for finite samples, only an approximation and the conditional 348

approach at worst only weakens this approximation a bit further. Another possible 349

limitation is that the stochastic process for daily intensities within a cluster is not time-350

reversible. 351

352

3.2. Hot Spell Model Fit 353

Method of moments estimates of the parameter θ of the geometric distribution for the hot 354

spell lengths as well as standard errors and the corresponding mean spell lengths are 355

given in Table 2, and Figure 4 shows histograms of hot spell lengths along with the fitted 356

geometric distributions for all three sites. Especially for Phoenix, the tail of the geometric 357

distribution seems to underestimate the observed frequency of longer spells 358

systematically indicating that it might not be heavy enough compared to the data. In spite 359

of this possible drawback, we favor the geometric distribution over more heavy-tailed 360

candidates such as the Zipf distribution (e.g., section 11.20 of Johnson et al., 1992), since 361

the effect due the observed underestimation should be small and the possibility to easily 362

introduce trends through a GLM approach more important. 363

Table 3 contains parameter estimates along with standard errors of the conditional 364

GP distribution for both choices of the scale parameter function and for all three sites. 365

Note that the estimates of the shape parameter are barely influenced at all by the choice 366


of this function and lie in an acceptable range (i.e., negative as expected for temperature 367

data but above −0.5). Figure 5 shows the conditional relationship between all consecutive 368

pairs of excesses (i.e., 1lE − and lE ) with respect to sample/observed (circles and black 369

vertical lines) and model/theoretical (colored lines) median and lower and upper quartiles: 370

linear function (top panels) and an exponential (bottom panels) for the scale parameter at 371

all three sites. For Paris we rounded the excesses to the nearest half degree centigrade in 372

order to be able to calculate stable conditional sample quantiles. Circles in the plots 373

without attached vertical lines correspond to a single pair of consecutive excesses for the 374

given value of the first excess, i.e., no measure of spread can be calculated. In general, 375

the circles that are the farther right in the plot are based on the fewer values in the sample 376

quantile calculation, i.e., they are less reliable. For Phoenix the model using the 377

exponential scale parameter function seems to provide a better fit to the last few points on 378

the right. Further to the left of the plots, both functional forms result in a similar fit of the 379

model to the data for all three stations. 380

Note that all of the model characteristics shown in these plots are derived from the 381

fitted model, with the displayed sample characteristics not being directly fitted explaining 382

at least some of the apparent less than ideal performance of the models. In view of the 383

fact that we only allow one parameter to vary (the scale parameter of the conditional GP 384

distribution) and that the right-hand side of the plots is naturally based on extremely few 385

observations, the fit of the conditional GP models seems adequate for all three stations. 386

So we choose to use the simpler linear function for the scale parameter in the following. 387

A major advantage over more conventional approaches like a conditional normal model 388

is that the conditional GP model is able to capture the effect of increasing variability with 389

increasing median or mean. 390

391

3.3. Trends in Hot Spells 392


We intentionally constructed our hot spell model such that the introduction of trends in 393

duration, frequency and intensity of hot spells, and later on indirectly for heat waves, is 394

easily possible. Technically these three characteristics correspond to the components of 395

the hot spell model: the geometric model for spell length, the Poisson model for number 396

of spells per season, and the (conditional) GP model for the sizes of the temperature 397

excesses within a spell. While duration and frequency are direct consequences of the 398

definition of a hot spell, intensity can be measured in different ways, e.g., by a mean, 399

maximum or total excess of a spell. In our case we concentrate on the first excess as an 400

indicator since it can be assessed easily in our modeling framework and since trends in 401

the first excess will induce changes in these other measures as well. 402

For all three model components we consider parameters fixed over the heat wave 403

season within a given year but allow shifts from one year to another, i.e., for each year y 404

of the record period 1,..., P we consider ( )yθ θ= for the geometric parameter, 405

( )yλ λ= for the Poisson parameter and ( )u u yσ σ= for the GP scale parameter. The GP 406

shape parameter is kept fixed since changes in shape are rarely observed and difficult to 407

model. For the geometric model trends are introduced through a GLM framework. For 408

the point process/Poisson-GP model, there are two possibilities: either (i) introducing 409

trends indirectly through covariate effects in the GEV parameters and then transforming 410

to the Poisson-GP parameterization; or (ii) introducing trends directly but separately 411

through a GLM framework in the Poisson model and through covariate effects in the GP 412

scale parameter. We prefer the second possibility because of the advantage that statistical 413

significance can be evaluated separately for number of spells and excesses. Covariate 414

effects in the parameters are obviously also possible for the conditional GP model, but in 415

view of the difficulties of fitting even our basic model for the dependence of excesses we 416

refrain from that possibility altogether. 417


Table 4 contains parameter estimates along with standard errors and p-values of the 418

likelihood ratio test, which indicate significant trends if smaller than a certain level, 419

usually taken as 5%, for all three components of the hot spell model, with and without 420

trends, and all three sites. Figure 6 contains the observed evolution over time of the same 421

three components compared to mean values of the respective model distribution with (red) 422

and without (black) trends at all three sites. As indicated in the table, there is a significant 423

trend in spell length for Phoenix, in number of spells for Fort Collins and in the first 424

excess and spell length for Paris, which are more or less all confirmed by the visual 425

impression of Figure 6. The downward trend in mean of first excesses in Paris is 426

surprising, but might simply be due to the fact that hot spell intensity is not adequately 427

represented by the mean first excess. The visual impression of the observed series 428

confirms a decreasing variance over the years, which matches the sign of the parameter 429

estimates in Table 4. Note that since we consider a season of fixed length it will be 430

difficult to allow for a trend in both spell length and number of spells. As mentioned 431

before, this at least somewhat unrealistic assumption will probably need to be relaxed in 432

an expanded version of the model. 433

434

4. Heat Wave Simulator 435

436

We introduce a “heat wave simulator”, i.e., a stochastic simulation algorithm that 437

generates temperature series from the discussed hot spell model in order to demonstrate 438

how characteristics of more full-fledged heat waves can be obtained. 439

440

4.1. Algorithm 441


The algorithm to simulate a time series of hot spells starts by generating the number of 442

hot spells for each year in the desired simulation period from the Poisson distribution 443

(which is derived from the point process model of the first excess of a spell). Then, for 444

each of these hot spells, a spell length is generated from the geometric distribution. From 445

the theory of Poisson processes, it follows that the distribution of the hot spells within the 446

season is uniform, and we use this fact to simulate the alternation between hot spells and 447

intervals between spells over the entire simulation period. The next step is to generate an 448

excess over the threshold for the first day of each hot spell from the GP distribution 449

(which is derived from the point process model), and finally to generate excesses for the 450

remaining days of each hot spell recursively using the conditional GP model. The 451

technical details for the implementation of the simulation algorithm are given in 452

Appendix A. 453

As a demonstration of the simulator applied to Phoenix, Figure 7 shows boxplots of 454

different characteristics of the observed temperature series of 67 years, along with 455

minimum/maximum, lower/upper quartile and median of 100 simulated temperature 456

series of length 67 years (with no trends in any parameters of the hot spell model). Here, 457

the mean excess per summer is shown as example of an indicator of the intensity of a hot 458

spell. This characteristic is calculated from all excesses in all spells of a season, i.e., the 459

simulated values are drawn from the GP model of the first excess and the conditional GP 460

model for the remaining excesses of a spell. The deviation in the central part of the 461

boxplot is a concern here, reflecting again that the conditional GP model is not a perfect 462

approximation of the true underlying process, see also Figure 5. 463

As a second demonstration of the simulator, this time including a trend in the 464

parameter of the geometric distribution for spell length, Figure 8 shows mean spell 465

lengths per season from the observed temperature series from Phoenix along with 466

pointwise 10% and 90% quantiles of 100 simulated temperature series of length 67 years. 467


This display emphasizes the positive effect of the introduction of a trend on the 468

simulation, but again indicating that the geometric model is probably not heavy tailed 469

enough, see also Figure 2. 470

471

4.2. Heat Waves 472

The extreme value methodology we apply requires, on the one hand, that the threshold be 473

high enough for the asymptotic theory to be valid but, on the other hand, low enough 474

such that enough data are available for the analysis to be stable. Another requirement is 475

that clusters/hot spells need to be approximately independent, which is usually achieved 476

by a declustering scheme of which we use the simplest one: two spells are separated by at 477

least one day of lower temperatures. To study heat waves, we will use the fact that at least 478

for some definitions, they can be indirectly derived from hot spells, for example by using 479

a higher threshold (see Figure 9 middle), using only longer hot spells, merging spells, i.e., 480

using 1r > (see Figure 9 right) or other functionals of the spell (e.g., mean or total 481

excess). The principal idea is to model hot spells using extreme value theory, and then 482

derive conclusions on heat waves, which themselves cannot so easily be analyzed for 483

various reasons: (i) there are too few data for direct models of heat waves to be as reliable; 484

(ii) different definitions of heat waves would require repeated model fitting, if not 485

different modeling approaches; and (iii) heat waves are, depending on the definition, 486

considerably more complex to model directly. 487

As an example, we use the fitted hot spell model (i.e., u = 40.8°C and 1r > ) for 488

Phoenix and accordingly simulated hot spell series with and without the detected trend in 489

spell length. From these simulated hot spells, we indirectly obtained simulations of heat 490

waves, defined as temperatures exceeding the higher threshold of 43.6°C (i.e., 110.5°F), 491

where two heat waves are separated by at least a day of lower temperatures (i.e., 1r = ). 492

Figure 10 shows observed series of number of heat waves (top), mean length (middle) 493


and mean excess (bottom, i.e., one measure of heat wave intensity) along with 494

corresponding pointwise 5% and 95% quantiles of 100 simulated temperature series of 495

the same length with and without trend. 496

In Figure 10 the observed number of heat waves in Phoenix seems to increase more 497

systematically than the observed number of hot spells (recall Figure 6), whereas the 498

length and mean excess during heat waves do not seem to show systematic changes over 499

time. Possibly the observed (and statistically significant, see Table 4) increasing trend in 500

hot spell length results in heat waves occurring more frequently. Most of the line 501

corresponding to number of heat waves is contained in the shaded area as it is rather wide. 502

The yellow area, corresponding to the hot spell model with a trend in the geometric spell 503

length distribution seems to reflect the potential trend in number of heat waves, more 504

convincingly so if bands with lower confidence (e.g. 10% and 90%) are used (not shown). 505

For mean heat wave length, it is clear that the simulations are not able to reproduce the 506

observed sudden spikes, and for the mean excess during heat waves the confidence bands 507

seem again rather wide compared to the observed values. 508

509

5. Discussion 510

511

A new technique has been proposed for the statistical modeling of hot spells. Unlike most 512

previous research on this topic, our method is based as much as feasible on the statistical 513

theory of extreme values. Given that hot spells are an extreme meteorological 514

phenomenon, this reliance on extreme value theory naturally produces an approach that 515

treats the basic characteristics (i.e., frequency, duration, and intensity) of such events in a 516

more realistic manner statistically than techniques heretofore applied. Perhaps less 517

obvious, the point process technique for extreme value analysis results in a more 518


powerful approach for systematically studying the statistical features of extreme high 519

temperatures. We have demonstrated how the statistical characteristics of more full-520

fledged heat waves can be derived from our statistical model for hot spells. In particular, 521

attention need no longer be restricted to a rigid definition of a heat wave, about which 522

there is not necessarily any consensus. 523

The proposed technique has been intentionally kept simple enough for trends in its 524

various components to be incorporated. Thus, there remain a number of respects in which 525

the technique could be extended, both to make its treatment of hot spells more realistic 526

statistically and of heat waves meteorologically. As already mentioned, it would seem 527

more reasonable to allow a trend in the length of the heat wave season, along with any 528

trends in other characteristics. A more appealing, but less parsimonious, approach would 529

consist in introducing seasonality into the parameters of the statistical model, rather than 530

holding them fixed over an entire season. A longer summer season could be modeled, 531

with hot spells being less likely at the beginning and end of the season. Concerning 532

meteorological realism, it would be straightforward to apply the technique to daily time 533

series of apparent temperature instead of maximum temperature, thus taking into account 534

humidity (e.g., as in Karl and Knight, 1997). Indices of atmospheric circulation patterns, 535

such as blocking, could be used as covariates instead of, or in addition to, a trend 536

component (e.g., Sillmann and Croci-Maspoli, 2009). Much more challenging would be 537

the simultaneous treatment of both daily maximum and minimum temperature, thus 538

taking into account night-time weather conditions as well. 539

This research has implications for the statistical modeling of temperature variables 540

more generally. Specifically, climate change scenarios are frequently produced by 541

stochastic weather generators (e.g., Semenov, 2008). Conventional weather generators 542

are based on autoregressive-type models for times series of daily minimum and 543

maximum temperature. As such, they cannot be expected to represent adequately the 544


statistical characteristics of extreme high temperatures, especially the temporal 545

dependence of excesses within a hot spell. How to modify such weather generators to 546

improve their performance in terms of simulating heat waves remains an open question. 547

548

Acknowledgements: We thank Philippe Naveau, Claudia Tebaldi and Yun Li for 549

advice on this research, and three anonymous reviewers for their comments. M.D. Walter 550

received support as a summer visitor in the NCAR program on Significant Opportunities 551

in Atmospheric Research and Science (SOARS). Research was partially supported by 552

NSF Grant DMS-0355474 to the NCAR Geophysical Statistics Project, and by NCAR’s 553

Weather and Climate Assessment Science Program. The National Center for Atmospheric 554

Research is managed by the University Corporation for Atmospheric Research under the 555

sponsorship of the National Science Foundation. 556

557

LITERATURE CITED 558

559

Abaurrea J, Asín J, Cebrin AC, Centelles A (2007) Modeling and forecasting extreme hot 560

events in the central Ebro valley, a continental-Mediterranean area. Global Planet Change 561

57:43–58 562

563

Balling RC Jr, Skindlov JA, Phillips DH (1990) The impact of increasing summer mean 564

temperatures on extreme maximum and minimum temperatures in Phoenix, Arizona. J 565

Clim 3:1491– 494 566

567

Brown BG, Katz RW (1995) Regional analysis of temperature extremes: Spatial analog 568

for climate change? J Clim 8:108–119 569


570

Brown SJ, Caesar J, Ferro CAT (2008) Global changes in extreme daily temperature 571

since 1950. Global Planet Change 113:D05115, doi:10.1029/2006JD008091 572

573

Coles S (2001) An Introduction to Statistical Modeling of Extreme Values. Springer, 574

London 575

576

Coles SG, Tawn JA, Smith RL (1994) A seasonal Markov model for extremely low 577

temperatures. Environmetrics 5:221–239 578

579

Cooley D, Nychka D, Naveau P (2007) Bayesian Spatial Modeling of Extreme 580

Precipitation Return Levels. Journal of the American Statistical Association 102:824–840 581

582

Einmahl JHJ, Magnus JR (2008) Records in athletics through extreme-value theory. 583

Journal of the American Statistical Association 103:1382–1391 584

585

Ferro CAT, Segers J (2003) Inference for clusters of extreme values. J R Stat Soc Ser B 586

65:545–556 587

588

Furrer EM, Katz RW (2008) Improving the simulation of extreme precipitation events by 589

stochastic weather generators. Water Resour Res 44:W12439, 590

doi:10.1029/2008WR007316 591

592

Gosling SN, Lowe JA, McGregor GR, Pelling M, Malamud BD (2009) Associations 593

between elevated atmospheric temperature and human mortality: a critical review of the 594

literature. Clim Change 92:299–341 595


596

Gumbel EJ (1958) Statistics of Extremes. Columbia University Press, New York 597

598

Johnson NL, Kotz S, Kemp AW (1992). Univariate Discrete Distributions (second 599

edition), John Wiley & Sons, New York 600

601

Karl TR, Knight RW (1997) The 1995 Chicago heat wave: How likely is a recurrence? 602

Bull Am Meteorol Soc 78:1107–1119 603

604

Katsoulis BD, Hatzianastassiou N (2005) Analysis of hot spell characteristics in the 605

Greek region. Clim Res 28:229–241 606

607

Katz RW, Parlange MB, Naveau P (2002) Statistics of extremes in hydrology. Adv Water 608

Resour 25:1287–1304 609

610

Khaliq MN, Ouarda TBMJ, St-Hilaire A, Gachon P (2007) Bayesian change-point 611

analysis of heat spell occurrences in Montreal, Canada. Int J Climatol 27:805–818 612

613

Khaliq MN, St-Hilaire A, Ouarda TBMJ, Bobée B (2005) Frequency analysis and 614

temporal pattern of occurrences of southern Quebec heatwaves. Int J Climatol 25:485–615

504 616

617

Klein Tank AMG et al. (2002) Daily dataset of 20th-century surface air temperature and 618

precipitation series for the European climate assessment. Int J Climatol 22:1441–1453 619

620


Koffi B, Koffi E (2008) Heat waves across Europe by the end of the 21st century: 621

multiregional climate simulations. Clim Res 36:153–168 622

623

Kysely J (2002) Probability estimates of extreme temperature events: Stochastic 624

modeling approach 625

vs. extreme value distributions. Studia Geophysica et Geodaetica 46:93–112 626

627

Mearns LO, Katz RW, Schneider SH (1984) Extreme high-temperature events: Changes 628

in their probabilities with changes in mean temperature. Journal of Climate and Applied 629

Meteorology 23:1601–1613 630

631

Meehl GA, Tebaldi C (2004) More intense, more frequent, and longer lasting heat waves 632

in the 21st century. Science 427:994–997 633

634

Meze-Hausken E (2008) On the (im-)possibilities of defining human climate thresholds. 635

Clim Change 89:299–324 636

637

R Development Core Team (2009) R: A Language and Environment for Statistical 638

Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-639

07-0, http://www.R-project.org 640

641

Rice JA (1995) Mathematical Statistics and Data Analysis. Duxbury Press, Belmont, CA 642

643

Robinson P (2001) On the definition of a heat wave. J Appl Meteorol 40:762–775 644

645


Schär C, Vidale P, Lüthi D, Frei C, Häberli C, Liniger MA, Appenzeller C (2004) The 646

role of increasing temperature variability in European summer heatwaves. Nature 647

427:332–336 648

649

Semenov MA (2008) Simulation of extreme weather events by a stochastic weather 650

generator. Clim Res 35:203–212 651

652

Sillmann J, Croci-Maspoli M (2009) Euro-Atlantic blocking and extreme events in 653

present and future climate simulations, Geophys Res Lett 36:L10702, 654

doi:10.1029/2009GL038259 655

656

Smith RL (1989) Extreme value analysis of environmental time series: An application to 657

trend detection in ground-level ozone (with discussion). Stat Sci 4:367–393 658

659

Smith RL, Tawn JA, Coles SG (1997) Markov chain models for threshold exceedances. 660

Biometrika 84:249–268 661

662

Tebaldi C, Hayhoe K, Arblaster JM, Meehl GA (2006) Going to the extremes: an 663

intercomparison of model-simulated historical and future changes in extreme events. 664

Clim Change 79:185–211 665


666

A Implementation of the Heat Wave Simulator 667

668

Recall that the discussed hot spell model is based on several parameters: λ for the 669

Poisson parameter of the number of hot spells, θ for the geometric parameter of hot spell 670

length, uσ and ξ for the GP parameters of the first excess of each spell and 2, ,a b ξ for 671

the parameters of the conditional GP distribution with scale parameter modeled with a 672

linear function for the excesses within a spell. Given a season length T , a number of 673

seasons S to be simulated and values (estimated or hypothetical) for the model 674

parameters, the algorithm to generate series of hot spells is given by the following 675

pseudo-code: 676

for y in 1,..., S repeat [1] to [8] 677

[1] generate ( ) ( )N y POIS λ 678

[2] if [ ]( ) / 2N y T> goto 1 679

[3] for 1,..., ( )i N y∈ generate ( ) 1 ( )iL y GEO θ+ 680

[4] if ( ) 2 ( ) 1i iL y N y T∑ + − > goto 3 681

[5] draw 1 ( ),..., N yu u without replacement from { }0,..., ( ) ( ) ,cold i iT T L y N y= − ∑ − 682

[6] divide { }1, 2,... coldT using [ ] [ ]( ),...,u N yu u 683

[7] for 1,..., ( )i N y∈ generate ,1( ) ( , )i uE y GP σ ξ 684

[8] for 1,..., ( ), 1,..., ( )ii N y j L y∈ ∈ generate , 1 , 2( ) ( ( ), )i j i jE y GP a bE y ξ+ + 685

Here ( )N y denotes the number of hot spells, ( ) 1iL y + is the length of the i th hot spell 686

(with spells being at least one day long), there are ( ) ( )cold i iT T L y N y= − ∑ − days not in 687

a hot spell, [ ] [ ]1 ( ),..., N yu u is the ordered sample and , ( )i jE y is the excess on day j of the 688


i th hot spell. In step [1] the notation ( ) ( )N y POIS λ refers to a random number ( )N y 689

distributed according to a Poisson distribution with parameter λ , similarly for the 690

geometric distribution in step [3] and the GP distribution in step [7] and [8]. Note that 691

steps [2] and [4] ensure that there are enough days in the season to fit the generated 692

number of hot spells with corresponding length and at least a day of colder temperature 693

between spells (i.e., at least ( ) 1N y − “cold” days). 694

The simulation algorithm is virtually the same if the model contains a trend in one or 695

several of the parameters, the only difference being that a different estimated parameter 696

value is used in the generation for each season. For example if the observed trend in the 697

spell length for Phoenix is included in the model the third step uses 698

( ) 1 ( ( ))iL y GEO yθ+ . 699


Figures 700

701

702

Figure 1: Time series of annual maximum temperature at the three sites. 703

704

705

706

707

Figure 2: Top row: histogram and estimated Poisson probability function for the number 708

of clusters per summer at three sites, bottom row: ( )Q Q− plots for the cluster maximum 709

excess under the GP hypothesis at the three sites. 710


711

Figure 3: Observed hot spells (red) during 1934 at Phoenix based on a threshold of 712

40.8u C= o and 1r = during June 16 (day 1) to September 15 (day 92). 713

714

715

716

717

718

719

720

Figure 4: Histogram and estimated geometric probability function for the hot spell 721

lengths per summer at the three sites. 722


723

Figure 5: Theoretical (red lines; linear function in top panels, exponential function in 724

bottom panels) and sample (dots and vertical lines) conditional medians ( C)o and lower 725

and upper quartiles ( C)o for the conditional GP model for all consecutive pairs of 726

excesses (i.e., the pair 1lE − and lE , with the threshold added to show original temperature 727

values rather than excesses) at the three sites. 728


729

730

731

Figure 6: Observed number of hot spells, mean first excess and mean length of hot spells 732

per summer (circles connected by black lines) compared to mean values of the GP, 733

Poisson and geometric distributions respectively with (red lines) and without (black 734

horizontal lines) trend at the three sites. 735


736

Figure 7: Boxplots of several hot spell characteristics of observed (67 years) temperature 737

series over the threshold of 40.8 Cu = o for Phoenix. Black vertical lines above boxplots 738

correspond with increasing length to minimum/maximum, lower/upper quartile and 739

median of 100 simulated temperature series of length 67 years (with no trends in any 740

parameters of the hot spell model). Green vertical lines extend quantiles of boxplot to 741

facilitate comparison with simulation. 742

743


744

Figure 8: Mean spell lengths: observed temperature series for Phoenix (black line), 745

pointwise 10% and 90% quantiles of 100 simulated temperature series of length 67 years 746

(hatched gray: without trend, solid yellow: with trend). 747

748

749

750

Figure 9: Observed hot spells (red) during 1934 at Phoenix based on a threshold of 751

40.8 Cu = o and 1r = (left), 43.6 Cu = o and 1r = (middle), and 40.8 Cu = o and 2r = 752

(right). 753


754

Figure 10: Number of heat waves ( 43.6 Cu = o , 1r = , mean heat wave length (middle) and 755

mean excess ( Co , bottom), solid lines correspond to the observed series at Phoenix of 756

length 67 years, areas between pointwise 5% and 95% quantiles of 100 simulated 757

temperature series of length 67 years are hatched gray for the hot spell model without 758

trend and solid yellow for the model including a trend. The gaps in the time series 759

(middle and bottom panel) are due to years in which no heat waves occurred (see top 760

panel). 761


Tables 762

763

Table 1: Parameter estimates of the GEV parameters and the corresponding GP and 764

Poisson parameters (with standard errors in parentheses), as well as p-values of the 765

Poisson dispersion test, for all three sites. 766

767

768

769

770

771

772

Table 2: Estimates of the parameter of the geometric distribution θ (with standard errors 773

in parentheses) and mean spell lengths for all three sites. 774

775


Table 3: Parameter estimates (with standard errors in parentheses) of the conditional GP 776

distribution for both choices of the scale parameter function and for all three sites 777

778


Table 4: Parameter estimates (with standard errors in parentheses) and p-values (bold 779

indicates statistical significance at 0.05 level) of the likelihood ratio test for the three 780

components of the hot spell model, number of spells, excesses and spell length, with and 781

without trends, for all three sites, where 1,...,y P= denotes the year in the record period. 782

783

Documents

Statistical modeling of hot spells and heat waves