Application of the Statistical Theory of
Extreme Values to Heat Waves
Marcus D. Walter
Academic Affiliation, Fall 2008: 1st-Year Graduate Student, Cornell University
SOARS® Summer 2008
Science Research Mentor: Richard Katz, Eric Gilleland
Writing and Communication Mentor: Tim Barnes
ABSTRACT
Heat waves can have devastating impacts on society, but a current weakness with the
analysis and modeling of heat waves is the negligible use of the Statistical Theory of Extreme
Values. This is a branch of statistics more appropriate for studying extreme events such as heat
waves, floods, etc. For this study, EVT was used to develop methods for analyzing heat waves
and their characteristics (frequency, intensity, duration, etc.). This analysis was performed using
temperature data from Phoenix, AZ and Fort Collins, CO. This study signaled how Statistical
Theory of Extreme Values can be applied to model certain features of heat waves. Results from
the analysis showed an increase in the summer highest temperature and in the number of heat
waves per year for both cities. This study also explored other characteristics of heat waves (heat
wave duration and individual maximum temperatures within heat waves), indicating how the
extreme value approach would need to be extended to fully model all features of heat waves. The
results show there hasn’t been a significant change in the intensity or duration of heat waves for
either city. The results as well descriptively imply a temperature dependence of daily
temperatures within a heat wave for both cities. More reliable quantification of return levels for
severe heat waves, including any trends in their characteristics, and other extreme events
involving spells will be achieved with the continual development and future use of these
methods.
The Significant Opportunities in Atmospheric Research and Science (SOARS) Program is managed by the University
Corporation for Atmospheric Research (UCAR) with support from participating universities. SOARS is funded by the National
Science Foundation, the National Oceanic and Atmospheric Administration (NOAA) Climate Program Office, the NOAA Oceans
and Human Health Initiative, the Center for Multi-Scale Modeling of Atmospheric Processes at Colorado State University, and
the Cooperative Institute for Research in Environmental Sciences. SOARS is a partner project with Research Experience in Solid
Earth Science for Student (RESESS).
1. Introduction
Heat waves, although fairly rare, have devastating impacts on society. The major heat
wave of 1995 in the Midwestern United States led to the more than 500 deaths, most of which
occurred in the city of Chicago (Karl et al. 1996). During the European heat wave of 2003,
thousands of people died across France, Italy, Spain, and Switzerland due to excessive heat
exposure (Koffi et al. 2008). Heat waves have the potential to affect millions of people around
the world, and because of their devastating impacts, extreme importance lies in understanding
their frequency and intensity. Today scientists and society have become even more concerned
with heat waves because of the likely increases in their occurrences due to climate change. If not
anticipated, more intense and longer heat waves could lead to even more devastating future
impacts on society.
Inquiries have gone in to understanding the occurrence of heat waves around the world.
Work performed by Karl et al. (1996) investigated the likelihood of future heat waves in Chicago
similar to the heat wave event that occurred there in 1995. From the analysis, little evidence was
found to support a trend or increase in heat waves in Chicago based on the historical climate
record. Counter to the Karl et al. finding, work performed by Schär et al. (2004) concluded that
there is an increasing trend in the frequency of heat waves in Europe, and this trend could be
explained by an increasing trend in temperature variability. Work performed by Meehl et al.
(2004) also concluded that projections of global climate models indicate future increases in the
frequency, duration, and severity of heat waves in both Europe and North America. For Europe
similar shifts in the characteristics of future heat waves were found by Koffi et al. (2008) as in
Meehl et al. (2004).
A current weakness with heat wave research is the little or negligible use of statistics of
extreme values in the analysis and modeling of these events. For example, in work of Karl et al.,
Schär et al., and Meehl et al., no statistics of extreme values were used. Specifically for Schär et
al., a Gaussian statistical approach was used.
By definition, a heat wave is an extreme event, and use of statistics of extreme values
would allow for more realistic analyses of these types of events. Through this understanding
there has been some use of extremes statistics in modeling simpler extreme temperature events
such as for single hot days (Nogaj et al. 2006), but very little other use has occurred. This is
especially true for more complex forms of extreme events, such as hot spells that necessarily last
more that than one day.
a. Study at Hand
For this study, we focus on using the Statistical Theory of Extreme Values (e.g. Coles
2001), EVT for short, to develop methods for analyzing the characteristics (frequency, intensity,
and duration) of heat waves. This will be done using two sample temperature data sets from
Phoenix, AZ and Fort Collins, CO, located in the United States obtained from the National
Climatic Data Center. Using statistics of extreme values should be an obvious tool in the
development of methods to study extreme events such as heat waves. We will build on the
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methods already developed to model simpler extreme events such for single hot days (Kharin et
al. 2005; Nogaj et al. 2006) and apply them to clusters of hot days (i.e. heat waves).
Such methods should enable us to more reliably quantify return levels for severe heat
waves, including any trends in their characteristics. We hope these efforts will help further the
science on heat waves and other extreme events, and allow us to better answer the many
questions about heat waves and their occurrences. This in the end should provide a way to help
society prepare and protect itself from these events.
II. Methods
All methods in this study were based on EVT, a branch of statistics used specifically to
analyze extreme events (e.g. Coles 2001). All methods were performed using a statistical
software package, R, and the extRemes Toolkit within R developed by Eric Gilleland. The three
approaches used in this study were derived from EVT and were used to analyze temperature data
from Phoenix, AZ and Fort Collins, CO. Phoenix was selected to be studied because of its
marked heat island effect; it was known that this data for Phoenix should contain trends in heat
waves even if not necessarily due to global warming. Fort Collins was selected based on its long
record of 100 years of temperature data available for the city. For Phoenix and Fort Collins, the
data used consisted of highest summer temperature for each year and daily maximum
temperature over a 62-day period, from July 1st to August 31st. The data for Phoenix spanned
year 1948 to 1990 and years 1900 to 1999 for Fort Collins.
The first approach derived from EVT, the Block Maxima Approach (e.g. Chapter 3 in
Coles 2001), was used to analyze the highest summer temperature for each year of the data. Then
trend models were fitted to the high summer temperature data using the Generalized Extreme
Value (GEV) distribution, a distribution also derived from EVT. This approach helped indicate
trends in the data but not any information about trends in the characteristics of heat waves. This
approach provided motivation to go on to the second approach.
The second approach, the Peaks over Threshold (POT) Approach (e.g. Chapters 4 and 5
in Coles 2001), was used to study the frequency of high temperature clusters (i.e. heat waves)
and the maximum intensity within clusters (measured by the highest temperature observed in
each cluster). To do this, a statistical analysis was performed on the daily high temperature data
for each city. Based solely on statistical criteria, a threshold for each city’s data was found that
satisfied the requirements of EVT and used as the value by which a heat wave was defined in
each city. With this definition, when the temperature exceeded the threshold that was start of a
heat wave and once the temperature fell below the threshold that was the end of a heat wave.
This allowed for multiple day heat wave events or single day heat wave events (events that
would not necessarily be viewed as heat waves from a societal point of view). From here, the
frequency of high temperature clusters per year was analyzed. Then a trend model was fit to this
clusters data using the Poisson Distribution (PD). As well, trend models were fit to maximum
intensity within a cluster data using the Generalized Pareto Distribution (GPD). The approach
provided information about the frequency and intensity of heat waves and motivated us to go on
to the third approach.
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The third approach in this study, an extension of Peaks over Thresholds Approach, was
used to study the duration of heats waves and the dependences of temperatures observed during a
heat wave. More specifically, R code was written to measure length of heat waves based on their
specified thresholds. Then trend models were fit to the duration cluster statistic. The maximum
temperatures on the first and second days of the heat waves for both Phoenix and Fort Collins
were plotted against each other in the form of a scatter plot, along with a scatter plot smoother, to
learn of any daily dependence of temperatures during heat waves.
III. Results
Figure 1. This is a plot of Phoenix’s annual highest
summer temperature using the Block Maxima
Approach. The summer is defined as a period of 62
days from July 1st to August 31st. The data spans
years 1948 to 1990. The trend lines in the figure
(solid, dashed) show there’s an increasing trend in
the highest summer temperature.
Figure 2. This is a plot of Fort Collins’ annual
highest summer temperature using the Block
Maxima Approach. The summer is defined as a
period of 62 days from July 1st to August 31st. The
data spans years 1948 to 1990. The trend lines in the
figure (solid, dashed) show there’s an increasing
trend in the highest summer temperature.
Figure 3. This is a plot of the Generalized Extreme Figure 4. This is a plot of the Generalized Extreme
®
Value Distributions for the highest
summer
SOARS
2008, Value
MarcusDistributions
D. Walter, 4for the highest summer
temperature. The first year’s distribution (1948) is temperature. The first year’s distribution (1900) is
shown by the solid line and the last year’s shown by the solid line and the last year’s
distribution (1990) is shown by the hashed lines. distribution (1999) is shown by the hashed lines.
Figure 5. This is a plot of Phoenix’s number of
clusters per year from 1948 to 1990. A cluster (i.e.
Heat Wave) in this study is defined by temperatures
exceeding a threshold for at least 1 day, and then
falling below the threshold. (see text for more
detail). There is a statistically significant trend in
the number of clusters per year (solid black line).
Figure 6. This is a plot of Fort Collins’ number of
clusters per year from 1900 to 1999. A cluster (i.e.
Heat Wave) in this study is defined by temperatures
exceeding a statistically significant threshold for at
least 1 day, and then falling below the threshold.
(see text for more detail). There is a statistically
significant trend in the number of clusters per year
(solid black line).
Figure 7. This is a plot of Phoenix’s Poisson
Distributions for the number of clusters per year in
the 1st year (1948, solid lines) and the last year
(1990, dashed lines) of the data. There is a
statistically significant shift in the distributions for
more clusters per year over time.
Figure 8. This is a plot of Fort Collins’s Poisson
Distributions for the number of clusters per year in
the 1st year (1900, solid line) and the last year
(1999, dashed lines) of the data. There is a
borderline statistically significant shift in the
distributions for more clusters per year over time.
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Figure 9. This is a plot of Phoenix’s maximum
temperatures within the clusters from 1948 to 1990.
The 50% quantile trend (solid black line) and 75%
quantile trend (dashed blue line) show a nonsignificant trend in the maximums over time.
Figure 10. This is a plot of Fort Collins’s maximum
temperatures within the clusters from 1900 to 1999.
The 50% quantile trend (solid black line) and 75%
quantile trend (dashed blue line) show a nonsignificant trend in the maximums over time.
Figure 11. This is a histogram of Phoenix’s
maximum temperature within clusters from 1948 to
1990 fitted to a Generalized Pareto Distribution.
The data fits well to GPD.
Figure 12. This is a plot of Fort Collins’s
Generalized Pareto Distribution of the maximum
temperatures within the clusters from 1900 to 1999.
The fits well to the GPD.
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Figure 13. This is a plot of Phoenix’s heat wave
durations from 1948 to 1990. The data in these
graphs have been jittered to clearly mark where the
data overlaps. No real trend was seen in the duration
of heat waves.
Figure 14. This is a plot of Fort Collin’s heat wave
durations from 1900 to 1999. The data in these
graphs have been jittered to clearly mark where the
data overlaps. No real trend was seen in the duration
of heat waves.
Figure 15. This is a plot of Phoenix’s 1st and 2nd
day temperatures during heat waves with a scatter
plot smoother (solid line). The data in these graphs
have been jittered to clearly mark where the data
overlaps. Descriptively there seems to be a
relationship between the 1st and 2nd day
temperatures of a heat wave.
Figure 16. This is a plot of Fort Collin’s 1st and 2nd
day temperatures during heat waves with a scatter
plot smoother (solid line). The data in these graphs
have been jittered to clearly mark where the data
overlaps. Descriptively there seems to be a
relationship between the 1st and 2nd day
temperatures of a heat wave.
a. Results 1: Block Maxima Approach for Phoenix and Fort Collins
As mentioned earlier the Block Maxima Approach would be used to analyze the annual
highest summer temperature for both Phoenix and Fort Collins. Figure 1 and Figure 2 displays
the results for each city.
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Figure 1 and Figure 2 are plots of Phoenix’s and Fort Collins’ annual highest summer
temperature with trend lines for the 50% quantile and 90% quantile of the data. The trend lines
for the quantiles were only two aspects of the GEV distribution that we displayed. Other trend
lines for quantiles could have been shown instead if desired. For each city there were statistically
significant trends (p-value ≈ 0 from likelihood ratio tests) in the highest summer temperatures.
For Phoenix, the median temperature (50% quantile trend) increased from approximately 112 F
in 1948 to 115 F in 1990. For Fort Collins, the median temperature (50% quantile trend)
increased from approximately 93 F to approximately 97 F from the year 1900 to 1999. Even
though Phoenix and Fort Collins summer highest temperature is increasing over time, the
variation of the temperature actually decreases over time. The 90% quantile trend is less rapid
than the 50% quantile trend, implying that they are beginning to converge. Further aspects of the
distributions fitted to the data can be seen by comparing the first year to the last year.
Figure 3 and Figure 4 were derived from the same information used to produce Figure 1
and Figure 2 and show the shift in the GEV distribution from the first year to last year of record.
Figure 3 and Figure 4 provide a more detailed look at just the first year and the last year of
record. There is a noticeable shift in the centers of the distributions for the last year for both
cities to the right, toward higher annual temperature maximums as already discussed in
conjunction with Figures 1 and 2. Phoenix’s first year distribution (solid line in graph) was
centered near 112 F and the last year’s distribution (dashed lines in graph) was centered near 115
F. Fort Collins’ first year’s distribution (solid line in graph) was centered near 94 F and for the
last year’s distribution (dashed lines in graph) the distribution was centered near 97 F. Also for
the first year distributions of both cities there was a larger variation in the highest summer
temperatures; the bulk of the data for both cities were spread out over a wider range of
temperatures. For the last year distributions of highest summer temperatures the variation in the
temperatures was much smaller; the highest summer temperature was spread over a smaller
range of temperatures.
Trends were found in the annual highest summer temperature and in the GEV
distributions for the temperature. Although these trends are for individual hot days, they suggest
that there may be possible trends in heat waves for these cities as well. This provided motivation
for us to use the next approach to analyze heat waves directly.
b. Results 2: Peak over Threshold Approach for Phoenix and Fort Collins
The POT Approach was used in this study to analyze the frequency of high temperature
clusters (i.e. heat waves) and the maximum intensity within clusters (measured by the highest
temperature observed in each cluster). Figure 5 through Figure 12 display the results for each
city.
Figure 5 and Figure 6 show plots of Phoenix’s and Fort Collins’ number of clusters per
year using a PD. For Phoenix, a statistically significant trend in the number of clusters per year
was found, displayed by the black line in the figure, with a cluster based on a threshold of 110.5
F. The likelihood-ratio test against the null model of no trend in the data produced a p-value =
0.0011. The mean number of clusters (trend line) increase from approximately 1.5 to 4.5 per year
over the course of 43 years. For Fort Collins, a statistically significant trend in the number of
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clusters per year was found, with a cluster based on a threshold of 93.5 F. The likelihood-ratio
test against the null model of no trend in the data produced a p-value = 0.0228. The mean
number of clusters (trend line) increase from approximately 1.8 to 2.9 per year over the course of
100 years.
Figure 7 and Figure 8 were derived from the same information used to produce Figure 5
and Figure 6 and show the shift in the PD of the number of clusters per year from the first year to
last year of record for Phoenix and Fort Collins. Figure 7 and Figure 8 provide a more detailed
look at just the first year and the last year of record. For Phoenix, the distributions for the first
and last years are very much difference from each other. For example on Figure 7 for Phoenix
the probability of having 4 or more clusters per year was approximately 7 % in the first year
according to the fitted PD. The probability of having more than 4 clusters per year for Phoenix
during the last year’s distribution was more than 50%. This is an extreme shift in the probability.
The same type of shift can be seen for Fort Collins. The probability of 4 or more clusters
occurring in a year, from the first year to the last year of the data, shifted from approximately 9
% to approximately 30%.
Trends were found in the number of the clusters per year using the PD fitted to the data.
Although such trends are important, they say nothing about the severity of heat waves. One way
to measure severity is in terms of the highest daily maximum temperature within a cluster.
Figure 9 and Figure 10 show that the temperature extremes seen within clusters for each
city, along with trends in two quantiles of the fitted GPD. For Fort Collins, there is really no
apparent trend in the extremes within clusters (not statistically significant according to likelihood
ratio test). For Phoenix, there is a slight apparent trend in the extremes within heat waves but not
statistically significant. This means that the severity of clusters stay basically the same over time
in both cities.
Because of no significant trend in the severity of heat waves, a GPD with no trend fitted
to the temperature extremes seen within clusters is shown instead. Figures 11 and 12 include
histograms of the temperature extremes seen within clusters from Figure 9 for Phoenix and
Figure 10 for Fort Collins along with the fitted Generalized Pareto Distribution. For both cities,
the actual data for the maximum temperature within a cluster fit fairly well to the Generalized
Pareto Distributions.
c. Results 3: Extension of Peak over Threshold Approach for Phoenix and Fort Collins
The extension of the POT Approach was used in this study to analyze the duration of heat
waves for the two cities. The extension of POT was needed because the regular POT Approach
does not provide any model of the probability distribution of the cluster length. Figure 13 and
Figure 14 display the results for the two cities.
Figure 13 and Figure 14 show the observed duration of heat waves for Phoenix and Fort
Collins, along with a scatter plot smoother. For Phoenix, descriptively, there seems to be no
apparent trend in the duration of heat waves. Average length of a heat wave is between 1 and 2
days. This is the same for Fort Collins in terms of length of heat wave. There is a apparent
SOARS® 2008, Marcus D. Walter, 9
decrease in the duration of heat waves early in the time period of the data for Fort Collins. Then
it levels off through the rest of the data.
The third approach also included examining the relationship between consecutive days of
heat waves because the regular POT Approach models only the cluster maxima. In this study we
only examine the 1st and 2nd day temperatures within a heat wave. We did not examine the
relationship between the 2nd and 3rd day, 3rd and 4th day, etc., because there was not as much
data available for these consecutive days. Figure 15 and Figure 16 show the results.
Figure 15 and Figure 16 are scatter plots of Phoenix’s and Fort Collins’ 1st and 2nd day
temperatures during heat waves along with a scatter plot smoother. For Phoenix and Fort Collins,
there seems to be an apparent positive relationship between 1st day’s and 2nd day’s temperature
during a heat wave. Both trend lines in each graph have positive slopes. For Phoenix, the trend
line increases to roughly a constant trend over time. For Fort Collins, the trend line increases
more gradually over time. These results indicate that the dependence between consecutive days
within clusters will need to be taken into account when developing statistical models to fully
characterize all features of heat waves.
IV. Discussion
With using the methods and approaches that incorporated EVT, we were able to learn
much about the data being studied and the methods used. From the analysis, we learned that both
Phoenix’s and Fort Collins’ highest temperature reached from July to August has steadily
increased over time. We found that the number of heat waves per year on average has increased
over the time for both cities based on our definition of a heat wave. It was detected that heat
waves for these cities have not become more intense, nor has the duration of heat waves
increased. It was discovered that there may be a positive relationship between the maximum
temperatures on individual days within a heat wave.
We also determined that the method and approaches used in this study have utility. They
provide more accurate ways to analyze heat waves and their interesting characteristics through
incorporating Extreme Value Theory.
In order to carry out this research we had to make several assumptions and develop
definitions for a heat wave and a summer. In the following it will be laid out why certain
definitions and assumption were made.
Defining a heat wave and what thresholds or threshold the temperature must cross to start
and end a heat wave was one of the first pieces to this research, but not a simple task. Heat waves
are different all over the world. A heat wave is Florida may not be the same as a heat wave in
Alaska. Objectively defining what a heat wave is is still being widely researched. Therefore we
chose to define it solely based on statistical criteria, a non biased or non rigged approach. Basing
it on statistics also allowed for flexibility in the thresholds we chose for the cities studied; it was
flexible enough for other statistically significant thresholds could be chosen that may work with
already established heat wave definitions. As it turned out, in this research a heat wave in
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Phoenix was not the same as a heat wave in Fort Collins. The ways we chose to define a heat
wave definitely influenced the results achieved.
We defined a summer based on the data we had, and we used data daily high temperature
data and highest summer temperature data for a period from July through August, from years
1948 to 1990 for Phoenix, and from years 1900 to 1999 for Fort Collins. We also based our
summer on having the probability of the temperature exceeding the threshold be the same
through out the entire period of the day. We acknowledge our definition of a “summer” was not a
summer in the real world. Because of our definition of a summer, it is understood that our results
were affected by our definition of a summer. For example if the summer was longer, we may
have found more high temperature clusters per summer.
It is understood that our research may not have produced the most accurate results based
on definitions and assumptions, but the use of EVT in this research was more important than the
actual results found.
V. Conclusion
The goal of this research was to develop methods that would be statistically more
appropriate for studying and analyzing heat waves. This research showed how Extreme Value
Theory and Statistics of Extreme Values can be applied to model certain features of heat waves
or high temperature clusters. For example, EVT was used to model the frequency and cluster
maxima of heat waves, including detection of trends. This research also explored other
characteristics of heat waves, e.g. cluster duration and individual maximum temperatures within
clusters, indicating how extreme value approach would need to be extended to fully model all
features of heat waves.
The implications of this research are clear. More reliable quantification of return levels
for severe heat waves, including any trends in their characteristics, will be achieved with the
continual development and future use of these methods that incorporate the Statistical Theory of
Extreme Values. Future research would involve applying these methods to analyzing output from
climate models. Future research would also involve the use of a more realistic definition for a
heat wave. These would provide a more accurate and reliable estimation of the likelihood of heat
waves and the other extreme events predicted by the climate models. This in the end should help
society have a better understanding of heat waves and other extreme events so that it can prepare
for these events, protect itself, and better adapt to future changes in these events due to climate
changes.
VI. Acknowledgements
The idea to study heat waves and apply the Statistical Theory of Extreme Values to these
events in hopes of better analyzing and understanding of these events was propose by my science
mentor Dr. Richard Katz. I thank Dr. Katz for all his help and support with this project. I would
not have been able to start or complete this project with him. I would not have learned as much
about EVT and the statistics of extremes in general if it was not for him. I am very appreciative
of the time he sat aside this summer to guide me through this research project and the hard work
SOARS® 2008, Marcus D. Walter, 11
he put in to teach me more about heat waves and statistics. I would like to thank my second
science mentor Dr. Eric Gilleland for the time he set aside to help me with the coding language
used in this research. I would also like to thank Eric and Rick for taking me out to eat several
times all over Boulder and making Boulder feel like my home a way from home. I thank Tim
Barnes for making time this summer to edit my papers and presentation and for giving me
positive feedback on my writing that I can keep with me as I continue to write in the future.
Lastly, but not least, I would like to thank the SOARS program and staff for providing me with
the opportunity to come out to Boulder and do research for a second summer. My participation in
this program has truly benefited me as a student and a professional.
VI. Reference:
Coles, S., 2001: An Introduction to Statistical Modeling of Extreme Values. Springer, London
Karl, Thomas R. and Richard W. Knight, 1996: “The 1995 Chicago Heat Wave:
How Likely Is a Recurrence?” Bulletin of the American Meteorological Society Vol. 78,
No. 6, June 1997, P. 1107 -1119
Kharin, V.V., and F.W. Zwiers, 2005: "Estimating extremes in transient climate change
simulations." Journal of Climate, 18, 1156-1173.
Koffi, Brigitte, and Ernest Koffi, 2008: “Heat waves across Europe by the end of the
21st century: multiregional climate simulations.” Climate Research, Vol. 36: 153–168,
2008
Meehl, Gerald A. and Claudia Tebaldi, 2004: “More Intense, More Frequent, and Longer Lasting
Heat Waves in the 21st Century.” Science, Vol. 305, 13 August 2004, P. 994 – 997
Nogaj, M., P. Yiou, S. Parey, F. Malek, and P. Naveau, 2006: "Amplitude and frequency of
temperature extremes over the North Atlantic region." Geophysical Research Letters, 33,
No. 10, 17 May
R Development Core Team (2007). R: A language and environment for statistical computing. R
Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL
http://www.R-project.org.
Schär, Christoph, Pier Luigi Vidale, Daniel Lüthi, Christoph Frei1, Christian Häberli, Mark A.
Liniger & Christ of Appenzeller, 2004: “The role of increasing temperature variability in
European summer heatwaves.” Nature, Vol. 427, 22 January 2004
VII. List of Figures
Figure 1. This is a plot of Phoenix’s annual highest summer temperature using the Block
Maxima Approach. The summer is defined as a period of 62 days from July 1st to August 31st.
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The data spans years 1948 to 1990. The trend lines in the figure (solid, dashed) show there’s an
increasing trend in the highest summer temperature.
Figure 2. This is a plot of Fort Collins’ annual highest summer temperature using the Block
Maxima Approach. The summer is defined as a period of 62 days from July 1st to August 31st.
The data spans years 1948 to 1990. The trend lines in the figure (solid, dashed) show there’s an
increasing trend in the highest summer temperature
Figure 3. This is a plot of the Generalized Extreme Value Distributions for the highest summer
temperature. The first year’s distribution (1948) is shown by the solid line and the last year’s
distribution (1990) is shown by the hashed lines. There is a statistically significant shift to higher
summer highest temperatures and a decrease in the variance. (Also see Figure 1)
Figure 4. This is a plot of the Generalized Extreme Value Distributions for the highest summer
temperature. The first year’s distribution (1900) is shown by the solid line and the last year’s
distribution (1999) is shown by the hashed lines. There is a statistically significant shift to higher
summer highest temperatures and a decrease in the variance. (Also see Figure 2)
Figure 5. This is a plot of Phoenix’s number of clusters per year from 1948 to 1990. A cluster
(i.e. Heat Wave) in this study is defined by temperatures exceeding a statistically significant
threshold for at least 1 day, and then falling below the threshold. (see text for more detail). There
is a statistically significant trend in the number of clusters per year (solid black line).
Figure 6. This is a plot of Fort Collins’ number of clusters per year from 1900 to 1999. A
cluster (i.e. Heat Wave) in this study is defined by temperatures exceeding a statistically
significant threshold for at least 1 day, and then falling below the threshold. (see text for more
detail). There is a borderline statistically significant trend in the number of clusters per year
(solid black line).
Figure 7. This is a plot of Phoenix’s Poisson Distributions for the number of clusters per year in
the 1st year (1948, solid lines) and the last year (1990, dashed lines) of the data. There is a
statistically significant shift in the distributions for more clusters per year over time.
Figure 8. This is a plot of Fort Collins’s Poisson Distributions for the number of clusters per
year in the 1st year (1900, solid line) and the last year (1999, dashed lines) of the data. There is a
borderline statistically significant shift in the distributions for more clusters per year over time.
Figure 9. This is a plot of Phoenix’s maximum temperatures within the clusters from 1948 to
1990. The 50% quantile trend (solid black line) and 75% quantile trend (dashed blue line) show a
non- significant trend in the maximums over time.
Figure 10. This is a plot of Fort Collins’s maximum temperatures within the clusters from 1900
to 1999. The 50% quantile trend (solid black line) and 75% quantile trend (dashed blue line)
show a non- significant trend in the maximums over time.
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Figure 11. This is a histogram of Phoenix’s maximum temperature within clusters from 1948 to
1990 fitted to a Generalized Pareto Distribution. The data fits well to GPD.
Figure 12. This is a plot of Fort Collins’s Generalized Pareto Distribution of the maximum
temperatures within the clusters from 1900 to 1999. The fits well to the GPD.
Figure 13. This is a plot of Phoenix’s heat wave durations from 1948 to 1990. The data in these
graphs have been jittered to clearly mark where the data overlaps. No real trend was seen in the
duration of heat waves.
Figure 14. This is a plot of Fort Collin’s heat wave durations from 1900 to 1999. The data in
these graphs have been jittered to clearly mark where the data overlaps. No real trend was seen in
the duration of heat waves.
Figure 15. This is a plot of Phoenix’s 1st and 2nd day temperatures during heat waves with a
scatter plot smoother (solid line). The data in these graphs have been jittered to clearly mark
where the data overlaps. Descriptively there seems to be a relationship between the 1st and 2nd
day temperatures of a heat wave.
Figure 16. This is a plot of Fort Collin’s 1st and 2nd day temperatures during heat waves with a
scatter plot smoother (solid line). The data in these graphs have been jittered to clearly mark
where the data overlaps. Descriptively there seems to be a relationship between the 1st and 2nd
day temperatures of a heat wave.
SOARS® 2008, Marcus D. Walter, 14