Statistics of Extreme Events with Application to Climate
Statistics of Extreme Events with Application to Climate
H. Abarbane1
S. Koonin
H. Levine
G.
MacDonald
O. Rothaus
Accesion For
NTIS CRA&I
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10;
January 1992
JSR9030S
Approved for public release; distribution WlIimIted.
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DATE
January 10, 1992
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4. TITLE AND SUlmu S. FUNDING NUMlIIlS
Statistics of Extreme Events with Application to
Climate
6. AUTHOl(S)
G. MacDonald, et al
PR  8503Z
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11.
AISTRACT
(1IM.,mumlOOwonlll.l
The statistical theory of extreme events is applied to observed global average temperature records and to simplified models of climate. Both hands of records exhibit behavior in the tails of the distribution that would be expected from a random variable having a normal distribution. A simple nonlinear model of climate due to Lorenz is used to demonstrate that the physical dimensions of the underlying attractor, determined by applicable conservation laws, limits the range of extremes. These limits are not reached in either observed series or in more complex models of climate. The effect of a shift in mean on the frequency of extremes is discussed with special reference to possible thresholds for damage to climate variability.
14. SU.,ECT TEAMS bernoulli trials, nonparametric order statistics, asymptotic distribution, lorenz 27variable
17. SICUInY CLASWICAT10III
OF llPOllT
UNCLASSIFIED
1 S. NUMlll OF PAGIS
11. PRlCI CDOI
,a.
SICUIllTY CLASSIfICATION 11. SlCUIUTY CLASS.ICATIOfII
OF THIS PAGE
OF AlSTUCT
UNCLASSIFIED
UNCLASSIFIED
ZO. UMlTATIOfII
Of
AISTUCT
SAR
Standard Form
198
(R.".
ll"
_ _ ., ...... 'to '"""
Contents
1 INTRODUCTION
1
2
RETURN PERIOD
5
3 FREQUENCY OF EXCEEDANCES
9
3.1 Bernoulli Trials . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4 NONPARAMETRIC ORDER STATISTICS
11
5 APPLICATIONS OF ORDER STATISTICS TO GLOBAL
AVERAGE TEMPERATURE
15
6 DISTRIBUTION OF EXTREMES
19
7 ASYMPTOTIC DISTRIBUTION OF EXTREMES FOR A
NORMAL PROCESS 23
7.1 Applications of Asymptotic Theory of Extremes to a Climate
Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
7.2 Physical Dimensions of an Attractor . . . . . . . . . . . . . . 29
7.3 Extremes for a Climate Model with a Linear Increase in Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48
7.4 Distribution of Extremes for Global Average Temperature Record 50
8 EFFECT OF A SHIFT OF THE MEAN ON THE FRE
QUENCY OF EXTREMES 63
9 SUMMARY
69
III
1 INTRODUCTION
The social and economic costs associated with global warming will be measured in terms of changes in the frequency and intensity of extreme events such as droughts, floods, hurricanes, tidal surges, etc. Small changes in the mean temperature are easily adjusted to, but successive years of drought or a sequence of storms are much less easily handled even by advanced societies.
The capability to forecast the frequency and intensity of extreme events in an altered atmosphere poses a great challenge.
In the statistical literature the large or small values assumed by a random variable from a finite set of measurements are termed extreme values.
The variable of interest could be sea level, atmospheric temperature, precipitation, stream flow, etc. The largest and smallest value of the extremes are often of most interest. However, other extremes such as the second or third largest or smallest may be of interest. The extreme values are random variables. For example, the minimum temperature in January in Bismarck,
North Dakota, exhibits random variations that are best described in terms of probabilities.
The application of extreme value statistics is uncommon in the environmental sciences except in the field of hydrology, where the concepts have been used in designing dams. As noted in Levine et al. (1990), extreme value theory has not received much attention in climate studies or in discussions of greenhouse warming despite the obvious importance of large deviations from
1
the mean. The theory and applications are rarely treated in books on probability and statistics and almost never covered in university courses. If there are references to the subject, it is generally to Gumbel's (1958) book which has long been out of print. Cramer and Leadbetter (1967) and Leadbetter,
Lindgren and Rootzen (1982) provide an extensive treatment of extremes in stationary sequences but focus primarily on certain of the difficult mathematical issues related to extreme values. The statistics of extremes is closely connected to order statistics which provides an approach to distribution free or robust statistics (David, 1981). The present paper provides a review of the theory with a number of applications that are related to climate change
ISSUes.
The applications are both to real data  time series of temperature derived from various sources  and to results obtained from much simplified computer models of climate. The latter application is primarily to provide an illustration of how the problem of extremes can be approached, rather than to give definite answers. Some of the examples could be extended to data derived from large scale global circulation models (GCMs) of the atmosphere, but since we do not have access to GCMs we have not done so.
Sections 2, 3 and 4 serve to introduce the nomenclature of order statistics with a few elementary examples of the calculation of the probability of extreme climatic events. In Section 5, results from order statistics are applied to calculate the probability that the observed global average surface air temperature records contain a trend. Unlike a previous analysis (Levine et aI.,
1990), no assumption is made as to the underlying statistical distribution.
2
Since 'extremes are rare events they can be approximated as independent with negligible correlation. In accord with the earlier analysis (Levine et al.,
1990), order statistics indicate very high odds (l05 10
6 to 1) in favor of the hypothesis that the data records contain a linearly increasing trend in global average surface air temperature.
Section 6 presents an introductory discussion of the distribution of extremes with no restricting assumptions as to the distribution of the parent population. While general results can be obtained, detailed applications require assumptions as to the underlying distribution. The theory is specialized in Section 7 to normal distributions. An application to the Lorenz
27variable model of climate shows that the temperature values calculated for the Lorenz model closely approximate normally distributed variates both about the mean and in the tails (extremes) of the distribution for a rUll of
1000 years.
For a normally distributed random variable, the range (maximum minus minimum) increases without limit as the number, n, of variates selected increases (as
(In n)1/2).
Section 7.2 provides an example where the physical dimension of the underlying attractor limits the growth in range with time. This limit is not observed in model runs or in observed temperature records (see Sections 7.3 and 7.4). Both model runs and observed temperature records from which a trend has been removed closely approximate normally distributed variates about the mean and in the tails of the distribution.
These results raise questions as to the predictability of climate.
The effect of a shift in mean on the frequency of extremes is discussed
3
in Section 8 with special reference to possible thresholds for damage due to climate variability. Section 9 provides a brief summary of the major findings of the report.
4
2 RETURN PERIOD
Engineers who design dams are interested in the time interval between two discharges of a river, each of which is greater than or equal to a given discharge. The probability p of a random variate
X
having a value greater than
x is p
=
1 
F(x) 
1 i~
f(y) dy
F(x) 
Pr(X
~
x)
where
F(x)
is the cumulative probability function and
f(x)
is the probability density function. We are interested in the number of independent trials
k
before the value x is exceeded. The variable
k
is an integer limited on the left but not on the right, since the value of
x
may never be surpassed. The probability that x is first exceeded at trial
k
is since the event failed for the first kl trials (Bernoulli trials). The mean for
k
is simply
E [k]
=
IIp
1 
1
F(x)
>
1.
The return period is defined by
T(x)
1
==
1 
F(x)'
If an event has probability p, on the a.verage we have to make 1
I
p trials in order for the event to happen once.
5
The standard deviation for
k
is which for a long return period becomes
0' '::!. v'2
1
(T  ).
2
The smaller the probability of an event, the larger the spread of the distribution. This general property lessens the usefulness of extremes in numerous situations.
The cumulative probability W
(k)
that the event happens before or at the kth trial is
W (k)
=
1  (1 _
p)k.
The probability that the event will happen before its return period
T
is
W(T)
_ 1  (1 _
.!.)T
T
~
1 
~ e
=
0.6231.
H the value to be exceeded, x, is large and the return period long, say greater than 10, then the cumulative probability
W(k)
can be approximated by
k
W(k)
'::!.
1  exp( 
T).
In hydrology the return period is usually measured in years. For example, a design engineer may wish to have a high probability (0.95) that an event does not happen in N years. The return period T for the design is then
6
approximately
N
T= lW
80 that in order to have a probability of 1 in 20 that the event does not happen in 10 years requires the design to have a much larger return period of 200 years.
These simple considerations illustrate certain features of the statistics of extremes. Under the assumption that the events are independent, results can be obtained that are free of any assumption with respect to the underlying distribution. The assumption of independence might appear to be highly restrictive. For dependent variables, their correlation as a function of time difference or lag is a partial measure of their dependence. Many of the results derived for independent variables apply to dependent variables provided that the correlation decreases rapidly enough with lag (Leadbetter, Lindgren and
Rootzen, 1983). The basic reason for applicability of results obtained from the theory of independent events to correlated series is that extreme events are rare. Rare events are expected to be separated by long time intervals
80 that the effects of correlation are negligible. If the correlation function decays at least as fast as 1/
In n
where
n
is the time, then the effects of the correlation can be neglected as is discussed by Leadbetter, Lindgren and
Rootzen (1983).
7
3 FREQUENCY OF EXCEEDANCES
3.1
Bernoulli Trials
The statistics of Bernoulli trials can easily be applied to extreme events.
The probability that during the next n years the maximum July temperature in Washington, D.C., will exceed 105°F exactly k times is where
p,
estimated on past observations, is the probability that the maximum
July temperature will not exceed 105°F and where
n n!
( k ) =
(n 
k)! k!
is the binomial coefficient. The probability of exceeding 105°F in Washington is about a 1 in 20 year event,
p
= 0.95. In the next decade, provided that there is no upward or downward trend in temperatures, the probability that the maximum July temperature will not exceed 105°F is or there is a 40 percent chance that the maximum July temperature will exceed 105°F in the period 19901999.
The probability that during the next ten years there will be fewer than three years for which the maximum temperature for July exceeds 105°F can
9
also be calculated. There are only three ways that the number of exceedances of 105°F cannot be greater than two namely when the number is zero, one or two. The probability of these mutually exclusive events is the sum of the probabilities of each
Po
+
PI
+
P2
=
(1~)
(0.95)10 (0.05)0
+ (
11
0
) (0.95)9 (0.05)
+
(1:) (0.95)8 (0.05)2
=
0.9885.
The probability of the maximum July temperature in Washington in the next decade exceeding 105°F three or more times is only 1 in a 100.
H in the next decade the temperature does exceed 105°F in July three or more times, the hypothesis that there is no upward trend in the temperature should be reexamined.
The number of exceedances depends on the interval over which the observations are made. From the results of the above example we can calculate the probability that over the next century every decade will have no more than two Julys in which the maximum temperature exceeds 105°F. With a shift in time scale the probability is
P = ( 1
0
0 ) (0.9885)10 (1  0.9885)0  0.89.
10
4 NONPARAMETRIC ORDER STATIS
TICS
The need to estimate the probability Pm of the mth ranked observation can be avoided by using past observations to determine the exceedances in future trials. Let n denote the number of past observations,
N
the number of future observations, m the rank of the ordered past observation and
k
the number of exceedances of the mth ranked observation in
N
future trials.
The probability of observing in
N
trials,
k
exceedances of the mth ranked observation seen in n trials is:
P(k;n,m,N)
( N )(
k
n )m fl m Jo
pNk
(1 _
p)k pnm
(1 _
p)mldp
( : )(
~
)m
(N
+
n) (
N
+
n 1 )
m+kl
since the integral is a special case of the Beta integral
For the largest value, m
=
1, the probability of
k
exceedances is
nN!(N
+
n  k
I)!
P(kj n, I,N)
= (N _ k)! (N
+
n) (N
+
n 1)'"
The probability that the maximum value observed in
n
trials is not exceeded in
N
future trials is
n
P(O;n,I,N)
= N
'
+n
11
and there is 0.5 probability that the maximum is not exceeded in an equal number,
N
= n, of future trials. The calculation of the probabilities for the exceedance of the largest value is aided by the recursion relation
(N
+
1 
k)
P(k+l;n,I,N)= (N+nk) P(k;n,I,N).
The probability that all values in N future trials will exceed the largest value observed in the first n trials is small. For the case
N
=
n,
this probability is which for
N
=
6 is 0.00108.
P(N; N,
1, n)
(N!)2
=
(2N)!
The various moments of the distribution function can be obtained from properties of the hypergeometric function (Gumbel, 1954). The mean number of exceedances is
E (n,m,N) 
N
L k P(k;n,m,N) k=l
mN/(n
+
1).
Similarly, the variance of the estimate of the number of exceedances is
2(
N)_m(nm+l)N(N+n+l) u n,m,

(n+l)2(n+2) .
The variance increases with the number,
N,
of future trials and decreases with the number of past observations used to set up the statistics. The variance for the number of exceedances of the maximum is
2
nN(N+n+l) u (n,l,N)
=
(n+
1)2
(n+2)"
12
The variance of the estimate for the median, m
= nil,
n
odd, is
2(
n+l N)=N(N+n+l)
u n, 2 '
4{n
+
2) so that the ratio of the variance of the estimated number of exceedances of the maximum to the number of exceedances of the median is
u
2
(n,
1,
N) _
4n u
2
(N, nr,N) 
(n
+
1)2·
In one sense, the extremes are more reliable than the median since the variance of the number of exceedances of the median is about
~n as large as the variance for the maximum. In the limit of large
N
with
N
= m the mean and the variance take on simple forms
E[k]
~ m and
The mean number of exceedances over the mth largest value is equal to the rank m itself. The distribution is then similar to a Poisson distribution for integers. However, the variance is twice that of a Poisson distribution.
The probability distribution of exceedances in this case is independent of the distribution of the random variable
X.
It does depend on the assumption that the observations are independent. The lack of dependence on the distribution is an a.ttractive feature of the theory, since the distribution may be known in the vicinity of the median but there will, in most cases, be very few observations with regard to extreme values.
13
5 APPLICATIONS OF ORDER STATIS
TICS TO GLOBAL AVERAGE TEMPER
ATURE
A 109year record of the annual global average temperature of the atmosphere has been determined by Hansen and Lebedetf (1987, 1988) (see
Figure 1). The data have been normalized to zero mean. The maximum temperature for the 100year period 18801979 is 0.2°C, observed in 1954.
The probability of one or more exceedances of the maximum of this 100year record in the nineyear period 19801988 is
P
(>
1; 100,1,9)
100
=
1  109
=
0.083 if the average temperature can be taken as an independent trial and there is no trend. The expected number of exceedances is E(100, 1,9)
=
9/100
=
0.089 with a variance of (7'2(100,1,9)
=
0.095. In fact there were five exceedances: 1988, 1981, 1987, 1983 and 1980. The probability under the independence assumption of five exceedances is
P
(5;100,1,9)
=
1.03 x
10
6
•
The observed five exceedances strongly contradict the assumption of independence and provide support for the hypothesis that there is an increasing trend in the annual global average surface temperature.
H a linear trend with a least square slopes of 0.55° per century is removed the maximum residual in the first 100 years is 0.28°C in 1926. There are no exceedances for the years 19801988 in the residual as would be expected from
15
0.60
r..
0.80
&.... _ _ _ _
1880 1900 1920
Date
1940 1960
Figure 1.
Variations of global average temperature. The solid curve is a filtered version corresponding to a fiveyear running average (after Hansen and Lebedeff,
1987; 1988).
'_I
1980
16
Table 1
Expected Number of Exceedances in SixYear Runs of Global
Annual Average Temperature and Observed Exceedances in 17
SixYear Runs
Number of
Exceedances k
Expected of Maximum Probability
Number of
Observed Number
Value of k Exceedances of Exceedances
Observed Number of Exceedances in Residual after Removal of Linear
Trend
0
1
2
3
4
5
6
0.5
0.273
0.136
0.061
0.027
0.006
0.001
8.5
4.64
2.31
1.03
0.46
0.11
0.02
2
1
12
0
1
1
0
2
3
3
4
2
1
2 the probability distribution for the maximum. In terms of odds, the ratio of the probability of the observations containing a trend to the probability that the temperatures are independent, identically distributed random variables
IS
0.917 =8.9xI0
5 •
1.03 x 106
As a further example of the use of the probability distribution, the 108year record, 19801987, is divided into 18 consecutive sixyear segments. The maximum value in the first sixyear period, 18801885 is  0.37°C. The number of exceedances in each of the subsequent sixyear segments is tabulated in Table 1 along with the number obtained for the probability distribution.
17
The expected number of exceedances in any sixyear period is
E
(6,1,6)
=
6/7
=
0.86 with a standard deviation of
iT
=
1.09.
Clearly the number of sixyear runs with all six values in excess of the maximum for the first sixyear segmeut is much larger, 16, than would be expected from a sequence of independent trials. The probability of obtaining 12 segments out of 17 in which all the values are in excess is
17!
( 1 2 5
= 1.5
32 x 10 .
Table 1 also tabulates the observed distribution of exceedance for the residuals after a linear trend has been removed. The observed distribution still differs from the calculated one. However, the probability of obtaining two segments in which all six values are greater than the maximum observed in the first sixyears while small, 1.5 x 10\ is much larger than the probability of obtaining 12 such sequences.
Levine et. al (1990) calculated that the odds for a linear trend in the data were 1.7 x 105 to one. In that analysis, the correlation between annual temperature was explicitly taken into account through the use of the full covariance matrix in the least squares analysis but the analysis was based on the assumption that the global average temperatures were normally distributed. The analysis presented above supports the hypothesis of a trend by similar odds, but is based on the assumption of independent trials without any assumption on the underlying statistical distribution.
18
6 DISTRIBUTION OF EXTREMES
The above analysis provides information with respect to the number of observations that would exceed the mth largest value of n initial observations.
The analysis provides no information on the value that would exceed the observed extremes. Such information requires that the underlying probability distribution,
F(x),
be known. As illustrated above, some of the analysis of the frequency of exceedances can be carried out without assumptions with respect to the underlying distribution of the random variable. The exact form of the distribution is generally not known, but an asymptotic theory that covers a wide variety of initial distribution has been developed (Cramer,
1946, Gumbel, 1958).
As before, let
k
denote the kth value from the top of a sample of n observations. The probability density
g(Xj n, k)
for the kth value is equal to the probability that among n samples n 
k are less than x, and k  1 are greater than
x
while the remaining value falls between
x
and
x
+
dx g(Xj n, k)
= n(
~
=
)[F(x),,k
(1  F(x))kl]f(x)dx where
F(x)
is the probability distribution function and
f(x)
is the density distribution function. For the maximum value
k =
1 and
g(xjn, 1) = [F(x)]" dx.
For a population having a normal distribution with zero mean and unit variance
F(x) =
~1% et
'/2
dt.
v211'
00
19
The dependence of the maximum value on the number of observations from a series of independent trials drawn from a normal population is illustrated in Figure 2. For a sample of ten, the maximum has a 0.05 probability of exceeding 2.57 standard deviations; for a sample of 1000, the maximum will exceed 3.89 standard deviations one in twenty times.
20
0.6
~ i
£
0.5
0.4
0.3
0.2
0.9
0.8
0.7
0.1
0
0
•••
••
••
••
••
•
•
•
•
•
• n 10'
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
••
••
••
••
•••••
n 1000,
I
~
I
I
I
,
,
I
I
,
I
I
,
I
I
•
•
I
I
•
,
•
I
•
•
I
•
, ,
,
•
•
•
•
,
,
I
•
•
I
•
I
•
•
•
~
! n 10.000
.
4
2 3
X Measured In Sid.
5
Figure 2. Variations of the probability of the maximum of n independent trials with n.
The distribution has zero mean and unit variance.
21
7 ASYMPTOTIC DISTRIBUTION OF EX
TREMES FOR A NORMAL PROCESS
The general asymptotic theory of the distribution of extremes deals with the conditions under which approaches a limiting distribution G(x) where a" and b" are suitable normalizing constants and
The theory states that every distribution G, has, up to scale and location changes, one of the following parametric forms commonly called the Three
Extreme Value distributions:
Type I:
oo<x<oo G(x)
= exp(
_er )
Type II:
x>O
x~O
G(x)
= exp(
_xa);
Q
G(x)
= 0
>
0
Type III:
x <0 x>O
G(x)
= exp( (
_x)O);
Q
> 0
=1
(Leadbetter, Lindgren and Rootzen,
1983).
The normal distribution, unlimited on the left and right, falls into the Type I domain of attraction. The density distribution for the kth ordered value becomes which is illustrated in Figure 3 for
k
=
1, 2, 3.
For a normal distribution with zero mean and unit variance, if x is the
23
i
~
0.8 ci:
I
0.6
1.4
1.2
0.4
,
,
,
..
,
,
,
I
•
•
•
•
•
•
•
•
•
•
,
,
,
,
•
figure 3. The probability density function for the double exponential.
24
kth value from the top, then { is
The normalizing constants can be obtained by a complex calculation.
The solution for x when n is large can be obtained by partial integration leading to
With bounded {, x is given by
x
=
(2
In
1/2 In In
n
+
In
411"
In { n) 
2(2
In n)1/2
(2
in n)1/2
1
+
O(
in n)·
For the general form of the normal distribution with arbitrary mean m and variance
(T2,
the form for
Mn =
x
is asymptotically
X 
_
m
112 In In
n
+
In
411"
(T v
+
0'(2 In
n) 
0'
2(2
in n)1/2
+
(2
In n)1/2
where v is a random variable having the frequency function
The expression for the kth value from the bottom is given by a similar expression with opposite sign for all terms but the mean, m.
The expected value Cor the kth value from the top is
=
m+O'
(2
I
nn
)1/2 _ In In n
+
In
411"
+
2(8
(2/nn)1/2
1 
C) 0(_1_))
+
Inn
with variance
D2
0'2
D2(x·
n
k)
= :
" 2
In
n
(
n2 _
6
8
2
)
+
0 (_1 )
In
2
n
25
where C is Euler's constant, C=0.57722 ... ,
1
+k1
1
+
(k
1)2'
Similar expressions can be calculated for the difference between the kth maximum and kth minimum values of the n independent trials drawn from a normal population.
E[ _ .
k] x 1/,n,
=
(4
In n In In n In
411" 
2(5
1 
C)
(1
{2
In n)1/2
0(_1_»)"
+
Inn
(12 11"2
D2[x
1/;
n, k]
=  (  52)
In n
6
1
+
0().
In2 n
The dependence of the range on the number of observations is indicated in Figure 4. The three standard deviation bounds are also shown. For a normal population of a 1000 variables there is about 1 in a 100 probability that the maximum range will exceed eight standard deviations. The expected value for the maximum of normal population is shown in Figure 5.
7.1 Applications of Asymptotic Theory of Extremes to a Climate Model
The variation of global average annual temperature depicted in Figure 1 shows irregular behavior with a nonvanishing correlation function for lags up to the length of the record and continuous power spectrum with abundant energy at low frequencies (Levine et.al, 1990). The irregular behavior, characteristic of chaotic systems, also shows clearly in complex models of the
26
6
10~~r~~~~
..

_._

......


2
4
6 o
Number of Observations
Figure 4. Variations of the maximum expected range for a random variate drawn from a normal population as a function of the number of observations. The distribution has zero mean and unit variance. The three standard deviations from the expected value are also shown.
27
3£" ; :
4 r,~~~~
3.5
'.5
I
I
,
,
,
•
,
,
,
,
"
"
"
,,"
,.
.

_
.
.


.


,

._
..

.

o
200
400
600
Number of Observations
800
'000
Figure
5. Variation in expected maximum value of an observation drawn from a normal population with zero mean and unit variance as a function of the number of observations. The three standard deviations from the expected value are also shown.
28
atmosphere such as General Circulation Models, which contain about
100,000
coupled nonlinear equations, and also in the simplest models. Lorenz (1984a) has developed a loworder model in 27 variables that has been analyzed in some detail by Levine et a1.,(1990). The physical variables include the mixing ratios of water vapor and liquid water as well as temperature, pressure and the usual dynamic variables. In the Lorenz model, the atmosphere and ocean exchange heat and water through evaporation and precipitation. The model also produces clouds, which reflect incoming solar radiation, while both phases of water absorb and reemit infrared radiation.
For a WOOyear run for the global average annual temperature, the observed standard deviation was 0.3583°C. The data scaled to unit variance and zero mean are shown in Figure 6. The observed maximum value is 3.407. The probability of obtaining this value with 1000 values is 0.712, or a probability of 0.29 that 3.4 would be exceeded in a WOOyear run of independent trials drawn from a normal population. The distribution of temperature deviations about the mean approximate a normal distribution as illustrated in Figure 7.
The variation of range with the number of observations is shown in Figure 8.
The variation closely follows the expected value of the range drawn from a normal population.
7.2 Physical Dimensions of an Attractor
For a truly random, nondeterministic process, such as a variate drawn from a distribution unlimited on the left and right, the range will grow with
29
4
3
2
0
0
~
11
~
E
~
0
1
2
3
4
0
200
400
Time (Years)
600
800
1000
Figure
6. Variation of globalmean. annualmean. sealevel air temperature for 1000 years in a numerical solution to the Lorenz 27variable model of the atmosphere.
Data are rated to unit variance and zero mean.
30
140r,
120
100
!
§
Z
80
60
40
20 oL__
~
4 3 2 1 o
Deviations (Unit Variance)
2 3
Figure 1. Distribution of globalmean, annual average temperatures for
1000 years of the
Lorenz model.
4
31
•
,4
; Q 0 ,
7

Q) u
.Iii
~
~
2
~
4
3
6
5


~
,,~
I
J
I
I
~
I
.'
....
1
2

f    


1 ~
0
0
I
200
I
400
Time (Years)
I
600
I
800
Figure 8. The variation with time of the range of globalmean, annual average temperature for the Lorenz model. The dotted curve gives the expected value for the range for samples drawn from a normal population.
1000
32
the number of variates selected at a rate approximately
For a time series
n,
the number of variates equals the number of time units.
In a deterministic system governed by a set of coupled nonlinear differential equations, one would expect that the embodied conservation equations would set limits to the maximum growth of any state variable. In terms of the language of nonlinear dynamical systems, the governing attractor has finite physical dimensions. In this context we are speaking of the physical dimensions of the attractor as contrasted to the usual embedding dimensions which constitute a measure of the denseness of the set making up the attractor.
In the limit of large times, the range of random variables grows without bound but very slowly as
(In n
)1/2.
For deterministic systems, the growth, as noted above, should be limited at a large enough n. For the Lorenz system, it is apparent that the limit has not been reached in 1000 years (see Figure 8).
A general problem is that of determining the limits to the range of a particular variable and the time required to explore the far reaches of the attractor. It would appear that these are difficult issues, particularly for a system as complex as the 27variable Lorenz system.
In order to further examine the time required for the Lorenz 27variable model to explore the outer regions of the attractor, the model was run for
10,000 years. The results are shown in Figure 9. It would appear that the times to attain the outer reaches of the attractor are on the order of
5,000 years. To distinguish between a random process with the events drawn from a normal population and a deterministic Lorenz system by examining
33
Range for Gaussian Distribution
1.60
7.00
6.40
5.80
5.20
:1
:I j
4.60
!
1
4.00
3.40
2.80
2.20
1.60
1.00
0
~:""'"1
:,:::::::.:;.:;4
Dew
Point Temperature
....
~~~¥~~~~.~
    
Sea Surface Temperature
2000 4000 itme in
Years
6000
8000
Figure 9.
The variation with time of the range of the globally averaged values of surface atmosphere temperature. sea surface temperatures and dew point temperature.
The values are averaged owr a month and reported each year. The model is the Lorenz 27 variable model. The data are normalized to zero mean and unit variance. The heavy curve is the expected value of the range for a sample drawn from a normal population.
100000
34

the tails of the distribution one would need to select on the order of 10
4 values. This observation suggests that before asymptotic behavior is reached by a dynamical system, transients of considerable duration can occur. Also shown in Figure 9 is the variation of the range of sea surface temperature and dew point temperature as calculated in the model. The tails of trails of the distribution of the parameters differ from that of a normal population, but very long times, on the order of 10
4 years, are required to reach the outer edges of the attractor. There is, of course, no guarantee that in any single integration the system will reach the boundary of the attractor. A further consideration is the random roundoff noise generated in the computer integration. This noise will cause the calculated range to mimic a random variable.
In a simple system, again one proposed by Lorenz (1984b), it does appear possible to obtain bounds on the maximum dimensions of the attractor.
Lorenz (1984b) labels the following set of three coupled nonlinear equations as the simplest possible general circulation model:
dX dt dY dt dZ dt



 y2 _ Z2  aX
+
aF
XY bXZ  Y
+G
bXY +XZ Z.
The variable X represents the strength of the largescale westerlywind current or the geostrophically equivalent equatorialpole temperature gradient, while
Y
and
Z
are the strengths of the cosine and sine phases of a chain of superposed waves (Lorenz, 1984b). The quadratic terms containing b represent the translation of the waves by the western current. The remaining
35
quadratic terms represent a continual transfer of energy, except when
X
is negative, from the westerly flow to the waves.
The principal external driving force, the contrast between equatorial and pole heating, acts directly on the zonal flow
X
and is represented by a
F.
A secondary driving force, dependent on the difference between oceans and continents, enters the equations through G.
A section of the attractor for the Lorenz 84 model is displayed in Figure 10, which shows the range for the
Z
variable projected onto a plane including the
Z
axis and at an angle 8 to the
X
axis. The motion is centered about
Z
= 0 but otherwise offset in
X
and
Y.
We treat
X, Y
and
Z,
as coordinates in threedimensional phase space.
A rate of change of energy equation follows from the governing set by multiplying.each by
2X,2Y,
and
2Z,
respectively and summing where
R2
=
X
2
+
y2
+
Z2
can be interpreted as the total energy  the sum of the kinetic, potential and internal energies. The righthand side of the equation for
Ii:" vanishes on an ellipsoid
a(X _ F/2)'l
+
(Y _ G/2)'l
+
Z2
= aF'l: G
2
•
The rate of change of energy outside the ellipsoid is negative. This implies that for any sphere centered on the origin in state space enclosing the ellipsoid, orbits passing through points outside the sphere will eventually pass through the sphere and remain inside. The ellipsoid is centered at (F /2, G /2,
36
2.1
0.9
0.6
0.3 c
0.0
N
0.3
0.6
1.8
1.5
1.2
0.9
1.2
1.5
1.8
____ __
__
__ __
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 x(n) c~s(t'J)
+
YIn) ::;:n(t'J)
1.0
1.5
 I
2.0
Figure 10. A section of the attractor for the Lorenz 3variable model at a plane containing the z axis and at an angle of 117 0 to the x axis. The parameters for the model are given in Table 2.
37
Table 2
Maxima and Range for Variables Defined in Lorenz
Lowest Order Central Circulation Model
(F=8.0, G=1.0, a=0.25, b=4)
Variable Maximum Minimum
X
Y
Z
12.24
3.06
2.06
0.12
1.06
2.06
0) with the corresponding major axes ((
aF:!G2
)1/2,
(aF2tG2
)1/2, (
aF2 t
G2
)1/2).
The maxima and minima for the state variables are given in Table 2 for initial conditions lying within the ellipsoid and for specific values of the constants
F,
G and a.
For initial conditions lying within the ellipsoid defined by
d:'2
= 0, aU orbits would be expected to remain within the ellipsoid, except for small inertial excursions, in the absence of noise. The ellipsoid provides the limits for values that can be taken on by the state variables. The constant b does not enter into the equation for the critical energy surface since
b defines the strength by which the waves are carried by the zonal westerly current and does not alter the energy of the system.
The above analysis provides bounds on the maxima, minima and range attained by the state variables but provides no estimate of the average time required for a state variable to reach the boundary of the attractor. The only time scale within the systems of equations is determined by the constant a, the inverse of the decay time for the zonal current. Lorenz (1984b) assumes
38
a decay time for the zonal current of 20 days by selecting a=0.25 and taking
5 days as the unit time.
A numerical trial employing the constants shown in Table 2, taking the time unit to be 6 hours, averaging over 2.5 days and integrating from 34.25 years (5,000 points) yielded a variation in the
X
coordinate shown in Figure 11.
The power spectrum and autocorrelation function for
X,
equivalent to the equatorpole temperature difference, are shown in Figures 12 and 13.
The power spectrum displays .nuch mo!"e structure than the Lorenz 27variable model even though there is only one time scale that enters through the constant a. The distribution of values of
X
departs markedly from that a normal distribution (see Figure 14) in contrast to the Lorenz 27variable model.
The maximum range observed in the data for
X
is 2.98, well within the limits shown in Table 2. In fact, in this particular run, the calculated orbits did not explore the outer edges of the attractor for the state variable
X.
The variation of the maximum range with time for
X normalized to zero mean and unit variance is shown in Figure 15. Unlike the the Lorenz 27variable model, the maximum range does not continue to climb with time but levels of well within the limits set by the critical energy surface.
The
Y
and
Z
state variables show similar behavior. The variation of
Z
with time is shown in Figure 16. The histogram (Figure 17) for the
Z
variable shows large deviation from a normal population. The range levels off with the maximum observed range of 4.17 (see Figure 18) slightly exceeding the maximum range expected for the
Z
variable of 4.12 (see Table 2). An
39
2.5~,
2
1.5 x
0.5 o
0.5
lLo1oLoo2~00~030~0040~0O~5~000
Time (unit
=
2.5 days)
figure
11. Variation of the Lorenz 3variable model X coordinate (X is the equatorialpole temperature gradient) with time for 12.500 days.
40
10'r__ o
0.2 0.4 0.6 1.4
FreQuency (Cycles Per Day)
figure 12. Power spectrum for state variable X in Lorenz 3variable model. The data are normalized to zero mean and unit variance.
41
0.2 a
0.2
0.4
0.8
0.6
0.4 o
200
250
Figure 13. Autocorrelation junction for state variable X in Lorenz 3variable model. The data are normalized to zero mean and unit variance.
42
500
450 ~
400 ~
350 ~
300
250


200

150

100

50
o
3 r
 

2 rr 

1
r rr r
~
o

I
.....
r
2
In
Figure 14. The distribution of value for the X state variable in the Lorenz 3variable model.
The data have been normalized to zero mean and unit variance.
3
43
5
&
.!
4
3
2
8~~~~~_
7
6 o
TIn1e (Unit  2.5 days) figure 15. Variation of the range of the state variable X in the Lorenz 3variable model.
The data are nonnaJized to zero mean and unit variance. The dotted line is the range expected for values chosen from a normal population with unit variance and zero mean.
44
z
3~~~~r~
2 o
Tune (Unit  2.5 Days)
Figure 16. Variation of the Z state variable with time for the Lorenz 3variable model.
The unit of time is
2.5 days.
45
· ;
iA
500
450
400

"'""
350
~
300
250
200



150

100
50

o
3
I
2 m
.
.
I
.
1
:r
I
.
I o

T
.
I 
I
Figure 17.
Histogram for the Z state variable in the Lorenz 1984 model.
' 

I 
' 
T
I
2









3
46
3
2
8rT,~~~
7
6
5 o
Time (Unit  2.5 Days)
Figure 18. Range for the Z state variable in the Lorenz 1984 model. Data have been normalized to unit variance and
ZE!"O mean. The curve represents expected range for a normaJly distributed var:able.
47
Ai
«'
0; observation of a value for the range slightly in excess of the ma.ximum predicted range is not unexpected, since inertia will carry the the orbit outward before the inward pull of the attractor brings the orbit back within the limits of the critical energy surface.
The above example illustrates that it is possible to predict the maximum range of variables in a complex nonlinear system provided the equations are known. The limitation to the range of values which variables can assume likely comes from the conservation equations governing the motion. The time scale over which the maximum value may be attained is not determined by the above analysis.
7.3 Extremes for a Climate Model with a Linear Increase in Temperat ure
Levine etal., (1990) describe a version of the Lorenz 27~variable model in which the solar insolation is increased linearly with time leading to an increase of global temperature of 10° over a
1000~year period. The resulting temperature record is shown in Figure 19. The calculated variance is 9.24, which can be broken down into the part due to the linear increase and
(72, the variance about the linearly increasing mean.
N
2 b
12
2
Var
~
_ _
+(72
(72 _
0.235 where
N
is the number of years and
b
is the rate of temperature increase.
48
8
6
4
2
0
0
S
i
~
0
2
4
6
8
0 200 400
Tme (Years)
600
800
Figure
19. Global average temperature for Lorenz 27variable model of the atmosphere in which the parameter corresponding to the insolation is increased linearly in time to provide an approximately tOO( increase in temperature over
1000 years.
1000
49
The residual temperature curve after removing a trend by least squares is shown in Figure 20 with a distribution of the residuals approximating a norm"'} distribution (see Figure 21). The maximum range after 1000 years for the residuals approximates the range expected by drawing variates from a normal population (see Figure 22). The removal of a trend before analyzing for extremes is a useful way of examining records of global average, surface air temperature records. There is a high probability that these records contain a linear trend (Levine et a1. 1990).
7.4 Distribution of Extremes for Global Average Temperature Record
Our final example of an application of extreme value theory is to an annual average, global surface air temperature record. Three research groups
(Jones, 1988; Hansen and Lebedeff, 1987, 1988; and Vinnikow et aI., 1990)
Ita\,(' produced similar analyses of hemispheric surface temperature variations from somewhat differing initial data sets. The longest of the data sets is that of Jones (1988) and we will use it in the analysis. An analysis of the Hansen
Lebedeff (1987,1988) series (see Figure 1) produced results similar to those obtained from the Jones record. The Jones global average temperature variation is shown in Figure 23, which can be compared with Figure 1. Both figures show a relatively cool late 19th century followed by warming interrupted by cooling in the 194070 period (see Levine et aI., 1990). The power spectrum for the Jones (1988) surface air temperature records is shown in
Figure 24. The distribution of values for the Jones record is shown in Figure 25.
50
1.5
2r~~_,~_r~
(3
0
GI
~
~
E
~
0.5
0
0.5
1 o
TIme (Years)
Figure
20. Residual temperatures after removing the trend by a least squares fit from
Figure 17.
51
160
140
120
100 j i
80
60
40
20
\)
1.5
1
0.5 0 0.5
Temperature
(Oel
1.5
Figure 21. Distribution of values of the temperature for record shown in Figure 18.
2
52
..................
~~
....... ...I
6
.
CD
~
~
E
{l!
.S
CD at
~
4
3
5
2
6
7 r~~~_r~
I
I
//'
'"
/",


~,.,,



OL________ o
200
400
Time (Years)
600 800
Figure
22. Range for the residuals for the Lorenz 27variable system shown in Figure 20.
The data have been normalized to zero mean and unit variance. The curve is the expected range for a normal population.
1000
53
0.3~¥~
0.2
0.1
0.0 o
I!!
~ 0.1
&
F
Iii
0.2
0.3
0.4
0.5~~~~~~~~
1850
1870 1890
\
1910 1930 1950 1970
1990
Date
Figure 23. The variation of annual average global surface air temperature according to
Jones (1988).
54
5
10
5~o
15
0.2
Frequency (cpyr)
Figure
24. Power spectrum for the global annual average surface air temperature record shown in Figure 23. The data have been normalized to zero mean and unit variance.
55
The distribution differs from a normal distribution by being Hatter. This is consistent with the observed trend of increasing temperature displayed in
Figure 23 (a straight line would exhibit a uniform distribution of values).
The range for the Jones record (see Figure 23) is shown in Figure 26. The deviations again are in the direction expected by a record containing a linear trend.
If a trend determined by least squares (slope
=
0.28°C/century) is removed from the record, the resulting temperature residuals exhibit a histogram that closely approximates a normal distribution (see Figure 27). The tails of the distribution also approximate those of a normal variate, as is illustrated in Figure 28, which shows the variation of the range with time for the record exhibited in Figure 23 from which a linear trend has been removed.
The above analysis leads to the conclusion that once the linear trend is removed from the observed record, the distribution of the residuals both about the mean and in the tails approximates a normal distribution. In the
Lorenz 27 variable system we observed chaotic behavior which led to a random (normally distributed) variation even over time scales of thousands of years. The observed data, even when temporally and spatially averaged, exhibit similar behavior. These observations suggest the possibility of chaotic noise at low frequency with important implications for predictability. In particular, records of less than 1000 years may show statistics indistinguishable from a random series (once trends assumed to be deterministic have been removed). Prediction, outside of the trend, may thus be impossible except in
56
25
20 ~
~
!
!5
10
~ z
.
5 ~
.. o
0.5
I
0.4
I I
0.3 0.2 0.1
J o
0.1
Temperature Deviations in °C
I
·0.2
I
0.3 0.4
Figure 25. Histogram for global annual average surface air temperature record shown in
Figure 23.
57
6r__
5~
,
/
/
/ o u
3
~
I
I
Q)
I
I
I
I
J
2H ."
'pol
.,,'"
","~




...

1 
O~~I~I~~~~ILI~~~ o
20 40 60 80 100 120 140
Time (Years)
Figure 26. Variation of range for the global annual awrage surface air temperature record shown in Figure 23. The data have been normalized to zero mean and unit variance. The dotted line gives the expected value of the range for a normal variate.
58
30
.
25
20
10



! 
5
o
....
3
""
~.;
10'1'
/
I
2
I
I
•
l
'I'
/
i
/
I
~.
/
,
~
\.
\
\
I'
\
\
\
\
,
~\
\
\
\
I
1
I o
Temperature Deviations in °C
I
1
,
I
'""
,
2
,..
3 figure 27. Histogram for the global annual average surface air temperature record after removal of a trend determined by a least squares fit. The residuals have been normalized to unit variance. The dotted line is the expected distribution of a normal variate.
59
6~'
1860 1880
1900 1920
Time (Date)
1940
1960 1980 2000
 figure 28. Variation of the range of the residual temperature obtained by removing the trend from the data displayed in figure 23. The curve represents the expected range for a normally distributed variate.
60
a statistical sense. Similar considerations hold for models. The models may generate series that for time scales of a 1000 years have statistics identical to those of a random series. Using the models for deterministic prediction would in this case not be possible.
61
8
EFFECT OF A SHIFT OF THE MEAN
ON THE FREQUENCY OF EXTREMES
The consensus view is that the greenhouse effect will bring about an overall warming of the planet, together with increased precipitation. But it is also anticipated that there will be strong regional variations with some regions cooling and others receiving less precipitation. The primary effect of these changes will be to shift the mean although there may also be changes in the variance other than those resulting from trends. The shift in mean can cause a large change in the probability of extreme events, as is evident from
Figure 2. If over 100 years a particular climate parameter shifts by onehalf a standard deviation toward lower or higher values, the initial probability of 0.5 becomes 0.05 or 0.95, respectively. The largescale wind, temperature and moisture patterns in the atmosphere and the land and ocean changes associated with them undergo variations at all time scales from hours to billions of years. Some longterm changes have their origin in events that are external to the atmosphereocean system, such as variation in the earth's orbital parameters, volcanic explosions and changes in atmospheric composition. These longterm changes can accentuate or ameliorate the impact of shorterterm fluctuations which have their origin in chaotic noise. Alternatively, longperiod chaotic noise might alter the local mean and carrys the short term fluctuations so that extreme values are reached. In this section we briefly consider the effect of shifts of the mean on the frequency of extremes.
The events of the Little Ice Age illustrate the impact of shifts in the
63
~
mean on extreme events. Parry and Carter (1985) have examined in detail the effect of shortterm cool spells during the Little Ice Age on marginal agriculture in southwestern Scotland.
In particular, they showed that small changes in the mean temperature associ::..ted with the Little Ice Age increased the frequency of damaging weather. Further, the probability of two successive bad years is even more sensitive to changes in the mean. The importance of successive extremes lies in their cumulative impact: a.n agricultural region may be able to understand a single shock, but if buffer stores are depleted by one bad season, a second one in succession may be far more devastating.
The longest available temperature record available to us is the one compiled by Manley (1974) for Central England from several discontinuous series.
The record shows a number of isolated cool years (1740, 1782, 1860, 1979,
1922) and a clustering of two, three, or even more extremes in successive years (167375, 168898,183840, 188788, 189192) as indicated in Figure 29.
During the clustering of several cool summers in a row, when the temperature was marginal for the growth of oats, the farms at higher elevations in Scotland suffered greatly. One cool period, which persisted from 1688 to
1698, led to the abandonment of most of the farms above 300 m (Parry and
Carter, 1985). Indeed, the "Seven Ice Years" of the 1690s caused catastrophe among rural populations in all of Northern Europe.
The effect of a change in the mean on the probability of an extreme event can be understood from a simple statistical model. The probability that some climate parameter (e.g., summer temperature, spring precipitation) falls above or below some threshold value for damage is
PI'
In a stable climate
64
a:q:rpc $iUCfLZfS
e qa:; ;.
~ ¥ :
> au.
( .
10.0
9.5
§
9.0
~
I
~
8.5
8.0
7.5
7.0
6.5~~~~~~
1650 1750 1850 1950
Date
Figure
29. Variations in yearly awrage temperature in Central England. After Manley
(1973).
65
regime the return period for the damaging event is
PiJ
years. Suppose that the climate shifts in such a way that the mean of the climate parameter shifts by X standard deviations toward the threshold value and that the climate parameter is normally distributed. The ratio of the probability in the shifted regime to that in the original regime,
PsI PI,
is a highly nonlinear function of the shift of the mean, as is shown in Figures 30 and 31.
If the initial probability is
PI
=
0.05 (a return period of 20 years) and the mean is shifted by one standard deviation toward the threshold, then the subsequent probability is
Ps
~
0.25, or a return period of four years. The probability of two successive years
0; an event with a return period of 20 years is only
1 in 400 in the initial state but moves to 1 in 16 in the shifted state. For example, the standard deviation for the temperature record of central England is 0.58°C, with a mean for the 16591977 period of 9.1°C.
A decrease in the mean to 8.5°C would alter the probability of a oncein
100 years cool year to a oncein12years event. The long~period fluctuation in climate that comprised the Little Ice Age increased the probability of a shortterm cool summer, making a sequence of successive cool summers much more likely than in the preceding or subsequent years.
Changes in the frequency of extreme high or low temperature, or of high or low precipitation, can be expected as climate shifts in response to greenhouse warming. The observed change in global average temperature of about 0.5° over the last century represents a shift of 2.3 standard deviations, making the probability of extremely warm global average years significantly higher.
66
40
35
~
30
~
25
~
20
~
15 ~
10
~
5~ o
Ratio of Probability for Mean
Shifted by
Xa
Units P,/P
1
P, o
i
I
.
.
=
0.005
0 i
I
. 1
0
1
.....
.. 
..
....
.
.
.
_dIU1
I
.............

....
.
"
..
.,..,.
..
.
"
.
/
/0 •••••••
....
/0 ••••

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Figure 30. Ratio of probability of an extreme event to the probability of the event in the original, stable climate. Shift in mean measure is in terms of standard deviation. A :lormal population is assumed.
67
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10'
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Figure 31. Shift in probability from a change in mean for initially rare events. Shift in mean is measured in terms of standard deviations, and a normal population is assumed.
68
9
SUMMARY
The statistical theory of extreme events can be applied in numerous ways to problems in climate change. Calculation of the range as a function of time can be used to detect the presence of a trend. The value of using extreme events in this application is that since they are rare, extreme events can be treated as independent without any assumption with respect to the underlying distribution. Analysis of the global annual average temperature record using extreme events and analysis based on an assumptions of an underlying normal distribution both confirm the existence of a trend with high odds (l05  10
6
) in favor of the trend hypothesis.
Analysis of observed longterm global annual average surface air temperature records and of model results show that both kinds of records are indistinguishable from a series of normally distributed variates once a trend is removed. For both kinds of records, the statistics approach those of normally distributed variates about the mean and in the tails of the distribution. For model results, this conclusion holds even for series that are several thousand years long. These observations cast doubt on the use of General Circulation
Models to predict future climate deterministically .
The behavior of extreme values in nonlinear systems can be understood in terms of the physical dimensions of the underlying attractor.
If the governing equations contain conservation laws, the outer boundary of the attractor lies within a manifold. The dimensions of the manifold define the limits to
69
~
. . .
                                    
the extreme values.
70
References
1. Cramer, H. (1946) Mathematical Methods of Statistics Princeton Univ.
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